And their convenient compactness has an ugly side: sometimes they're easier to lose.
|
|
Speaking of compactness, the implications for mobile VR with this interface technology are huge.
|
|
Given that it's not trying for compactness, the range of spec options is pretty wide.
|
|
The digital display is bold, clear, and well-sized for the compactness of the unit.
|
|
And yet, Theranos leadership was married to the compactness of the Edison despite pushback from employees.
|
|
Lightness and compactness are important too if you're going to be lugging this around with your phone.
|
|
This key holder offers ultimate compactness and portability, ensuring bulky pockets are a thing of the past.
|
|
The shared benefit for DJs and bankers is the compactness and flexibility of these over-ear headphones.
|
|
Over all, it admirably adheres to traditional nonpartisan redistricting criteria, like compactness and the avoidance of unnecessary county splits.
|
|
"By late November, we had found it had problems with stability, removal of obstructions, compactness and elevation," Mr. Bian said.
|
|
The devices will be judged on performance, compactness, noise, user experience, and ability to achieve "near-vertical" takeoff and landing.
|
|
Those tiny dwellings are a narrative coup, their compactness allowing them to exist as storytelling interludes in the main game.
|
|
"We focused so much on size and compactness with Here One that we kind of compromised battery life," Lanman told Wired.
|
|
I was able to do most of my exploration of Xian solo, though, thanks to the compactness of the city center.
|
|
The goal is not to come up with a single compactness measure, but to bring order to the jostling crowd of contenders.
|
|
A compactness rule would limit the range of tactics available for drawing unfair maps, but it would be far from a panacea.
|
|
In Vogue is successful in its compactness, a runtime sitting at around half an hour, and several songs being under two minutes.
|
|
Good jazz music requires the architectural compactness of a jazz club, not the airiness of a huge auditorium designed for symphony orchestras.
|
|
Its compactness also allows the elderly easy access to shops, clinics, banks and transport networks, with most places located within walking distance.
|
|
The Tonal spokesperson said that the company has a number of customers who are renters that enjoy the compactness of the machine.
|
|
Flexibility and compactness are coming into vogue, and this prototype satellite radiator is inspired by that most compact and flexible of arts: origami.
|
|
Rechargeable lithium-ion batteries still provide the best combination of compactness, power and efficiency for products ranging from drones to smartphones and cars.
|
|
Redistricting is just a process that weighs competing values, including but not limited to compactness, competitiveness, preserving communities of interest, and racial representation.
|
|
Merely creating districts randomly or optimized for unbiased parameters (such as compactness or equal-population) is no guarantee that those districts are fair.
|
|
After a few seconds, the 'chute separated from the drone, the robot's wings—which had folded into the body for compactness—extended outward.
|
|
Technically, the most impressive thing about the Feniks Essence, once you get over their looks and compactness, is the stereo imaging they provide.
|
|
The dilemma I had at the Apple Store a decade ago â€" deciding between compactness versus power â€" would essentially be a solved problem.
|
|
He said the compactness of a district — and how squiggly or rectangular it was — was rarely a good indication of how gerrymandered it is.
|
|
The plaintiffs could then rebut that claim by producing a less biased plan that performed as well as the existing map on measures like compactness.
|
|
Probably. But Svensson says the compactness of this engine let the design and engineering teams identify "voids" in the car that they could work around.
|
|
When I held the device at Native Instruments' booth, I was pleasantly surprised both at how incredibly light the M32 was and also its compactness.
|
|
The commission had, he said, considered compactness, contiguity, respect for "communities of interest," geography, local boundaries, political competitiveness and compliance with the Voting Rights Act.
|
|
What Pound missed was Housman's music, which so lent itself to composers—the intensity of his tone and the tautness and compactness of his expression.
|
|
The Nokia 3310 candy bar I had in high school and the LG VX3100 flip phone I had in college were adorable paragons of compactness.
|
|
"Quality tests are not in place, and there has not been monitoring of the compactness of the filling on the inclined area," said a report from October.
|
|
The grounds collect in the machine's hefty commercial portafilter, then an internal fan blows for a few seconds to tamp them to the ideal level of compactness.
|
|
Many states require that districts should be reasonably "compact" wherever possible, but there's no one mathematical measure of compactness that fully captures what these shapes should look like.
|
|
Despite the book's laconic compactness, Enia manages to fuse into it a fully realized personal narrative: that of a beloved uncle lost and a reticent father finally understood.
|
|
The number and severity of problems I've heard of with them suggest the "feats of engineering heroics" of which Adams spoke, but directed singlemindedly in the direction of compactness.
|
|
State lawyers contend that Democrats have no legal right to bring the challenge and that the maps are in compliance with traditional redistricting principles such as compactness and contiguity.
|
|
My one grudge with the Gear S3 was that it lost the elegance and compactness of the Gear S3493, and it was trying to appear small when it truly wasn't.
|
|
Compactness was the guiding ethos behind the Eggo design; the headphones were introduced the same month as the very first portable MiniDisc player and were intended to be used outside.
|
|
Since the districts could be justified under traditional redistricting criteria like compactness, contiguity, incumbency protection and political considerations, the court said, race could not have been the predominant reason for drawing them.
|
|
The new map "is composed of congressional districts which follow the traditional redistricting criteria of compactness, contiguity, equality of population, and respect for the integrity of political subdivisions," the court's order said.
|
|
The existing literature on compactness by nonmathematicians is filled with elementary errors and oversights, Duchin said, such as comparing two measures statistically without realizing that they are essentially the same measure in disguise.
|
|
The new state map "is composed of congressional districts which follow the traditional redistricting criteria of compactness, contiguity, equality of population, and respect for the integrity of political subdivisions," a court order said.
|
|
The RX18818893 VII's compactness isn't just impressive because of the large sensor it packs inside, however; you also get an EVF, an integrated flash, an external microphone jack and an articulating LCD display.
|
|
Justice Kagan laid those complaints to rest, pointing to "baseline" districting criteria that some states have adopted to guide judicial decision-making — contiguity and compactness of districts, no dividing of counties, the state's geography.
|
|
It gives both parties a similar chance to translate their votes to seats, and makes no compromises to do so; it still admirably adheres to standard nonpartisan criteria like compactness or minimizing county splits.
|
|
During a quick two-night trip, taking the ferry across the Pearl River Estuary, I found it was the perfect place for a getaway from the noise and intense urban compactness of Hong Kong.
|
|
As Fedor detailed in a Medium post, algorithmically optimizing district boundaries for compactness (such as Bdistricting) or randomly generating the boundaries actually leads to districts that are more gerrymandered than the boundaries drafted up by politicians.
|
|
The Automat focused on compactness, but it could do that because it shot on Instax Mini; I've grown to love the little ones but I can't say I'd mind a little extra space to work with either.
|
|
Oppo Digital, sister company to Oppo the smartphone maker, has the well regarded PM3 headphones, which cost $399, but don't come with the same aesthetic flair and compactness as the Audeze Sine, which were designed by BMW DesignWorks.
|
|
I'm probably the most vocal supporter of small smartphones outside of HP Veer fan forums, and Apple's new 4-inch device has rekindled my love of small things and served as a reminder of my preference for compactness.
|
|
"There were maps available to this committee on the various factors that this committee claimed to be pursuing," such as compactness of districts and respect for geographic features, Senator Natasha Marcus, a suburban Charlotte Democrat, said in an interview.
|
|
When FiveThirtyEight's Gerrymandering Project drew seven sets of nationwide U.S. House maps, using such neutral criteria as partisan proportionality, compactness, racial and ethnic fairness, and competitiveness, improvements in one criterion inevitably came at the price of declines in the other.
|
|
If these two requirements are met, Stephanopoulos and McGhee propose, the burden then falls to the state to explain why it created such a biased plan; perhaps, the state could argue, other considerations such as compactness and preservation of boundaries tied its hands.
|
|
Memory sticks and storage drivesImage: AppleAs you would expect, external storage drives and memory sticks have been quick to move over to the USB-C standard for its compactness and associated speed boosts (the latest USB 3.1 standard always uses USB-C).
|
|
As Altman and McDonald acknowledge, the computational limits of the redistricting problem could be addressed if the non-partisan criteria for redistricting was greatly simplified and ranked (for instance, equal populations are more important than compactness, contiguity more important than the rest, etc.).
|
|
Everything contributed to the dreamlike atmosphere: her soft voice and sure touch, the fact that I was lying on the bed, and the compactness of Tokyo hotel rooms, which meant that she periodically had to move things around to make enough room to stand.
|
|
In its press release, the company pointed out that despite its compactness, the ZenBook 3 still has a backlit keyboard with a 19.8mm key pitch, comparable to most desktop keyboards (the Apple MacBook's keyboard has a a butterfly mechanism, and is significantly thinner than regular keyboards).
|
|
With the approval of 85033 percent of voters, Missouri adopted a different approach, authorizing a non-partisan state demographer (selected by the state auditor, state Senate majority and minority leaders) to draw state legislative districts, taking into consideration partisan fairness and competitiveness, contiguousness, compactness, and the boundaries of political subdivisions.
|
|
But by spreading their algorithm's tasks across a massive number of processors, Cho's team found a way to create millions or even billions of what they call "reasonably imperfect" maps—ones that perform at least as well as the original map on whatever nonpartisan measures (such as compactness) a court might be interested in.
|
|
A topological space X is said to be limit point compact if every infinite subset of X has a limit point in X, and countably compact if every countable open cover has a finite subcover. In a metric space, the notions of sequential compactness, limit point compactness, countable compactness and compactness are all equivalent (if one assumes the axiom of choice). In a sequential (Hausdorff) space sequential compactness is equivalent to countable compactness.Engelking, General Topology, Theorem 3.10.
|
|
Various definitions of compactness may apply, depending on the level of generality. A subset of Euclidean space in particular is called compact if it is closed and bounded. This implies, by the Bolzano–Weierstrass theorem, that any infinite sequence from the set has a subsequence that converges to a point in the set. Various equivalent notions of compactness, such as sequential compactness and limit point compactness, can be developed in general metric spaces.
|
|
Another example is the definition of distributions, which uses the space of smooth functions that are zero outside of some (unspecified) compact space. Various equivalent notions of compactness, including sequential compactness and limit point compactness, can be developed in general metric spaces. In general topological spaces, however, different notions of compactness are not necessarily equivalent. The most useful notion, which is the standard definition of the unqualified term compactness, is phrased in terms of the existence of finite families of open sets that "cover" the space in the sense that each point of the space lies in some set contained in the family.
|
|
In mathematics, Mumford's compactness theorem states that the space of compact Riemann surfaces of fixed genus g > 1 with no closed geodesics of length less than some fixed ε > 0 in the Poincaré metric is compact. It was proved by as a consequence of a theorem about the compactness of sets of discrete subgroups of semisimple Lie groups generalizing Mahler's compactness theorem.
|
|
Likewise, it is analogous to the finite intersection property characterization of compactness in topological spaces: a collection of closed sets in a compact space has a non-empty intersection if every finite subcollection has a non-empty intersection. The compactness theorem is one of the two key properties, along with the downward Löwenheim–Skolem theorem, that is used in Lindström's theorem to characterize first-order logic. Although, there are some generalizations of the compactness theorem to non-first-order logics, the compactness theorem itself does not hold in them.
|
|
In second-countable spaces--as in metric spaces--compactness, sequential compactness, and countable compactness are all equivalent properties. Urysohn's metrization theorem states that every second-countable, Hausdorff regular space is metrizable. It follows that every such space is completely normal as well as paracompact. Second-countability is therefore a rather restrictive property on a topological space, requiring only a separation axiom to imply metrizability.
|
|
In mathematical logic, the Barwise compactness theorem, named after Jon Barwise, is a generalization of the usual compactness theorem for first-order logic to a certain class of infinitary languages. It was stated and proved by Barwise in 1967.
|
|
In probability theory, the result implies compactness of a tight family of measures.
|
|
A common use of compactness measures is in redistricting. The goal is to maximize the compactness of electoral districts, subject to other constraints, and thereby to avoid gerrymandering.Rick Gillman "Geometry and Gerrymandering", Math Horizons, Vol. 10, #1 (Sep, 2002) 10-13.
|
|
The effects among people were that it instilled isolationism, xenophobia, slavophobia, linguistic uniformity and ethnic compactness..
|
|
Like many compactness conditions, compactly generated spaces are often assumed to be Hausdorff or weakly Hausdorff.
|
|
The Reock degree of compactness, or Reock compactness score, is a ratio that quantifies how compact is the geographic area of a voting district. The score is sometimes used as an indication of the extent to which a voting district may be considered gerrymandered. The Reock compactness score is computed by dividing the area of the voting district by the area of the smallest circle that would completely enclose it. Since the circle encloses the district, its area cannot be less than that of the district, and so the Reock compactness score will always be a number between zero and one (which may be expressed as a percentage).
|
|
His first work on this subject, in his thesis, concerned the compactness degree of a space: this is a number, defined to be −1 for a compact space, and 1 + x if every point in the space has a neighbourhood the boundary of which has compactness degree x. He made an important conjecture, only solved much later in 1982 by Pol and 1988 by Kimura, that the compactness degree was the same as the minimum dimension of a set that could be adjoined to the space to compactify it. Thus, for instance the familiar Euclidean space has compactness degree zero; it is not compact itself, but every point has a neighborhood bounded by a compact sphere. This compactness degree, zero, equals the dimension of the single point that may be added to Euclidean space to form its one-point compactification.
|
|
With this configuration, the Linutop 6 computer targets a varied use where compactness and power are required.
|
|
52 (for R), p. 300 (for Rn). The theorem is sometimes called the sequential compactness theorem.Fitzpatrick 2006, p. xiv.
|
|
Moreover, the compactness of the device's dimensions drove the propellant reaction to complete outside of the rocket, misemploying energy.
|
|
There is evidence that compactness is one of the basic dimensions of shape features extracted by the human visual system.
|
|
As noted in the previous paragraph, second- order comprehension axioms easily generalize to the higher-order framework. However, theorems expressing the compactness of basic spaces behave quite differently in second- and higher-order arithmetic: on one hand, when restricted to countable covers/the language of second-order arithmetic, the compactness of the unit interval is provable in WKL0 from the next section. On the other hand, given uncountable covers/the language of higher-order arithmetic, the compactness of the unit interval is only provable from (full) second-order arithmetic (). Other covering lemmas (e.g.
|
|
Second-order logic, however, fails to retain many desirable properties of first-order logic, such as the completeness and compactness theorems.
|
|
In 1889, Italian mathematician Cesare Arzelà generalized Ascoli's Theorem into the Arzelà–Ascoli theorem, a practical sequential compactness criterion of functions.See .
|
|
Because the Reock compactness score is defined in terms of a circle that must enclose all points of a district, it is sensitive to the orientations of the district's extremities. Even a very unnatural shape (for example, a "coiled snake") may have a high Reock compactness score so long as it fits compactly within the circumscribing circle.
|
|
The PEMFC is a prime candidate for vehicle and other mobile applications of all sizes down to mobile phones, because of its compactness.
|
|
The theorem depends crucially upon the precise definitions of compactness and of the product topology; in fact, Tychonoff's 1935 paper defines the product topology for the first time. Conversely, part of its importance is to give confidence that these particular definitions are the most useful (i.e. most well-behaved) ones. Indeed, the Heine–Borel definition of compactness—that every covering of a space by open sets admits a finite subcovering—is relatively recent. More popular in the 19th and early 20th centuries was the Bolzano–Weierstrass criterion that every sequence admits a convergent subsequence, now called sequential compactness.
|
|
A metric space M is compact if every sequence in M has a subsequence that converges to a point in M. This is known as sequential compactness and, in metric spaces (but not in general topological spaces), is equivalent to the topological notions of countable compactness and compactness defined via open covers. Examples of compact metric spaces include the closed interval [0,1] with the absolute value metric, all metric spaces with finitely many points, and the Cantor set. Every closed subset of a compact space is itself compact. A metric space is compact if and only if it is complete and totally bounded.
|
|
The Polsby–Popper test is a mathematical compactness measure of a shape developed to quantify the degree of gerrymandering of political districts. The method was developed by lawyers Daniel D. Polsby and Robert Popper, Polsby, Daniel D., and Robert D. Popper. 1991. “The Third Criterion: Compactness as a procedural safeguard against partisan gerrymandering.” Yale Law & Policy Review 9 (2): 301–353.
|
|
In contrast, the different notions of compactness are not equivalent in general topological spaces, and the most useful notion of compactness--originally called bicompactness--is defined using covers consisting of open sets (see Open cover definition below). That this form of compactness holds for closed and bounded subsets of Euclidean space is known as the Heine–Borel theorem. Compactness, when defined in this manner, often allows one to take information that is known locally--in a neighbourhood of each point of the space--and to extend it to information that holds globally throughout the space. An example of this phenomenon is Dirichlet's theorem, to which it was originally applied by Heine, that a continuous function on a compact interval is uniformly continuous; here, continuity is a local property of the function, and uniform continuity the corresponding global property.
|
|
The air-solar structure allows an acceptable heat input also during cloudy days, maintaining the compactness of the system and the easiness to install it.
|
|
The Delta-compactness theorem of T. C. Lim states that if (X,d) is an asymptotically complete metric space, then every bounded sequence in X has a Delta-convergent subsequence. The Delta- compactness theorem is similar to the Banach–Alaoglu theorem for weak convergence but, unlike the Banach-Alaoglu theorem (in the non-separable case) its proof does not depend on the Axiom of Choice.
|
|
Lindström's theorem implies that the only extension of first-order logic satisfying both the compactness theorem and the downward Löwenheim–Skolem theorem is first-order logic.
|
|
A folding chain progresses toward lower intra-chain free-energies by increasing its compactness. The chain's conformational options become increasingly narrowed ultimately toward one native structure.
|
|
The placement of the transmission in the rear allowed for the driver and passenger seats to be placed closer together, adding to the compactness of the Cobra.
|
|
It is a central tool in model theory, providing a fundamental method for constructing models. The compactness theorem has a limiting effect on which collections of first-order structures are elementary classes. For example, the compactness theorem implies that any theory that has arbitrarily large finite models has an infinite model. Thus the class of all finite graphs is not an elementary class (the same holds for many other algebraic structures).
|
|
All the methods below satisfy the first four core properties of GDM (coercivity, GD-consistency, limit-conformity, compactness), and in some cases the fifth one (piecewise constant reconstruction).
|
|
Conversely, for many deductive systems, it is possible to prove the completeness theorem as an effective consequence of the compactness theorem. The ineffectiveness of the completeness theorem can be measured along the lines of reverse mathematics. When considered over a countable language, the completeness and compactness theorems are equivalent to each other and equivalent to a weak form of choice known as weak König's lemma, with the equivalence provable in RCA0 (a second- order variant of Peano arithmetic restricted to induction over Σ01 formulas). Weak König's lemma is provable in ZF, the system of Zermelo–Fraenkel set theory without axiom of choice, and thus the completeness and compactness theorems for countable languages are provable in ZF. However the situation is different when the language is of arbitrary large cardinality since then, though the completeness and compactness theorems remain provably equivalent to each other in ZF, they are also provably equivalent to a weak form of the axiom of choice known as the ultrafilter lemma.
|
|
This expressiveness comes at a metalogical cost, however: by Lindström's theorem, the compactness theorem and the downward Löwenheim–Skolem theorem cannot hold in any logic stronger than first-order.
|
|
Information about China Refrigeration Exhibition. Accessed on 9 November 2014. Secop presented the X Series for R600a. The XV received an Innovation Award for its compactness and low weight.
|
|
However, the compactness theorem can be used to show that connected graphs are not an elementary class in first-order logic, and there is no formula φ(x,y) of first-order logic, in the logic of graphs, that expresses the idea that there is a path from x to y. Connectedness can be expressed in second-order logic, however, but not with only existential set quantifiers, as \Sigma_1^1 also enjoys compactness.
|
|
For any metric space , the following are equivalent (assuming countable choice): # is compact. # is complete and totally bounded (this is also equivalent to compactness for uniform spaces). # is sequentially compact; that is, every sequence in has a convergent subsequence whose limit is in (this is also equivalent to compactness for first-countable uniform spaces). # is limit point compact (also called countably compact); that is, every infinite subset of has at least one limit point in .
|
|
In mathematics, a topological space X is sequentially compact if every sequence of points in X has a convergent subsequence converging to a point in X. Every metric space is naturally a topological space, and for metric spaces, the notions of compactness and sequential compactness are equivalent (if one assumes countable choice). However, there exist sequentially compact topological spaces that are not compact, and compact topological spaces that are not sequentially compact.
|
|
Historically, these sentences were proved in reverse order. In 1957, James had proved the reflexivity criterion for separable Banach spaces and 1964 for general Banach spaces. Since the reflexivity is equivalent to the weak compactness of the unit sphere, Victor L. Klee reformulated this as a compactness criterion for the unit sphere in 1962 and assumes that this criterion characterizes any weakly compact quantities. This was then actually proved by RC James in 1964.
|
|
The Jordan–Hölder theorem also holds in the context of operator groups. The requirement that a group have a composition series is analogous to that of compactness in topology, and can sometimes be too strong a requirement. It is natural to talk about "compactness relative to a set", i.e. talk about composition series where each (normal) subgroup is an operator-subgroup relative to the operator set X, of the group in question.
|
|
The 2024 Budapest Olympic bid sought to take advantage of the compactness of the city and co-location of venues along the River Danube and others around the Budapest area.
|
|
Researchers have also considered Hilbert's fifth problem without supposing finite dimensionality. The last chapter of Benyamini and Lindenstrauss discuss the thesis of Per Enflo, on Hilbert's fifth problem without compactness.
|
|
Marveled at for its compactness, "Fire and Ice" signaled for Frost "a new style, tone, manner, [and] form." Its casual tone masks the serious question it poses to the reader.
|
|
In particular, no theory extending ZF can prove either the completeness or compactness theorems over arbitrary (possibly uncountable) languages without also proving the ultrafilter lemma on a set of same cardinality.
|
|
It initially began with cassettes and over a period of time, has switched to CDs keeping in mind the durability, the better sound quality and their compactness that enables easy storage.
|
|
The completeness theorem and the compactness theorem are two cornerstones of first-order logic. While neither of these theorems can be proven in a completely effective manner, each one can be effectively obtained from the other. The compactness theorem says that if a formula φ is a logical consequence of a (possibly infinite) set of formulas Γ then it is a logical consequence of a finite subset of Γ. This is an immediate consequence of the completeness theorem, because only a finite number of axioms from Γ can be mentioned in a formal deduction of φ, and the soundness of the deductive system then implies φ is a logical consequence of this finite set. This proof of the compactness theorem is originally due to Gödel.
|
|
Similarly, thermostable proteins are rich in salt bridges or/and extra disulfide bridges stabilizing the structure. Other factors of protein thermostability are compactness of protein structure, oligomerization, and strength interaction between subunits.
|
|
However, the sensitivity of the fiber optic gyro is enhanced by having a long optical fiber, coiled for compactness, in which the Sagnac effect is multiplied according to the number of turns.
|
|
Neither of these results are provable in ZFC alone. Finally, some questions arising from model theory (such as compactness for infinitary logics) have been shown to be equivalent to large cardinal axioms.
|
|
Watercolor on card. Sultan's drawings, such as his self-portrait, are characterized by their economy and compactness. The lines are powerful and fully developed. His early paintings were influenced by the Impressionists.
|
|
A SAT file has a .sat file extension. SAB files cannot be viewed with a simple text editor and are meant for compactness and not for human readability. A SAB file has a .
|
|
Kurt Gödel proved the countable compactness theorem in 1930. Anatoly Maltsev proved the uncountable case in 1936.Vaught, Robert L.: "Alfred Tarski's work in model theory". Journal of Symbolic Logic 51 (1986), no.
|
|
Since Prokhorov's theorem expresses tightness in terms of compactness, the Arzelà–Ascoli theorem is often used to substitute for compactness: in function spaces, this leads to a characterization of tightness in terms of the modulus of continuity or an appropriate analogue—see tightness in classical Wiener space and tightness in Skorokhod space. There are several deep and non-trivial extensions to Prokhorov's theorem. However, those results do not overshadow the importance and the relevance to applications of the original result.
|
|
These fundamental open sets (for all possible configurations ) form an open cover of the shift space. However, the shift space is a compact space: it is a product of finite topological spaces with the alphabet as their points, so compactness follows from Tychonoff's theorem. By compactness, every open cover has a finite subcover. The finite set of positions appearing in this finite subcover may be used as the neighborhood of position zero in a description of as a cellular automaton rule.
|
|
In 1995, Hamilton extended Jeff Cheeger's compactness theory for Riemannian manifolds to give a compactness theorem for sequences of Ricci flows. Given a Ricci flow on a closed manifold with a finite-time singularity, Hamilton developed methods of rescaling around the singularity to produce a sequence of Ricci flows; the compactness theory ensures the existence of a limiting Ricci flow, which models the small-scale geometry of a Ricci flow around a singular point. Hamilton used his maximum principles to prove that, for any Ricci flow on a closed three-dimensional manifold, the smallest value of the sectional curvature is small compared to its largest value. This is known as the Hamilton-Ivey estimate; it is extremely significant as a curvature inequality which holds with no conditional assumptions beyond three-dimensionality.
|
|
The compactness measure of a shape is a numerical quantity representing the degree to which a shape is compact. The meaning of "compact" here is not related to the topological notion of compact space.
|
|
In the mathematical field of functional analysis, the Eberlein–Šmulian theorem (named after William Frederick Eberlein and Witold Lwowitsch Schmulian) is a result that relates three different kinds of weak compactness in a Banach space.
|
|
An important consequence is that, in three dimensions, a limiting Ricci flow as produced by the compactness theory automatically has nonnegative curvature. As such, Hamilton's Harnack inequality is applicable to the limiting Ricci flow. These methods were extended by Grigori Perelman, who due to his "noncollapsing theorem" was able to apply Hamilton's compactness theory in a number of extended contexts. In 1997, Hamilton was able to combine the methods he had developed to define "Ricci flow with surgery" for four-dimensional Riemannian manifolds of positive isotropic curvature.
|
|
There is a similarity between the definitions of compactness and paracompactness: For paracompactness, "subcover" is replaced by "open refinement" and "finite" by is replaced by "locally finite". Both of these changes are significant: if we take the definition of paracompact and change "open refinement" back to "subcover", or "locally finite" back to "finite", we end up with the compact spaces in both cases. Paracompactness has little to do with the notion of compactness, but rather more to do with breaking up topological space entities into manageable pieces.
|
|
One says that a group has the complete reducibility property if every representation decomposes as a direct sum of irreducible representations. But, the non-compactness of the Lorentz group, in combination with lack of simple connectedness, cannot be dealt with in all the aspects as in the simple framework that applies to simply connected, compact groups. Non-compactness implies, for a connected simple Lie group, that no nontrivial finite-dimensional unitary representations exist. Lack of simple connectedness gives rise to spin representations of the group.
