LONDON (Reuters) - London-based bitcoin start-up Elliptic and the Internet Watch Foundation, a charity that monitors online child sex abuse, will cooperate to clamp down on the use of bitcoin in online child pornography, Elliptic said on Wednesday.
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The translation from integers to elliptic curves is a common one in mathematics.
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And Elliptic claims its ultrasonic sensors are actually more reliable than optical ones.
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Tom Robinson, COO and co-founder of bitcoin specialists Elliptic says it is.
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"Blood Moon" search frequencyScreenshot: Google TrendsThe Moon also has an elliptic orbit around the Earth.
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But law enforcement agencies, working with companies like Elliptic, have figured out ways to trace this.
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He declines to comment on how far off Elliptic is from achieving breakeven or profitability yet.
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Instead of a square, they are an algebraic structure extracted from a special kind of elliptic curve.
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Mochizuki expressed much of the data from Szpiro's conjecture—which concerns elliptic curves—in terms of Frobenioids.
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But actually it's still perfectly possible to express the solution in terms of elliptic or hypergeometric functions.
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" On the aesthetics side, Android phones will benefit the most from the Elliptic sensors, aptly called "Beauty.
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Do we really need elliptic curve levels of encryption to secure what might be $200 in assets?
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Juniper said this month it would remove Dual Elliptic Curve entirely in future versions of its products.
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That is how Dual Elliptic Curve made it into a software kit distributed by security company RSA.
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But a different wallet for each donation makes this so-called tagging far more complicated, Elliptic said.
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Front Back "Xiaomi came to us about a year ago," Laila Danielsen, CEO of Elliptic Labs, told Mashable.
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The payload will be an original Tesla Roadster, playing Space Oddity, on a billion year elliptic Mars orbit.
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GPS can't really do its job detecting distances, and machines like the elliptic tend to be even trickier.
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The last withdrawal of the virtual cash was made at 3:25 am on Thursday, according to Elliptic.
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So it is — hopefully — with Elliptic Labs, creators of some of the smoothest smartphone gesture controls I've ever seen.
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Going forward, Elliptic is opening offices in Singapore and Japan as part of a push into the Asian market.
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Nor, therefore, the business opportunity for Elliptic to sell support services to help others avoid touching the hot stuff.
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Elliptic managed to bag Wells Fargo as an investor after building a "relationship" with the U.S. bank, Smith said.
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Elliptic curve cryptography, or ECC, is a robust approach to crypto, used in everything from securing websites to messages.
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Elliptic doesn't have any consumer products available right now, but was able to show off a handful of demo interactions.
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Elliptic CEO Laila Danielsen said the company wants to make gesture controls that are as easy to use as possible.
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Luke Wilson at 4iq, formerly of the FBI, and the currency tracker Elliptic, is an example of a cryptocurrency investigator.
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Elliptic uses a database of information linking digital coin addresses to exchanges, darkweb marketplaces, and proscribed groups to track cryptocurrencies.
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Instead, Musk will launch an original Tesla Roadster playing David Bowie's "Space Oddity" into "a billion year elliptic Mars orbit."
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In 220, Yutaka Taniyama, a colleague and friend of Dr. Shimura's, posed some questions about mathematical objects called elliptic curves.
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The translation, which was well known before Mochizuki's work, is simple — you associate each abc equation with the elliptic curve whose graph crosses the x-axis at a, b and the origin — but it allows mathematicians to exploit the rich structure of elliptic curves, which connect number theory to geometry, calculus and other subjects.
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UK fintech startup Elliptic has pulled in a $5 million Series A round to keep building out its blockchain forensics tool.
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To completely hide the sensors underneath the screen, Xiaomi turned to the experts at Elliptic Labs to utilize their ultrasound technology.
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This was confirmed by Elliptic, a London-based start-up that helps law enforcement authorities track down criminals using the cryptocurrency.
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Elliptic Labs' tech is based on ultrasound, which allows the company to monitor movement in a 360-degree dome surrounding your smartphone.
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"It shows how far the industry has come that a hack of this scale isn't really an issue," said Robinson at Elliptic.
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Meanwhile, another startup called Elliptic Labs has actually figured out how to use ultrasonic sensors to replace the light and proximity sensors altogether.
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Elliptic Labs promises its solution is "at par or better" than infrared in terms of accuracy and latency when detecting the user's head.
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Wells Fargo has invested $5 million into U.K. start-up Elliptic, which helps banks manage the risks associated with being exposed to cryptocurrencies.
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Some of its users' communications were protected both with elliptic-curve encryption and New Hope, a post-quantum protocol developed as part of PQCRYPTO.
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Last month Elliptic said it was working with the Internet Watch Foundation to clamp down on the use of bitcoin for online child pornography.
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Thirteen of the donations were made from a separate exchange, also from Asia, said Elliptic, which declined to give further details of the exchanges.
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Based in London, Elliptic sells blockchain analytics tools to some of the world's largest cryptocurrency platforms — including Binance and Circle — as well as banks.
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Elliptic shows a different side of the crypto industry, in that its technology is seen as more favorable to financial services businesses and regulators.
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"They are still in experimentation stage - trying it out, seeing how much they can raise, and whether it works," said Elliptic co-founder Tom Robinson.
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Robinson's company Elliptic, along with others like Chainalysis, works closely with U.S. law enforcement, providing tools for agents to track bitcoin payments through the blockchain.
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Rather than working with the famously simple but constraining formulation of the problem (which states that there is no solution in positive integers to the equation an +bn = cn for any integer value of n greater than 2), he translated it twice over: once into a statement about elliptic curves and then into a statement about another type of mathematical object called "Galois representations" of elliptic curves.
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Elliptic is able to trace bitcoin transactions and link them to a real world address, helping law enforcement around the globe track illicit activity involving bitcoin.
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Conversely, the elliptic-curve algorithm Google is normally using might be worthless against future's quantum computers, but it's the best option against the computers of today.
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But what these researchers are really doing, finding points on elliptic curves, is a fundamental mathematical idea used in the cryptography that secures things like bitcoin.
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Elliptic provides banks with insight into the licences and KYC policies of crypto exchanges and the transactions happening on that exchange that may be in sanctioned countries.
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"This is the first time anybody has started identifying these crimes in bitcoin and flagging them up in a system like ours," said Elliptic CEO James Smith.
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Mochizuki's approach to the abc conjecture translates the problem into a question about elliptic curves, a special type of cubic equation in two variables, x and y.
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Keys for elliptic-curve cryptography, another current standard, are just 32 bytes long; any post-quantum solution needs to aim for a similar ratio of brevity to security.
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Elliptic Labs attracted a lot of attention at CES a couple of years ago when it demonstrated some pretty cool 3D gestural interactions based on ultrasonic wave technology.
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Flush with a sizeable injection of Series B capital, Elliptic is especially targeting business growth at Asia — with a plan to open new offices in Japan and Singapore.
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We are seeing a growing demand for their services across our portfolio of crypto-assets related companies and view Elliptic as best-placed to meet this considerable opportunity.
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So far, roughly $80,000 has been deposited into the bitcoin addresses linked to the attack, according to Elliptic, a company that tracks online financial transactions involving virtual currencies.
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"You have to have these holes on your device, and [manufacturers] are trying to make their devices symmetric and having a difficult time," says Laila Danielsen, CEO of Elliptic.
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On the competition side, he names Chainalysis and Block Seer as main rivals but argues they are more focused on law enforcement vs the core compliance imperative driving Elliptic.
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The new Beauty software from Elliptic Labs seeks to obviate the omnipresent infrared-based proximity sensor with the help of hardware we already have on most of our phones.
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The investment into Elliptic from Wells Fargo's venture unit, Wells Fargo Strategic Capital, is an extension of the start-up's $23 million Series B funding round announced in September.
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Elliptic was founded back in 2013, amid much Bitcoin hype, and initially launched a Bitcoin storage product, in July 2014, taking in $2 million from Octopus Investments at the time.
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At the moment, Elliptic is working on trying to trace the payments, but Smith said this would become clearer when the hackers try to withdraw their bitcoin in fiat currency.
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Elliptic will integrate IWF's data set into its transaction-monitoring systems and will then alert clients when it sees money moving from the addresses identified as bad actors by IWF.
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A recent report by Elliptic, in partnership with the Foundation for Defense of Democracies' Center on Sanctions and Illicit Finance, found that Europe was the "wild west" of cryptocurrency regulation.
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As of Saturday afternoon, several Bitcoin accounts associated with the ransomware had received the equivalent of $22016,21, according to Elliptic, a firm that tracks online financial transactions involving virtual currencies.
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Funds totaling the equivalent of about $33,000 were deposited into several Bitcoin accounts associated with the ransomware, according to Elliptic, a company that tracks online financial transactions involving virtual currencies.
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"This certainly doesn't look like a typical ICO, given the low level of transaction activity," said Tom Robinson, chief data officer and co-founder of Elliptic, a London-based blockchain data company.
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Though the academic team looking at Juniper has not named a suspect in the 2008, 2012 or 2014 changes, 2008 was one year after veteran cryptographers raised questions about Dual Elliptic Curve.
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However, according to research shared with Reuters by leading blockchain analysis firm Elliptic, in recent weeks it has changed the mechanism, with its website generating a new digital wallet with every transaction.
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Some ISO delegates said much of their skepticism stemmed from the 2000s, when NSA experts invented a component for encryption called Dual Elliptic Curve and got it adopted as a global standard.
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But at least, in eliminating one of the uglier necessities of smartphone design, Elliptic Labs appears to have succeeded in making the Jony Ives of the world sleep a little better at night.
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The random number generator used in the Juniper products, known as Dual Elliptic Curve, has long been suspected by security professionals of containing a back door engineered by the U.S. National Security Agency.
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Non-profit organization the Internet Watch Foundation (IWF) has teamed up with blockchain start-up Elliptic to combat the use of bitcoin to buy child sex abuse images online, the two said on Wednesday.
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Elliptic manages to trawl through this to identify suspicious activity and then can use this to link transactions together and get an idea of where the money is moving and which payments are linked.
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Elliptic works with top U.S. and European bitcoin exchanges – the entity that processes transactions in the cryptocurrency – as well as law enforcement to identify activity such as money laundering, drug sales, extortion and theft.
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But London-based start-up Elliptic – which raised $5 million earlier this year – has developed a tool which can trace suspicious transaction patterns of bitcoin due to the open nature of the cryptocurrency platform.
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London-based Elliptic and U.S. rival Chainalysis are the most prominent blockchain analysis firms, and have gained traction as watchdogs, cryptocurrency companies and firms such as hedge funds seek tools to track digital coins.
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Elliptic said it would also use the fresh funds to further develop its product to support new digital currencies including Libra, Link — a token being developed by Japan's Line Corporation — and central bank cryptocurrencies.
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It's not as simple as just running the software on existing devices, and some integration is required, but Elliptic expects to have devices with its new ultrasound proximity sensor out on the market this year.
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The Silicon Valley maker of networking gear said it would ship new versions of security software in the first half of this year to replace those that rely on numbers generated by Dual Elliptic Curve technology.
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Fintech startups like Block Seer and Chainalysis in the U.S. and Elliptic in the UK provide blockchain analysis software for law enforcement and financial institutions to identify illicit bitcoin transactions and potentially the criminals behind them.
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"It looks like the attackers had no intent in decrypting anything," said Tom Robinson, a founder of Elliptic, a company in London that tracks online financial transactions involving virtual currencies and helps organizations respond to digital attacks.
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Until now, the most influential adopter of Dual Elliptic Curve was believed to be RSA, part of storage company EMC, which Reuters reported received a $10-million federal contract to distribute it in a software kit for others.
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But the academics who studied the code said that while Juniper had not disavowed the 2008 code, it had not explained how that constant was picked or why it was using the widely faulted Dual Elliptic Curve at all.
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It has shared that database with Elliptic, which monitors bitcoin transactions and can alert its clients - ranging from bitcoin exchanges to U.S. and European intelligence agencies - when money moves from bitcoin addresses that have been identified as bad actors.
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Just as well then that Elliptic has not hitched its wagon to the fortunes of Bitcoin per se, with Smith talking up the potential of the underlying blockchain technology — and envisaging future applications for its technology that don't involved Bitcoin itself.
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By sending out small waves of sound from the phone's earpiece and listening out for their reflections with the handset's microphones, Elliptic is able to detect when a person brings up the phone for a call and dim the display appropriately.
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The UK-based IWF, which aims to eliminate child pornography on the internet, has given Elliptic - which identifies illicit activity on bitcoin's public ledger of transactions, the blockchain - a database of bitcoin addresses that it has associated with the pornography.
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"When people look to launder these types of funds, they sometimes spread it into smaller transactions because it's less likely to trigger (exchanges') anti-money laundering (mechanisms)," said Tom Robinson, co-founder of Elliptic, a cryptocurrency security firm in London.
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The battery pack and electric motor, along with the windows, were all removed before it was loaded onto the Falcon Heavy rocket and blasted into a "billion year elliptic Mars orbit," that should keep her safe and sound for the time being.
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James Smith, CEO of Elliptic, a London-based start-up that helps law enforcement agencies track criminals using the cryptocurrency, said his company had uncovered that since Friday, around $50,000 worth of bitcoin payments have been made to the hackers by 7 a.m.
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"That was one of the pieces that we added that took a pretty significant amount of research to develop for us — to get proxy re-encryption to work with things like ECIES, which is a standard elliptic curve, NIST-certified," he notes.
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Whereas the abc conjecture describes an underlying mathematical phenomenon in terms of relationships between integers, Szpiro's conjecture casts that same underlying relationship in terms of elliptic curves, which give a geometric form to the set of all solutions to a type of algebraic equation.
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Monero's use on darknet marketplaces - sites used for buying illicit goods from drugs to stolen credit cars - is on the rise, said Tom Robinson, chief data officer of Elliptic, a London-based firm that provides blockchain-tracking software to law enforcement agencies and private companies.
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The surface of a sphere in Euclidean solid geometry is a non-Euclidean surface in the sense of elliptic geometry. Spherical geometry is the simplest form of elliptic geometry. In elliptic geometry, an elliptic rectangle is a figure in the elliptic plane whose four edges are elliptic arcs which meet at equal angles greater than 90°. Opposite arcs are equal in length.
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In mathematics -- specifically, in the theory of partial differential equations -- a semi-elliptic operator is a partial differential operator satisfying a positivity condition slightly weaker than that of being an elliptic operator. Every elliptic operator is also semi-elliptic, and semi- elliptic operators share many of the nice properties of elliptic operators: for example, much of the same existence and uniqueness theory is applicable, and semi-elliptic Dirichlet problems can be solved using the methods of stochastic analysis.
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The brown to black coloured seeds within have an elliptic to widely elliptic shape.
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They also find applications in elliptic curve cryptography (ECC) and integer factorization. An elliptic curve is not an ellipse: see elliptic integral for the origin of the term. Topologically, a complex elliptic curve is a torus, while a complex ellipse is a sphere.
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In algebraic geometry, supersingular elliptic curves form a certain class of elliptic curves over a field of characteristic p > 0 with unusually large endomorphism rings. Elliptic curves over such fields which are not supersingular are called ordinary and these two classes of elliptic curves behave fundamentally differently in many aspects. discovered supersingular elliptic curves during his work on the Riemann hypothesis for elliptic curves by observing that in positive characteristic elliptic curves could have endomorphism rings of unusually large rank 4, and developed their basic theory. The term "supersingular" has nothing to do with singular points of curves, and all supersingular elliptic curves are non-singular.
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Since these are elliptic integrals, the coordinates ξ and η can be expressed as elliptic functions of u.
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They contain glossy dark brown coloured seeds that have an elliptic to broadly elliptic shape and a length of about .
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Camellia yunnanensis is a 1.3–7 m tall shrub or small tree. Its leaves are elliptic to broad-elliptic or ovate-elliptic, bluntly acute. They are deep green.Camellia yunnanensis Cohen Stuart Its flowers is white, perulate and solitary.
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In mathematics, Legendre's relation can be expressed in either of two forms: as a relation between complete elliptic integrals, or as a relation between periods and quasiperiods of elliptic functions. The two forms are equivalent as the periods and quasiperiods can be expressed in terms of complete elliptic integrals. It was introduced (for complete elliptic integrals) by .
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Elliptic curve cryptography is a popular form of public key encryption that is based on the mathematical theory of elliptic curves. Points on an elliptic curve can be added and form a group under this addition operation. This article describes the computational costs for this group addition and certain related operations that are used in elliptic curve cryptography algorithms.
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The glossy seeds inside have an elliptic to oblong-elliptic shape with a length of and a have a cream coloured aril.
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Elliptic curve scalar multiplication is the operation of successively adding a point along an elliptic curve to itself repeatedly. It is used in elliptic curve cryptography (ECC) as a means of producing a one-way function. The literature presents this operation as scalar multiplication, as written in Hessian form of an elliptic curve. A widespread name for this operation is also elliptic curve point multiplication, but this can convey the wrong impression of being a multiplication between two points.
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In mathematics, elliptic cohomology is a cohomology theory in the sense of algebraic topology. It is related to elliptic curves and modular forms.
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In cryptography, the Elliptic Curve Digital Signature Algorithm (ECDSA) offers a variant of the Digital Signature Algorithm (DSA) which uses elliptic curve cryptography.
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Graphs of elliptic curves y2 = x3 − x and y2 = x3 − x + 1. If we consider these as curves over the rationals, then the modularity theorem asserts that they can be parametrized by a modular curve. A modular elliptic curve is an elliptic curve E that admits a parametrisation X0(N) → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, something that could be called an elliptic modular curve.
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During a key exchange entities A and B will each transmit information of 2 coefficients (mod p2) defining an elliptic curve and 2 elliptic curve points. Each elliptic curve coefficient requires log2p2 bits. Each elliptic curve point can be transmitted in log2p2+1 bits, hence the transmission is 4log2p2 \+ 4 bits. This is 6144 bits for a 768-bit modulus p (128-bit security).
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The Lenstra elliptic-curve factorization or the elliptic-curve factorization method (ECM) is a fast, sub-exponential running time, algorithm for integer factorization, which employs elliptic curves. For general-purpose factoring, ECM is the third-fastest known factoring method. The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra.
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The facility of versors illustrate elliptic geometry, in particular elliptic space, a three-dimensional realm of rotations. The versors are the points of this elliptic space, though they refer to rotations in 4-dimensional Euclidean space. Given two fixed versors u and v, the mapping q \mapsto u q v is an elliptic motion. If one of the fixed versors is 1, then the motion is a Clifford translation of the elliptic space, named after William Kingdon Clifford who was a proponent of the space.
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3, 635–681. Tom Ilmanen made progress on understanding the theory of such elliptic equations, via approximations by elliptic equations of a more standard character.Tom Ilmanen.
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Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations.
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The suspension used semi-elliptic leaf springs at the front and three-quarter elliptic at the rear. Several of the cars were used in motor sport competition.
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This is proved using integration by parts. These operators are elliptic although in general elliptic operators may not be non- negative. They are however bounded from below.
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The arithmetic–geometric mean can be used to compute – among others – logarithms, complete and incomplete elliptic integrals of the first and second kind, and Jacobi elliptic functions.
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Sepals narrowly elliptic; petals ovate, elliptic or rhomboid; lip folded to form a tube, with very wavy front margin. Pollinia 4, with curved appendages. Cattleya rex. Habit.
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Legendre's relation stated using complete elliptic integrals is : K'E + KE' - KK' = \frac \pi 2 where K and K′ are the complete elliptic integrals of the first kind for values satisfying , and E and E′ are the complete elliptic integrals of the second kind. This form of Legendre's relation expresses the fact that the Wronskian of the complete elliptic integrals (considered as solutions of a differential equation) is a constant.
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The theory of elliptic surfaces is analogous to the theory of proper regular models of elliptic curves over discrete valuation rings (e.g., the ring of p-adic integers) and Dedekind domains (e.g., the ring of integers of a number field). In finite characteristic 2 and 3 one can also get quasi-elliptic surfaces, whose fibers may almost all be rational curves with a single node, which are "degenerate elliptic curves".
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In mathematics, an elliptic divisibility sequence (EDS) is a sequence of integers satisfying a nonlinear recursion relation arising from division polynomials on elliptic curves. EDS were first defined, and their arithmetic properties studied, by Morgan WardMorgan Ward, Memoir on elliptic divisibility sequences, Amer. J. Math. 70 (1948), 31-74.
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The pods are flat and in length and wide. The brown to dark-brown seeds have a narrowly oblong-elliptic or oblong-elliptic shape with a length of .
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Example: If E is an elliptic curve and B is a curve of genus at least 2, then E×B is an elliptic surface of Kodaira dimension 1.
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Also the relative position of one body with respect to the other follows an elliptic orbit. Examples of elliptic orbits include: Hohmann transfer orbit, Molniya orbit, and tundra orbit.
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The use of elliptic curves in cryptography was suggested independently by Neal Koblitz and Victor S. Miller in 1985. Elliptic curve cryptography algorithms entered wide use in 2004 to 2005.
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There are two families of 'canonical' elliptic functions: those of Jacobi and those of Weierstrass. Although Jacobi's elliptic functions are older and more directly relevant to applications, modern authors mostly follow Weierstrass when presenting the elementary theory, because his functions are simpler, and any elliptic function can be expressed in terms of them.
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Note that the half-period ratio can be thought of as a simple number, namely, one of the parameters to elliptic functions, or it can be thought of as a function itself, because the half periods can be given in terms of the elliptic modulus or in terms of the nome. This follows because Klein's j-invariant is surjective onto the complex plane; it gives a bijection between isomorphism classes of elliptic curves and the complex numbers. See the pages on quarter period and elliptic integrals for additional definitions and relations on the arguments and parameters to elliptic functions.
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For example, a pseudoinverse of an elliptic differential operator of positive order is not a differential operator, but is a pseudodifferential operator. Also, there is a direct correspondence between data representing elements of K(B(X), S(X)) (clutching functions) and symbols of elliptic pseudodifferential operators. Pseudodifferential operators have an order, which can be any real number or even −∞, and have symbols (which are no longer polynomials on the cotangent space), and elliptic differential operators are those whose symbols are invertible for sufficiently large cotangent vectors. Most version of the index theorem can be extended from elliptic differential operators to elliptic pseudodifferential operators.
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Historically, elliptic functions were first discovered by Niels Henrik Abel as inverse functions of elliptic integrals, and their theory was improved by Carl Gustav Jacobi; these in turn were studied in connection with the problem of the arc length of an ellipse, whence the name derives. Jacobi's elliptic functions have found numerous applications in physics, and were used by Jacobi to prove some results in elementary number theory. A more complete study of elliptic functions was later undertaken by Karl Weierstrass, who found a simple elliptic function in terms of which all the others could be expressed. Besides their practical use in the evaluation of integrals and the explicit solution of certain differential equations, they have deep connections with elliptic curves and modular forms.
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The fibers that are not elliptic curves are called the singular fibers and were classified by Kunihiko Kodaira. Both elliptic and singular fibers are important in string theory, especially in F-theory. Elliptic surfaces form a large class of surfaces that contains many of the interesting examples of surfaces, and are relatively well understood in the theories of complex manifolds and smooth 4-manifolds. They are similar to (have analogies with, that is), elliptic curves over number fields.
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The slightly shiny black seeds have an elliptic to oblong-elliptic shape and a length of and a width of and a pitted surface with a white to pale cream coloured aril.
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Elliptic curve point operations: Addition (shown in facet 1), doubling (facets 2 and 4) and negation (facet 3). There are three commonly defined operations for elliptic curve points, addition, doubling and negation.
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In mathematics, a hyperelliptic surface, or bi-elliptic surface, is a surface whose Albanese morphism is an elliptic fibration. Any such surface can be written as the quotient of a product of two elliptic curves by a finite abelian group. Hyperelliptic surfaces form one of the classes of surfaces of Kodaira dimension 0 in the Enriques–Kodaira classification.
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The only 'easy' cases are those for d = 1, for an elliptic curve with linear span the projective plane or projective 3-space. In the plane, every elliptic curve is given by a cubic curve. In P3, an elliptic curve can be obtained as the intersection of two quadrics. In general abelian varieties are not complete intersections.
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The Elementary Properties of the Elliptic Functions, with Examples by Alfred Cardew Dixon, Palala Press 2016, Certain elliptic functions (meromorphic doubly periodic functions) denoted cm and sm satisfying the identity cm(z)3 + sm(z)3 = 1 are known as Dixon's elliptic functions. Dixon's identity is any of several closely related identities involving binomial coefficients and hypergeometric functions.
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The chassis was a conventional ladder frame, with solid axles sprung on semi-elliptic front and three- quarter-elliptic rear leaf springs. The brakes were on the transmission and on the rear wheels.
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Each pod is to around in length and a width of . The glossy black seeds within the pods are longitudinal with a broadly elliptic to oblong-elliptic shape and have a length of .
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Notices 1993, no. 1, 29–39. in projective spaces and developing the theory of elliptic genus of singular algebraic varieties.L.Borisov, A.Libgober, McKay correspondence for elliptic genera, Annals of Mathematics (2) 161 (2005),no.
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Hull painted black. > Artificial quarter galleries. Elliptic stern. Straight stem.
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They are of a flattened, elliptic shape, with lateral veins.
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In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids. Projective polyhedra are also referred to as elliptic tessellations or elliptic tilings, referring to the projective plane as (projective) elliptic geometry, by analogy with spherical tiling, a synonym for "spherical polyhedron". However, the term elliptic geometry applies to both spherical and projective geometries, so the term carries some ambiguity for polyhedra.
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To ensure non-divergence of the flow field, an elliptic equation is solved. The elliptic equation is derived from the continuity equation wherein velocity components are expressed in terms of p_. Since the elliptic equation is derived from the discrete form of the continuity equation and the discrete form of the pressure gradient, conservativity is guaranteed (Flassak and Moussiopoulos, 1988). The discrete pressure equation is solved numerically with a fast elliptic solver in conjunction with a generalized conjugate gradient method.
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In mathematics, an elliptic hypergeometric series is a series Σcn such that the ratio cn/cn−1 is an elliptic function of n, analogous to generalized hypergeometric series where the ratio is a rational function of n, and basic hypergeometric series where the ratio is a periodic function of the complex number n. They were introduced by Date-Jimbo-Kuniba-Miwa-Okado (1987) and in their study of elliptic 6-j symbols. For surveys of elliptic hypergeometric series see , or .
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Elliptic geometry is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point (rather than two). Because of this, the elliptic geometry described in this article is sometimes referred to as single elliptic geometry whereas spherical geometry is sometimes referred to as double elliptic geometry.
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Every surface of Kodaira dimension 1 is an elliptic surface (or a quasielliptic surface in characteristics 2 or 3), but the converse is not true: an elliptic surface can have Kodaira dimension -\infty, 0, or 1. All Enriques surfaces, all hyperelliptic surfaces, all Kodaira surfaces, some K3 surfaces, some abelian surfaces, and some rational surfaces are elliptic surfaces, and these examples have Kodaira dimension less than 1. An elliptic surface whose base curve B is of genus at least 2 always has Kodaira dimension 1, but the Kodaira dimension can be 1 also for some elliptic surfaces with B of genus 0 or 1. Invariants: c_1^2 =0, c_1 \geqslant 0.
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The Jacobi elliptic functions are used more often in practical problems than the Weierstrass elliptic functions as they do not require notions of complex analysis to be defined and/or understood. They were introduced by .
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The Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It was published independently by Nagell and by Élisabeth Lutz.
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Landen's transformation is a mapping of the parameters of an elliptic integral, useful for the efficient numerical evaluation of elliptic functions. It was originally due to John Landen and independently rediscovered by Carl Friedrich Gauss.
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The seeds are yellow-brown, and elliptic to rhomboidal in shape.
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The pods contain oblong to elliptic seeds that are in length.
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In mathematics, an elliptic Gauss sum is an analog of a Gauss sum depending on an elliptic curve with complex multiplication. The quadratic residue symbol in a Gauss sum is replaced by a higher residue symbol such as a cubic or quartic residue symbol, and the exponential function in a Gauss sum is replaced by an elliptic function. They were introduced by , at least in the lemniscate case when the elliptic curve has complex multiplication by , but seem to have been forgotten or ignored until the paper .
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Absolute geometry is, in fact, the intersection of hyperbolic geometry and Euclidean geometry when these are regarded as sets of propositions. Absolute geometry is inconsistent with elliptic geometry: in elliptic geometry there are no parallel lines at all, but in absolute geometry parallel lines do exist. Also, in elliptic geometry, the sum of the angles in any triangle is greater than 180°.
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Helmut Hasse conjectured that L(E, s) could be extended by analytic continuation to the whole complex plane. This conjecture was first proved by for elliptic curves with complex multiplication. It was subsequently shown to be true for all elliptic curves over Q, as a consequence of the modularity theorem. Finding rational points on a general elliptic curve is a difficult problem.
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In mathematics, in particular in partial differential equations and differential geometry, an elliptic complex generalizes the notion of an elliptic operator to sequences. Elliptic complexes isolate those features common to the de Rham complex and the Dolbeault complex which are essential for performing Hodge theory. They also arise in connection with the Atiyah- Singer index theorem and Atiyah-Bott fixed point theorem.
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This means that Laplace's equation describes a steady state of the heat equation. In parabolic and hyperbolic equations, characteristics describe lines along which information about the initial data travels. Since elliptic equations have no real characteristic curves, there is no meaningful sense of information propagation for elliptic equations. This makes elliptic equations better suited to describe static, rather than dynamic, processes.
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The (strong) torsion conjecture first posed by has been completely resolved in the case of elliptic curves. Barry Mazur proved uniform boundedness for elliptic curves over the rationals. His techniques were generalized by and , who obtained uniform boundedness for quadratic fields and number fields of degree at most 8 respectively. Finally, Loïc Merel () proved the conjecture for elliptic curves over any number field.
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The Tate pairing via elliptic nets. In Pairing-Based Cryptography (Tokyo, 2007), volume 4575 of Lecture Notes in Comput. Sci. Springer, Berlin, 2007. has applied EDS and their higher rank generalizations called elliptic nets to cryptography.
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In mathematics, the Nagell–Lutz theorem is a result in the diophantine geometry of elliptic curves, which describes rational torsion points on elliptic curves over the integers. It is named for Trygve Nagell and Élisabeth Lutz.
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Leaves are pinnate and alternate on the stem. There are three to eight leaflets, mostly 5 to 12 cm long, 2 to 5 cm wide. Elliptic or narrowly elliptic. With a short blunt point at the tip.
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It has compound pinnate leaves from long, with 7-25 , elliptic leaflets.
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Birkhoff showed that a billiard system with an elliptic table is integrable.
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The spores are ovoid-elliptic with warts up to 1 micrometre high.
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The mature leaves are glabrescent, and lanceolate to broadly elliptic in shape.
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In mathematics, the Jacobi elliptic functions are a set of basic elliptic functions, and auxiliary theta functions, that are of historical importance. They are found in the description of the motion of a pendulum (see also pendulum (mathematics)), as well as in the design of the electronic elliptic filters. While trigonometric functions are defined with reference to a circle, the Jacobi elliptic functions are a generalization which refer to other conic sections, the ellipse in particular. The relation to trigonometric functions is contained in the notation, for example, by the matching notation sn for sin.
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The phyllodes have a narrowly elliptic to narrowly elliptic, sometimes narrowly oblanceolate shape. The flowers between January or April and September are yellow, and held in cylindrical clusters in length. The pods are papery, about long and wide.
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B. tokioi has creeping rhizomes (0.5 mm diam.), lacking pseudobulbs. Its fleshy, sessile, glabrous leaves are minute (5–6 mm long by 3-4.5 mm wide), elliptic or elliptic-orbicular, acute or obtuse, some having tiny, membranaceous sheaths at base. It has axillary scapes (1–3 cm long) that are slender and erect, with a few sheaths close by its base. Bracts are elliptic and acute.
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Alveolina is an extinct genus of foraminifera with an elliptic or spheric form.
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The dark brown seeds have an elliptic-lenticular shape with a length of .
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The Peirce quincuncial projection is a map projection based on Jacobian elliptic functions.
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In elliptic curve cryptography, the multiplication in a finite field is replaced by the operation of adding points on an elliptic curve, i.e., the solutions of an equation of the form :. Finite fields are also used in coding theory and combinatorics.
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The 4-torsion subgroup of an elliptic curve over the complex numbers. The torsion elements of an abelian variety are torsion points or, in an older terminology, division points. On elliptic curves they may be computed in terms of division polynomials.
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The usual foot (transmission) and hand (back wheels) brake levers are provided. The pressed steel frame is suspended by three-quarter elliptic springs at the rear and semi-elliptic in front. The steering gear incorporates a provision to take up wear.
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The lower glume by itself is elliptic just like lemma, with an erose apex. The species palea is elliptic too, is long and have 2 veines. Paleas keels are ciliated and adorned. Flowers are fleshy, oblong, truncate, and grow together.
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Hyperelliptic curve cryptography is similar to elliptic curve cryptography (ECC) insofar as the Jacobian of a hyperelliptic curve is an abelian group in which to do arithmetic, just as we use the group of points on an elliptic curve in ECC.
