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The work of Lubotzky and Mann, combined with Michel Lazard's solution to Hilbert's fifth problem over the p-adic numbers, shows that a pro-p group is p-adic analytic if and only if it has finite rank, i.e. there exists a positive integer r {\displaystyle r} such that any closed subgroup has a topological generating set with no more than r {\displaystyle r} elements. More generally it was shown that a finitely generated profinite group is a compact p-adic Lie group if and only if it has an open subgroup that is a uniformly powerful pro-p-group. The Coclass Theorems have been proved in 1994 by A. Shalev and independently by C. R. Leedham-Green. Theorem D is one of these theorems and asserts that, for any prime number p and any positive integer r, there exist only finitely many pro-p groups of coclass r. This finiteness result is fundamental for the classification of finite p-groups by means of directed coclass graphs.
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https://en.wikipedia.org/wiki/Pro-p_group
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In mathematics, a pro-simplicial set is an inverse system of simplicial sets. A pro-simplicial set is called pro-finite if each term of the inverse system of simplicial sets has finite homotopy groups. Pro-simplicial sets show up in shape theory, in the study of localization and completion in homotopy theory, and in the study of homotopy properties of schemes (e.g. étale homotopy theory).
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https://en.wikipedia.org/wiki/Pro-simplicial_set
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In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events; for example, the value assigned to "1 or 2" in a throw of a dice should be the sum of the values assigned to "1" and "2". Probability measures have applications in diverse fields, from physics to finance and biology.
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https://en.wikipedia.org/wiki/Probability_measure
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In mathematics, a product is the result of multiplication, or an expression that identifies objects (numbers or variables) to be multiplied, called factors. For example, 21 is the product of 3 and 7 (the result of multiplication), and x ⋅ ( 2 + x ) {\displaystyle x\cdot (2+x)} is the product of x {\displaystyle x} and ( 2 + x ) {\displaystyle (2+x)} (indicating that the two factors should be multiplied together). When one factor is an integer, the product is called a multiple. The order in which real or complex numbers are multiplied has no bearing on the product; this is known as the commutative law of multiplication.
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https://en.wikipedia.org/wiki/Mathematical_product
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When matrices or members of various other associative algebras are multiplied, the product usually depends on the order of the factors. Matrix multiplication, for example, is non-commutative, and so is multiplication in other algebras in general as well. There are many different kinds of products in mathematics: besides being able to multiply just numbers, polynomials or matrices, one can also define products on many different algebraic structures.
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https://en.wikipedia.org/wiki/Mathematical_product
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In mathematics, a product of rings or direct product of rings is a ring that is formed by the Cartesian product of the underlying sets of several rings (possibly an infinity), equipped with componentwise operations. It is a direct product in the category of rings. Since direct products are defined up to an isomorphism, one says colloquially that a ring is the product of some rings if it is isomorphic to the direct product of these rings. For example, the Chinese remainder theorem may be stated as: if m and n are coprime integers, the quotient ring Z / m n Z {\displaystyle \mathbb {Z} /mn\mathbb {Z} } is the product of Z / m Z {\displaystyle \mathbb {Z} /m\mathbb {Z} } and Z / n Z . {\displaystyle \mathbb {Z} /n\mathbb {Z} .}
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https://en.wikipedia.org/wiki/Direct_product_of_rings
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In mathematics, a profinite group is a topological group that is in a certain sense assembled from a system of finite groups. The idea of using a profinite group is to provide a "uniform", or "synoptic", view of an entire system of finite groups. Properties of the profinite group are generally speaking uniform properties of the system. For example, the profinite group is finitely generated (as a topological group) if and only if there exists d ∈ N {\displaystyle d\in \mathbb {N} } such that every group in the system can be generated by d {\displaystyle d} elements.
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https://en.wikipedia.org/wiki/Profinite_groups
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Many theorems about finite groups can be readily generalised to profinite groups; examples are Lagrange's theorem and the Sylow theorems.To construct a profinite group one needs a system of finite groups and group homomorphisms between them. Without loss of generality, these homomorphisms can be assumed to be surjective, in which case the finite groups will appear as quotient groups of the resulting profinite group; in a sense, these quotients approximate the profinite group. Important examples of profinite groups are the additive groups of p {\displaystyle p} -adic integers and the Galois groups of infinite-degree field extensions.
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https://en.wikipedia.org/wiki/Profinite_groups
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Every profinite group is compact and totally disconnected. A non-compact generalization of the concept is that of locally profinite groups. Even more general are the totally disconnected groups.
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https://en.wikipedia.org/wiki/Profinite_groups
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In mathematics, a profinite integer is an element of the ring (sometimes pronounced as zee-hat or zed-hat) Z ^ = lim ← Z / n Z = ∏ p Z p {\displaystyle {\widehat {\mathbb {Z} }}=\varprojlim \mathbb {Z} /n\mathbb {Z} =\prod _{p}\mathbb {Z} _{p}} where lim ← Z / n Z {\displaystyle \varprojlim \mathbb {Z} /n\mathbb {Z} } indicates the profinite completion of Z {\displaystyle \mathbb {Z} } , the index p {\displaystyle p} runs over all prime numbers, and Z p {\displaystyle \mathbb {Z} _{p}} is the ring of p-adic integers. This group is important because of its relation to Galois theory, étale homotopy theory, and the ring of adeles. In addition, it provides a basic tractable example of a profinite group.
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https://en.wikipedia.org/wiki/Profinite_integer
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In mathematics, a progressive function ƒ ∈ L2(R) is a function whose Fourier transform is supported by positive frequencies only: s u p p f ^ ⊆ R + . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{+}.} It is called super regressive if and only if the time reversed function f(−t) is progressive, or equivalently, if s u p p f ^ ⊆ R − . {\displaystyle \mathop {\rm {supp}} {\hat {f}}\subseteq \mathbb {R} _{-}.}
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https://en.wikipedia.org/wiki/Progressive_function
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The complex conjugate of a progressive function is regressive, and vice versa. The space of progressive functions is sometimes denoted H + 2 ( R ) {\displaystyle H_{+}^{2}(R)} , which is known as the Hardy space of the upper half-plane. This is because a progressive function has the Fourier inversion formula f ( t ) = ∫ 0 ∞ e 2 π i s t f ^ ( s ) d s {\displaystyle f(t)=\int _{0}^{\infty }e^{2\pi ist}{\hat {f}}(s)\,ds} and hence extends to a holomorphic function on the upper half-plane { t + i u: t , u ∈ R , u ≥ 0 } {\displaystyle \{t+iu:t,u\in R,u\geq 0\}} by the formula f ( t + i u ) = ∫ 0 ∞ e 2 π i s ( t + i u ) f ^ ( s ) d s = ∫ 0 ∞ e 2 π i s t e − 2 π s u f ^ ( s ) d s .