|
|
Alternatively, expansive working can also be provided by means of Woolf compounding, just as compactly if the two expansion stages are provided on either side of a single piston within a single cylinder, all arranged as a trunk engine (however, an oscillating cylinder engine does not benefit from the advantage of compactness that a trunk engine would otherwise provide, as it has that already from its own design features). Each approach to expansive working compromises the advantage of simplicity but still retains the advantage of compactness.
|
|
Gromov formulated an important compactness theorem in this setting, giving a condition under which a sequence of pointed and "proper" metric spaces must have a subsequence which converges. This was later reformulated by Gromov and others into the more flexible notion of an ultralimit. Gromov's compactness theorem had a deep impact on the field of geometric group theory. He applied it to understand the asymptotic geometry of the word metric of a group of polynomial growth, by taking the limit of well-chosen rescalings of the metric.
|
|
For raster shapes, i.e. shapes composed of pixels or cells, some tests involve distinguishing between exterior and interior edges (or faces). More sophisticated measures of compactness include calculating the shape's moment of inertia or boundary curvature.
|
|
Lindström's theorem states that first-order logic is the strongest (subject to certain constraints) logic satisfying both compactness and completeness. A completeness theorem can be proved for modal logic or intuitionistic logic with respect to Kripke semantics.
|
|
This diagram shows the Ananke group on the same scale as the other diagram, illustrating its wide dispersion in comparison with the compact Carme group (see related diagram). This diagram shows the compactness of the Carme group.
|
|
Another significant change was the exhaust system, which was no longer a center-up underseat design. The new exhaust was a side-slung design in order to increase mass centralization and compactness while mimicking a MotoGP-style.
|
|
The wz.37 was dubbed "Szczeniak" (Polish for pup) due to its lightness and compactness compared with previous Vickers E and Vickers F machine guns. The wz.37 was used mostly in Polish PZL.37 Łoś bombers.
|
|
It was published after several important applications of those definitions had already been discovered, and has been called a "classical reference paper". Her only publication with her advisor, Alfred Tarski, was a brief announcement of related research using reduced products in connection with the compactness theorem in mathematical logic. Among other results, it provided a proof of the compactness theorem using ultraproducts. With Chen Chung Chang, she also used reduced products to show that a sufficient condition for properties to be preserved under direct products, derived by Alfred Horn, was not also a necessary condition.
|
|
In a non-Hausdorff space, such as the particular point topology on an infinite set, the closure of a compact subset is not necessarily compact; said differently, a compact subset of a non-Hausdorff space is not necessarily relatively compact. In the case of a metric topology, or more generally when sequences may be used to test for compactness, the criterion for relative compactness becomes that any sequence in has a subsequence convergent in . Some major theorems characterize relatively compact subsets, in particular in function spaces. An example is the Arzelà–Ascoli theorem.
|
|
According to the Brouwer fixed-point theorem, every compact and convex subset of a Euclidean space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP. In 1932 Borsuk asked whether compactness together with contractibility could be a necessary and sufficient condition for the FPP to hold. The problem was open for 20 years until the conjecture was disproved by Kinoshita who found an example of a compact contractible space without the FPP.
|
|
The search claim for the mountain bike was its weight, the experience claim was its ease of control. The search claim for the camera was its compactness and the experience claim was its photo quality. The results indicated that consumers were more likely to believe (and be satisfied with) claims if they thought the source was trustworthy or had experience with the product. They were more likely to believe advertisements with concrete evidence behind the claims, such as the weight of the bike or the compactness of the camera.
|
|
In mathematical logic, the compactness theorem states that a set of first- order sentences has a model if and only if every finite subset of it has a model. This theorem is an important tool in model theory, as it provides a useful method for constructing models of any set of sentences that is finitely consistent. The compactness theorem for the propositional calculus is a consequence of Tychonoff's theorem (which says that the product of compact spaces is compact) applied to compact Stone spaces,See Truss (1997). hence the theorem's name.
|
|
Review of . Duchin's expertise in geometry has led her to conduct research on the mathematics of gerrymandering. A key aspect of this research is the geometric notion of the compactness of a given political district, a numerical measure that attempts to quantify how extensively gerrymandered it is. “What courts have been looking for is one definition of compactness that they can understand, that we can compute, and that they can use as a kind of go-to standard”, she said in an interview with The Chronicle of Higher Education.
|
|
The study also found that the animal had reached sexual maturity at the age of 13 to 15 years, due to a decrease in growth rate at this point. In 2018, the Brazilian palaeontologist Tito Aureliano and colleagues reported a spinosaurid tibia from Brazil which exhibited high compactness of the bone, a feature which is correlated with semi-aquatic habits in tetrapods; it is used for ballast to reduce buoyancy caused by the air-filled lungs. Mammal groups with such bone compactness are adapted for living in shallow water.
|
|
The following is a list of champions, along with runner-ups and semifinalists, within the Evolution Championship Series. For table compactness, only the top 4 are included. Additional details are available in the individual event articles by year.
|
|
Still, the compactness and total performance make some good reputation for this kit lens. This lens is one of the few not billed by Olympus as Movie and Still Compatible ("MSC"), due to noise generated by its focus motor.
|
|
The brakes need light pedal pressure. They tended to grab and affect the steering but this might have been on just this particular car. The synchromesh works well. "The outstanding characteristic in the appearance of this car is compactness".
|
|
Enchanting brings NXT programming into the popular Scratch IDE, designed by the Lifelong Kindergarten Group at MIT to make programming intuitive even for young children. The resulting NXT programs have the compactness and clarity offered by that programming environment.
|
|
The designers of the Java programming language try to take advantage of the compactness of interpreted code, because a Java program may need to be transmitted over the Internet before execution can start on the target's Java Virtual Machine.
|
|
In mathematics, the Bishop–Gromov inequality is a comparison theorem in Riemannian geometry, named after Richard L. Bishop and Mikhail Gromov. It is closely related to Myers' theorem, and is the key point in the proof of Gromov's compactness theorem.
|
|
In mathematics, particularly in functional analysis, the Krein-Smuilian theorem can refer to two theorems relating the closed convex hull and compactness in the weak topology. They are named after Mark Krein and Vitold Shmulyan, who published them in 1940.
|
|
Assuming compactness (of the given space, and therefore also the function space) one may extend it to a regular Borel measure. After completion one gets a probability space that is not necessarily standard.Its standardness is investigated in: . See Theorem 1(c).
|
|
There are various ways of encoding causation, which form somewhat of a continuum of "compactness."Dixon, R.M.W. 2000. “A typology of causatives: form, syntax and meaning”. In Changing Valency: Case Studies in Transitivity, Dixon, R.M.W. and Alexandra Y. Aikhenvald, eds.
|
|
The protein encoded by this gene, a member of the HMGN protein family, is thought to reduce the compactness of the chromatin fiber in nucleosomes, thereby enhancing transcription from chromatin templates. Transcript variants utilizing alternative polyadenylation signals exist for this gene.
|
|
Sarkar, pp.556–559. Early stripline directly coupled resonator filters were end-coupled, but the length was reduced and the compactness successively increased with the introduction of parallel-coupled line filters,Cohn, 1958. interdigital filters,Matthaei, 1962. and comb-line filters.
|
|
These first generation free- piston engines were without exception opposed piston engines, in which the two pistons were mechanically linked to ensure symmetric motion. The free-piston engines provided some advantages over conventional technology, including compactness and a vibration-free design.
|
|
A view of the Rajasthani Paintings Sections Rajasthani Miniatures flourished mainly in Mewar, Bundi, Kota, Kishangarh, Jaipur, Jodhpur and Bikaner. Mewar Miniatures are illustrating Hindu mythological themes. Bundi and Kota Miniatures excel in composition compactness. Hunting scenes are Kota's speciality.
|
|
For example, ultrapowers can be used to construct new fields from given ones. The hyperreal numbers, an ultrapower of the real numbers, are a special case of this. Some striking applications of ultraproducts include very elegant proofs of the compactness theorem and the completeness theorem, Keisler's ultrapower theorem, which gives an algebraic characterization of the semantic notion of elementary equivalence, and the Robinson–Zakon presentation of the use of superstructures and their monomorphisms to construct nonstandard models of analysis, leading to the growth of the area of nonstandard analysis, which was pioneered (as an application of the compactness theorem) by Abraham Robinson.
|
|
The fact that there is no standard definition of what a domain really is has meant that domain assignments have varied enormously, with each researcher using a unique set of criteria. A structural domain is a compact, globular sub-structure with more interactions within it than with the rest of the protein. Therefore, a structural domain can be determined by two visual characteristics: its compactness and its extent of isolation. Measures of local compactness in proteins have been used in many of the early methods of domain assignment and in several of the more recent methods.
|
|
In mathematics, especially topology, a perfect map is a particular kind of continuous function between topological spaces. Perfect maps are weaker than homeomorphisms, but strong enough to preserve some topological properties such as local compactness that are not always preserved by continuous maps.
|
|
Torri Superiore has been called by some a labyrinth or a fortress, and offers a remarkable interest for its compactness and its state of preservation. Today it is mostly restored and open to ecologically minded visitors for cultural initiatives and group activities.
|
|
The side is commented to fare well in most big games, with compactness and discipline springing surprises. Instead of being burdened with keeping possession, they stifled opponents with a rehearsed repertoire of both defensive and pressing moves, topped off with lethal counter-attacks.
|
|
People in the hamlets were impressed by "compactness, dignity, and simple churchly beauty" of the Chapel Car. They were also impressed by Walker what did. He lived in the car, did his own cooking, and cleaned both his living quarters and the chapel.
|
|
Examples of geons include cones and spheres. A wide range of other shape representations have also been investigated. Shape features seem to boil down to three basic dimensions: segmentability, compactness, and spikiness. There is also clear evidence that shapes guide human attention.
|
|
Svendsen, 12.1 The asymmetrical massing and the compactness of the structure emphasized the strong verticality of the structure. The rather plain brick exterior was punctuated with gothic-arched windows that were enhanced with narrow hood molds. An addition to the building was constructed in 1919.
|
|
Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995), 319–337.(Theorem 1), or by ad-hoc monikers such as vanishing lemma or inverse embedding.
|
|
The King is a dual purpose breed that originated in the United States. They were developed during the 1890s by crossing four older varieties. The Duchess for grace; The Homer for alertness; The Maltese for compactness and style: the Runt for body and size.
|
|
Such a homeomorphism is given by using ternary notation of numbers. (See Cantor space.) In the foundations of mathematics, the study of compactness properties of products of {0,1} is central to the topological approach to the ultrafilter principle, which is a weak form of choice.
|
|
The dots can be discarded for compactness without loss of information. As with typical FASTA used in alignments, the gap ("") is taken to mean exactly one position. A3M is similar to A2M, with the added rule that gaps aligned to insertions can too be discarded.
|
|
The (downward) Löwenheim–Skolem theorem is one of the two key properties, along with the compactness theorem, that are used in Lindström's theorem to characterize first-order logic. In general, the Löwenheim–Skolem theorem does not hold in stronger logics such as second-order logic.
|
|
In the calculus of variations, where one is typically interested in infinite-dimensional function spaces, the condition is necessary because some extra notion of compactness beyond simple boundedness is needed. See, for example, the proof of the mountain pass theorem in section 8.5 of Evans.
|
|
Pyrrolidine is a base. Its basicity is typical of other dialkyl amines. Relative to many secondary amines, pyrrolidine is distinctive because of its compactness, a consequence of its cyclic structure. Pyrrolidine is used as a building block in the synthesis of more complex organic compounds.
|
|
This is one instance of the geometric unsuitability of the Zariski topology. Grothendieck solved this problem by defining the notion of properness of a scheme (actually, of a morphism of schemes), which recovers the intuitive idea of compactness: Proj is proper, but Spec is not.
|
|
One of the earliest examples of a non-quasiprojective algebraic variety were given by Nagata. Nagata's example was not complete (the analog of compactness), but soon afterwards he found an algebraic surface that was complete and non- projective. Since then other examples have been found.
|
|
Physical soap films are more accurately modeled by the (M, 0, \Delta)-minimal sets of Frederick Almgren, but the lack of a compactness theorem makes it difficult to prove the existence of an area minimizer. In this context, a persistent open question has been the existence of a least-area soap film. Ernst Robert Reifenberg solved such a "universal Plateau's problem" for boundaries which are homeomorphic to single embedded spheres. In his book Almgren claimed to use varifolds to solve the problem for more than one sphere, as well as more general boundaries, but Allard's compactness theorem for integral varifolds produces a minimal surface, not necessarily an area minimizer.
|
|
The cost of this expressiveness is that second-order and higher-order logics have fewer attractive metalogical properties than first-order logic. For example, the Löwenheim–Skolem theorem and compactness theorem of first-order logic become false when generalized to higher-order logics with full semantics.
|
|
Condition of asymptotic completeness in the Delta-compactness theorem is satisfied by uniformly convex Banach spaces, and more generally, by uniformly rotund metric spaces as defined by J. Staples.J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976), 181–192.
|
|
Clearly every basic elementary class is an elementary class, and every elementary class is a pseudo-elementary class. Moreover, as an easy consequence of the compactness theorem, a class of σ-structures is basic elementary if and only if it is elementary and its complement is also elementary.
|
|
Ariadne was, in the main, well received. "An exception to the generalisation that all modern epics are tedious," declared The Cambridge Review. "Its plot is a model of epic construction in its compactness, directness and speed." Julian Bell, review of Ariadne in The Cambridge Review, 10 February 1933.
|
|
The grape bunch is large in size and has an average compactness. It can grow in two different shapes: cylindrical or as a long cone. The grapes are large and spherical and have a yellowish color. The grapes are late to bud burst and also late to ripen.
|
|
The compactness theorem first appeared as a lemma in Gödel's proof of the completeness theorem, and it took many years before logicians grasped its significance and began to apply it routinely. It says that a set of sentences has a model if and only if every finite subset has a model, or in other words that an inconsistent set of formulas must have a finite inconsistent subset. The completeness and compactness theorems allow for sophisticated analysis of logical consequence in first-order logic and the development of model theory, and they are a key reason for the prominence of first-order logic in mathematics. Gödel's incompleteness theorems (Gödel 1931) establish additional limits on first- order axiomatizations.
|
|
Often, an object is dualizable only when it satisfies some finiteness or compactness property. A category in which each object has a dual is called autonomous or rigid. The category of finite- dimensional vector spaces with the standard tensor product is rigid, while the category of all vector spaces is not.
|
|
Elegance (compactness) versus goodness (speed): With only six core instructions, "Elegant" is the clear winner, compared to "Inelegant" at thirteen instructions. However, "Inelegant" is faster (it arrives at HALT in fewer steps). Algorithm analysiscf Knuth 1973:7 (Vol. I), and his more-detailed analyses on pp. 1969:294–313 (Vol II).
|
|
Not only for lack of power of character but the diversion of my mental attitude. The novel ends with Shaker's migration from Kolkata to Dhaka. In the novel Karim has delineated the subconscious mind of the humans more acutely than the picture of the society and the compactness of the story.
|
|
In the fifth chapter, the same concepts are used to give a non-standard semantics to type theory. After these chapters on other types of logic, the final two chapters introduce many-sorted logic, prove its soundness, completeness, and compactness, and describe how to translate the other forms of logic into it.
|
|
If one global condition, namely compactness, is added, the surface is necessarily algebraic. This feature of Riemann surfaces allows one to study them with either the means of analytic or algebraic geometry. The corresponding statement for higher-dimensional objects is false, i.e. there are compact complex 2-manifolds which are not algebraic.
|
|
It was Maurice Fréchet who, in 1906, had distilled the essence of the Bolzano–Weierstrass property and coined the term compactness to refer to this general phenomenon (he used the term already in his 1904 paperFrechet, M. 1904. Generalisation d'un theorem de Weierstrass. Analyse Mathematique. which led to the famous 1906 thesis).
|
|
Other widely used techniques for proving inexpressibility results, such as the compactness theorem, do not work in finite models. Ehrenfeucht–Fraïssé-like games can also be defined for other logics, such as fixpoint logics and pebble games for finite variable logics; extensions are powerful enough to characterise definability in existential second-order logic.
|
|
NDS separates navigation software from navigation data, thus enhancing flexibility for creating various navigation products for end users. In addition to this interoperability, NDS databases support incremental updates, protection against illegal use, and compactness. NDS products have been available in the market since 2012, among others in BMW, Daimler, and Volkswagen cars.
|
|
These changes improved the frame's rigidity by 30%. The cooling system was redesigned for better performance and compactness. The exhaust system was changed from a 4-into-1 to a new titanium 4-into-2-into-1 design. The rear end of the motorcycle was updated and streamlined with a LED taillight.
|
|
Congress has allowed those requirements to lapse,See Wood v. Broom, 287 U.S. 1 (1932). but the Supreme Court has re-imposed the population requirement on the States under the Equal Protection Clause and is suspicious of districts that do not meet the other "traditional" districting criteria of compactness and contiguity.See, e.g.
|
|
Leonidas (Leon) Alaoglu (; March 19, 1914 - August 1981) was a mathematician, known for his result, called Alaoglu's theorem on the weak-star compactness of the closed unit ball in the dual of a normed space, also known as the Banach–Alaoglu theorem.American Men & Women of Science. 14th edition. New York: R.R. Bowker, 1979.
|
|
Other New York City Jewish Christmas Eve events have included parties for "the pro-Israel crowd, Jewish gays and lesbians, and downtown Jewish hipsters." The sheer number of events, combined with the compactness of Manhattan, means that events may be held within a short walking distance, if not eyesight of one another.
|
|
He worked on Diophantine approximation and geometry of numbers, where he used both classical and p-adic analytic methods.special issue of Annales de l'Institut Fourier (vol. XXIX, Fasc. 1), March 1979, for Chabauty's retirement He introduced the Chabauty topology to generalise Mahler's compactness theorem from Euclidean lattices to more general discrete subgroups.
|
|
Technologists stated that for such compactness and purity, it will take millions and millions of strokes. It has greater purity than the iron standard forged in space, which was brought from the United States to compare it to the needle. In 2005, another needle-shaped object was discovered, also dated to the 14th century BC.
|
|
Another interesting feature about the weapon is that it was bullpup in design, which means that the magazine is located behind the trigger, and also utilized the blowback system of operation. The bullpup design was a significant tactical advantage because it increased the firearm's compactness and allowed for a smaller, lighter and more efficient firearm.
|
|
McGraw- Hill, Inc., October 24, 1991. MPP was developed into cores by the Western Electric Company and the Bell Telephone Laboratory (formerly known as AT&T;) in the early 1940s. It has made its largest impact in the power conversion field by permitting increased frequency, resulting in weight reduction and increased compactness in computer systems.
|
|
First sold in 1925, the Leica was the first commercially successful camera to use 35 mm film. Its compactness and bright viewfinder, matched to lenses of quality (changeable on Leicas sold from 1930) helped photographers move through busy streets and capture fleeting moments.Michael Pritchard, "Leica I–III models", pp. 358–359 of Robin Lenman, ed.
|
|
A statement by Penn in a 1966 article has been widely quoted in recent years as an example of faulty prognostication. "Despite the trend to compactness and lower costs," wrote Penn, "it is unlikely everyone will have his own computer any time soon." In 1969 and 1970 he reported on the Newark, New Jersey Mafia.
|
|
In the category of sets, the inverse limit of any inverse system of non-empty finite sets is non-empty. This may be seen as a generalization of Kőnig's lemma and can be proved with Tychonoff's theorem, viewing the finite sets as compact discrete spaces, and then using the finite intersection property characterization of compactness.
|
|
In 1926, Witold Hurewicz introduced the above property of topological spaces that is formally stronger than the Menger property. He didn't know whether Menger's conjecture is true, and whether his property is strictly stronger than the Menger property, but he conjectured that in the class of metric spaces his property is equivalent to \sigma-compactness.
|
|
Binary Ordered Compression for Unicode (BOCU) is a MIME compatible Unicode compression scheme. BOCU-1 combines the wide applicability of UTF-8 with the compactness of Standard Compression Scheme for Unicode (SCSU). This Unicode encoding is designed to be useful for compressing short strings, and maintains code point order. BOCU-1 is specified in a Unicode Technical Note.
|
|
White matter carries signals between regions of the brain, and its compactness is linked to faster nerve activity. It generally deteriorates with age. Moreover, exercise, even in older adults, has been shown to increase white matter. Kramer has also been a lead investigator on studies of the relationships between brain structure and function and fluid intelligence.
|
|
As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues. If the compactness assumption is removed, it is not true that every self-adjoint operator has eigenvectors.
|
|
The company specializes in large aperture truss-pole Dobsonian telescopes. The OT telescopes have a set standard for telescopes among amateur astronomer telescopes. One of the advantages is the ultra-compact style, which allows higher mobility, with a compact size, relative to the diameter of the main mirror. The compactness is achieved by making the traditional Dobsonian design collapsible.
|
|
The grape is classified as a berry. On the vine, grapes are organized through systems known as clusters. Grape clusters can vary in compactness which can result in long clusters (resulting in the grapes spreading out) or short clusters (resulting in grapes packed together). In some grape species, clusters ripen collectively which allow them to be harvested together.
|
|
The Godement compactness criterion in the theory of arithmetic groups was a conjecture of his. He later worked with Jacquet on the zeta function of a simple algebra. He was an active member of the Bourbaki group in the early 1950s, and subsequently gave a number of significant Bourbaki seminars. He also took part in the Cartan seminar.
|
|
The specimens Willis discovered had a 'fastigiated' or vertical habit in contrast to the wider spread of the common yew seen throughout the island of Ireland, the British Isles and Europe. The characteristics of the young seedlings survived into maturity and the tree attracted interest from visitors and horticulturists due to its compactness and attractive columnar shape.
|
|
In measure theory Prokhorov's theorem relates tightness of measures to relative compactness (and hence weak convergence) in the space of probability measures. It is credited to the Soviet mathematician Yuri Vasilyevich Prokhorov, who considered probability measures on complete separable metric spaces. The term "Prokhorov’s theorem" is also applied to later generalizations to either the direct or the inverse statements.
|
|
The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel yields a compact operator. Compactness may be shown by invoking equicontinuity. As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.
|
|
An infinitary logic is a logic that allows infinitely long statements and/or infinitely long proofs. Some infinitary logics may have different properties from those of standard first-order logic. In particular, infinitary logics may fail to be compact or complete. Notions of compactness and completeness that are equivalent in finitary logic sometimes are not so in infinitary logics.
|
|
The most wide spread sources are the pulsed diode lasers. They have a FWHM of around 100 ps and repetition frequency of up to 100 MHz and an average power of about a few milliwatts. Even though they lack tunability, their low cost and compactness allows for multiple modules to be used in a single system.
|
|
Asuman Güven Aksoy is a Turkish-American mathematician whose research concerns topics in functional analysis, metric geometry, and operator theory including Banach spaces, measures of non-compactness, fixed points, Birnbaum–Orlicz spaces, real trees, injective metric spaces, and tight spans. She works at Claremont McKenna College, where she is Crown Professor of Mathematics and George R. Roberts Fellow.
|
|
Topological spaces can be broadly classified, up to homeomorphism, by their topological properties. A topological property is a property of spaces that is invariant under homeomorphisms. To prove that two spaces are not homeomorphic it is sufficient to find a topological property not shared by them. Examples of such properties include connectedness, compactness, and various separation axioms.
|
|
The space }, along with the double origin topology is an example of a Hausdorff space, although it is not completely Hausdorff. In terms of compactness, the space }, along with the double origin topology fails to be either compact, paracompact or locally compact, however, X is second countable. Finally, it is an example of an arc connected space.
|
|
In mathematics, Helly's selection theorem states that a sequence of functions that is locally of bounded total variation and uniformly bounded at a point has a convergent subsequence. In other words, it is a compactness theorem for the space BVloc. It is named for the Austrian mathematician Eduard Helly. The theorem has applications throughout mathematical analysis.
|
|
For example, if X = (0, 1) and for each positive integer i, Xi is the set of elements of X having a decimal expansion with digit 0 in the ith decimal place, then any finite intersection is non- empty (just take 0 in those finitely many places and 1 in the rest), but the intersection of all Xi for i ≥ 1 is empty, since no element of (0, 1) has all zero digits. The finite intersection property is useful in formulating an alternative definition of compactness: a space is compact if and only if every collection of closed sets having the finite intersection property has non- empty intersection. This formulation of compactness is used in some proofs of Tychonoff's theorem and the uncountability of the real numbers (see next section).
|
|
Lindberg JS, Zobitz MM, Poindexter JR, Pak CY: "Magnesium bioavailability from magnesium citrate and magnesium oxide.", J Am Coll Nutr. 1990 Feb;9(1):48-55. The ligand of choice for large-scale manufacturers of multivitamins and minerals containing magnesium is the magnesium oxide due to its compactness, high magnesium content by weight, low cost, and ease-of-use in manufacturing.
|
|
An action of a group G on a locally compact space X is cocompact if there exists a compact subset A of X such that . For a properly discontinuous action, cocompactness is equivalent to compactness of the quotient space X/G. The action of G on X is said to be proper if the mapping that sends is a proper map.
|
|
Ten of the twelve townhouses have eight units, while the remaining two have four and six respectively. Quinn planned each unit according to minimum house design, a technique popular with housing reformers that emphasized compactness and eliminated less essential rooms such as hallways and dining rooms. The district was added to the National Register of Historic Places on November 16, 2005.
|
|
In mathematics, cocompact embeddings are embeddings of normed vector spaces possessing a certain property similar to but weaker than compactness. Cocompactness has been in use in mathematical analysis since the 1980s, without being referred to by any name E. Lieb, On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74 (1983), 441–448.(Lemma 6),V.
|
|
The house feels spacious despite its compactness. An open plan kitchen, dining and living area are on the upper level and this space is overlooked by a partially screened loft bedroom. To the north are views of a Victorian roof-scape and distant mountains. Services for hydronic heating are tucked under the floors, while outside, room is also made for a water tank.
|
|
Crisman, Karl-Dieter, and Jones, Michael A. The Mathematics of Decisions, Elections, and Games pg. 3 A district's Polsby–Popper score will always fall within the interval of [0,1], with a score of 0 indicating complete lack of compactness and a score of 1 indicating maximal compactness.Miller, William J., and Walling, Jeremy D. The Political Battle Over Congressional Redistricting pg.
|
|
His work Tezkiretü'ş-Şuara was the one that made him famous. It was the last of the great bibliographical works on poets of the 16th century. It was structured in three sections. According to Mustafa ben Abd Allah Hadjidji Khalifa, it surpasses all previous works of its kind both in kindness of language and compactness of information which is embedded in it.