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Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme. They are also used in several integer factorization algorithms based on elliptic curves that have applications in cryptography, such as Lenstra elliptic-curve factorization.
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The evergreen phyllodes are ascending and have an obliquely oblong-elliptic to narrowly elliptic shape with a length of and a width of with a mid-nerve that is quite prominent. It blooms from May to August producing yellow flowers.
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Note that Ribet's theorem does not guarantee that if one begins with an elliptic curve E of conductor qN, there exists an elliptic curve E' of level N such that ρE, p ≈ ρE′, p. The newform g of level N may not have rational Fourier coefficients, and hence may be associated to a higher-dimensional abelian variety, not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation :E: y^2 + xy + y = x^3 - 663204x + 206441595 with conductor 43×97 and discriminant 437 × 973 does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod p Galois representation is isomorphic to the mod p Galois representation of an irrational newform g of level 97.
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The chassis had half elliptic leaf springs at the front and quarter elliptic at the rear and the brakes used rather primitive metal shoes. The 65 mph Sports version with a Coventry-Simplex engine of 1498 cc capacity had quarter elliptic leaf springs all round. A Dorman engined model was introduced in 1921 but it is believed that only two were made. The last cars were made in 1921.
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Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide equivalent security.Commercial National Security Algorithm Suite and Quantum Computing FAQ U.S. National Security Agency, January 2016. Elliptic curves are applicable for key agreement, digital signatures, pseudo- random generators and other tasks.
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In mathematics, a Shioda modular surface is one of the elliptic surfaces studied by .
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By carefully studying the spherical and elliptic flow, experimentalists put the theory to test.
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QUICK is most appropriate for steady flow or quasi-steady highly convective elliptic flow.
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The fruit is approximately elliptic, 9 mm (0.36 in) long, almost hairless and shining.
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Pascal was remembered for his work on elliptic functions based on Jacobi theta function.
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In elliptic geometry, two lines are Clifford parallel or paratactic lines if the perpendicular distance between them is constant from point to point. The concept was first studied by William Kingdon Clifford in elliptic space and appears only in spaces of at least three dimensions. Since parallel lines have the property of equidistance, the term "parallel" was appropriated from Euclidean geometry, although the "lines" of elliptic geometry are geodesic curves and, unlike the lines of Euclidean geometry, are of finite length. The algebra of quaternions provides a descriptive geometry of elliptic space in which Clifford parallelism is made explicit.
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The size of the elliptic curve determines the difficulty of the problem. The U.S. National Institute of Standards and Technology (NIST) has endorsed elliptic curve cryptography in its Suite B set of recommended algorithms, specifically elliptic-curve Diffie–Hellman (ECDH) for key exchange and Elliptic Curve Digital Signature Algorithm (ECDSA) for digital signature. The U.S. National Security Agency (NSA) allows their use for protecting information classified up to top secret with 384-bit keys. However, in August 2015, the NSA announced that it plans to replace Suite B with a new cipher suite due to concerns about quantum computing attacks on ECC.
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Graphs of curves y2 = x3 − x and y2 = x3 − x + 1 Although the formal definition of an elliptic curve requires some background in algebraic geometry, it is possible to describe some features of elliptic curves over the real numbers using only introductory algebra and geometry. In this context, an elliptic curve is a plane curve defined by an equation of the form :y^2 = x^3 + ax + b where a and b are real numbers. This type of equation is called a Weierstrass equation. The definition of elliptic curve also requires that the curve be non-singular.
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An elliptic surface is a surface equipped with an elliptic fibration (a surjective holomorphic map to a curve B such that all but finitely many fibers are smooth irreducible curves of genus 1). The generic fiber in such a fibration is a genus 1 curve over the function field of B. Conversely, given a genus 1 curve over the function field of a curve, its relative minimal model is an elliptic surface. Kodaira and others have given a fairly complete description of all elliptic surfaces. In particular, Kodaira gave a complete list of the possible singular fibers.
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In mathematics the division polynomials provide a way to calculate multiples of points on elliptic curves and to study the fields generated by torsion points. They play a central role in the study of counting points on elliptic curves in Schoof's algorithm.
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In mathematics, the supersingular isogeny graphs are a class of expander graphs that arise in computational number theory and have been applied in elliptic-curve cryptography. Their vertices represent supersingular elliptic curves over finite fields and their edges represent isogenies between curves.
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The original construction of tmf uses the obstruction theory of Hopkins, Miller, and Paul Goerss, and is based on ideas of Dwyer, Kan, and Stover. In this approach, one defines a presheaf Otop ("top" stands for topological) of multiplicative cohomology theories on the etale site of the moduli stack of elliptic curves and shows that this can be lifted in an essentially unique way to a sheaf of E-infinity ring spectra. This sheaf has the following property: to any etale elliptic curve over a ring R, it assigns an E-infinity ring spectrum (a classical elliptic cohomology theory) whose associated formal group is the formal group of that elliptic curve. A second construction, due to Jacob Lurie, constructs tmf rather by describing the moduli problem it represents and applying general representability theory to then show existence: just as the moduli stack of elliptic curves represents the functor that assigns to a ring the category of elliptic curves over it, the stack together with the sheaf of E-infinity ring spectra represents the functor that assigns to an E-infinity ring its category of oriented derived elliptic curves, appropriately interpreted.
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The glossy black seeds within have a broadly elliptic shape and are about in length.
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Since the Cauchy-Riemann equations form an elliptic operator, it follows that f is smooth.
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Indigo Mist. Vu/Karpen Project. Rarenoise Records, CD and LP, 2014. Aperture II and Elliptic.
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Study was careful in 1905 to distinguish the hyperbolic and elliptic cases in Hermitian geometry.
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The oblong to elliptic shiny black seeds found within the pods have a length of .
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Karl Joseph Bobek (1855–1899) was a German mathematician working on elliptic functions and geometry.
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The performance of the proposed floating immittance is demonstrated on a fifth order elliptic filter.
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An elliptic curve over the complex numbers is obtained as a quotient of the complex plane by a lattice Λ, here spanned by two fundamental periods ω1 and ω2. The four-torsion is also shown, corresponding to the lattice 1/4 Λ containing Λ. The formulation of elliptic curves as the embedding of a torus in the complex projective plane follows naturally from a curious property of Weierstrass's elliptic functions. These functions and their first derivative are related by the formula :\wp'(z)^2 = 4\wp(z)^3 -g_2\wp(z) - g_3 Here, g2 and g3 are constants; \wp(z) is the Weierstrass elliptic function and \wp'(z) its derivative. It should be clear that this relation is in the form of an elliptic curve (over the complex numbers).
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Since they are based on elliptic integrals, they were the first examples of elliptic functions. Similar functions were shortly thereafter defined by Carl Gustav Jacobi. In spite of the Abel functions having several theoretical advantages, the Jacobi elliptic functions have become the standard. This can have to do with the fact that Abel died only two years after he presented them while Jacobi could continue with his exploration of them throughout his lifetime.
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In mathematics, elliptic units are certain units of abelian extensions of imaginary quadratic fields constructed using singular values of modular functions, or division values of elliptic functions. They were introduced by Gilles Robert in 1973, and were used by John Coates and Andrew Wiles in their work on the Birch and Swinnerton-Dyer conjecture. Elliptic units are an analogue for imaginary quadratic fields of cyclotomic units. They form an example of an Euler system.
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Kolyvagin later used Heegner points to construct Euler systems, and used this to prove much of the Birch-Swinnerton-Dyer conjecture for rank 1 elliptic curves. Brown proved the Birch–Swinnerton-Dyer conjecture for most rank 1 elliptic curves over global fields of positive characteristic .
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The chassis was of 'tubular trussed construction' with half elliptic front springing and quarter elliptic rear. The brakes comprised a foot-operated band brake operating on the transmission countershaft, and two internally expanding brakes on the rear axle operated by the hand brake lever.
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He was awarded a National Science Foundation grant to work on Elliptic Fibrations and String Theory in 2014. This allowed him to investigate F-theory and elliptic fibrations. In 2017 Esole was named a NextEinstein Forum Fellow. This award celebrates the best young African scientists.
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The separate body was built on a conventional ladder frame; fore and aft there were solid axles, on semi- elliptic springs at the front and three-quarter elliptic springs at the rear. Braking was by drums on the transmission and on the rear wheels.
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It spikelets are elliptic and are long. The glumes are purple in colour with pale green florets that have 2-3 fertile florets. The stem itself is with its lemma being elliptic and long. It is also herbaceous, granular- scaberulous and is 5–7-veined.
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Magnolia rimachii has chartaceous elliptic leaves 12–26 cm long and 5–10 cm broad. Flowers are fragrant and can have 6 or 7 obovate petals 2–4.5 cm long and 1–2 cm wide. The elliptic fruit can be ca. 3.5 cm long.
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For elliptic curves, potential good reduction is equivalent to the j-invariant being an algebraic integer.
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In the case of an elliptic curve there is an algorithm of John Tate describing it.
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This is a timeline of the theory of abelian varieties in algebraic geometry, including elliptic curves.
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The cocoon is 7–9 mm, white, flat elliptic, with some white grains attached on surface.
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L. A. Sohnke was a German mathematician who worked on the complex multiplication of elliptic functions.
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In mathematics, Raynaud's isogeny theorem, proved by , relates the Faltings heights of two isogeneous elliptic curves.
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Sun Labs reported implementations of RSA and elliptic curve cryptography (ECC) optimized for small embedded devices.
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Leaves are compound with 2–6 obovate to oblong-elliptic, smooth, somewhat glossy, somewhat thick leaflets.
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Spruck is well known in the field of elliptic partial differential equations for his series of papers "The Dirichlet problem for nonlinear second-order elliptic equations," written in collaboration with Luis Caffarelli, Joseph J. Kohn, and Louis Nirenberg. These papers were among the first to develop a general theory of second-order elliptic differential equations which are fully nonlinear, with a regularity theory that extends to the boundary. Caffarelli, Nirenberg & Spruck (1985) has been particularly influential in the field of geometric analysis since many geometric partial differential equations are amenable to its methods. With Basilis Gidas, Spruck studied positive solutions of subcritical second-order elliptic partial differential equations of Yamabe type.
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This allows fast arithmetic in these groups, which can speed up the calculations with a factor 3 compared with elliptic curves and hence speed up the cryptosystem. Another advantage is that for groups of cryptographically relevant size, the order of the group can simply be calculated using the characteristic polynomial of the Frobenius endomorphism. This is not the case, for example, in elliptic curve cryptography when the group of points of an elliptic curve over a prime field is used for cryptographic purpose. However to represent an element of the trace zero variety more bits are needed compared with elements of elliptic or hyperelliptic curves.
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Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems based their security on the assumption that it is difficult to factor a large integer composed of two or more large prime factors. For later elliptic-curve- based protocols, the base assumption is that finding the discrete logarithm of a random elliptic curve element with respect to a publicly known base point is infeasible: this is the "elliptic curve discrete logarithm problem" (ECDLP). The security of elliptic curve cryptography depends on the ability to compute a point multiplication and the inability to compute the multiplicand given the original and product points.
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Silverman has written two graduate texts on elliptic curves, The Arithmetic of Elliptic Curves (1986) and Advanced Topics in the Arithmetic of Elliptic Curves (1994). For these two books he received a Steele Prize for Mathematical Exposition from the American Mathematical Society, which cited them by saying that “Silverman's volumes have become standard references on one of the most exciting areas of algebraic geometry and number theory.” Silverman has also written three undergraduate texts: Rational Points on Elliptic Curves (1992, co-authored with John Tate), A Friendly Introduction to Number Theory (3rd ed. 2005), and An Introduction to Mathematical Cryptography (2008, co-authored with Jeffrey Hoffstein and Jill Pipher).
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It comes from the phrase "singular values of the j-invariant" used for values of the j-invariant for which a complex elliptic curve has complex multiplication. The complex elliptic curves with complex multiplication are those for which the endomorphism ring has the maximal possible rank 2. In positive characteristic it is possible for the endomorphism ring to be even larger: it can be an order in a quaternion algebra of dimension 4, in which case the elliptic curve is supersingular. The primes p such that every supersingular elliptic curve in characteristic p can be defined over the prime subfield F_p rather than F_{p^m} are called supersingular primes.
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The white lip is accompanied by dark yellow lateral lobes with dark red stripes at the base. The dorsal sepal is elliptic to elliptic-ovate and obtuse-rounded. The lateral sepals are obliquely ovate, sub-acute and divergent. Petals are rhomboid, cuneate-clawed, obtuse and broadly rounded.
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It is a small, bushy and glabrous shrub that typically grows to in height and across. It has smooth grey coloured bark. The distinctive red branches are angled upward and have prominent ridges. The green slightly curved phyllodes have an elliptic to narrowly elliptic or oblanceolate shape.
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Here is an image showing the elliptic filter next to other common kind of filters obtained with the same number of coefficients: upright=3.6 As is clear from the image, elliptic filters are sharper than all the others, but they show ripples on the whole bandwidth.
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In his doctoral dissertation, submitted in 1904 to the Sorbonne, Bernstein solved Hilbert's nineteenth problem on the analytic solution of elliptic differential equations. His later work was devoted to Dirichlet's boundary problem for non-linear equations of elliptic type, where, in particular, he introduced a priori estimates.
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In mathematics, the Schneider–Lang theorem is a refinement by of a theorem of about the transcendence of values of meromorphic functions. The theorem implies both the Hermite–Lindemann and Gelfond–Schneider theorems, and implies the transcendence of some values of elliptic functions and elliptic modular functions.
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If Y is a sphere and X is some point embedded in Y, then any elliptic operator on Y is the image under i! of some elliptic operator on the point. This reduces the index theorem to the case of a point, where it is trivial.
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Ochna lanceolata is a species of plant in the family Ochnaceae. It is native to India and Sri Lanka. It is an 8m tall plant with greyish bark and reddish blaze. Leaves are simple, alternate; lamina narrow elliptic, elliptic- lanceolate; apex acute; base acute with serrate margin.
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With Morrey, Nirenberg proved that solutions of elliptic systems with analytic coefficients are themselves analytic, extending to the boundary earlier known work. These contributions to elliptic regularity are now considered as part of a "standard package" of information, and are covered in many textbooks. The Douglis-Nirenberg and Agmon-Douglis-Nirenberg estimates, in particular, are among the most widely-used tools in elliptic partial differential equations.Morrey, Charles B., Jr. Multiple integrals in the calculus of variations.
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Chiloschista segawae is an epiphytic, leafless herb that forms clumps with many flattened greenish, photosynthetic roots up to long radiating from inconspicuous stems. Between six and fifteen slightly fleshy, whitish green or yellow resupinate flowers are arranged along a pendulous flowering stem long. The dorsal sepal is broadly elliptic, long, wide, the lateral sepals are broadly elliptic to egg- shaped, long, wide and the petals are elliptic, long, wide. The labellum is long with three lobes.
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In mathematics, Weierstrass's elliptic functions are elliptic functions that take a particularly simple form; they are named for Karl Weierstrass. This class of functions are also referred to as p-functions and generally written using the symbol ℘ (a calligraphic lowercase p). The ℘ functions constitute branched double coverings of the Riemann sphere by the torus, ramified at four points. They can be used to parametrize elliptic curves over the complex numbers, thus establishing an equivalence to complex tori.
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He solved 19th Hilbert problem on the regularity of solutions of elliptic partial differential equations. Before his results, mathematicians were not able to venture beyond second order nonlinear elliptic equations in two variables. In a major breakthrough, De Giorgi proved that solutions of uniformly elliptic second order equations of divergence form, with only measurable coefficients, were Hölder continuous. His proof was proved in 1956/57 in parallel with John Nash's, who was also working on and solved Hilbert's problem.
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In the mathematical field of algebraic geometry, an elliptic curve E over a field K has an associated quadratic twist, that is another elliptic curve which is isomorphic to E over an algebraic closure of K. In particular, an isomorphism between elliptic curves is an isogeny of degree 1, that is an invertible isogeny. Some curves have higher order twists such as cubic and quartic twists. The curve and its twists have the same j-invariant.
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Prof. Smart is best known for his work in elliptic curve cryptography, especially work on the ECDLP.S. D. Galbraith and N. P. Smart, A cryptographic application of the Weil descent, Cryptography and Coding, 1999.P. Gaudry, F. Hess, and N. P. Smart, Constructive and destructive facets of Weil descent on elliptic curves, Hewlett Packard Laboratories Technical Report, 2000.N. Smart, The discrete logarithm problem on elliptic curves of trace one, Journal of Cryptology, Volume 12, 1999.
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This was extended to the case where F is any finite abelian extension of K by . # showed that if a modular elliptic curve has a first-order zero at s = 1 then it has a rational point of infinite order; see Gross–Zagier theorem. # showed that a modular elliptic curve E for which L(E, 1) is not zero has rank 0, and a modular elliptic curve E for which L(E, 1) has a first-order zero at s = 1 has rank 1. # showed that for elliptic curves defined over an imaginary quadratic field K with complex multiplication by K, if the L-series of the elliptic curve was not zero at s = 1, then the p-part of the Tate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p > 7.
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Suspension was also typical, with half-elliptic springs all round damped by Andre Hartford friction shock absorbers.
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This essentially follows because the three-wave interaction has exact solutions that are given by elliptic functions.
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The cocoon is 6–8 mm, white, flat elliptic, with some white grains attached on the surface.
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The glossy black seeds have a broadly elliptic shape and a length of with an apical aril.
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Her work in number theory includes the invention with Joe Kilian of primality proving using elliptic curves.
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In mathematics, the Brumer bound is a bound for the rank of an elliptic curve, proved by .
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Therefore, Cosset roughly claims that using hyperelliptic curves for factorization is no worse than using elliptic curves.
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The dark brown swollen seeds have an elliptic shape and a length of with an orbicular areole.
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Hardt, Robert; Simon, Leon. Nodal sets for solutions of elliptic equations. J. Differential Geom. 30 (1989), no.
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The stamens are found inside. The grey-green leaves are oblong- elliptic, crenate-denate and usually long.
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Melicope glabra grows up to tall. The fruits are round to elliptic and measure up to long.
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The four petals are pink, oblong to elliptic and about long and the eight stamens are hairy.
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A hyperelliptic curve is a particular kind of algebraic curve. There exist hyperelliptic curves of every genus g \geq 1. If the genus of a hyperelliptic curve equals 1, we simply call the curve an elliptic curve. Hence we can see hyperelliptic curves as generalizations of elliptic curves.
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The inflorescence consists of a tight grouping of usually three pale yellow flowers on a more or less smooth peduncle. The calyx long, smooth, lobes triangular shaped, petals narrowly elliptic, long, smooth and the stamens long. The elliptic-shaped fruit are long ending with a beak long at maturity.
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X, 275–379, Surv. Differ. Geom., 10, Int. Press, Somerville, MA, 2006. He is especially known for his collaboration with Shmuel Agmon and Avron Douglis in which they extended the Schauder theory, as previously understood for second- order elliptic partial differential equations, to the general setting of elliptic systems.
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Both the lower and upper glumes are elliptic, keelless, membranous, and have acute apexes. Their size is different; Lower glume is long while the upper one is long. Palea is elliptic, have scabrous surface and is 2-veined. Flowers are fleshy, oblong, truncate, have 2 lodicules, and grow together.
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It is a shrub up to 1m tall. The leaves are elliptic to oblong- elliptic, rounded to obtuse at the apex and rounded to cuneate at the base. Domatia are absent and the petioles are up to 5mm long. Flowers are carried in 3 to 8-flowered cymes.
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Quercus helferiana is a tree up to 20 m. tall, with a trunk up to 0.3 m in diameter. Leaves oblong-elliptic, to elliptic-lanceolate, 120-150 (up to 220) × 40-80 (up to 95) mm, with wavy edges but no teeth or lobes.Flora of China, Cyclobalanopsis helferiana (A.
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Fertile spikelets are pediceled and have rhachilla stems that are long. Florets are diminished at the apex. Its lemma have scaberulous surface and emarginated apex with fertile lemma being chartaceous elliptic, keelless, and long. Both the lower and upper glumes are elliptic, keelless, membranous, and have acute apexes.
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Ardisia elliptica is a tropical understory shrub that can reach heights of up 5 meters. Undamaged plants in forest habitats are characterized by a single stem, producing short, perpendicular branches. Leaves are elliptic to elliptic-obovate, entire, leathery and alternate. Umbellate inflorescences develop in leaf axils of branch leaves.
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Clifford elaborated elliptic space geometry as a non-Euclidean metric space. Equidistant curves in elliptic space are now said to be Clifford parallels. John Collier Clifford's contemporaries considered him acute and original, witty and warm. He often worked late into the night, which may have hastened his death.
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Historically, elliptic cohomology arose from the study of elliptic genera. It was known by Atiyah and Hirzebruch that if S^1 acts smoothly and non-trivially on a spin manifold, then the index of the Dirac operator vanishes. In 1983, Witten conjectured that in this situation the equivariant index of a certain twisted Dirac operator is at least constant. This led to certain other problems concerning S^1-actions on manifolds, which could be solved by Ochanine by the introduction of elliptic genera.
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Alternative proofs use the smoothness of the fundamental solution of the Laplacian or suitable a priori elliptic estimates.
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Using these for imaginary values of the modulus, one can therefore also calculate the corresponding Abel elliptic functions.
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Buneman advanced elliptic equation solver methods and their associated applications (as well as for the fast Fourier transforms).
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Agmon's contributions to partial differential equations include Agmon's method for proving exponential decay of eigenfunctions for elliptic operators.
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The four petals are elliptic and about long and the eight stamens are hairy. Flowering occurs in September.
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Rather, the case g = 1 (if we choose a distinguished point) is an elliptic curve. Hence the terminology.
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The shiny black seeds within are arranged longitudinally and have an oblong-elliptic shape with a length of .
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The glossy black seeds have an elliptic shape with a length of and a sub-conical terminal aril.
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The petals are broadly elliptic, about long with a prominent keel. Flowering mainly occurs in spring and autumn.
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When fully written out, the EFE are a system of ten coupled, nonlinear, hyperbolic-elliptic partial differential equations.
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Hoberman mechanism. Same as the crank-driven elliptic trammel, Hoberman mechanisms move because of their particular geometric configurations.
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The mature leaves are elliptic, 7.5 cm long by 2.5 cm wide, green above and grey- green below.
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The slightly glossy black seeds have a broadly elliptic shape with a length of and an apical aril.
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The dark brown seeds are obliquely arranged with a narrowly oblong to elliptic shape with a length of .
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This agrees with the elliptic curve case, because it can be shown that the Jacobian of an elliptic curve is isomorphic with the group of points on the elliptic curve. The use of hyperelliptic curves in cryptography came about in 1989 from Neal Koblitz. Although introduced only 3 years after ECC, not many cryptosystems implement hyperelliptic curves because the implementation of the arithmetic isn't as efficient as with cryptosystems based on elliptic curves or factoring (RSA). The efficiency of implementing the arithmetic depends on the underlying finite field K, in practice it turns out that finite fields of characteristic 2 are a good choice for hardware implementations while software is usually faster in odd characteristic.
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More generally, any algebraic curve of genus one, for example the intersection of two quadric surfaces embedded in three-dimensional projective space, is called an elliptic curve, provided that it is equipped with a marked point to act as the identity. Using the theory of elliptic functions, it can be shown that elliptic curves defined over the complex numbers correspond to embeddings of the torus into the complex projective plane. The torus is also an abelian group, and this correspondence is also a group isomorphism. Elliptic curves are especially important in number theory, and constitute a major area of current research; for example, they were used in Andrew Wiles's proof of Fermat's Last Theorem.
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The lack of the notion of prime elements in the group of points on elliptic curves makes it impossible to find an efficient factor base to run index calculus method as presented here in these groups. Therefore this algorithm is incapable of solving discrete logarithms efficiently in elliptic curve groups. However: For special kinds of curves (so called supersingular elliptic curves) there are specialized algorithms for solving the problem faster than with generic methods. While the use of these special curves can easily be avoided, in 2009 it has been proven that for certain fields the discrete logarithm problem in the group of points on general elliptic curves over these fields can be solved faster than with generic methods.
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Frits van Sold, Obadiah Elliott & the Elliptic Spring, The Carriage Journal: Vol 53 No 2 March 2015 Coachbuilder Obadiah Elliott obtained a patent covering the use of elliptic springs - which were not his invention. His patent lasted 14 years delaying development because Elliott allowed no others to license and use his patent. Elliott mounted each wheel with two durable elliptic steel leaf springs on each side and the body of the carriage was fixed directly to the springs attached to the axles. After the expiry of his patent most British horse carriages were equipped with elliptic springs; wooden springs in the case of light one-horse vehicles to avoid taxation, and steel springs in larger vehicles.
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The elliptic functions thus have a double periodicity. Equivalently one can say that they have two complex periods . Their zeros and poles will thus form a regular, two-dimensional lattice. Corresponding properties of the lemniscatic elliptic functions had also been established by Gauss, but not published before after his death.
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Median sepal apiculate, galea 8–10 mm deep; spur slender, grading into the galea, 7–20 mm long; lateral sepals projecting away;elliptic to narrowly elliptic, with apiculi up to 4 mm long;petals spear-shaped, 5–7 mm long; lip narrowly egg- to spear-shaped, 10–12 mm long.
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Aside from the boundary condition, boundary value problems are also classified according to the type of differential operator involved. For an elliptic operator, one discusses elliptic boundary value problems. For a hyperbolic operator, one discusses hyperbolic boundary value problems. These categories are further subdivided into linear and various nonlinear types.
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Acronychia eungellensis, commonly known as Eungella aspen, is a species of small rainforest tree that is endemic to a restricted area in east-central Queensland. It has simple, elliptic leaves on cylindrical stems, flowers in small groups in leaf axils, and fleshy fruit that is elliptic to egg-shaped in outline.
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Spikelets are long and are both elliptic and solitary. They also carry both a pediceled fertile spikelet and one fertile floret which have a hairless callus. The glumes are long, lanceolate, membranous and have acute apexes. Fertile lemma is of the same size as glumes and is both elliptic and hyaline.
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Adam Langley has posted speed tests on his personal website showing Curve25519, used by DNSCurve, to be the fastest among elliptic curves tested. According to the U.S. National Security Agency (NSA), elliptic curve cryptography offers vastly superior performance over RSA and Diffie–Hellman at a geometric rate as key sizes increase.
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The pods have a narrowly oblong to oblong-elliptic shape and are narrower at the base. The light brown, straight or shallowly curved pods are in length and have obliquely longitudinal nerves. The broen seeds inside the pods have an oblong- elliptic shape and are in length with a dark pleurogram.
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This species is a shrub, growing up to 3 metres in height. Its leaves are narrow-obovate to round or elliptic to narrow-elliptic . The flowers which are red, or occasionally pink, appear predominantly from late winter to late spring (August to November in Australia) but appear sporadically throughout the year.
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Criteria to be a product of two elliptic curves (up to isogeny) were a popular study in the nineteenth century. Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1 × S1 × S1 × S1 so the fundamental group is Z4. Hodge diamond: : Examples: A product of two elliptic curves.
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Droogmansia megalantha grows as a shrub up to tall. The elliptic or oblong leaves measure up to long and are glabrescent to pilose. Inflorescences measure up to long and have many flowers with bright red petals. The oblong or elliptic fruits are hairy and yellowish and measure up to long.
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For examples, elliptic geometry (no parallels) and hyperbolic geometry (many parallels). Both elliptic and hyperbolic geometry are consistent systems, showing that the parallel postulate is independent of the other axioms.Harold Scott Macdonald Coxeter Non-Euclidean Geometry, pages 1-15 Proving independence is often very difficult. Forcing is one commonly used technique.
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Elliptic in shape, toothed, glossy green and almost without leaf stalks. Leaf veins raised and noticeable on both surfaces.
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The brown seeds inside have an oblong elliptic to orbicular shape with a length of and a conical aril.
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By standard elliptic theory, this is possible because the integral of over is zero, by the Gauss–Bonnet theorem.
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Pyrola elliptica, known as shinleaf, shinleaf pyrola, waxflower shinleaf, elliptic shineleaf and white wintergreen is a species of heath.
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Elliptic-Cylindrical Wavelets: The Mathieu Wavelets,IEEE Signal Processing Letters, vol.11, n.1, January, pp. 52-55, 2004.
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The shiny black seeds have an oblong- elliptic shape with a length of around and with a black aril.
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The four petals are broadly elliptic, long and thickened- glandular along the mid-line. The eight stamens are hairy.
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Shor's algorithm can also efficiently solve the discrete logarithm problem, which is the basis for the security of Diffie–Hellman, elliptic curve Diffie–Hellman, elliptic curve DSA, Curve25519, ed25519, and ElGamal. Although quantum computers are currently in their infancy, the ongoing development of quantum computers and their theoretical ability to compromise modern cryptographic protocols (such as TLS/SSL) has prompted the development of post-quantum cryptography. SIDH was created in 2011 by De Feo, Jao, and Plut. It uses conventional elliptic curve operations and is not patented.
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The modularity theorem (formerly called the Taniyama–Shimura conjecture, Taniyama-Weil conjecture or modularity conjecture for elliptic curves) states that elliptic curves over the field of rational numbers are related to modular forms. Andrew Wiles proved the modularity theorem for semistable elliptic curves, which was enough to imply Fermat's Last Theorem. Later, a series of papers by Wiles's former students Brian Conrad, Fred Diamond and Richard Taylor, culminating in a joint paper with Christophe Breuil, extended Wiles's techniques to prove the full modularity theorem in 2001.
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A semistable elliptic curve may be described more concretely as an elliptic curve that has bad reduction only of multiplicative type.Husemöller (1987) pp.116-117 Suppose E is an elliptic curve defined over the rational number field Q. It is known that there is a finite, non-empty set S of prime numbers p for which E has bad reduction modulo p. The latter means that the curve Ep obtained by reduction of E to the prime field with p elements has a singular point.
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Elliptic flow is a fundamental observable since it directly reflects the initial spatial anisotropy, of the nuclear overlap region in the transverse plane, directly translated into the observed momentum distribution of identified particles. Since the spatial anisotropy is largest at the beginning of the evolution, elliptic flow is especially sensitive to the early stages of system evolution. A measurement of elliptic flow thus provides access to the fundamental thermalization time scale and many more things in the early stages of a relativistic heavy-ion collision.
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Leaves persistent, coriaceous, blades 1–3 cm wide; calyx lobes neither foliaceous nor overlapping in bud (2). 2\. Plants green, glabrous or glandular; leaves 4–9 cm long, elliptic to narrowly elliptic; calyx lobes lanceolate and longer than the tube at anthesis; HI exc, Ni & Ka .….2. Vaccinium dentatum 2\. Plants pubescent or glaucous, or both; leaves 1–3 cm long, ovate to obovate or rarely elliptic; calyx lobes deltate, usually not as long as the tube at tnthesis; K, O, Mo, M, H ….. 3.
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André Néron (November 30, 1922, La Clayette, France - April 6, 1985, Paris, France) was a French mathematician at the Université de Poitiers who worked on elliptic curves and abelian varieties. He discovered the Néron minimal model of an elliptic curve or abelian variety, the Néron differential, the Néron–Severi group, the Néron–Ogg–Shafarevich criterion, the local height and Néron–Tate height of rational points on an abelian variety over a discrete valuation ring or Dedekind domain, and classified the possible fibers of an elliptic fibration.
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Leaves are elliptic, narrowly ovate-round or obovate-elliptic 4.2-10.5 cm long and 2.2-4.0 cm wide, and glabrous; the petioles are 5–8 mm long. The fruit has one seed in it, the seed is only 8 mm long. Flowers have five petals and they are yellow or yellowish-green.
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Viola stipularis at Guadeloupe. Herb 20–30 cm tall, spreading by creeping rhizomes. Petioles up to 8 mm long, surrounded by fringed triangular stipules up to 2 cm long. Leaves elliptic to lanceolate-elliptic,up to 9.5 cm long and 3.4 cm wide, margin serrate or crenate, sometimes dentate, apex acuminate, base cuneate.
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Fertile spikelets are pediceled, the pedicels of which are filiform and are long. Florets are diminished at the apex. Its lemma have scabrous surface and acute apex with fertile lemma is being chartaceous, elliptic, keelless, and is long. Both the lower and upper glumes are elliptic, keelless, membranous, and have acute apexes.
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Instead of just one elliptic operator, one can consider a family of elliptic operators parameterized by some space Y. In this case the index is an element of the K-theory of Y, rather than an integer. If the operators in the family are real, then the index lies in the real K-theory of Y. This gives a little extra information, as the map from the real K theory of Y to the complex K theory is not always injective. Atiyah's former student Graeme Segal (in 1982), who worked with Atiyah on equivariant K-theory With Bott, Atiyah found an analogue of the Lefschetz fixed-point formula for elliptic operators, giving the Lefschetz number of an endomorphism of an elliptic complex in terms of a sum over the fixed points of the endomorphism. As special cases their formula included the Weyl character formula, and several new results about elliptic curves with complex multiplication, some of which were initially disbelieved by experts.
