154 Sentences With "mathematical problem" | Random Sentence Generator

But the number of receptors they discovered instantly posed a mathematical problem.

But investing for retirement is an incredibly difficult mathematical problem with a great many variables.

Look, that's just a hypothetical and ultimately specious mathematical problem — that's not what we're going to do.

The answer is a complex mathematical problem involving wave dynamics — and under-appreciated mathematician Sophie Germain is the original code-cracker.

In trying to find an efficient solution to an arbitrary mathematical problem, they arrived at a shape as functional-looking as a crowbar.

That reference is part of the mathematical problem that needs to be solved in order to bring the following block into the network and the chain.

Affleck plays Christian Wolff, a meticulous mathematical problem solver who leads an isolated life due to autism in "The Accountant," out in U.S. theaters on Friday.

This is what Alan Turing discovered in his pioneering work: Simple rules for turning those switches on and off can be used to solve any mathematical problem.

The anonymously authored proof (which was recently reposted on a Fandom wiki) is currently the most elegant solution to part of a mathematical problem involving something called superpermutations.

In this essay, Sebastian Zagler compares the ways that both a famous mathematical problem and the issue of climate change will require new innovation and collaboration to solve.

Under the circumstances, and given McConnell's determination to freeze liberals and the media out of the legislative process, there's nothing concrete Democrats can do to change the mathematical problem.

This led to an awkward situation in the 1990s, when Andrew Wiles finished proving Fermat's last theorem — surely the most famous mathematical problem in history — just after turning 40.

At the time, he felt stymied by a mathematical problem involving simplexes—a simplex is the polygon with the fewest vertices in any given dimension—and he wanted a break.

Students saw parallels to climate change in Plato's "Allegory of the Cave" and Beckett's "Waiting for Godot," but also in a Rachmaninoff concerto and a mathematical problem called the Collatz conjecture.

For Paul Darley, 53, it was a mathematical problem: The business could not support the people in the next generation of his family if they joined the company at the same rate as his generation.

A team led by John Martinis from Google and the University of California, Santa Barbara, used a 85033 qubit quantum computer to solve a mathematical problem in just three minutes that would take the fastest current computer 10,000 years to calculate.

"Originally, I thought of how to solve data protection from the developer side as a mathematical problem, but it was only having done the research and getting exposed to others in that space that it became something interesting to me," he added.

Most quantum researchers take a softer public stance than McKay, but they too have begun to voice their anxieties, particularly in response to a specific hyped announcement: Google's quantum supremacy demonstration, in which the company's researchers performed a largely useless mathematical problem on a quantum computer faster than a supercomputer.

They're weird because he writes in mirror script because he's lefthanded, and paper is sort of a premium, so on any page of the notebook you see a mind that's beautifully dancing with nature, because he'll go from a sketch of people at a table that might help him with The Last Supper, to a set design for a play he's doing, to a flying machine that's both part of the play but actually might become a real flying machine, to the mathematical problem of squaring the circle, all crammed onto a page.

However, its mathematical problem can be solved by easy exponential algorithm.

The Berlekamp switching game is a mathematical problem proposed by American mathematician Elwyn Berlekamp.

Three bridges of the famous Seven Bridges of Königsberg mathematical problem connected to Lomse.

A corresponding statement for the case that k is not a divisor of n is an open mathematical problem.

Professor Balbus, named after a hero with "anecdotes whose vagueness in detail was more than compensated by their sensational brilliance", is given a problem by students. The number of guests for a party is described in puzzling terms. He in turn creates a mathematical problem for them: two answers are required of readers. :Solution: The mathematical problem is solved with the aid of a diagram.

Langley’s Adventitious Angles Solution to Langley’s 80-80-20 triangle problem Langley’s Adventitious Angles is a mathematical problem posed by Edward Mann Langley in The Mathematical Gazette in 1922..

The word seems to have entered the French language around the 17th century. The oldest known reference is found in the Dictionnaire des Arts et des sciences (1694), II, p. 615, wherein the Methode zetetique is the method to resolve a mathematical problem. In the Littré dictionary of 1872, zététique is referred to as an educational term about research, especially as a method to resolve a mathematical problem, and in general a method to "penetrate the reason of things".

A mathematical problem is a problem that is amenable to being represented, analyzed, and possibly solved, with the methods of mathematics. This can be a real-world problem, such as computing the orbits of the planets in the solar system, or a problem of a more abstract nature, such as Hilbert's problems. It can also be a problem referring to the nature of mathematics itself, such as Russell's Paradox. The result of mathematical problem solved is demonstrated and examined formally.

Science is a way of seeking only new mathematics, if all of that correspond. On the view of modern mathematics, It have thought that to solve a mathematical problem be able to reduced formally to an operation of symbol that restricted by the certain rules like chess (or shogi, or go). translated from On this meaning, Wittgenstein interpret the mathematics to a language game (:de:Sprachspiel). So a mathematical problem that not relation to real problem is proposed or attempted to solve by mathematician.

The ECOH hash functions are based on concrete mathematical functions. They were designed such that the problem of finding collisions should be reducible to a known and hard mathematical problem (the subset sum problem). It means that if one could find collisions, one would be able to solve the underlying mathematical problem which is assumed to be hard and unsolvable in polynomial time. Functions with these properties are known provably secure and are quite unique among the rest of hash functions.

Koppelman completed a master of arts and an all but dissertation in mathematics at Yale University. For two years, she conducted doctoral research on a mathematical problem before uncovering that an obscure mathematics journal in Poland had already published the solution.

Treasure MathStorm! is an educational computer game intended to teach children ages five to nine mathematical problem solving. This sequel to Treasure Mountain! is the sixth installment of The Learning Company's Super Seekers games and the second in its "Treasure" series.

The diagram shows the angles formed by the hands of an analog clock showing a time of 2:20 Clock angle problems are a type of mathematical problem which involve finding the angle between the hands of an analog clock.

San José State University. p. 3. Retrieved 2019-04-13. The knight's tour problem is the mathematical problem of finding a knight's tour. Creating a program to find a knight's tour is a common problem given to computer science students.

It is among the most notable theorems in the history of mathematics and prior to its proof was in the Guinness Book of World Records as the "most difficult mathematical problem" in part because the theorem has the largest number of unsuccessful proofs.

The whole equations system is the public key. To use a mathematical problem for cryptography, it must be modified. The computing of the n variables would need a lot of resources. A standard computer isn't able to compute this in an acceptable time.

Chern's conjecture for affinely flat manifolds was proposed by Shiing-Shen Chern in 1955 in the field of affine geometry. As of 2018, it remains an unsolved mathematical problem. Chern's conjecture states that the Euler characteristic of a compact affine manifold vanishes.

Ellina Grigorieva is a Russian mathematician and mathematics educator known for her books on mathematical problem solving. She is a professor in the Texas Woman's University Department of Mathematics and Computer Science, and an expert on control theory and its applications to the spread of disease.

In cryptography, the unbalanced oil and vinegar (UOV) scheme is a modified version of the oil and vinegar scheme designed by J. Patarin. Both are digital signature protocols. They are forms of multivariate cryptography. The security of this signature scheme is based on an NP-hard mathematical problem.

The Canadian Mathematical OlympiadCanadian Mathematical Olympiad (CMO) official web site (CMO) is Canada's top mathematical problem-solving competition. It is run by the Canadian Mathematical Society. The Olympiad plays several roles in Canadian mathematics competitions, most notably being Canada's main team selection process for the International Mathematical Olympiad.