|
|
These theorems are for Banach spaces with the Radon–Nikodym property. A theorem of Joram Lindenstrauss states that, in a Banach space with the Radon–Nikodym property, a nonempty closed and bounded set has an extreme point. (In infinite-dimensional spaces, the property of compactness is stronger than the joint properties of being closed and being bounded). Edgar's theorem implies Lindenstrauss's theorem.
|
|
This number is not always followed, and individual keys or whole sections are commonly skipped for the sake of compactness or user preference. The most common choice is to not include the numpad, which can usually be fully replaced by the alphanumeric section. Laptops and wireless peripherals often lack duplicate keys and ones seldom used. Function- and arrow keys are nearly always present.
|
|
He has also been the first to develop a rigorously size-extensive state-specific multi-reference coupled cluster formalism, and its perturbative counterpart which is getting increasingly recognized as a very promising methodological advance. The attractive aspects of Mukherjee's formalisms are compactness and high accuracy. These are now accepted as pioneering and standard works in the field. which has attracted wide international attention.
|
|
The method of constructing infinitesimals of the kind used in nonstandard analysis depends on the model and which collection of axioms are used. We consider here systems where infinitesimals can be shown to exist. In 1936 Maltsev proved the compactness theorem. This theorem is fundamental for the existence of infinitesimals as it proves that it is possible to formalise them.
|
|
Advantages of double-row bearings as compared to single-row include that they can bear radial and axial loads in both directions. Double-row angular contact ball bearings have a steep mounting, which also can bear tilting effects. Other advantages of double-row bearings are their rigidity and compactness. Their disadvantage is they need better alignment than single-row bearings.
|
|
Compact cameras are usually designed to be easy to use. Almost all include an automatic mode, or "auto mode", which automatically makes all camera settings for the user. Some also have manual controls. Compact digital cameras typically contain a small sensor which trades-off picture quality for compactness and simplicity; images can usually only be stored using lossy compression (JPEG).
|
|
Also when the air passes through these openings, its velocity increases giving profound diffusion. It has been observed that humid areas like Kerala and Konkan have larger holes with overall lower opacity than compared with the dry climate regions of Gujarat and Rajasthan. With compactness of the residential areas in the modern India, jalis became less frequent for privacy and security matters.
|
|
They may use suitable human-built structures, such as house eaves and streetlights. The nest is an open structure made of grass, lined with feathers and hairs, packed densely for compactness. Some ornithologists have reported this bird building domed nests in acacia trees, but these records may reflect confusion with the Spanish sparrow. The average clutch contains three to five eggs.
|
|
If a sequence of manifolds converge in the Lipschitz sense to a limit Lipschitz manifold then the SWIF limit exists and has the same limit. Wenger's compactness theorem states that if a sequence of compact Riemannian manifolds, Mj, has a uniform upper bound on diameter, volume and boundary volume, then a subsequence converges SWIF-ly to an integral current space.
|
|
In particular, they are locally compact, locally connected, first countable, locally contractible, and locally metrizable. Being locally compact Hausdorff spaces, manifolds are necessarily Tychonoff spaces. Adding the Hausdorff condition can make several properties become equivalent for a manifold. As an example, we can show that for a Hausdorff manifold, the notions of σ-compactness and second-countability are the same.
|
|
M32's unusual characteristics of dense compactness and burst of star formation 2 billion years ago would be explained by this theory as a remnant of an earlier large galaxy, given its unlikeness to other similarly sized elliptical galaxies. It was described in 2018 by scientists at the University of Michigan. It is thought to have been 2.5×1010 M☉ in size.
|
|
One of the most important properties of first-countable spaces is that given a subset A, a point x lies in the closure of A if and only if there exists a sequence {xn} in A which converges to x. (In other words, every first-countable space is a Fréchet-Urysohn space.) This has consequences for limits and continuity. In particular, if f is a function on a first-countable space, then f has a limit L at the point x if and only if for every sequence xn → x, where xn ≠ x for all n, we have f(xn) → L. Also, if f is a function on a first-countable space, then f is continuous if and only if whenever xn → x, then f(xn) → f(x). In first-countable spaces, sequential compactness and countable compactness are equivalent properties.
|
|
In mathematics, the Aubin–Lions lemma (or theorem) is the result in the theory of Sobolev spaces of Banach space-valued functions, which provides a compactness criterion that is useful in the study of nonlinear evolutionary partial differential equations. Typically, to prove the existence of solutions one first constructs approximate solutions (for example, by a Galerkin method or by mollification of the equation), then uses the compactness lemma to show that there is a convergent subsequence of approximate solutions whose limit is a solution. The result is named after the French mathematicians Jean-Pierre Aubin and Jacques-Louis Lions. In the original proof by Aubin, the spaces X0 and X1 in the statement of the lemma were assumed to be reflexive, but this assumption was removed by Simon, so the result is also referred to as the Aubin–Lions–Simon lemma.
|
|
If two spaces have different number of cut-points, they are not homeomorphic. A classic example is using cut-points to show that lines and circles are not homeomorphic. Cut-points are also useful in the characterization of topological continua, a class of spaces which combine the properties of compactness and connectedness and include many familiar spaces such as the unit interval, the circle, and the torus.
|
|
Acmeism, or the Guild of Poets, was a transient poetic school, which emerged in 1912 in Russia under the leadership of Nikolay Gumilev and Sergei Gorodetsky. Their ideals were compactness of form and clarity of expression.Poem for the Day, Two, The Nicholas Albery Foundation, Chatto and Windus, London The term was coined after the Greek word άκμη (ákmē), i.e., "the best age of man".
|
|
Since writing permutations elementwise, that is, as piecewise functions, is cumbersome, several notations have been invented to represent them more compactly. Cycle notation is a popular choice for many mathematicians due to its compactness and the fact that it makes a permutation's structure transparent. It is the notation used in this article unless otherwise specified, but other notations are still widely used, especially in application areas.
|
|
The evaluation Discrete Global Grid consists of many aspects, including area, shape, compactness, etc. Evaluation methods for map projection, such as Tissot's indicatrix, are also suitable for evaluating map projection based Discrete Global Grid. Z-curve indexes, because have no "jumps", preserving nearest cells as neighbours. In addition, averaged ratio between complementary profiles (AveRaComp) gives a good evaluation of shape distortions for quadrilateral-shaped Discrete Global Grid.
|
|
The Hebbel Requiem was first performed in Heidelberg on 16July 1916 as part of a memorial concert for Reger, conducted by Philipp Wolfrum. Reger thought the Hebbel Requiem was "among the most beautiful things" he ever wrote. It has been described as of "lyrical beauty, a dramatic compactness, and [of] economy of musical means" in which the composer's "mastery of impulse, technique, and material is apparent".
|
|
This made it necessary to revise the design with side rods to transmit the motion forward to the lever mechanism mounted near the front of the frames. Thus, some of the advantage of 'neatness' and compactness of the original was lost."Locomotive Adventure" Harold Holcroft, Ian Allan. London The gear was also used on the sole K1 and the first of the U1 class.
|
|
The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space. #Gromov's compactness theorem. The set of all Riemannian manifolds with positive Ricci curvature and diameter at most D is pre-compact in the Gromov-Hausdorff metric.
|
|
338 Winchester Magnum and .458 Winchester Magnum for the U.S. market, as well as 6.5×68mm, 8×68mm S, and others for the world market. The rifle remains popular due to its aesthetic qualities, compactness, the smoothness of its action and its precision and quality of manufacture. The rifle is also known for its low recoil when chambered for the original 6.5×54 cartridge.
|
|
RoboMind is educational software that is specially developed to teach students about logic, programming and robotics. The strength of RoboMind is the compactness of the learning environment, which allows to quickly develop and test scripts in a virtual environment. The scripts can then directly be transferred to a Lego Mindstorms NXT robot, to see the result in real life. RoboMind script run on the standard firmware.
|
|
Modular homes are becoming increasingly common in Japanese urban areas, due to improvements in design and quality, speed and compactness of onsite assembly, as well as due to lowering costs and ease of repair after earthquakes. Recent innovations allow modular buildings to be indistinguishable from site-built structures. Surveys have shown that individuals can rarely tell the difference between a modular home and a site- built home.
|
|
Ten years later, the contemporary English snooker professional Fred Davis said of Marshall, "Most noticeable about his style is his compactness, so like Walter Lindrum, and the shortness of his back-swing, hardly more than a couple of inches." Marshall dominated amateur billiards before and after the war with a career that spanned six decades, broken by retirements in 1963 and 1970 followed by come-backs.
|
|
Wilson, 44-48. Locke also wrote guiding principles for regional development that were remarkably similar to principles of modern planning, including aspects of sustainable development and smart growth. Such aspects include, a) consistency of development practices with the general plan (Fundamental Constitutions); b) concurrent provision of infrastructure with land development; and compactness of development to promote efficient use of land and access to markets.Wilson, 44-48.
|
|
The method proceeds by the eigenvalue decomposition of the kernel matrix. It will then analyze the eigenvalues and eigenvectors to obtain a measure of the compactness of the input distribution. Finally, a plot will be drawn, where the elbow of that plot indicates the optimal number of clusters in the data set. Unlike previous methods, this technique does not need to perform any clustering a-priori.
|
|
The Big Joe 5 crossbow (BJ5) was a shoulder-fired weapon developed by the Office of Strategic Services (OSS) (predecessor to Central Intelligence Agency) to be used for assassinations up to 200 metres. The weapon was created with compactness in mind. It is described in John Brunner's book OSS Weapons.OSS Weapons book It's ammunition consisted of 8 ounce aluminum darts or incendiary flare bolts.
|
|
A systematic study of these planes began in 1954 with a paper by Skornyakov. Earlier, the topological properties of the real plane had been introduced via ordering relations on the affine lines, see, e.g., Hilbert, Coxeter, and O. Wyler. The completeness of the ordering is equivalent to local compactness and implies that the affine lines are homeomorphic to \R and that the point space is connected.
|
|
Pachyostosis is a non-pathological condition in vertebrate animals in which the bones experience a thickening, generally caused by extra layers of lamellar bone. It often occurs together with bone densification (osteosclerosis), reducing inner cavities. This joint occurrence is called pachyosteosclerosis. However, especially in the older literature, "pachyostosis" is often used loosely, referring to all osseous specializations characterized by an increase in bone compactness and/or volume.
|
|
The Macintosh IIcx is a personal computer designed, manufactured and sold by Apple Computer, Inc. from March 1989 to March 1991. Introduced six months after the Macintosh IIx, the IIcx resembles the IIx and provides the same performance, but is seven inches narrower, ten pounds lighter, and quieter due to a smaller internal fan. The relative compactness results in three NuBus slots being available, compared with six on the IIx.
|
|
A space is called ω-bounded if the closure of every countable set is compact. For example, the long line and the closed long ray are ω-bounded but not compact. When restricted to a metric space ω-boundedness is equivalent to compactness. The bagpipe theorem states that every ω-bounded connected surface is the connected sum of a compact connected surface and a finite number of long pipes.
|
|
Doug Drexler took four months to design the eponymous Enterprise for the fifth spinoff, Star Trek: Enterprise (2001–2005). A predecessor to Jeffries' original Enterprise, some elements of this ship were inspired by the Akira class in First Contact, and its overall compactness was inspired by Deep Space Nines Defiant.Herman Zimmerman, Deep Space Nine season 3 commentary Eden FX created computer-generated models for all four seasons of Enterprise.
|
|
Harmonic gearing A harmonic gear or strain wave gear is a specialized gearing mechanism often used in industrial motion control, robotics and aerospace for its advantages over traditional gearing systems, including lack of backlash, compactness and high gear ratios. Though the diagram does not demonstrate the correct configuration, it is a "timing gear," conventionally with far more teeth than a traditional gear to ensure a higher degree of precision.
|
|
Evolution of central pressure against compactness (radius over mass) for a uniform density 'star'. This central pressure diverges at the Buchdahl bound. In general relativity, Buchdahl's theorem, named after Hans Adolf Buchdahl, makes more precise the notion that there is a maximal sustainable density for ordinary gravitating matter. It gives an inequality between the mass and radius that must be satisfied for static, spherically symmetric matter configurations under certain conditions.
|
|
5, 2587–2855. Their article was most recently updated in 2013, to fix an incorrect statement of Hamilton's compactness theorem. In 2015, Kleiner and Lott were awarded the Award for Scientific Reviewing from the National Academy of Sciences of the United States for their work. Other well-known expositions of Perelman's work are due to Huai-Dong Cao and Xi-Ping Zhu, and to John Morgan and Gang Tian.
|
|
They also form the background for parameter estimation. In the case of extremum estimators for parametric models, a certain objective function is maximized or minimized over the parameter space. Theorems of existence and consistency of such estimators require some assumptions about the topology of the parameter space. For instance, compactness of the parameter space, together with continuity of the objective function, suffices for the existence of an extremum estimator.
|
|
A diode-pumped solid-state laser (DPSSL) is a solid-state laser made by pumping a solid gain medium, for example, a ruby or a neodymium-doped YAG crystal, with a laser diode. DPSSLs have advantages in compactness and efficiency over other types, and high power DPSSLs have replaced ion lasers and flashlamp-pumped lasers in many scientific applications, and are now appearing commonly in green and other color laser pointers.
|
|
John Bligh Conway (born September 22, 1939) is an American mathematician. He is currently a professor emeritus at the George Washington University. His specialty is functional analysis, particularly bounded operators on a Hilbert space. Conway earned his B.S. from Loyola University and Ph.D. from Louisiana State University under the direction of Heron Collins in 1965, with a dissertation on The Strict Topology and Compactness in the Space of Measures.
|
|
Then is unitary with respect to . See Another consequence is that every compact Lie group has the complete reducibility property, meaning that all its finite-dimensional representations decompose as a direct sum of irreducible representations. It is also true that there are no infinite-dimensional irreducible unitary representations of compact Lie groups, stated, but not proved in . This is due to the non-compactness of the Lorentz group.
|
|
Advantages of such catamarans are light weight, compactness and convenience in transportation (the whole product is packed in one pack- backpack, suitable for air traffic standards) and the speed of assembly (10–15 minutes for the inflation). All-inflatable models are available in North America. A cata-raft design has been used on the Colorado River to handle heavy whitewater, yet maintain a good speed through the water.
|
|
Therefore for infinitary logics, notions of strong compactness and strong completeness are defined. This article addresses Hilbert-type infinitary logics, as these have been extensively studied and constitute the most straightforward extensions of finitary logic. These are not, however, the only infinitary logics that have been formulated or studied. Considering whether a certain infinitary logic named Ω-logic is complete promises to throw light on the continuum hypothesis.
|
|
By homogeneity, local compactness of the underlying space for a topological group need only be checked at the identity. That is, a group G is a locally compact space if and only if the identity element has a compact neighborhood. It follows that there is a local base of compact neighborhoods at every point. A topological group is Hausdorff if and only if the trivial one-element subgroup is closed.
|
|
Edmund Fuller of Saturday Review wrote that the book > is effective, striking, in many ways. Yet in spite of the sparseness and > compactness of the book [Waltari] manages to make it almost lush, or purple, > in some of its passages. It is good of its kind, but I am afraid it is more > of an interesting item in relation to Waltari's body of work than an > enhancement of his reputation.
|
|
Pachyophis Cladogram Pachyophis is considered a hind-limbed snake despite presence of any indication of forelimbs or shoulder girdle due to its other shared general morphology. All Simoliophiidae display evident pachyostosis (increase in bone massiveness) and osteosclerosis (increase in inner bone compactness). Pachyophis evidently displays such in their vertebrae specifically. Other shared characteristics include a smaller head and distally straight ribs for a lateral compression of the trunk.
|
|
Ferrand had nearly 100 mathematical publications, including ten books, and was active in mathematical research into her late 70s. One of her accomplishments, in 1971, was to prove the compactness of the group of conformal mappings of a non- spherical compact Riemannian manifold, resolving a conjecture of André Lichnerowicz, and on the basis of this work she became an invited speaker at the 1974 International Congress of Mathematicians in Vancouver.
|
|
Critics have disagreed about whether the protagonist's point of view at the end of the story coincides with Chekhov's perspective. Other critical interpretation has focused on the symmetrical structure of "The Student" as well as the significance of the language used both to tell the story of the Denial of Peter and to tell the story as a whole. "The Student" has been praised by critics for its compactness and delicacy.
|
|
As the first generation of the 'J Mover' Series, the Honda Capa was developed on the Honda Logo platform. Its concept was based on a 'Small is Smart' way of thinking. Ease of use in everyday life, compactness, lightness and friendliness to the environment were some of the criteria built into the design. The proposal was that the Capa was to be 'Joyful' however this translates better as 'fun.
|
|
Altman's contributions to electoral districting and redistricting have been both theoretical and implementational. He established that the computational complexity of the districting problem is NP-hard and hence optimal redistricting is likely to be intractable. The undesirable implications of this result are that redistricting cannot be fully automated in practice and the choice of constraints and manual selection of the winning, "optimal" plan from a group of auto-generated plans, reintroduce value-laden and politically biased decision making back into the redistricting process (something that the use of "objective" computer programs was hoped to avoid), while potentially also legitimizing such undercover gerrymandering for the less knowledgeable public. Further, computational simulations that he performed showed also that even the constraints that have been traditionally considered politically non-preferential, such as the overall compactness of the district, are not necessarily non-preferential because compactness requirements have different effects on political groups if the groups are distributed in geographically different ways.
|
|
What fascinated him was this beautiful new style, the rhythmically arranged packing of sentences, the rigidity and purity of the language with a dramatic compactness, and the French strangeness which made the translation truly incomprehensible for simple litterateurs. It showed Kazinczy a sample for the cultivation of Hungarian belles-lettres. Thus after being primarily inspired by Bessenyi he found himself a life goal as a translator and cultivator of the Hungarian language.
|
|
There are also more subtle limitations of first-order logic that are implied by the compactness theorem. For example, in computer science, many situations can be modeled as a directed graph of states (nodes) and connections (directed edges). Validating such a system may require showing that no "bad" state can be reached from any "good" state. Thus one seeks to determine if the good and bad states are in different connected components of the graph.
|
|
These conditions are equivalent for metrizable spaces, but neither one implies the other in the class of all topological spaces. It is almost trivial to prove that the product of two sequentially compact spaces is sequentially compact—one passes to a subsequence for the first component and then a subsubsequence for the second component. An only slightly more elaborate "diagonalization" argument establishes the sequential compactness of a countable product of sequentially compact spaces.
|
|
With the Red Army now on the offensive, Salogor called on Nikolai Mikhailovich Frolov, a seasoned participant in Komsomol resistance, to take up command over Bessarabian partisan troops. As Frolov reports, no such units actually existed at that stage, mainly because "hostile elements" and "Boyar Romania" held the region in full control. The effort to land partisan commanders directly into Bessarabian territory was hampered by the region's compactness and lack of suitable airfields.
|
|
To see this consider the set {F(1), F(2),...}. Clearly the formula ∀xF(x) is a logical consequence of the set, but it is not a consequence of any finite subset of it (and hence it is not deducible from it). It follows immediately that both compactness and the strong completeness theorem fail for truth-value semantics. This is rectified by a modified definition of logical consequence as given in Dunn and Belnap 1968.
|
|
Adams was presented with the problem of a greatly increasing volume of commuter traffic experienced with the suburbanisation of London during the 1880s.Burtt, F. (1949). This was exacerbated by the fact that there were few locomotive classes in the LSWR stable that could undertake commuter traffic at the desired level of efficiency. The LSWR therefore required a locomotive with attributes of power and compactness, with a small wheel size to gain acceleration on intensive timetables.
|
|
Ikarus is committed to provide engines for aircraft and weapons. It was originally planned to use a 552 kW Hispano-Suiza 14AB because of its compactness. It was intended to be well-armed with a forward firing 20 mm Hispano Suiza HS.404 cannon and a rear-firing 20 mm Oerlikon FF for defence. It was also envisaged that the aircraft would have three 7.92 mm machine guns (two front and one on the back).
|
|
So there is no utilitarian-optimal division. The problem with the above example is that the value measure of partner 2 is finitely-additive but not countably-additive. The compactness part of the DS theorem immediately implies that: :::If all value measures are countably- additive and nonatomic, :::then a utilitarian-optimal division exists. In this special case, non-atomicity is not required: if all value measures are countably-additive, then a utilitarian-optimal partition exists.
|
|
Located in the Saragozza district, about 3.5 km from the center of the city, it regularly hosts Bologna's home matches. With around 36,000 seats it is the eleventh Italian stadium for capacity, which can increase up to 55,000 for concerts. It is considered one of the best fields in Italy and in Europe, together with the Artemio Franchi stadium in Florence, due to the extraordinary draining quality and compactness of the turf.
|
|
Pavel Urysohn Pavel Samuilovich Urysohn () (February 3, 1898 – August 17, 1924) was a Soviet mathematician who is best known for his contributions in dimension theory, and for developing Urysohn's metrization theorem and Urysohn's lemma, both of which are fundamental results in topology. His name is also commemorated in the terms Urysohn universal space, Fréchet–Urysohn space, Menger–Urysohn dimension and Urysohn integral equation. He and Pavel Alexandrov formulated the modern definition of compactness in 1923.
|
|
Washington, DC: Island Press. The key to increasing the amount of walking, cycling and public transport use is compactness: if housing is built near to existing facilities then travel time is reduced, and sustainable methods of transport are encouraged. Various options include the development of brown field sites and car parks, redevelopment of council estates, conversion of empty commercial space, intensification of existing housing areas, better use of empty homes, and subdivision of larger houses.
|
|
The bags quickly became very popular due to their low price, light weight, and compactness. Krčil soon extended the range of designs, including bags to be carried at the elbow or on the shoulder, and bags for sporting equipment. In the late 1920s string bags were being produced in Switzerland and Italy, and were distributed around the world. Krčil himself exported the bags to Canada, France, Switzerland, Germany, Austria and North African countries..
|
|
The compact finite difference formulation, or Hermitian formulation, is a numerical method to compute finite difference approximations. Such approximations tend to be more accurate for their stencil size (i.e. their compactness) and, for hyperbolic problems, have favorable dispersive error and dissipative error properties when compared to explicit schemes. A disadvantage is that compact schemes are implicit and require to solve a diagonal matrix system for the evaluation of interpolations or derivatives at all grid points.
|
|
When two of the three Mercedes-Benz engines were replaced, the compactness of the Napier engines was graphically illustrated—they were half the size of the original engines and approximately one fifth the weight. Proving successful, Deltic Diesel engines became a common power plant in small and fast naval craft. The Royal Navy used them first in the Dark class fast attack craft. Subsequently they were used in a number of other smaller attack craft.
|
|
The origin of the theory of compact operators is in the theory of integral equations, where integral operators supply concrete examples of such operators. A typical Fredholm integral equation gives rise to a compact operator K on function spaces; the compactness property is shown by equicontinuity. The method of approximation by finite-rank operators is basic in the numerical solution of such equations. The abstract idea of Fredholm operator is derived from this connection.
|
|
PMODE was originally designed to be used with demoscene demos, and this was the area of application where it became extremely popular because of its performance and compactness. These features of PMODE also made it relevant for the developers of games and embedded systems. PMODE/W is one of the four DOS extenders included in the Open Watcom compiler distribution. PMODE/W is also the default DOS extender used by TMT Pascal.
|
|
Piezoelectric direct discharge (PDD) plasma is a type of cold non-equilibrium plasma, generated by a direct gas discharge of a high voltage piezoelectric transformer. It can be ignited in air or other gases in a wide range of pressures, including atmospheric. Due to the compactness and the efficiency of the piezoelectric transformer, this method of plasma generation is particularly compact, efficient and cheap. It enables a wide spectrum of industrial, medical and consumer applications.
|
|
It was believed that turbojet or turboprop engines could power all aircraft, from the largest to smallest designs. The Wankel engine did not find many applications in aircraft, but was used by Mazda in a popular line of sports cars. The French company Citroën had developed Wankel powered helicopter in 1970's. In modern times the Wankel engine has been used in motor gliders where the compactness, light weight, and smoothness are crucially important.
|
|
No. 86 (1997), 115–197. Another consequence is Gromov's compactness theorem, stating that the set of compact Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric. The possible limit points of sequences of such manifolds are Alexandrov spaces of curvature ≥ c, a class of metric spaces studied in detail by Burago, Gromov and Perelman in 1992. Along with Eliyahu Rips, Gromov introduced the notion of hyperbolic groups.
|
|
Scientific American, 94. Instead of taking the traditional top- down approach, the bottom-up approach will probably soon have to be adopted because of the sheer size of these nanocircuits. Not everything in the circuit will probably work because at the nano level, nanocircuits will be more defective and faulty because of their compactness. Scientists and engineers have created all of the essential components of nanocircuits such as transistors, logic gates and diodes.
|
|
In particular, projective spaces satisfy a condition called properness which is analogous to compactness. Affine schemes cannot be proper (except in trivial situations like when the scheme has only a single point), and hence no projective space is an affine scheme (except for zero-dimensional projective spaces). Projective schemes, meaning those that arise as closed subschemes of a projective space, are the single most important family of schemes. Several generalizations of schemes have been introduced.
|
|
In conditions of high humidity, P. niger can form striking drops of black liquid on the actively growing caps. Spines of Phellodon tomentosus The fibrous flesh is single to double-layered; duplex layering results from differences in compactness or in the alignment of the constituent hyphae. Tough and leathery when fresh, the flesh develops a corklike texture when dry. In the dried state it often has an odor of fenugreek or curry powder.
|
|
In 1925 he started work in the area of almost periodic functions, simplifying the approach of Harald Bohr by use of compactness and approximate identity arguments. In 1933 he defined the Bochner integral, as it is now called, for vector-valued functions. Bochner's theorem on Fourier transforms appeared in a 1932 book. His techniques came into their own as Pontryagin duality and then the representation theory of locally compact groups developed in the following years.
|
|
Early submarine cross-section, before saddle tanks The first effective submarines, those of World War I, had hulls that were broadly circular in cross-section, with a deck plate mounted midway. Their heavy battery tanks were mounted beneath this deck, for stability. The ballast tanks were mounted inside the pressure hull. For compactness the ballast tanks were wrapped around the batteries, low down and sharing the flat surfaces of the battery tank.
|
|
The weapon is equipped with side-folding buttstock, which is adjustable for length of pull, and can be completely removed if maximum compactness is required. Additional equipment also includes a new, specially designed 40 mm underbarrel grenade launcher and a bayonet. The fire control unit includes ambidextrous safety/fire selector switch, which permits semi-automatic, 2-round bursts and full automatic fire. The charging handle can also be installed on either side of the weapon, depending on user preferences.
|
|
The two lur blowers The monument stands tall. The sculpture at the top of the column consists of two male figures standing very close together, wearing capes and helmets, each holding a raised lur. The compactness of the composition is a result of a design change late in the process, since the original design featured only a single lur blower. The decision to change the design was based on the fact that lurs are always tuned in pairs.