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https://en.wikipedia.org/wiki/Progressive_function
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{\displaystyle f(t+iu)=\int _{0}^{\infty }e^{2\pi is(t+iu)}{\hat {f}}(s)\,ds=\int _{0}^{\infty }e^{2\pi ist}e^{-2\pi su}{\hat {f}}(s)\,ds.} Conversely, every holomorphic function on the upper half-plane which is uniformly square-integrable on every horizontal line will arise in this manner. Regressive functions are similarly associated with the Hardy space on the lower half-plane { t + i u: t , u ∈ R , u ≤ 0 } {\displaystyle \{t+iu:t,u\in R,u\leq 0\}} . This article incorporates material from progressive function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
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https://en.wikipedia.org/wiki/Progressive_function
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In mathematics, a projection is an idempotent mapping of a set (or other mathematical structure) into a subset (or sub-structure). In this case, idempotent means that projecting twice is the same as projecting once. The restriction to a subspace of a projection is also called a projection, even if the idempotence property is lost. An everyday example of a projection is the casting of shadows onto a plane (sheet of paper): the projection of a point is its shadow on the sheet of paper, and the projection (shadow) of a point on the sheet of paper is that point itself (idempotency).
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https://en.wikipedia.org/wiki/Projection_(mathematics)
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The shadow of a three-dimensional sphere is a closed disk. Originally, the notion of projection was introduced in Euclidean geometry to denote the projection of the three-dimensional Euclidean space onto a plane in it, like the shadow example. The two main projections of this kind are: The projection from a point onto a plane or central projection: If C is a point, called the center of projection, then the projection of a point P different from C onto a plane that does not contain C is the intersection of the line CP with the plane.
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https://en.wikipedia.org/wiki/Projection_(mathematics)
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The points P such that the line CP is parallel to the plane does not have any image by the projection, but one often says that they project to a point at infinity of the plane (see Projective geometry for a formalization of this terminology). The projection of the point C itself is not defined. The projection parallel to a direction D, onto a plane or parallel projection: The image of a point P is the intersection with the plane of the line parallel to D passing through P. See Affine space § Projection for an accurate definition, generalized to any dimension.The concept of projection in mathematics is a very old one, and most likely has its roots in the phenomenon of the shadows cast by real-world objects on the ground.
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https://en.wikipedia.org/wiki/Projection_(mathematics)
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This rudimentary idea was refined and abstracted, first in a geometric context and later in other branches of mathematics. Over time different versions of the concept developed, but today, in a sufficiently abstract setting, we can unify these variations.In cartography, a map projection is a map of a part of the surface of the Earth onto a plane, which, in some cases, but not always, is the restriction of a projection in the above meaning. The 3D projections are also at the basis of the theory of perspective.The need for unifying the two kinds of projections and of defining the image by a central projection of any point different of the center of projection are at the origin of projective geometry. However, a projective transformation is a bijection of a projective space, a property not shared with the projections of this article.
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https://en.wikipedia.org/wiki/Projection_(mathematics)
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In mathematics, a projectionless C*-algebra is a C*-algebra with no nontrivial projections. For a unital C*-algebra, the projections 0 and 1 are trivial. While for a non-unital C*-algebra, only 0 is considered trivial.
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https://en.wikipedia.org/wiki/Projectionless_C*-algebra
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The problem of whether simple infinite-dimensional C*-algebras with this property exist was posed in 1958 by Irving Kaplansky, and the first example of one was published in 1981 by Bruce Blackadar. For commutative C*-algebras, being projectionless is equivalent to its spectrum being connected. Due to this, being projectionless can be considered as a noncommutative analogue of a connected space.
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https://en.wikipedia.org/wiki/Projectionless_C*-algebra
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In mathematics, a projective bundle is a fiber bundle whose fibers are projective spaces. By definition, a scheme X over a Noetherian scheme S is a Pn-bundle if it is locally a projective n-space; i.e., X × S U ≃ P U n {\displaystyle X\times _{S}U\simeq \mathbb {P} _{U}^{n}} and transition automorphisms are linear. Over a regular scheme S such as a smooth variety, every projective bundle is of the form P ( E ) {\displaystyle \mathbb {P} (E)} for some vector bundle (locally free sheaf) E.
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https://en.wikipedia.org/wiki/Projectivized_vector_bundle
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In mathematics, a projective line is, roughly speaking, the extension of a usual line by a point called a point at infinity. The statement and the proof of many theorems of geometry are simplified by the resultant elimination of special cases; for example, two distinct projective lines in a projective plane meet in exactly one point (there is no "parallel" case). There are many equivalent ways to formally define a projective line; one of the most common is to define a projective line over a field K, commonly denoted P1(K), as the set of one-dimensional subspaces of a two-dimensional K-vector space. This definition is a special instance of the general definition of a projective space. The projective line over the reals is a manifold; see real projective line for details.
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https://en.wikipedia.org/wiki/Projective_line
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In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect at exactly one point.
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https://en.wikipedia.org/wiki/Desarguesian_plane
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Renaissance artists, in developing the techniques of drawing in perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by PG(2, R), RP2, or P2(R), among other notations.
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https://en.wikipedia.org/wiki/Desarguesian_plane
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There are many other projective planes, both infinite, such as the complex projective plane, and finite, such as the Fano plane. A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.
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https://en.wikipedia.org/wiki/Desarguesian_plane
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In mathematics, a projective range is a set of points in projective geometry considered in a unified fashion. A projective range may be a projective line or a conic. A projective range is the dual of a pencil of lines on a given point. For instance, a correlation interchanges the points of a projective range with the lines of a pencil.