In the United States, the book was released as Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem. The book was released in the United States in October 1998 to coincide with the US release of Singh's documentary The Proof about Wiles's proof of Fermat's Last Theorem.

MD5 and SHA-1 in particular both have published techniques more efficient than brute force for finding collisions. However, some hash functions have a proof that finding collisions is at least as difficult as some hard mathematical problem (such as integer factorization or discrete logarithm). Those functions are called provably secure.

"How to Support Special Needs Students". PhdinSpecialEducation.com Research generally demonstrates neutral or positive effects of inclusive education. A study by Kreimeyer et al. showed that a group of deaf/hard-of-hearing students in an inclusive classroom scored better than the national averages on reading comprehension, vocabulary, and mathematical problem solving measures.

Schoenfeld's study found that the strategies alone are weak, and need to be strengthened by complementary domain-specific tactics. He also showed the importance of students' monitoring their work on a problem and adjusting their tactical and technical moves accordingly. This work was published as Mathematical Problem Solving (1985). On models of teaching.

It turns out that this formulation provides extreme convergence in representation-to-representation transformations. This completely mathematical problem has a direct physical application. One can apply the cluster-expansion transformation to robustly project classical measurement into a quantum-optical measurement.Kira, M.; Koch, S. W.; Smith, R. P.; Hunter, A. E.; Cundiff, S. T. (2011).

The main difference is that where MuHASH applies a random oracle , ECOH applies a padding function. Assuming random oracles, finding a collision in MuHASH implies solving the discrete logarithm problem. MuHASH is thus a provably secure hash, i.e. we know that finding a collision is at least as hard as some hard known mathematical problem.

His main fields of research include methodology of mathematical thought, teaching theory of mathematics, curriculum evaluation, education of mathematics, mathematical problem solving and mathematical communication. On 16 November 2012, Ge Jun took office as president of High School Affiliated to Nanjing Normal University. He is in charge of school administration, alumni and senior three students.

Plateau also studied the phenomena of capillary action and surface tension. vol. 1 and vol. 2. The mathematical problem of existence of a minimal surface with a given boundary is named after him. He conducted extensive studies of soap films and formulated Plateau's laws, which describe the structures formed by such films in foams.

Historically, this was the mathematical problem that led Airy to develop this special function. A different function that is also named after Airy is important in microscopy and astronomy; it describes the pattern, due to diffraction and interference, produced by a point source of light (one which is much smaller than the resolution limit of a microscope or telescope).

A signature scheme has a signing key, which is kept private, and a verification key, which is publicly revealed. For instance, in signature schemes based on RSA the keys are both exponents. In the UOV scheme, and in every other multivariate signature scheme the keys are more complex. The mathematical problem is to solve m equations with n variables.

Knot I, Excelsior. Two knights discuss the distance they will have travelled that day, uphill and downhill at different speeds. The older knight obscurely explains the mathematical problem. :Carroll's Solution: As with most of the Knots, the solution includes: a simplified restatement of the problem, a method to arrive at the solution, the solution, a discussion of readers' solutions, then readers' grades.

It then uses Monte Carlo simulations to analyze buckling phase transition behavior and critical phenomena, drawing comparisons with the Ising model and XY spin-glass model. Finally it introduces a continuum elastic theory for certain hexatic molecules.Mathematics Genealogy Project During his Ph.D studies he briefly interned at the Bell Labs in 1989. There he was introduced to the mathematical problem of neural networks.

Spectral element method is a finite element type method. It requires the mathematical problem (the partial differential equation) to be cast in a weak formulation. This is typically done by multiplying the differential equation by an arbitrary test function and integrating over the whole domain. Purely mathematically, the test functions are completely arbitrary - they belong to an infinite-dimensional function space.

Köhler published another book, Dynamics in Psychology, in 1940 but thereafter the Gestalt movement suffered a series of setbacks. Koffka died in 1941 and Wertheimer in 1943. Wertheimer's long-awaited book on mathematical problem- solving, Productive Thinking, was published posthumously in 1945, but Köhler was left to guide the movement without his two long-time colleagues.For more on the history of Gestalt psychology, see .

Anne Lester Hudson is an American mathematician and mathematics educator. Her research specialty is the theory of topological semigroups; she is also known for her skill at mathematical problem-solving, and has coached students to success in both the International Mathematical Olympiad and the William Lowell Putnam Mathematical Competition. She is a professor emeritus at the Georgia Southern University-Armstrong Campus (formerly Armstrong State College).

Lunch bills are muddled due to the aunt's reluctance in writing down numbers that could "easily" be memorised. :Solution: Carroll gives a solution which "universally" produces an answer, then gives detailed critiques of several other approaches that only "accidentally" give a solution. Knot VIII, De Omnibus Rebus. The travellers of Knot VI are leaving Kgovjni with relief, when a mathematical problem occurs to one of them.

C. M. Bowra (1957). The Greek experience, p. 166. The Doric order dominated during the 6th and the 5th century BC but there was a mathematical problem regarding the position of the triglyphs, which couldn't be solved without changing the original forms. The order was almost abandoned for the Ionic order, but the Ionic capital also posed an insoluble problem at the corner of a temple.

The patent examiner rejected all 11 of the claims on the grounds that "the invention is not implemented on a specific apparatus and merely manipulates [an] abstract idea and solves a purely mathematical problem without any limitation to a practical application, therefore, the invention is not directed to the technological arts." Ex parte Bilski, No. 2002-2257 (B.P.A.I. Sept. 26, 2006) (last viewed July 3, 2015).

A mathematical problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity. Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram).

Because "overtone" makes the upper partials seem like such a distinct phenomena, it leads to the mathematical problem where the first overtone is the second partial. Also, unlike discussion of "partials", the word "overtone" has connotations that have led people to wonder about the presence of "undertones" (a term sometimes confused with "difference tones" but also used in speculation about a hypothetical "undertone series").

Five binary trees on three vertices, an example of Catalan numbers. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of certain combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, many of the problems that arise in applications have a relatively simple combinatorial description. Fibonacci numbers is the basic example of a problem in enumerative combinatorics.

Semantically secure encryption algorithms include Goldwasser-Micali, El Gamal and Paillier. These schemes are considered provably secure, as their semantic security can be reduced to solving some hard mathematical problem (e.g., Decisional Diffie-Hellman or the Quadratic Residuosity Problem). Other, semantically insecure algorithms such as RSA, can be made semantically secure (under stronger assumptions) through the use of random encryption padding schemes such as Optimal Asymmetric Encryption Padding (OAEP).

In cryptography, Very Smooth Hash (VSH) is a secure cryptographic hash function invented in 2005 by Scott Contini, Arjen Lenstra and Ron Steinfeld. Provably secure means that finding collisions is as difficult as some known hard mathematical problem. Unlike other secure collision-resistant hashes, VSH is efficient and usable in practice. Asymptotically, it only requires a single multiplication per log(n) message-bits and uses RSA-type arithmetic.

In mathematical problem solving, the solution to a problem (such as a proof of a mathematical theorem) exhibits mathematical elegance if it is surprisingly simple and insightful yet effective and constructive. Such solutions might involve a minimal amount of assumptions and computations, while outlining an approach that is highly generalizable. Similarly, a computer program or algorithm is elegant if it uses a small amount of code to great effect.

In a 2012 study, Anderson and Jon Fincham, a colleague at Carnegie Mellon, examined the cognitive stages participants engaged in when solving mathematical problems. These stages included encoding, planning, solving, and response. The study determined how much time participants spent in each problem solving stage when presented with a mathematical problem. Multi-voxel pattern recognition techniques and Hidden Markov models were used to determine participants' problem solving stages.