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They are arranged in dense clusters. The fruits are elliptic and deeply ridged, becoming light brown and buoyant when ripe.
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Werner is the author of a German- language book on elliptic curve cryptography, Elliptische Kurven in der Kryptographie (Springer, 2002).
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Lamina membr., glab., ovate to ovate-elliptic to lanceolate, acuminate, tapering to petiole; ± 60-(75) × 20- (35) mm.; margins ± waved.
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The warty or wrinkled fruit are broadly elliptic to circular long and wide ending in a coarse short oblique beak.
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The seeds insode are arranged transversely within the pod and have a broadly elliptic shape with a length of around .
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274, number 5 8\. Optimal sliding mode in systems described by the equations of elliptic type Izv.AN Azerbaijani SSR, ser.fiz.mat.
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As an example, Fig. 13 shows elliptic LCSs identified as closed \lambda- lines within the Great Red Spot of Jupiter.
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Bi- elliptic transfer from blue to red circular orbit In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver. The bi-elliptic transfer consists of two half elliptic orbits. From the initial orbit, a delta-v is applied boosting the spacecraft into the first transfer orbit with an apoapsis at some point r_b away from the central body. At this point, a second delta-v is applied sending the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third delta-v is performed, injecting the spacecraft into the desired orbit.
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While the RSA patent expired in 2000, there may be patents in force covering certain aspects of ECC technology. However some argue that the US government elliptic curve digital signature standard (ECDSA; NIST FIPS 186-3) and certain practical ECC-based key exchange schemes (including ECDH) can be implemented without infringing them, including RSA Laboratories and Daniel J. Bernstein. The primary benefit promised by elliptic curve cryptography is a smaller key size, reducing storage and transmission requirements, i.e. that an elliptic curve group could provide the same level of security afforded by an RSA-based system with a large modulus and correspondingly larger key: for example, a 256-bit elliptic curve public key should provide comparable security to a 3072-bit RSA public key.
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Graphical illustration of an elliptic function where its values are indicated by colours. These are periodically repeated in the two directions of the complex plane. Abel elliptic functions are holomorphic functions of one complex variable and with two periods. They were first established by Niels Henrik Abel and are a generalization of trigonometric functions.
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Christoffel contributed to complex analysis, where the Schwarz–Christoffel mapping is the first nontrivial constructive application of the Riemann mapping theorem. The Schwarz–Christoffel mapping has many applications to the theory of elliptic functions and to areas of physics. In the field of elliptic functions he also published results concerning abelian integrals and theta functions.
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The straight and narrowly elliptic to oblong-elliptic shaped phyllodes with a length of and a width of . The semi-pungent phyllodes are thinly-coriaceous and have three distant raised main nerves with many parallel secondary nerves. It blooms from April to June and produces yellow flowers. The simple inflorescences occur singly in the axils.
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Edwards curves of equation x2 + y2 = 1 − d ·x2·y2 over the real numbers for d = 300 (red), d = (yellow) and d = −0.9 (blue) In mathematics, the Edwards curves are a family of elliptic curves studied by Harold Edwards in 2007. The concept of elliptic curves over finite fields is widely used in elliptic curve cryptography. Applications of Edwards curves to cryptography were developed by Daniel J. Bernstein and Tanja Lange: they pointed out several advantages of the Edwards form in comparison to the more well known Weierstrass form.
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The other oval, if it exists, does not contain any real inflection point and appears either as an oval or as two infinite branches. Like for conic sections, a line cuts this oval at, at most, two points. A non-singular plane cubic defines an elliptic curve, over any field K for which it has a point defined. Elliptic curves are now normally studied in some variant of Weierstrass's elliptic functions, defining a quadratic extension of the field of rational functions made by extracting the square root of a cubic.
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Lagrangian mechanics and Hamiltonian mechanics, when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes or dimensions (known as generalized coordinates in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth. In the early 19th century the theory of elliptic functions succeeded in giving a basis for the theory of elliptic integrals, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved the square roots of cubic and quartic polynomials.
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In mathematics, Hodge–Arakelov theory of elliptic curves is an analogue of classical and p-adic Hodge theory for elliptic curves carried out in the framework of Arakelov theory. It was introduced by . The main comparison in his theory remains unpublished as of 2019. Mochizuki's main comparison theorem in Hodge–Arakelov theory states (roughly) that the space of polynomial functions of degree less than d on the universal extension of a smooth elliptic curve in characteristic 0 is naturally isomorphic (via restriction) to the d2-dimensional space of functions on the d-torsion points.
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However, for p large enough compared to the level N of the level-lowered newform, a rational newform (e.g. an elliptic curve) must level-lower to another rational newform (e.g. elliptic curve). In particular for p ≫ NN1+ε, the mod p Galois representation of a rational newform cannot be isomorphic to that of an irrational newform of level N. Similarly, the Frey-Mazur conjecture predicts that for p large enough (independent of the conductor N), elliptic curves with isomorphic mod p Galois representations are in fact isogenous, and hence have the same conductor.
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Later, it widens into an elliptic blotch. Pupation takes place in the blotch.bladmineerders.nl Larvae can be found from March to April.
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The Gauss- Legendre algorithm can be proven without elliptic modular functions. This is done here and here using only integral calculus.
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For more information about the running-time required in a specific case, see Table of costs of operations in elliptic curves.
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Rivière is the author of the book Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation (SIAM, 2008).
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The seeds inside are arranged longitudinally and have an oblong to elliptic shpe with a length of and a conical aril.
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Without idempotency, total symmetric quasigroups correspond to the geometric notion of extended Steiner triple, also called Generalized Elliptic Cubic Curve (GECC).
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It has large elliptic leaves to 2 cm wide that are convex, which are papery to leathery in texture. The flowers are relatively larger than other forms and markedly hairy. The distinctive 'Picton form' has narrow elliptic leaves and smaller flower heads. This form resembles G. kedumbrensis and may be reclassified as a different taxon with future study.
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The concept can be generalized to more complex problems, for example with masses in elliptic orbits,Szenkovits, Z. M. F., & Csillik, I. (2004). Polynomial representation of the zero velocity surfaces in the spatial elliptic restricted three-body problem. Pure Mathematics and Application, 15(2-3), 323-322. the general planar three-body problem,Bozis, G. (1976).
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Rhododendron fuyuanense (富源杜鹃) is a rhododendron species native to eastern Yunnan, China, where it grows at altitudes of about 2000 meters. It is a shrub that grows to 0.5–2.5 m in height, with leaves that are elliptic or narrowly elliptic, 1.2–3.5 by 0.6–1.2 cm in size. Flowers are purplish red.
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The leaves are spreading to erect, and are more or less glaucous, and are in size. They are elliptic or rarely lanceolate-elliptic, are concolorous and thinly coriaceous. Their apex is acute to subacute or rounded-obtuse, with a rounded or cuneate base.They have 0-3 pairs of lateral veins and are unbranched (at least visibly).
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The bushy, rounded shrub typically grows to a height of . It has glabrous and resin-ribbed branchlets that are angled towards the apices. Like most species of Acacia it has phyllodes rather than true leaves. The evergreen slightly asymmetric phyllodes have a narrowly elliptic to oblong-elliptic shape and a length of and a width of .
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The spikelets carry 2–3 sterile florets which are cuneate, clumped, and long. Both the upper and lower glumes are elliptic, keelless, membranous, and have an acute apex. The lower glume is long while the upper one is long. Just like the lower glume, the fertile lemma is elliptic, keelless, and is 4–8 mm long.
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Kunzea aristulata is an erect, spreading shrub which grows to a height of up to with its branches silky hairy when young. The leaves are elliptic to broad elliptic, long, about wide and covered with soft hairs when young. The leaves often abruptly taper to a sharp point. Only the midvein of the leaf is prominent.
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The spreading multi-stemmed shrub typically grows to a height of and a width of around . The glabrous branchlets are commonly sericeous at the extremities. Like most species of Acacia it has phyllodes rather than true leaves. The patent to ascending phyllodes have a narrowly elliptic to oblong- elliptic shape and are straight or shallowly curved.
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Leaf blade ovate, elliptic, obovate-elliptic, oblong, or oblong-lanceolate; leathery to thinly leathery, pale green or glaucescent green and reddish brown glandular punctate. Axillary flowers are pale yellow. There are up to more than 10 in a corymb. The fruit is ellipsoid 2–3.5 cm in diameter and contains long obovate seeds, with a fleshy red outer layer.
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A terminal leaflet is seen on the end of the compound leaf. Leaf stalks vary between 20 mm and no leaf stalk in sub species leptophylla. Leaf shape varies between ovate or elliptic to broad-elliptic in sub species sambucifolia. However the sessile leaflets of sub species leptophylla are oblong linear and somewhat curved (falcate) in shape.
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The bushy glabrous shrub has a rounded to spreading habit and normally in height, sometimes reaching and usually to a width of . The bark is smooth and a light grey colour. The narrowly elliptic to oblong- elliptic or obovate to oblanceolate, phyllodes have a length of and a width of . It produces yellow flowers from April to November.
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Hakea elliptica is a dense, rounded, erect non-lignotuberous shrub or small tree typically grows to a height of . The smaller branches are covered with densely matted reddish brown hairs near flowering. The dark green leaves are alternately arranged with an elliptic to broadly elliptic shape ending in a sharp point. The leaves are flat, long and wide.
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The blackish coloured seed pods that form after flowering have an oblong to narrowly oblong shape and are raised over seeds. The glabrous pods have a length of up to and a width of and are firmly chartaceous. The seeds within the pod have an elliptic to widely elliptic shape with a length of and a width of .
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Sterculia balanghas is a species of plant in the family Malvaceae. It is native to India and Sri Lanka. Leaves are simple, alternate; swollen at base and tipped; lamina elliptic, obovate, oblong, elliptic-ovate or oblong-ovate; base subcordate or round; apex acuminate; with entire margin. Flowers may be unisexual or polygamous are yellow or greenish-purple in color.
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Saddle tower minimal surface. Minimal surfaces are among the objects of study in geometric analysis. Geometric analysis is a mathematical discipline where tools from differential equations, especially elliptic partial differential equations are used to establish new results in differential geometry and differential topology. The use of linear elliptic PDEs dates at least as far back as Hodge theory.
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In an article published in 1986, they made use of similar techniques to give a new proof of the classification of complete parabolic or elliptic affine hyperspheres. By adapting Jürgen Moser's method of proving Caccioppoli inequalities,Moser, Jürgen. A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math.
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The spreading viscid shrub or tree typically grows to a height of and to a width of around . It blooms from September to November and produces yellow flowers. The obliquely widely elliptic to elliptic phyllodes are long and wide. The simple inflorescences have globular flower heads with a diameter of containing 54 to 60 golden flowers.
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In 2008, Taylor, following the ideas of Michael Harris and building on his joint work with Laurent Clozel, Michael Harris, and Nick Shepherd-Barron, announced a proof of the Sato–Tate conjecture, for elliptic curves with non-integral j-invariant. This partial proof of the Sato–Tate conjecture uses Wiles's theorem about modularity of semistable elliptic curves.
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The coriaceous, silvery-green phyllodesa have a very narrowly elliptic to elliptic shape and are flat and slightly sickle shaped. They have a length of and a width of and can be glabrous or slightly haired with three prominent main nerves. It blooms between June and September producing flower-spikes that are in length and packed with golden flowers.
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The flowers of the C. mercadoi are greenish-yellow and include in terminal or subterminal panicles up to 15 centimeters long. The fruits are smooth, shiny, steel blue, elliptic-shaped, seated on a bowl-shaped perianth cup, and are usually 12 x 8 millimeters in dimension. The seeds are smooth and are narrow to elliptic-shaped.
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Eugenia mooniana, is a species of plant in the family Myrtaceae which is native to Western Ghats of India and Sri Lanka. It is an 8m tall tree with terete branchlets. Leaves are simple, opposite; lamina elliptic to narrow elliptic; apex caudate-acuminate with blunt tip; base acute to rounded with entire margin. Flowers are white colored.
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There is a single, hard, oval or oblate, rough central stone which contains 2 elliptic, brown seeds, 1/4 in (6mm) long.
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The sign of an elliptic divisibility sequence. J. Ramanujan Math. Soc., 21(1):1-17, 2006. for and in terms of , , , and .
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Flowering occurs from October to November and the fruit is a smooth, narrow elliptic drupe, long and wide containing a single seed.
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The suspension used quarter elliptic springs all round. The body had two seats plus a dickey seat and cost £200 in 1920.
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Encyclopedia of Earth. Eds.M.Pidwirny & C.J.Cleveland. National Council for Science and Environment. Washington DC. Its shape is elliptic and resembles that of Rhodes.
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In modern presentations, Kellogg's theorem is usually covered as a specific case of the boundary Schauder estimates for elliptic partial differential equations.
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The dull dark brown to black coloured seeds in the pods have an oblong to elliptic shape and are around in length.
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This came about as a modification of an alternate solution for the six-vertex model; it makes use of elliptic theta functions.
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The modularity theorem, also known as the Taniyama–Shimura conjecture, asserts that every elliptic curve defined over the rational numbers is modular.
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This specification includes key agreement, signature, and encryption schemes using several mathematical approaches: integer factorization, discrete logarithm, and elliptic curve discrete logarithm.
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In mathematical analysis, the Agranovich–Dynin formula is a formula for the index of an elliptic system of differential operators, introduced by .
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Each pod contains severalglossy dark brown seeds with an oblong to elliptic shape and a length of . The shrub is dieback resistant.
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The light brown seeds inside are arranged longitudinally and have an elliptic shape with a length of and have a thin pleurogram.
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Forced vibrations in isotropic, elastic, solid spheres, and spherical shells. Camb: Phil. Trans XVI. 9. Rotating, elastic, solid cylinders of elliptic section.
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Only in the 1970s was it realized that conjugacy based methods work very well for partial differential equations, especially the elliptic type.
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Scuola Norm. Sup. Pisa Cl. Sci. (4) 2 (1975), no. 4 to the ABC conjecture,Goldfeld, Dorian Modular elliptic curves and Diophantine problems.
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The leaves are ovate or elliptic, 4–7 cm long, with a cinnamon-like odour. Flowers are star-shaped and borne in panicles.
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Rostral very large, with obtuse horizontal edge, concave below. Eye very small, with vertically elliptic pupil. Nostril in a semidivided nasal. No loreal.
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The fruits of R. lobbii are 12–15 mm long, round to elliptic berries. They are reddish-brown, roughly bristled, glandular and edible.
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The pods are in length and wide. The shiny, black seeds within the pods have an oblong to elliptic and are in length.
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The pairing is used in number theory and algebraic geometry, and has also been applied in elliptic curve cryptography and identity based encryption.
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The seeds inside the fruit have a narrowly ovate or elliptic shape and are in length with a narrow wing down one side.
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The dull black seeds within the pods are arranged longitudinally and have an oblique oblong-elliptic to ovate shape with a length of .
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Flowering occurs from January to December and the follicles are elliptic, in heads of eighty or more, each follicle long, high and wide.
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This species is a broad-leafed shrub with grayish stems and elliptic leaves that are approximately 8 cm. It has purple-pink flowers.
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In mathematics, the Mestre bound is a bound on the analytic rank of an elliptic curve in terms of its conductor, introduced by .
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The pods are twisted and have a length of and contain narrowly oblong-elliptic shaped seeds. The dark brown seeds are in length.
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When he then sees the next publication by Jacobi where he makes use of elliptic functions to prove his results without referring to Abel, the Norwegian mathematician finds himself to be in a struggle with Jacobi over priority. He finishes several new articles about related issues, now for the first time dating them, but dies less than a year later. In the meantime Jacobi completes his great work Fundamenta nova theoriae functionum ellipticarum on elliptic functions which appears the same year as a book. It ended up defining what would be the standard form of elliptic functions in the years that followed.
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If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, if the rank of the curve is greater than 0, then the curve has an infinite number of rational points. Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effective method for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numerical methods but (in the current state of knowledge) it is unknown if these methods handle all curves.
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In mathematics, an elliptic surface is a surface that has an elliptic fibration, in other words a proper morphism with connected fibers to an algebraic curve such that almost all fibers are smooth curves of genus 1. (Over an algebraically closed field such as the complex numbers, these fibers are elliptic curves, perhaps without a chosen origin.) This is equivalent to the generic fiber being a smooth curve of genus one. This follows from proper base change. The surface and the base curve are assumed to be non-singular (complex manifolds or regular schemes, depending on the context).
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Elliptic equations have no real characteristic curves, curves along which it is not possible to eliminate at least one second derivative of u from the conditions of the Cauchy problem. Since characteristic curves are the only curves along which solutions to partial differential equations with smooth parameters can have discontinuous derivatives, solutions to elliptic equations cannot have discontinuous derivatives anywhere. This means elliptic equations are well suited to describe equilibrium states, where any discontinuities have already been smoothed out. For instance, we can obtain Laplace's equation from the heat equation u_t=\Delta u by setting u_t=0.
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A curve , over is called a modular curve if for some there exists a surjective morphism , given by a rational map with integer coefficients. The famous modularity theorem tells us that all elliptic curves over are modular. Mappings also arise in connection with since points on it correspond to some -isogenous pairs of elliptic curves. An isogeny between two elliptic curves is a non-trivial morphism of varieties (defined by a rational map) between the curves which also respects the group laws, and hence which sends the point at infinity (serving as the identity of the group law) to the point at infinity.
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Integrated Encryption Scheme (IES) is a hybrid encryption scheme which provides semantic security against an adversary who is allowed to use chosen- plaintext and chosen-ciphertext attacks. The security of the scheme is based on the computational Diffie–Hellman problem. Two incarnations of the IES are standardized: Discrete Logarithm Integrated Encryption Scheme (DLIES) and Elliptic Curve Integrated Encryption Scheme (ECIES), which is also known as the Elliptic Curve Augmented Encryption Scheme or simply the Elliptic Curve Encryption Scheme. These two incarnations are identical up to the change of an underlying group and so to be concrete we concentrate on the latter.
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In many models of projective geometry, the representation of a line rarely conforms to the notion of the "straight curve" as it is visualised in Euclidean geometry. In elliptic geometry we see a typical example of this.Faber, Part III, p. 108. In the spherical representation of elliptic geometry, lines are represented by great circles of a sphere with diametrically opposite points identified.
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The Schoof–Elkies–Atkin algorithm (SEA) is an algorithm used for finding the order of or calculating the number of points on an elliptic curve over a finite field. Its primary application is in elliptic curve cryptography. The algorithm is an extension of Schoof's algorithm by Noam Elkies and A. O. L. Atkin to significantly improve its efficiency (under heuristic assumptions).
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Rhododendron rigidum (基毛杜鹃) is a rhododendron species native to Sichuan and Yunnan, China, where it grows at altitudes of 2000–3400 meters. It is a shrub that grows to 1–2 m in height, with leaves that are elliptic, oblong elliptic, oblong-lanceolate or oblanceolate, 2.5–6.8 by 1–3.2 cm in size. Flowers are white to reddish purple.
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The glabrous shrub has an erect habit and typically grows to a height of around . It has angled to flattened brownish grey coloured branchlets that are resin ribbed. The dull green phyllodes become greyish with age. The phyllodes have an elliptic to ovate-elliptic shape with a length of and a width of and have four to seven longitudinal nerves.
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Finding criteria for a complex torus of dimension 2 to be a product of two elliptic curves (up to isogeny) was a popular subject of study in the nineteenth century. Invariants: The plurigenera are all 1. The surface is diffeomorphic to S1×S1×S1×S1 so the fundamental group is Z4. Hodge diamond: Examples: A product of two elliptic curves.
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Clifford parallels were first described in 1873 by the English mathematician William Kingdon Clifford.William Kingdon Clifford (1882) Mathematical Papers, 189-93, Macmillan & Co. In 1900 Guido Fubini wrote his doctoral thesis on Clifford's parallelism in elliptic spaces.Guido Fubini (1900) D.H. Delphenich translator Clifford Parallelism in Elliptic Spaces, Laurea thesis, Pisa. In 1931 Heinz Hopf used Clifford parallels to construct the Hopf map.
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In particular, if f is infinitely-often differentiable, then so is u. Any differential operator exhibiting this property is called a hypoelliptic operator; thus, every elliptic operator is hypoelliptic. The property also means that every fundamental solution of an elliptic operator is infinitely differentiable in any neighborhood not containing 0. As an application, suppose a function f satisfies the Cauchy–Riemann equations.
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It is a shrub or small tree growing to 4 m in height. The elliptic-ovate to narrowly elliptic leaves are 20–60 mm long, 13–25 mm wide, with a slightly foetid odour when bruised. The small green flowers are 8 mm long.. The ellipsoidal, reddish-orange fruits are 10 mm long. The flowering season is from May to July.
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The modular group and its subgroups were first studied in detail by Richard Dedekind and by Felix Klein as part of his Erlangen programme in the 1870s. However, the closely related elliptic functions were studied by Joseph Louis Lagrange in 1785, and further results on elliptic functions were published by Carl Gustav Jakob Jacobi and Niels Henrik Abel in 1827.
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It is a perennial herb, tufted and sparsely branched, growing to 5–15 cm in height. The linear or narrowly elliptic to elliptic-oblanceolate leaves are usually 5–20 mm long, 1–6 mm wide. It has blue, bell shaped flowers 6.5–10 mm long. The growth habit varies according to the substrate, being more stunted and tufty in exposed clefts in rocks.
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These surfaces have positive second Betti number. They have spherical shells, and can be deformed into a blown-up Hopf surface. Parabolic Inoue surfaces contain a cycle of rational curves with 0 self-intersection and an elliptic curve. They are a particular case of Enoki surfaces which have a cycle of rational curves with zero self-intersection but without elliptic curve.
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They are oval elliptic or oblong elliptic in shape, 9 to 20 cm (3.6–8 in) long, and 3 to 6 cm (1.2-2.4 in) wide. Shiny green above, more yellow-green underneath, the leaves are somewhat leathery to touch. The leaf stalk is between 8 and 25 mm long. Leaf veins not easily seen on the top surface, but more clear underneath.
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In algebra, an elliptic algebra is a certain regular algebra of a Gelfand–Kirillov dimension three (quantum polynomial ring in three variables) that corresponds to a cubic divisor in the projective space P2. If the cubic divisor happens to be an elliptic curve, then the algebra is called a Sklyanin algebra. The notion is studied in the context of noncommutative projective geometry.
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The plant grows up to 0.8m high. The stems are freely branched and densely pubescent with short incurved (or appressed) ascending trichomes. The leaves are elliptic to elliptic-lanceolate which are 2–6 cm long and 0.5–2 cm wide, obtuse, crenate. The base of the leaves are cuneate to rounded, with pubescence of both surfaces (more or less glabrate).
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Once the thorns have been on the tree for two years they are a shiny purplish black, and 4 to 7cm long. Typically older branches and the trunk do not have thorns. Its deciduous leaves are glabrous and coriaceous. The dark green leaf blades are more or less narrowly obovate to broadly elliptic or rhombic- elliptic, 4 to 5cm long, with serrate margins.
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The spikelets are also elliptic, are long, and have 2 fertile florets which are diminished at the apex. Lemma is chartaceous, lanceolated, and is long and wide. Its lemma have an obtuse apex while the fertile lemma itself is chartaceous, elliptic, keelless, and is long. It is also 7-9 veined while the surface of the lemma is villous with ciliated margins.
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Rhododendron polycladum (多枝杜鹃) is a rhododendron species native to central and northwestern Yunnan in China, where it grows at altitudes of 3000–4300 meters. It is a shrub that grows to 1.2 m in height, with leaves that are narrowly elliptic, elliptic, oblong, or lanceolate, 0.6–1.5 by 0.2–0.4 cm in size. Flowers are lavender to purple-blue.
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Rhododendron selense (多变杜鹃) is a rhododendron species native to southwestern Sichuan, eastern Xizang, and western Yunnan in China, where it grows at altitudes of 2700–4000 meters. It is a shrub that grows to 1–2 m in height, with leaves that are oblong-elliptic or obovate to elliptic, 4–8 by 2–4 cm in size. Flowers are pink.
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Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli. Viscosity solutions have become a central concept in the study of elliptic PDE. In particular, Viscosity solutions are essential in the study of the infinity Laplacian. In the modern approach, the existence of solutions is obtained most often through the Perron method.
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Rhododendron xanthocodon (黄铃杜鹃) is a rhododendron species native to northeastern India, Tibet, and Bhutan, where it grows at altitudes of 2900–4100 meters. It is an evergreen shrub or small tree that grows to 1.5–7.5 meters in height, with leathery leaves that are elliptic or oblong-elliptic, and 3–7.5 × 1.5–4.5 cm in size. Flowers are creamy yellow.
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The shrub typically grows to a height of and has a spreading habit that can be flat-topped. The glabrous and resinous branchlets with prominent ribbing. Like most species of Acacia it has phyllodes rather than true leaves. The patent to ascending phyllodes usually have an ovate to elliptic or oblong-elliptic shape that straight to slightly recurved at the apices.
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Boronia capitata is a slender, spreading shrub that grows to a height of . It has simple, thick, linear to club-shaped leaves long. The flowers are pink and are arranged in clusters on the ends of the branches, each on a pedicel long. The four sepals are broadly elliptic to narrow triangular, and the four petals are broadly elliptic, about long.
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Trees up to 25 m tall. Leaves lanceolate, elliptic or ovate, with acuminate or acute apex. Figs edible, globose, 0.8-1.2 cm in diameter.
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The bracts are ciliate, long, and have elliptic nutlets. The flowers bloom from June to July and the fruits ripe from July to August.
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Flowers have three stamens, two stigmas, and are hairy. The fruits have caryopses which have an additional pericarp, a hairy apex, and elliptic hilum.
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This leads to a nested certificate where at each level we have an elliptic curve E, an m and the prime in doubt, q.
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But when displaced from the origin by η, this axis intersects every sheet. In the elliptic case, the normal cone has no real sheets.
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Smooth grey to black fruit that are obliquely ovate or elliptic, dilated apically and approximately long and wide. It blooms from August to December.
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Conformal Projections Based on Elliptic Functions. Supplement No. 1 to Canadian Cartographer, Vol 13. (Designated as Monograph 16). Toronto: Department of Geography, York University.
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FISH works in any case and is supported by Konqueror. Dropbear supports elliptic curve cryptography for key exchange, as of version 2013.61test and beyond.
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However, if \Delta_4 e 0 and \Delta_3 = 0, the surface is a paraboloid, which is elliptic of hyperbolic, depending on the sign of \Delta_4.
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The shiny brown seeds found in the pods have a narrowly elliptic or oblong shape have a length of with a creamy while aril.
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A catalog of elliptic curves. Region shown is [−3,3]2 (For (a, b) = (0, 0) the function is not smooth and therefore not an elliptic curve.) In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point O. Every elliptic curve over a field of characteristic different from 2 and 3 can be described as a plane algebraic curve given by an equation of the form :y^2 = x^3 + ax + b. The curve is required to be non-singular, which means that the curve has no cusps or self- intersections. (This is equivalent to the condition 4a^3+27b^2 e0.) It is always understood that the curve is really sitting in the projective plane, with the point O being the unique point at infinity.
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The petals are elliptic, spreading, about long and smooth. The stamens are slightly longer than the petals. Flowering occurs from late spring to early summer.
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These properties make SIDH a natural candidate to replace Diffie-Hellman (DHE) and elliptic curve Diffie- Hellman (ECDHE), which are widely used in Internet communication.
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The method of continuity is used in conjunction with a priori estimates to prove the existence of suitably regular solutions to elliptic partial differential equations.
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Seed7 has its own implementation of Transport Layer Security.A Transport Layer Security (TLS) library written in Seed7 The library includes AES and elliptic-curve cryptography.
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Fertile lamina narrower, elliptic, obtuse and base unequal. Dark brown to black sporangia covers almost the entirety of the underside of the leaf surface (acrostichoid).
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The firmly chartaceous pods are up to in length and wide and contain dull dark brown coloured seeds with a broadly elliptic to ovate shape.
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Often elliptic curves in Edwards form are defined having c=1, without loss of generality. In the following sections, it is assumed that c=1.
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Bark splits into narrow vertical strips. Leaves broadly elliptic to lanceolate, lacking glandular hairs. Staminate (male) catkins are 3.5–5 cm long.Correll, Donovan Stewart. 1965.
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Sergio Campanato (17 February 1930 - 1 March 2005) was an Italian mathematician who studied the theory of regularity for elliptic and parabolic partial differential equations.
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The blackish seeds have a yellow centre and have an oblong to elliptic shape with a length of about and have a club-shaped aril.
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I was observing it. Suddenly a hand began to > write on the screen. I became all attention. That hand wrote a number of > elliptic integrals.
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Kunoth is the author of the monograph Wavelet Methods — Elliptic Boundary Value Problems and Control Problems (Springer, 2001), a book version of her habilitation thesis.
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The shiny black seeds inside the pods have an oblong to elliptic shape with a length of with a dark red-brown club shaped aril.
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In three dimensions, the wave equation, when written in elliptic cylindrical coordinates, may be solved by separation of variables, leading to the Mathieu differential equation.
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Even the case of elliptic curves (abelian varieties of dimension 1) is central to number theory, with applications including the proof of Fermat's last theorem.
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Pentachlaena orientalis grows as a tree up to tall with a trunk diameter of up to . Its leaves are elliptic and measure up to long.
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The semi-glossy drak brown seeds within are arranged longitudinally and have an oblong-elliptic shape that is in length with an apical white aril.
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The drum brakes operated on all four wheels. Suspension was traditional, involving rigid axles front and back with semi-elliptic leaf springs and “friction dampers”.
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The four petals are pale red or pale mauve, elliptic and about long. The eight stamens are about long a have a few soft hairs.
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Felix Klein saw screw theory as an application of elliptic geometry and his Erlangen Program.Felix Klein (1902) (D.H. Delphenich translator) On Sir Robert Ball's Theory of Screws He also worked out elliptic geometry, and a fresh view of Euclidean geometry, with the Cayley–Klein metric. The use of a symmetric matrix for a von Staudt conic and metric, applied to screws, has been described by Harvey Lipkin.
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He then worked with Peter Swinnerton-Dyer on computations relating to the Hasse–Weil L-functions of elliptic curves. Their subsequently formulated conjecture relating the rank of an elliptic curve to the order of zero of an L-function was an influence on the development of number theory from the mid-1960s onwards. He introduced modular symbols in about 1971. only partial results have been obtained.
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Rhododendron campylocarpum (弯果杜鹃) is a rhododendron species native to eastern Nepal, Sikkim, Bhutan, Arunachal Pradesh, southeastern Tibet, and southwestern China, where it grows at altitudes of 3000–4000 meters. It is a shrub that grows to 2–3 m in height, with leathery leaves that are suborbicular or ovate- elliptic to oblong-elliptic, 4–8.5 by 2.5–4 cm in size. Flowers are yellow.
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Rhododendron siderophyllum (锈叶杜鹃) is a rhododendron species native to Guizhou, Sichuan and Yunnan, China, where it grows at altitudes of 1800–3000 meters. It is a shrub that grows to 1–4 m in height, with leaves that are elliptic or elliptic-lanceolate, 3–7 by 1.2–3.5 cm in size. Flowers range from white to pink to pale purple or red.
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The poles correspond to those complex numbers which are mapped to ∞. On a non-compact Riemann surface, every meromorphic function can be realized as a quotient of two (globally defined) holomorphic functions. In contrast, on a compact Riemann surface, every holomorphic function is constant, while there always exist non-constant meromorphic functions. Meromorphic functions on an elliptic curve are also known as elliptic functions.
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Dual_EC_DRBG (Dual Elliptic Curve Deterministic Random Bit Generator) is an algorithm that was presented as a cryptographically secure pseudorandom number generator (CSPRNG) using methods in elliptic curve cryptography. Despite wide public criticism, including a backdoor, for seven years it was one of the four (now three) CSPRNGs standardized in NIST SP 800-90A as originally published circa June 2006, until it was withdrawn in 2014.
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Leaves are sessile and chartaceous in texture. The shape of the lamina (leaf blade) is variable: it may be linear, lanceolate, or slightly elliptic. In the case of rosettes and short stems, the lamina is typically oblanceolate to oblong-elliptic and measures up to 7.5 cm in length by 2.5 cm in width. It has an acute apex and does not exhibit a peltate tendril attachment.
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Engelhardia serrata is a tree growing up to 12 m tall. The leaves are pinnate, rarely unpaired, and 150–250 mm long. The petiole is 10–20 mm long and hairy; the rhachis is also hairy. The 6 to 14 leaflets are seated or short stalked, the blade is elliptic to elliptic-lanceolate, 60–130 mm long and 25–45 mm wide, the underside is hairy.
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Leaves are sparsely bristly or have a variable number of bristles; bristles are not dark at the base. Lamina are elliptic-lanceolate to elliptic-oblanceolate, narrow-oblanceolate, or more rarely linear- lanceolate. Petioles are 1.5–9 cm in length. Flowers: 2.6–8.2 cm across, with 4– 8– 11 satiny deep-blue to violet, to indigo-purple, more rarely pinkish, or very rarely light blue petals.