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https://en.wikipedia.org/wiki/Harmonic_range
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A projectivity is said to act from one range to another, though the two ranges may coincide as sets. A projective range expresses projective invariance of the relation of projective harmonic conjugates. Indeed, three points on a projective line determine a fourth by this relation.
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https://en.wikipedia.org/wiki/Harmonic_range
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Application of a projectivity to this quadruple results in four points likewise in the harmonic relation. Such a quadruple of points is termed a harmonic range. In 1940 Julian Coolidge described this structure and identified its originator: Two fundamental one-dimensional forms such as point ranges, pencils of lines, or of planes are defined as projective, when their members are in one-to-one correspondence, and a harmonic set of one ... corresponds to a harmonic set of the other. ... If two one-dimensional forms are connected by a train of projections and intersections, harmonic elements will correspond to harmonic elements, and they are projective in the sense of Von Staudt.
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https://en.wikipedia.org/wiki/Harmonic_range
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In mathematics, a proof by infinite descent, also known as Fermat's method of descent, is a particular kind of proof by contradiction used to show that a statement cannot possibly hold for any number, by showing that if the statement were to hold for a number, then the same would be true for a smaller number, leading to an infinite descent and ultimately a contradiction. It is a method which relies on the well-ordering principle, and is often used to show that a given equation, such as a Diophantine equation, has no solutions.Typically, one shows that if a solution to a problem existed, which in some sense was related to one or more natural numbers, it would necessarily imply that a second solution existed, which was related to one or more 'smaller' natural numbers. This in turn would imply a third solution related to smaller natural numbers, implying a fourth solution, therefore a fifth solution, and so on. However, there cannot be an infinity of ever-smaller natural numbers, and therefore by mathematical induction, the original premise—that any solution exists—is incorrect: its correctness produces a contradiction.
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https://en.wikipedia.org/wiki/Proof_by_infinite_descent
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An alternative way to express this is to assume one or more solutions or examples exists, from which a smallest solution or example—a minimal counterexample—can then be inferred. Once there, one would try to prove that if a smallest solution exists, then it must imply the existence of a smaller solution (in some sense), which again proves that the existence of any solution would lead to a contradiction.
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https://en.wikipedia.org/wiki/Proof_by_infinite_descent
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The earliest uses of the method of infinite descent appear in Euclid's Elements. A typical example is Proposition 31 of Book 7, in which Euclid proves that every composite integer is divided (in Euclid's terminology "measured") by some prime number.The method was much later developed by Fermat, who coined the term and often used it for Diophantine equations. Two typical examples are showing the non-solvability of the Diophantine equation r 2 + s 4 = t 4 {\displaystyle r^{2}+s^{4}=t^{4}} and proving Fermat's theorem on sums of two squares, which states that an odd prime p can be expressed as a sum of two squares when p ≡ 1 ( mod 4 ) {\displaystyle p\equiv 1{\pmod {4}}} (see Modular arithmetic and proof by infinite descent).
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https://en.wikipedia.org/wiki/Proof_by_infinite_descent
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In this way Fermat was able to show the non-existence of solutions in many cases of Diophantine equations of classical interest (for example, the problem of four perfect squares in arithmetic progression). In some cases, to the modern eye, his "method of infinite descent" is an exploitation of the inversion of the doubling function for rational points on an elliptic curve E. The context is of a hypothetical non-trivial rational point on E. Doubling a point on E roughly doubles the length of the numbers required to write it (as number of digits), so that a "halving" a point gives a rational with smaller terms. Since the terms are positive, they cannot decrease forever.
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https://en.wikipedia.org/wiki/Proof_by_infinite_descent
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In mathematics, a proof of impossibility is a proof that demonstrates that a particular problem cannot be solved as described in the claim, or that a particular set of problems cannot be solved in general. Such a case is also known as a negative proof, proof of an impossibility theorem, or negative result. Proofs of impossibility often are the resolutions to decades or centuries of work attempting to find a solution, eventually proving that there is no solution. Proving that something is impossible is usually much harder than the opposite task, as it is often necessary to develop a proof that works in general, rather than to just show a particular example.
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https://en.wikipedia.org/wiki/Impossibility_proof
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Impossibility theorems are usually expressible as negative existential propositions or universal propositions in logic. The irrationality of the square root of 2 is one of the oldest proofs of impossibility.
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https://en.wikipedia.org/wiki/Impossibility_proof
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It shows that it is impossible to express the square root of 2 as a ratio of two integers. Another consequential proof of impossibility was Ferdinand von Lindemann's proof in 1882, which showed that the problem of squaring the circle cannot be solved because the number π is transcendental (i.e., non-algebraic), and that only a subset of the algebraic numbers can be constructed by compass and straightedge. Two other classical problems—trisecting the general angle and doubling the cube—were also proved impossible in the 19th century, and all of these problems gave rise to research into more complicated mathematical structures.
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https://en.wikipedia.org/wiki/Impossibility_proof
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A problem that arose in the 16th century was creating a general formula using radicals to express the solution of any polynomial equation of fixed degree k, where k ≥ 5. In the 1820s, the Abel–Ruffini theorem (also known as Abel's impossibility theorem) showed this to be impossible, using concepts such as solvable groups from Galois theory—a new sub-field of abstract algebra. Some of the most important proofs of impossibility found in the 20th century were those related to undecidability, which showed that there are problems that cannot be solved in general by any algorithm, with one of the more prominent ones being the halting problem.
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https://en.wikipedia.org/wiki/Impossibility_proof
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Gödel's incompleteness theorems were other examples that uncovered fundamental limitations in the provability of formal systems.In computational complexity theory, techniques like relativization (the addition of an oracle) allow for "weak" proofs of impossibility, in that proofs techniques that are not affected by relativization cannot resolve the P versus NP problem. Another technique is the proof of completeness for a complexity class, which provides evidence for the difficulty of problems by showing them to be just as hard to solve as any other problem in the class. In particular, a complete problem is intractable if one of the problems in its class is.