|
|
The compactness of the cyclotron reduces other costs as well, such as foundations, radiation shielding, and the enclosing building. Cyclotrons have a single electrical driver, which saves both money and power. Furthermore, cyclotrons are able to produce a continuous stream of particles at the target, so the average power passed from a particle beam into a target is relatively high. M. Stanley Livingston and Ernest O. Lawrence (right) in front of Lawrence's cyclotron at the Lawrence Radiation Laboratory.
|
|
This map is closed, continuous (by the pasting lemma), and surjective and therefore is a perfect map (the other condition is trivially satisfied). However, p is not open, for the image of under p is which is not open relative to (the range of p). Note that this map is a quotient map and the quotient operation is 'gluing' two intervals together. 8\. Notice how, to preserve properties such as local connectedness, second countability, local compactness etc.
|
|
This implies in particular that is compact. There is a more direct proof of compactness using symmetry groups. Given a Jordan frame (ei) in E, for every a in A there is a k in U = Γu(A) such that with (and if a is invertible). In fact, if (a,b) is in X then it is equivalent to k(c,d) with c and d in the unital Jordan subalgebra , which is the complexification of .
|
|
Other cases of interest relate to uniform integrability, and the concept of normal family in complex analysis. Mahler's compactness theorem in the geometry of numbers characterizes relatively compact subsets in certain non-compact homogeneous spaces (specifically spaces of lattices). The definition of an almost periodic function at a conceptual level has to do with the translates of being a relatively compact set. This needs to be made precise in terms of the topology used, in a particular theory.
|
|
However, Bloom filters do not store the data items at all, and a separate solution must be provided for the actual storage. Linked structures incur an additional linear space overhead for pointers. A Bloom filter with a 1% error and an optimal value of , in contrast, requires only about 9.6 bits per element, regardless of the size of the elements. This advantage comes partly from its compactness, inherited from arrays, and partly from its probabilistic nature.
|
|
The optical design for CELT is a Ritchey-Chretien two-mirror system, with a segmented mirror mosaic with 1080 segments. This rather naturally provides a large, 20 arcminute field of view with less than 0.5 arcsecond images (100% enclosed energy). This focus is free of coma and only suffers from astigmatism, which grows quadratically with field angle. The primary was planned to be in diameter, and for compactness, the primary f-number will be f/1.5.
|
|
The distance transform is also useful for determining the center of the object, and for matching in image recognition. Another class of operations is gathering orientation-free metrics. This is often important in image recognition where the orientation of the camera needs to be removed. Orientation-free metrics of a group of connected or surrounded pixels include the Euler number, the perimeter, the area, the compactness, the area of holes, the minimum radius, the maximum radius.
|
|
This is known as the Heine–Borel theorem. Note that compactness depends only on the topology, while boundedness depends on the metric. Lebesgue's number lemma states that for every open cover of a compact metric space M, there exists a "Lebesgue number" \delta such that every subset of M of diameter r<\delta is contained in some member of the cover. Every compact metric space is second countable, and is a continuous image of the Cantor set.
|
|
This page focuses on finitary first order model theory of infinite structures. Finite model theory, which concentrates on finite structures, diverges significantly from the study of infinite structures in both the problems studied and the techniques used. Model theory in higher-order logics or infinitary logics is hampered by the fact that completeness and compactness do not in general hold for these logics. However, a great deal of study has also been done in such logics.
|
|
The development of the fundamentals of model theory (such as the compactness theorem) rely on the axiom of choice, or more exactly the Boolean prime ideal theorem. Other results in model theory depend on set-theoretic axioms beyond the standard ZFC framework. For example, if the Continuum Hypothesis holds then every countable model has an ultrapower which is saturated (in its own cardinality). Similarly, if the Generalized Continuum Hypothesis holds then every model has a saturated elementary extension.
|
|
In order to better provide stark, contrasting, analogies, the Futurist literature promoted a kind of hyper-conciseness. It was dubbed essential and synthetic lyricism. The former refers to a paring down of any and all superfluous objects while the latter expresses an unnatural compactness of the language unseen elsewhere.Clough. This idea explains where poetry became the preferred literary medium of Futurism and why there are no Futurist novels (since novels are neither pared down nor compressed).
|
|
Researcher using a diamond anvil cell to study materials under deep Earth conditions. Prior to the invention of the diamond anvil cell, static high-pressure apparatus required large hydraulic presses which weighed several tons and required large specialized laboratories. The simplicity and compactness of the DAC meant that it could be accommodated in a wide variety of experiments. Some contemporary DACs can easily fit into a cryostat for low-temperature measurements, and for use with a superconducting electromagnet.
|
|
The FPP is also preserved by any retraction. According to Brouwer fixed point theorem every compact and convex subset of a Euclidean space has the FPP. More generally, according to the Schauder-Tychonoff fixed point theorem every compact and convex subset of a locally convex topological vector space has the FPP. Compactness alone does not imply the FPP and convexity is not even a topological property so it makes sense to ask how to topologically characterize the FPP.
|
|
The deformations that are considered in topology are homeomorphisms and homotopies. A property that is invariant under such deformations is a topological property. Basic examples of topological properties are: the dimension, which allows distinguishing between a line and a surface; compactness, which allows distinguishing between a line and a circle; connectedness, which allows distinguishing a circle from two non- intersecting circles. The ideas underlying topology go back to Gottfried Leibniz, who in the 17th century envisioned the and .
|
|
The basic object of study is topological spaces, which are sets equipped with a topology, that is, a family of subsets, called open sets, which is closed under finite intersections and (finite or infinite) unions. The fundamental concepts of topology, such as continuity, compactness, and connectedness, can be defined in terms of open sets. Intuitively, continuous functions take nearby points to nearby points. Compact sets are those that can be covered by finitely many sets of arbitrarily small size.
|
|
C. J. Cherryh is best known for her science fiction and fantasy novels. Short story writing is an art she never considered until she had several novels published. The compactness and limited characterisation of the short story form did not lend itself well to Cherryh's precise and complex novel construction methodologies. In spite of this, she has proved herself quite capable of working in this medium and this collection as a whole has been well received by critics.
|
|
Simon Brendle has solved major open problems regarding the Yamabe equation in conformal geometry. This includes his counterexamples to the compactness conjecture for the Yamabe problem, and the proof of the convergence of the Yamabe flow in all dimensions (conjectured by Richard Hamilton). In 2007, he proved the differentiable sphere theorem (in collaboration with Richard Schoen), a fundamental problem in global differential geometry. In 2012, he proved the Hsiang–Lawson's conjecture, a longstanding problem in minimal surface theory.
|
|
Sometimes both the integrated graphics and a dedicated graphics card can be used simultaneously to feed separate displays. The main advantages of integrated graphics include cost, compactness, simplicity and low energy consumption. The performance disadvantage of integrated graphics arises because the graphics processor shares system resources with the CPU. A dedicated graphics card has its own random access memory (RAM), its own cooling system, and dedicated power regulators, with all components designed specifically for processing video images.
|
|
Here 'own-nature' would mean characteristic nature, which is not something inherent in a dhamma as a separate ultimate reality, but arise due to the supporting conditions both of other dhammas and previous occurrences of that dhamma."Harvey, in his excellent INTRODUCTION TO BUDDHISM, page 87 wrote: The Visuddhimagga of Buddhaghosa, the most influential classical Theravada treatise, states that not-self does not become apparent because it is concealed by "compactness" when one does not give attention to the various elements which make up the person.Ñāṇamoli Bhikkhu (trans), Buddhaghosa, The Path of Purification: Visuddhimagga, Buddhist Publication Society, 1991, p 668. The Paramatthamañjusa Visuddhimaggatika of Acariya Dhammapala, a later Theravada commentary on the Visuddhimagga, refers to the fact that we often assume unity and compactness in phenomena and functions which are instead made up of various elements, but when one sees that these are merely empty dhammas, one can understand the not-self characteristic: > "when they are seen after resolving them by means of knowledge into these > elements, they disintegrate like froth subjected to compression by the hand.
|
|
The compactness of the site and relatively intact state of the stamp battery, haulage gear and water tanks, is rare in North Queensland. The close spatial relationship between the mine and battery is typical of early mining operations conducted on a small intermittently worked scale. Location of a treatment plant on top of a potential mineral resource is today unusual. The very intact timbering in the mine shaft, including the ladder, and the intact whip pole, are rare in Queensland.
|
|
Camp bed A camp bed, or cot in North America, is a small portable, lightweight bed used in situations where larger permanent beds cannot be used. The main advantage of this bed is its portability and compactness. Camp beds are generally used by armies or organizations, in tourism, and in emergency situations when there is a need to quickly provide accommodation for victims. Camp beds generally consist of a foldable lightweight wood or metal frame, covered with canvas, linen or nylon.
|
|
He further showed that the Souslin-Kleene closure of the open set quantifier logic fails Craig Interpolation which implies that it is strictly weaker than the interior operator logic. His later research concentrated on proving the existence of maximal extensions of first order logic which satisfy Łoś's theorem on ultraproducts and have the Souslin-Kleene property. Also this was extended to ultraproduct extensions of first order logic which satisfied both the Łoś's theorem and an extended form of the compactness theorem.
|
|
There are many situations where another condition of topological spaces (such as paracompactness or local compactness) will imply regularity if preregularity is satisfied. Such conditions often come in two versions: a regular version and a Hausdorff version. Although Hausdorff spaces are not, in general, regular, a Hausdorff space that is also (say) locally compact will be regular, because any Hausdorff space is preregular. Thus from a certain point of view, it is really preregularity, rather than regularity, that matters in these situations.
|
|
IBBCEAS has been used in conjunction with LEDs and superluminescent LEDs in a number of gas phase and liquid analyte studies. Simultaneous concentration measurements of NO2 and NO3 have been achieved using LED based IBBCEAS within the ppbv detection limit. The advantages of using LEDs as light source are their compactness, long life, power efficiency and price. Also, due to the small area emission, the emitted power per unit area at the peak wavelength can approach that of Xe arc lamps.
|
|
Haydon also invented a printed circuit motor that used a printed circuit that comprised the entire winding and commutator in a single flat printed circuit disc. The motor field is a simple doughnut- shaped piece of ferrite material. This motor is covered by U.S. Patent No. RE25,305. Because of its compactness, simplicity, reliability, and apparent low cost in volume production, the North American Phillips Company took exclusive royalty-paying licenses throughout the world except in the United States and Canada.
|
|
The compactness of the tournament has given it the nickname "college basketball's toughest tournament". It began when Dr. James Naismith, Emil S. Liston, Frank Cramer, and local leaders formed the National College Basketball Tournament, which was staged at Municipal Auditorium in Kansas City, Missouri. The goal of the tournament was to establish a forum for small colleges and universities to determine a national basketball champion. NAIA history The first championship game featured Central Missouri State University defeating Morningside College (Iowa) 35–24.
|
|
The large dispersion and small footprint of the device make the chromo-modal dispersion device potentially useful for on-chip dispersion compensation using optical components such as integrated gratings and planar multimode waveguides. The advantage of physical compactness, combined with the magnitude and tunability of its dispersion suggest its potential use as a versatile tool for pulse stretching or compression in a variety of applications in which the capabilities of singlemode fiber or diffraction grating-based dispersive elements will not suffice.
|
|
If this were not enough, much of the income not granted to Inchaffray had since been given to other religious institutions; some revenue was even controlled by the Bishop of Dunkeld.Cynthia Neville, Native Lordship in Medieval Scotland: The Earldoms of Strathearn and Lennox, c. 1140–1365, (Dublin, 2005), pp. 150–51. Dunblane had its origins in an older Gaelic monastic establishment, that is, in an institution with an abbot-bishop heading a relatively informal establishment of smaller cells with little geographical compactness.
|
|
Gorgon's had two engines of 160 horse power each, built by Seaward and Company. It were direct-acting engines, and as such they were a novel construction remarkable for compactness, strength and lightness. It placed the crankshaft directly above the center line of the cylinders and these were connected to the shaft by means of a simple connecting rod. The absence of the usual cast-iron framing, sway-beams, side-rods and crossheads saved upwards of 60 tons in weight.
|
|
Each index has one possible value per dimension of the underlying vector space. The number of indices equals the degree (or order) of the tensor. For compactness and convenience, the notational convention implies summation over indices repeated within a term and universal quantification over free indices. Expressions in the notation of the Ricci calculus may generally be interpreted as a set of simultaneous equations relating the components as functions over a manifold, usually more specifically as functions of the coordinates on the manifold.
|
|
The cross-flow fan, or transverse fan, is thus a two-stage partial admission machine. The popularity of the crossflow fan in HVAC comes from its compactness, shape, quiet operation, and ability to provide a high pressure coefficient. Effectively a rectangular fan in terms of inlet and outlet geometry, the diameter readily scales to fit the available space, and the length is adjustable to meet flow rate requirements for the particular application. Common household tower fans are also cross-flow fans.
|
|
Also, a subset of a normal space need not be normal (i.e. not every normal Hausdorff space is a completely normal Hausdorff space), since every Tychonoff space is a subset of its Stone–Čech compactification (which is normal Hausdorff). A more explicit example is the Tychonoff plank. The only large class of product spaces of normal spaces known to be normal are the products of compact Hausdorff spaces, since both compactness (Tychonoff's theorem) and the T2 axiom are preserved under arbitrary products.
|
|
Consider a probability density function p(x)=\exp(-\phi(x)). This probability density function p(x) is said to be a log-concave measure if the \phi(x) function is convex. Such probability density functions have tails which decay exponentially fast, so most of the probability mass resides in a small region around the mode of p(x) . The Brascamp–Lieb inequality gives another characterization of the compactness of p(x) by bounding the mean of any statistic S(x).
|
|
On- highway vehicle operators look for straight-six diesels, which are smooth- operating and quiet. Likewise, off-highway applications such as tractors, marine engines, and electric generators need an engine that is rugged and powerful. Unlike passenger cars, compactness is not an important criterion for these applications, while the additional reliability and maintainability of a straight-six diesel engines is. As with everyday passenger vehicles, the smooth running characteristics of the straight-six engine are what make it desirable for industrial use.
|
|
In more realistic examples, the 3rd form could have problems due to operator precedence, perhaps returning an unexpected type, where the prior forms would in some languages report an error. Thus, "simplicity" is not merely a matter of length, but of logical and conceptual structure; making code shorter may make it less or more complex. For large, long lived programs using verbose alternatives could contribute to bloat. Compactness can allow coders to view more code per page, reducing scrolling gestures and keystrokes.
|
|
The lower wing was much smaller in area than the upper wing. The concept was intended to combine the strength, compactness and stability of the biplane's wire braced wing cell with the speed and ease of handling of the monoplane.Spooner, 1917, p.884 Many were built or converted as single-seat fighters by covering the front cockpit, and adding a Lewis Gun or Vickers machine gun either to fire through the center section of the top wing or mounted over it, firing forwards.
|
|
On September 14, 2015, OneVirginia2021 filed a suit in Vesilind v. Virginia State Board of Elections claiming 11 Virginia state districts fail the criteria of compactness outlined by the Virginia State Constitution. In an official release, OneVirginia2021 executive director Brian Cannon says, “Far from having a standard, the legislature effectively ignored the Constitution on this point, and gave us distorted, weirdly shaped districts that break up communities and rig elections by depriving voters of meaningful competition”. The Virginia supreme court upheld the districts.
|
|
This advantage of compactness comes with some disadvantages. The capacitance temperature dependence of polyester film capacitors is relatively high compared to other film capacitors, ±5% over the entire temperature range. The capacitance frequency dependence of polyester film capacitors compared with the other film capacitors is -3% in the range from 100 Hz to 100 kHz at the upper limit. Also, the temperature and frequency dependence of the dissipation factor are higher for polyester film capacitors compared with the other film capacitor types.
|
|
Corrosion can cause unprotected magnets to spall off a surface layer or to crumble into a powder. Use of protective surface treatments such as gold, nickel, zinc, and tin plating and epoxy-resin coating can provide corrosion protection; the majority of neodymium magnets use nickel plating to provide a robust protection. Originally, the high cost of these magnets limited their use to applications requiring compactness together with high field strength. Both the raw materials and the patent licenses were expensive.
|
|
Large black pillars support a low, slightly coffered ceiling. Circular light fixtures are set into the ceiling of the ticket lobby, within each of the slight indentations. The advance sales lobby, accessible from 50th Street just east of Sixth Avenue, contains a single ticket booth on the eastern wall. To the ticket lobby's east, and the advance sales lobby's northeast, is the elliptical grand foyer, whose four-story-high ceiling and dramatic artwork contrast with the compactness of the lobby.
|
|
The Bristol T.T.A was designed in 1915 to a War Office requirement for a local defence aircraft. The T.T.A was a two-seat, twin-engine biplane with T.T. standing for twin tractor; the Bristol Type number 6 was added retrospectively in 1923. The guiding principles in the design were compactness and a wide field of fire from both cockpits. The T.T.A was an unswept biplane with slight stagger, the wings having constant chord and carrying long ailerons on the upper planes.
|
|
As a product of compact Hausdorff spaces, the Hilbert cube is itself a compact Hausdorff space as a result of the Tychonoff theorem. The compactness of the Hilbert cube can also be proved without the Axiom of Choice by constructing a continuous function from the usual Cantor set onto the Hilbert cube. In ℓ2, no point has a compact neighbourhood (thus, ℓ2 is not locally compact). One might expect that all of the compact subsets of ℓ2 are finite-dimensional.
|
|
The short travel distances within cells serve to quicken the flows. Moreover, the compactness of a cell minimizes space that might allow build-ups of inventory between cell stations. To formalize that advantage, cells often have designed-in rules or physical devices that limit the amount of inventory between stations. Such a rule is known, in JIT/lean parlance, as kanban (from the Japanese), which establishes a maximum number of units allowable between a providing and a using work station.
|
|
This aids vehicle control and reduces wear that could cause difficulties in steering precisely. Worm drives are a compact means of substantially decreasing speed and increasing torque. Small electric motors are generally high-speed and low-torque; the addition of a worm drive increases the range of applications that it may be suitable for, especially when the worm drive's compactness is considered. Worm drives are used in presses, rolling mills, conveying engineering, mining industry machines, on rudders, and worm drive saws.
|
|
The first successful application of the free-piston engine concept was as air compressors. In these engines, air compressor cylinders were coupled to the moving pistons, often in a multi-stage configuration. Some of these engines utilised the air remaining in the compressor cylinders to return the piston, thereby eliminating the need for a rebound device. Free-piston air compressors were in use among others by the German Navy, and had the advantages of high efficiency, compactness and low noise and vibration.
|
|
The original proof of the De Bruijn–Erdős theorem, by De Bruijn, used transfinite induction. provided the following very short proof, based on Tychonoff's compactness theorem in topology. Suppose that, for the given infinite graph , every finite subgraph is -colorable, and let be the space of all assignments of the colors to the vertices of (regardless of whether they form a valid coloring). Then may be given a topology as a product space , where denotes the set of vertices of the graph.
|
|
By Tychonoff's theorem this topological space is compact. For each finite subgraph of , let be the subset of consisting of assignments of colors that validly color . Then the system of sets is a family of closed sets with the finite intersection property, so by compactness it has a nonempty intersection. Every member of this intersection is a valid coloring of .. Gottschalk states his proof more generally as a proof of the theorem of that generalizes the De Bruijn–Erdős theorem.
|
|
Wymore received both his B.Sc. and M.Sc. in mathematics from the Iowa State University early 1950s. In 1956 he received his PhD in mathematics from the University of Wisconsin–Madison under the Supervision of Associate Professor William F. Eberlein for the thesis "On the Weak Compactness in Functional Analysis".The University of Wisconsin Press (1956) Summaries of Doctoral Dissertations, University of Wisconsin, Volume 16. p. 356. With his PhD in pure mathematics, Wymore started his career in industry for two years.
|
|
He was trained in Argos as a student of Dorotheos, with whom he worked at Delphi and Hermione.Giuliano 1987, p. 686 Between 450 and 420 BC he worked mainly in Athens, as a follower of Myron's school and in the post- Phidias period he brought elements of compactness due to the Peloponnesian period. Roman writer Pliny the Elder wrote of a competition between the four sculptors Polykleitos, Phidias, Kresilas, and Phradmon, on the best statues of Amazons for the Temple of Artemis at Ephesus.
|
|
The nonaesthetic properties associated with cuteness - smallness, compactness, formal simplicity, softness or pliancy thus also index minor negative affects such as helplessness, pitifulness and even despondency. Ngai also argues that the term cuteness is a way of sexualizing beings while simultaneously rendering them unthreatening. She illustrates this by providing several examples of poems that deploy ‘cuteness’ as a means of rendering the overtly aggressive and sexual dimension of the theme unthreatening. If "cuteness" is symptomatic of the aesthetics of contemporary consumption, zaniness is about production.
|
|
High mobility group nucleosome-binding domain-containing protein 3 is a protein that in humans is encoded by the HMGN3 gene. Thyroid hormone receptors are hormone-dependent transcription factors that regulate expression of a variety of specific target genes. The protein encoded by this gene binds thyroid hormone receptor beta, but only in the presence of thyroid hormone. The encoded protein, a member of the HMGN protein family, is thought to reduce the compactness of the chromatin fiber in nucleosomes, thereby enhancing transcription from chromatin templates.
|
|
A thickening of the trefoil knot Topology is the field concerned with the properties of continuous mappings, and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness. The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'.
|
|
Fagin's proof uses the compactness theorem,; , Theorem 2.4.4, pp. 51–52. Based on this equivalence, the theory of sentences modeled by the Rado graph has been called "the theory of the random graph" or "the almost sure theory of graphs". Because of this 0-1 law, it is possible to test whether any particular first-order sentence is modeled by the Rado graph in a finite amount of time, by choosing a large enough value of and counting the number of -vertex graphs that model the sentence.
|
|
1) Tychonoff's 1930 proof used the concept of a complete accumulation point. 2) The theorem is a quick corollary of the Alexander subbase theorem. More modern proofs have been motivated by the following considerations: the approach to compactness via convergence of subsequences leads to a simple and transparent proof in the case of countable index sets. However, the approach to convergence in a topological space using sequences is sufficient when the space satisfies the first axiom of countability (as metrizable spaces do), but generally not otherwise.
|
|
This article gives a table of some common Lie groups and their associated Lie algebras. The following are noted: the topological properties of the group (dimension; connectedness; compactness; the nature of the fundamental group; and whether or not they are simply connected) as well as on their algebraic properties (abelian; simple; semisimple). For more examples of Lie groups and other related topics see the list of simple Lie groups; the Bianchi classification of groups of up to three dimensions; and the list of Lie group topics.
|
|
Fold designs may use larger wheels, even the same size as in non-folders, for users prioritizing ride over fold compactness. Bikes that use this kind of fold include, Dahon, and Montague, and Tern. ;Vertical Fold Instead of folding horizontally, this style of bike has one or two hinges along the main tube and/or chain and seat stays that allow the bike to fold vertically. The result leaves the two wheels side by side but is often more compact than a horizontally hinged design.
|
|
The then Pawi-Lakher Regional Council (PLRC) comprised all of the present Lai Autonomous District Council, Mara Autonomous District Council (MADC), and Chakma Autonomous District Council (CADC). The PLRC also had international boundaries with Myanmar and Bangladesh. Beside the Pawi(Lai) and the Lakher (Mara), other tribes, namely Bawm, Tlanglau, Pang, Riang(Bru), and the Chakma, also inhabited the PLRC area. The PLRC covered all the Lakher inhabited areas within Mizoram due to its compactness as well as the ethnic identity consciousness of the Lakher.
|
|
A relatively refined theory is available for pseudocompact topological groups.See, for example, Mikhail Tkachenko, Topological Groups: Between Compactness and \aleph_0-boundedness, in Mirek Husek and Jan van Mill (eds.), Recent Progress in General Topology II, 2002 Elsevier Science B.V. In particular, W. W. Comfort and Kenneth A. Ross proved that a product of pseudocompact topological groups is still pseudocompact (this might fail for arbitrary topological spaces).Comfort, W. W. and Ross, K. A., Pseudocompactness and uniform continuity in topological groups, Pacific J. Math. 16, 483-496, 1966.
|
|
The simplicity, lightness, compactness and reliability might seem ideal for an aero-engine, but because of their low efficiency, early flathead engines were deemed unsuitable. Two notable exceptions were the American Aeronca E-107 opposed twin aero engine of 1930 and the Continental A40 flat four of 1931, which became one of the most popular light aircraft engines of the 1930s. Two modern flatheads are the Belgian D-Motor flat-fours and flat-sixes.Although very small and compact, the D-Motor flat-six displaces nearly 4 litres.
|
|
For use as a data structure, the main alternative to the adjacency list is the adjacency matrix. Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only bytes of contiguous space, where is the number of vertices of the graph. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a sparse graph, adjacency lists require less space, because they do not waste any space to represent edges that are not present.
|
|
The main advantage of washer dryer combination units is their compactness. The small size of these machines compared to the total space consumed by a separate washer and dryer suits them to small homes, apartments, condominiums, and any place where space is an issue. Aside from having a small footprint, combo units also have a small height, allowing them to fit into confined places, like under a kitchen counter or in a closet. Another potential benefit of combo washer dryers, compared to larger, separate machines, is portability.
|
|
His earliest work was on the aim to develop a regularity theory for minimal hypersurfaces, changing how we view the advanced theory of minimal surfaces and calculus of variations forever. The proof required De Giorgi to develop his own version of geometric measure theory along with a related key compactness theorem. With these results, he was able to conclude that a minimal hypersurface is analytic outside a closed subset of codimension at least two. He also established regularity theory for all minimal surfaces in a similar manner.
|
|
There were other public rooms: > Adjoining the theatre and near the inclosed part appropriated to the > lecturer, is the chemical laboratory, in which convenience, compactness, and > elegance are united. Contiguous to it is the committee-room. On the other > side of the theatre is the library, which is sixty feet in length, with a > gallery on three sides, and an easy access to it by a flight of > steps.William Henry Pyne, William Combe, Rudolph Ackermann, Thomas > Rowlandson, Augustus Pugin, Microcosm of London; or, London in miniature > vol.
|
|
Leon Henkin (1950) defined these semantics and proved that Gödel's completeness theorem and compactness theorem, which hold for first-order logic, carry over to second-order logic with Henkin semantics. This is because Henkin semantics are almost identical to many-sorted first-order semantics, where additional sorts of variables are added to simulate the new variables of second-order logic. Second-order logic with Henkin semantics is not more expressive than first- order logic. Henkin semantics are commonly used in the study of second-order arithmetic.
|
|
The surface gravity of a white dwarf is very high, and of a neutron star even higher. The neutron star's compactness gives it a surface gravity of up to 7 m/s² with typical values of order 1012 m/s² (that is more than 1011 times that of Earth). One measure of such immense gravity is that neutron stars have an escape velocity of around 100,000 km/s, about a third of the speed of light. For black holes, the surface gravity must be calculated relativistically.