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The latter three methods can be found in Ramanujan's theory of elliptic functions to alternative bases. The inversion applied in high-precision calculations of elliptic function periods even as their ratios become unbounded. A related result is the expressibility via quadratic radicals of the values of at the points of the imaginary axis whose magnitudes are powers of 2 (thus permitting compass and straightedge constructions).
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The species' spikelets are long and are both elliptic and solitary with pedicelled fertile spikelets and one fertile floret which have a hairy callus. The glumes are long and are lanceolate, membranous and have one keel. They also have scaberulous veins and acute apexes. It have a hairy and long rhachilla and elliptic long and keelless fertile lemma while the lemma itself have a dentated apex.
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Only in a few examples have mathematicians been able to verify the conjecture. In his seminal address, Kontsevich commented that the conjecture could be proved in the case of elliptic curves using theta functions. Following this route, Alexander Polishchuk and Eric Zaslow provided a proof of a version of the conjecture for elliptic curves. Kenji Fukaya was able to establish elements of the conjecture for abelian varieties.
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Carlo Miranda (15 August 1912 – 28 May 1982) was an Italian mathematician, working on mathematical analysis, theory of elliptic partial differential equations and complex analysis: he is known for giving the first proof of the Poincaré–Miranda theorem,. for Miranda's theorem in complex analysis,. and for writing an influential monograph in the theory of elliptic partial differential equations.See and its revised and translated second edition .
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Acacia prominens usually grows to a height of , sometimes to a height of . It has glabrous branchlets that are angled at the extremeties and has smooth grey coloured bark. Like most species of Acacia it has phyllodes rather than true leaves. The grey-green to grey-blue, glabrous to sparsely hairy phyllodes have a narrowly elliptic to narrowly oblong- elliptic shape and are more or less straight.
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In the 1950s and 1960s a connection between elliptic curves and modular forms was conjectured by the Japanese mathematician Goro Shimura based on ideas posed by Yutaka Taniyama. In the West it became well known through a 1967 paper by André Weil. With Weil giving conceptual evidence for it, it is sometimes called the Taniyama–Shimura–Weil conjecture. It states that every rational elliptic curve is modular.
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Following flowering one to six stalked fruits will form per axil. Fruits have an obliquely elliptic shape that is sometimes curved with a length of and a width of . The light to dark brown seeds within have blackish patches. Each seed has an obliquely ovate to elliptic shape and a length of and a width of with a wing down both sides of the body.
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5 in normalized units. Frequency responses are shown for the Butterworth, Chebyshev, inverse Chebyshev, and elliptic filters. center As is clear from the image, the elliptic filter is sharper than the others, but at the expense of ripples in both its passband and stopband. The Butterworth filter has the poorest transition but has a more even response, avoiding ripples in either the passband or stopband.
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The tree or shrub typically grows to a height of and has smooth bark that is rougher at the base. The stout and angular branchlets are grey in colour and densely covered in silky hairs. Like most species of Acacia it has phyllodes rather than true leaves. The flat and straight, elliptic to narrowly elliptic phyllodes have a length of and a width of and thinly coriaceous.
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The configurations correspondingly responsible for higher, i.e. excited, states are periodic instantons defined on a circle of Euclidean time which in explicit form are expressed in terms of Jacobian elliptic functions (the generalization of trigonometric functions). The evaluation of the path integral in these cases involves correspondingly elliptic integrals. The equation of small fluctuations about these periodic instantons is a Lamé equation whose solutions are Lamé functions.
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Such a map is always surjective and has a finite kernel, the order of which is the degree of the isogeny. Points on correspond to pairs of elliptic curves admitting an isogeny of degree with cyclic kernel. When has genus one, it will itself be isomorphic to an elliptic curve, which will have the same -invariant. For instance, has -invariant , and is isomorphic to the curve .
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On the relationship between the solution sets the basic and advanced tasks for managing tasks in elliptic equations. International Scientific and Technical Journal "Problems of control and informatics", №4, Kiev, 2010, 43–52. 17\. Elliptic equation slippery regime described in the management of properties. Students, undergraduates, graduate students and young researchers, "Actual problems of mathematics and mechanics", a traditional conference, Baku, 2010. p. 53–55. 18\.
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Rhododendron mucronulatum, the Korean rhododendron or Korean rosebay (), is a rhododendron species native to Korea, Mongolia, Russia, and parts of northern China. It is a deciduous shrub that grows to in height, with elliptic or elliptic-lanceolate leaves, long by wide. The reddish-purple flowers appear in late winter or early spring, often on the bare branches before the foliage unfurls. It inhabits forested regions at .
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Rhododendron rubiginosum (红棕杜鹃) is a rhododendron species native to Myanmar, and Sichuan, Xizang, and Yunnan in China, where it grows at altitudes of 2800–3600 meters. It is a shrub that grows to 1–3 m in height, with leaves that are elliptic or elliptic-lanceolate or oblong-ovate, 3.5–8 by 1.3–3.5 cm in size. Flowers are pink, red, or pale purple.
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Atiyah and Singer's first published proof used K-theory rather than cobordism. If i is any inclusion of compact manifolds from X to Y, they defined a 'pushforward' operation i! on elliptic operators of X to elliptic operators of Y that preserves the index. By taking Y to be some sphere that X embeds in, this reduces the index theorem to the case of spheres.
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In China the nominate variety grows as shrub or small tree, some 2 to 8 m tall. Its light olive to greyish-green leaves are elliptic, oblong-elliptic, oblong- lanceolate, even obovate, some 3.5-13 cm by 1.5-4.5 cm in size. The inflorescences grow terminally or axillary. The drupes are a laterally compressed ellipsoid shape, 5-6 by 4-6 mm in size.
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Like many species of Acacia it has phyllodes rather than true leaves. The green pyllodes are also often covered in a white powdery coating and have a straight and dimidiately elliptic shape that are sometimes straight and symmetrically broad-elliptic. The glabrous phyllodes have a length of and a width of and have three to five prominent longitudinal veins. It blooms in September producing yellow flowers.
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The intricate shrub typically grows to a height of but can reach as high as and has a dense spreading habit. It has glabrous and lenticellular obscurely ribbed branchlets. Like most species of Acacia it has phyllodes rather than true leaves. The glabrous, rigid, green to grey-green to blue-green phyllodes have a narrowly elliptic to oblong-elliptic or somewhat lanceolate shape are a little asymmetric.
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Eucalyptus alipes is a mallet that grows to a height of up to and lacks a lignotuber. It has smooth grey to light brown or bronze bark. The leaves on young plants and on coppice regrowth under tall are linear to narrow elliptic, long and wide. Adult leaves are linear to narrow elliptic or lance-shaped, long and wide with a petiole up to long.
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The shrub or tree has a bushy habit and typically grows to a height of less than but can reach as high as . The shrub often has over four primary erect branches that diverge at the base. The terete brown-green to brown branchlets are ribbed and hairy. It has elliptic or occasionally ovate-elliptic shaped phyllodes with a length of and a width of .
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L. grandiflorum has broadly elliptic to elliptic oblong leaves, of 5–8 cm (2–3¼ in) long, mostly with three teeth near the tip, with fine grey cringy hairs and the perianth 4½–5 cm (1.8–2.0 in) long, and bright yellow bright yellow when opening (later turning orange). It differs from its look-a-like Leucospermum gueinzii, which has eventually hairless, pointy lance-shaped to elliptic leaves of 7½–10 cm (3–4 in) long, with an entire margin or seldomly with two or three teeth near the tip, with the perianth 5½–6 cm (2.2–2.4 in) long and the flower is deep orange when opening (later turning crimson).
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It is assumed that spheres A and B are the same size. In any elliptic hexlet, such as the one shown at the top of the article, there are two tangent planes to the hexlet. In order for an elliptic hexlet to exist, the radius of C must be less than one quarter that of A. If C's radius is one quarter of A's, each sphere will become a plane in the journey. The inverted image shows a normal elliptic hexlet, though, and in the parabolic hexlet, the point where a sphere turns into a plane is precisely when its inverted image passes through the centre of inversion.
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An elliptic, parabolic, and hyperbolic Kepler orbit: Elliptic orbit by eccentricity The orbital eccentricity of an astronomical object is a dimensionless parameter that determines the amount by which its orbit around another body deviates from a perfect circle. A value of 0 is a circular orbit, values between 0 and 1 form an elliptic orbit, 1 is a parabolic escape orbit, and greater than 1 is a hyperbola. The term derives its name from the parameters of conic sections, as every Kepler orbit is a conic section. It is normally used for the isolated two-body problem, but extensions exist for objects following a Klemperer rosette orbit through the galaxy.
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He initially worked on elliptic curves. After a period when he worked on geometry of numbers and diophantine approximation, he returned in the later 1950s to the arithmetic of elliptic curves, writing a series of papers connecting the Selmer group with Galois cohomology and laying some of the foundations of the modern theory of infinite descent. His best- known single result may be the proof that the Tate-Shafarevich group of an elliptic curve, if it is finite, must have order that is a square; the proof being by construction of an alternating form. Cassels often studied individual Diophantine equations by algebraic number theory and p-adic methods.
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Abel wrote a fundamental work on the theory of elliptic integrals, containing the foundations of the theory of elliptic functions. While travelling to Paris he published a paper revealing the double periodicity of elliptic functions, which Adrien-Marie Legendre later described to Augustin-Louis Cauchy as "a monument more lasting than bronze" (borrowing a famous sentence by the Roman poet Horatius). The paper was, however, misplaced by Cauchy. While abroad Abel had sent most of his work to Berlin to be published in the Crelle's Journal, but he had saved what he regarded as his most important work for the French Academy of Sciences, a theorem on addition of algebraic differentials.
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For any e, a multiple of and any a relatively prime to p, by Fermat's little theorem we have . Then is likely to produce a factor of n. However, the algorithm fails when has large prime factors, as is the case for numbers containing strong primes, for example. ECM gets around this obstacle by considering the group of a random elliptic curve over the finite field Zp, rather than considering the multiplicative group of Zp which always has order The order of the group of an elliptic curve over Zp varies (quite randomly) between and by Hasse's theorem, and is likely to be smooth for some elliptic curves.
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Eupatorium japonicum is a herbaceous perennial growing 50–200 cm tall from short rhizomes with many fibrous roots. The stems are upright and marked with purplish red, ending with simple or corymbose, (flat) inflorescence that branch near their ends. The leaves are oppositely arranged on the stems and have short but rather thick petioles that are 1–2 cm long. The leaves midway up the stems are elliptic, narrowly elliptic, ovate-elliptic, or lanceolate in shape and 6 to 20 long and 2 to 6.5 cm wide. The leaves are pinnately veined, with lateral veins 7-paired, the undersides of the leaves have prominent veining.
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The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other (called antipodal points) are identified (considered the same). This is also one of the standard models of the real projective plane. The difference is that as a model of elliptic geometry a metric is introduced permitting the measurement of lengths and angles, while as a model of the projective plane there is no such metric. In the elliptic model, for any given line and a point A, which is not on , all lines through A will intersect .
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During the early stages of his career, he developed and compiled several mathematical tables such as the Standard Four Figure Mathematical Tables jointly constructed with L. J. Comrie and published in 1931, Standard Table of Square Roots (1932), and Jacobian Elliptic Function Tables (1932). Later, Milne- Thomson wrote the chapters on Elliptic Integrals and Jacobian Elliptic Functions in the classic NBS AMS 55 handbook. In 1933 Milne-Thomson published his first book, The Calculus of Finite Differences which became a classic textbook and the original text was reprinted in 1951. In the mid 1930s, Milne- Thomson developed an interest in hydrodynamics and later in aerodynamics.
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Plants of M. burchellii without leaves resemble M. coriacea plants without leaves and it is almost impossible to separate the two taxa in this condition. M. burchellii can be distinguished from M. coriacea by the linear or narrowly elliptic or very narrowly obovate leaves with a single main vein and entire margins; in M. coriacea, the leaves are mostly obovate, sometimes elliptic to broadly elliptic with 3 to 5 main veins and entire or apically few, broad dentate margins. Although M. coriacea mostly has solitary or 2-headed synflorescences while M. burchellii usually has corymbosely arranged capitula, solitary capitula are also sometimes found in M. burchellii.
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In October 2014, Jao and Soukharev from the University of Waterloo presented an alternative method of creating undeniable signatures with designated verifier using elliptic curve isogenies.
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The species also have an elliptic and hairless nutlet which have membranous wings. Flowers bloom from June to July while fruits are from July to August.
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The shrub typically grows to a height of and has a spreading habit. patent to reflexed phyllodes that have a narrowly oblong-elliptic to lanceolate shape.
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Species in the genus Oxyrhopus share the following characters: Head distinct from neck. Eye moderate or small. Pupil vertically elliptic. Body cylindrical or slightly laterally compressed.
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In algebraic geometry, an elliptic singularity of a surface, introduced by , is a surface singularity such that the arithmetic genus of its local ring is 1\.
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The bracts shed early, peduncles long and calyx lobes long. The seed capsule is elliptic to cone shaped and long. Flowering occurs in spring and summer.
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The central frieze above the windows present symbols of worship and an incense boat. The lateral façades reveal the almost elliptic floorplan of the church nave.
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By 1925 Siegfried had designed first "Buzzing Wind" airplane for the Deutscher Rundflug 1925 competition, which featured the first elliptic design based on Prandtl's 1918 theory.
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The glossy mottled seeds are longitudinally arranged inside the pods. The seeds have an elliptic shape with a length of with a blunt white terminal aril.
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This cryptographic system relies on the properties of supersingular elliptic curves and supersingular isogeny graphs to create a Diffie-Hellman replacement with forward secrecy. This cryptographic system uses the well studied mathematics of supersingular elliptic curves to create a Diffie-Hellman like key exchange that can serve as a straightforward quantum computing resistant replacement for the Diffie- Hellman and elliptic curve Diffie–Hellman key exchange methods that are in widespread use today. Because it works much like existing Diffie–Hellman implementations, it offers forward secrecy which is viewed as important both to prevent mass surveillance by governments but also to protect against the compromise of long term keys through failures. In 2012, researchers Sun, Tian and Wang of the Chinese State Key Lab for Integrated Service Networks and Xidian University, extended the work of De Feo, Jao, and Plut to create quantum secure digital signatures based on supersingular elliptic curve isogenies.
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Later, she made important contributions to the study of the porous medium equation,See . : \partial_t u = \Delta_x u^m, \,\, m > 1, and to non-linear elliptic equations.See .
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The plant grows slowly, reaching in height, with a spread of up to . It has elliptic–oval-shaped leaves long and arranged in a basal rosette pattern.
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In differential geometry there are a number of second-order, linear, elliptic differential operators bearing the name Laplacian. This article provides an overview of some of them.
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More classical topics, on which there is still much active research, include elliptic and parabolic partial differential equations, fluid mechanics, Boltzmann equations, and dispersive partial differential equations.
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The siliques are 7–8 mm long, elliptic and glabrous. Flowers from May to July.Wild flowers of Cyprus, George Sfikas, Efstathiadis Group S.A. 1993 Anixi, Attikis, Greece.
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Status: Fibonacci number, Elliptic Curve Primality Proof. Retrieved 2018-04-05. The largest known probable Fibonacci prime is F3340367. It was found by Henri Lifchitz in 2018.
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The first elliptic functions were found by Carl Friedrich Gauss around 1795 in connection with his calculation of the lemniscate arc length, but first published after his death.J. Stillwell, Mathematics and Its History, Springer, New York (2010). . These are special cases of the general, elliptic functions which were first investigated by Abel in 1823 when he still was a student.A. Stubhaug, Niels Henrik Abel and his Times, Springer-Verlag, Berlin (2000). .
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He has developed new methods of adaptive meshes, which are used in adaptive mesh refinement for coupled elliptic- hyperbolic systems.Pretorius, Choptuik Adaptive Mesh Refinement for Coupled Elliptic-Hyperbolic Systems, J. Comput. Phys., 218, 2006, 246-274, Arxiv Pretorius has numerically investigated the possibilities and the signatures of small black holes in particle colliders such as the LHC.William E. East, Frans Pretorius Ultrarelativistic Black Hole Formation, Phys. Rev. Lett.
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Rhododendron ciliatum (睫毛杜鹃) is a rhododendron species native to eastern Nepal, Sikkim, Bhutan, southern Tibet, and Xizang in China, where it grows at altitudes of 2700–3500 meters. It is a shrub that grows to 0.3–2 m in height, with leathery leaves that are elliptic or oblong-elliptic to oblong- lanceolate, 3–8 by 1.6–3.7 cm in size. Flowers are white tinged with pink.
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In mathematics, the Atiyah–Bott fixed-point theorem, proven by Michael Atiyah and Raoul Bott in the 1960s, is a general form of the Lefschetz fixed-point theorem for smooth manifolds M, which uses an elliptic complex on M. This is a system of elliptic differential operators on vector bundles, generalizing the de Rham complex constructed from smooth differential forms which appears in the original Lefschetz fixed-point theorem.
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On June 30, 2020, blockchain analysis company Elliptic added support for ZCash to their crypto transaction monitoring and wallet screening compliance solutions. According to the developer of ZCash, The Electric Coin Company, Zcash privacy remains strongest of any cryptocurrency, even with Chainalysis, Elliptic support. In Fall 2020, the ownership of Electric Coin Company is in the process of transitioning to a US non-profit called the Bootstrap Project.
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The coriaceous and evergreen phyllodes have an elliptic to obliquely narrowly elliptic shape that narrows swiftly into the broad pulvinus,. The flat and falcate phyllodes have a length of and a width of and have a hooked apex with two or three prominent main veins. It blooms between July and September producing golden flowers. The cylindrical flower-spikes have a length of and are covered with bright yellow flowers.
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The spreading often dense shrub typically grows to a height of and branches from near ground level. The grey-green phyllodes have a linear to linear-elliptic to narrowly oblong-elliptic shape and are straight to shallowly curved. Phyllodes have a length of and a width of with red to brown margins with numerous, fine, closely parallel veins. It blooms from July to September and produces yellow flowers.
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MQV (Menezes–Qu–Vanstone) is an authenticated protocol for key agreement based on the Diffie–Hellman scheme. Like other authenticated Diffie–Hellman schemes, MQV provides protection against an active attacker. The protocol can be modified to work in an arbitrary finite group, and, in particular, elliptic curve groups, where it is known as elliptic curve MQV (ECMQV). MQV was initially proposed by Alfred Menezes, Minghua Qu and Scott Vanstone in 1995.
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Martin Maximilian Emil Eichler (29 March 1912 - 7 October 1992) was a German number theorist. Eichler received his Ph.D. from the Martin Luther University of Halle-Wittenberg in 1936. Eichler and Goro Shimura developed a method to construct elliptic curves from certain modular forms. The converse notion that every elliptic curve has a corresponding modular form would later be the key to the proof of Fermat's last theorem.
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Rhododendron wardii (黄杯杜鹃) is a rhododendron species native to southwestern Sichuan, southeastern Xizang, and northwestern Yunnan in China, where it grows at altitudes of 3000–4600 meters. It is a shrub that grows to 3 m in height, with leathery leaves that are narrowly ovate to oblong-elliptic or broadly ovate-elliptic, 5–8 by 3–4.5 cm in size. Flowers are yellow to white.
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The erect dense shrub typically grows to a height of . It is often has multiple slender stems and has a woody rootstock with hairy branchlets and narrowly triangular stipules with a length of . It has green elliptic to broadly elliptic or obovate shaped phyllodes with a length of and a width of and prominent midrib and marginal nerves. It blooms from March to September and produces white-cream-yellow flowers.
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It has 2 pairs of petals, 3 large sepals (outer petals), known as the 'falls' and 3 inner, smaller petals (or tepals), known as the 'standards'. The elliptic to oblanceolate falls are long, and 1.3 cm wide, with a long blue beard, in the centre of the fall. The elliptic to oblanceolate standards are long, they also have a thin beard. It has short pedicels and a long perianth tube.
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The Epsilon was powered by a Tipo 58 side valve monobloc inline-four, displacing 4,080 cc, which produced 60 hp at 1,500 rpm. Top speed was . The separate body was built on a ladder frame; front and rear there were solid axles on semi-elliptic springs at the front and three-quarter elliptic springs at the rear. The brakes were on the transmission and on the rear wheels.
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The motion of a fluid at subsonic speeds can be approximated with elliptic PDEs, and the Euler–Tricomi equation is elliptic where . # (parabolic partial differential equation): Equations that are parabolic at every point can be transformed into a form analogous to the heat equation by a change of independent variables. Solutions smooth out as the transformed time variable increases. The Euler–Tricomi equation has parabolic type on the line where .
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The inflorescences are found on racemes in groups of three to six. The spherical flower-heads have a diameter of and contain 20 to 30 golden flowers. Following flowering seed pods that resemble a string of beads form and have a length of up to and a width of . The shiny black seeds have a length of and have an elliptic to narrowly elliptic or narrowly oblong shape.
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Hence both curves are modular of level , having mappings from . By a theorem of Henri Carayol, if an elliptic curve is modular then its conductor, an isogeny invariant described originally in terms of cohomology, is the smallest integer such that there exists a rational mapping . Since we now know all elliptic curves over are modular, we also know that the conductor is simply the level of its minimal modular parametrization.
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Eucalyptus annuliformis is a mallee that typically grows to a height of and has smooth bark and a dull green crown. The leaves on young plants are arranged alternately, broadly elliptic to egg- shaped, dull greyish green, long and wide. The adult leaves are elliptic to broadly lance-shaped, up to long and wide on a petiole long. The leaves are the same dull green on both surfaces.
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Karl Cooper Rubin (born January 27, 1956) is an American mathematician at University of California, Irvine as Thorp Professor of Mathematics. Between 1997 and 2006, he was a professor at Stanford, and before that worked at Ohio State University between 1987 and 1999. His research interest is in elliptic curves. He was the first mathematician (1986) to show that some elliptic curves over the rationals have finite Tate–Shafarevich groups.
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The shrub or tree typically grows to a height of and can be found to . It has terete and glabrous branchlets with many red, resinous micro-hairs. Phyllodes are spreading to erect with leaves that are linear, narrowly elliptic or narrowly oblong-elliptic shape that is straight to recurved, terete to flat, in length and wide. Leaves are hairy when young, becoming hairless, edges smooth, with a straight often sharp point.
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In a stricter sense, it is a Kepler orbit with the eccentricity greater than 0 and less than 1 (thus excluding the circular orbit). In a wider sense, it is a Kepler's orbit with negative energy. This includes the radial elliptic orbit, with eccentricity equal to 1. In a gravitational two-body problem with negative energy, both bodies follow similar elliptic orbits with the same orbital period around their common barycenter.
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The shrub has an open to spindly habit and typically grows to a height of . The dull grey-green phyllodes are flat or slightly twisted with an elliptic to broadly elliptic shape that can sometimes be broadly obovate. The phyllodes have a length of and a width of . The shrub blooms between September and November producing up to 20 inflorescences on axillary racemes along an axis of around in length.
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They mine the leaves of their host plant. The mine starts as an undulating epidermal corridor. Later, it widens into an elliptic blotch. The blotch has two levels.
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Shell minute to small (adult length 1–6 mm). Color white, hyaline; surface smooth, glossy. Shape usually elliptic, obovate, or subtriangular; weakly shouldered. Spire completely immersed to low.
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The plane curves of degree 3 are called cubic plane curves and, if they are non-singular, elliptic curves. Those of degree 4 are called quartic plane curves.
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Leaves are 25 to 30cm. The fruit, c.5.5 by 3-3.5cm is brown and elliptic, 1 to 3 seeds. It flowers and fruits from November to January.
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Delicate, erect herb to 15 cm, with spreading hairs. Leaves broadly oblong-elliptic to orbicular, 0.5-1 x 0.3–07 cm. Flowers lilac-blue, <0.8 cm long, sessile.
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Stipules are red, usually about long. Leaf blades are ovate to elliptic, up to long. Figs are red, in diameter, borne in the axils of the leaves.S.S. Chang.
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The species' lemma of fertile floret elliptic to oblong and is long. Lemma is also obtuse or subacute, 7-nerved, hairless and scaberulous. The species' anthers are long.
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Kolyvagin constructed an Euler system from the Heegner points of an elliptic curve, and used this to show that in some cases the Tate-Shafarevich group is finite.
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The seed pods that form later are straight to narrowly oblong. They have a length of around and a width of and contain glossy brown oblong-elliptic seeds.
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Schizolaena microphylla grows as a tree up to tall, exceptionally up to . Its leaves are elliptic to ovate or roundish in shape and are hairy on the underside.
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These tiny epiphytic and rarely lithophytic orchids lack pseudobulbs. The erect, thick, leathery leaf is elliptic-ovate in shape. The aerial roots seem like fine hairs.Luer, C.A. (1996).
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Weil also suggested that the conductor was equal to the level of a modular form corresponding to the elliptic curve. extended the theory to conductors of abelian varieties.
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The protocol combines the Double Ratchet algorithm, prekeys, and a triple Elliptic-curve Diffie–Hellman (3-DH) handshake, and uses Curve25519, AES-256, and HMAC-SHA256 as primitives.
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Shubin has written over 140 papers and books, and supervised almost twenty doctoral theses. He has published results in convolution equations, factorization of matrix functions and Wiener–Hopf equations, holomorphic families of subspaces of Banach spaces, pseudo-differential operators, quantization and symbols, method of approximate spectral projection, essential self-adjointness and coincidence of minimal and maximal extensions, operators with almost periodic coefficients, random elliptic operators, transversally elliptic operators, pseudo-differential operators on Lie groups, pseudo-difference operators and their Green function, complete asymptotic expansion of spectral invariants, non-standard analysis and singular perturbations of ordinary differential equations, elliptic operators on manifolds of bounded geometry, non-linear equations, Lefschetz- type formulas, von Neumann algebras and topology of non-simply connected manifolds, idempotent analysis, the Riemann-Roch theorem for general elliptic operators, spectra of magnetic Schrödinger operators, and geometric theory of lattice vibrations and specific heat. In 2012 he became a fellow of the American Mathematical Society. He died in May 2020 at the age of 75.
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L. gueinzii differs from its look-a-like Leucospermum grandiflorum because it has eventually hairless, pointy lance-shaped to elliptic leaves of 7½–10 cm (3–4 in) long, with an entire margin or seldomly with two or three teeth near the tip, with the perianth 5½–6 cm (2.2–2.4 in) long and the flower is deep orange when opening (later turning crimson). L grandiflorum on the other hand has broadly elliptic to elliptic oblong leaves, of 5–8 cm (2–3¼ in) long, mostly with three teeth near the tip, with fine grey cringy hairs and the perianth 4½–5 cm (1.8–2.0 in) long, and bright yellow bright yellow when opening (later turning orange).
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In the 1970s Feng developed embedding theories in discontinuous finite element space, and generalized classical theory on elliptic partial differential equations to various dimensional combinations, which provided a mathematical foundation for elastic composite structures. He also worked on reducing elliptic PDEs to boundary integral equations, which led to the development of the natural boundary element method, now regarded as one of three main boundary element methods. Since 1978 he had given lectures and seminars on finite elements and natural boundary elements in more than ten universities and institutes in France, Italy, Japan and United States. From 1984 Feng changed his research field from elliptic PDEs to dynamical systems such as Hamiltonian systems and wave equations.
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Unlike unauthenticated Diffie-Hellman, SPEKE prevents man-in-the- middle attack by the incorporation of the password. An attacker who is able to read and modify all messages between Alice and Bob cannot learn the shared key K and cannot make more than one guess for the password in each interaction with a party that knows it. In general, SPEKE can use any prime order group that is suitable for public key cryptography, including elliptic-curve cryptography. However, when SPEKE is realized by using Elliptic-curve cryptography, the protocol is essentially changed by requiring an additional primitive that must securely map a password onto a random point on the designated elliptic curve.
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In astronautics and aerospace engineering, the bi-elliptic transfer is an orbital maneuver that moves a spacecraft from one orbit to another and may, in certain situations, require less delta-v than a Hohmann transfer maneuver. The bi-elliptic transfer consists of two half-elliptic orbits. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an apoapsis at some point r_b away from the central body. At this point a second burn sends the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit.
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Let K be a global field, i.e. a number field or a function field in one variable over a finite field and let E be an elliptic curve. If v is a non-archimedean place of norm qv of K and a ∈ K, with v(a) = 0 then ≥ 1. v is called a Wieferich place for base a, if > 1, an elliptic Wieferich place for base P ∈ E, if NvP ∈ E2 and a strong elliptic Wieferich place for base P ∈ E if nvP ∈ E2, where nv is the order of P modulo v and Nv gives the number of rational points (over the residue field of v) of the reduction of E at v.
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The first example of a K3 surface with Picard number 22 was given by , who observed that the Fermat quartic :w4 \+ x4 \+ y4 \+ z4 = 0 has Picard number 22 over algebraically closed fields of characteristic 3 mod 4. Then Shioda showed that the elliptic modular surface of level 4 (the universal generalized elliptic curve E(4) → X(4)) in characteristic 3 mod 4 is a K3 surface with Picard number 22, as is the Kummer surface of the product of two supersingular elliptic curves in odd characteristic. showed that all K3 surfaces with Picard number 22 are double covers of the projective plane. In the case of characteristic 2 the double cover may need to be an inseparable covering.
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Both the elliptic functions of Abel and of Jacobi can be derived from a more general formulation which was later given by Karl Weierstrass based on their double periodicity.
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Petals twice as long as sepals. Sepals 5, ovate, acute, 2 x 1.5 mm. Petals 5, elliptic, 3.5 x 2 mm. Disc large, cushion- like, with 5 globose lobes.
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David Hilbert is said to have remarked that the theory of complex multiplication of elliptic curves was not only the most beautiful part of mathematics but of all science.
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Rhododendron rarum is a rhododendron species native to the Central and Bismarck Ranges of Papua New Guinea. It is a shrub with tubular, red flowers and narrowly elliptic leaves.
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Melicope latifolia grows up as a shrub or tree to tall. Inflorescences are often dense and measure up to long. The fruits are elliptic and measure up to long.
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The hairy pods are firmly chartaceous with glabrous yellow coloured margins. The glossy, mottled grey-brown to brown seeds have an oblong-elliptic shape and a length of around .
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In mathematics, an Igusa curve is (roughly) a coarse moduli space of elliptic curves in characteristic p with a level p Igusa structure, where an Igusa structure on an elliptic curve E is roughly a point of order p of E(p) generating the kernel of V:E(p) → E. An Igusa variety is a higher-dimensional analogue of an Igusa curve. Igusa curves were studied by and Igusa varieties were introduced by .
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Rhododendron cinnabarinum (朱砂杜鹃) is a rhododendron species native to eastern Nepal, Bhutan, Sikkim, southeastern Tibet, and southwest China, where it grows at altitudes of 1900–4000 meters. It is a shrub that grows to 1–3 m in height, with leathery leaves that are broadly elliptic, oblong-elliptic to oblong- lanceolate or ovate, 3–6 by 1.5–2.5 cm in size. Flowers are yellow to cinnabar red, sometimes ranging to plum colors.
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In turn, Witten related these to (conjectural) index theory on free loop spaces. Elliptic cohomology, invented in its original form by Landweber, Stong and Ravenel in the late 1980s, was introduced to clarify certain issues with elliptic genera and provide a context for (conjectural) index theory of families of differential operators on free loop spaces. In some sense it can be seen as an approximation to the K-theory of the free loop space.
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Eucalyptus horistes is a mallee or small tree that typically grows to a height of and forms a lignotuber. It has smooth greyish bark, often with rough, firm fibrous bark on the base or all of the trunk. Young plants and coppice regrowth have sessile, heart-shaped, more or less round or elliptic leaves long and wide. Adult leaves are glossy green, narrow lance- shaped to elliptic, long and wide on a petiole long.
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Chiloglottis longiclavata is a terrestrial, perennial, deciduous, herb with two elliptic leaves long and wide on a petiole long. A single pinkish green flower long and wide is borne on a flowering stem high. The dorsal sepal is narrow egg-shaped to elliptic with a narrow base, long and about wide. The lateral sepals are linear but tapered, long, about wide and erect near the base before curving downwards and spreading apart from each other.
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The modularity theorem is a special case of more general conjectures due to Robert Langlands. The Langlands program seeks to attach an automorphic form or automorphic representation (a suitable generalization of a modular form) to more general objects of arithmetic algebraic geometry, such as to every elliptic curve over a number field. Most cases of these extended conjectures have not yet been proved. However, proved that elliptic curves defined over real quadratic fields are modular.
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The bark is grey to brown in colour and sheds in coarse flakes. Leaf blades are narrowly elliptic to oblanceolate in shape with length of and a width of . The inflorescences are long producing dry winged fruit with a flattened, elliptic to obovate shape and a length of . It is in a variety of habitats over laterite or sandstone in the Kimberley region of Western Australia and the Northern Territory growing in sandy-stony soils.
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Rhododendron macgregoriae is a rhododendron species native to Indonesia and Papua New Guinea at altitudes of 500 to 3350 m. It is a shrub or small tree that grows to 15 m in height, with leaves that are ovate-elliptic or obovate- elliptic, 40–140 × 25–50 mm in size. Flowers are light yellow to orange. R. macgregoriae is relatively easy to grow in cultivation and a popular parent for hybrid cultivars.