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https://en.wikipedia.org/wiki/Impossibility_proof
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In mathematics, a proof without words (or visual proof) is an illustration of an identity or mathematical statement which can be demonstrated as self-evident by a diagram without any accompanying explanatory text. Such proofs can be considered more elegant than formal or mathematically rigorous proofs due to their self-evident nature. When the diagram demonstrates a particular case of a general statement, to be a proof, it must be generalisable.A proof without words is not the same as a mathematical proof, because it omits the details of the logical argument it illustrates. However, it can provide valuable intuitions to the viewer that can help them formulate or better understand a true proof.
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https://en.wikipedia.org/wiki/Proof_without_words
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In mathematics, a proper ideal of a commutative ring is said to be irreducible if it cannot be written as the intersection of two strictly larger ideals.
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https://en.wikipedia.org/wiki/Irreducible_ideal
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In mathematics, a property is any characteristic that applies to a given set. Rigorously, a property p defined for all elements of a set X is usually defined as a function p: X → {true, false}, that is true whenever the property holds; or equivalently, as the subset of X for which p holds; i.e. the set {x | p(x) = true}; p is its indicator function. However, it may be objected that the rigorous definition defines merely the extension of a property, and says nothing about what causes the property to hold for exactly those values.
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https://en.wikipedia.org/wiki/Property_(mathematics)
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In mathematics, a protorus is a compact connected topological abelian group. Equivalently, it is a projective limit of tori (products of a finite number of copies of the circle group), or the Pontryagin dual of a discrete torsion-free abelian group. Some examples of protori are given by solenoid groups.
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https://en.wikipedia.org/wiki/Protorus
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In mathematics, a prototile is one of the shapes of a tile in a tessellation.
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https://en.wikipedia.org/wiki/Prototile
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In mathematics, a pseudo-canonical variety is an algebraic variety of "general type".
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https://en.wikipedia.org/wiki/Pseudo-canonical_variety
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In mathematics, a pseudo-finite field F is an infinite model of the first-order theory of finite fields. This is equivalent to the condition that F is quasi-finite (perfect with a unique extension of every positive degree) and pseudo algebraically closed (every absolutely irreducible variety over F has a point defined over F). Every hyperfinite field is pseudo-finite and every pseudo-finite field is quasifinite. Every non-principal ultraproduct of finite fields is pseudo-finite. Pseudo-finite fields were introduced by Ax (1968).
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https://en.wikipedia.org/wiki/Pseudofinite_field
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In mathematics, a pseudo-monotone operator from a reflexive Banach space into its continuous dual space is one that is, in some sense, almost as well-behaved as a monotone operator. Many problems in the calculus of variations can be expressed using operators that are pseudo-monotone, and pseudo-monotonicity in turn implies the existence of solutions to these problems.
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https://en.wikipedia.org/wiki/Pseudo-monotone_operator
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In mathematics, a pseudo-reductive group over a field k (sometimes called a k-reductive group) is a smooth connected affine algebraic group defined over k whose k-unipotent radical (i.e., largest smooth connected unipotent normal k-subgroup) is trivial. Over perfect fields these are the same as (connected) reductive groups, but over non-perfect fields Jacques Tits found some examples of pseudo-reductive groups that are not reductive. A pseudo-reductive k-group need not be reductive (since the formation of the k-unipotent radical does not generally commute with non-separable scalar extension on k, such as scalar extension to an algebraic closure of k).
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https://en.wikipedia.org/wiki/Pseudo-reductive_algebraic_group
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Pseudo-reductive groups arise naturally in the study of algebraic groups over function fields of positive-dimensional varieties in positive characteristic (even over a perfect field of constants). Springer (1998) gives an exposition of Tits' results on pseudo-reductive groups, while Conrad, Gabber & Prasad (2010) builds on Tits' work to develop a general structure theory, including more advanced topics such as construction techniques, root systems and root groups and open cells, classification theorems, and applications to rational conjugacy theorems for smooth connected affine groups over arbitrary fields. The general theory (with applications) as of 2010 is summarized in Rémy (2011), and later work in the second edition Conrad, Gabber & Prasad (2015) and in Conrad & Prasad (2016) provides further refinements.
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https://en.wikipedia.org/wiki/Pseudo-reductive_algebraic_group
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In mathematics, a pseudofunctor F is a mapping between 2-categories, or from a category to a 2-category, that is just like a functor except that F ( f ∘ g ) = F ( f ) ∘ F ( g ) {\displaystyle F(f\circ g)=F(f)\circ F(g)} and F ( 1 ) = 1 {\displaystyle F(1)=1} do not hold as exact equalities but only up to coherent isomorphisms. The Grothendieck construction associates to a pseudofunctor a fibered category.
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https://en.wikipedia.org/wiki/Pseudo-functor
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In mathematics, a pseudogamma function is a function that interpolates the factorial. The gamma function is the most famous solution to the problem of extending the notion of the factorial beyond the positive integers only. However, it is clearly not the only solution, as, for any set of points, an infinite number of curves can be drawn through those points.
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https://en.wikipedia.org/wiki/Pseudogamma_function
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Such a curve, namely one which interpolates the factorial but is not equal to the gamma function, is known as a pseudogamma function. The two most famous pseudogamma functions are Hadamard's gamma function: H ( x ) = ψ ( 1 − x 2 ) − ψ ( 1 2 − x 2 ) 2 Γ ( 1 − x ) {\displaystyle H(x)={\frac {\psi \left(1-{\frac {x}{2}}\right)-\psi \left({\frac {1}{2}}-{\frac {x}{2}}\right)}{2\Gamma (1-x)}}} and the Luschny factorial: Γ ( x + 1 ) ( 1 − sin ( π x ) π x ( x 2 ( ψ ( x + 1 2 ) − ψ ( x 2 ) ) − 1 2 ) ) {\displaystyle \Gamma (x+1)\left(1-{\frac {\sin \left(\pi x\right)}{\pi x}}\left({\frac {x}{2}}\left(\psi \left({\frac {x+1}{2}}\right)-\psi \left({\frac {x}{2}}\right)\right)-{\frac {1}{2}}\right)\right)} where Γ(x) denotes the classical gamma function and ψ(x) denotes the digamma function. == References ==
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https://en.wikipedia.org/wiki/Pseudogamma_function
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In mathematics, a pseudogroup is a set of diffeomorphisms between open sets of a space, satisfying group-like and sheaf-like properties. It is a generalisation of the concept of a group, originating however from the geometric approach of Sophus Lie to investigate symmetries of differential equations, rather than out of abstract algebra (such as quasigroup, for example). The modern theory of pseudogroups was developed by Élie Cartan in the early 1900s.