In cryptography research, it is desirable to prove the equivalence of a cryptographic algorithm and a known hard mathematical problem. These proofs are often called "security reductions", and are used to demonstrate the difficulty of cracking the encryption algorithm. In other words, the security of a given cryptographic algorithm is reduced to the security of a known hard problem. Researchers are actively looking for security reductions in the prospects for post quantum cryptography.

Feferman 1999, p. 1 Alonzo Church and Alan Turing proposed formal models of computability, giving independent negative solutions to Hilbert's Entscheidungsproblem in 1936 and 1937, respectively. The Entscheidungsproblem asked for a procedure that, given any formal mathematical statement, would algorithmically determine whether the statement is true. Church and Turing proved there is no such procedure; Turing's paper introduced the halting problem as a key example of a mathematical problem without an algorithmic solution.

How would a navigator know when the ship had reached its old route and should turn north again? How to avoid overshooting or undershooting the old course? The traverse problem: intended course AB (bearing N), actual course AC (bearing NW). Calculating the ritorno (distance on return course CD, bearing NE) and avanzo (distance made good on intended course) is a matter of solving the triangle ACD This is a mathematical problem of solving a triangle.

In January 2009, Gowers chose to start a social experiment on his blog by choosing an important unsolved mathematical problem and issuing an invitation for other people to help solve it collaboratively in the comments section of his blog. Along with the math problem itself, Gowers asked a question which was included in the title of his blog post, "is massively collaborative mathematics possible?" This post led to his creation of the Polymath Project.

Both are seen in terms of some difficulty or barrier that is encountered.Bernd Zimmermann, On mathematical problem solving processes and history of mathematics, University of Jena. Empirical research shows many different strategies and factors influence everyday problem solving. Rehabilitation psychologists studying individuals with frontal lobe injuries have found that deficits in emotional control and reasoning can be re-mediated with effective rehabilitation and could improve the capacity of injured persons to resolve everyday problems.

The name is based on the quadrature of the circle, an unsolvable mathematical problem. According to the collective, it is “impossible to effectively control the flow of information in the digital age by law and technology without harming public freedoms and damaging economic and social development”. The collective makes an analogy with the squaring of the circle as a problem that could take ages for people to realize is impossible to solve, as initially intended.

Phenomenology at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation. This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the Yang–Mills existence and mass gap problem is a Millennium Prize Problem.

In the 100 prisoners problem each prisoner has to find his number in one of 100 drawers, but may open only 50 of the drawers. The 100 prisoners problem is a mathematical problem in probability theory and combinatorics. In this problem, 100 numbered prisoners must find their own numbers in one of 100 drawers in order to survive. The rules state that each prisoner may open only 50 drawers and cannot communicate with other prisoners.

The classification of tensors is a purely mathematical problem. In GR, however, certain tensors that have a physical interpretation can be classified with the different forms of the tensor usually corresponding to some physics. Examples of tensor classifications useful in general relativity include the Segre classification of the energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants.

The Diffie–Hellman problem (DHP) is a mathematical problem first proposed by Whitfield Diffie and Martin Hellman in the context of cryptography. The motivation for this problem is that many security systems use one-way functions: mathematical operations that are fast to compute, but hard to reverse. For example, they enable encrypting a message, but reversing the encryption is difficult. If solving the DHP were easy, these systems would be easily broken.

Working memory (WM) can be described as a limited capacity system that allows one to temporarily store and process information. This temporary store enables one to complete or work on complex tasks while being able to keep information in mind. For instance, the ability to work on a complicated mathematical problem utilizes one's working memory. One highly influential theory of WM is the Baddeley and Hitch multi-component model of working memory.

In networks-based biocomputation, self-propelled biological agents, such as molecular motor proteins or bacteria, explore a microscopic network that encodes a mathematical problem of interest. The paths of the agents through the network and/or their final positions represent potential solutions to the problem. For instance, in the system described by Nicolau et al. , mobile molecular motor filaments are detected at the "exits" of a network encoding the NP-complete problem SUBSET SUM.

Edward Mann Langley (22 January 1851 – 9 June 1933Obituary: Edward Mann Langley, by E. T. Bell and J. P. Kirkman, The Mathematical Gazette Vol. 17, No. 225 (Oct., 1933), pp. 225-229) was a British mathematician, author of mathematical textbooks and founder of the Mathematical Gazette.The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, by Chris Pritchard, Cambridge University Press, 2003 He created the mathematical problem known as Langley’s Adventitious Angles.

Map of Königsberg from 1651. Shows Seven Bridges of Königsberg, a mathematical problem solved by Leonhard Euler When Imperial and then Swedish armies over-ran Brandenburg during the Thirty Years' War of 1618–1648, the Hohenzollern court fled to Königsberg. On 1 November 1641, Elector Frederick William persuaded the Prussian diet to accept an excise tax. In the Treaty of Königsberg of January 1656, the elector recognised his Duchy of Prussia as a fief of Sweden.

The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (c. 1650 BC) is said to be based on an older mathematical text from the 12th dynasty. The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are so called mathematical problem texts.

The technique evolved from techniques of electrical prospecting that predate digital computers, where layers or anomalies were sought rather than images. Early work on the mathematical problem in the 1930s assumed a layered medium (see for example Langer, Slichter). Andrey Nikolayevich Tikhonov who is best known for his work on regularization of inverse problems also worked on this problem. He explains in detail how to solve the ERT problem in a simple case of 2-layered medium.

Demonstration of the primary operation. The spatula is flipping over the top three pancakes, with the result seen below. In the burnt pancake problem, their top sides would now be burnt instead of their bottom sides. Pancake sorting is the colloquial term for the mathematical problem of sorting a disordered stack of pancakes in order of size when a spatula can be inserted at any point in the stack and used to flip all pancakes above it.

Mental health professionals study the human problem solving processes using methods such as introspection, behaviorism, simulation, computer modeling, and experiment. Social psychologists look into the person- environment relationship aspect of the problem and independent and interdependent problem-solving methods. Problem solving has been defined as a higher-order cognitive process and intellectual function that requires the modulation and control of more routine or fundamental skills. Problem solving has two major domains: mathematical problem solving and personal problem solving.

The story of this celebrated mathematical problem was also the subject of Singh's first book, Fermat's Last Theorem. In 1997, he began working on his second book, The Code Book, a history of codes and codebreaking. As well as explaining the science of codes and describing the impact of cryptography on history, the book also contends that cryptography is more important today than ever before. The Code Book has resulted in a return to television for him.

For example, when MIT students surreptitiously put a fake police car atop the dome on MIT's Building 10, that was a hack in this sense, and the students involved were therefore hackers. Other types of hacking are reality hackers, wetware hackers ("hack your brain"), and media hackers ("hack your reputation"). In a similar vein, a "hack" may refer to a math hack, that is, a clever solution to a mathematical problem. All of these uses have spread beyond MIT.

John Canny considered the mathematical problem of deriving an optimal smoothing filter given the criteria of detection, localization and minimizing multiple responses to a single edge.J. Canny (1986) "A computational approach to edge detection", IEEE Transactions on Pattern Analysis and Machine Intelligence, vol 8, pages 679–714. He showed that the optimal filter given these assumptions is a sum of four exponential terms. He also showed that this filter can be well approximated by first-order derivatives of Gaussians.