|
|
Therefore, exon shuffling became a major role in the construction of younger proteins. Moreover, to define more precisely the time when exon shuffling became significant in eukaryotes, the evolutionary distribution of modular proteins that evolved through this mechanism were examined in different organisms (i.e., Escherichia coli, Saccharomyces cerevisiae, Arabidopsis thaliana, etc.) These studies suggested that there was an inverse relationship between the genome compactness and the proportion of intronic and repetitive sequences. As well as the fact that exon shuffling became significant after metazoan radiation.
|
|
David Browne of Entertainment Weekly described Insomniac as "14 slices of hearty anarchy, played with a follow-the-bouncing-spitball compactness and vigor." Ian Winwood of Kerrang! wrote of the album's "master class in buzzsaw efficiency, the songs are so economical, not to mention harsh, that the removal of even a single chord would cause each composition to collapse in on itself," calling it "the album on which its creators lost their baby teeth." The album features bleaker, more pessimistic lyrics than those of Dookie.
|
|
Classical Chinese lexicon is the lexicon of Classical Chinese, a language register marked by a vocabulary that greatly differs from the lexicon of modern vernacular Chinese. In terms of conciseness and compactness, Classical Chinese rarely uses words composed of two Chinese characters; nearly all words are of one syllable only. This stands directly in contrast with modern Chinese dialects, in which two-syllable words are extremely common. This phenomenon exists, in part, because polysyllabic words evolved in Chinese to disambiguate homophones that result from sound changes.
|
|
The infrared light on a sample is reflected and transmitted at different amounts depending on the bulk properties of the material. Diffuse reflection of the incident light produced by the sample's rough surface reflection in all directions is collected by use of an ellipsoid or paraboloid mirror. Shape, compactness, refractive index, reflectivity and absorption of the particles are all characteristic of the material being analyzed. If the sample is too absorbent, then it can be diluted with a nonabsorbent material such as potassium bromide, potassium chloride, etc.
|
|
This goes in the direction of developing a more efficient engine. During development the Duke has been tested at MAHLE Powertrain in the United Kingdom and in the United States; test results show that is has multi-fuel capabilities. The Duke engine's benefits of lightness and compactness should render this design ideal for motorcycles engines; and these benefits might make the powerplant suitablle for light aircraft as well. (There is little data on whether the Duke engine is smooth; the mainshaft has a large counterweight attached).
|
|
Originally, the Oklahoma Constitution-based the Senate on counties. The 19 most populous counties, as determined by the most recent federal census, were each to elect one senator. The 58 less populous counties were to be joined into 29 two-county districts, each of which was to elect one senator. In apportioning the Senate, the Oklahoma Constitution required that consideration be given to population, compactness, area, political units, historical precedents, economic and political interests, contiguous territory and other major factors, to the extent feasible.
|
|
Alexandrov spaces with curvature ≥ k are important as they form the limits (in the Gromov-Hausdorff metric) of sequences of Riemannian manifolds with sectional curvature ≥ k, as described by Gromov's compactness theorem. Alexandrov spaces with curvature ≥ k were introduced by the Russian mathematician Aleksandr Danilovich Aleksandrov in 1948 and should not be confused with Alexandrov-discrete spaces named after the Russian topologist Pavel Alexandrov. They were studied in detail by Burago, Gromov and Perelman in 1992 and were later used in Perelman's proof of the Poincaré conjecture.
|
|
The Palais–Smale compactness condition, named after Richard Palais and Stephen Smale, is a hypothesis for some theorems of the calculus of variations. It is useful for guaranteeing the existence of certain kinds of critical points, in particular saddle points. The Palais-Smale condition is a condition on the functional that one is trying to extremize. In finite-dimensional spaces, the Palais–Smale condition for a continuously differentiable real-valued function is satisfied automatically for proper maps: functions which do not take unbounded sets into bounded sets.
|
|
Examples of structures possible with air stripline: directional coupler (top left), branch-line coupler (top right), coupled-line bandpass filter (bottom left), and hybrid ring power splitter (bottom right) Air stripline finds its greatest use at microwave frequencies in the C band (). At these frequencies and belowPradhan & Barrow, 1977 for instance it has the advantage of compactness over waveguide. Air stripline can be used outside the C band, but at the higher Ku band () waveguide tends to dominate because of its lower loss.Han & Hwang, pp.
|
|
A common compactness measure is the isoperimetric quotient, the ratio of the area of the shape to the area of a circle (the most compact shape) having the same perimeter. In the plane, this is equivalent to the Polsby–Popper test. Alternatively, the shape's area could be compared to that of its bounding circle, its convex hull, or its minimum bounding box. Similarly, a comparison can be made between the perimeter of the shape and that of its convex hull, its bounding circle, or a circle having the same area.
|
|
Gödel used the completeness theorem to prove the compactness theorem, demonstrating the finitary nature of first- order logical consequence. These results helped establish first-order logic as the dominant logic used by mathematicians. In 1931, Gödel published On Formally Undecidable Propositions of Principia Mathematica and Related Systems, which proved the incompleteness (in a different meaning of the word) of all sufficiently strong, effective first-order theories. This result, known as Gödel's incompleteness theorem, establishes severe limitations on axiomatic foundations for mathematics, striking a strong blow to Hilbert's program.
|
|
Detailed drawing. The Swiss military was in need of a cavalry carbine for their mounted units, so they tried shortening the existing Schmidt–Rubin 1889, but its action proved to be too long to be effective to maneuver with while mounted, so the Swiss government began trials for a new carbine. Two turning- bolt designs were submitted by SIG, a turning-bolt and a straight-pull design were submitted by Ferdinand Mannlicher, a modified Mauser design and a straight-pull design by Vogelsang and Krauser. The Mannlicher straight-pull design was chosen for its compactness.
|
|
A Canonical S-expression (or csexp) is a binary encoding form of a subset of general S-expression (or sexp). It was designed for use in SPKI to retain the power of S-expressions and ensure canonical form for applications such as digital signatures while achieving the compactness of a binary form and maximizing the speed of parsing. The particular subset of general S-expressions applicable here is composed of atoms, which are byte strings, and parentheses used to delimit lists or sub-lists. These S-expressions are fully recursive.
|
|
The compactness theorem states that a set of first-order sentences has a model if and only if every finite subset of it has a model.Hodel, R. E., An Introduction to Mathematical Logic (Mineola NY: Dover, 1995), p. 199. This implies that if a formula is a logical consequence of an infinite set of first-order axioms, then it is a logical consequence of some finite number of those axioms. This theorem was proved first by Kurt Gödel as a consequence of the completeness theorem, but many additional proofs have been obtained over time.
|
|
When the strings of a keyboard instrument are laid out in the simplest way (ascending in pitch from left to right, as in full-size harpsichords), the resulting triangular shape is space-consuming and inefficient. The spinet harpsichord, which saves space by arranging the strings in slanted pairs, is still much longer than it is wide. Virginals, which enclose their triangle of strings in a rectangular box, have a great deal of unused space. In comparison with these designs, Cristofori's quasi-oval layout stands out for its compactness and efficiency.
|
|
The largest external and ergonomic difference in marker bodies is in the trigger and barrel position. Designers of expensive models attempt to position the trigger frame forward towards the center, or slightly forward of center of the body on speed-ball-oriented markers. This allows the HPA tank to be mounted in a position allowing compactness and balance without requiring any additional modifications that allow the tank to fall down and forwards. Such aftermarket "drop forwards", may create a larger gun profile, which can result in eliminations due to hopper hits.
|
|
The message format was designed for compactness rather than flexibility, consistent with the slow dialup modems used in 1980. The initial "A" dictated the layout of header and message information, and expansions would require changing the initial character. This scheme was abandoned after A news for the more verbose but expandable format seen . Because Usenet grew rapidly, the limited capabilities and simplistic article storage scheme (all articles were placed in a single disk directory and there was no facility for expiring old articles) quickly made A News impractical to use.
|
|
The theorem states, somewhat surprisingly, that the total integral of all curvatures will remain the same, no matter how the deforming is done. So for instance if you have a sphere with a "dent", then its total curvature is 4π (the Euler characteristic of a sphere being 2), no matter how big or deep the dent. Compactness of the surface is of crucial importance. Consider for instance the open unit disc, a non-compact Riemann surface without boundary, with curvature 0 and with Euler characteristic 1: the Gauss–Bonnet formula does not work.
|
|
She received positive critical attention for her paintings, but Sparhawk-Jones did not enjoy and avoided promoting her work. Her painting Shop Girls was exhibited at the Art Institute of Chicago as part of an intention to pick "eye-popping" or "little masterpieces of compactness and stimulation" from American artists' works of the late 19th century and early 20th-century. Now in the Art Institute's collection, Shop Girls was the first woman's painting bought by the Friends of American Art of the Art Institute. It was purchased in 1913 from Sparhawk-Jones for $550.
|
|
The compactness in coverage has been linked to broad audience attrition. According to the Pew Research Center, the circulation for U.S. newspapers has fallen sharply in the 21st century. The digital era also introduced journalism that ordinary citizens play a greater role in the process, with the rise of citizen journalism being possible through the Internet. Using video camera-equipped smartphones, active citizens are now enabled to record footage of news events and upload them onto channels like YouTube (which is often discovered and used by mainstream news media outlets).
|
|
', according to Deborah Hauptmann in De Architect. ;Mobile Forces (1994) For this book the authors have chosen to employ the very format of the book to elucidate their architectural approach, differentiating four themes that together constitute a modest repertoire of new architectural definitions: Mobile Forces, Crossing Points, Storing the Detail and Corporate Compactness. All projects are headed under one of these categories. In our electronic age the old architectural definitions have lost much relevance; even the building processes themselves are changing, making it necessary to look afresh at the potential meanings of architecture.
|
|
Forty works have been attributed to Sri Raghavendra swamy. Sharma notes that his works are characterised by their compactness, simplicity and their ability to explain the abstruse metaphysical concepts of Dvaita in understandable terms. His Tantradipika is an interpretation of the Brahma Sutra from the standpoint of Dvaita incorporating elements from Jayatirtha's Nyaya Sudha, Vyasatirtha's Tatparya Chandrika and the glosses by Vijayendra Tirtha. Bhavadipa is a commentary on Jayatirtha's Tattva Prakasika which, apart from elucidating the concepts of the source text, criticises the allegations against Madhva raised by Appaya Dikshita and grammarian Bhattoji Dikshita.
|
|
The advantages include: no backlash, high compactness and light weight, high gear ratios, reconfigurable ratios within a standard housing, good resolution and excellent repeatability (linear representation) when repositioning inertial loads, high torque capability, and coaxial input and output shafts. High gear reduction ratios are possible in a small volume (a ratio from 30:1 up to 320:1 is possible in the same space in which planetary gears typically only produce a 10:1 ratio). Disadvantages include a tendency for 'wind-up' (a torsional spring rate) in the low torque region.
|
|
Serge synthesizers are known for their flexibility, audio quality and relative compactness. Other well-known musicians using Serge synthesizers include: Malcolm Cecil, whose studio was used in Stevie Wonder albums; Gary Chang, movie composer; Roger Powell, keyboard player for Todd Rundgren; John Adams, composer; Ingram Marshall, composer; and many experimental and electronic musicians such as Jim O'Rourke, Sarah Davachi, R. Luke DuBois, Keith Fullerton Whitman and Paolo Tofani of the italian free-jazz and experimental group Area. Commercial builds and DIY kits of Serge synthesizers are currently available under license from a number of manufacturers.
|
|
In Detroit, this allowed them to use the softer Goodyear compound, while all the others had to use the harder compound. Also, the compactness of the cars, in comparison with the turbos, was an advantage in threading between the concrete walls, especially through the chicane leading on to the pit straight. Most importantly, Brundle and Bellof were putting in much better performances in their first year than anyone could have expected. Here, with things threatening to deteriorate at half distance, it was Brundle who would bring the race back to life.
|
|
Cigarette cases were a way to avoid the invasive labels (another way being various satirical stickers, with slogans such as "You could be hit by a bus tomorrow"). The United States Census Bureau, for the purposes of industry statistics, includes manufacturing or adorning of cigarette cases in the category NAICS 339914 "Costume jewelry and novelty manufacturing".NAICS 339914 Census.gov Due to the compactness of a cigarette case, being just small enough to conveniently fit in a pocket, they can also be used to store or conceal other small items.
|
|
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed. Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection. A topological space X is disconnected if there exist disjoint, nonempty, open subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.
|
|
This machine originally had only the rudder at the end of the boom, but a horizontal tailplane bearing end-plate fins was later added. A less obvious change was the addition of hydraulically-assisted cyclic pitch control. The second prototype had a modified cabin which allowed for stretcher-borne casualties to be carried. A pair of Ultra-lights were equipped with a slightly larger diameter rotor (32 ft/9.75 m rather than 28 ft/8.6 m) for the purpose of improving performance in situations where compactness was not at a premium.
|
|
Unlike Unicode, purposely avoids Han unification; no attempt at compactness of the encoding is made, nor is there an attempt to keep all common characters below U+FFFF as there is in Unicode. Unicode, on the other hand, sorts its CJK into blocks based on how common they are: the most common are generally put into the Basic Multilingual Plane, while those that are rare or obscure are put into the Astral Planes. For example, Radical 9 has two characters where Unicode has one: MJ054435 (), and MJ059031 (), both represented in Unicode as .
|
|
Welded bellows can be fabricated from a variety of exotic metals and alloys, whereas formed bellows are limited to alloys with good elongation – brass being a prime example. Welded bellows are not fabricated from brass because of its fundamentally poor weldability. Other advantages to welded bellows include compactness (higher performance in a smaller package), ability to be compressed to solid height with no damage, resistance to nicks and dents, and dramatically greater flexibility. The welding of metal bellows is a microscopic welding process, typically performed under laboratory conditions at high magnification.
|
|
The Hodge decomposition is closely related to the Helmholtz decomposition, generalizing from vector fields on R3 to differential forms on a Riemannian manifold M. Most formulations of the Hodge decomposition require M to be compact. Since this is not true of R3, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem.
|
|
In general compactness of the space does not imply completeness of a (G,X)-structure. For example, an affine structure on the torus is complete if and only if the monodromy map has its image inside the translations. But there are many affine tori which do not satisfy this condition, for example any quadrilateral with its opposite sides glued by an affine map yields an affine structure on the torus, which is complete if and only if the quadrilateral is a parallelogram. Interesting examples of complete, noncompact affine manifolds are given by the Margulis spacetimes.
|
|
Its value reflects both the importance of the ice edge location in general and the difficulty of accurately locating the ice edge with remotely sensed data. It is also useful to provide a description of the ice edge in terms of indications of freezing or thawing, wind-driven advance or retreat, and compactness or diffuseness. Other important auxiliary information includes the location of the icebergs, floebergs, ice islands, old ice, ridging and hummocking. These ice features are poorly monitored by remote sensing techniques but are very important aspects of the ice cover.
|
|
Although the usual natural numbers satisfy the axioms of PA, there are other models as well (called "non- standard models"); the compactness theorem implies that the existence of nonstandard elements cannot be excluded in first-order logic. The upward Löwenheim–Skolem theorem shows that there are nonstandard models of PA of all infinite cardinalities. This is not the case for the original (second-order) Peano axioms, which have only one model, up to isomorphism. This illustrates one way the first-order system PA is weaker than the second-order Peano axioms.
|
|
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed "size" (where the meaning of "size" depends on the structure of the ambient space.) The term precompact (or pre-compact) is sometimes used with the same meaning, but precompact is also used to mean relatively compact. These definitions coincide for subsets of a complete metric space, but not in general.
|
|
Dental Records also show that the L. lufengensis has molars with thick enamel, peripheralized cusp apices with expansive basin and a dense, complex pattern of occlusal crenulations. The pattern of compactness of the small transverse ridges in the enamel of permanent teeth of L. lufengensis are very similar to that of modern humans. The L. lufengensis upper central incisors are high- crowned and labiolingually thick in relation to mesiodistal length, with a distinct, high relief median lingual pillar. In contrast the lower incisors are high crowned and relatively narrow mesiodistal and moderately procumbent.
|
|
Continue this process whereby choosing a neighbourhood Un+1 ⊂ Un whose closure does not contain xn+1. Then the collection {Ui : i ∈ N} satisfies the finite intersection property and hence the intersection of their closures is non-empty by the compactness of X. Therefore, there is a point x in this intersection. No xi can belong to this intersection because xi does not belong to the closure of Ui. This means that x is not equal to xi for all i and f is not surjective; a contradiction. Therefore, X is uncountable.
|
|
All the conditions in the statement of the theorem are necessary: 1\. We cannot eliminate the Hausdorff condition; a countable set (with at least two points) with the indiscrete topology is compact, has more than one point, and satisfies the property that no one point sets are open, but is not uncountable. 2\. We cannot eliminate the compactness condition, as the set of rational numbers shows. 3\. We cannot eliminate the condition that one point sets cannot be open, as any finite space with the discrete topology shows. Corollary.
|
|
Although active components play an important role in the use of plasmonic circuitry, passive circuits are just as integral and, surprisingly, not trivial to make. Many passive elements such as prisms, lenses, and beam splitters can be implemented in a plasmonic circuit, however fabrication at the nano scale has proven difficult and has adverse effects. Significant losses can occur due to decoupling in situations where a refractive element with a different refractive index is used. However, some steps have been taken to minimize losses and maximize compactness of the photonic components.
|
|
One such step relies on the use of Bragg reflectors, or mirrors composed of a succession of planes to steer a surface plasmon beam. When optimized, Bragg reflectors can reflect nearly 100% of the incoming power. Another method used to create compact photonic components relies on CPP waveguides as they have displayed strong confinement with acceptable losses less than 3 dB within telecommunication wavelengths. Maximizing loss and compactness with regards to the use of passive devices, as well as active devices, creates more potential for the use of plasmonic circuits.
|
|
Most modern smartphones only have a menu choice to start a camera application program and an on-screen button to activate the shutter. Some also have a separate camera button, for quickness and convenience. A few camera phones are designed to resemble separate low-end digital compact cameras in appearance and to some degree in features and picture quality, and are branded as both mobile phones and cameras. The principal advantages of camera phones are cost and compactness; indeed for a user who carries a mobile phone anyway, the addition is negligible.
|
|
PC/104 systems often require small, non-volatile storage, such as that afforded by compact flash and solid state disk (SSD) devices. These are often more popular than mechanical (rotating) hard drives. Compared to rotating disks, flash-based storage devices have limited lifetimes in terms of write cycles, but they are faster and draw less power. Additionally, their compactness and physical durability is often better-suited to rugged PC/104 applications; the size of magnetic hard drives can be cumbersome and their many delicate parts are more susceptible to failure in harsh environments.
|
|
There are many results for topological spaces that hold for both regular and Hausdorff spaces. Most of the time, these results hold for all preregular spaces; they were listed for regular and Hausdorff spaces separately because the idea of preregular spaces came later. On the other hand, those results that are truly about regularity generally don't also apply to nonregular Hausdorff spaces. There are many situations where another condition of topological spaces (such as normality, pseudonormality, paracompactness, or local compactness) will imply regularity if some weaker separation axiom, such as preregularity, is satisfied.
|
|
The hypothesis of the statement can be weakened a bit: a sequence in C(X) converges uniformly if it is equicontinuous and converges pointwise on a dense subset to some function on X (not assumed continuous). This weaker version is typically used to prove Arzelà–Ascoli theorem for separable compact spaces. Another consequence is that the limit of an equicontinuous pointwise convergent sequence of continuous functions on a metric space, or on a locally compact space, is continuous. (See below for an example.) In the above, the hypothesis of compactness of X cannot be relaxed.
|
|
In subsequent years, Congress expanded on the requirements, successively adding contiguity, compactness, and substantial equality of population to the districting requirements. These standards were all later deleted in the Reapportionment Act of 1929. Congress subsequently reinstated the requirement that districts be composed of contiguous territory, be "compact," and have equal populations within each State. (1934) ("In each State entitled under this apportionment to more than one Representative,... [such Representatives] shall be elected by districts composed of a contiguous and compact territory, and containing as nearly as practicable an equal number of inhabitants.").
|
|
This manuscript consists of a bunch of marginal articles, written during the past six years and slung together with the minimum of care."Hart-Davis, Volume 3, Letter of 13 April 1958 Some critics agreed. The New Yorker commented, "his methods and the point of view behind them don't seem as funny or as sharp as they once did, possibly because they are no longer surprising, or possibly because he is getting a little tired of his own joke." But Edmund Wilson remained a fan of Potter, praising "the brevity and compactness of the presentation.
|
|
His constant habit of taking notes of cases coming before him gave his mind a compactness and clearness in summing up facts which was the parent of practical views in theory and successful decision in action. On 13 March 1828 he was elected a fellow of the Royal Society. His only contribution to literature was a series of papers on cholera, printed in The Lancet on 10 and 17 February and 3 March 1849. He married, 10 February 1821, Mary, daughter of William Mackinen Fraser, MD, of Lower Grosvenor Street, London.
|
|
The OG-43, designed by Giovanni Oliani in the factory based in Cremona named Società Anonima Revelli Manifattura Armiguerra, was an advanced submachine gun, designed for compactness and ease of mass production. It is part of a family of weapons produced under emergency measures during the time of the Italian Social Republic, which include the TZ-45 machine gun, FNAB-43 and Isotta Fraschini. The next version, the OG-44, instead had a more traditional shape. The Ministry of War of the RSI had approved the construction, but it was never initiated as events unfolded.
|
|
That is, by the Peter–Weyl theorem the irreducible unitary representations ρ of G are into a unitary group (of finite dimension) and the image will be a closed subgroup of the unitary group by compactness. Cartan's theorem states that Im(ρ) must itself be a Lie subgroup in the unitary group. If G is not itself a Lie group, there must be a kernel to ρ. Further one can form an inverse system, for the kernel of ρ smaller and smaller, of finite- dimensional unitary representations, which identifies G as an inverse limit of compact Lie groups.
|
|
Geotechnical engineers classify the soil particle types by performing tests on disturbed (dried, passed through sieves, and remolded) samples of the soil. This provides information about the characteristics of the soil grains themselves. Classification of the types of grains present in a soil does not account for important effects of the structure or fabric of the soil, terms that describe compactness of the particles and patterns in the arrangement of particles in a load carrying framework as well as the pore size and pore fluid distributions. Engineering geologists also classify soils based on their genesis and depositional history.
|
|
It is often not possible to join two metal components by welding, i.e. to melt part of both in the vicinity of the joint, if the two materials have very different properties from their alloy, due to the creation of brittle, inter-metallic compounds. This situation cannot be changed, even by electron-beam heating in vacuum, but this nevertheless makes it possible to realize joints meeting high demands for mechanical compactness and that are perfectly vacuum-tight. The principal approach is not to melt both parts, but only the one with the lower melting point, while the other remains solid.
|
|
Design work on the M09 started in about 2008 with work proceeding secretly until it was publicly announced in mid-September 2012. The design goals were compactness and quick installation. The M09 was designed by Franco Lambertini, who previously designed Moto Morini motorcycle engines, although the engine has no commonality with other Lambertini motorcycle engines and is a clean- sheet design for aviation use. The engine is a four stroke, 90° V twin cylinder, four valve, powerplant that delivers a maximum of at 7000 rpm and is capable of inverted flight as well as tractor and pusher configuration installations.
|
|
The statue is based upon compactness and solidity with few projecting parts; Khafre's block-like body is attached to the throne to last for eternity, creating one single structure. His arms rest on his thighs, directly facing the viewer in a rigid, frontal pose. The bilaterally symmetric statue, symbolizing order and control in the pharaoh, is the same on either side of the vertical axis of the statue, only differing in Khafre's clenched right fist. The tight profile and block-like aspect represent Khafre as a permanent being and part of the stone to keep his ka safe.
|
|
Because of their compactness, bit arrays have a number of applications in areas where space or efficiency is at a premium. Most commonly, they are used to represent a simple group of boolean flags or an ordered sequence of boolean values. Bit arrays are used for priority queues, where the bit at index k is set if and only if k is in the queue; this data structure is used, for example, by the Linux kernel, and benefits strongly from a find-first-zero operation in hardware. Bit arrays can be used for the allocation of memory pages, inodes, disk sectors, etc.
|
|
The wings were fitted with slotted flaps and full length leading edge slats and could be folded for compact, ship-board stowage. They also had a variable angle of incidence, pivoting at the front spar and actuated by an electrically driven jackscrew attached to the rear spar. This arrangement reduced stalling speed and increased lift, allowing the aircraft to use a smaller wing – compactness being an important feature for a ship-borne aircraft. Supermarine had tested this arrangement in the Type 322 and its capability was demonstrated when test pilot Mike Lithgow flew a Seagull at only .
|
|
Barnes, G. C., Norfolk's Biodiversity Habitat Action Plans - Traditional orchards at norfolkbiodiversity.org The Norfolk Biffin is also mentioned by Charles Dickens, first in A Christmas Carol and in Martin Chuzzlewit (1843), later in Dombey and Son (1846-1848)Dickens, Charles, Dombey and Son (1846-1848), Chapter 60, online at wikisource.org and in Boots at the Holly-tree Inn (1858). The first of these says: "Norfolk Biffins, squab and swarthy, setting off the yellow of oranges and lemons, and in the great compactness of their juicy persons, urgently entreating and beseeching to be carried home in paper bags and eaten after dinner".
|
|
744,000 of the jobs in the city are held by residents of other municipalities, while 242,000 residents in the capital have jobs outside. Thus passenger flows are predominantly into and out of the city centre, although further decentralisation of economic activity to the outskirts is altering this pattern. From the point of view of sustainable transport, Madrid has performed well from the compactness of the city centre and middle-to-high-density peripheral nuclei, favouring public transport and pedestrian movement. The weak points appear in the “new peripheries”, with low-density residential developments and dispersal of journey destinations, leading to higher car use.
|
|
In mathematics, in particular in algebraic geometry, a complete algebraic variety is an algebraic variety X, such that for any variety Y the projection morphism : is a closed map, i.e. maps closed sets onto closed sets.Here the product variety X × Y does not carry the product topology, in general; the Zariski topology on it will have more closed sets (except in very simple cases). This can be seen as an analogue of compactness in algebraic geometry: a topological space X is compact if and only if the above projection map is closed with respect to topological products.
|
|
We prove the finite version, using Radon's theorem as in the proof by . The infinite version then follows by the finite intersection property characterization of compactness: a collection of closed subsets of a compact space has a non-empty intersection if and only if every finite subcollection has a non-empty intersection (once you fix a single set, the intersection of all others with it are closed subsets of a fixed compact space). The proof is by induction: Base case: Let . By our assumptions, for every there is a point that is in the common intersection of all with the possible exception of .