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https://en.wikipedia.org/wiki/Local_Lie_group
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In mathematics, a pseudomanifold is a special type of topological space. It looks like a manifold at most of its points, but it may contain singularities. For example, the cone of solutions of z 2 = x 2 + y 2 {\displaystyle z^{2}=x^{2}+y^{2}} forms a pseudomanifold. A pseudomanifold can be regarded as a combinatorial realisation of the general idea of a manifold with singularities. The concepts of orientability, orientation and degree of a mapping make sense for pseudomanifolds and moreover, within the combinatorial approach, pseudomanifolds form the natural domain of definition for these concepts.
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https://en.wikipedia.org/wiki/Pseudomanifold
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In mathematics, a pseudometric space is a generalization of a metric space in which the distance between two distinct points can be zero. Pseudometric spaces were introduced by Đuro Kurepa in 1934. In the same way as every normed space is a metric space, every seminormed space is a pseudometric space. Because of this analogy the term semimetric space (which has a different meaning in topology) is sometimes used as a synonym, especially in functional analysis. When a topology is generated using a family of pseudometrics, the space is called a gauge space.
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https://en.wikipedia.org/wiki/Pseudometrizable_space
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In mathematics, a pseudoreflection is an invertible linear transformation of a finite-dimensional vector space such that it is not the identity transformation, has a finite (multiplicative) order, and fixes a hyperplane. The concept of pseudoreflection generalizes the concepts of reflection and complex reflection and is simply called reflection by some mathematicians. It plays an important role in Invariant theory of finite groups, including the Chevalley-Shephard-Todd theorem.
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https://en.wikipedia.org/wiki/Pseudoreflection
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In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle π: E → B and a continuous map f: B′ → B one can define a "pullback" of E by f as a bundle f*E over B′. The fiber of f*E over a point b′ in B′ is just the fiber of E over f(b′). Thus f*E is the disjoint union of all these fibers equipped with a suitable topology.
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https://en.wikipedia.org/wiki/Induced_bundle
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In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
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https://en.wikipedia.org/wiki/Pullback
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In mathematics, a pyramid number, or square pyramidal number, is a natural number that counts the number of stacked spheres in a pyramid with a square base. The study of these numbers goes back to Archimedes and Fibonacci. They are part of a broader topic of figurate numbers representing the numbers of points forming regular patterns within different shapes. As well as counting spheres in a pyramid, these numbers can be described algebraically as a sum of the first n {\displaystyle n} positive square numbers, or as the values of a cubic polynomial.
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https://en.wikipedia.org/wiki/Square_pyramidal_number
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They can be used to solve several other counting problems, including counting squares in a square grid and counting acute triangles formed from the vertices of an odd regular polygon. They equal the sums of consecutive tetrahedral numbers, and are one-fourth of a larger tetrahedral number. The sum of two consecutive square pyramidal numbers is an octahedral number.
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https://en.wikipedia.org/wiki/Square_pyramidal_number
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In mathematics, a q-analog of a theorem, identity or expression is a generalization involving a new parameter q that returns the original theorem, identity or expression in the limit as q → 1. Typically, mathematicians are interested in q-analogs that arise naturally, rather than in arbitrarily contriving q-analogs of known results. The earliest q-analog studied in detail is the basic hypergeometric series, which was introduced in the 19th century.q-analogs are most frequently studied in the mathematical fields of combinatorics and special functions. In these settings, the limit q → 1 is often formal, as q is often discrete-valued (for example, it may represent a prime power).
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https://en.wikipedia.org/wiki/Q-analog
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q-analogs find applications in a number of areas, including the study of fractals and multi-fractal measures, and expressions for the entropy of chaotic dynamical systems. The relationship to fractals and dynamical systems results from the fact that many fractal patterns have the symmetries of Fuchsian groups in general (see, for example Indra's pearls and the Apollonian gasket) and the modular group in particular. The connection passes through hyperbolic geometry and ergodic theory, where the elliptic integrals and modular forms play a prominent role; the q-series themselves are closely related to elliptic integrals. q-analogs also appear in the study of quantum groups and in q-deformed superalgebras. The connection here is similar, in that much of string theory is set in the language of Riemann surfaces, resulting in connections to elliptic curves, which in turn relate to q-series.
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https://en.wikipedia.org/wiki/Q-analog
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In mathematics, a quadratic algebra is a filtered algebra generated by degree one elements, with defining relations of degree 2. It was pointed out by Yuri Manin that such algebras play an important role in the theory of quantum groups. The most important class of graded quadratic algebras is Koszul algebras.
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https://en.wikipedia.org/wiki/Quadratic_algebra
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In mathematics, a quadratic differential on a Riemann surface is a section of the symmetric square of the holomorphic cotangent bundle. If the section is holomorphic, then the quadratic differential is said to be holomorphic. The vector space of holomorphic quadratic differentials on a Riemann surface has a natural interpretation as the cotangent space to the Riemann moduli space, or Teichmüller space.
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https://en.wikipedia.org/wiki/Quadratic_differential
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In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is a x 2 + b x + c = 0 , {\displaystyle ax^{2}+bx+c=0,} where a ≠ 0. The quadratic equation on a number x {\displaystyle x} can be solved using the well-known quadratic formula, which can be derived by completing the square.
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https://en.wikipedia.org/wiki/Solving_quadratic_equations_with_continued_fractions
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That formula always gives the roots of the quadratic equation, but the solutions are expressed in a form that often involves a quadratic irrational number, which is an algebraic fraction that can be evaluated as a decimal fraction only by applying an additional root extraction algorithm. If the roots are real, there is an alternative technique that obtains a rational approximation to one of the roots by manipulating the equation directly. The method works in many cases, and long ago it stimulated further development of the analytical theory of continued fractions.