Which point on the surface of the spherical mirror can reflect a ray of light from the candle to the observer's eye? Alhazen's problem, also known as Alhazen's billiard problem, is a mathematical problem in geometrical optics first formulated by Ptolemy in 150 AD. It is named for the 11th-century Arab mathematician Alhazen (Ibn al-Haytham) who presented a geometric solution in his Book of Optics. The algebraic solution involves quartic equations and was found in 1965 by Jack M. Elkin.

In cryptography, NewHope is a key-agreement protocol by Erdem Alkim, Léo Ducas, Thomas Pöppelmann, and Peter Schwabe that is designed to resist quantum computer attacks. NewHope is based on a mathematical problem known as Ring learning with errors (RLWE) that is believed to be difficult to solve. NewHope has been selected as a round-two contestant in the NIST Post-Quantum Cryptography Standardization competition, and was used in Google's CECPQ1 experiment as a quantum-secure algorithm, alongside the classical X25519 algorithm.

This protocol allows two parties to generate a key only known to them, under the assumption that a certain mathematical problem (e.g., the Diffie–Hellman problem in their proposal) is computationally infeasible (i.e., very very hard) to solve, and that the two parties have access to an authentic channel. In short, that an eavesdropper—conventionally termed 'Eve', who can listen to all messages exchanged by the two parties, but who can not modify the messages—will not learn the exchanged key.

Soap bubbles are physical examples of the complex mathematical problem of minimal surface. They will assume the shape of least surface area possible containing a given volume. A true minimal surface is more properly illustrated by a soap film, which has equal pressure on inside as outside, hence is a surface with zero mean curvature. A soap bubble is a closed soap film: due to the difference in outside and inside pressure, it is a surface of constant mean curvature.

Beeckman's most influential teachers in Leiden probably were Snellius and Simon Stevin. He himself was a teacher to Johan de Witt and a teacher and friend of René Descartes. Beeckman had met the young Descartes in November 1618 in Breda, where Beeckman then lived and Descartes was then garrisoned as a soldier. It is said that they met when both were looking at a placard that was set up in the Breda marketplace, detailing a mathematical problem to be solved.

In an interview, Henney recalled that at one point during this period, she found twenty bombshells scattered on her front lawn. As a Jewish child, Henney was not allowed to enroll in a formal school during the war years. Her father taught her chess and mathematics at home, rewarding her with a mathematical problem set if she won a game. At age 10, Henney took the admittance exam for admission to the Abitur High School in Hamburg, Germany, from which she would graduate.

The discourse's central chapter features examples and observations of the quincunx in botany. The Belgian physicist Joseph Plateau (1801–1883) formulated the mathematical problem of the existence of a minimal surface with a given boundary, which is now named after him. He studied soap films intensively, formulating Plateau's laws which describe the structures formed by films in foams. Ernst Haeckel (1834–1919) painted beautiful illustrations of marine organisms, in particular Radiolaria, emphasising their symmetry to support his faux-Darwinian theories of evolution.

Calculator spelling is the formation of words and phrases by displaying a number and turning the calculator upside down. The jest may be formulated as a mathematical problem where the result, when read upside down, appears to be an identifiable phrase like "ShELL OIL" or "Esso" using seven-segment display character representations where the open-top "4" is an inverted 'h' and '5' looks like 'S'. Other letters can be used as numbers too with 8 and 9 representing B and G, respectively.

Now under suspicion, Blake approaches Forbin, who is devastated by his wife's arrest. Explaining the details of their plot, Blake convinces Forbin to help after explaining the details of Cleo's captivity. Forbin travels in disguise with the requested information, first to St. John's, then to New York City, where he receives an incomprehensible mathematical problem that the transmission claims will destroy Colossus once it is fed into the computer. Upon his return, Forbin slips the problem to Blake, who enters it into Colossus.

The following are summaries of the more famous characterizations (Kleene, Markov, Knuth) together with those that introduce novel elements—elements that further expand the definition or contribute to a more precise definition. [ A mathematical problem and its result can be considered as two points in a space, and the solution consists of a sequence of steps or a path linking them. Quality of the solution is a function of the path. There might be more than one attribute defined for the path, e.g.

Speaking to his class of a certain mathematical problem that Cairns had solved, Professor Kelland said that it had been solved by only one other of his thousands of students. Cairns was associated with Alexander Campbell Fraser, David Masson, and other leading students in organising the Metaphysical Society for weekly philosophical discussions. He graduated MA in 1841, being facile princeps in classics and philosophy, and equal first in mathematics. Having entered the Presbyterian Secession Hall in 1840, Cairns continued his brilliant career as a student.

According to the fluency account, this is because infants share perceptual equipment that make them process consonance in music more easily than dissonance. When children grow up, they are exposed to the music of their culture, resulting in culture-specific musical fluency. This familiarization explains why individuals from different cultures have different musical tastes. In addition, the theory helps explain why beauty (in a wide sense; perhaps the term elegance is more apt) is a cue for truth in mathematical problem solving and scientific discovery.

In this case study of two epileptic seizure patients, participants reported feeling determined to overcome an approaching challenge; this emotion was reported to feel pleasant, rather than unpleasant. Following electrical stimulation, participants exhibited elevated cardiovascular activity and reported a warm feeling in their upper chest and neck. This work supports the idea that determination is a positive emotion that prepares an individual to overcome obstacles. Another study compared determination and pride to see how these two positive emotions differentially influenced perseverance in the context of a mathematical problem-solving task.

The frame problem is the problem of finding adequate collections of axioms for a viable description of a robot environment. John McCarthy and Patrick J. Hayes defined this problem in their 1969 article, Some Philosophical Problems from the Standpoint of Artificial Intelligence. In this paper, and many that came after, the formal mathematical problem was a starting point for more general discussions of the difficulty of knowledge representation for artificial intelligence. Issues such as how to provide rational default assumptions and what humans consider common sense in a virtual environment.

Children with spina bifida are more likely than their peers without spina bifida to be dyscalculic. Individuals with spina bifida have demonstrated stable difficulties with arithmetic accuracy and speed, mathematical problem- solving, and general use and understanding of numbers in everyday life. Mathematics difficulties may be directly related to the thinning of the parietal lobes (regions implicated in mathematical functioning) and indirectly associated with deformities of the cerebellum and midbrain that affect other functions involved in mathematical skills. Further, higher numbers of shunt revisions are associated with poorer mathematics abilities.

In theories of quantum gravity, the graviton is the hypothetical quantum of gravity, an elementary particle that mediates the force of gravity. There is no complete quantum field theory of gravitons due to an outstanding mathematical problem with renormalization in general relativity. In string theory, believed to be a consistent theory of quantum gravity, the graviton is a massless state of a fundamental string. If it exists, the graviton is expected to be massless because the gravitational force is very long range and appears to propagate at the speed of light.

In this approach, we base the security of hash function on some hard mathematical problem and we prove that finding collisions of the hash functions is as hard as breaking the underlying problem. This gives a somewhat stronger notion of security than just relying on complex mixing of bits as in the classical approach. A cryptographic hash function has provable security against collision attacks if finding collisions is provably polynomial-time reducible from problem P which is supposed to be unsolvable in polynomial time. The function is then called provably secure, or just provable.

Her contributions in cryptography include Nonmalleable Cryptography with Danny Dolev and Moni Naor in 1991, the first lattice-based cryptosystem with Miklós Ajtai in 1997, which was also the first public-key cryptosystem for which breaking a random instance is as hard as solving the hardest instance of the underlying mathematical problem ("worst-case/average-case equivalence"). With Naor she also first presented the idea of, and a technique for, combating e-mail spam by requiring a proof of computational effort, also known as proof-of-work - a key technology underlying hashcash and bitcoin.