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The elliptic to narrowly elliptic shiny black seeds within are long. It is native to an area in the Wheatbelt and the Mid West regions of Western Australia. It is found as far north as an area in between Denham and Kalbarri to around Piawaning in the south on sand plains and gentle rises where it grows in sandy lateritic soils. The shrub is usually part of the understorey in woodland or tall shrubland communities.
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A solution to Laplace's equation defined on an annulus. The Laplace operator is the most famous example of an elliptic operator. In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions.
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They travel along the characteristics of the equation. This feature qualitatively distinguishes hyperbolic equations from elliptic partial differential equations and parabolic partial differential equations. A perturbation of the initial (or boundary) data of an elliptic or parabolic equation is felt at once by essentially all points in the domain. Although the definition of hyperbolicity is fundamentally a qualitative one, there are precise criteria that depend on the particular kind of differential equation under consideration.
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Rhododendron heliolepis (亮鳞杜鹃) is a rhododendron species native to Myanmar, and Sichuan, Xizang, and Yunnan in China, where it grows at altitudes of 3000–3700 meters. It is a shrub or small tree that grows to 2–5 m in height, with leaves that are elliptic or elliptic-lanceolate, 5–12 by 1.7–4 cm in size. Flowers are pink, pale purple-red, or rarely white, with purple or brown spots inside.
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With the Rover 8 and Rover 9/20 chassis and enlarged 9/20 engine the 10/25 chassis was conventional with rigid axles and leaf spring suspension all round, half elliptic at the front and quarter elliptic behind. The four cylinder, overhead valve engine's capacity had been increased by ten per cent to 1185 cc. Drive was to the rear wheels through a three speed gearbox. There were internally expanding brakes on all four wheels.
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Since the US government and US companies have also used the vulnerable BSAFE, NSA can potentially have made US data less safe, if NSA's secret key to the backdoor had been stolen. It is also possible to derive the secret key by solving a single instance of the algorithm's elliptic curve problem (breaking an instance of elliptic curve cryptography is considered unlikely with current computers and algorithms, but a breakthrough may occur).
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The erect tree typically grows to a height of less than and has fissured grey coloured bark. It has light green to brown coloured branchlets that are angular toward the apices but otherwise terete that are sometimes pruinose or scurfy. Like most species of Acacia it has phyllodes rather than true leaves. The phyllodes are flat and falcate with an elliptic to narrowly elliptic shape and a length of and a width of .
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Rhododendron longipes (长柄杜鹃) is a rhododendron species native to Chongqing, eastern Guizhou, southwestern Sichuan, and northeastern Yunnan in China, where it grows at altitudes of 1700–2500 meters. It is a shrub or small tree that grows to 2–4 meters in height, with leathery leaves that are elliptic or elliptic-lanceolate, and 5–13 × 1.5–3.5 cm in size. Flowers are rose-colored with dark red spots inside.
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Eucalyptus coccifera is a tree that typically grows to a height of but is sometimes a mallee to . The bark is smooth and light grey to white, with streaks of tan. Young plants and coppice regrowth have sessile, blue-green, elliptic to heart- shaped leaves long and wide. Adult leaves are arranged alternately, elliptic to lance-shaped, the same glossy green to bluish on both sides, long and wide on a petiole long.
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The specific epithet ' is from the Latin meaning "elliptic leaves". Habitat is mixed dipterocarp forests from sea-level to altitude. D. elliptifolia is found in Sumatra, Borneo and the Philippines.
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The four petals are elliptic, long with a rounded end and their bases overlapping. The eight stamens are hairy and similar to each other. Flowering occurs in June or September.
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The forward brakes operate on the Rubury principle. Steering is by worm and wheel. Suspension is by half elliptic springs fitted with shock absorbers. Gaiters are fitted back and front.
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Rhododendron degronianum is a shrub that grows to in height, with leaves that are narrowly to broadly elliptic, or linear lanceolate. Its flowers are funnel-shaped and pink to white.
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Their sepals have an oblong, elliptic, or narrowly lanceolate form. The species in the tribe Vanilleae are long plants characterized by long, thick, succulent vines and a lip without spur.
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Thus digital IIR filters can be based on well-known solutions for analog filters such as the Chebyshev filter, Butterworth filter, and elliptic filter, inheriting the characteristics of those solutions.
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The petals are broadly elliptic, about long with a prominent keel. The stamens are free from each other and hairy near the base. Flowering mainly occurs in spring and autumn.
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Leaves are elliptic to lanceolate, up to long. Flower heads are 1-4 per plant, with yellow flowers.Gray, Asa. Notes on Compositae and characters of certain genera and species, etc.
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Leaves are ovate to elliptic in shape. 4 to 10 cm long, and 1.2 to 4.5 cm wide. Opposite on the stem, with smooth edges. Thick, not glossy and hairless.
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S. reticulata is a climbing, perennial, woody shrub. The plant has dichotomous branching pattern. Bark is smooth, greenish grey in colour, thin, and white internally. Leaves: opposite and elliptic-oblong.
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The flowers are pink to white, with yellow marks. Fruits are samaras with three spreading, papery oblanceolate to elliptic wings, 2–5 cm long, and propagate via wind or by cuttings.
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Following flowering it forms glabrous flat seed pods that are up to in length with a width of . The dull black seeds within have an elliptic shape and are in length.
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Standard fittings include a spare (steel artillery) wheel and tyre. Shock absorbers are provided. There are semi-elliptic springs all round, flat set at the front. The rear springs are underhung.
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Following flowering blacking coloured firmly chartaceous to thinly coriaceous-crustaceous seed pods form that resemble a string of beads. The coiled pods conain elliptic shaped seeds with a length of around .
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Leaves can be ovate, lanceolate, elliptic or oval in shape. 4 to 13 cm long, 2.5 to 6 cm wide. Oil dots not or seldom visible. Leaves three veined in appearance.
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The pods are in length and wide containing longitudinally arranged seeds. The sub-shiny dark brown seeds are flattened and have an oblong to broadly elliptic shape and a length of .
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Epacris sparsa, is a small upright shrub with creamy-white flowers, elliptic to egg-shaped leaves and reddish new growth. It is endemic to New South Wales with a restricted distribution.
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The flower contains tepals, stamens, and stigmas. The tepals are oblong to elliptic with the apex either obtuse or acute. The stamens can be exserted, with dark purple or black anthers.
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Trees 10–25 m high; with narrowly elliptic or oblong leaves, 5–15 cm long and 1–2 cm wide; male cones 2–3 cm long; seeds 5–6 mm long.
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The pods have a length of and have pale margins. The dark brown to black seeds within the pods have a narrowly oblong to elliptic shape and have a length of .
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After flowering linear seed pods that are raised over and constricted between each seed that are in length and wide. The dark brown seeds with an elliptic to oblong-ovate shape.
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In complex analysis, an elliptic function is a meromorphic function that is periodic in two directions. Just as a periodic function of a real variable is defined by its values on an interval, an elliptic function is determined by its values on a fundamental parallelogram, which then repeat in a lattice. Such a doubly periodic function cannot be holomorphic, as it would then be a bounded entire function, and by Liouville's theorem every such function must be constant. In fact, an elliptic function must have at least two poles (counting multiplicity) in a fundamental parallelogram, as it is easy to show using the periodicity that a contour integral around its boundary must vanish, implying that the residues of all simple poles must cancel.
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In a plane cubic model three points sum to zero in the group if and only if they are collinear. For an elliptic curve defined over the complex numbers the group is isomorphic to the additive group of the complex plane modulo the period lattice of the corresponding elliptic functions. The intersection of two quadric surfaces is, in general, a nonsingular curve of genus one and degree four, and thus an elliptic curve, if it has a rational point. In special cases, the intersection either may be a rational singular quartic or is decomposed in curves of smaller degrees which are not always distinct (either a cubic curve and a line, or two conics, or a conic and two lines, or four lines).
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The importance of this concept was realised first in the analytic theory of theta functions, and geometrically in the theory of bitangents. In the analytic theory, there are four fundamental theta functions in the theory of Jacobian elliptic functions. Their labels are in effect the theta characteristics of an elliptic curve. For that case, the canonical class is trivial (zero in the divisor class group) and so the theta characteristics of an elliptic curve E over the complex numbers are seen to be in 1-1 correspondence with the four points P on E with 2P = 0; this is counting of the solutions is clear from the group structure, a product of two circle groups, when E is treated as a complex torus.
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The Gross–Zagier theorem describes the height of Heegner points in terms of a derivative of the L-function of the elliptic curve at the point s = 1\. In particular if the elliptic curve has (analytic) rank 1, then the Heegner points can be used to construct a rational point on the curve of infinite order (so the Mordell–Weil group has rank at least 1). More generally, showed that Heegner points could be used to construct rational points on the curve for each positive integer n, and the heights of these points were the coefficients of a modular form of weight 3/2. Shou-Wu Zhang generalized the Gross–Zagier theorem from elliptic curves to the case of modular abelian varieties (, ).
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Relativistic heavy-ion collisions produce very large numbers of subatomic particles in all directions. In such collisions, flow refers to how energy, momentum, and number of these particles varies with direction, and elliptic flow is a measure of how the flow is not uniform in all directions when viewed along the beam-line. Elliptic flow is strong evidence for the existence of quark–gluon plasma, and has been described as one of the most important observations measured at the Relativistic Heavy Ion Collider (RHIC). Elliptic flow describes the azimuthal momentum space anisotropy of particle emission from non-central heavy-ion collisions in the plane transverse to the beam direction, and is defined as the second harmonic coefficient of the azimuthal Fourier decomposition of the momentum distribution.
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Let L be an elliptic operator of order 2k with coefficients having 2k continuous derivatives. The Dirichlet problem for L is to find a function u, given a function f and some appropriate boundary values, such that Lu = f and such that u has the appropriate boundary values and normal derivatives. The existence theory for elliptic operators, using Gårding's inequality and the Lax–Milgram lemma, only guarantees that a weak solution u exists in the Sobolev space Hk. This situation is ultimately unsatisfactory, as the weak solution u might not have enough derivatives for the expression Lu to even make sense. The elliptic regularity theorem guarantees that, provided f is square-integrable, u will in fact have 2k square-integrable weak derivatives.
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A period of recuperation from an illness in 1940 gave him the opportunity to put several years of lecture notes into publishable form. The result was his best-known work: Jacobian Elliptic Functions (1944). By starting with the Weierstrass p-function and associating with it a group of doubly periodic functions with two simple poles, he was able to give a simple derivation of the Jacobian elliptic functions, as well as modifying the existing notation to provide a more systematic approach to the subject. Unfortunately, it failed to achieve its author's stated intention "to restore the Jacobian functions to the elementary curriculum" (NEVILLE 1951, vi) and its appearance came too late to have any real effect on the dominance of the classical approach to elliptic functions.
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Just as one classifies conic sections and quadratic forms into parabolic, hyperbolic, and elliptic based on the discriminant , the same can be done for a second-order PDE at a given point. However, the discriminant in a PDE is given by due to the convention of the term being rather than ; formally, the discriminant (of the associated quadratic form) is , with the factor of 4 dropped for simplicity. # (elliptic partial differential equation): Solutions of elliptic PDEs are as smooth as the coefficients allow, within the interior of the region where the equation and solutions are defined. For example, solutions of Laplace's equation are analytic within the domain where they are defined, but solutions may assume boundary values that are not smooth.
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Formally, an elliptic function is a function meromorphic on for which there exist two non-zero complex numbers and with , such that and for all . Denoting the "lattice of periods" by , this can be rephrased as requiring that for all . In terms of complex geometry, an elliptic function consists of a genus-one Riemann surface and a holomorphic mapping . From this perspective, one is treating two lattices and as equivalent if there is a nonzero complex number with .
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The mangrove may grow as a single-stemmed tree or multi-stemmed shrub. It has short buttresses at the base of the trunk, and knee-like air-breathing roots, or pneumatophores. The bark is a smooth grey- brown colour. The smooth, glossy green leaves are simple and opposite, elliptic to elliptic-oblong, 9.5–20 cm long, 3–7 cm wide, with a pointed apex and a 6 cm petiole, occurring in clusters at the end of the branches.
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Rhododendron facetum (绵毛房杜鹃) is a species of flowering plant in the Ericaceae family. It is native to northeast Myanmar, northern Vietnam, and western Yunnan, China, where it grows at altitudes of 2100–3600 meters. It is a shrub or small tree that grows to 3–7 m in height, with leathery leaves that are oblong-elliptic to obovate-elliptic, 8.5–20 by 3–6 cm in size. Flowers are red with deeper colored spots.
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Rhododendron pulchrum (锦绣杜鹃), also identified as Rhododendron x pulchrum is a rhododendron endemic to China. It grows as a semi-evergreen shrub, 1.5–2.5 meters in height, with leaf blades leathery, elliptic-oblong to elliptic- lanceolate or oblong-oblanceolate, 2–5(–7) × 1–2.5 cm in size. Flowers are rose-purple with dark red flecks. Hirsutum describes it as "a natural hybrid; seed x pollen= R. mucronatum var mucronatum x R. indicum var formosanum".
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The dorsal sepal is elliptic to narrow egg-shaped, about long, wide and greenish with narrow, dark purplish bands. The lateral sepals are linear to lance-shaped, about long, wide and spread widely apart from each other. The petals are linear to egg-shaped, about long and wide with dark purplish bands. The labellum is elliptic to broadly oblong, about long, wide and dark purplish-black with its edges densely covered with short, coarse purplish hairs.
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With the restriction to only this exponential, as shown by Galois theory, only compositions of Abelian extensions may be constructed, which suffices only for equations of the fourth degree and below. Something more general is required for equations of higher degree, so to solve the quintic, Hermite, et al. replaced the exponential by an elliptic modular function and the integral (logarithm) by an elliptic integral. Kronecker believed that this was a special case of a still more general method.
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Goro Shimura and Taniyama worked on improving its rigor until 1957. rediscovered the conjecture, and showed that it would follow from the (conjectured) functional equations for some twisted L-series of the elliptic curve; this was the first serious evidence that the conjecture might be true. Weil also showed that the conductor of the elliptic curve should be the level of the corresponding modular form. The Taniyama–Shimura–Weil conjecture became a part of the Langlands program.
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His main areas of interest are in Computational Number Theory, Combinatorics, Data Compression and Cryptography. He is one of the co-inventors of Elliptic Curve Cryptography.V. Miller, Use of elliptic curves in cryptography, Advances in cryptology---CRYPTO 85, Springer Lecture Notes in Computer Science vol 218, 1985. He is also one of the co-inventors, with Mark Wegman, of the LZW data compression algorithm, and various extensions, one of which is used in the V.42bis international modem standard.
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British mathematician Andrew Wiles. Ribet's proof of the epsilon conjecture in 1986 accomplished the first of the two goals proposed by Frey. Upon hearing of Ribet's success, Andrew Wiles, an English mathematician with a childhood fascination with Fermat's Last Theorem, and who had worked on elliptic curves, decided to commit himself to accomplishing the second half: proving a special case of the modularity theorem (then known as the Taniyama–Shimura conjecture) for semistable elliptic curves.Singh, p.
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For a brief description of this and related researches, see . found necessary and sufficient conditions for the validity of maximum principles for elliptic and parabolic systems of PDEs and introduced the so–called approximate approximations. He also contributed to the development of the theory of capacities, nonlinear potential theory, the asymptotic and qualitative theory of arbitrary order elliptic equations, the theory of ill-posed problems, the theory of boundary value problems in domains with piecewise smooth boundary.
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The shrub usually has a bushy habit and grows to a height of less that but can reach as high as . It is often composed of five to six main branches diverging at the base of the plant. The branches are erect or arched and split into ribbed, brown to green, smooth, hairy branchlets. The dark grey-green to green coloured phyllodes are flat or convex with an elliptic to broadly elliptic or slightly orbicular shape.
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Elliptic PDEs generally have very smooth solutions leading to smooth contours. Using its smoothness as an advantage Laplace's equations can preferably be used because the Jacobian found out to be positive as a result of maximum principle for harmonic functions. After extensive work done by Crowley (1962) and Winslow (1966) on PDEs by transforming physical domain into computational plane while mapping using Poisson’s equation, Thompson et al. (1974) have worked extensively on elliptic PDEs to generate grids.
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Rhododendron laetum is a rhododendron species native to the Anggi Lakes area of the Arfak Mountains in Indonesia and western New Guinea, where it grows at forest edges, in open marsh, and in swamps at the edge of lakes. It is a shrub that grows to 3 m in height, with leaves that are broadly elliptic or sub- ovate-elliptic, 40–95 × 25–53 mm in size. Flowers are deep yellow, flushing with red or orange as they age.
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An elliptic operator D on a compact smooth manifold defines an class in K homology. One invariant of this class is the analytic index of the operator. This is seen as the pairing of the class [D], with the element 1 in HC(C(M)). Cyclic cohomology can be seen as a way to get higher invariants of elliptic differential operators not only for smooth manifolds, but also for foliations, orbifolds, and singular spaces that appear in noncommutative geometry.
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Verticordia apecta, commonly known as scruffy verticordia or Hay River featherflower, is a flowering plant in the myrtle family, Myrtaceae and is endemic to the south-west of Western Australia. It is a slender shrub with linear lower stem leaves, narrow elliptic upper stem leaves and elliptic to egg-shaped leaves near the flowers. There are only a few flowers in the upper leaf axils on relatively long stalks and the sepals are deep pink with fine, white fringes.
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Lions, Joachim Nitsche and Miloš Zlámal. DG methods for elliptic problems were already developed in a paper by Garth Baker in the setting of 4th order equations in 1977. A more complete account of the historical development and an introduction to DG methods for elliptic problems is given in a publication by Arnold, Brezzi, Cockburn and Marini. A number of research directions and challenges on DG methods are collected in the proceedings volume edited by Cockburn, Karniadakis and Shu.
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It was a small car with a 1018 cc four-cylinder side-valve engine with fixed cylinder head from White & Poppe. Ignition was by a Bosch magneto. The chassis made by Rubery Owen was of pressed-steel construction and suspension was by leaf springs, semi-elliptic at the front and longer three-quarter elliptic at the rear slung above the axle. The welded single piece banjo rear axle with splined half shafts was driven by a Wrigley Worm.
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Overconvergent p-adic modular forms are similar to the modular forms defined by Katz, except that the form has to be defined on a larger collection of elliptic curves. Roughly speaking, the value of the Eisenstein series Ek–1 on the form is no longer required to be invertible, but can be a smaller element of R. Informally the series for the modular form converges on this larger collection of elliptic curves, hence the name "overconvergent".
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Four ECOH algorithms were proposed, ECOH-224, ECOH-256, ECOH-384 and ECOH-512. The number represents the size of the message digest. They differ in the length of parameters, block size and in the used elliptic curve. The first two uses the elliptic curve B-283: X^{283} + X^{12} + X^7 + X^5 + 1 , with parameters (128, 64, 64). ECOH-384 uses the curve B-409: X^{409} + X^{87} + 1 , with parameters (192, 64, 64).
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The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher- dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it).
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Rhododendron lapponicum (Lapland rosebay, 高山杜鹃) is a rhododendron species found in subarctic regions around the world, where it grows at altitudes ranging from sea level to 1900 meters. It is a shrub that grows to 0.2–0.45 m in height, with leaves that are oblong-elliptic or ovate-elliptic to oblong- obovate, 0.4–1.5 by 0.2–0.5 cm in size. Flowers are reddish or purple. And fragrant, scent somewhat like Lily-of-the-Valley.
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Philotheca myoporoides subsp. acuta is a shrub that typically grows to a height of with glabrous, densely glandular-warty stems. The leaves are sessile, oblong-elliptic to elliptic or rarely lance-shaped with the narrower end towards the base, long and wide and there is a small point on the tip. The flowers are arranged singly or in twos or threes in leaf axils on a peduncle up to long, each flower on a pedicel long.
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Rhododendron impeditum (粉紫杜鹃) is a species of flowering plant in the Ericaceae family. It is native to southwestern Sichuan and northwest Yunnan in China, where it grows at altitudes of 2500–4600 meters. It is a shrub that grows to 0.8 m in height, with leaves that are ovate, elliptic or broadly elliptic to oblong, 0.5–1.4 by 0.3–0.6 cm in size. Flowers are purple, violet, or rose- lavender, or rarely white.
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This work also gave rise to the ideas of an algebraic space and algebraic stack, and has proved very influential in moduli theory. Additionally, he has made important contributions to the deformation theory of algebraic varieties. With Peter Swinnerton-Dyer, he provided a resolution of the Shafarevich-Tate conjecture for elliptic K3 surfaces and the pencil of elliptic curves over finite fields. Artin contributed to the theory of surface singularities which are both fundamental and seminal.
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The erect non-lignotuberous dense rounded shrub typically grows to a height of . It blooms from May to September and produces white flowers and have woolly white or yellowish brown perianths with a deep red style in clusters in the leaf axils. The leaves are flat, elliptic or obovate,about long by young leaves and branchlets are clothed in rusty-woolly hairs. The smooth narrowly elliptic fruit are normally 2.5-3cm (1 inch) long and only a slight beak.
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This is because there are no antipodal points in elliptic geometry. For example, this is achieved in the hyperspherical model (described below) by making the "points" in our geometry actually be pairs of opposite points on a sphere. The reason for doing this is that it allows elliptic geometry to satisfy the axiom that there is a unique line passing through any two points. Every point corresponds to an absolute polar line of which it is the absolute pole.
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Any point on this polar line forms an absolute conjugate pair with the pole. Such a pair of points is orthogonal, and the distance between them is a quadrant.Duncan Sommerville (1914) The Elements of Non-Euclidean Geometry, chapter 3 Elliptic geometry, pp 88 to 122, George Bell & Sons The distance between a pair of points is proportional to the angle between their absolute polars. As explained by H. S. M. Coxeter :The name "elliptic" is possibly misleading.
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Elliptic, parabolic, and hyperbolic partial differential equations of order two have been widely studied since the beginning of the twentieth century. However, there are many other important types of PDE, including the Korteweg–de Vries equation. There are also hybrids such as the Euler–Tricomi equation, which vary from elliptic to hyperbolic for different regions of the domain. There are also important extensions of these basic types to higher-order PDE, but such knowledge is more specialized.
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The tree or tall shrub typically grows to a height of and has sparsely arranged whippy branches. It has smooth red-brown to grey- brown coloured bark that becomes fibrous with age. The evergreen narrowly elliptic or elliptic shaped phyllodes are straight with a length of and a width of and have between two and four prominent nerves. It blooms between June and August producing flower-spikes with a length of that are densely packed with bright yellow flowers.
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Salvia plebeia is an annual or biennial herb that is native to a wide region of Asia. It grows on hillsides, streamsides, and wet fields from sea level to . S. plebeia grows on erect stems to a height of tall, with elliptic-ovate to elliptic-lanceolate leaves. Inflorescences are 6-flowered verticillasters in racemes or panicles, with a distinctly small corolla () that comes in a wide variety of colors: reddish, purplish, purple, blue-purple, to blue, and rarely white.
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In mathematics, the Lubin–Tate formal group law is a formal group law introduced by to isolate the local field part of the classical theory of complex multiplication of elliptic functions. In particular it can be used to construct the totally ramified abelian extensions of a local field. It does this by considering the (formal) endomorphisms of the formal group, emulating the way in which elliptic curves with extra endomorphisms are used to give abelian extensions of global fields.
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He provided also detailed studies of elliptic fibrations of surfaces over a curve, or in other language elliptic curves over algebraic function fields, a theory whose arithmetic analogue proved important soon afterwards. This work also included a characterisation of K3 surfaces as deformations of quartic surfaces in P4, and the theorem that they form a single diffeomorphism class. Again, this work has proved foundational. (The K3 surfaces were named after Ernst Kummer, Erich Kähler, and Kodaira).
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Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration. Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters (\lambda,\mu) of the Hessian form may belong to an extension field of the field of definition of the original curve.
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Casearia graveolens grows as a 3 to 15m tall tree. Its trunk, with dark-grey rough, fissured bark with white specks, grows to a dbh of 20 cm when the tree is between 3-6m. The green, smooth branches have grey-white patches, with glabrous branchlets, twig tips and terminal buds. The leaves are broadly elliptic to elliptic-oblong, 6–15 cm by 4–8 cm, with reddish brown dots and streaks visible at low magnification.
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The silvery-grey shrub typically grows to a height of . It has smooth, grey to brown coloured bark with two or three angled branchlets covered in dense silky hairs. Like most species of Acacia it has phyllodes rather than true leaves. The flat and evergreen phyllodes have a narrowly elliptic to elliptic shape with a length of and a width of and have three to six prominent longitudinal nerves present of the face if the phyllode.
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She shows how EDS can be used to compute the value of the Weil and Tate pairings on elliptic curves over finite fields. These pairings have numerous applications in pairing-based cryptography.
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E. Cauer et al., p. 5 A special case of elliptic rational functions is the Chebyshev polynomials due to Pafnuty Chebyshev (1821–1894) and is an important part of approximation theory.Swanson, p.
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The four sepals are thick, glabrous and egg- shaped, long. The petals are white with blue or pale green backs, broadly elliptic, long and prominently glandular. Flowering occurs from May to October.
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In mathematics, the Quillen metric is a metric on a determinant line bundle. It was introduced by for certain elliptic operators over a Riemann surface, and generalized to higher-dimensional manifolds by .
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Part 2. Florida Orchidist 34: 195–211. Basiphyllaea corallicola is a terrestrial herb up to 40 cm tall, with underground tubers. Leaves are narrowly linear to elliptic, up to 25 cm long.
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It grows as an erect shrub to 3 metres high. Leaves 5 to 12 cm long, 1 to 5 cm wide. Elliptic in shape, occasionally lanceolate or ovate. Flowers form on panicles.
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The leaves are alternate, not toothed. 6 to 10 cm long, 2 to 5 cm wide. Blunt or bluntly pointed at the end of the leaf. Ovate to ovate elliptic in shape.
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He found a proof of several generalizations using elliptic operators; this new proof used an idea that he used to give a particularly short and easy proof of Bott's original periodicity theorem.
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Pentachlaena latifolia grows as a shrub or small tree up to tall. Its coriaceous leaves are elliptic to circular in shape. The flowers are either almost sessile or borne on short peduncles.
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It is a perennial growing tall. 2 to 7 lanceolate to narrow elliptic leaves should be present. The inflorescence is a terminal racemic structure, long with 15 to 60 whitish-green flowers.
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The pods are raised and constricted between seeds and have a length of around and a width of . The glossy mottled brown linear-elliptic shaped seeds within the pods are in length.
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In 1998, Frey proposed the idea of Weil descent attack for elliptic curves over finite fields with composite degree. As a result of this attack, cryptographers lost their interest in these curves...
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The white, shining shell has a smooth sculpture. Its length measures 4–5 mm. The four whorls of the teleoconch are rather convex, subangulated at the suture. The aperture is ovate-elliptic.
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In mathematics, Gårding's inequality is a result that gives a lower bound for the bilinear form induced by a real linear elliptic partial differential operator. The inequality is named after Lars Gårding.
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The group law of an abelian variety is necessarily commutative and the variety is non- singular. An elliptic curve is an abelian variety of dimension 1. Abelian varieties have Kodaira dimension 0.
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Main scientific papers are dedicated to differential equations, elliptic and hypoelliptic equations, the study of the properties of functions in different multianisotropic spaces, integral representations and embedding theorems for functions in multianisotropic spaces.
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Camelina plants are annual or biennial herbs. Their leaves are simple, lanceolate to narrowly elliptic. The flowers are hermaphroditic actinomorphic, grouped in racemes, and yellowish colored. The seeds are formed in dehiscent siliques.
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Lasianthus kilimandscharicus is a shrub or tree found in Kenya. It becomes tall; bark smooth, grey. Leaves (narrowly) elliptic, base cuneate, apex acuminate, by , glabrous or nearly so. Flowers white or pale purple.
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The five sepals are more or less round, fleshy and about long. The five petals are white, elliptic and about long and the ten stamens are hairy. Flowering occurs from August to October.
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The papers , and identified their type in the classification of algebraic surfaces. Most of them are surfaces of general type, but several are rational surfaces or blown up K3 surfaces or elliptic surfaces.
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The leaves are alternate, simple, long and broad, oblong-elliptic, densely hairy on the underside, and with a coarsely serrated margin and a petiole. The flowers are white, long, produced on panicles long.
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Ling Long is a Chinese mathematician whose research concerns modular forms, elliptic surfaces, and dessins d'enfants, as well as number theory in general. She is a professor of mathematics at Louisiana State University.
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It is conjectured that a nonsingular EDS contains only finitely many primes M. Einsiedler, G. Everest, and T. Ward. Primes in elliptic divisibility sequences. LMS J. Comput. Math., 4:1-13 (electronic), 2001.
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These are not toothed, and are egg-shaped to elliptic-oblong, and long. The tips are often notched or blunt. Leaf veins are evident on both sides. The veins are mostly raised underneath.
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Its spikelets are elliptic and are long. Fertile spikelets are pediceled, the pedicels of which are curved, filiform and are long. Florets are diminished. Its lemma have long hairs and have villous surface.
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Fruit grow at an angle on the stalk are egg-shaped long and wide. The surface has spiky toothed ridges, fruit may remain green even at maturity. The winged elliptic seeds are long.
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The proof of Fermat's Last Theorem proceeds by first reinterpreting elliptic curves and modular forms in terms of these Galois representations. Without this theory there would be no proof of Fermat's Last Theorem.
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In noncommutative geometry, a Fredholm module is a mathematical structure used to quantize the differential calculus. Such a module is, up to trivial changes, the same as the abstract elliptic operator introduced by .
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Cam steering is by Marles and brakes by Bendix- Cowdrey. Suspension is by half-elliptic springs, those in the back are underslung, dampened by hydraulic shock absorbers. There is an easy jacking system.
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Pimelea angustifolia is a small shrub high with smooth stems. The leaves are arranged in opposite pairs on a short petiole, mostly linear or narrowly elliptic, smooth, mid-green throughout, long and wide.
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Since elliptic and parabolic partial differential equations are particularly smooth when the domain is two dimensions, the Chang-Ding-Ye result is considered to be indicative of the general character of the flow.
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The leaves are simple ovate-lanceolate, long, , and are attached to short petioles. They are opposite, ovate to elliptic-lanceolate, and have entire or undulating margins with small hairs, which can irritate skin.
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In cryptography, CECPQ1 (combined elliptic-curve and post-quantum 1) is a post-quantum cipher developed by Google to make web browsers secure via Transport Layer Security (TLS). It was succeeded by CECPQ2.
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These short hairs are lost during development. Perithecia (fruiting bodies) are elliptic, 160-280 µm long and 100-185 µm wide. The perithecial neck has a length of 53-90 µm. Asci (sing.
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However, whereas the spectral method is based on the eigendecomposition of the particular boundary value problem, the finite element method does not use that information and works for arbitrary elliptic boundary value problems.
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The thinly coriaceous, glabrous seed pods have a length of and a width of . The dark brown seeds found within the pods are longitudinally arranged with an elliptic shape and a length of .
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The pollen is also white. The white stigma is notched, or serrated. After the iris has flowered, it produces a trigonal, or elliptic seed capsule, is long. It has a beak (curved ending).
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The seed pods that form after flowering have a length of around and a width of . The shiny black seeds within the pods have an oblong to elliptic shape and a length of .
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It is a cultivar of medium vigour, with a spreading growth form, elliptic leaves, an expansive global crown, and large size. The olives are of medium-to-high weight (5-6 g), elliptic in shape with a rounded tip and slightly asymmetrical. The stone is ovoid, rounded on both ends, with a rough surface and a mucro. The fruits can be harvested at smaller size in late November, while for larger olives it is better to wait until December or January.
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These integrals are defined using the higher Haar measure and objects from higher class field theory. Fesenko generalized the Iwasawa-Tate theory from 1-dimensional global fields to 2-dimensional arithmetic surfaces such as proper regular models of elliptic curves over global fields. His theory led to three further developments. The first development is the study of functional equation and meromorphic continuation of the Hasse zeta function of a proper regular model of an elliptic curve over a global field.
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Pancratium maximum is a perennial glabrous herb up to 20 cm tall arising from a bulb. The bulb is globose, 4–6 cm in diameter, narrowed above into a cylindrical neck, covered with several layers of dark reddish brown papery tunics. Leaves 2–7 cm long, variable in width, linear-elliptic to narrowly elliptic or ovate and abruptly narrowed into a petiole below, 10–30 cm long x 2.3–18 cm across. The flower is white with yellow anthers and black angular seeds.
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It has 2 pairs of petals, 3 large sepals (outer petals), known as the 'falls' and 3 inner, smaller petals (or tepals, known as the 'standards'). The falls are elliptic or obovate, with a spreading limb and blue or purple/violet blotching, spots, (or dots) around a central yellow signal patch around a visible yellow, or orange crest. They are long and 1.4–2 cm wide. The standards are elliptic or narrowly obovate. They are long and 1.5–2.1 cm wide.