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https://en.wikipedia.org/wiki/Solving_quadratic_equations_with_continued_fractions
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In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, is a quadratic form in the variables x and y. The coefficients usually belong to a fixed field K, such as the real or complex numbers, and one speaks of a quadratic form over K. If K = R {\displaystyle K=\mathbb {R} } , and the quadratic form equals zero only when all variables are simultaneously zero, then it is a definite quadratic form; otherwise it is an isotropic quadratic form. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal groups), differential geometry (the Riemannian metric, the second fundamental form), differential topology (intersection forms of four-manifolds), Lie theory (the Killing form), and statistics (where the exponent of a zero-mean multivariate normal distribution has the quadratic form − x T Σ − 1 x {\displaystyle -\mathbf {x} ^{\mathsf {T}}{\boldsymbol {\Sigma }}^{-1}\mathbf {x} } ) Quadratic forms are not to be confused with a quadratic equation, which has only one variable and includes terms of degree two or less. A quadratic form is one case of the more general concept of homogeneous polynomials.
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https://en.wikipedia.org/wiki/Quadratic_space
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In mathematics, a quadratic form over a field F is said to be isotropic if there is a non-zero vector on which the form evaluates to zero. Otherwise the quadratic form is anisotropic. More explicitly, if q is a quadratic form on a vector space V over F, then a non-zero vector v in V is said to be isotropic if q(v) = 0.
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https://en.wikipedia.org/wiki/Hyperbolic_plane_(quadratic_forms)
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A quadratic form is isotropic if and only if there exists a non-zero isotropic vector (or null vector) for that quadratic form. Suppose that (V, q) is quadratic space and W is a subspace of V. Then W is called an isotropic subspace of V if some vector in it is isotropic, a totally isotropic subspace if all vectors in it are isotropic, and an anisotropic subspace if it does not contain any (non-zero) isotropic vectors. The isotropy index of a quadratic space is the maximum of the dimensions of the totally isotropic subspaces.A quadratic form q on a finite-dimensional real vector space V is anisotropic if and only if q is a definite form: either q is positive definite, i.e. q(v) > 0 for all non-zero v in V ; or q is negative definite, i.e. q(v) < 0 for all non-zero v in V.More generally, if the quadratic form is non-degenerate and has the signature (a, b), then its isotropy index is the minimum of a and b. An important example of an isotropic form over the reals occurs in pseudo-Euclidean space.
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https://en.wikipedia.org/wiki/Hyperbolic_plane_(quadratic_forms)
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In mathematics, a quadratic integral is an integral of the form It can be evaluated by completing the square in the denominator.
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https://en.wikipedia.org/wiki/Quadratic_integral
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In mathematics, a quadratic irrational number (also known as a quadratic irrational or quadratic surd) is an irrational number that is the solution to some quadratic equation with rational coefficients which is irreducible over the rational numbers. Since fractions in the coefficients of a quadratic equation can be cleared by multiplying both sides by their least common denominator, a quadratic irrational is an irrational root of some quadratic equation with integer coefficients. The quadratic irrational numbers, a subset of the complex numbers, are algebraic numbers of degree 2, and can therefore be expressed as a + b c d , {\displaystyle {a+b{\sqrt {c}} \over d},} for integers a, b, c, d; with b, c and d non-zero, and with c square-free. When c is positive, we get real quadratic irrational numbers, while a negative c gives complex quadratic irrational numbers which are not real numbers.
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https://en.wikipedia.org/wiki/Quadratic_irrationalities
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This defines an injection from the quadratic irrationals to quadruples of integers, so their cardinality is at most countable; since on the other hand every square root of a prime number is a distinct quadratic irrational, and there are countably many prime numbers, they are at least countable; hence the quadratic irrationals are a countable set. Quadratic irrationals are used in field theory to construct field extensions of the field of rational numbers Q. Given the square-free integer c, the augmentation of Q by quadratic irrationals using √c produces a quadratic field Q(√c). For example, the inverses of elements of Q(√c) are of the same form as the above algebraic numbers: d a + b c = a d − b d c a 2 − b 2 c .
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https://en.wikipedia.org/wiki/Quadratic_irrationalities
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{\displaystyle {d \over a+b{\sqrt {c}}}={ad-bd{\sqrt {c}} \over a^{2}-b^{2}c}.} Quadratic irrationals have useful properties, especially in relation to continued fractions, where we have the result that all real quadratic irrationals, and only real quadratic irrationals, have periodic continued fraction forms. For example 3 = 1.732 … = {\displaystyle {\sqrt {3}}=1.732\ldots =} The periodic continued fractions can be placed in one-to-one correspondence with the rational numbers.
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https://en.wikipedia.org/wiki/Quadratic_irrationalities
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The correspondence is explicitly provided by Minkowski's question mark function, and an explicit construction is given in that article. It is entirely analogous to the correspondence between rational numbers and strings of binary digits that have an eventually-repeating tail, which is also provided by the question mark function. Such repeating sequences correspond to periodic orbits of the dyadic transformation (for the binary digits) and the Gauss map h ( x ) = 1 / x − ⌊ 1 / x ⌋ {\displaystyle h(x)=1/x-\lfloor 1/x\rfloor } for continued fractions.
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https://en.wikipedia.org/wiki/Quadratic_irrationalities
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In mathematics, a quadratic polynomial is a polynomial of degree two in one or more variables. A quadratic function is the polynomial function defined by a quadratic polynomial. Before the 20th century, the distinction was unclear between a polynomial and its associated polynomial function; so "quadratic polynomial" and "quadratic function" were almost synonymous. This is still the case in many elementary courses, where both terms are often abbreviated as "quadratic".
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https://en.wikipedia.org/wiki/Quadratic_functions
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For example, a univariate (single-variable) quadratic function has the form f ( x ) = a x 2 + b x + c , a ≠ 0 , {\displaystyle f(x)=ax^{2}+bx+c,\quad a\neq 0,} where x is its variable. The graph of a univariate quadratic function is a parabola, a curve that has an axis of symmetry parallel to the y-axis. If a quadratic function is equated with zero, then the result is a quadratic equation.