Types of cryptarithm include the alphametic, the digimetic, and the skeletal division. ;Alphametic : A type of cryptarithm in which a set of words is written down in the form of a long addition sum or some other mathematical problem. The object is to replace the letters of the alphabet with decimal digits to make a valid arithmetic sum. ;Digimetic : A cryptarithm in which digits are used to represent other ;Skeletal division : A long division in which most or all of the digits are replaced by symbols (usually asterisks) to form a cryptarithm.

If a filter is implemented using, for instance, biquad stages using op-amps, N/2 stages are needed. In a digital implementation, the number of computations performed per sample is proportional to N. Thus the mathematical problem is to obtain the best approximation (in some sense) to the desired response using a smaller N, as we shall now illustrate. Below are the frequency responses of several standard filter functions that approximate a desired response, optimized according to some criterion. These are all fifth-order low-pass filters, designed for a cutoff frequency of .

An Eppendorf laboratory centrifuge The load in a laboratory centrifuge must be carefully balanced. This is achieved by using a combination of samples and balance tubes which all have the same weight or by using various balancing patterns without balance tubes.It is an interesting mathematical problem to solve the balance pattern for given n slots and k tubes with the same weight. It is known that the solution exist if and only if both n and n-k can be both expressed as a sum of prime factors of n.

Foreman (2003) does not reject Woodin's argument outright but urges caution. Solomon Feferman (2011) has argued that CH is not a definite mathematical problem. He proposes a theory of "definiteness" using a semi-intuitionistic subsystem of ZF that accepts classical logic for bounded quantifiers but uses intuitionistic logic for unbounded ones, and suggests that a proposition \phi is mathematically "definite" if the semi-intuitionistic theory can prove (\phi \lor eg\phi). He conjectures that CH is not definite according to this notion, and proposes that CH should, therefore, be considered not to have a truth value.

The problem has, however, also been extended to even values of n by asking, for those n, whether all of the edges of the complete graph except for a perfect matching can be covered by copies of the given 2-regular graph. Like the ménage problem (a different mathematical problem involving seating arrangements of diners and tables), this variant of the problem can be formulated by supposing that the n diners are arranged into n/2 married couples, and that the seating arrangements should place each diner next to each other diner except their own spouse exactly once.

The PTO decision ruled that the claims could not be patented, on three grounds: # "The claimed process involves only information exchange and data processing and does not involve a process of transforming or reducing an article to a different state or thing...." # The claimed method "involves a mathematical algorithm or mathematical calculation steps, as the method includes a procedure for solving a given type of mathematical problem. . . . [T]he mathematical computations of the summation of the possible bidding combinations is at the heart of the invention." # The claimed subject matter is a method of doing business, which cannot be patented, under § 101\.

It can be shown that finding collisions in SWIFFT is at least as difficult as finding short vectors in cyclic/ideal lattices in the worst case. By giving a security reduction to the worst-case scenario of a difficult mathematical problem, SWIFFT gives a much stronger security guarantee than most other cryptographic hash functions. Unlike many other provably secure hash functions, the algorithm is quite fast, yielding a throughput of 40Mbit/s on a 3.2 GHz Intel Pentium 4. Although SWIFFT satisfies many desirable cryptographic and statistical properties, it was not designed to be an "all- purpose" cryptographic hash function.

A mathematical problem, discussed on Eric W. Weisstein's MathWorld and Brady Haran's YouTube channel "Numberphile," is that of determining the greatest number of McNuggets which cannot be made from any combination of pack sizes on offer. For example, in the UK, McNuggets are sold in boxes of 6, 9 or 20 (excluding Happy Meals). Consequently, the greatest number of McNuggets which cannot be purchased exactly is 43, the Frobenius number of the set {6,9,20}. This means that all natural numbers greater than 43 can be expressed, in some way, as the sum of some multiple of each of 6, 9, and 20.

For the portions of the ranking that were determined by polls and computer-generated rankings, the BCS used a series of Borda counts to arrive at its overall rankings. This was an example of using a voting system to generate a complete ordered list of winners from both human and computer-constructed votes. Obtaining a fair ranking system was a difficult mathematical problem and numerous algorithms were proposed for ranking college football teams in particular. One example was the "random-walker rankings" studied by applied mathematicians Thomas Callaghan, Peter Mucha, and Mason Porter that employed the science of networks.

The mathematical problem represented is typically ill-posed because it has an infinitude of solutions. In fact, in any three-dimensional solid body one may have infinitely many (and infinitely complicated) non-zero stress tensor fields that are in stable equilibrium even in the absence of external forces. These stress fields are often termed hyperstatic stress fields and they co- exist with the stress fields that balance the external forces. In linear elasticity, their presence is required to satisfy the strain/displacement compatibility requirements and in limit analysis their presence is required to maximise the load carrying capacity of the structure or component.

After asking on his blog whether "massively collaborative mathematics" was possible, he solicited comments on his blog from people who wanted to try to solve mathematical problems collaboratively. The first problem in what is called the Polymath Project, Polymath1, was to find a new combinatorial proof to the density version of the Hales–Jewett theorem. After seven weeks, Gowers wrote on his blog that the problem was "probably solved". In 2009, with Olof Sisask and Alex Frolkin, he invited people to post comments to his blog to contribute to a collection of methods of mathematical problem solving.

Finding the roots of a given polynomial has been a prominent mathematical problem. Solving linear, quadratic, cubic and quartic equations by factorization into radicals can always be done, no matter whether the roots are rational or irrational, real or complex; there are formulae that yield the required solutions. However, there is no algebraic expression (that is, in terms of radicals) for the solutions of general quintic equations over the rationals; this statement is known as the Abel–Ruffini theorem, first asserted in 1799 and completely proved in 1824. This result also holds for equations of higher degrees.

Ambient networks of intelligent objects and, sooner or later, a new generation of information systems which are even more diffused and based on nanotechnology, will profoundly change this concept. Small devices that can be compared to insects do not dispose of a high intelligence on their own. Indeed, their intelligence can be classed as fairly limited. It is, for example, impossible to integrate a high performance calculator with the power to solve any kind of mathematical problem into a biochip that is implanted into the human body or integrated in an intelligent tag which is designed to trace commercial articles.

140px100px With only 2 pence and 5 pence coins, one cannot make 3 pence, but one can make any higher integral amount. The coin problem (also referred to as the Frobenius coin problem or Frobenius problem, after the mathematician Ferdinand Frobenius) is a mathematical problem that asks for the largest monetary amount that cannot be obtained using only coins of specified denominations. For example, the largest amount that cannot be obtained using only coins of 3 and 5 units is 7 units. The solution to this problem for a given set of coin denominations is called the Frobenius number of the set.

Teachers customarily do not ask questions of individual students; rather, a standard teaching technique is to narrate a historical event or to describe a mathematical problem, pausing at key junctures to allow the students to call out responses that "fill in the blanks". By not identifying individual problems of students and retaining an emotionally distanced demeanor, teachers are said to show themselves to be patient, which is considered admirable. Children ages 6–12 attend primary school, called Sekolah Dasar (SD). As of 2014, most elementary schools are government-operated public schools, accounting for 90.29% of all elementary schools in Indonesia.