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Separately from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between elliptic curves and modular forms. These were mathematical objects with no known connection between them. Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways. They conjectured that every rational elliptic curve is also modular.
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The cylindrical flower-spikes have a length of with densely packed yellow to pale golden coloured flowers. Following flowering seed pods form that have a narrowly oblong shape and can be constricted between the seeds. The woody and grooved pods are sub-terete to slightly flattened and can be straight to slightly curved with a length of with a width of The glossy black seeds inside are elliptic to irregularly elliptic with a length of around and a width of .
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The lateral sepals are broadly elliptic in shape and spread apart from each other, turning slightly downwards. The petals are also elliptic in shape but often sickle-shaped and have a pointed tip. The labellum is pale pink, white near its edges and has dark red bars. The sides of the labellum curve upwards, partly surrounding the column, the tip is yellow with notched edges and there are two rows of stalked calli with bright yellow heads along the centre of the labellum.
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Plot of the absolute value of the third order elliptic rational function with ξ=1.4. There is a zero at x=0 and the pole at infinity. Since the function is antisymmetric, it is seen there are three zeroes and three poles. Between the zeroes, the function rises to a value of 1 and, between the poles, the function drops to the value of the discrimination factor Ln Plot of the absolute value of the fourth order elliptic rational function with ξ=1.4.
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The horizontally spreading petals are egg-shaped to elliptic or oblong elliptic, 1.2 to 1.8 inches long by 0.7 to 1.5 inches wide, and have margins that are wavy and fairly deeply notched or toothed. The lip is oblong or somewhat fiddle-shaped. It is 0.8 to 1.2 inches long by 0.5 to 0.6 inches wide, has toothed margins and is rather sharply pointed at the apex. The callus is fleshy with a pair of diverging lobes at the apex.
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Alina Carmen Cojocaru is a Romanian mathematician who works in number theory and is known for her research on elliptic curves, arithmetic geometry, and sieve theory. She is a professor of mathematics at the University of Illinois at Chicago and a researcher in the Institute of Mathematics of the Romanian Academy. Cojocaru earned her Ph.D. from Queen's University in Kingston, Ontario, in 2002. Her dissertation, Cyclicity of Elliptic Curves Modulo p, was jointly supervised by M. Ram Murty and Ernst Kani.
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The shrub typically grows to a height of and has a rounded, bushy and spreading habit. The branchlets are covered with a dense layer of fine hairs velvety citron hairs on older shoots and silvery white hairs on new shoots. It has small, grayish rounded phyllodes and spherical flower-heads of bright golden flowers on long stalks. The grey-green to silvery coloured phyllodes have an elliptic to oblong-elliptic or obovate shape with a length of and a width of .
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The shrub has an erect to spreading habit and typically grows to a height of with angled branchlets that are minutely hairy. It has grey-green phyllodes that can have white to grey hairs. The phyllodes have a narrowly elliptic to narrowly oblong-elliptic shape and have a length of and a width of with a prominent mid-vein and fainter lateral veins. t blooms between September and November producing groups of 3 to 16 inflorescences found in the axillary racemes.
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Leaves are elliptic-oblong to elliptic-lanceolate in shape. Apex is acute to acuminate with blunt tip, base is acute to attenuate, coriaceous, glabrous; midrib of the leaf is canaliculate above, stout beneath; secondary nerves usually 5-9 pairs, where lower pairs closer than above ones; tertiary nerves are strongly reticulate on both surfaces. Fruits of the plant are usually as berries, and are globose, up to 7 cm in diameter, usually rusty brown in color and fruit bear about 8 seeds.
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The Cox–Zucker machine is an algorithm created by David A. Cox and Steven Zucker. This algorithm determines if a given set of sections provides a basis (up to torsion) for the Mordell–Weil group of an elliptic surface E → S where S is isomorphic to the projective line. The algorithm was first published in the 1979 paper "Intersection numbers of sections of elliptic surfaces" by Cox and Zucker, and it was later named the "Cox–Zucker machine" by Charles Schwartz in 1984.
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Philotheca myoporoides subsp. myoporoides is a shrub, sometimes a small tree, that typically grows to a height of with glabrous, slightly to moderately glandular-warty stems. The leaves are variable in shape, oblong to elliptic or broadly elliptic to egg- shaped with the narrower end towards the base, long, wide and glandular-warty with a prominent midrib. The flowers are mostly arranged in groups of three to eight in leaf axils on a peduncle long, each flower on a pedicel long.
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Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research. He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in applications of elliptic curves to integer factorization and other problems, and the Montgomery ladder, which is used to protect against side-channel attacks in elliptic curve cryptography.
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There are recent developments in using hyperelliptic curves to factor integers. Cosset shows in his article (of 2010) that one can build a hyperelliptic curve with genus two (so a curve y^2 = f(x) with of degree 5), which gives the same result as using two "normal" elliptic curves at the same time. By making use of the Kummer surface, calculation is more efficient. The disadvantages of the hyperelliptic curve (versus an elliptic curve) are compensated by this alternative way of calculating.
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Isogenous elliptic curves to E can be obtained by quotienting E by finite subgroups, here subgroups of the 4-torsion subgroup. For abelian varieties, such as elliptic curves, this notion can also be formulated as follows: Let E1 and E2 be abelian varieties of the same dimension over a field k. An isogeny between E1 and E2 is a dense morphism f : E1 → E2 of varieties that preserves basepoints (i.e. f maps the identity point on E1 to that on E2).
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Pimelea pagophila is a small shrub high with smooth stems and prominent leaf nodes. The mid green leaves are arranged in opposite pairs along the branches and are narrowly egg-shaped to elliptic, long, wide, smooth and mostly paler on the underside. The inflorescence is a pendulous spherical head containing numerous individual flowers. The 4, 6 or 8 overlapping flower bracts are sessile, elliptic or egg-shaped, long, wide, thin, smooth, light green or yellow-green, occasionally a reddish colour.
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Rhododendron sikangense (川西杜鹃) is a rhododendron species native to western Sichuan and northeastern Yunnan in China (the area of the former province of Sikang for which it is named), where it grows at altitudes of 2800–4500 meters. It is a shrub or small tree that grows to 3–5 m in height, with leathery leaves that are oblong-elliptic or elliptic-lanceolate, 7–12 by 2.5–5.5 cm in size. Flowers are white, purple, or pink, with purple flecks.
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In 1999, David published a paper with Francesco Pappalardi which proved that the Lang–Trotter conjecture holds in most cases. She has shown that for several families of curves over finite fields, the zeroes of zeta functions are compatible with the Katz–Sarnak conjectures. She has also used random matrix theory to study the zeroes in families of elliptic curves. David and her collaborators have exhibited a new Cohen–Lenstra phenomenon for the group of points of elliptic curves over finite fields.
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This species has often been confused with A. muscaria, some subspecies of which are also orange-colored. It also bears some resemblance to A. frostiana and A. flavorubescens. One 1982 study concluded that a "large majority" of herbarium specimens labeled as A. frostiana were actually A. flavoconia. The use of microscopic features is necessary to distinguish clearly among the species: A. flavoconia has elliptic, amyloid spores, while A. frostiana has round, non-amyloid spores; A. muscaria has nonamyloid, elliptic spores.
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This is because the Dedekind zeta functions factorize as a product of powers of Artin L-functions, so zeros of Artin L-functions sometimes give rise to multiple zeros of Dedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some elliptic curves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecture predicts that the multiplicity of this zero is the rank of the elliptic curve.
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The tree or shrub typically grows to a height of but can be as high as . It is generally V-shaped with an open and usually spindly form. It usually divides above ground level to form some main stems that are straight, diagonally spreading to erect and covered in smooth light grey bark except toward the base where it can become longitudinally fissured. The phyllodes are usually obliquely elliptic to narrowly elliptic in shape that becomes narrowed at both ends.
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An abelian function is a meromorphic function on an abelian variety, which may be regarded therefore as a periodic function of n complex variables, having 2n independent periods; equivalently, it is a function in the function field of an abelian variety. For example, in the nineteenth century there was much interest in hyperelliptic integrals that may be expressed in terms of elliptic integrals. This comes down to asking that J is a product of elliptic curves, up to an isogeny.
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Bromheadia finlaysoniana is a terrestrial, evergreen herb with flattened yellowish stems long with tough, stiffly spreading elliptic to egg-shaped leaves long and wide. The flowering stems is long with a short zig-zag end where up to seventy five single white flowers open in succession. The flowers are long, wide and are pinkish on the outside. The sepals are elliptic to egg-shaped, long and wide and the petals are egg-shaped and a similar length but broader than the sepals.
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In mathematics, the Jacobi curve is a representation of an elliptic curve different from the usual one (Weierstrass equation). Sometimes it is used in cryptography instead of the Weierstrass form because it can provide a defence against simple and differential power analysis style (SPA) attacks; it is possible, indeed, to use the general addition formula also for doubling a point on an elliptic curve of this form: in this way the two operations become indistinguishable from some side-channel information.Olivier Billet, The Jacobi Model of an Elliptic Curve and Side-Channel Analysis The Jacobi curve offers also faster arithmetic compared to the Weierstrass curve. The Jacobi curve can be of two types: the Jacobi intersection, that is given by an intersection of two surfaces, and the Jacobi quartic.
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For example, the first and fourth of Euclid's postulates, that there is a unique line between any two points and that all right angles are equal, hold in elliptic geometry. Postulate 3, that one can construct a circle with any given center and radius, fails if "any radius" is taken to mean "any real number", but holds if it is taken to mean "the length of any given line segment". Therefore any result in Euclidean geometry that follows from these three postulates will hold in elliptic geometry, such as proposition 1 from book I of the Elements, which states that given any line segment, an equilateral triangle can be constructed with the segment as its base. Elliptic geometry is also like Euclidean geometry in that space is continuous, homogeneous, isotropic, and without boundaries.
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An important example of a compact operator is compact embedding of Sobolev spaces, which, along with the Gårding inequality and the Lax–Milgram theorem, can be used to convert an elliptic boundary value problem into a Fredholm integral equation.William McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, 2000 Existence of the solution and spectral properties then follow from the theory of compact operators; in particular, an elliptic boundary value problem on a bounded domain has infinitely many isolated eigenvalues. One consequence is that a solid body can vibrate only at isolated frequencies, given by the eigenvalues, and arbitrarily high vibration frequencies always exist. The compact operators from a Banach space to itself form a two-sided ideal in the algebra of all bounded operators on the space.
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The bi-elliptic transfer consists of two half-elliptic orbits. From the initial orbit, a first burn expends delta-v to boost the spacecraft into the first transfer orbit with an apoapsis at some point r_b away from the central body. At this point a second burn sends the spacecraft into the second elliptical orbit with periapsis at the radius of the final desired orbit, where a third burn is performed, injecting the spacecraft into the desired orbit. While they require one more engine burn than a Hohmann transfer and generally require a greater travel time, some bi-elliptic transfers require a lower amount of total delta-v than a Hohmann transfer when the ratio of final to initial semi-major axis is 11.94 or greater, depending on the intermediate semi-major axis chosen.
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Models of non-Euclidean geometry are mathematical models of geometries which are non-Euclidean in the sense that it is not the case that exactly one line can be drawn parallel to a given line l through a point that is not on l. In hyperbolic geometric models, by contrast, there are infinitely many lines through A parallel to l, and in elliptic geometric models, parallel lines do not exist. (See the entries on hyperbolic geometry and elliptic geometry for more information.) Euclidean geometry is modelled by our notion of a "flat plane." The simplest model for elliptic geometry is a sphere, where lines are "great circles" (such as the equator or the meridians on a globe), and points opposite each other are identified (considered to be the same).
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Finding the points on an elliptic curve modulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check. However, for large primes it is computationally intensive.
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The florets are long and are elliptic. Flowers have 3 anthers which are in length. Glumes are thinner than fertile lemma with the lower one being of which is one length of upper one.
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Cinchona officinalis is a shrub or tree with rugose bark and branchlets covered in minute hairs. Stipules lanceolate or oblong, acute or obtuse, glabrous. Leaves lanceolate to elliptic or ovate, usually about . long and .
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Brownea coccinea trees have compound leaves which are 10–35 cm. long, containing 4-10 leaflets. Leaflets are oblong or elliptic, pointed at the apex and 4–23 cm. long and 1.5-6.5 cm.
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53, no. 8, August 2005, pp. 3172-3182. . Babuška, I., F. Nobile and R. Tempone, 2005, Worst case scenario analysis for elliptic problems with uncertainty, Numerische Mathematik (in English) vol.101 pp.185–219.
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The seeds inside the pods are arranged longitudinally and have a narrowly oblong to slightly elliptic shape. The slightly shiny black seeds have a length of and are minutely pitted with a clavate aril.
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Schizolaena pectinata grows as a tree up to tall. Its elliptic to ovate leaves measure up to long. The small flowers are white or pink. The involucre of the flowers is fleshy and laciniate.
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Many other characterizations of the Bring radical have been developed, the first of which is in terms of elliptic modular functions by Charles Hermite in 1858, and further methods later developed by other mathematicians.
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The independent front suspension to Alvis's own design is by transverse leaf springs. The semi- elliptic rear springs are underslung. Luvax hydraulic shock absorbers control the springing. A fully floating rear axle is fitted.
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The soft, dull, brown coloured seeds within the pods are arranged longitudinally and have an oblong or broadly elliptic shape and are flattened but thick with a length of and have a filiform funicle.
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Its leaves are odd-pinnate, coriaceous, 15–50 cm long, comprising 9-17 leaflets, each of which is 3–15 cm long by 1–5 cm wide, and ovate to elliptic-lanceolate in shape.
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Available as a 4-door saloon, the Sixteen had a Morris QH engine the same size as its predecessor, at 2062cc. It achieved 15.94 hp (RAC hp). It had semi-elliptic leaf-spring suspension.
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Rhodolaena humblotii grows as a shrub or small to medium-sized tree. The twigs have dense hairs. Its leaves are small and elliptic in shape. The inflorescences bear two flowers on a short peduncle.
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Leaf blade elliptic or ovate-lanceolate, 6–17 × 2–6 cm, leathery, margin sharply coarsely-serrate. Stamen baculate to terete; thecae shorter than connective. Stigma subcapitate. Fruit globose or ovoid, 3–4 mm in diam.
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While several steps of SIDH involve complex isogeny calculations, the overall flow of SIDH for parties A and B is straightforward for those familiar with a Diffie-Hellman key exchange or its elliptic curve variant.
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The species was first formally described by botanist Jacques Labillardière in 1806 in Novae Hollandiae Plantarum Specimen. The specific epithet (elliptica) is from the Latin ellipticus meaning "elliptic", referring to the shape of the leaves.
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Baum, H. R., & Atreya, A. (2015). The Elliptic Emmons Problem. In ICHMT DIGITAL LIBRARY ONLINE. Begel House Inc.. The flame is of diffusion flame type because it separates fuel and oxygen by a flame sheet.
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It is not usually confused with other fig species, being distinctive by the green (or yellowish-green) of its mature syconia, and by its narrowly elliptic or lanceolate leaves. It is not a strangling fig.
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The pods have a narrowly oblong shape and are uo to in length and wide. The shiny black seeds inside have an ovate to oblong-elliptic shape and are in length with a clavate aril.
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They also have scabrous bottom, are pubescent and a bit hairy. The panicle is open, is linear and is long. The main panicle branches are spread out. It spikelets are elliptic, solitary and are long.
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For example, if the shaft extends from surface to surface a closed orbit is possible consisting of parts of two cycles of simple harmonic motion and parts of two different (but symmetric) radial elliptic orbits.
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Not toothed, elliptic, 5 to 7 cm long, pointed at the tip. Glossy green both sides, paler beneath. Five to seven parallel and longitudinal veins on the leaf. New leaves brilliant dark pink or red.
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Following flowering thin walled seed pods that resemble a string of beads and are curved or coiled with a length of and a width of . The pods contain narrowly elliptic seeds with a length of .
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The official comments on ECOH included a proposal called ECOH2 that doubles the elliptic curve size in an effort to stop the Halcrow-Ferguson second preimage attack with a prediction of improved or similar performance.
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L'vov V.N., Smekhacheva R.I., Smirnov S.S., Tsekmejster S.D. Some peculiarities in the Hildas motion. Izv. Pulkovo Astr. Obs., 2004, 217, 318-324 (in Russian) Each of the Hilda objects moves along its own elliptic orbit.
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Palea is elliptic and is long and have 2 veines with puberulous surface. Flowers are growing together and have 3 anthers that are long with 2 lodicules. Fruits are fusiform and have an additional pericarp.
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Adenodolichos exellii grows as a shrub, measuring up to tall, rarely to . The leaves consist of three elliptic leaflets, measuring up to long, pubescent on both surfaces. Inflorescences are terminal, featuring purple to red flowers.
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Sepals are elliptic and up to 5 mm long. Most parts of the plant are virtually glabrous, although a short, dense indumentum of velvety brown hairs is present on the stem, inflorescences, and lamina midribs.
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Camu camu has small flowers with waxy white petals and a sweet-smelling aroma. It has bushy, feathery foliage. The evergreen, opposite leaves are lanceolate to elliptic. Individual leaves are m in length and wide.
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Rhodolaena bakeriana grows as a medium sized tree. Its twigs are hairy. It has small to medium leaves, obovate, elliptic or oblong in shape. The inflorescences have one or two flowers on a long stem.
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He derived an existence and regularity theory for a class of constrained variational problems. Parks has discovered, and characterized, a type of minimal surface with surprising properties, defined in terms of the Jacobi elliptic functions.
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Ulmer did his undergraduate study at Princeton University. In 1987, he received his PhD at Brown University, where his advisor was Benedict Hyman Gross; his thesis was titled The Arithmetic of Universal Elliptic Modular Curves.
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The flower-spikes are around in length. The linear, slightly moniliform, semicircular seed pods that form after flowering are in length. The pods contain brownish black seeds with an elliptic shape that are in length.
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N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 2, 101–181 (1827). At the end of the same year he became aware of Carl Gustav Jacobi and his works on new transformations of elliptic integrals. Abel finishes then a second part of his article on elliptic functions and shows in an appendix how the transformation results of Jacobi would easily follow.N.H. Abel, Recherches sur les fonctions elliptiques, Journal für die reine und angewandte Mathematik, 3, 160–190 (1828).
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R. kelticum in flower in north Wales Rhododendron keleticum (独龙杜鹃) is a species of flowering plant in the Ericaceae family. It is native to southeast Xizang and northwest Yunnan, China, as well as Myanmar, where it grows at altitudes of 3000–3900 meters. It is a small shrub that grows 0.05–0.3 m in height, with leathery leaves, elliptic-lanceolate, elliptic, or ovate in shape, 0.6–2 by 0.3–1 cm in size. Flowers are pale purplish red or green tinged with red.
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A predecessor to the SIDH was published in 2006 by Rostovtsev and Stolbunov. They created the first Diffie-Hellman replacement based on elliptic curve isogenies. Unlike the method of De Feo, Jao, and Plut, the method of Rostovtsev and Stolbunov used ordinary elliptic curves and was found to have a subexponential quantum attack. In March 2014, researchers at the Chinese State Key Lab for Integrated Service Networks and Xidian University extended the security of the SIDH to a form of digital signature with strong designated verifier.
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The spikelets have 2 fertile florets which are diminished at the apex while the sterile florets are barren, lanceolate, clumped and are long. Its rhachilla have scaberulous internodes while the floret callus is glabrous. Both the upper and lower glumes are keelless, membranous, and have acute apexes but have different size and description; Lower glume is obovate and is long while upper one is elliptic and is long. The species' lemma have eciliated margins while its fertile one is chartaceous, elliptic, and is long by wide.
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The hermaphrodite shrub, having both the male and female reproductive organs in the same individual, is 3–5 m tall with a 5 cm diameter at breast height. Twigs are free from hair and somewhat tetragonal, while the angle ridges are prominent. The leaves are elliptic to narrowly elliptic-ovate while being wedge-shaped or round at the base. The leaves are 5–12 cm long, 2–5 cm wide having the sub-3-veined from the base with the outer 2 nerves forming submarginal veins.
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Its outer margin may undulate slightly. The operculum or lid varies in shape from elliptic to ovate or broadly ovate. It has a rounded apex and may have a somewhat cordate base. It measures up to 4 cm in length by 3.5 cm in width. No appendages are present on the lower surface of the lid, although it bears a small number (5 or 6) of sparsely scattered nectar glands. These nectaries are transversely elliptic to circular in shape and measure 0.2–0.4 mm in length.
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Mairia coriacea is a geophytic perennial herb of about high, with dense, silky, orange brown hairs on its growing points. It has a rizome with succulent, dark brown to black roots of up to about long and thick. It usually has up to about six leathery, bright lime-green leaves per growing point, which are seated or have a leaf stalk of up to long, and are flat or curve downwards. The leaf blade is mostly inverted egg-shaped, sometimes elliptic or broadly elliptic, long and wide.
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The samaras of A. rousei have two indistinct flanges medially along the notably inflated nutlet. The overall shape of the nutlet is circular to elliptic with the average length of the samara up to and a wing width of . The paired samaras of the species have a notably high attachment angle of 80° to 90°. While very similar in morphology to species in the modern section Palmata, A. rousei differs in the presence of the flanges on the nutlet and circular to elliptic outline of the nutlet.
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Muiriantha hassellii is a small under shrub to high with branchlets sparsely covered in soft, thin, separated, star-shaped hairs. The leaves are arranged alternately, aromatic, upright, narrowly elliptic, long, leathery, smooth and sparsely covered in soft hairs. The fragrant inflorescence are terminal on branches, tubular long, pendulous with small to medium sized bracts. The 5 yellowish-green petals are narrowly oblong to elliptic, rounded at the end, with a purple or green centre stripe, pedicels long and soft and weak hairs toward the petals apex.
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The Microsoft provider that implements CNG is housed in Bcrypt.dll. CNG also supports elliptic curve cryptography which, because it uses shorter keys for the same expected level of security, is more efficient than RSA.The Case for Elliptic Curve Cryptography, NSA The CNG API integrates with the smart card subsystem by including a Base Smart Card Cryptographic Service Provider (Base CSP) module which encapsulates the smart card API. Smart card manufacturers just have to make their devices compatible with this, rather than provide a from-scratch solution.
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Nematolepis phebalioides is an upright shrub to high. The leaves are on ascending branches on a short petiole, elliptic to broadly elliptic shaped, about long, leathery, smooth, glossy on the upper surface, grey scales on underside and rounded at the apex. The flowers are borne singly in leaf axils, corolla tubular about spreading, pendulous, on a pedicel about long with small bracts, boat-shaped and close to the base of the calyx. The sepals are triangular or rounded, about long, smooth or with occasional scales.
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Certicom Corp. is a cryptography company founded in 1985 by Gordon Agnew, Ron Mullin and Scott Vanstone. The Certicom intellectual property portfolio includes over 350 patents and patents pending worldwide that cover key aspects of elliptic- curve cryptography (ECC): software optimizations, efficient hardware implementations, methods to enhance the security, and various cryptographic protocols. The National Security Agency (NSA) has licensed 26 of Certicom's ECC patents as a way of clearing the way for the implementation of elliptic curves to protect U.S. and allied government information.
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Clerodendrum infortunatum is a flowering shrub or small tree, and is so named because of its rather ugly leaf. The stem is erect, high, with no branches and produce circular leaves with diameter. Leaves are simple, opposite; both surfaces sparsely villous- pubescent, elliptic, broadly elliptic, ovate or elongate ovate, wide, long, dentate, inflorescence in terminal, peduncled, few-flowered cyme; flowers white with purplish pink or dull-purple throat, pubescent. Fruit berry, globose, turned bluish-black or black when ripe, enclosed in the red accrescent fruiting-calyx.
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Eupatorium fortunei is herbaceous perennial that grows 40 to 100 centimeters tall, growing from procumbent rhizomes. Plants are upright growing with green stems that are often tinted with reddish or purple dots. The stems have few branches and the inflorescence is apically branched. Cauline foliage is large with short petioles, the 5 to 10 cm long and 1.5 to 2.5 cm wide leaves are 3-sected or 3-partite and the terminal lobe of the leaves is large and narrowly elliptic to elliptic-lanceolate or oblanceolate shaped.
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Born in Pula, a native of Zadar, Dujella got his M.Sc. and Ph.D. in mathematics from the University of Zagreb with a dissertation titled "Generalized Diophantine–Davenport problem". His main area of research is number theory, in particular Diophantine equations, elliptic curves, and applications of number theory in cryptography. Dujella has shown that there doesn't exist a Diophantine 6-tuple and that there exist at most a finite number of Diophantine 5-tuples. He applied Diophantine tuples to construct elliptic curves with high rank.
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Animation of Orbit by eccentricity barycenter with elliptic orbits. gravitational potential well of the central mass shows potential energy, and the kinetic energy of the orbital speed is shown in red. The height of the kinetic energy decreases as the orbiting body's speed decreases and distance increases according to Kepler's laws. In astrodynamics or celestial mechanics, an elliptic orbit or elliptical orbit is a Kepler orbit with an eccentricity of less than 1; this includes the special case of a circular orbit, with eccentricity equal to 0.
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The tall shrub reaching in height or tree to in height has an erect to spreading habit with grey-black or grey-brown coloured bark that can be smooth or rough. The glabrous branchlets are angled toward the apices. It has phyllodes instead of true leaves which have two prominent veins (giving the plant its species name binervata). The evergreen phyllodes have a narrowly elliptic to broadly elliptic or occasionally lanceolate shape and are straight or sometimes subfalcate with a length of and a width of .
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Leptospermum luehmannii is a species of shrub or small tree that is endemic to Queensland. It has glossy green elliptic leaves, white flowers and fruit that falls from the plant shortly after the seeds are released.
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Astragalus centralpinus can reach a height of . The hairy stem has a diameter of about 10 mm. Leaves are petiolated, long, with rachis covered with ascending hairs. Leaflets are ovate to elliptic, in 20-25 pairs.
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Suspension was provided by half-elliptic springs at both front and back, with hydraulic shock absorbers. Brakes were a Bendix- Perrot duo-servo series on all four wheels, operated through armoured cables by pedal or lever.
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The tree grows to be eight to ten meters tall. It is dimorphic. The sterile branches have longer spines, and the fertile branches have shorter spines or no spines. The alternate leaves are ovate and elliptic.
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Guido Stampacchia (26 March 1922 – 27 April 1978) was a 20th-century Italian mathematician, known for his work on the theory of variational inequalities, the calculus of variation and the theory of elliptic partial differential equations..
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The spherical flower-heads contain 20 to 25 bright yellow flowers. The seed pods that form after flowering are up to in length and in width and contain oblong-elliptic shaped seeds that are in length.
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The longitudinally arranged dark brown to black coloured seeds have an oblong to elliptic shape and are quite flattened. They have a length of and a width of with a small cream to orange coloured aril.
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In cryptography, the Standards for Efficient Cryptography Group (SECG) is an international consortium founded by Certicom in 1998. The group exists to develop commercial standards for efficient and interoperable cryptography based on elliptic curve cryptography (ECC).
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The leaves are elliptic to oblanceolate, about 25 mm long and 8 mm wide, on a petiole about 2 mm long. The veins are prominent and end near the margin. The margins are serrate or crenate.
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The species has elliptic leaf blades long by wide on petioles up to long. The inflorescence is a panicle of flowers. Each flower has a fuzzy, tubular, cream or yellowish corolla just under a centimeter long.
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The Bretherton equation is a model equation for studying weakly-nonlinear wave dispersion. It has been used to study the interaction of harmonics by nonlinear resonance. Bretherton obtained analytic solutions in terms of Jacobi elliptic functions.
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The anthers are also red or yellow. Flowers are borne from June to August. The seeds are dark brown, glabrous, long, and wide. They are elliptic, ovoid or reniform in shape, with longitudinal ribs bearing spines.
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Rhopalocarpus randrianaivoi grows as a tree up to tall. The coriaceous leaves are elliptic in shape and measure up to long. The species is not known to have any flowers. The fleshy fruits are coloured brown.
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The fertile lemma is chartaceous, elliptic, keelless, and is long. The species' palea have ciliolated keels and is 2-veined. Flowers are fleshy, oblong and truncate. They also grow together, have 2 lodicules and 3 anthers.
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Melhania virescens grows as a small shrub tall. The leaves are silver-grey stellate tomentose, shaped oblong elliptic and measure up to long. Inflorescences generally have solitary flowers, occasionally two-flowered. The flowers feature yellow petals.
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The perianth is long and the pistil curved and long. Flowering occurs from late October to early December and the there are up to fifteen elliptic follicles in each head, the follicles long, high and wide.
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It is a perennial herb. The narrowly oblanceolate-elliptic leaves are 7–20 cm long and 1.5–4 cm wide. The scape is 7–25 cm tall. The inflorescence is cylindrical and 2–10 cm long.
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The seeds are dull brown in colour, longitudinal and elliptic in shape and around long. It is closely related to Acacia cana which has silvery young phyllodes as well as Acacia coriacea which has longer phyllodes.
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Pogostemon purpurascens is an erect branched herb with a 20 cm tall, hairy stem. The leaves are elliptic, opposite with serrated margins. It bears tiny whitish flowers which blooms during the months of January and February.
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Note: Newton simply describes the arcs IH and KL as 'minimally small' and the areas traced out by the lines IL and HK can be any shape, not necessarily elliptic, but they will always be similar.
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The car's track grew from 42 to 43.5 inches (1105 mm), the rear springs were changed from quarter to semi elliptic on the Mk II Saloons, and the de-luxe models got a four-speed gearbox.
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In mathematics, Dolgachev surfaces are certain simply connected elliptic surfaces, introduced by . They can be used to give examples of an infinite family of homeomorphic simply connected compact 4-manifolds, no two of which are diffeomorphic.
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Many of the preceding results remain valid when the field of definition of E is a number field K, that is to say, a finite field extension of Q. In particular, the group E(K) of K-rational points of an elliptic curve E defined over K is finitely generated, which generalizes the Mordell–Weil theorem above. A theorem due to Loïc Merel shows that for a given integer d, there are (up to isomorphism) only finitely many groups that can occur as the torsion groups of E(K) for an elliptic curve defined over a number field K of degree d. More precisely, there is a number B(d) such that for any elliptic curve E defined over a number field K of degree d, any torsion point of E(K) is of order less than B(d). The theorem is effective: for d > 1, if a torsion point is of order p, with p prime, then :p < d^{3d^2} As for the integral points, Siegel's theorem generalizes to the following: Let E be an elliptic curve defined over a number field K, x and y the Weierstrass coordinates.
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In arithmetic geometry, the Weil–Châtelet group or WC-group of an algebraic group such as an abelian variety A defined over a field K is the abelian group of principal homogeneous spaces for A, defined over K. named it for who introduced it for elliptic curves, and , who introduced it for more general groups. It plays a basic role in the arithmetic of abelian varieties, in particular for elliptic curves, because of its connection with infinite descent. It can be defined directly from Galois cohomology, as H^1(G_K,A), where G_K is the absolute Galois group of K. It is of particular interest for local fields and global fields, such as algebraic number fields. For K a finite field, proved that the Weil–Châtelet group is trivial for elliptic curves, and proved that it is trivial for any connected algebraic group.
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Many sources define an elliptic curve to be simply a curve given by an equation of this form. (When the coefficient field has characteristic 2 or 3, the above equation is not quite general enough to include all non-singular cubic curves; see below.) An elliptic curve is an abelian variety – that is, it has a group law defined algebraically, with respect to which it is an abelian group – and O serves as the identity element. If y2 = P(x), where P is any polynomial of degree three in x with no repeated roots, the solution set is a nonsingular plane curve of genus one, an elliptic curve. If P has degree four and is square-free this equation again describes a plane curve of genus one; however, it has no natural choice of identity element.
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It is a rustic deciduous tree that defies hard, dry or poor soils. Therefore, its roots require well drained terrain. Its height ranges 6 to 12m. Leaves are opposite and petiolate, elliptic and lanceolate, with pinnate venation.
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The pods are in length and wide and have loosely matted hairs. The dark brown seeds inside are arranged longitudinallyand have a narrow oblong-elliptic shape with a length of approximately and a depressed grey-brown areole.
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Its lemma have a toothed apex which is also truncate and awned. The fertile lemma is long and is both membranous and oblong. The species also have an elliptic and hyaline palea which is long of lemma.
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Erythrina coralloides is a tree reaching a height of 5 m. Its seeds are elliptic, smooth, glossy, coral-red, with a salient longitudinal line on the back, and with a white hilum surrounded by a black border.
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Rhododendron crassifolium is a tropical rhododendron native to mountainside forests of Borneo at . It is a medium size evergreen shrub about tall. Leaves are dark green, broad, ribbed and elliptic. The bell-shaped flowers are red-orange.
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Its palea is elliptic, 2 veined, and have puberulous surface. Flowers are fleshy, oblong and truncate. They also grow together, and have 3 anthers that are long. The fruits are caryopsis with additional pericarp and linear hilum.