|
|
Another simple and very useful result in Riemannian geometry is Gromov's compactness theorem, which states that the set of Riemannian manifolds with Ricci curvature ≥ c and diameter ≤ D is relatively compact in the Gromov–Hausdorff metric. The limit spaces are metric spaces. Additional properties on the length spaces have been proven by Cheeger and Colding.Cheeger-Colding: On the structure of spaces with Ricci curvature bounded below I The Gromov–Hausdorff distance metric has been applied in the field of computer graphics and computational geometry to find correspondences between different shapes Mémoli, F., & Sapiro, G. (2004, July).
|
|
For high output and low fuel consumption, the MIVEC system and other measures (including optimized shaping of the intake and exhaust manifolds and cylinder head) were used to promote intake and exhaust efficiency. Optimally shaped cooling passages in the cylinder head and optimal control of the flow of coolant into the cylinder head help to suppress engine knocking. Comprehensive measures to engine components were taken to minimize friction. For lightness and compactness; design optimization, material optimization, and component integration were identified as effective means of minimizing weight and bulk, so they were comprehensively effected in combination with each other.
|
|
Mukherjee has been the earliest developer of a class of many- body methods for electronic structure which are now standard and highly acclaimed works in the field. These methods, collectively called multireference coupled cluster (MRCC) formalisms, are versatile and powerful methods for predicting with quantitative accuracy the energetics of a vast range of molecular excitations and ionization. The attractive aspects of the formalisms are size-extensivity, compactness and high accuracy. He also developed a linear response theory based on coupled cluster formalism (CCLRT), which is similar in scope to the SAC-CI and done independently of it.
|
|
Stanine (STAndard NINE) is a method of scaling test scores on a nine-point standard scale with a mean of five and a standard deviation of two. Some web sources attribute stanines to the U.S. Army Air Forces during World War II. Psychometric legend has it that a 1-9 scale was used because of the compactness of recording the score as a single digit but ThorndikeThorndike, R. L. (1982). Applied Psychometrics. Boston, MA: Houghton Mifflin claims that by reducing scores to just nine values, stanines "reduce the tendency to try to interpret small score differences (p. 131)".
|
|
The Northern Survey has 2,712 clusters, and in order for a cluster to be put into the catalogue it must pass four criteria. The first criterion is richness, and Abell divided the clusters into 6 different richness groups. Group 0 was 30-49 galaxies, Group 1 was 50-79 galaxies, Group 2 was 80-129 galaxies, Group 3 was 130 to 199 galaxies, Group 4 was 200-299 galaxies, and Group 5 was more than 299 galaxies. The second criterion was compactness. To meet this requirement a cluster must have 50 or more members in one counting radius of the cluster's center.
|
|
Finite model theory (FMT) is the subarea of model theory (MT) that deals with its restriction to interpretations on finite structures, which have a finite universe. Since many central theorems of model theory do not hold when restricted to finite structures, FMT is quite different from MT in its methods of proof. Central results of classical model theory that fail for finite structures under FMT include the compactness theorem, Gödel's completeness theorem, and the method of ultraproducts for first-order logic. The main application areas of FMT are descriptive complexity theory, database theory and formal language theory.
|
|
Though unsupervised learning encompasses other domains involving summarizing and explaining data features. Cluster analysis is the assignment of a set of observations into subsets (called clusters) so that observations within the same cluster are similar according to one or more predesignated criteria, while observations drawn from different clusters are dissimilar. Different clustering techniques make different assumptions on the structure of the data, often defined by some similarity metric and evaluated, for example, by internal compactness, or the similarity between members of the same cluster, and separation, the difference between clusters. Other methods are based on estimated density and graph connectivity.
|
|
The compactness of the K-ration made it the preferred choice of the foot soldiers, but troops with access to transport and heating preferred the C-ration. Finally, there was the 10-in-1, an American version of the British 12-in-1, which was intended to feed ten men. It could be used by field kitchens, and offered the variety of five different menus. The lemon powder in the C- and K-rations was their primary sources of vitamin C, but it was particularly unpopular with the troops, who frequently discarded it, or used it for tasks like scrubbing floors.
|
|
Publius (or Gaius) Cornelius Tacitus ( , ; – ) was a Roman historian and politician. Tacitus is considered by modern scholars to be one of the greatest Roman historians. Compare: He lived in what has been called the Silver Age of Latin literature, and has a reputation for the brevity and compactness of his Latin prose, as well as for his penetrating insights into the psychology of power politics. The surviving portions of his two major works—the Annals and the Histories—examine the reigns of the emperors Tiberius, Claudius, Nero, and those who reigned in the Year of the Four Emperors (69 AD).
|
|
Prior to its adoption by Honda, the V4 design had been used only rarely for motorcycles, most notably by Matchless. Ian Chadwick Histories Retrieved 21 July 2015 In modern times, the V4 configuration has been embraced by Aprilia, Ducati, Suzuki, Yamaha & Norton. Compared to an in-line four, the advantages of a 90° V4 engine include compactness, narrow width, and perfect primary balance giving a smooth and nearly vibration-free operation. A disadvantage is that it is more expensive to manufacture, and, just like a V-twin, more difficult to locate ancillaries and inlet & exhaust systems.
|
|
Most early musket bayonets were of this type. Beginning in the early 19th century, knife and/or sword bayonets began to appear, which could also be wielded by hand. In the early to mid-20th century, spike bayonets reappeared, often folding or stowed under the barrel for compactness, such as on the French Lebel M1886 and MAS-36, Russian SKS and Mosin-Nagant, German FG 42, and British Lee–Enfield. The Lee-Enfield Rifle No.4 bayonet, took the form of a short spike (but fixed conventionally), and was unpopular due to its length and lack of utility.
|
|
A different proof using Zorn's lemma was given by Lajos Pósa, and also in the 1951 Ph.D. thesis of Gabriel Andrew Dirac. If is an infinite graph in which every finite subgraph is -colorable, then by Zorn's lemma it is a subgraph of a maximal graph with the same property (one to which no more edges may be added without causing some finite subgraph to require more than colors). The binary relation of nonadjacency in is an equivalence relation, and its equivalence classes provide a -coloring of . However, this proof is more difficult to generalize than the compactness proof.
|
|
A cellular automaton is said to be reversible when every configuration of the automaton has exactly one predecessor. It follows by a compactness argument that the function mapping each configuration to its predecessor is itself continuous in the shift space, and it is clearly also shift-invariant. Therefore, by the Curtis–Hedlund–Lyndon theorem, the time-reversed dynamics of the cellular automaton may itself be generated using a different cellular automaton rule. However, the neighborhood of a cell in the reverse automaton may be significantly larger than the neighborhood of the same cell in the forward automaton...
|
|
The notion of an open set provides a fundamental way to speak of nearness of points in a topological space, without explicitly having a concept of distance defined. Once a choice of open sets is made, the properties of continuity, connectedness, and compactness, which use notions of nearness, can be defined using these open sets. Each choice of open sets for a space is called a topology. Although open sets and the topologies that they comprise are of central importance in point-set topology, they are also used as an organizational tool in other important branches of mathematics.
|
|
Although higher-order logics are more expressive, allowing complete axiomatizations of structures such as the natural numbers, they do not satisfy analogues of the completeness and compactness theorems from first- order logic, and are thus less amenable to proof-theoretic analysis. Another type of logics are s that allow inductive definitions, like one writes for primitive recursive functions. One can formally define an extension of first- order logic -- a notion which encompasses all logics in this section because they behave like first-order logic in certain fundamental ways, but does not encompass all logics in general, e.g. it does not encompass intuitionistic, modal or fuzzy logic.
|
|
The Kochen–Specker theorem refines this statement by constructing a specific finite subset of rays on which no such probability measure can be defined. The fact that such a finite subset of rays must exist follows from Gleason's theorem by way of a logical compactness argument, but this method does not construct the desired set explicitly. In the related no-hidden-variables result known as Bell's theorem, the assumption that the hidden-variable theory is noncontextual instead is replaced by the assumption that it is local. The same sets of rays used in Kochen–Specker constructions can also be employed to derive Bell-type proofs.
|
|
The standard model of long-duration gamma- ray bursts (GRBs) holds that these explosions are ultra-relativistic (initial \gamma greater than approximately 100), which is invoked to explain the so- called "compactness" problem: absent this ultra-relativistic expansion, the ejecta would be optically thick to pair production at typical peak spectral energies of a few 100 keV, whereas the prompt emission is observed to be non- thermal.Cenko, S. B. et al., iPTF14yb: The First Discovery of a Gamma-Ray Burst Afterglow Independent of a High-Energy Trigger, Astrophysical Journal Letters 803, 2015, L24 (6 pp). Subatomic particles called muons, have a relatively high Lorentz factor and therefore experience extreme time dilation.
|
|
Richard Palais (right) and Chuu-Lian Terng (wife) at Oberwolfach, 2010 Richard Sheldon Palais (born May 22, 1931) is a mathematician working in geometry who introduced the Principle of Symmetric Criticality, the Mostow–Palais theorem, the Lie–Palais theorem, the Morse–Palais lemma, and the Palais–Smale compactness condition. From 1965 to 1967 Palais was a Sloan Fellow. In 1970 he was an invited speaker (Banach manifolds of fiber bundle sections) at the International Congress of Mathematicians in Nice. From 1965 to 1982 he was an editor for the Journal of Differential Geometry and from 1966 to 1969 an editor for the Transactions of the American Mathematical Society.
|
|
One of the interesting properties of periodic groups is that the definition cannot be formalized in terms of first- order logic. This is because doing so would require an axiom of the form :\forall x.\, ((x=e) \lor (x\circ x=e) \lor ((x\circ x)\circ x=e) \lor \cdots) which contains an infinite disjunction and is therefore inadmissible: First order logic permits quantifiers over one type and can't capture properties or subsets of that type. It is also not possible to get around this infinite disjunction by using an infinite set of axioms: the compactness theorem implies that no set of first-order formulae can characterize the periodic groups.
|
|
However, the product of continuum many copies of the closed unit interval (with its usual topology) fails to be sequentially compact with respect to the product topology, even though it is compact by Tychonoff's theorem (e.g., see ). This is a critical failure: if X is a completely regular Hausdorff space, there is a natural embedding from X into [0,1]C(X,[0,1]), where C(X,[0,1]) is the set of continuous maps from X to [0,1]. The compactness of [0,1]C(X,[0,1]) thus shows that every completely regular Hausdorff space embeds in a compact Hausdorff space (or, can be "compactified".) This construction is the Stone–Čech compactification.
|
|
Once we have this fact, Tychonoff's theorem can be applied; we then use the finite intersection property (FIP) definition of compactness. The proof itself (due to J. L. Kelley) follows: Let {Ai} be an indexed family of nonempty sets, for i ranging in I (where I is an arbitrary indexing set). We wish to show that the cartesian product of these sets is nonempty. Now, for each i, take Xi to be Ai with the index i itself tacked on (renaming the indices using the disjoint union if necessary, we may assume that i is not a member of Ai, so simply take Xi = Ai ∪ {i}).
|
|
Reviewers generally praised the image quality and wide range of customization it offers, although there were some criticisms of its performance. The small size also divided reviewers, with some praising its compactness and others criticising it as difficult to handle and noting that the lack of external controls making it hard to quickly change settings. DP Review rated it 71% and a silver award, praising its image quality, small size, and wide range of customisable features. What Digital Camera called it "an easy to use camera" while criticising the auto white balance accuracy, weak performance in low light, and lack of physical grip on the camera body.
|
|
Data is protected by a built-in rechargeable (Ni-Cd) battery when the AA batteries discharge or are removed for replacement. There are several simple programs available on the Internet for transferring files between a Model 100 and a modern personal computer (or a vintage one). The Model 100 was also used for industrial applications and in science laboratories as a programming terminal for configuration of control systems and instruments. Its compactness (ease of handling and small space requirements), low maintenance needs, lack of air vents (a plus for dusty or dirty environments), full complement of ports, and easy portability made it very well suited for these applications.
|
|
Although Lindström had only partially developed the hierarchy of quantifiers which now bear his name, it was enough for him to observe that some nice properties of first-order logic are lost when it is extended with certain generalized quantifiers. For example, adding a "there exist finitely many" quantifier results in a loss of compactness, whereas adding a "there exist uncountably many" quantifier to first-order logic results in a logic no longer satisfying the Löwenheim–Skolem theorem. In 1969 Lindström proved a much stronger result now known as Lindström's theorem, which intuitively states that first-order logic is the "strongest" logic having both properties.
|
|
The statement that every partial order can be extended to a total order is known as the order-extension principle. A proof using the axiom of choice was first published by Edward Marczewski in 1930. Marczewski writes that the theorem had previously been proven by Stefan Banach, Kazimierz Kuratowski, and Alfred Tarski, again using the axiom of choice, but that the proofs had not been published.. In modern axiomatic set theory the order-extension principle is itself taken as an axiom, of comparable ontological status to the axiom of choice. The order-extension principle is implied by the Boolean prime ideal theorem or the equivalent compactness theorem,.
|
|
The Bates Automatic Numbering-Machine or Bates stamper is named after the late 19th century inventor Edwin G. Bates of New York City. He obtained several United States patents for the device from 1891 to 1901, and the patents were assigned to the Bates Manufacturing Company. — Consecutive- Numbering Machine (October 18, 1892) — Automatic Numbering-Machine (June 11, 1901) — Automatic Numbering-Machine (June 11, 1901) The earliest patent claimed a "new and useful improvement in Consecutive-Numbering Machines", indicating that Bates was not the originator of the idea. Bates' objective was "generally to increase the effectiveness, simplicity and compactness of machines of this general character".
|
|
During the ban, mourners carried out false cremations by burning firewood atop graves. Advocates argued that cremation was not unfilial as the compactness of the resulting ashes made it easier for people to fulfil the filial task of interring family members together in ancestral graves. The Meiji government was less lenient than the contemporary Qing government in China, which made an exception for those who had died far from home. Advocates argued that burning bodies were better than rotting ones, citing European studies on the detrimental effect of decomposing bodies on public health, as well as the fact that cremation was being promoted in the West as a hygienic practice.
|
|
Compare this with the standard semantics, which says that a universal (existential) formula is true if and only if for all (some) members of the domain, the formula holds for all (some) of them; e.g. ∀xA is true (under an interpretation) if and only if for all k in the domain D, A(k/x) is true (where A(k/x) is the result of substituting k for all occurrences of x in A). (Here we are assuming that constants are names for themselves—i.e. they are also members of the domain.) Truth-value semantics is not without its problems. First, the strong completeness theorem and compactness fail.
|
|
Edward Witten came up with a related construction in the early 1980s sometimes known as Morse–Witten theory. Morse homology can be extended to finite-dimensional non-compact or infinite-dimensional manifolds where the index remains finite, the metric is complete and the function satisfies the Palais–Smale compactness condition, such as the energy functional for geodesics on a Riemannian manifold. The generalization to situations in which both index and coindex are infinite, but the relative index of any pair of critical points is finite, is known as Floer homology. Sergei Novikov generalized this construction to a homology theory associated to a closed one-form on a manifold.
|
|
They can be found in Japan, Asia, Africa, Europe, the Middle East, New Zealand and South America. The short-wheelbase models are not quite as common in Australia, but they are very popular in the homeland of Japan, where compactness and rugged 4X4 are a welcome combination; they are also used by the Syrian state. Today, few short-wheelbase 70 series can be purchased new—some are still in production today in some select South American countries and is still being produced by Toyota Auto Body at the Yoshiwara Plant in Toyota City. The 70 series was only sold in small numbers in the North American market.
|
|
There are at least two basic directions for embedding term to term relatedness, other than exact keyword matching, into a retrieval model: # compute semantic correlations between terms # compute frequency co-occurrence statistics from large corpora Recently Tsatsaronis focused on the first approach. They measure semantic relatedness (SR) using a thesaurus (O) like WordNet. It considers the path length, captured by compactness (SCM), and the path depth, captured by semantic path elaboration (SPE). They estimate the t_i \cdot t_j inner product by: t_i \cdot t_j = SR((t_i, t_j), (s_i, s_j), O) where si and sj are senses of terms ti and tj respectively, maximizing SCM \cdot SPE.
|
|
The Gould GR51 refined the concept of the GR37. The chassis, whilst based on a Ralt carbon fibre tub, was further customised for its application in hill climbing. The chassis was mated to a DTM-derived Opel- Cosworth V6 engine enlarged to 2.8 litres from the original 2.5 litres and mated to an Arrows F1 gearbox. Whilst down on absolute power compared to the 4-litre V8 monsters of its competitors, the compactness and light weight of the car, engine and gearbox gave the GR51 a greater power-to-weight ratio, which resulted in a car perfectly suited to the tortuously twisty hillclimbs of the British Championship.
|
|
The concept of the surface-barrier particle detector that Mayer first developed served as a cornerstone for the rapid development of numerous research areas. Because of its small size and compactness, the surface-barrier particle detector almost immediately started replacing many of the cumbersome detectors in use at that time, i.e. magnetic spectrometers and ionization chambers, revolutionizing low energy nuclear structure physics almost overnight. These semiconductor spectrometers led to the practical development of many modern materials analysis techniques that have wide spread use today, such as X-ray fluorescence and ion beam analysis of materials, including Rutherford backscattering, ion channeling, and X-ray spectrometry based on alpha particle sources.
|
|
On the other hand, the set of first-order sentences valid in the reals has arbitrarily large models due to the compactness theorem. Thus the least-upper-bound property cannot be expressed by any set of sentences in first-order logic. (In fact, every real- closed field satisfies the same first-order sentences in the signature \langle +,\cdot,\le\rangle as the real numbers.) In second-order logic, it is possible to write formal sentences which say "the domain is finite" or "the domain is of countable cardinality." To say that the domain is finite, use the sentence that says that every surjective function from the domain to itself is injective.
|
|
Existing electron accelerators such as SLAC could use electron-beam afterburners to greatly increase the energy of their particle beams, at the cost of beam intensity. Electron systems in general can provide tightly collimated, reliable beams; laser systems may offer more power and compactness. Thus, plasma wakefield accelerators could be used – if technical issues can be resolved – to both increase the maximum energy of the largest accelerators and to bring high energies into university laboratories and medical centres. Higher than 0.25 GeV/m gradients have been achieved by a dielectric laser accelerator, which may present another viable approach to building compact high-energy accelerators.
|
|
The one exception to the compactness of the quartet comes in the minuet. The minuet's first half is a standard 12 measures, but the second half is an unusually long 44 measures and involves harmonic excursions deep into the flat-side keys. The trio echoes the first and fourth movements by commencing with a B upbeat leading to an E. The finale completes the interconnectedness of the work with a theme audibly similar to those of the first movement and the minuet and trio. The movement is the most economical of all: the exposition and development commence with nearly identical passages and conclude in exactly the same way.
|
|
The Baptistery of San Giovanni ad Fontes in Lomello, also departed from the typical Palaeo-Christian compactness in the use of a tall central octagon. As it had been in Roman times, the commission of lay and religious buildings was used by the Lombard elite to express their prestige and to legitimate their authority. San Salvatore, Brescia In the 7th and 8th centuries, the Lombard architecture evolved in an original direction, with increasing references to Classical architecture. This trend, characterized by the co-presence of different influences and the adoption of new techniques, culminated in the reign of king Liutprand (712–744), in particular at Cividale del Friuli.
|
|
Since the operator norm of L(x) gives an equivalent norm on E, a standard compactness argument shows that, passing to a subsequence if necessary, the spectral idempotents of the a(n) and their corresponding eigenvalues are convergent. The limit of Jordan frames is a Jordan frame, since a limit of non-zero idempotents yields a non-zero idempotent by continuity of the operator norm. It follows that every Jordan frame is made up of r minimal idempotents. If e and f are orthogonal idempotents, the spectral theorem shows that e and f are polynomials in a = e − f, so that L(e) and L(f) commute.
|
|
The compositional type of the Madonna with an open cloak is derived from French cathedral sculpture (the cathedral portal in Rheims, 1230-1233), from where it spread through the Rhineland and Germany to central Europe. This connection with cathedral sculpture is the basis for the Strakonice Madonna’s clearly arranged and perfectly balanced composition, for the purity and sophistication of its forms as well as for the refinement and cultivated character of its physical gestures. With the compactness of its mass, simplicity and abstract plasticity, the Strakonice Madonna anticipates a new conception of sculpture in the 14th century.Jaromír Homolka: Madona strakonická, in: Jiří Kuthan (ed.), 1982, pp.
|
|
For example, compactness and connectedness are topological properties, whereas boundedness and completeness are not. Algebraic topology is the study of topologically invariant abstract algebra constructions on topological spaces. ;Topological space: A topological space (X, T) is a set X equipped with a collection T of subsets of X satisfying the following axioms: :# The empty set and X are in T. :# The union of any collection of sets in T is also in T. :# The intersection of any pair of sets in T is also in T. :The collection T is a topology on X. ;Topological sum: See Coproduct topology. ;Topologically complete: Completely metrizable spaces (i. e.
|
|
The type by virtue of its compactness also remained in use with the ROCA's outlying island garrisons (as coastal artillery) as well as mountain troops. 153 M116s were supplied to Japan (Japan Ground Self-Defense Force) and they were used until the 1980s. The Croatian Army fields M116 howitzers as ceremonial cannon, 12 of which are still kept in service for that purpose; an additional 45 guns have been retired and are no longer utilized by the army, some being stored in local army museums. In 2010 the M116 75 mm pack howitzer was used by the Turkish Army in operations against Kurdish separatists in southeastern Turkey.
|
|
Floer homology is the homology of this chain complex. The gradient flow line equation, in a situation where Floer's ideas can be successfully applied, is typically a geometrically meaningful and analytically tractable equation. For symplectic Floer homology, the gradient flow equation for a path in the loopspace is (a perturbed version of) the Cauchy–Riemann equation for a map of a cylinder (the total space of the path of loops) to the symplectic manifold of interest; solutions are known as pseudoholomorphic curves. The Gromov compactness theorem is then used to show that the differential is well-defined and squares to zero, so that the Floer homology is defined.
|
|
However, while standard Australian wall mounting plates have the same dimensions as those used in North America, they have different dimensions from those used in the UK or Europe. Switches (and other mechanisms) of this Australian design series are currently available in the UK (and other countries), together with wall-plates appropriate to the mounting standards of the countries concerned. For a short time, Australian rocker switches were exported to the United States in the 1960s. Although the switches had adequate ratings for usage on 120 V circuits and had advantages of compactness and distinctive appearance, they failed to establish themselves in the American market.
|
|
Daphnia pulex was the first crustacean to have its genome sequenced. Its genome contains 31,000 genes – 8,000 more than are present in the human genome – as a result of extensive gene duplication. One of the most astonishing features of the D. pulex genome is its compactness: despite being around 200 Mb in size (around 16-fold smaller than the human genome which is 3,200 Mb in size); its 12 chromosomes contain a minimum set of 30,907 predicted protein-coding genes, more than the 20,000–25,000 contained in the human counterpart. Such an efficient gene packaging is achieved by means of a small intron size.
|
|
The remaining nationalities had no right of self-government at all. Joseph Stalin's 1913 definition of a nation as "a historically constituted and stable community of people formed on the basis of common language, territory, economic life, and psychological makeup revealed in a common culture" was retained by Soviet authorities throughout the 1980s. However, in granting nationalities union republic status, three additional factors were considered: a population of at least 1 million, territorial compactness, and location on the borders of the Soviet Union. Although Lenin believed that eventually all nationalities would merge into one, he insisted that the Soviet Union be established as a federation of formally equal nations.
|
|
The group formed as a power trio with Hammond organ as the main instrument. Their first and only album sold itself through "...compactness, wealth of ideas, forceful lead vocals and complicated arrangements, enriched by pianist Robinson's tasteful use of classical strings which are on display along with spacious keyboard passages at their height in the mold of The Nice."Graf, Christian "Rock Musik Lexikon" (Taurus Press, Hamburg) One track, "Laughin' Tackle", includes 16 violins, 6 violas, 6 cellos, and 3 double bass, arranged by Robinson, and a drum solo by Underwood. Underwood remained in close contact with Blackmore, and visited Deep Purple in the studio while they were recording In Rock.
|
|
Kari Astala (born 26 July 1953 in Helsinki) is a Finnish mathematician, specializing in analysis. Astala graduated from the University of Helsinki with M.Sc. in 1977 and received his doctorate there in 1980 with thesis On Measures of compactness and ideal variations in Banach spaces. In the 1980s and 1990s he held academic appointments at the University of Helsinki and the Academy of Finland. He was a full professor at the University of Jyväskylä from 1995 to 2002, a full professor at the University of Helsinki from 2002 to 2017, and an Academy Professor from 2006 to 2011 at the Academy of Finland. Since 2017 he is an adjunct professor at Aalto University.
|
|
Let G be a group of isometries on a normed vector space X. One says that a sequence (x_k)\subset X converges to x\in X G-weakly, if for every sequence (g_k)\subset G, the sequence g_k(x_k-x) is weakly convergent to zero. A continuous embedding of two normed vector spaces, X\hookrightarrow Y is called cocompact relative to a group of isometries G on X if every G-weakly convergent sequence (x_k)\subset X is convergent in Y.C. Tintarev, Concentration analysis and compactness, in: Adimuri, K. Sandeep, I. Schindler, C. Tintarev, editors, Concentration Analysis and Applications to PDE ICTS Workshop, Bangalore, January 2012, , Birkhäuser, Trends in Mathematics (2013), 117–141.
|
|
The same set of points would not accumulate to any point of the open unit interval ; so the open unit interval is not compact. Euclidean space itself is not compact since it is not bounded. In particular, the sequence of points , which is not bounded, has no subsequence that converges to any real number. Apart from closed and bounded subsets of Euclidean space, typical examples of compact spaces are encountered in mathematical analysis, where the property of compactness of some topological spaces arises in the hypotheses or in the conclusions of many fundamental theorems, such as the Bolzano–Weierstrass theorem, the extreme value theorem, the Arzelà–Ascoli theorem, and the Peano existence theorem.
|
|
The Heine–Borel theorem, as the result is now known, is another special property possessed by closed and bounded sets of real numbers. This property was significant because it allowed for the passage from local information about a set (such as the continuity of a function) to global information about the set (such as the uniform continuity of a function). This sentiment was expressed by , who also exploited it in the development of the integral now bearing his name. Ultimately, the Russian school of point-set topology, under the direction of Pavel Alexandrov and Pavel Urysohn, formulated Heine–Borel compactness in a way that could be applied to the modern notion of a topological space.
|
|
However, the spaces in which every convergent filter has a unique limit are precisely the Hausdorff spaces. In general we must select, for each element of the index set, an element of the nonempty set of limits of the projected ultrafilter base, and of course this uses AC. However, it also shows that the compactness of the product of compact Hausdorff spaces can be proved using (BPI), and in fact the converse also holds. Studying the strength of Tychonoff's theorem for various restricted classes of spaces is an active area in set-theoretic topology. The analogue of Tychonoff's theorem in pointless topology does not require any form of the axiom of choice.