Pfungst asked subjects to stand on his right and think "with a high degree of concentration" about a particular number, or a simple mathematical problem. Pfungst would then tap out the answer with his right hand. He frequently observed "a sudden slight upward jerk of the head" when reaching the final tap, and noted that this corresponded to the subject resuming the position they had adopted before thinking of the question. Even after this official debunking, von Osten, who was never persuaded by Pfungst's findings, continued to show Hans around Germany, attracting large and enthusiastic crowds.

Small groups of people can engage in activities such as mathematical problem solving and can accomplish intellectual achievements. These accomplishments often proceed by means of interactions in which ideas emerge from the discourse between multiple perspectives and cannot be credited to any one person. An utterance by one person is elicited by and responds to the previous discussion and group context in ways that would otherwise not have arisen, and the utterance is structured so as to elicit specific kinds of responses from other participants. Through a sequence of complexly and subtly interwoven interactions, cognitive results are achieved.

The vibrations of the membrane are given by the solutions of the two-dimensional wave equation with Dirichlet boundary conditions which represent the constraint of the frame. It can be shown that any arbitrarily complex vibration of the membrane can be decomposed into a possibly infinite series of the membrane's normal modes. This is analogous to the decomposition of a time signal into a Fourier series. The study of vibrations on drums led mathematicians to pose a famous mathematical problem on whether the shape of a drum can be heard, with an answer being given in 1992 in the two-dimensional setting.

In the mathematical problem Tower of Hanoi, solving a puzzle with an -disc tower requires steps, assuming no mistakes are made. The number of rice grains on the whole chessboard in the wheat and chessboard problem is . The asteroid with minor planet number 8191 is named 8191 Mersenne after Marin Mersenne, because 8191 is a Mersenne prime (3 Juno, 7 Iris, 31 Euphrosyne and 127 Johanna having been discovered and named during the 19th century). In geometry, an integer right triangle that is primitive and has its even leg a power of 2 ( ) generates a unique right triangle such that its inradius is always a Mersenne number.

Gerhard Ringel Gerhard Ringel (October 28, 1919 in Kollnbrunn, Austria – June 24, 2008 in Santa Cruz, California) was a German mathematician. He was one of the pioneers in graph theory and contributed significantly to the proof of the Heawood conjecture (now the Ringel-Youngs theorem), a mathematical problem closely linked with the Four Color Theorem. Although born in Austria, Ringel was raised in Czechoslovakia and attended Charles University before being drafted into the Wehrmacht in 1940 (after Germany had taken control of much of what had been Czechoslovakia). After the war Ringel served for over four years in a Soviet prisoner of war camp.

At Wiseman's suggestion, Singh directed a segment about politicians lying in different mediums, and getting the public's opinion on whether the person was lying or not. Simon Singh signing a book for a fan, Brisbane, 23 May 2005 After attending some of Wiseman's lectures, Singh came up with the idea to create a show together, and Theatre of Science was born. It was a way to deliver science to normal people in an entertaining manner. Richard Wiseman has influenced Singh in such a way that Singh states: Singh directed his BAFTA award-winning documentary about the world's most notorious mathematical problem entitled Fermat's Last Theorem in 1996.

A primary advantage is that the mathematical problem to be solved in the algorithm is quantum-resistant. So when a quantum computer is built that can handle enough states to break commercial signature schemes like RSA or ElGamal, the unbalanced oil and vinegar signature scheme should remain secure, as no algorithm currently exists that gives a quantum computer a great advantage in solving these multivariate systems. The second advantage is that the operations used in the equations are relatively simple. Signatures get created and validated only with addition and multiplication of "small" values, making this signature viable for low-resource hardware as found in smart cards.

A Quantum Digital Signature (QDS) refers to the quantum mechanical equivalent of either a classical digital signature or, more generally, a handwritten signature on a paper document. Like a handwritten signature, a digital signature is used to protect a document, such as a digital contract, against forgery by another party or by one of the participating parties. As e-commerce has become more important in society, the need to certify the origin of exchanged information has arisen. Modern digital signatures enhance security based on the difficulty of solving a mathematical problem, such as finding the factors of large numbers (as used in the RSA algorithm).

In discussing his opinion that every mathematical problem should have a solution, Hilbert allows for the possibility that the solution could be a proof that the original problem is impossible.This issue that finds its beginnings in the "foundational crisis" of the early 20th century, in particular the controversy about under what circumstances could the Law of Excluded Middle be employed in proofs. See much more at Brouwer–Hilbert controversy. He stated that the point is to know one way or the other what the solution is, and he believed that we always can know this, that in mathematics there is not any "ignorabimus" (statement whose truth can never be known).

Working memory capacity is correlated with learning outcomes in literacy and numeracy. Initial evidence for this relation comes from the correlation between working-memory capacity and reading comprehension, as first observed by Daneman and Carpenter (1980) and confirmed in a later meta-analytic review of several studies. Subsequent work found that working memory performance in primary school children accurately predicted performance in mathematical problem solving. One longitudinal study showed that a child's working memory at 5 years old is a better predictor of academic success than IQ. In a large- scale screening study, one in ten children in mainstream classrooms were identified with working memory deficits.

The book is written for a general audience, unlike a follow-up work published by Knorr, Textual Studies in Ancient and Medieval Geometry (1989), which is aimed at other experts in the close reading of Greek mathematical texts. Nevertheless, reviewer Alan Stenger calls The Ancient Tradition of Geometric Problems "very specialized and scholarly". Reviewer Colin R. Fletcher calls it "essential reading" for understanding the background and content of the Greek mathematical problem- solving tradition. In its historical scholarship, historian of mathematics Tom Whiteside writes that the book's occasionally speculative nature is justified by its fresh interpretations, well-founded conjectures, and deep knowledge of the subject.

The hexagonal tortoise problem () was invented by Korean aristocrat and mathematician Choi Seok-jeong, who lived from 1646 to 1715. It is a mathematical problem that involves a hexagonal lattice, like the hexagonal pattern on some tortoises' shells, to the (N) vertices of which must be assigned integers (from 1 to N) in such a way that the sum of all integers at the vertices of each hexagon is the same. The problem has apparent similarities to a magic square although it is a vertex-magic format rather than an edge-magic form or the more typical rows-of-cells form. His book, Gusuryak, contains many interesting mathematical discoveries.

This introduction reveals the central role played by Professor Gardner's just-published time-average theory in understanding the relationship between Einstein's and Norbert Wiener's (1930) contributions to statistical spectral analysis. Gardner won the international IEEE Stephen O. Rice Prize Paper award in communication theory in 1988 and the International EURASIP Best Paper of the Year Award in 1987; both papers treated his theory of cyclostationarity. Gardner and his students went on to further prove the uses of his theory of cyclostationarity in applications in communications and signals intelligence. Together with his doctoral student Chi Kang Chen, he wrote the book of mathematical problem solving, The Random Processes Tutor: A Comprehensive Solutions Manual for Independent Study in 1989.

Simon Lehna Singh, (born 19 September 1964) is a British popular science author, theoretical and particle physicist whose works largely contain a strong mathematical element. His written works include Fermat's Last Theorem (in the United States titled Fermat's Enigma: The Epic Quest to Solve the World's Greatest Mathematical Problem), The Code Book (about cryptography and its history), Big Bang (about the Big Bang theory and the origins of the universe), Trick or Treatment? Alternative Medicine on Trial (about complementary and alternative medicine, co-written by Edzard Ernst) and The Simpsons and Their Mathematical Secrets (about mathematical ideas and theorems hidden in episodes of The Simpsons and Futurama). In 2012 Singh founded the Good Thinking Society.