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Preserved stamens which were dislodged from the flower during entombment in the resin show two rows of bilocular anthers on their upper surfaces. The possibly elliptic-ovate petals distinguish the species from the living species Hymenaea courbaril.
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Tectonic processes may also be responsible for the elliptic shape of the Cerro Blanco caldera. There is evidence of intense earthquakes during the Quaternary and some faults such as the El Peñón Fault have been recently active.
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Thus they form a natural higher-dimensional generalization of modular curves viewed as moduli spaces of elliptic curves with level structure. In many cases, the moduli problems to which Shimura varieties are solutions have been likewise identified.
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The cap is flabby and has brown spores. The upper layer of the cap is elastic and can be stretched slightly at the margin. The spores are short elliptic and smooth. The spore print is snuff brown.
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Boronia grandisepala is a plant in the citrus family Rutaceae and is endemic to northern parts of the Northern Territory. It is an erect shrub with elliptic leaves and white, pink or burgundy-coloured, four-petalled flowers.
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Aerial shoots reach up to 50 cm tall. Leaves are broadly elliptic, wider than long, up to 12 cm long and 14 cm wide. Flowers are white, about 90 mm in diameter. Chromosome number: 2n (4x) = 20.
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The pseudobulbs are reduced. The obtuse, fleshy leaves are 9 cm long. They are broadly elliptic to ovate- lanceolate. The large, showy flowers grow basally on a short peduncle in a single-flowered to few-flowered raceme.
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Trifolium siskiyouenseis a glabrous, perennial herb with thickened roots but no rhizomes. Leaves are trifoliate with lanceolate stipules; leaflets are elliptic to oblanceolate, up to long. Flowers are white to cream- colored.Jepson Flora ProjectGillett, John Montague. 1980.
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The mature female head is egg-shaped, long, and nearly across. The mature bracts are covered with densely matted woolly hairs on the outside. The fruits are elliptic, about long, compressed, with a hairless and wrinkled surface.
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The chassis had beam axles front and rear and suspension by half elliptic leaf springs and hydraulic dampers.Graham Robson, A-Z British Cars 1945-80, Herridge & Sons, 2006, page 388 The brakes used a Lockheed hydraulic system.
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The Heliodorus pillar was erected to Vāsudeva by the Greek Heliodorus in 115 BCE. It was crowned by a Garuda capital. Excavation of the Vrishni Temple, with elliptic plan. The Heliodorus pillar appears in the immediate background.
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In mathematics the Montgomery curve is a form of elliptic curve, different from the usual Weierstrass form, introduced by Peter L. Montgomery in 1987. It is used for certain computations, and in particular in different cryptography applications.
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Leaves are usually 10 to 25 mm long, 3 to 7 mm wide. Paler below with longitudinal leaf veins. A sharp prickle is on the leaf end. The leaf shape may be elliptic, oblong or reverse lanceolate.
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J. Comput. Phys., 257-Part B:1163–1227, 2014. It allows the approximation of elliptic problems using a large class of polyhedral meshes. The proof that it enters the GDM framework is done in [Droniou et al].
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Giulio Carlo, Count Fagnano, and Marquis de Toschi (December 6, 1682 – September 26, 1766) was an Italian mathematician. He was probably the first to direct attention to the theory of elliptic integrals. Fagnano was born in Senigallia.
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For D= n\cdot P with n>0, the degree of K-D is strictly negative, so that the correction term is 0. The sequence of dimensions can also be derived from the theory of elliptic functions.
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They also have smooth surface and peduncle. The panicle is linear, open, inflorescenced and is long. Spikelets are elliptic and solitary with pedicelled fertile spikelets that carry 4-6 fertile florets. The main panicle branches are hairy.
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Legendre's relation stated using elliptic functions is : \omega_2 \eta_1 - \omega_1 \eta_2 = 2\pi i \, where ω1 and ω2 are the periods of the Weierstrass elliptic function, and η1 and η2 are the quasiperiods of the Weierstrass zeta function. Some authors normalize these in a different way differing by factors of 2, in which case the right hand side of the Legendre relation is i or i / 2\. This relation can be proved by integrating the Weierstrass zeta function about the boundary of a fundamental region and applying Cauchy's residue theorem.
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Verticordia habrantha is a shrub which grows to high and wide and which has a few main stems with many short, leafy side-branches. The leaves on the side branches are linear to narrow elliptic in shape, roughly triangular in cross-section, long, while those on the flowering stems are elliptic to egg-shaped and up to long. The flowers are arranged in rounded or corymb-like groups near the ends of the long flowering stems, each flower on an erect stalk, long. The floral cup is about long and covered with short, soft hairs.
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In mathematics, k-Hessian equations (or Hessian equations for short) are partial differential equations (PDEs) based on the Hessian matrix. More specifically, a Hessian equation is the k-trace, or the kth elementary symmetric polynomial of eigenvalues of the Hessian matrix. When k ≥ 2, the k-Hessian equation is a fully nonlinear partial differential equation.. Much like differential equations often study the actions of differential operators (e.g. elliptic operators and elliptic equations), Hessian equations can be understood as simply eigenvalue equations acted upon by the Hessian differential operator.
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Buddleja blattaria is a dioecious shrub, < 1 m tall, with brown fissured bark. The young branches are quadrangular and covered with thick tomentum. The leaves are sessile elliptic or oblong-elliptic, 4-10 cm long by 1.5-3 cm wide, lanose on both surfaces. The white or cream inflorescence is 3-8 cm long, comprising sessile flowers borne on one terminal and 1-3 pairs of globose heads below, in the axils of small leaves, each head 1-2 cm diameter with 20-40 flowers, the corolla 5 mm long.
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These constructions work over the moduli stack of smooth elliptic curves, and they also work for the Deligne- Mumford compactification of this moduli stack, in which elliptic curves with nodal singularities are included. TMF is the spectrum that results from the global sections over the moduli stack of smooth curves, and tmf is the spectrum arising as the global sections of the Deligne–Mumford compactification. TMF is a periodic version of the connective tmf. While the ring spectra used to construct TMF are periodic with period 2, TMF itself has period 576.
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The most efficient identity-based encryption schemes are currently based on bilinear pairings on elliptic curves, such as the Weil or Tate pairings. The first of these schemes was developed by Dan Boneh and Matthew K. Franklin (2001), and performs probabilistic encryption of arbitrary ciphertexts using an Elgamal-like approach. Though the Boneh-Franklin scheme is provably secure, the security proof rests on relatively new assumptions about the hardness of problems in certain elliptic curve groups. Another approach to identity-based encryption was proposed by Clifford Cocks in 2001.
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That reflects a good understanding of their Tate modules as Galois modules. It also makes them harder to deal with in terms of the conjectural algebraic geometry (Hodge conjecture and Tate conjecture). In those problems the special situation is more demanding than the general. In the case of elliptic curves, the Kronecker Jugendtraum was the programme Leopold Kronecker proposed, to use elliptic curves of CM-type to do class field theory explicitly for imaginary quadratic fields – in the way that roots of unity allow one to do this for the field of rational numbers.
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Sir Andrew John Wiles Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to proof by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge. Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".
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Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero a, b, c and n greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. If the link identified by Frey could be proven, then in turn, it would mean that a proof or disproof of either Fermat's Last Theorem or the Taniyama–Shimura–Weil conjecture would simultaneously prove or disprove the other.Singh, pp. 194–198; Aczel, pp. 109–114.
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In spherical geometry, a direct motion of the -sphere (an example of the elliptic geometry) is the same as a rotation of -dimensional Euclidean space about the origin (). For odd , most of these motions do not have fixed points on the -sphere and, strictly speaking, are not rotations of the sphere; such motions are sometimes referred to as Clifford translations. Rotations about a fixed point in elliptic and hyperbolic geometries are not different from Euclidean ones. Affine geometry and projective geometry have not a distinct notion of rotation.
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It was a two-door convertible that weighed under . Initially offered at US$325 for a two-passenger coupe or $350 for a four-passenger sedan,The two models were essentially the same car, except the sedan had rear side windows. the Crosley cars were cheaper than the nearest competition, the American Austin Car Company's American Bantam, which sold for $449 to $565. The Crosley car's chassis had an wheelbase and used beam axles with leaf-springs (half-elliptic springs in front, and quarter- elliptic springs in the rear).
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The pubescence is located on the main vein on the bottom of the leave. The leaf shape can vary from oval to elliptic and present coriaceous leaves; leaf base and apex are rounded. Solitary flowers located at the end of the branches, colored from yellowish green to beige, with 3 to 5 deciduous floral bracts; 3 obovate thick fresh sepals; 6 to 7 obovate and fleshy petals with truncate base and acute apex. Woody fruit, elliptic, measuring from 6,9 to 8,5 cm long and 3,3 to 4,5 cm broad; the carpels split open irregularly.
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This allowed for a sloping front and good legroom when combined with lowered seating. This also allowed Fiat to lower the roofline. Although nominally a two-seater more were often squeezed in behind the seats. Initially it had quarter elliptic leaf spring rear suspension, but with an axle locating trailing arm, that was upgraded to stronger semi-elliptic to cope with overloading by customers. The front suspension was independent and was used as the basis of the suspension of the first English Cooper racing cars in the 1940s that became successful in the 1950s.
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In mathematics, the Weil pairing is a pairing (bilinear form, though with multiplicative notation) on the points of order dividing n of an elliptic curve E, taking values in nth roots of unity. More generally there is a similar Weil pairing between points of order n of an abelian variety and its dual. It was introduced by André Weil (1940) for Jacobians of curves, who gave an abstract algebraic definition; the corresponding results for elliptic functions were known, and can be expressed simply by use of the Weierstrass sigma function.
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In mathematics, the Néron–Ogg–Shafarevich criterion states that if A is an elliptic curve or abelian variety over a local field K and ℓ is a prime not dividing the characteristic of the residue field of K then A has good reduction if and only if the ℓ-adic Tate module Tℓ of A is unramified. introduced the criterion for elliptic curves. used the results of to extend it to abelian varieties, and named the criterion after Ogg, Néron and Igor Shafarevich (commenting that Ogg's result seems to have been known to Shafarevich).
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The RLWE Key Exchange is designed to be a "quantum safe" replacement for the widely used Diffie–Hellman and elliptic curve Diffie–Hellman key exchanges that are used to secure the establishment of secret keys over untrusted communications channels. Like Diffie–Hellman and Elliptic Curve Diffie–Hellman, the Ring-LWE key exchange provides a cryptographic property called "forward secrecy"; the aim of which is to reduce the effectiveness of mass surveillance programs and ensure that there are no long term secret keys that can be compromised that would enable bulk decryption.
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Let E and D be elliptic curves over a field k. An isogeny between E and D is a finite morphism f : E → D of varieties that preserves basepoints (in other words, maps the given point on E to that on D). The two curves are called isogenous if there is an isogeny between them. This is an equivalence relation, symmetry being due to the existence of the dual isogeny. Every isogeny is an algebraic homomorphism and thus induces homomorphisms of the groups of the elliptic curves for k-valued points.
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Eremophila demissa is a low, compact, spreading shrub which grows to a height of less than with leaves and branches covered with fine hairs giving the surface a felty texture. The leaves are densely clustered near the ends of the branches and are elliptic to egg-shaped, long and wide. The flowers are borne singly, rarely in pairs, in leaf axils on a densely hairy, straight stalk long. There are 5 slightly overlapping, lance-shaped to elliptic sepals which are hairy on the outer surface and mostly long.
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The case of elliptic curves was worked out by Hasse in 1934. Since the genus is 1, the only possibilities for the matrix H are: H is zero, Hasse invariant 0, p-rank 0, the supersingular case; or H non-zero, Hasse invariant 1, p-rank 1, the ordinary case. Here there is a congruence formula saying that H is congruent modulo p to the number N of points on C over F, at least when q = p. Because of Hasse's theorem on elliptic curves, knowing N modulo p determines N for p ≥ 5.
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Diplolaena dampieri is a spreading, rounded shrub that typically grows to a height of . It has strongly aromatic, elliptic to oblong- elliptic shaped, leathery leaves to long, the upper surface dark olive green and hairless when mature, the lower surface thickly covered in cream to grey weak hairs. The pendulous flowers are borne at the end of branches, about in diameter, outer bracts narrowly triangular to oval shaped, long with thick, grey to reddish star-shaped hairs. The inner bracts narrowly oblong, about long and densely covered with short, matted, star shaped hairs.
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The evergreen phyllodes have an elliptic to broadly elliptic shape and are straight to slightly curved with a length of and a width of and have three to four prominent veins. It usually flowers in the spring and produces inflorescences that appear singly on the raceme axis. The spherical flower- heads have a diameter of and contain 15 to 30 pale yellow to cream-coloured flowers. The firmly papery to thinly leathery seed pods that form after flowering are straight or curved and flat but can be constricted between the seeds.
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The primary difference between Goodyera and Spiranthes (A similar genus in the family Orchidaceae) is that Goodyera have elliptic leaves with white or pale green markings. Goodyera pubescens flowers in mid July-early September with a small spike inflorescence of between 10 and 57 cylindric flowers. The leaves have the white-green marbling in the form of veins throughout, broadly elliptic to broadly ovate (2.1-6.2 x 1.3–3 cm), with either an acute or obtuse apex. The peduncle (stem that connects the stalk to a floret) is 11–35 cm long.
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Eucalyptus conglomerata is a straggly tree or a mallee, that typically grows to a height of and forms a lignotuber. It has greyish brown, fibrous stringybark over the trunk and most of the branches, sometimes smooth bark on the thinnest branches. Young plants and coppice regrowth have leaves that are glossy green on the upper surface, paler below, narrow elliptic to narrow lance-shaped, long, wide on a short petiole. Adult leaves are lance- shaped to elliptic, the same glossy green on both sides, long and wide on a petiole long.
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R. Firoozjaee, M.H. Afshar, Discrete least squares meshless method with sampling points for the solution of elliptic partial differential equations. Engineering Analysis with Boundary Elements 33 (2009) 83–92. proposed the collocated discrete least squares meshless (CDLSM) method to solve elliptic partial differential equations, and studied the effect of the collocation points on the convergence and accuracy of the method. The method can be considered as an extension the earlier method of DLSM by the introduction of a set of collocation points for the calculation of the least squares functional.
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RFC 8301 was issued in January 2018. It bans SHA-1 and updates key sizes (from 512-2048 to 1024-4096). RFC 8463 was issued in September 2018. It adds an elliptic curve algorithm to the existing RSA.
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Leionema ceratogynum is a dense shrub, it grows on the south coast of New South Wales, Australia. It has oval-elliptic shaped leaves, scented foliage and lemon flowers usually in groups of three arising from the leaf axils.
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Any quasi-hyperelliptic surface is a quotient (E×F)/G, where E is a rational curve with one cusp, F is an elliptic curve, and G is a finite subgroup scheme of F (acting on F by translations).
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Charles Auguste Briot (19 July 1817 St Hippolyte, Doubs, Franche-Comté, France – 20 September 1882 Bourg-d'Ault, France) was a French mathematician who worked on elliptic functions. The Académie des Sciences awarded him the Poncelet Prize in 1882.
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The glossy, glabrous leaves are 12 x 5 cm in length, simple, alternate, elliptic, entire, apiculate, acute and lanceolate with prominent stipules, a scar encircling each leaf's petiole. The bark is smooth, reddish brown with a gray cast.
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"Chip firms form prpl foundation to keep MIPS architecture in the pink". Gigacom, Stacey Higginbotham May 22, 2014 The security PEG includes several of the above, as well as CUPP Computing, Elliptic Technologies, Imperas Software, Kernkonzept, and Seltech.
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Leptospermum spectabile is a species of shrub that is endemic to a small area of New South Wales. It has thin bark, narrow elliptic leaves, dark red flowers arranged singly on short side shoots and relatively large fruit.
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The rest of the mechanicals were akin to the Kappa's: ladder frame, solid axles on semi-elliptic leaf springs front and rear, transmission brake and rear-wheel drum brakes, 4-speed gearbox and a multi-plate dry clutch.
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Agouticarpa is characterized by being dioecious, having elliptic to obovate, membranaceous stipules, male flowers in a branched dichasial or thyrse-like inflorescence, a poorly developed cup-shaped calyx, pollen grains with 3-7 apertures, and large globose fruits.
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Alphonsea maingayi is a middling to tall tree, whose branches are black. It has elliptic/oblong/lanceolate leaves which are shiny on the upper surface and whose lower surface has a dense covering of rusty, short, soft hairs.
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Atsuko Miyaji (, born 1965) is a Japanese cryptographer and number theorist known for her research on elliptic-curve cryptography and software obfuscation. She is a professor in the Division of Electrical, Electronic and Information Engineering, at Osaka University.
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The four petals are white or pale pink, broadly elliptic and about long and the eight stamens are about long and hairy. Flowering occurs from May to December and the fruit is about long with a short beak.
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The petals are broadly elliptic, white to pink and about long and the stamens are free from each other and hairy. Flowering occurs from July to January and the fruit is about long with a beak about long.
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The plant is an epiphytic fern. It has a stout, erect rhizome with light brown, lanceolate scales. Its simple fronds combine a short stipe with a narrowly elliptic lamina 3–15 cm long and 0.4–0.8 cm wide.
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A small tree, up to 5 metres tall, often encountered as a shrub half that size. This plant features grey furry leaves. 2 to 5 cm long, 1 to 2 cm wide. Reverse ovate or elliptic in shape.
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See for further details. He also applied the methods of functional analysis, at the same time as Mark Vishik but independently of him, to the investigation of boundary value problems for degenerate second order elliptic partial differential equations.
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Oval, elliptic or narrow-ovate in shape. Pale violet flowers occur throughout the year, but are most often seen in spring or autumn. The red berry is around 7 mm in diameter, mostly covered by the calyx lobes.
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Artin describes GL(n, k) group structure. More details are given about vector spaces over finite fields. Chapter five is "The Structure of Sympletic and Orthogonal Groups". It includes sections on elliptic spaces, Clifford algebra, and spinorial norm.
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Shrubs are up to 3m. The elliptic-lanceolate opposite leaves are up to 15cm long. Terminal inflorescences have up to 15 conspicuous 5-lobed bell shaped flowers, which are up to 4cm long and purplish-red.Fischer, E. (1996).
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The pods are straight to slightly curved with a length of up to and a width of . The slightly glossy light to dark brown seeds within the pods have a broadly elliptic or oblong shape and are long.
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Astilbe chinensis is a herbaceous perennial growing in clumps. The species reaches a height of . The leaves are predominantly basal and ternately compound with sharply-toothed (often biserrated) leaflets. Most leaflets are elliptic to oval in shape and hairy.
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White, vanilla scented flowers form from August to November. The fruit capsule is 4 to 5 cm long, in an oblong/elliptic shape. Inside are many flat, winged seeds. This plant was cultivated as early as 1800 in England.
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Bjorn Poonen B. Poonen. Using elliptic curves of rank one towards the undecidability of Hilbert's tenth problem over rings of algebraic integers. In Algorithmic number theory (Sydney, 2002), volume 2369 of Lecture Notes in Comput. Sci., pages 33-42\.
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Both the leaf-sheaths and leaf-blades have a glabrous surface. The membrane is eciliated and is long. The panicle is open, linear and is long. Spikelets are elliptic, solitary, are long and have fertile spikelets that are pediceled.
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The erect, flat and straight-sided seed pods have a length of and a width of The brown seeds are arranged obliquely in the pods. The oblong-elliptic shaped seeds have a length of with a narrowly conical aril.
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The pods have a length of and a width of and are tapered at base and apex. The dark brown seeds inside are arranged obliquely and have an elliptic shape but are dorsoventrally flattened with a length of around .
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Xerochlamys itremoensis grows as a prostrate shrub. Its leaves are glossy green above, green to yellow on the underside. They are elliptic in shape and measure up to long. The tree's flowers are solitary with pink to white petals.
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Nuxia glomerulata has a restricted range between Pretoria and Zeerust, South Africa, and differs by its more elliptic, leathery and glabrous leaves. Nuxia floribunda carries the leaves on long and slender petioles, and has larger and less dense inflorescences.
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In June 2012 the National Institute of Information and Communications Technology (NICT), Kyushu University, and Fujitsu Laboratories Limited improved the previous bound for successfully computing a discrete logarithm on a supersingular elliptic curve from 676 bits to 923 bits.
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Universidad de Antioquia, Medellín. Trixis inula is a much-branched herb up to 300 cm (10 feet) tall. It has lanceolate to elliptic leaves up to 17 cm (7 inches) long. Yellow flower heads are borne in paniculate arrays.
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Power was transmitted via a four-speed manual transmission to the rear wheels which were fixed to a rigid axle suspended from semi-elliptic leaf springs. The braking applied to all four wheels, mechanically controlled using rod linkages.Oswald, p.
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Power was transmitted via a three-speed manual transmission to the rear wheels, which were fixed to a rigid axle suspended from semi-elliptic leaf springs. The braking applied to all four wheels, mechanically controlled using rod linkages.Oswald, p.
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Salix denticulata can reach a height of . The shoots are downy when young. The dull green leaves are paler underneath, obovate, lanceolate or elliptic, with toothed margins, long, with very short petioles. Like all willows this species is dioecious.
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A new proof of De Giorgi's theorem concerning the regularity problem for elliptic differential equations. Comm. Pure Appl. Math. 13 (1960), 457–468.Pogorelov, A.V. On the improper convex affine hyperspheres. Geometriae Dedicata 1 (1972), no. 1, 33–46.
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All of the stamens are fertile. The filament is 0.28mm long, broad and thin. The anthers are linear-elliptic and 4.2mm long. The apical glands are 0.28mm in length, ovate in shape and with a somewhat rounded tip (subobtuse).
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The engine had a bore of 68mm and a stroke of 100mm. A Solex carburettor was used. This drove the rear wheels via a cone clutch and four speed gearbox. The chassis had rigid axles and half elliptic springs.
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Barystethus tropicus reaches about in length. This species is usually black but it is quite variable in coloration. The body is elliptic, the legs and the rostrum are smooth and glossy, elytra are striated and rostrum is slightly arched.
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R (2002) Elliptic calderas in the Ethiopian Rift: control of pre-existing structures. Journal of Volcanology and Geothermal Research. 119:189–203 Three Silicic phases and then a basalt phase. The most recent basalt phase is dated around 1820.
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Given a curve, E, defined along some equation in a finite field (such as E: ), point multiplication is defined as the repeated addition of a point along that curve. Denote as for some scalar (integer) n and a point that lies on the curve, E. This type of curve is known as a Weierstrass curve. The security of modern ECC depends on the intractability of determining n from given known values of Q and P if n is large (known as the elliptic curve discrete logarithm problem by analogy to other cryptographic systems). This is because the addition of two points on an elliptic curve (or the addition of one point to itself) yields a third point on the elliptic curve whose location has no immediately obvious relationship to the locations of the first two, and repeating this many times over yields a point nP that may be essentially anywhere.
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Pottery furnaces are elliptic and have two halves. On both furnaces, ceramic plates were produced. The brick cooking area is the only building material found in the archaeological excavations in Transcaucasia. The most unique remnants in Torpaggala are rare glass furnaces.
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Polygonum cognatum is a perennial, prostrate or ascending branched herb, 15–30 cm long with a thick stout root stock. Stems are prostrate, green like the leaves. Leaves oblong-elliptic, petiolate, often slightly mucronate. Flowers in bundles in the leaf axils.
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Phebalium whitei is a small shrub that is endemic to south-east Queensland. It has branchlets covered with silvery and rust-coloured scales, leathery, oblong to elliptic leaves and bright yellow flowers arranged in sessile umbels on the ends of branchlets.
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Such equations first arose in the theory of multiplication of elliptic functions (geometrically, the n2-fold covering map from a 2-torus to itself given by the mapping x → n·x on the underlying group) expressed in terms of complex analysis.
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Flowers during July - November.Greeting Cards, Tropical Botanic Garden & Research Institute, Pacha, Palode, Trivandrum - 695 562. 1997 Leaves dense, fleshy, 1-1.5 x 0.6-0.8 cm, elliptic-ovate, obtuse to round at apex, attenuate at base; margins thin; midrib prominent.Exacum travancoricum Bedd.
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The rear suspension comprised semi-elliptic leaf springs, wide, mounted on the chassis and shackled to the rear axle. The 10–12 cwt version had seven leaves, and the 15–17 cwt version had eight leaves. Each leaf was thick.
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Androsace lactea can reach a height of . This plant produces rosettes of leaves with a diameter of about . The leaves are shining dark green, linear or lightly elliptic. Flowers are white with a yellow centre, in diameter, with broadly notched petals.
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Aspasia psittacina is the only species found in Ecuador. It is vegetatively close to A. epidendroides, with large elliptic and highly flat pseudobulbs, it shows the same colors of the later, but has narrower flowers with the labellum proportionally much smaller.
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Heegner points can be used to compute very large rational points on rank 1 elliptic curves (see for a survey) that could not be found by naive methods. Implementation of the algorithm is available in Magma, PARI/GP, and Sage.
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Inula spiraeifolia reaches a height of . The middle leaves are oblong-elliptic and they measure of length and of width. The base of the leaves is not embracing the stem. The upper cauline leaves are erect, with a round base.
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Suspension was independent at the front using MacPherson struts, and at the rear the live axle used semi elliptic leaf springs. A contemporary road tester was impressed, noting that "probably the most impressive thing about the Classic is its road holding".
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The suspension was independent at the front using coil springs with semi elliptic leaf springs at the rear. The brakes used drums all round and were operated hydraulically at the front and mechanically at the rear via a gearbox driven servo.
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Henderson augen gneiss Augen gneiss, from the , meaning "eyes", is a coarse-grained gneiss resulting from metamorphism of granite, which contains characteristic elliptic or lenticular shear-bound feldspar porphyroclasts, normally microcline, within the layering of the quartz, biotite and magnetite bands.
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The yellow to greenish- yellow petals are upright, lance to elliptic shaped, long, smooth and stamens twice the length of the petals. The fruit are a capsule, each segment high ending with a beak long. Flowering occurs from April to September.
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The upper glandular stalk is stalk-round, sometimes woody to the middle. The opposite leaves are simple, elliptic or ovate to broad- lanceolate, sometimes linear and usually bleak. Leaflets are missing.Erich Oberdorfer: Plant sociology excursion flora for Germany and adjacent areas .
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Springer, Berlin, 2002. has applied EDS to logic. He uses the existence of primitive divisors in EDS on elliptic curves of rank one to prove the undecidability of Hilbert's tenth problem over certain rings of integers. Katherine Stange K. Stange.
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Leaves are alternate, elliptic, 6 to 12 cm long, with a blunt tip. Both leaf sides green and glossy. Leaf margins wavy, leaf stalks 5 to 10 mm long. Leaf veins visible on both surfaces, more evident above the leaf.
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The ovary is subglobose with 2 styles. The stigmas are divergent or curved. The 3 mm wide, rounded seed capsule, is capped by the withered corolla. Each capsule often has 4, pale brown, elliptic, seeds that are 1 mm long.
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Like most species of Acacia it has phyllodes rather than true leaves. The evergreen, moderately coriaceous to sub-rigid phyllodes have a linear to narrowly elliptic shape with a length of and a width of with many fine parallel longitudinal nerves.
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According to the International Mathematical Union citation, he was awarded the prize "for developing powerful new methods in the geometry of numbers, which he applied to count rings of small rank and to bound the average rank of elliptic curves".
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Following flowering firmly chartaceous, glabrous seed pods form with a white dusty covering. The pods have a length of up to and a width of . The shiny blackish seeds found within the pod have a circular to widely elliptic shape.
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Palea have scaberulous keels and surface. Rhachilla is in length and is extended. Lower glumes are elliptic and are long while the upper glumes are lanceolate and are long. Both the lower and upper glumes are obtuse and have asperulous surfaces.
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The undulate brown seed pods that form after flowering are prominently rounded over seeds. The pods have a length of up to and a width of . The mottles seeds within have an irregularly oblong to elliptic and are around in length.
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Colubrina pedunculata is a thorny, sometimes straggling, shrub or small tree. Its thorns are 5–20 mm long. Its leaves are alternate, narrowly elliptic, and deciduous after fruiting. It bears many yellow-green flowers, 5–6 mm across and clustered.
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The internodes are long, exceeding all leaves except some of the uppermost. The sessile, spreading, persistent leaves are long and wide. The leaves are papery and membranous. Lower leaves are ovate or elliptic and upper leaves are ovate or suborbicular.
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Private Internet Access supports the AES (128-bit or 256-bit) specifications, SHA1 and SHA256 authentication, and RSA (2048, 3072, 4096) or elliptic curve cryptography (ECC) (256R1, 256K1, 521) handshakes; however, the default settings are AES-128, SHA1, RSA-2048.
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It peduncle is long with the bracts length being . The nutlet itself is elliptic and is long and wide. It also have membranous wings and it blooms from June to August while the flowers come out from May to June.
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The firmly chartaceous seed pods that form after flowering have a narrowly oblong shape with a length up to containing longitudinally arranged seeds. The black seeds have an oblong-elliptic shape with a length of and a cream coloured clavate aril.
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The shiny brown seeds are longitudinally arranged in the pod. They have an oblong to elliptic shape and are long. Like many other Acacia species, A. truncata has phyllodes rather than true leaves. The triangular phyllodes range from long and wide.
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Studies of non-Euclidean (hyperbolic and elliptic) and other spaces such as complex spaces, discovered over the preceding century, led to the discovery of more new polyhedra such as complex polyhedra which could only take regular geometric form in those spaces.
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Analogously to the quartic twist case, an elliptic curve over K with j-invariant equal to zero can be twisted by cubic characters. The curves obtained are isomorphic to the starting curve over the field extension given by the twist degree.
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The lid or operculum is ovate- elliptic. An unbranched spur, up to 5 mm long, is inserted at the base of the lid. Nepenthes hispida has a racemose inflorescence. The peduncle is up to 5 cm long and 1.5 cm thick.
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Shoots and stems hairy. The elliptic or reverse lanceolate shaped leaves are alternate and not toothed, 8 to 10 cm long and 2 to 4 cm wide. Bluntly pointed or sometimes notched at the tip. Leaf stalks 5 mm long.
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ASTRO-G (also known as VSOP-2, and very rarely called VSOP-B) was a planned radio telescope satellite by JAXA. It was expected to be launched into elliptic orbit around Earth (apogee height 25,000 km, perigee height 1,000 km).
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The trees have characteristic straight, smooth barked stems. Leaves are narrow-ovate to elliptic in shape, and have slightly serated margins. Leaves are also stalked and alternately arranged. On the adaxial surface, leaves are dark green with deeply impressed veins.
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The W07 had a contemporary boxed chassis suspended by semi-elliptic leaf springs onto beam axles front and rear. Dimensions would vary with coachwork, but the chassis had a wheelbase of and a front track equal to the rear track of .
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The palpebral fissure is the elliptic space between the medial and lateral canthi of the two open eyelids. In simple terms, it is the opening between the eyelids. In adult humans, this measures about 10 mm vertically and 30 mm horizontally.
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The flat evergreen phyllodes have a falcate narrowly elliptic shape that tapers gradually towards apex and base. They are in length with a width of with three 3 main conspicuous nerves. The tree flowers between May and June, producing yellow inflorescences.
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It is a smooth-stemmed tree, growing to 15 m in height. The oblanceolate-elliptic leaves are 5–7 cm long and 1.7–2.5 cm wide. The flowers are tiny. The round purple fruits are 4–5 mm in diameter.
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Leaves are elliptic to cordate (heart-shaped), up to 13 cm (5.2 inches) long. The plant is monoecious, with staminate (male, pollen-producing) and pistillate (female, seed-producing) flowers separate on the same plant. Sepals are whitish, petals pink or red.
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His smallest counterexample was ::. A particular case of Elkies' solutions can be reduced to the identity :: where ::. This is an elliptic curve with a rational point at . From this initial rational point, one can compute an infinite collection of others.
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The glabrous pods have a length of up to and a width of and have fine longitudinal divisions. The shiny dark brown seeds have an oblong to elliptic shape with a length of up to and have a white coloured aril.
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Hodge diamond: : : Examples: Take a non-trivial line bundle over an elliptic curve, remove the zero section, then quotient out the fibers by Z acting as multiplication by powers of some complex number z. This gives a primary Kodaira surface.
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The leaves form between the spines and are alternate, oblong to broadly elliptic (30-80 × 15–25 mm), greyish- olive green, covered in velvet hairs (or smooth); margins entire, rolled under. The petiole is 4–12 mm long and velvety.
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Mature trees can grow to over in height and are evergreen. It is frost sensitive but can tolerate light frosts. The leaves are elliptic-oblong, tapered at both ends, dark green, glossy and long. Some trees bear only male flowers.
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Gerhard Frey (; born 1944) is a German mathematician, known for his work in number theory. His Frey curve, a construction of an elliptic curve from a purported solution to the Fermat equation, was central to Wiles's proof of Fermat's Last Theorem.
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Diplolaena mollis, is a species of flowering plant in the family Rutaceae and is endemic to the west coast of Western Australia. It has broadly elliptic or egg-shaped, leathery leaves that are densely covered in hairs and reddish, pendulous flowers.