|
|
Neutrons in a degenerate neutron gas are spaced much more closely than electrons in an electron-degenerate gas because the more massive neutron has a much shorter wavelength at a given energy. In the case of neutron stars and white dwarfs, this phenomenon is compounded by the fact that the pressures within neutron stars are much higher than those in white dwarfs. The pressure increase is caused by the fact that the compactness of a neutron star causes gravitational forces to be much higher than in a less compact body with similar mass. The result is a star with a diameter on the order of a thousandth that of a white dwarf.
|
|
The language, Berthoff continues, reflects a "controlling intelligence, of right judgment and completed understanding". The sense of free inquiry and exploration which infused his earlier writing and accounted for its "rare force and expansiveness," tended to give way to "static enumeration". By comparison to the verbal music and kinetic energy of Moby-Dick, Melville's subsequent writings seem "relatively muted, even withheld" in his later works. Melville's paragraphing in his best work Berthoff considers to be the virtuous result of "compactness of form and free assembling of unanticipated further data", such as when the mysterious sperm whale is compared with Exodus's invisibility of God's face in the final paragraph of Chapter 86 ("The Tail").
|
|
Brompton folding bicycle Folded Brompton, side view A folding bicycle is a bicycle designed to fold into a compact form, facilitating transport and storage. When folded, the bikes can be more easily carried into buildings, on public transportation (facilitating mixed-mode commuting and bicycle commuting), and more easily stored in compact living quarters or aboard a car, boat or plane. Folding mechanisms vary, with each offering a distinct combination of folding speed, folding ease, compactness, ride, weight, durability, and price. Distinguished by the complexities of their folding mechanism, more demanding structural requirements, greater number of parts, and more specialized market appeal, folding bikes may be more expensive than comparable non-folding models.
|
|
The original goal was to develop algorithms for energy optimization of several biopolymers with respect to an arbitrary subset of internal coordinates such as bond lengths, bond angles torsion angles and phase angles. The efficient and general global optimization method which evolved from the original ICM method is still the central piece of the program. It is this basic algorithm which is used for peptide prediction, homology modeling and loop simulations, flexible macromolecular docking and energy refinement. However the complexity of problems related to structure prediction and analysis, as well as the desire for perfection, compactness and consistency, led to the program's expansion into neighboring areas such as graphics, chemistry, sequence analysis and database searches, mathematics, statistics and plotting.
|
|
Its young lemon scented leaves are used in herbal teas and the leaves are also often used in flavoring foods. This plant can thrive in arid or semi-arid regions, but it is necessary for the seed to pass a cold and humid season which break dormancy in order to produce germination. For a proper development it requires alkaline or neutral, aerated and light soil, otherwise the excessive humidity, compactness and damping will putrify the root system. This species requires direct sunlight or partial shade, but doesn't do well in interior corners or under taller plants; these situations will make the plant tend to yellowing and loss of leaves, absence of flowering and even death.
|
|
The Arzelà–Ascoli theorem is a fundamental result of mathematical analysis giving necessary and sufficient conditions to decide whether every sequence of a given family of real-valued continuous functions defined on a closed and bounded interval has a uniformly convergent subsequence. The main condition is the equicontinuity of the family of functions. The theorem is the basis of many proofs in mathematics, including that of the Peano existence theorem in the theory of ordinary differential equations, Montel's theorem in complex analysis, and the Peter–Weyl theorem in harmonic analysis and various results concerning compactness of integral operators. The notion of equicontinuity was introduced in the late 19th century by the Italian mathematicians Cesare Arzelà and Giulio Ascoli.
|
|
A weak form of the theorem was proven by , who established the sufficient condition for compactness, and by , who established the necessary condition and gave the first clear presentation of the result. A further generalization of the theorem was proven by , to sets of real-valued continuous functions with domain a compact metric space . Modern formulations of the theorem allow for the domain to be compact Hausdorff and for the range to be an arbitrary metric space. More general formulations of the theorem exist that give necessary and sufficient conditions for a family of functions from a compactly generated Hausdorff space into a uniform space to be compact in the compact-open topology; see .
|
|
The notion of local compactness is important in the study of topological groups mainly because every Hausdorff locally compact group G carries natural measures called the Haar measures which allow one to integrate measurable functions defined on G. The Lebesgue measure on the real line R is a special case of this. The Pontryagin dual of a topological abelian group A is locally compact if and only if A is locally compact. More precisely, Pontryagin duality defines a self-duality of the category of locally compact abelian groups. The study of locally compact abelian groups is the foundation of harmonic analysis, a field that has since spread to non-abelian locally compact groups.
|
|
In later civilian use in the US, the Stearman became particularly associated with stunt flying such as wing-walking, and with crop dusting, where its compactness worked well at low levels, where it had to dodge obstacles. Polikarpov Po-2, of which over 20,000 were built by the Soviet Union Modern biplane designs still exist in specialist roles such as aerobatics and agricultural aircraft with the competition aerobatics role and format for such a biplane well-defined by the mid-1930s by the Udet U 12 Flamingo and Waco Taperwing. The Pitts Special dominated aerobatics for many years after World War II and is still in production. The vast majority of biplane designs have been fitted with reciprocating engines.
|
|
The first Contaflex SLR was introduced in 1953 as one of the earliest 35mm SLR cameras equipped with a leaf shutter, but with a fixed lens. The Mecaflex was presented at photokina in 1951 and launched two years later with a leaf shutter behind the removable lens, while the Hasselblad 500C of 1957 has a leaf shutter in its interchangeable lens. This design, using a leaf shutter in an SLR, involves a complex sequence of events when the shutter is released, more looked upon as a challenge than a drawback at Zeiss Ikon, but no Contaflex model ever got a rapid return mirror. The advantages are low manufacturing costs, compactness and flash synchronization at all shutter speeds.
|
|
This saves space for intense cultivation. It could provide the opportunity of greenhouse-produced fruits and vegetable to be available for the market more quickly and less expensively due to the effect of LED lighting on earliness, compactness and quality of products . Incandescent and fluorescent lamps currently available for lighting greenhouse, phytotrones and plant incubators emit color bands that may cause unwanted stem elongation and low quality in plant species, are electrically inefficient, short-lived, and particularly not eco-friendly for having hazardous waste disposal issues. On the other side, high-pressure sodium (HPS) discharge lamps have been well established in the greenhouses for their sufficient light intensities which support transplants and seedlings growth and development.
|
|
Pilotka of the Red Army A man in Russia dressed as a veteran of the Great Patriotic War wearing both a gymnastyorka and a pilotka. In the Soviet Union, the garrison cap was known as pilotka (пилотка, from "pilot" — the original cap was a part of the air force pilots' uniform in World War I). It was the most common type of cap used by the Red Army during WWII and after until the 1980s. The pilotka was worn during the summer season instead of the winter ushanka. It continues to be worn in modern Russia, although more in the Air Force and the Navy, especially among submarine personnel, where its compactness is inherently practical.
|
|
A widespread application of pneumatic motors is in hand-held tools, impact wrenches, pulse tools, screwdrivers, nut runners, drills, grinders, sanders and so on. Pneumatic motors are also used stationary in a wide range of industrial applications. Though overall energy efficiency of pneumatics tools is low and they require access to a compressed-air source, there are several advantages over electric tools. They offer greater power density (a smaller pneumatic motor can provide the same amount of power as a larger electric motor), do not require an auxiliary speed controller (adding to its compactness), generate less heat, and can be used in more volatile atmospheres as they do not require electric power Air motors.
|
|
Any affine variety can be completed, in a unique way, into a projective variety by adding its points at infinity, which consists of homogenizing the defining polynomials, and removing the components that are contained in the hyperplane at infinity, by saturating with respect to the homogenizing variable. An important property of projective spaces and projective varieties is that the image of a projective variety under a morphism of algebraic varieties is closed for Zariski topology (that is, it is an algebraic set). This is a generalization to every ground field of the compactness of the real and complex projective space. A projective space is itself a projective variety, being the set of zeros of the zero polynomial.
|
|
Mathematically LQG is local gauge theory of the self-dual subgroup of the complexified Lorentz group, which is related to the action of the Lorentz group on Weyl spinors commonly used in elementary particle physics. This is partly a matter of mathematical convenience, as it results in a compact group SO(3) or SU(2) as gauge group, as opposed to the non-compact groups SO(3,1) or SL(2.C). The compactness of the Lie group avoids some thus- far unsolved difficulties in the quantization of gauge theories of noncompact lie groups, and is responsible for the discreteness of the area and volume spectra. The theory involving the Immirzi parameter is necessary to resolve an ambiguity in the process of complexification.
|
|
In the 19th century, several disparate mathematical properties were understood that would later be seen as consequences of compactness. On the one hand, Bernard Bolzano (1817) had been aware that any bounded sequence of points (in the line or plane, for instance) has a subsequence that must eventually get arbitrarily close to some other point, called a limit point. Bolzano's proof relied on the method of bisection: the sequence was placed into an interval that was then divided into two equal parts, and a part containing infinitely many terms of the sequence was selected. The process could then be repeated by dividing the resulting smaller interval into smaller and smaller parts—until it closes down on the desired limit point.
|
|
However, a different notion of compactness altogether had also slowly emerged at the end of the 19th century from the study of the continuum, which was seen as fundamental for the rigorous formulation of analysis. In 1870, Eduard Heine showed that a continuous function defined on a closed and bounded interval was in fact uniformly continuous. In the course of the proof, he made use of a lemma that from any countable cover of the interval by smaller open intervals, it was possible to select a finite number of these that also covered it. The significance of this lemma was recognized by Émile Borel (1895), and it was generalized to arbitrary collections of intervals by Pierre Cousin (1895) and Henri Lebesgue (1904).
|
|
To prevent a negative impact on water bodies and to assist environmental recovery and conservation of the ecological balance of the Baltic Sea region, in accordance with the international agreement HELCOM (Baltic Marine Environment Protection Convention, adopted in Helsinki in 1992) JSC "Rosterminalugol" finished the reconstruction of sewage treatment plants and water purification systems in December 2016. A new automatic wastewater treatment system installed in the port uses dispersion-reagent nanoflotation technology. It brings mixed-utility domestic and rainwater flow to optimal indexes of the maximum permitted norms for wastewater, and it fully cleanses drains of coal dust and organic materials. The use of chemical reagents for cleaning in the system is significantly reduced, and its high effectiveness and compactness of structures allows for reduced power consumption.
|
|
Compactness, or density, plays an important yet limited role in sustainable urban development because it can support reductions in per- capita transport energy use by increasing walking, cycling, active transport and public transit use. The relatively low density of some urban and especially suburban and exurban development is too low to support efficient transit and walk-to destinations. Such low-density development is a characteristic of urban sprawl, which is the major cause of high dependence on private automobiles, inefficient infrastructure, increased obesity, loss of farmlands and natural habitats, pollution, and so on. For these reasons, sustainable urbanism tends to promote more compact development with greater intensities of use and greater variety of uses and activities in a given urban area.
|
|
In a Hilbert system, the premises and conclusion of the inference rules are simply formulae of some language, usually employing metavariables. For graphical compactness of the presentation and to emphasize the distinction between axioms and rules of inference, this section uses the sequent notation (\vdash) instead of a vertical presentation of rules. The formal language for classical propositional logic can be expressed using just negation (¬), implication (→) and propositional symbols. A well-known axiomatization, comprising three axiom schemata and one inference rule (modus ponens), is: (CA1) ⊢ A → (B → A) (CA2) ⊢ (A → (B → C)) → ((A → B) → (A → C)) (CA3) ⊢ (¬A → ¬B) → (B → A) (MP) A, A → B ⊢ B It may seem redundant to have two notions of inference in this case, ⊢ and →.
|
|
The basis sets in the product topology have almost the same definition as the above, except with the qualification that all but finitely many Ui are equal to the component space Xi. The product topology satisfies a very desirable property for maps fi : Y → Xi into the component spaces: the product map f: Y → X defined by the component functions fi is continuous if and only if all the fi are continuous. As shown above, this does not always hold in the box topology. This actually makes the box topology very useful for providing counterexamples--many qualities such as compactness, connectedness, metrizability, etc., if possessed by the factor spaces, are not in general preserved in the product with this topology.
|
|
The work of Laura P. Spinadel is marked by the pursuit of comprehensiveness in ideas, processes and results undoubtedly influenced by the anthroposophical education. Since starting out with a humanistic, theosophical and sensualist vision, Laura P. Spinadel has moved towards a holistic and ecological position, that seeks to put maximum emphasis on the health of open and closed spaces, as well as on the requirements and principles of bio-construction. Holistic thinking implies profoundly replacing rational and analytical thought with a more inclusive thinking, in which all factors are taken into account, even those that seem trivial or invisible, such as perception, health or freedom. The Compact City in Vienna, Austria (1995-2001), was based on superimposed layers, compactness and high density structures.
|
|
The action is also engineered for compactness. In early designs, the action was very simple and comprised a sprung arrangement where air was compressed in the lower chest of the organ, and depressing a key would open a pallet that would release the air up to the pipe ranks. Later designs, as technology progressed, started to encompass more of what could be found in church and other organs and more complex mechanisms, including rollerboards, pedalboards, reed organs (rather than pipes), and eventually electric rather than mechanical actions. Such residence organs were the province of professional house organ makers (who continue to exist even today) in the main, with a notable exception of Toggenburg where (at that time) residence organs were often constructed by amateurs and enthusiasts themselves.
|
|
Zames’s research focused on imprecisely modelled systems using the input-output method, an approach that is distinct from the state space representation that dominated control theory for several decades. At the core of much of his work is the objective of complexity reduction through organization: > For the purposes of control design, gross qualitative properties such as > robustness can be analyzed and predicted without depending on accurate > models or syntheses. Mathematical analysis provides topological tools that > are very well suited for this purpose, such as compactness, contraction, and > fixed-point methods. Furthermore, in control design, where there is lots of > model uncertainty, it is often more important to be able to gauge > qualitative behaviour (robustness, stability, existence of oscillations) > than to compute exactly.
|
|
To say that the domain has countable cardinality, use the sentence that says that there is a bijection between every two infinite subsets of the domain. It follows from the compactness theorem and the upward Löwenheim–Skolem theorem that it is not possible to characterize finiteness or countability, respectively, in first-order logic. Certain fragments of second order logic like ESO are also more expressive than first-order logic even though they are strictly less expressive than the full second-order logic. ESO also enjoys translation equivalence with some extensions of first-order logic which allow non-linear ordering of quantifier dependencies, like first-order logic extended with Henkin quantifiers, Hintikka and Sandu's independence- friendly logic, and Väänänen's dependence logic.
|
|
First, one extends the signature by adding a new constant symbol for every element of M. The complete theory of M for the extended signature σ' is called the elementary diagram of M. In the next step one adds κ many new constant symbols to the signature and adds to the elementary diagram of M the sentences c ≠ c' for any two distinct new constant symbols c and c'. Using the compactness theorem, the resulting theory is easily seen to be consistent. Since its models must have cardinality at least κ, the downward part of this theorem guarantees the existence of a model N which has cardinality exactly κ. It contains an isomorphic copy of M as an elementary substructure.
|
|
Eventually Asahi Optical and Pentax slowly shifted much of their lens production under the Pentax name and transitioned some of the successful designs that were first introduced under the Takumar name to use Asahi/Pentax badging as well as beginning to use the "smc" abbreviation. Eventually the Asahi partnership disappeared and the Pentax name became solely used. Pentax lenses saw many feature changes to answer the market, such as: incorporating "Auto-Aperture" with the M42, the light weight and compactness with the 'M' series, Aperture Priority overrides with the 'A' series, and Auto-Focus with the 'F' series. Modern Pentax lenses for digital SLR cameras have seen the elimination of the aperture ring completely as found on Pentax DA and D-FA series lenses.
|
|
Compared to pistons, the rotors of a Wankel engine are massive and need time to warm up; yet, as NSU found to their cost with their Ro80 car, neither bikers nor car drivers have proved patient enough to give the engines time to reach operating temperature, and this impatience adversely affected reliability. The Wankel's virtues of smoothness, compactness and lightness should be particularly suitable for aircraft, especially since the start-up procedure and the taxi to the runway gives ample time for warm-up. However, although wankels have been fitted into gliders and light aircraft such as the ARV Super2, such engines are still uncommon in aviation. Perhaps inevitably, the Norton Classic's Wankel engine was further developed at Staverton into the MidWest aero-engine.
|
|
Most tourists visit the historic area, which is a UNESCO World Heritage Site, between the river Tomebamba and the street Gran Colombia to the north, General Torres to the west, and Hermano Miguel to the east. This area's compactness, grid-like layout, and numerous readily identifiable monuments make it easy to navigate. Outside this area the city can be confusing, as there are dozens of narrow colonial streets with similar buildings. Major fiestas of Cuenca come at the time of the "Mass of Children" that is carried out the day of the Arrival of Kings (January 6 - Epiphany Day), or in the commemoration of the independence of the city (November 3), during which processions, cultural acts and dances are organized.
|
|
In mathematics, the simplicial approximation theorem is a foundational result for algebraic topology, guaranteeing that continuous mappings can be (by a slight deformation) approximated by ones that are piecewise of the simplest kind. It applies to mappings between spaces that are built up from simplices-- that is, finite simplicial complexes. The general continuous mapping between such spaces can be represented approximately by the type of mapping that is (affine-) linear on each simplex into another simplex, at the cost (i) of sufficient barycentric subdivision of the simplices of the domain, and (ii) replacement of the actual mapping by a homotopic one. This theorem was first proved by L.E.J. Brouwer, by use of the Lebesgue covering theorem (a result based on compactness).
|
|
Gromov showed that certain moduli spaces of pseudoholomorphic curves (satisfying additional specified conditions) are compact, and described the way in which pseudoholomorphic curves can degenerate when only finite energy is assumed. (The finite energy condition holds most notably for curves with a fixed homology class in a symplectic manifold where J is \omega-tame or \omega-compatible). This Gromov compactness theorem, now greatly generalized using stable maps, makes possible the definition of Gromov–Witten invariants, which count pseudoholomorphic curves in symplectic manifolds. Compact moduli spaces of pseudoholomorphic curves are also used to construct Floer homology, which Andreas Floer (and later authors, in greater generality) used to prove the famous conjecture of Vladimir Arnol'd concerning the number of fixed points of Hamiltonian flows.
|
|
This is lower than the separation between the two components of 61 Cygni, and so the two most likely do not share a common atmosphere. The compactness of the astrosphere is likely due to the low mass outflow and the relatively high velocity through the local interstellar medium. 61 Cygni B displays a more chaotic pattern of variability than A, with significant short-term flares. There is an 11.7-year periodicity to the overall activity cycle of B. Both stars exhibit stellar flare activity, but the chromosphere of B is 25% more active than for 61 Cygni A. As a result of differential rotation, the period of rotation varies by latitude from 32 to 47 days, with an average period of 38 days.
|
|
2018 Olympic venues The compactness of the venue locations for the 2018 Winter Olympics and 2018 Winter Paralympics, hosted by the county of Pyeongchang, was one of the winning arguments of the bid. The Games were gathered around two main venues: these were the mountain resort of Alpensia in Pyeongchang for the outdoor (snow) sports (Nordic and alpine skiing, bobsleigh, etc.) and the coastal city of Gangneung for the indoor (ice) sports (figure skating, ice hockey, etc.) There were also two stand-alone mountain venues. The Games had a total of 13 competition venues, six of which were constructed especially for the Games. The last venue to start being constructed was the long track speed skating oval in Gangneung; work began on this in October 2014.
|
|
An actual image displayed by a VR headset, showing compensation for lens distortion and chromatic aberration The lenses of the headset are responsible for mapping the up-close display to a wide field of view, while also providing a more comfortable distant point of focus. One challenge with this is providing consistency of focus: because eyes are free to turn within the headset, it is important to avoid having to refocus to prevent eye strain. Fresnel lenses are commonly used in virtual reality headsets due to their compactness and lightweight structure. The lenses do not use multiple pieces of material in their lenses like other lenses, but the lens will be broken down into sections, allowing the individual to have a wider range of view.
|
|
Every finite planar graph can be colored with four colors, by the four-color theorem. The De Bruijn–Erdős theorem then shows that every graph that can be drawn without crossings in the plane, finite or infinite, can be colored with four colors. More generally, every infinite graph for which all finite subgraphs are planar can again be four-colored.. states the same result for the five-color theorem for countable planar graphs, as the four-color theorem had not yet been proven when he published his survey, and as the proof of the De Bruijn–Erdős theorem that he gives only applies to countable graphs. For the generalization to graphs in which every finite subgraph is planar (proved directly via Gödel's compactness theorem), see .
|
|
One can generalize the definition of cellular automaton to those maps that are defined by rules for computing the new value of each position in a configuration based on the values of cells in a finite but variable neighborhood surrounding the position. In this case, as in the classical definition, the local rule is the same for all cells, but the neighborhood is also a function of the configuration around the position. The counterexample given above for a continuous and shift-equivariant map which is not a classical cellular automaton, is an example of a generalized cellular automaton. When the alphabet is finite, the definition of generalized cellular automata coincides with the classical definition of cellular automata due to the compactness of the shift space.
|
|
The NC.420 was a twin-engine aircraft, an uncommon arrangement for ship-borne reconnaissance aircraft where a premium was placed on compactness. It could nevertheless be reduced to a width of with the wings folded, which made it comparable in size (and weight) to the Loire 130 it was slated to have replaced. Work on the prototype was nearing completion in June 1940, but the German victory in the Battle of France ended many French military programs, including that for the NC.420. The German Armistice Commission approved the construction of up to fifteen aircraft to replace Loire 130s that were damaged or destroyed through normal operation, but work thereafter proceeded slowly and the prototype was only completed in mid-1943.
|
|
Netflix has also struck deals with many consumer-electronics makers to make their interface available in the device's menus, for their streaming subscribers. This symbiotic relationship between Netflix and consumer electronics makers has helped propel Netflix to become the largest subscription video service in the U.S., using up to 20% of U.S. bandwidth at peak times. Media players are often designed for compactness and affordability, and tend to have small or non-existent hardware displays other than simple LED lights to indicate whether the device is powered on. Interface navigation on the television is usually done with an infrared remote control, while more-advanced digital media players come with high-performance remote controls which allow control of the interface using integrated touch sensors.
|
|
According to The New Princeton Encyclopedia of Poetry and Poetics, the defining traits of the prose poem are "unity even in brevity and poetic quality even without the line breaks of free verse: high patterning, rhythmic and figural repetition, sustained intensity, and compactness."Alex Preminger and T. V. F. Brogan, co-editors, The New Princeton Encyclopedia of Poetry and Poetics (Princeton, NJ: Princeton University Press, 1993), p. xlvi Invented in the nineteenth century, the modern prose poem form is largely indebted to Charles Baudelaire's experiments in the genre, notably in his Petits poèmes en prose (1869), which created the subsequent interest in France exemplified by later writers such as Stéphane Mallarmé and Arthur Rimbaud. In English literature, Edgar Allan Poe and Charles Kingsley were progenitors of the form.
|
|
The gudgeon pin is typically a forged short hollow rod made of a steel alloy of high strength and hardness that may be physically separated from both the connecting rod and piston or crosshead. The design of the gudgeon pin, especially in the case of small, high-revving automotive engines is challenging. The gudgeon pin has to operate under some of the highest temperatures experienced in the engine, with difficulties in lubrication due to its location, while remaining small and light so as to fit into the piston diameter and not unduly add to the reciprocating mass. The requirements for lightness and compactness demand a small diameter rod that is subject to heavy shear and bending loads, with some of the highest pressure loadings of any bearing in the whole engine.
|
|
Rototoms are furnished with "Controlled Sound" Black Dot drumheads as standard, where the reinforced black central sound area is said to provide ”compactness of sound with minimum ‘overring’ plus great durability.” Other heads, each of which may offer distinctive tonal differences, are also available, including Pinstripe, which offers “a dampened tom-tom sound — wet, flat and funky”, FiberSkyn, which produces “a round, dark tom-tom sound...especially suited for recording use and in tuned tom-toms for general orchestral use, and Timpani, intended for “purity of sound and consistent performance.” Rototoms can be tuned quickly by rotating the drumhead, which sits in a threaded metal ring. Rotation raises or lowers the tension hoop relative to the rim, which increases or decreases the pitch of the drum by increasing or decreasing the tension of the drumhead.
|
|
The fine topology was introduced in 1940 by Henri Cartan to aid in the study of thin sets and was initially considered to be somewhat pathological due to the absence of a number of properties such as local compactness which are so frequently useful in analysis. Subsequent work has shown that the lack of such properties is to a certain extent compensated for by the presence of other slightly less strong properties such as the quasi-Lindelöf property. In one dimension, that is, on the real line, the fine topology coincides with the usual topology since in that case the subharmonic functions are precisely the convex functions which are already continuous in the usual (Euclidean) topology. Thus, the fine topology is of most interest in \R^n where n\geq 2.
|
|
In January 2016, a letter was published in the journal Nature called: "Eight per cent leakage of Lyman continuum photons from a compact, star-forming dwarf galaxy" by authors: Y.I. Izotov, I. Orlitová, D. Schaerer, T.X. Thuan, A. Verhamme, N.G. Guseva & G. Worseck. The abstract states: "One of the key questions in observational cosmology is the identification of the sources responsible for ionisation of the Universe after the cosmic Dark Ages". It also states: "Here we present far-ultraviolet observations of a nearby low-mass star-forming galaxy, J0925+1403, selected for its compactness and high excitation... The galaxy is 'leaking' ionising radiation, with an escape fraction of 7.8%." These levels of radiation are thought to be similar to those of the first galaxies in the universe, which emerged in a time known as reionization.
|
|
Since the length of the ground loop system is minimized, DX systems require smaller drill rigs - in comparison with water well rigs - which can get into smaller spaces and yards. This provides a greater flexibility of installation and makes it an available option in more areas and for more properties. The compactness of the earth loop systems, which require less drilling and smaller borehole, makes up for a simpler system that is cheaper and quicker to install.Typical drill rig for water source installation, length 22 ftThe relative simplicity of refrigerant-based systems – relying on a one-step heat exchange process – translates not only into a simpler and less expensive installation but also into less maintenance as well: unlike water-based systems, they do not require maintenance to top up water and glycol levels.
|
|
The two-stage supercharged Allison was essentially developed as an "add on" to the single-stage engine, and required minimal changes to the base engine. While it lacked the refinement, compactness and after-cooler of the two-stage Merlin, the Allison used a pressure-altitude governed variable-speed first stage. Various configurations of this auxiliary supercharger were used in production versions of the V-1710 that powered aircraft such as the Bell P-63 and North American P-82E/F/G series. In addition, it was tried or studied as the powerplant for many experimental and test aircraft such as the Curtiss XP-55 Ascender, North American XP-51J "lightweight Mustang", Boeing XB-38 Flying Fortress, and Republic XP-47A (AP-10), both of the latter with turbo-superchargers.