Artistic representation of a Turing machine The first design for a program-controlled computer was Charles Babbage's Analytical Engine in the 1830s. A century later, in 1936, mathematician Alan Turing published his description of what became known as a Turing machine, a theoretical concept intended to explore the limits of mechanical computation. Turing was not imagining a physical machine, but a person he called a "computer", who acted according to the instructions provided by a tape on which symbols could be read and written sequentially as the tape moved under a tape head. Turing proved that if an algorithm can be written to solve a mathematical problem, then a Turing machine can execute that algorithm.

Venda thought that astrologers mistakenly believed that they were studying the influence of stars on people's lives, but, in fact, as they were observing the starry sky as a complex clock and simultaneously recording events on Earth, they were accumulating valuable data on cyclic processes in nature. He proposed a strategy for winning the game "Sportloto 6 of 49", trying to solve the problem not as a traditional mathematical problem, but as psychological one, arguing that although the balls pop up chaotically, after studying the for the majority of players’ strategy in playing sportloto, one can see there is a chance for a constant probability of winning much more than prescribed by probability theory.

300 AD), recorded the use of the odometer, providing description (attributing it to the Western Han era, from 202 BC–9 AD). The passage in the Jin Shu expanded upon this, explaining that it took a similar form to the mechanical device of the south-pointing chariot invented by Ma Jun (200–265, see also differential gear). As recorded in the Song Shi of the Song Dynasty (960-1279 AD), the odometer and south-pointing chariot were combined into one wheeled device by engineers of the 9th century, 11th century, and 12th century. The Sunzi Suanjing (Master Sun's Mathematical Manual), dated from the 3rd century to 5th century, presented a mathematical problem for students involving the odometer.

MIT mathematics faculty There he had been involved in many scholarly extracurricular activities, including running SPUR (Summer Program in Undergraduate Research) for MIT undergraduates, overseeing the mathematics section of RSI (Research Science Institute) for advanced high school students, and coaching the MIT Putnam exam team for nearly two decades starting in 1990, including the years 2003 and 2004 when MIT won for the first time since 1979. He also ran a seminar called 18.S34: Mathematical Problem Solving for MIT freshmen. Rogers is known within the MIT undergraduate community also for having developed a multivariable calculus course (18.022: Multivariable Calculus with Theory) with the explicit goal of providing a firm mathematical foundation for the study of physics.

In January 1918, Griffith was appointed by Prime Minister Billy Hughes as head of a Royal Commission into the recruitment levels needed to maintain the Australian Imperial Force's fighting strength overseas. This came only a month after a second referendum on overseas conscription had returned a vote in the negative. Griffith was given such narrow terms of reference that his report took only a single week, and was effectively little more than a mathematical problem relating to the "existing size of the AIF, likely future losses of men, the numbers required to replace them, and so on". After the report was released, Hughes used it as vindication of his statements during the referendum debate.

Most mathematics questions, or calculation questions from subjects such as chemistry, physics, or economics employ a style which does not fall into any of the above categories, although some papers, notably the Maths Challenge papers in the United Kingdom employ multiple choice. Instead, most mathematics questions state a mathematical problem or exercise that requires a student to write a freehand response. Marks are given more for the steps taken than for the correct answer. If the question has multiple parts, later parts may use answers from previous sections, and marks may be granted if an earlier incorrect answer was used but the correct method was followed, and an answer which is correct (given the incorrect input) is returned.

In the mathematical field of graph theory, the pancake graph Pn or n-pancake graph is a graph whose vertices are the permutations of n symbols from 1 to n and its edges are given between permutations transitive by prefix reversals. Pancake sorting is the colloquial term for the mathematical problem of sorting a disordered stack of pancakes in order of size when a spatula can be inserted at any point in the stack and used to flip all pancakes above it. A pancake number is the minimum number of flips required for a given number of pancakes. Obtaining the pancake number is equivalent to the problem of obtaining the diameter of the pancake graph.

The earliest of those described on this subject in the scientific literature should be recognized, perhaps, the result presented in 1973, which was obtained under the guidance of , a Soviet academician, and which initiated then a number of foreign studies of the mathematical problem known as "stability loss delay for dynamical bifurcations". A new wave of interest in the study of the strange behaviour of dynamic systems in a certain region of the state space has been caused by the desire to explain the non-linear effects revealed during the getting out of controllability of ships. Subsequently, similar phenomena were also found in biological systems — in the system of blood coagulation and in one of the mathematical models of myocardium.

Upon visiting the Tetra Pak factory in Lund in the 1950s, Danish Nobel Prize winner and Physics professor Niels Bohr called the tetrahedron "a perfect practical application of a mathematical problem", and the invention lay the foundation of the Tetra Pak success saga.Andersson, Peter and Larsson, Tommy, Tetra. Historien om dynastin Rausing, Stockholm: Norstedts 1998 (), p. 23 The Royal Swedish Academy of Engineering Sciences has called the Tetra Pak packaging system one of Sweden's most successful inventions of all times. The development of the triangular tetrahedron carton package, the rectangular Tetra Brik, was represented at the 2011 exhibition Hidden Heroes – The Genius of Everyday Things at the London Science Museum/Vitra Design Museum, celebrating "the miniature marvels we couldn’t live without".

An ansatz is the establishment of the starting equation(s), the theorem(s), or the value(s) describing a mathematical or physical problem or solution. It typically provides an initial estimate or framework to the solution of a mathematical problem, and can also take into consideration the boundary conditions (in fact, an ansatz is sometimes thought of as a "trial answer" and an important technique in solving differential equations). After an ansatz, which constitutes nothing more than an assumption, has been established, the equations are solved more precisely for the general function of interest, which then constitutes a confirmation of the assumption. In essence, an ansatz makes assumptions about the form of the solution to a problem—so as to make the solution easier to find.

Hilbert's twenty-fourth problem is a mathematical problem that was not published as part of the list of 23 problems known as Hilbert's problems but was included in David Hilbert's original notes. The problem asks for a criterion of simplicity in mathematical proofs and the development of a proof theory with the power to prove that a given proof is the simplest possible.Hilbert’s twenty-fourth problem Rüdiger Thiele, American Mathematical Monthly, January 2003 The 24th problem was rediscovered by German historian Rüdiger Thiele in 2000, noting that Hilbert did not include the 24th problem in the lecture presenting Hilbert's problems or any published texts. Hilbert's friends and fellow mathematicians Adolf Hurwitz and Hermann Minkowski were closely involved in the project but did not have any knowledge of this problem.

School students during the Kangaroo in Germany in 2006 Mathematicians in Australia came up with the idea to organize a competition that underlines the joy of mathematics and encourages mathematical problem-solving. A multiple-choice competition was created, which has been taking place in Australia since 1978. At the same time, both in France and all over the world, a widely supported movement emerged towards the popularization of mathematics. The idea of a multiple-choice competition then sprouted from two French teachers, André Deledicq and Jean Pierre Boudine, who visited their Australian colleagues Peter O’HolloranObituary: Peter Joseph O'Halloran (1931-1994) at AMT website and Peter Taylor and witnessed their competition. In 1990, they decided to start a challenge in France under the name Kangourou des Mathématiques in order to pay tribute to their Australian colleagues.