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Leptospermum crassifolium is a species of shrub that is endemic to the Budawang Range in New South Wales. It has thin, rough bark that is shed annually, broadly elliptic leaves, white flowers borne singly on short side branches, and woody fruit.
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Schizolaena isaloensis grows as a tree up to tall. The bark is thick and spongy. Its subcoriaceous leaves are elliptic to ovate or obovate in shape and coloured dark green above and pale green below. They measure up to long.
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Melhania latibracteolata grows as a suffrutex (subshrub) up to tall. The elliptic to ovate leaves are tomentose and measure up to long. Inflorescences are two to five-flowered, on a stalk measuring up to long. The flowers have pale yellow petals.
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Persoonia marginata, commonly known as the Clandulla geebung, is a plant in the family Proteaceae and is endemic to New South Wales. It is low, spreading shrub with elliptic to egg-shaped leaves and small groups of cylindrical yellow flowers.
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The dry, one-seeded, indehiscent fruits called cypsellae are elliptic, about long and wide, yellowish brown in colour, with a pale, densely hairy marginal ridge, the surface in the upper half with few silky hairs or hairless, without other adornment.
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The leaves measures 1–4½ cm. Leaflets are 7–15, narrowly elliptic, lanceolate to oblong to oblanceolate; tips acute, subacute, or exceptionally emarginate; sparingly appressed-pubescent, 2–11 mm. The terminal leaflet is generally much broader than the subfiliform rachis.
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Salix retusa can reach a height of . This plant usually develops creeping stems, rarely erect. The dull green leaves are obovate, lanceolate or elliptic, with entire margins, 2 × 1 cm, with very short petioles. Like all willows this species is dioecious.
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A. japonica is a perennial plant growing to tall with thickened roots. Stems are glabrous or slightly pubescent and shape quadrangular and branched. Its nodes are dilated. The leaves opposite and shape elliptic or oval and slightly pubescent and have petiolate.
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In number theory, Szpiro's conjecture relates the conductor and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known abc conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.
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Persoonia cornifolia is a plant in the family Proteaceae and is endemic to eastern Australia. It is a shrub with elliptic to egg-shaped leaves and hairy yellow flowers, and grows in northern New South Wales and south-eastern Queensland.
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Prunus microcarpa is a deciduous bushy shrub with rigid branchlets. Its glabrous leaves are ovate to elliptic. Prunus microcarpa produces white to pale pink hermaphrodite flowers in April. The flowers are solitary or in pairs and are 1 cm across.
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The brittle seed pods that follow break easily into single seed units at the constrictions. The pods are to about in length and wide and contain longitudinally arranged mottled tan seeds with a narrowly oblong-elliptic shape and a length of .
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The plant is perennial and caespitose with long culms that grow in a clump. The ligule is long and is going around the eciliate membrane. Leaf-blades are broad with scabrous margins. The panicle is elliptic, open, inflorescenced and is long.
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The leaf axils are sometimes densely woolly. The leaf blade varies in outline between narrowly or broadly inverted egg-shaped and narrowly elliptic to elliptic, mostly 4–10 cm (1–4 in) long (full range 3–12 cm) and 1–3 cm (0.6–1.4 in) wide (full range –5 cm). The leaves have a blunt to pointy tip and a margin that is rolled under, with rounded or pointy teeth or is sometimes almost entire with some peg-like extensions. The upper surface shows a distinct main vein, is hairless or has some dispersed woolly hairs.
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Designed by T. C. Pullinger from Darracq, Sunbeam and Humber with Fred Neale from Hillman, and heavily influenced by the Fiat 501, the 10/20 used a straight four, side valve engine of 1460 cc driving the rear wheels through either a three or four speed gearbox in unit with the engine. Suspension was by semi elliptic leaf springs at the front and quarter elliptic springs at the rear. However, the Galloway car had several adaptations to appeal to women drivers. Some, like the introduction of a rear-view mirror and more reliable engine, would be appreciated by all drivers.
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Klein is responsible for the terms "hyperbolic" and "elliptic" (in his system he called Euclidean geometry "parabolic", a term which has not survived the test of time and is used today only in a few disciplines.) His influence has led to the common usage of the term "non-Euclidean geometry" to mean either "hyperbolic" or "elliptic" geometry. There are some mathematicians who would extend the list of geometries that should be called "non-Euclidean" in various ways. In other disciplines, most notably mathematical physics, where Klein's influence was not as strong, the term "non-Euclidean" is often taken to mean not Euclidean.
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The Jacobian of the modular curve can (up to isogeny) be written as a product of irreducible Abelian varieties, corresponding to Hecke eigenforms of weight 2. The 1-dimensional factors are elliptic curves (there can also be higher- dimensional factors, so not all Hecke eigenforms correspond to rational elliptic curves). The curve obtained by finding the corresponding cusp form, and then constructing a curve from it, is isogenous to the original curve (but not, in general, isomorphic to it). stated a preliminary (slightly incorrect) version of the conjecture at the 1955 international symposium on algebraic number theory in Tokyo and Nikkō.
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186–187 (text condensed). On 24 October 1994, Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem" and "Ring theoretic properties of certain Hecke algebras", the second of which was co-authored with Taylor and proved that certain conditions were met that were needed to justify the corrected step in the main paper. The two papers were vetted and published as the entirety of the May 1995 issue of the Annals of Mathematics. These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured.
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A logarithmic transformation (of order m with center p) of an elliptic surface or fibration turns a fiber of multiplicity 1 over a point p of the base space into a fiber of multiplicity m. It can be reversed, so fibers of high multiplicity can all be turned into fibers of multiplicity 1, and this can be used to eliminate all multiple fibers. Logarithmic transformations can be quite violent: they can change the Kodaira dimension, and can turn algebraic surfaces into non-algebraic surfaces. Example: Let L be the lattice Z+iZ of C, and let E be the elliptic curve C/L.
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Calyx dark-vivid red, narrow infundibular, tube 16–22 mm long, 3–5 mm basally expanding to 6–8 mm wide at throat, lobes deltoid-ovate, subulate-acuminate, 8–12 mm long; persistent in fruit. Standard petal brilliant red, paler toward spotted center, blade oblong-lanceolate, 25–33 mm long x 14–17 mm wide, claw 21–24 mm long. Wing petals shorter than keel, red, flaring apically, blade elliptic-oblong 25–33 mm long x 14–17 mm wide, claw 21–24 mm long. Keel petals red, blade elliptic- oblong, weakly falcate, 17–23 mm long x 2.5–5 mm wide.
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Obadiah Elliott (1763 – 13 January 1838) was a British inventor from Tonbridge, Kent who patented in 1804 the method of mounting coach bodies on elliptical springs attached directly to the axles, replacing the traditional heavy perch. Elliptic springs The elliptic spring consisted of steel plates piled on top of one another and pinned together; it is the same method still used in rear suspensions. His invention was a major breakthrough in carriage design and it inspired a boom in the construction and sale of lightweight private carriages. Ultimately, there was greater investment in roads and the beginnings of a national network.
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In cryptography, the Boneh-Lynn-Shacham (BLS) signature scheme allows a user to verify that a signer is authentic. The scheme uses a bilinear pairing for verification, and signatures are elements of an elliptic curve group. Working in an elliptic curve group provides some defense against index calculus attacks (with the caveat that such attacks are still possible in the target group G_T of the pairing), allowing shorter signatures than FDH signatures for a similar level of security. Signatures produced by the BLS signature scheme are often referred to as short signatures, BLS short signatures, or simply BLS signatures.
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The green phyllodes have a narrowly lanceolate to broadly ovate to broadly elliptic shape with a length of and a width of . It blooms from March to July or in September and produces yellow flowers. The inflorescences occur singly or in pairs with obloid or spike shaped flower- heads that are in length although they can also be spherical with a diameter of around (0.5–3.5 cm long) and are made up of yellow to bright yellow or orange-yellow coloured flowers. The thin brown woody seed pods that form after flowering have a narrowly oblanceolate to narrowly elliptic to linear shape.
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A second-order filter decreases at −12 dB per octave, a third-order at −18 dB and so on. Butterworth filters have a monotonically changing magnitude function with ω, unlike other filter types that have non-monotonic ripple in the passband and/or the stopband. Compared with a Chebyshev Type I/Type II filter or an elliptic filter, the Butterworth filter has a slower roll-off, and thus will require a higher order to implement a particular stopband specification, but Butterworth filters have a more linear phase response in the pass-band than Chebyshev Type I/Type II and elliptic filters can achieve.
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The leaves are elliptic to oblong-elliptic, long and wide. The flowers are in diameter, or larger in some cultivars, soft-pink to deep-pink and rarely almost white, with 5–7 petals or more in some cultivars, and are produced in sub-terminal or axillary positions on the branch. The fruit is a light brown, three-segmented capsule, about in diameter that ripens in the fall This Camellia is very susceptible to cold weather and has a late blooming season; August through October in the southern hemisphere and March through May in the northern hemisphere.
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He also did pioneering work on the distribution of primes, and on the application of analysis to number theory. His 1798 conjecture of the prime number theorem was rigorously proved by Hadamard and de la Vallée-Poussin in 1896. Legendre did an impressive amount of work on elliptic functions, including the classification of elliptic integrals, but it took Abel's stroke of genius to study the inverses of Jacobi's functions and solve the problem completely. He is known for the Legendre transformation, which is used to go from the Lagrangian to the Hamiltonian formulation of classical mechanics.
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His observations in the 1960s on analogies between primes and knots were taken up by others in the 1990s giving rise to the field of arithmetic topology. Coming under the influence of Alexander Grothendieck's approach to algebraic geometry, he moved into areas of diophantine geometry. Mazur's torsion theorem, which gives a complete list of the possible torsion subgroups of elliptic curves over the rational numbers, is a deep and important result in the arithmetic of elliptic curves. Mazur's first proof of this theorem depended upon a complete analysis of the rational points on certain modular curves.
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Diplolaena drummondii is a small, spreading shrub to high with papery, elliptic to oblong-elliptic leaves long, margins flat, wedge shaped at the base, rounded at the apex on a petiole long. The leaf upper surface is covered sparsely with short, soft hairs, the underside sparsely to moderately covered with star-shaped hairs. The flowerheads about in diameter, the outer green to reddish brown bracts are egg-shaped to narrowly triangular, about long, covered in star-shaped, soft, short hairs. The inner bracts are about long, narrowly oblong, covered in soft, short, star-shaped hairs that taper gradually to a point.
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Because spherical elliptic geometry can be modeled as, for example, a spherical subspace of a Euclidean space, it follows that if Euclidean geometry is self-consistent, so is spherical elliptic geometry. Therefore it is not possible to prove the parallel postulate based on the other four postulates of Euclidean geometry. Tarski proved that elementary Euclidean geometry is complete: there is an algorithm which, for every proposition, can show it to be either true or false.Tarski (1951) (This does not violate Gödel's theorem, because Euclidean geometry cannot describe a sufficient amount of arithmetic for the theorem to apply.
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Here, the ratios considered of successive terms, instead of a rational function of n, are a rational function of qn. Another generalization, the elliptic hypergeometric series, are those series where the ratio of terms is an elliptic function (a doubly periodic meromorphic function) of n. During the twentieth century this was a fruitful area of combinatorial mathematics, with numerous connections to other fields. There are a number of new definitions of general hypergeometric functions, by Aomoto, Israel Gelfand and others; and applications for example to the combinatorics of arranging a number of hyperplanes in complex N-space (see arrangement of hyperplanes).
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The leaves of C. tenuipes are typically 2-2.5 cm (occasionally up to 3.5 cm) long, and 1.5–2 cm wide, and range in shape from ovate or elliptic-ovate to narrowly elliptic-ovate. The undersides are grayish with raised veins, and covered with short, woolly hairs which lie flatly to the surface; the uppersides are green, with slightly impressed veins, and sparsely covered with long, thin, soft, weak, hairs when young, but nearly hairless with age. Both the leaf-stems (3–5 mm long) and their stipules (2.5–5 mm long, lanceolate) are hairy, but the stipules are much less so.
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Thus public key systems require longer key lengths than symmetric systems for an equivalent level of security. 3072 bits is the suggested key length for systems based on factoring and integer discrete logarithms which aim to have security equivalent to a 128 bit symmetric cipher. Elliptic curve cryptography may allow smaller-size keys for equivalent security, but these algorithms have only been known for a relatively short time and current estimates of the difficulty of searching for their keys may not survive. As early as 2004, a message encrypted using a 109-bit key elliptic curve algorithm had been broken by brute force.
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If p and q are two prime divisors of n, then implies the same equation also and These two smaller elliptic curves with the \boxplus-addition are now genuine groups. If these groups have Np and Nq elements, respectively, then for any point P on the original curve, by Lagrange's theorem, is minimal such that kP=\infty on the curve modulo p implies that k divides Np; moreover, N_p P=\infty. The analogous statement holds for the curve modulo q. When the elliptic curve is chosen randomly, then Np and Nq are random numbers close to and respectively (see below).
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In the case of Riemann surfaces, one can explain that the complex structure on the Riemann sphere is isolated (no moduli). For genus 1, an elliptic curve has a one-parameter family of complex structures, as shown in elliptic function theory. The general Kodaira–Spencer theory identifies as the key to the deformation theory the sheaf cohomology group : H^1(\Theta) \, where Θ is (the sheaf of germs of sections of) the holomorphic tangent bundle. There is an obstruction in the H2 of the same sheaf; which is always zero in case of a curve, for general reasons of dimension.
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The theorems of absolute geometry hold in hyperbolic geometry, which is a non-Euclidean geometry, as well as in Euclidean geometry.Indeed, absolute geometry is in fact the intersection of hyperbolic geometry and Euclidean geometry when these are regarded as sets of propositions. Absolute geometry is inconsistent with elliptic geometry: in that theory, there are no parallel lines at all, but it is a theorem of absolute geometry that parallel lines do exist. However, it is possible to modify the axiom system so that absolute geometry, as defined by the modified system, will include spherical and elliptic geometries, that have no parallel lines.
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A ballistic trajectory is a parabola with homogeneous acceleration, such as in a space ship with constant acceleration in absence of other forces. On Earth the acceleration changes magnitude with altitude and direction with latitude/longitude. This causes an elliptic trajectory, which is very close to a parabola on a small scale. However, if an object was thrown and the Earth was suddenly replaced with a black hole of equal mass, it would become obvious that the ballistic trajectory is part of an elliptic orbit around that black hole, and not a parabola that extends to infinity.
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One of the influential examples, both for the history of the more general L-functions and as a still-open research problem, is the conjecture developed by Bryan Birch and Peter Swinnerton-Dyer in the early part of the 1960s. It applies to an elliptic curve E, and the problem it attempts to solve is the prediction of the rank of the elliptic curve over the rational numbers (or another global field): i.e. the number of free generators of its group of rational points. Much previous work in the area began to be unified around a better knowledge of L-functions.
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The first problem with this is that the coefficient field for a Weil cohomology theory cannot be the rational numbers. To see this consider the case of a supersingular elliptic curve over a finite field of characteristic . The endomorphism ring of this is an order in a quaternion algebra over the rationals, and should act on the first cohomology group, which should be a 2-dimensional vector space over the coefficient field by analogy with the case of a complex elliptic curve. However a quaternion algebra over the rationals cannot act on a 2-dimensional vector space over the rationals.
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In analogy with the conic sections, the set of points whose Cartesian coordinates satisfy the general equation of the second degree, namely, :Ax^2 + By^2 + Cz^2 + Fxy + Gyz + Hxz + Jx + Ky + Lz + M = 0, where and are real numbers and not all of and are zero, is called a quadric surface. There are six types of non-degenerate quadric surfaces: # Ellipsoid # Hyperboloid of one sheet # Hyperboloid of two sheets # Elliptic cone # Elliptic paraboloid # Hyperbolic paraboloid The degenerate quadric surfaces are the empty set, a single point, a single line, a single plane, a pair of planes or a quadratic cylinder (a surface consisting of a non-degenerate conic section in a plane and all the lines of through that conic that are normal to ). Elliptic cones are sometimes considered to be degenerate quadric surfaces as well. Both the hyperboloid of one sheet and the hyperbolic paraboloid are ruled surfaces, meaning that they can be made up from a family of straight lines.
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In mathematics, complex multiplication (CM) is the theory of elliptic curves E that have an endomorphism ring larger than the integers; and also the theory in higher dimensions of abelian varieties A having enough endomorphisms in a certain precise sense (it roughly means that the action on the tangent space at the identity element of A is a direct sum of one-dimensional modules). Put another way, it contains the theory of elliptic functions with extra symmetries, such as are visible when the period lattice is the Gaussian integer lattice or Eisenstein integer lattice. It has an aspect belonging to the theory of special functions, because such elliptic functions, or abelian functions of several complex variables, are then 'very special' functions satisfying extra identities and taking explicitly calculable special values at particular points. It has also turned out to be a central theme in algebraic number theory, allowing some features of the theory of cyclotomic fields to be carried over to wider areas of application.
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In 2017, Balakrishnan led a team of mathematicians in solving the "cursed curve" X_s(13). This curve is modeled by the equation :y^4+5x^4-6x^2y^2+6x^3z+26x^2yz+10xy^2z-10y^3z-32x^2z^2-40xyz^2+24y^2z^2+32xz^3-16yz^3=0 and, as a Diophantine equation, the problem is to identify all the combinations of rational numbers for the variables x, y, and z for which the equation is true. Although as an explicit equation this curve has a complicated form, it is significant in the theory of elliptic curves, as a modular curve whose solutions characterize the one remaining unsolved case of a theorem of on the Galois representations of elliptic curves without complex multiplication. Computations by and had previously identified seven solutions to the cursed curve (six corresponding to elliptic curves with complex multiplication, and one cusp), but their computational methods were unable to show that the list of solutions was complete.
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Schizolaena parvipetala grows as a shrub or small tree up to tall. Its twigs are glabrous, occasionally pubescent with small lenticels. The leaves are elliptic to ovate in shape. They are coloured medium brown above and light brown below, measuring up to long.
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Hans Heinrich Wilhelm Magnus known as Wilhelm Magnus (February 5, 1907, Berlin, Germany – October 15, 1990, New Rochelle, NY) was a German-American mathematician. He made important contributions in combinatorial group theory, Lie algebras, mathematical physics, elliptic functions, and the study of tessellations.
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Nepenthes of Borneo. Natural History Publications (Borneo), Kota Kinabalu. A rosette plant from Sulawesi The leaves of this species are sessile. The lamina or leaf blade is lanceolate to elliptic in shape and up to 15 cm long by 3 cm wide.
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First Wilson associates a bivector with an ellipse. The product of the bivector with a complex number on the unit circle is then called an elliptical rotation. Wilson continues with a description of elliptic harmonic motion and the case of stationary waves.
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Following flowering linear woody brown seed pods are formed that are raised over the seeds and constricted between. The pods have a length of around and a width of . The black seeds within have a broadly elliptic shape and are in length.
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The lip is three-lobed. The lateral lobes are elliptic- obovate, obtuse-rounded and erect-incurved forming a cylinder. The mid-lobe is oblong-ovate with the base hastate to subauriculate. The apex is notched with the lanceolate lobules elongate and recurved.
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Glossy green above, somewhat duller below. An occasional orange senescent leaf will be seen in the canopy. Leaves 5 to 12 cm long, elliptic in shape, toothed or sometimes not toothed. Leaf stalks 5 to 13 mm long, channeled on the upper surface.
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The calyx is hemispherical, about high, wide and covered with silvery and rust-coloured scales. The petals are bright yellow, elliptic to egg-shaped, about long and wide with silvery and rust-coloured scales on the back. Flowering occurs from June to September.
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When mature the spikelets (2.5–3 mm long ) fall entirely. The upper glume has five nerves. The lower lemma (similar to the upper glume), has seven nerves and is sterile. The fertile florets are elliptic to lanceolate, with nerves which are obscure.
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Pyrenaria buisanensis is an evergreen tree that can grow tall. Bark is brown-reddish with thin and irregular slices. The leaves are alternate, more or less clustered, thick-coriaceous, elliptic or obovate and typically measure , occasionally longer. The flowers are axillary and solitary.
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They mine the leaves of their host plant. The mine has the form of a short corridor that widens into a long, elliptic blotch. The blotch is upper-surface and whitish to yellow-brown. The inside of the mine is lined with silk.
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The stadium, designed on an elliptic form, is constructed on a 232,485 m² (330m x 704.5m) rectangular site. The design introduces innovative solutions adopting high technology principles for operational management, interaction with the environment and especially with harsh climatic conditions of the geography.
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Compound leaves are 35 to 45 cm long with 16 to 24 leaflets. Oblong-elliptic or reverse lanceolate in shape. Leaflets 6 to 13 cm long, 2.5 to 5 cm wide, sub opposite or alternate on the stem. Leaflets sharply and prominently toothed.
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The simple leaves are alternate, ovate or elliptic, and long. Flowers are in lateral cymes and are in diameter. The five-lobed corolla is white and the five stamens have yellow anthers. The fruit is a yellow berry in diameter with many seeds.
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Male flowers yellowish-white, 3.0-4.0 mm long, tepals 6, c. 3.0 mm long, c. 2. or 4 mm wide, elliptic to obovate slightly similar, externally glabrous or sparsely pubescent on the central portion, stamens usually 9 They are all similar, filaments c.
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Joel Spruck (born 1946) is a mathematician, J. J. Sylvester Professor of Mathematics at Johns Hopkins University, whose research concerns geometric analysis and elliptic partial differential equations. He obtained his PhD from Stanford University with the supervision of Robert S. Finn in 1971.
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Front shock absorbers were hydraulic of a Houdaille type. At the rear was a live axle with semi-elliptic springs and Houdaille hydraulic shock absorbers. Brakes were of a hydraulic drum type, all-round. The transmission was 5-speed and non-synchronised.
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The dull green to grey green, coriaceous, sub-rigid and erect phyllodes have a narrowly linear to narrowly elliptic shape and a length of and a width of . They are flat and straight to shallowly incurved with many parallel longitudinal fine nerves.
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The species rhizomes are elongated. The culms are long with leaf-blades being of in length and wide. The leaf-blade bottom is pubescent, rough and scaberulous. It has an open panicle which is both effuse and elliptic and is long and wide.
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Despite great efforts being put into minimisation,Lee, pp. 756-757 no general theory of minimisation has ever been discovered as it has for the Boolean algebra of digital circuits.Kalman, p. 10 Cauer used elliptic rational functions to produce approximations to ideal filters.
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The subspecies only occurs in scattered populations in low grounds and coastal mud from New Jersey to Florida and west to Texas. H. mutilum subsp. boreale has shorter or absent apical internodes. The broadly ovate to elliptic leaves have no pale undersides.
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The species is an evergreen shrub that is tall. It have leaves that are by long and are elliptic and obovate to oblong. They are also green in colour and have long petioles. Females' peduncles are long and are located on the flowers.
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Bossiaea armitii grows to about 3 m, with cladodes up to about 40 mm wide. The inflorescence bearing cladodes are smooth except for hairs on the margin immediately above the axil. Cladodes are green/greyish at flowering. In profile new growth is elliptic.
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The leaves are large and elliptic in shape. It has white fragrant flowers in a cymose inflorescence with trichotomous branches. The calyx and corolla each have five lobes, and there are five stamens. The two locules of the ovary each contain many seeds.
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The erect viscid shrub typically grows to a height of . It has obscurely ribbed, terete branchlets. The thin, evergreen phyllodes have a narrowly elliptic shape that can be shallowly recurved. The phyllodes have a length of and that dry to a light brown.
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Knema kostermansiana is a species of plant in the family Myristicaceae. It is a tree endemic to Borneo. These trees typically vary from about six to 20 meters in height. The leaves are membranous (thin and transparent), chartaceous (paper-like), and elliptic.
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In particular, quasi-projective varieties are Noetherian schemes. This class includes algebraic curves, elliptic curves, abelian varieties, calabi-yau schemes, shimura varieties, K3 surfaces, and cubic surfaces. Basically all of the objects from classical algebraic geometry fit into this class of examples.
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The Temple measured 30x30 meters, and the walls were 2.4 meters thick. Pottery finds confirmed that the Temple dated to the 2nd century BCE. An earlier and smaller elliptic temple structure underneath probably dates to the end of the 3rd century BCE.
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10–20 per fruit, embryo coiled. Trompettia cardenasiana is similar to Brugmansia species in bearing both pendant flowers and fleshy, indehiscent fruits. It is, however, readily distinguishable by its much smaller flowers, small, narrowly elliptic leaves, small, round fruits, and tetrahedral seeds.
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Leaves form on zig-zagging branchlets, the branchlets have small brown dots. Leaves alternate, elliptic or egg shaped. 7 to 15 cm long, with a short point at the tip. Leaf veins are raised on the underside and more prominent than above.
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Luigi Amerio (15 August 1912 – 28 September 2004), was an Italian electrical engineer and mathematician. He is known for his work on almost periodic functions, on Laplace transforms in one and several dimensions, and on the theory of elliptic partial differential equations.
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Its palea have thick keels and is elliptic and 2-veined. Flowers are fleshy, oblong and truncate with 2 lodicules. They also grow together and have 3 anthers which are long. The fruits have caryopsis with additional pericarp and have linear hilum.
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The sepals triangular shaped, about long, fleshy and mostly smooth. The white petals elliptic shaped, slightly overlap, about long and smooth. The dry seed capsule is almost square, about high with a very small triangular point. Flowering occurs from September to November.
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Ilex guayusa is an evergreen dioecious tree which grows tall. The leaves are ovate, elliptic, oblong or lanceolate; long, wide; with serrate or dentate margin. The flowers are small and white, arranged in thyrses. The fruit is spherical and red, in diameter.
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The peristome is greatly reduced and bears a row of tiny teeth. The pitcher lid is elliptic to oblong and has no appendages. An unbranched, 1 mm long spur is inserted at the base of the lid. N. campanulata is wholly glabrous.
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A contemporary example of using bilinear pairings is exemplified in the Boneh-Lynn-Shacham signature scheme. Pairing-based cryptography relies on hardness assumptions separate from e.g. the Elliptic Curve Discrete Logarithm Problem, which is older and has been studied for a longer time.
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The stem is woody, sparsely prickly, and long. Petiole is long; leaf blade is elliptic to orbicular, long and wide, sometimes wider. Berries are red, globose, and in diameter. Kaempferol 7-O-glucoside, a flavonol glucoside, can be found in S. china.
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The petals are elliptic in shape, long and about wide. The labellum is long, wide, has deep purplish red veins and three lobes. The middle lobe turns downwards and is wavy but the side lobes are upright. Flowering occurs between July and November.
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The pods have a length of up to and a width of and are covered in a fine white powdery coating with longitudinally arranged seeds inside. The dull dark brown seeds have an oblong to elliptic shape with a length of around .
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This species is a short, tender perennial shrub growing tall. Tomentose branches extend radially from a central stem. Leaves are dull green, elliptic, usually up to 10–12 cm (4 to 5 in) long. The flowers are small, green and bell-shaped.
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Like most species of Acacia it has phyllodes rather than true leaves. Th smooth green and glabrous phyllodes are in length and wide with an asymmetrical elliptic to obtriangular shape ending with a rigid, pungent, straight, brown point with a length of .
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The smooth calyx lobes are triangular shaped, and smooth. The petals are narrowly elliptic, spreading, long and pale yellow and stamens marginally longer than petals. The dry fruit has occasional hairs, rounded at the apex, about long and a very small beak.
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Henshilwood, C. S. (2012). The Still Bay and Howiesons Poort: ‘Palaeolithic’ techno-traditions in southern Africa. Journal of World Prehistory, 25, 205–237. Still Bay points have bifacially retouched sides, are elliptic to lanceolate shaped and most often they have two pointed apices.
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The research themes of Xiaonan Ma encompass global analysis and local Atiyah–Singer–index theory (analytic Ray–Singer torsion, Eta forms, elliptic genera), Bergman kernels und geometric quantization. He is editor of Science in China A (Mathematics) and of International Journal of Mathematics.
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In mathematics, a Frey curve or Frey–Hellegouarch curve is the elliptic curve ::y^2 = x(x - a^\ell)(x + b^\ell) associated with a (hypothetical) solution of Fermat's equation :a^\ell + b^\ell = c^\ell. The curve is named after Gerhard Frey.
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They have applications in many fields of the physical sciences, such as optics, quantum mechanics, and general relativity. They tend to occur in problems involving periodic motion, or in the analysis of partial differential equation boundary value problems possessing elliptic symmetry.Gutiérrez-Vega (2015).
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Diplolaena cinerea, is a species of flowering plant in the family Rutaceae and is endemic to the west coast of Western Australia. It has pale orange flowers, papery, elliptic shaped leaves that are covered in star-shaped hairs on the upper surface.
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Mortar discretizations lend themselves naturally to the solution by iterative domain decomposition methods such as FETI and balancing domain decompositionM. Dryja, A Neumann-Neumann algorithm for a mortar discretization of elliptic problems with discontinuous coefficients, Numer. Math., 99 (2005), pp. 645--656.
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There are examples where the minimal free resolution is known explicitly. For a rational normal curve it is an Eagon–Northcott complex. For elliptic curves in projective space the resolution may be constructed as a mapping cone of Eagon–Northcott complexes.Eisenbud, Ch. 6.
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Melhania dehnhardtii grows as a suffrutex (subshrub) up to tall. The elliptic to ovate leaves are tomentose and measure up to long. Inflorescences are solitary or two or three-flowered, on a stalk measuring up to long. The flowers have bright yellow petals.
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Schizolaena masoalensis grows as a liana or tree. Its papery leaves are elliptic to ovate in shape and are coloured grayish green above, tinted orangish below. They measure up to long. The inflorescences bear many flowers, each with three sepals and five petals.
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The Tonelli–Shanks algorithm can (naturally) be used for any process in which square roots modulo a prime are necessary. For example, it can be used for finding points on elliptic curves. It is also useful for the computations in the Rabin cryptosystem.
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Rhodolaena acutifolia grows as a small tree or shrub. Its leaves are small, subcoriaceous, elliptic in shape, tapering to a point and sharp at the base. They measure up to wide. The flowers are paired in solitary inflorescences on a long stem.
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Magnolia amazonica grows up to high, in terra firma tropical lowland forests. Leaves are elliptic, 11 - 28.5 cm long and 4.2 - 10.5 cm broad. The creamy white fragrant flowers reportedly open at night, petals can be 6 – 7 cm long.Ducke, A. Talauma amazonica.
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Restrepia falkenbergii, commonly called the Falkenberg's Restrepia, is an epiphytic orchid, found at altitudes between 1,000-2,000 m in Colombia. This large orchid lacks pseudobulbs. The erect, thick, leathery leaf is elliptic- ovate in shape. The aerial roots seem like fine hairs.
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In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form which gives rise to the same Galois representation).
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Schizolaena charlotteae grows as a shrub or tree up to tall. Its twigs are glabrous with small lenticels. The subcoriaceous leaves are elliptic to ovate or obovate in shape. They are coloured chocolate brown above and more orangish below, measuring up to long.
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The milk stains fabric a sulphur yellow colour in minutes. The spore print is cream, while the spores themselves are broadly elliptic with large warts. Most warts are joined by thin to thickish ridges, forming a very incomplete network. Other warts are isolated.
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Correa eburnea, commonly known as the Deep Creek correa, is a species of shrub that is endemic to the Fleurieu Peninsula in South Australia. It has papery, elliptic to egg-shaped leaves, and up to five green, nodding flowers arranged in leaf axils.
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The linear brown seed pods that form after flowering have a linear shape with straight sides. The pods are in length and wide with prominent pale margins. The brown seeds found inside the pods have an oblong-elliptic shape and around in length.
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161 (I-ME, Fondo SS. Salvatore, Ms. gr. 161 ff.71–74), as of Constantinopolitan origin. According to him the dramaturgy of the doors were not those of the choir screen, but of an elliptic ambo under the dome of the Hagia Sophia.
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There is considerable variation in the form, leaves and flowers of the species. Plants are between in height. The leaves may be linear, oblong or elliptic and are generally between long and wide. Both surfaces of leaves may or may not have hairs.
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Eucalyptus alipes is a mallet that is endemic to the south-west of Western Australia. It has smooth grey to light brown or bronze bark, linear to narrow elliptic leaves, oval to spindle-shaped buds with a long, narrow operculum and conical fruits.
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Boronia amplectens is a plant in the citrus family Rutaceae and is only known from two specimens collected from the Arnhem Land plateau in the Northern Territory of Australia. It is a sprawling shrub with narrow elliptic leaves and four-petalled flowers.
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L. rodolentum can be distinguished from other species by its tall, up to 3.0 m high, upright habit and its grey felty elliptic to wedge-shaped leaves of ¾–1½ cm wide. It can be distinguished from L. parile, which has red pointed bracts.
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In mathematical analysis, the Hilbert–Schmidt theorem, also known as the eigenfunction expansion theorem, is a fundamental result concerning compact, self-adjoint operators on Hilbert spaces. In the theory of partial differential equations, it is very useful in solving elliptic boundary value problems.
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