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https://en.wikipedia.org/wiki/Quadratic_functions
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The solutions of a quadratic equation are the zeros of the corresponding quadratic function. The bivariate case in terms of variables x and y has the form f ( x , y ) = a x 2 + b x y + c y 2 + d x + e y + f , {\displaystyle f(x,y)=ax^{2}+bxy+cy^{2}+dx+ey+f,} with at least one of a, b, c not equal to zero. The zeros of this quadratic function is, in general (that is, if a certain expression of the coefficients is not equal to zero), a conic section (a circle or other ellipse, a parabola, or a hyperbola).
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https://en.wikipedia.org/wiki/Quadratic_functions
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A quadratic function in three variables x, y, and z contains exclusively terms x2, y2, z2, xy, xz, yz, x, y, z, and a constant: f ( x , y , z ) = a x 2 + b y 2 + c z 2 + d x y + e x z + f y z + g x + h y + i z + j , {\displaystyle f(x,y,z)=ax^{2}+by^{2}+cz^{2}+dxy+exz+fyz+gx+hy+iz+j,} where at least one of the coefficients a, b, c, d, e, f of the second-degree terms is not zero. A quadratic function can have an arbitrarily large number of variables. The set of its zero form a quadric, which is a surface in the case of three variables and a hypersurface in general case.
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https://en.wikipedia.org/wiki/Quadratic_functions
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In mathematics, a quadratic set is a set of points in a projective space that bears the same essential incidence properties as a quadric (conic section in a projective plane, sphere or cone or hyperboloid in a projective space).
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https://en.wikipedia.org/wiki/Quadratic_set
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In mathematics, a quadratic-linear algebra is an algebra over a field with a presentation such that all relations are sums of monomials of degrees 1 or 2 in the generators. They were introduced by Polishchuk and Positselski (2005, p.101). An example is the universal enveloping algebra of a Lie algebra, with generators a basis of the Lie algebra and relations of the form XY – YX – = 0.
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https://en.wikipedia.org/wiki/Quadratic-linear_algebra
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In mathematics, a quadratically closed field is a field in which every element has a square root.
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https://en.wikipedia.org/wiki/Quadratically_closed_field
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In mathematics, a quadric or quadric hypersurface is the subspace of N-dimensional space defined by a polynomial equation of degree 2 over a field. Quadrics are fundamental examples in algebraic geometry. The theory is simplified by working in projective space rather than affine space.
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https://en.wikipedia.org/wiki/Quadric_(algebraic_geometry)
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An example is the quadric surface x y = z w {\displaystyle xy=zw} in projective space P 3 {\displaystyle {\mathbf {P} }^{3}} over the complex numbers C. A quadric has a natural action of the orthogonal group, and so the study of quadrics can be considered as a descendant of Euclidean geometry. Many properties of quadrics hold more generally for projective homogeneous varieties. Another generalization of quadrics is provided by Fano varieties.
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https://en.wikipedia.org/wiki/Quadric_(algebraic_geometry)
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In mathematics, a quadric or quadric surface (quadric hypersurface in higher dimensions), is a generalization of conic sections (ellipses, parabolas, and hyperbolas). It is a hypersurface (of dimension D) in a (D + 1)-dimensional space, and it is defined as the zero set of an irreducible polynomial of degree two in D + 1 variables; for example, D = 1 in the case of conic sections. When the defining polynomial is not absolutely irreducible, the zero set is generally not considered a quadric, although it is often called a degenerate quadric or a reducible quadric. In coordinates x1, x2, ..., xD+1, the general quadric is thus defined by the algebraic equation ∑ i , j = 1 D + 1 x i Q i j x j + ∑ i = 1 D + 1 P i x i + R = 0 {\displaystyle \sum _{i,j=1}^{D+1}x_{i}Q_{ij}x_{j}+\sum _{i=1}^{D+1}P_{i}x_{i}+R=0} which may be compactly written in vector and matrix notation as: x Q x T + P x T + R = 0 {\displaystyle xQx^{\mathrm {T} }+Px^{\mathrm {T} }+R=0\,} where x = (x1, x2, ..., xD+1) is a row vector, xT is the transpose of x (a column vector), Q is a (D + 1) × (D + 1) matrix and P is a (D + 1)-dimensional row vector and R a scalar constant.
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https://en.wikipedia.org/wiki/Quadric
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The values Q, P and R are often taken to be over real numbers or complex numbers, but a quadric may be defined over any field. A quadric is an affine algebraic variety, or, if it is reducible, an affine algebraic set. Quadrics may also be defined in projective spaces; see § Normal form of projective quadrics, below.
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https://en.wikipedia.org/wiki/Quadric
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In mathematics, a quantaloid is a category enriched over the category Sup of suplattices. In other words, for any objects a and b the morphism object between them is not just a set but a complete lattice, in such a way that composition of morphisms preserves all joins: ( ⋁ i f i ) ∘ ( ⋁ j g j ) = ⋁ i , j ( f i ∘ g j ) {\displaystyle (\bigvee _{i}f_{i})\circ (\bigvee _{j}g_{j})=\bigvee _{i,j}(f_{i}\circ g_{j})} The endomorphism lattice H o m ( X , X ) {\displaystyle \mathrm {Hom} (X,X)} of any object X {\displaystyle X} in a quantaloid is a quantale, whence the name. == References ==
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https://en.wikipedia.org/wiki/Quantaloid
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In mathematics, a quantum affine algebra (or affine quantum group) is a Hopf algebra that is a q-deformation of the universal enveloping algebra of an affine Lie algebra. They were introduced independently by Drinfeld (1985) and Jimbo (1985) as a special case of their general construction of a quantum group from a Cartan matrix. One of their principal applications has been to the theory of solvable lattice models in quantum statistical mechanics, where the Yang–Baxter equation occurs with a spectral parameter. Combinatorial aspects of the representation theory of quantum affine algebras can be described simply using crystal bases, which correspond to the degenerate case when the deformation parameter q vanishes and the Hamiltonian of the associated lattice model can be explicitly diagonalized.