The SJS Academic Bowl Team won the NAQT High School National Championship in 2002, placed third in 2003 and 2004, and advanced to the semi-finals of the PACE NSC in 2004. Most recently, St. John's placed 2nd in the 2014 HSNCT National Championships Dozens of other student organizations, from the Yearbook to Model United Nations to "Pots and Pans" (a moral/spirit group), are active throughout the academic year. Other examples of clubs include sports based clubs (baseball, hockey, soccer, curling), science (Science and Math Club, Faraday), cinematography (MavTV), academic (Speech and Debate Team, Quiz Bowl/Academic Challenge, Mathematical Problem Solving Club), government (Junior Statesmen, Model UN, Young Political Organization), international interests (Spanish Club, Italian Club, International Club), and general interests (Bread Club, Auto club, Anime Club et al.).

In computer science, an interchangeability algorithm is a technique used to more efficiently solve constraint satisfaction problems (CSP). A CSP is a mathematical problem in which objects, represented by variables, are subject to constraints on the values of those variables; the goal in a CSP is to assign values to the variables that are consistent with the constraints. If two variables A and B in a CSP may be swapped for each other (that is, A is replaced by B and B is replaced by A) without changing the nature of the problem or its solutions, then A and B are interchangeable variables. Interchangeable variables represent a symmetry of the CSP and by exploiting that symmetry, the search space for solutions to a CSP problem may be reduced.

Most textbooks of astronomy written in the medieval Islamic World contain a chapter on the determination of the qibla, considered one of the many things connecting astronomy with Islamic law (sharia). According to David A. King, various medieval solutions for the determination of the qibla "bear witness to the development of mathematical methods from the 3rd/9th to the 8th/14th centuries and to the level of sophistication in trigonometry and computational techniques attained by these scholars". The first mathematical methods developed in the early 9th century were approximate solutions to the mathematical problem, usually using a flat map or two-dimensional geometry. Since in reality the earth is spherical, the directions found were inexact, but they were sufficient for locations relatively close to Mecca (including as far away as Egypt and Iran) because the errors were less than 2°.

Anderson's transreal arithmetic, and concept of "nullity" in particular, were introduced to the public by the BBC with its report in December 2006 where Anderson was featured on a BBC television segment teaching schoolchildren about his concept of "nullity". The report implied that Anderson had discovered the solution to division by zero, rather than simply attempting to formalize it. The report also suggested that Anderson was the first to solve this problem, when in fact the result of zero divided by zero has been expressed formally in a number of different ways (for example, NaN). The BBC was criticized for irresponsible journalism, but the producers of the segment defended the BBC, stating that the report was a light-hearted look at a mathematical problem aimed at a mainstream, regional audience for BBC South Today rather than at a global audience of mathematicians.

The problem of unit commitment involves finding the least-cost dispatch of available generation resources to meet the electrical load. Generating resources can include a wide range of types: #Nuclear #Thermal (using coal, gas, other fossil fuels, or biomass) #Renewables (including hydro, wind, wave-power, and solar) The key decision variables that are decided by the computer program are: #Generation level (in megawatts) #Number of generating units on The latter decisions are binary {0,1}, which means that the mathematical problem is not continuous. In addition, generating plants are subject to a number of complex technical constraints, including: #Minimum stable operating level #Maximum rate of ramping up or down #Minimum time period the unit is up and/or down These constraints have many different variants; all this gives rise to a large class of mathematical optimization problems.

Essai sur les données immédiates de la conscience (Dissertation, 1889) Quid Aristoteles de loco senserit (Dissertation, 1889) Bergson attended the Lycée Fontanes (known as the Lycée Condorcet 1870–1874 and 1883–present) in Paris from 1868 to 1878. He had previously received a Jewish religious education.Lawlor, Leonard and Moulard Leonard, Valentine, "Henri Bergson", The Stanford Encyclopedia of Philosophy (Summer 2016 Edition), Edward N. Zalta (ed.), URL = Between 14 and 16, however, he lost his faith. According to Hude (1990), this moral crisis is tied to his discovery of the theory of evolution, according to which humanity shares common ancestry with modern primates, a process sometimes construed as not needing a creative deity.Henri Hude, Bergson, Paris, Editions Universitaires, 1990, 2 volumes, quoted by Anne Fagot-Largeau in her 21 December 2006 course at the College of France While at the lycée Bergson won a prize for his scientific work and another, in 1877 when he was eighteen, for the solution of a mathematical problem.

An artificially produced word problem is a genre of exercise intended to keep mathematics relevant. Stephen Leacock described this type:Stephen Leacock "A,B,C – The Human Element in Mathematics", pages 131 to 55 in The Mathematical Magpie (1962) by Clifton Fadiman (editor) Simon & Schuster :The student of arithmetic who has mastered the first four rules of his art and successfully striven with sums and fractions finds himself confronted by an unbroken expanse of questions known as problems. These are short stories of adventure and industry with the end omitted and, though betraying a strong family resemblance, are not without a certain element of romance. A distinction between an exercise and a mathematical problem was made by Alan H. Schoenfeld:Alan H. Schoenfeld (1988) "Problem Solving",(see page 85), chapter 5 of Mathematics Education in Secondary Schools and Two-Year Colleges by Paul J. Campbell and Louis S. Grinstein, Garland Publishing, :Students must master the relevant subject matter, and exercises are appropriate for that.

Typically, there are 12 tones in the western scale. Computing the number of unique trichords is a mathematical problem. A computer program can quickly iterate all the triads and remove the ones that are merely transpositions of others, leaving (as noted above) nineteen or, to within inversional equivalence, twelve. As an example, the following list contains all trichords that can be made including the note C, but includes 36 that are merely transpositions or transposed inversions of others: # C D♭ D [0,1,2] – this combination has no name (half step cluster, with doubly diminished third and quintuply diminished fifth, spelled enharmonically) # C D♭ E♭ [0,1,3] – this combination has no name # C D♭ E [0,1,4] – Eaug with sus6 # C D♭ F [0,1,5] – Dmaj seventh (omit 5th) # C D♭ G♭ [0,1,6] – Gsus#4 # C D♭ G [0,5,6] (= inv. of [0,1,6]) # C D♭ A♭ [0,4,5] (= inv.

In 1936, mathematician Alan Turing published a definition of a theoretical "universal computing machine", a computer which held its program on tape, along with the data being worked on. Turing proved that such a machine was capable of solving any conceivable mathematical problem for which an algorithm could be written.. During the 1940s, Turing and others such as Konrad Zuse developed the idea of using the computer's own memory to hold both the program and data, instead of tape, but it was mathematician John von Neumann who became widely credited with defining that stored-program computer architecture, on which the Manchester Mark 1 was based. The practical construction of a von Neumann computer depended on the availability of a suitable memory device. The University of Manchester's Baby, the world's first electronic stored-program computer, had successfully demonstrated the practicality of the stored-program approach and of the Williams tube, an early form of computer memory based on a standard cathode ray tube (CRT), by running its first program on 21 June 1948.

Some instances of the smallest bounding circle. The smallest-circle problem (also known as minimum covering circle problem, bounding circle problem, smallest enclosing circle problem) is a mathematical problem of computing the smallest circle that contains all of a given set of points in the Euclidean plane. The corresponding problem in n-dimensional space, the smallest bounding sphere problem, is to compute the smallest n-sphere that contains all of a given set of points. The smallest-circle problem was initially proposed by the English mathematician James Joseph Sylvester in 1857.. The smallest-circle problem in the plane is an example of a facility location problem (the 1-center problem) in which the location of a new facility must be chosen to provide service to a number of customers, minimizing the farthest distance that any customer must travel to reach the new facility.. Both the smallest circle problem in the plane, and the smallest bounding sphere problem in any higher-dimensional space of bounded dimension are solvable in worst-case linear time.