|
|
The advantages of a sidevalve engine include: simplicity, reliability, low part count, low cost, low weight, compactness, responsive low-speed power, low mechanical engine noise, and insensitivity to low-octane fuel. The absence of a complicated valvetrain allows a compact engine that is cheap to manufacture, since the cylinder head may be little more than a simple metal casting. These advantages explain why side valve engines were used for economy cars, trucks, and agricultural engines for many years, while OHV designs came to be specified only for high-performance applications such as aircraft, luxury cars, sports cars, and some motorcycles. At top dead centre, the piston gets very close to the flat portion of the cylinder head above, and the resultant squish turbulence produces excellent fuel/air mixing.
|
|
Antimatter annihilates with regular matter on contact, so it would be necessary to prevent contact, for example by producing antimatter in the form of solid charged or magnetized particles, and suspending them using electromagnetic fields in a near-perfect vacuum. The obvious solution of confining a charged object inside a similarly charged container is not feasible as the electric field inside is uniform. For this reason it is necessary to have charged objects moving relative to the container which can be confined to a central region by magnetic fields; for example, in the form of a toroid or Penning trap (see below). In order to achieve compactness given macroscopic weight, the overall electric charge of the antimatter weapon core would have to be very small compared to the number of particles.
|
|
This argument allows us to discuss specific features of the model that we may otherwise miss—for example, a bound on a specific increasing sequence cn can be expressed as realizing the type which uses countably many parameters. If the sequence is not definable, this fact about the structure cannot be described using the base language, so a weakly saturated structure may not bound the sequence, while an ω-saturated structure will. The reason we only require parameter sets that are strictly smaller than the model is trivial: without this restriction, no infinite model is saturated. Consider a model M, and the type Each finite subset of this type is realized in the (infinite) model M, so by compactness it is consistent with M, but is trivially not realized.
|
|
Urban planning is the mechanism by which much of the New Urban Agenda can be accomplished. In particular NUA ¶51 calls for: > urban planning and design instruments that support sustainable management > and use of natural resources and land, appropriate compactness and density, > polycentrism, and mixed uses, through infill or planned urban extension > strategies as applicable, to trigger economies of scale and agglomeration, > strengthen food system planning, enhance resource efficiency, urban > resilience, and environmental sustainability. In ¶66 the parties endorse a smart city model: > We commit ourselves to adopting a smart-city approach that makes use of > opportunities from digitization, clean energy and technologies, as well as > innovative transport technologies, thus providing options for inhabitants to > make more environmentally friendly choices and boost sustainable economic > growth and enabling cities to improve their service delivery.
|
|
Because of this, a lead (high Z) shield is 20–30% better as a gamma shield than an equal mass of another low-Z shielding material, such as aluminium, concrete, water, or soil; lead's major advantage is not in lower weight, but rather its compactness due to its higher density. Protective clothing, goggles and respirators can protect from internal contact with or ingestion of alpha or beta emitting particles, but provide no protection from gamma radiation from external sources. The higher the energy of the gamma rays, the thicker the shielding made from the same shielding material is required. Materials for shielding gamma rays are typically measured by the thickness required to reduce the intensity of the gamma rays by one half (the half value layer or HVL).
|
|
A metric appears in the definition of total boundedness only to ensure that each element of the finite cover is of comparable size, and can be weakened to that of a uniform structure. A subset S of a uniform space X is totally bounded if and only if, for any entourage E, there exists a finite cover of S by subsets of X each of whose Cartesian squares is a subset of E. (In other words, replaces the "size" , and a subset is of size E if its Cartesian square is a subset of E.) C.f. definition 39.7 and lemma 39.8. The definition can be extended still further, to any category of spaces with a notion of compactness and Cauchy completion: a space is totally bounded if and only if its (Cauchy) completion is compact.
|
|
Active speakers have several advantages, the most obvious being their compactness and simplicity. Additionally the amplifier(s) can be designed to closely match the optimal requirements of the speaker it will power; and the speaker designer is not required to include a passive crossover, decreasing production cost and possibly sound quality. Some also claim that the shorter distances between components can decrease external interference and increase fidelity; although this is highly dubious, and the reciprocal argument can also be made. Disadvantages include heavier loudspeaker enclosures; reduced reliability due to active electronic components within; and the need to supply both the audio signal and power to every unit separately, typically requiring two cables to be run to each speaker (as opposed to the single cable required with passive speakers and an external amplifier).
|
|
In mathematics, the Kodaira embedding theorem characterises non-singular projective varieties, over the complex numbers, amongst compact Kähler manifolds. In effect it says precisely which complex manifolds are defined by homogeneous polynomials. Kunihiko Kodaira's result is that for a compact Kähler manifold M, with a Hodge metric, meaning that the cohomology class in degree 2 defined by the Kähler form ω is an integral cohomology class, there is a complex-analytic embedding of M into complex projective space of some high enough dimension N. The fact that M embeds as an algebraic variety follows from its compactness by Chow's theorem. A Kähler manifold with a Hodge metric is occasionally called a Hodge manifold (named after W. V. D. Hodge), so Kodaira's results states that Hodge manifolds are projective.
|
|
Almaty 2022 was an unsuccessful bid for the 2022 Winter Olympics by the city of Almaty and the National Olympic Committee of the Republic of Kazakhstan. The IOC selected the host city for the 2022 Winter Olympics at the 128th IOC Session in Kuala Lumpur, Malaysia on July 31, 2015, in which Beijing won. Under the slogan “Keeping it Real,” Almaty had emphasized their traditional winter setting with tall mountains and plenty of natural snow coverage as well as compactness — most venues are within half an hour’s travel through Alpine scenery.Almaty offers traditional winter setting in 2022 Olympic bid, The Washington Post, 2015/7/28Almaty 2022 Winter Olympic bid at a glance, The Washington Times, 2015/7/26 Almaty was the only contender to have never hosted the Olympics before.
|
|
The compactness of the original Betza notation can be partly preserved by choice of convenient defaults: the default modality on non-final legs is m (against mc on final legs), while the default direction set for a continuation step is all directions except exactly backwards. This way a chu shogi lion can be written as KNADcaKmcabK, where the KNAD direct leaps to the 24 possible target squares are supplemented by caK for the hit-and-run and double captures, and mcabK for the rifle captures and conditional turn pass. Some convenience extensions of XBetza are e as extra modality for e.p. capture, O as an atom to specify castling, (the range indicating how far the king moves), and i as a modifier for indicating moves that only virgin pieces have (so that the complete FIDE pawn becomes fmWfceFifmnD).
|
|
Classical Chinese is distinguished from written vernacular Chinese in its style, which appears extremely concise and compact to modern Chinese speakers, and to some extent in the use of different lexical items (vocabulary). An essay in Classical Chinese, for example, might use half as many Chinese characters as in vernacular Chinese to relate the same content. In terms of conciseness and compactness, Classical Chinese rarely uses words composed of two Chinese characters; nearly all words are of one syllable only. This stands directly in contrast with modern Northern Chinese varieties including Mandarin, in which two-syllable, three- syllable, and four-syllable words are extremely common, whilst although two- syllable words are also quite common within modern Southern Chinese varieties, they are still more archaic in that they use more one-syllable words than Northern Chinese varieties.
|
|
This method is a combination of three characteristics of the image: partition of the image based on histogram analysis is checked by high compactness of the clusters (objects), and high gradients of their borders. For that purpose two spaces have to be introduced: one space is the one- dimensional histogram of brightness H = H(B); the second space is the dual 3-dimensional space of the original image itself B = B(x, y). The first space allows to measure how compactly the brightness of the image is distributed by calculating a minimal clustering kmin. Threshold brightness T corresponding to kmin defines the binary (black-and-white) image – bitmap b = φ(x, y), where φ(x, y) = 0, if B(x, y) < T, and φ(x, y) = 1, if B(x, y) ≥ T. The bitmap b is an object in dual space.
|
|
The family is built in a bullpup layout (the action is behind the trigger group), with a forward-mounted pistol grip. The main advantage of this type of arrangement is the overall compactness of the weapon, which can be achieved without compromising the barrel length, hence the overall length of the L85 rifle is shorter than a carbine, but the barrel length is that of an assault rifle. While left-handed conversion kits were trialled, the weapon as finally issued must be used exclusively right- handed since the ejection port and cocking handle (which reciprocates during firing) are on the right side of the receiver, making aimed fire from the left shoulder difficult. This can also give rise to a tactical disadvantage when firing around the left side of cover, where the shooter must expose the majority of his body.
|
|
This added to the engines' compactness which was needed to allow its use in front-wheel drive applications, as well as rear-wheel drive cars. Other features of these engines include: an oil to water heat exchanger is mounted within the V of the engine block, Bosch Motronic engine management system with full sequential fuel injection, knock control on each bank, Distributorless Ignition System (DIS), and closed loop lambda control. Minor changes were made to the unit during its production, including an uprated oil pump around mid-1997, with the addition of larger oilways in the head machining (T-Vents), and modified valve lifters to reduce top-end valve noise, at which point the cam belt arrangement also changed with the lower idler moving. Around 1998, the spin-on metal canister oil filter was changed to a disposable paper element.
|
|
The first model was presented at photokina in Cologne in 1972 and was called the Olympus M-1. Thirteen years earlier, the release of the Nikon F had made the 35 mm SLR the standard choice for professionals accustomed to Leica and other rangefinders, but it had driven the market towards heavy and bulky cameras. The Olympus M-1 changed this and with it began a reduction of size, weight and noise of the 35 mm SLRs. It was designed by a team led by Yoshihisa Maitani, who had already created the Pen and Pen F cameras, noted for their compactness. Olympus OM-1n with 50mm / f1.8 lens Since Leica's flagship rangefinder cameras are known as the M Series, Leica complained about the name of the M-1, forcing Olympus to rename it as the OM-1 to further clarify between the brands.
|
|
In mathematics, a weakly compact cardinal is a certain kind of cardinal number introduced by ; weakly compact cardinals are large cardinals, meaning that their existence cannot be proven from the standard axioms of set theory. (Tarski originally called them "not strongly incompact" cardinals.) Formally, a cardinal κ is defined to be weakly compact if it is uncountable and for every function f: [κ] 2 → {0, 1} there is a set of cardinality κ that is homogeneous for f. In this context, [κ] 2 means the set of 2-element subsets of κ, and a subset S of κ is homogeneous for f if and only if either all of [S]2 maps to 0 or all of it maps to 1. The name "weakly compact" refers to the fact that if a cardinal is weakly compact then a certain related infinitary language satisfies a version of the compactness theorem; see below.
|
|
In mathematical logic, Lindström's theorem (named after Swedish logician Per Lindström, who published it in 1969) states that first-order logic is the strongest logicIn the sense of Heinz-Dieter Ebbinghaus Extended logics: the general framework in K. J. Barwise and S. Feferman, editors, Model-theoretic logics, 1985 page 43 (satisfying certain conditions, e.g. closure under classical negation) having both the (countable) compactness property and the (downward) Löwenheim–Skolem property.A companion to philosophical logic by Dale Jacquette 2005 page 329 Lindström's theorem is perhaps the best known result of what later became known as abstract model theory, the basic notion of which is an abstract logic; the more general notion of an institution was later introduced, which advances from a set-theoretical notion of model to a category-theoretical one. Lindström had previously obtained a similar result in studying first-order logics extended with Lindström quantifiers.
|
|
The CLC had two CdS cells connected in series that purportedly offered superior metering, especially in high contrast lighting. Also, the 7S had a hot shoe instead of the cold shoe of the 7. The 9 was the same as the 7S with the addition of a slightly faster f/1.7 lens, additional shutter speeds of 1/2 and 1 sec., and Minolta's "Easy- Flash" system, which simplified flash photography. The Hi-Matic 11 of 1969 was similar to the 9, but the 11 had shutter priority automatic exposure, aperture and shutter speed displayed in the viewfinder, and no aperture ring. In 1969 Minolta came out with a new, smaller model, the Hi-Matic C. In the interest of compactness, it had a smaller 40 mm f/2.7 lens (which was collapsible), reduced aperture and shutter speed ranges, and no longer featured a rangefinder.
|
|
The style of Remo Rossi was modified over the years from the pure academic level of the 1930s and 1940s which was maximally expressed in the feminine nudes and with the rounded shapes of large animals. Remo then progressed to more geometric and rigid shapes in several Opere di arte sacra, in the Acrobati, and in scenes portraying humans and animals. The de-materialisation observable in the latest works seem to be inspired by the shapes typical of Alberto Giacometti in which the plasticity loses compactness and where the empty spaces prevail over the filled spaces. Remo Rossi was mostly appreciated for his cemeterial art as well as for his highly visible art pieces that decorated several public buildings in Ticino and in Switzerland and ultimately also for the numerous commemoration coins that were mandated to him by the Federal, Cantonal and Communal public authorities.
|
|
In an opinion by Justice William J. Brennan joined partially by Justices Byron White, Thurgood Marshall, Harry Blackmun, and John Paul Stevens the Court used the term "vote dilution through submergence" to describe claims that a jurisdiction's use of an at-large/multimember election system or gerrymandered redistricting plan diluted minority votes, and it established a legal framework for assessing such claims under Section 2. Under the Gingles test, plaintiffs must show the existence of three preconditions: # The racial or language minority group "sufficiently large and geographically compact to constitute a majority in a single-member district"; # The minority group is "politically cohesive" (meaning its members tend to vote similarly); and # The "majority votes sufficiently as a bloc to enable it ... usually to defeat the minority's preferred candidate."Thornburg v. Gingles, The first precondition is known as the "compactness" requirement and concerns whether a majority-minority district can be created.
|
|
French Wikisource: Hubert Pierquin] 1912 Tolkien comments on the risk of using a translation as a substitute for study with grammar and dictionary, calling it an abuse, and writing that He notes that a readable translation cannot always translate an Old English (OE) word the same way; thus eacen is rendered 'stalwart', 'broad', 'huge', and 'mighty', correctly in each case to fit the context, but losing the clue to the word's special meaning, "not 'large' but 'enlarged'". The word implies, in fact, supernatural or superhuman power, like Beowulf's gift from God of "thirtyfold strength". And this is just an example, Tolkien points out, of a minor challenge to the translator. A second issue (in his view) is the compactness of Old English words, which often have no modern equivalents, and phrases which are "inevitably weakened even in prose by transference to our looser modern language".
|
|
By joining the single arrows together and the double arrows together, one obtains a torus with seven mutually touching regions; therefore seven colors are necessary This construction shows the torus divided into the maximum of seven regions, each one of which touches every other. The four-color theorem applies not only to finite planar graphs, but also to infinite graphs that can be drawn without crossings in the plane, and even more generally to infinite graphs (possibly with an uncountable number of vertices) for which every finite subgraph is planar. To prove this, one can combine a proof of the theorem for finite planar graphs with the De Bruijn–Erdős theorem stating that, if every finite subgraph of an infinite graph is k-colorable, then the whole graph is also k-colorable . This can also be seen as an immediate consequence of Kurt Gödel's compactness theorem for first-order logic, simply by expressing the colorability of an infinite graph with a set of logical formulae.
|
|
Gödel's second incompleteness theorem (see Gödel's incompleteness theorems), another celebrated result, shows that there are inherent limitations in what can be achieved with formal proofs in mathematics. The name for the incompleteness theorem refers to another meaning of complete (see model theory – Using the compactness and completeness theorems): A theory T is complete (or decidable) if for every formula f in the language of T either T\vdash f or T\vdash eg f. Gödel's second incompleteness theorem states that in any consistent effective theory T containing Peano arithmetic (PA), a formula CT like CT = eg (0 = 1) expressing the consistency of T cannot be proven within T. The completeness theorem implies the existence of a model of T in which the formula CT is false. Such a model (precisely, the set of "natural numbers" it contains) is necessarily a non-standard model, as it contains the code number of a proof of a contradiction of T. But T is consistent when viewed from the outside.
|
|
The retreat of the Sienese troops must have seemed to the troops led by Dino della Rocca a real retreat, so much so that he launched himself eagerly against the enemy. This decision made the Pisan-Massetano army lose compactness and made their troops more exposed to the counterattack of the Sienese army which had now taken up its position. The Massetans were defeated by the battle, losing 200 soldiers, 6 military banners and Captain Dino della Rocca, who was taken prisoner together with 200 other soldiers. The plain of Giuncarico, where on 14 December 1332 the field battle took place between the Pisan-Massetan militias led by Dino della Rocca and the Sienese troops led by Guidoriccio da Fogliano The defeat of Giuncarico greatly damaged the Republic of Massa, which was forced to ask the enemy for a two- month truce for the great famine and shortage of goods that followed.
|
|
The November class were double-hulled submarines with streamlined stern fins and nine compartments (I – bow torpedo, II – living and battery, III – central station, IV – diesel-generator, V – reactor, VI – turbine, VII – electromechanic, VIII – living, IX – stern). Three compartments equipped with bulkheads to withstand 10 atm pressure could be used as emergency shelters. The November-class attack submarines were considerably noisier than diesel submarines and the early American nuclear-powered submarines, despite the streamlined torpedo-like hull, limited number of holes in the hull, special low-noise variable-pitch propellers, vibration dampening of main equipment, and antisonar coating of the hull (used for the first time on nuclear-powered submarines). Soviet reactors were superior to American ones in compactness and power-to-weight ratio, but the vibrations of Soviet reactors were much more pronounced. Novembers detected submarine targets during active service (for example, there were 42 detections in 1965 when regular cruises of Soviet nuclear-powered submarines began).
|
|
Since holomorphic functions are much more rigid than smooth functions, the theories of smooth and complex manifolds have very different flavors: compact complex manifolds are much closer to algebraic varieties than to differentiable manifolds. For example, the Whitney embedding theorem tells us that every smooth n-dimensional manifold can be embedded as a smooth submanifold of R2n, whereas it is "rare" for a complex manifold to have a holomorphic embedding into Cn. Consider for example any compact connected complex manifold M: any holomorphic function on it is constant by Liouville's theorem. Now if we had a holomorphic embedding of M into Cn, then the coordinate functions of Cn would restrict to nonconstant holomorphic functions on M, contradicting compactness, except in the case that M is just a point. Complex manifolds that can be embedded in Cn are called Stein manifolds and form a very special class of manifolds including, for example, smooth complex affine algebraic varieties.
|
|
Subaru also wanted to reduce engine noise by placing the engine at the front and improve interior space by implementing front wheel drive, thereby eliminating a centrally mounted drive shaft powering the rear wheels, and utilizing an independent suspension at all four wheels. The only other Japanese company to use an air-cooled, horizontally opposed engine at the time was in the Toyota Publica with the Toyota U engine. Subaru 1000 coupe In 1963, Subaru tried again, with a new project code A-4, with a smaller 923 cc engine, front wheel drive, and an overall length of , a wheelbase of , a front wheel width of and a rear wheel width of , weighing . The model was put into production; it was assigned production code A-63 and was eventually introduced as the Subaru 1000. For compactness and to ensure quietness of operation with vibration kept to a minimum, the engine was developed with water cooling instead of the originally intended air cooling in the A-5 concept.
|
|
This star- forming region is located in the 5-kpc ring, a ring with that radius that encircles the central bar of our galaxy and that contains most of its molecular hydrogen as well as most of its star formation. It is associated with a very massive complex of molecular clouds with a total mass of more than 7 million times the one of our Sun and is forming stars of all masses within star clusters that are less massive versions of those found on starburst galaxies; it still has capacity to form more clusters. There are also massive protostars as well as stellar clusters in formation embedded within the nebula, with this star formation region likened to NGC 3603 W43's center, finally, contains a dense and massive star cluster with several O stars and Wolf-Rayet stars that has been compared due to its compactness to NGC 3603 or even Large Magellanic Cloud's R136.
|
|
This use of immediate—sometimes incongruous—contrasts has an exaggerative effect on the foreground's strong colour saturation; itself a component in the effect of aerial perspective. In addition to this local differentiating of the range and compactness of contrasting pigments, Pissarro also variegated the work in a more general way; the predominant colour at the foreground is green, and as the picture recedes, the use of greens is gradually discounted and hues of red are subtly amplified. Just after the completion of this painting, Pissarro made prints such as his Paysage en long, and Paysage sous bois, à l’Hermitage (both 1879) wherein he experimented with similar methods of deploying degrees of light and contrast to denote depth.Camille Pissarro, Impressionism, Landscape and Rural Labour, Richard Thompson, The South Bank Centre and The Herbert Press Limited, London, 1990, isbn 1853320528\. p. 40. The surface of The Côte des Bœufs at L’Hermitage is heavily worked.
|
|
The sail may be an analog of the sail of the Permian synapsid Dimetrodon, which lived before the dinosaurs even appeared, produced by convergent evolution. Reconstructed replicas of the holotype vertebrae, National Geographic Museum, Washington, D. C. The structure may also have been more hump-like than sail-like, as noted by Stromer in 1915 ("one might rather think of the existence of a large hump of fat [German: Fettbuckel], to which the [neural spines] gave internal support") and by Jack Bowman Bailey in 1997. In support of his "buffalo-back" hypothesis, Bailey argued that in Spinosaurus, Ouranosaurus, and other dinosaurs with long neural spines, the spines were relatively shorter and thicker than the spines of pelycosaurs (which were known to have sails); instead, the dinosaurs' neural spines were similar to the neural spines of extinct hump-backed mammals such as Megacerops and Bison latifrons. In 2014, Ibrahim and colleagues instead posited that the spines were covered tightly by skin, similar to a crested chameleon, given their compactness, sharp edges, and likely poor blood flow.
|
|
Cyberformance refers to live theatrical performances in which remote participants are enabled to work together in real time through the medium of the internet, employing technologies such as chat applications or purpose- built, multiuser, real-time collaborative software (for example, UpStage, Visitors Studio, the Waterwheel Tap, MOOs, and other platforms). Cyberformance is also known as online performance, networked performance, telematic performance, and digital theatre; there is as yet no consensus on which term should be preferred, but cyberformance has the advantage of compactness. For example, it is commonly employed by users of the UpStage platform to designate a special type of Performance art activity taking place in a cyber-artistic environment. Cyberformance can be created and presented entirely online, for a distributed online audience who participate via internet-connected computers anywhere in the world, or it can be presented to a proximal audience (such as in a physical theatre or gallery venue) with some or all of the performers appearing via the internet; or it can be a hybrid of the two approaches, with both remote and proximal audiences and/or performers.
|
|
A collection of brass instruments The high malleability and workability, relatively good resistance to corrosion, and traditionally attributed acoustic properties of brass, have made it the usual metal of choice for construction of musical instruments whose acoustic resonators consist of long, relatively narrow tubing, often folded or coiled for compactness; silver and its alloys, and even gold, have been used for the same reasons, but brass is the most economical choice. Collectively known as brass instruments, these include the trombone, tuba, trumpet, cornet, baritone horn, euphonium, tenor horn, and French horn, and many other "horns", many in variously-sized families, such as the saxhorns. Other wind instruments may be constructed of brass or other metals, and indeed most modern student-model flutes and piccolos are made of some variety of brass, usually a cupronickel alloy similar to nickel silver/German silver. Clarinets, especially low clarinets such as the contrabass and subcontrabass, are sometimes made of metal because of limited supplies of the dense, fine-grained tropical hardwoods traditionally preferred for smaller woodwinds.
|
|
The numerous items specifically made for travel include a variety of types of bed from four poster or tent beds to chairs that would extend for sleeping; large dining tables, dining chairs, easy chairs, sofas and couches, chests of drawers, book cabinets, washstands, wardrobes, shelves, desks, mirrors, lanterns and candlesticks, canteens of silver, cooking equipment, toiletry equipment and thunderboxes were all made to be portable. By the mid-19th century the demand for campaign furniture encouraged many makers to be inventive in their design thus creating many unusual and interesting pieces that offer a surprise in the ease in which they dismantle or the compactness of their storage. Tables were cleverly hinged to fold down into a box the size of a briefcase, chess board boxes would contain tripod legs and a telescopic column to convert into a table; chairs that would break down to a minimal size and could also be altered to become a sedan were all designed. The need to quickly be able to pack a piece of furniture to make it portable with minimal use of bolts etc.
|
|
Additionally, although originally intended to replace the oboe and the bassoon, the practical ranges of the corresponding sarrusophones, the soprano and bass, as per famed band conductor Edwin Franko Goldman and organologist Anthony Baines, did not lend themselves to proper playing of oboe and bassoon parts, especially in orchestra transcriptions for wind band. The need for a contrabass pitched woodwind has existed since at least the 19th century. During the 19th century and into the 20th there were sporadic attempts by Sax, Buffet, Besson and others to build a successful contrabass clarinet in either E or B. In the early 1930s, upon the suggestion of the American Bandmaster's Association, the French firm Selmer succeeded when they introduced their E contrabass model (the popular E and B contrabass models by the French firm LeBlanc not being placed into production until the late 1940s, although invented earlier). It can be conjectured that the compactness and musical qualities of these instruments may have contributed to the non-use of the sarrusophone, as they are now very common in musical circles ranging from junior high school through professional.
|
|
The strong reducibilities include: ;One-one reducibility: A is one-one reducible (or 1-reducible) to B if there is a total computable injective function f such that each n is in A if and only if f(n) is in B. ;Many-one reducibility: This is essentially one-one reducibility without the constraint that f be injective. A is many-one reducible (or m-reducible) to B if there is a total computable function f such that each n is in A if and only if f(n) is in B. ;Truth-table reducibility: A is truth-table reducible to B if A is Turing reducible to B via an oracle Turing machine that computes a total function regardless of the oracle it is given. Because of compactness of Cantor space, this is equivalent to saying that the reduction presents a single list of questions (depending only on the input) to the oracle simultaneously, and then having seen their answers is able to produce an output without asking additional questions regardless of the oracle's answer to the initial queries. Many variants of truth-table reducibility have also been studied.
|
|
Because each entry in the adjacency matrix requires only one bit, it can be represented in a very compact way, occupying only 2/8 bytes to represent a directed graph, or (by using a packed triangular format and only storing the lower triangular part of the matrix) approximately 2/16 bytes to represent an undirected graph. Although slightly more succinct representations are possible, this method gets close to the information- theoretic lower bound for the minimum number of bits needed to represent all -vertex graphs.. For storing graphs in text files, fewer bits per byte can be used to ensure that all bytes are text characters, for instance by using a Base64 representation.. Besides avoiding wasted space, this compactness encourages locality of reference. However, for a large sparse graph, adjacency lists require less storage space, because they do not waste any space to represent edges that are not present. An alternative form of adjacency matrix (which, however, requires a larger amount of space) replaces the numbers in each element of the matrix with pointers to edge objects (when edges are present) or null pointers (when there is no edge).
|
|