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https://en.wikipedia.org/wiki/Quantum_affine_algebra
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In mathematics, a quantum groupoid is any of a number of notions in noncommutative geometry analogous to the notion of groupoid. In usual geometry, the information of a groupoid can be contained in its monoidal category of representations (by a version of Tannaka–Krein duality), in its groupoid algebra or in the commutative Hopf algebroid of functions on the groupoid. Thus formalisms trying to capture quantum groupoids include certain classes of (autonomous) monoidal categories, Hopf algebroids etc.
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https://en.wikipedia.org/wiki/Quantum_groupoid
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In mathematics, a quantum or quantized enveloping algebra is a q-analog of a universal enveloping algebra. Given a Lie algebra g {\displaystyle {\mathfrak {g}}} , the quantum enveloping algebra is typically denoted as U q ( g ) {\displaystyle U_{q}({\mathfrak {g}})} . The notation was introduced by Drinfeld and independently by Jimbo.Among the applications, studying the q → 0 {\displaystyle q\to 0} limit led to the discovery of crystal bases.
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https://en.wikipedia.org/wiki/Quantized_enveloping_algebra
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In mathematics, a quartic equation is one which can be expressed as a quartic function equaling zero. The general form of a quartic equation is a x 4 + b x 3 + c x 2 + d x + e = 0 {\displaystyle ax^{4}+bx^{3}+cx^{2}+dx+e=0\,} where a ≠ 0. The quartic is the highest order polynomial equation that can be solved by radicals in the general case (i.e., one in which the coefficients can take any value).
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https://en.wikipedia.org/wiki/Quartic_equation
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In mathematics, a quasi-Frobenius Lie algebra ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} over a field k {\displaystyle k} is a Lie algebra ( g , ) {\displaystyle ({\mathfrak {g}},)} equipped with a nondegenerate skew-symmetric bilinear form β: g × g → k {\displaystyle \beta :{\mathfrak {g}}\times {\mathfrak {g}}\to k} , which is a Lie algebra 2-cocycle of g {\displaystyle {\mathfrak {g}}} with values in k {\displaystyle k} . In other words, β ( , Z ) + β ( , Y ) + β ( , X ) = 0 {\displaystyle \beta \left(\left,Z\right)+\beta \left(\left,Y\right)+\beta \left(\left,X\right)=0} for all X {\displaystyle X} , Y {\displaystyle Y} , Z {\displaystyle Z} in g {\displaystyle {\mathfrak {g}}} . If β {\displaystyle \beta } is a coboundary, which means that there exists a linear form f: g → k {\displaystyle f:{\mathfrak {g}}\to k} such that β ( X , Y ) = f ( ) , {\displaystyle \beta (X,Y)=f(\left),} then ( g , , β ) {\displaystyle ({\mathfrak {g}},,\beta )} is called a Frobenius Lie algebra.
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https://en.wikipedia.org/wiki/Quasi-Frobenius_Lie_algebra
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In mathematics, a quasi-Lie algebra in abstract algebra is just like a Lie algebra, but with the usual axiom = 0 {\displaystyle =0} replaced by = − {\displaystyle =-} (anti-symmetry).In characteristic other than 2, these are equivalent (in the presence of bilinearity), so this distinction doesn't arise when considering real or complex Lie algebras. It can however become important, when considering Lie algebras over the integers. In a quasi-Lie algebra, 2 = 0. {\displaystyle 2=0.} Therefore, the bracket of any element with itself is 2-torsion, if it does not actually vanish.
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https://en.wikipedia.org/wiki/Quasi-Lie_algebra
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In mathematics, a quasi-analytic class of functions is a generalization of the class of real analytic functions based upon the following fact: If f is an analytic function on an interval ⊂ R, and at some point f and all of its derivatives are zero, then f is identically zero on all of . Quasi-analytic classes are broader classes of functions for which this statement still holds true.
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https://en.wikipedia.org/wiki/Denjoy–Carleman_theorem
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In mathematics, a quasi-finite field is a generalisation of a finite field. Standard local class field theory usually deals with complete valued fields whose residue field is finite (i.e. non-archimedean local fields), but the theory applies equally well when the residue field is only assumed quasi-finite.
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https://en.wikipedia.org/wiki/Quasi-finite_field
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In mathematics, a quasi-invariant measure μ with respect to a transformation T, from a measure space X to itself, is a measure which, roughly speaking, is multiplied by a numerical function of T. An important class of examples occurs when X is a smooth manifold M, T is a diffeomorphism of M, and μ is any measure that locally is a measure with base the Lebesgue measure on Euclidean space. Then the effect of T on μ is locally expressible as multiplication by the Jacobian determinant of the derivative (pushforward) of T. To express this idea more formally in measure theory terms, the idea is that the Radon–Nikodym derivative of the transformed measure μ′ with respect to μ should exist everywhere; or that the two measures should be equivalent (i.e. mutually absolutely continuous): μ ′ = T ∗ ( μ ) ≈ μ . {\displaystyle \mu '=T_{*}(\mu )\approx \mu .} That means, in other words, that T preserves the concept of a set of measure zero.
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https://en.wikipedia.org/wiki/Quasi-invariant_measure
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Considering the whole equivalence class of measures ν, equivalent to μ, it is also the same to say that T preserves the class as a whole, mapping any such measure to another such. Therefore, the concept of quasi-invariant measure is the same as invariant measure class. In general, the 'freedom' of moving within a measure class by multiplication gives rise to cocycles, when transformations are composed. As an example, Gaussian measure on Euclidean space Rn is not invariant under translation (like Lebesgue measure is), but is quasi-invariant under all translations. It can be shown that if E is a separable Banach space and μ is a locally finite Borel measure on E that is quasi-invariant under all translations by elements of E, then either dim(E) < +∞ or μ is the trivial measure μ ≡ 0.
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https://en.wikipedia.org/wiki/Quasi-invariant_measure
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In mathematics, a quasi-polynomial (pseudo-polynomial) is a generalization of polynomials. While the coefficients of a polynomial come from a ring, the coefficients of quasi-polynomials are instead periodic functions with integral period. Quasi-polynomials appear throughout much of combinatorics as the enumerators for various objects. A quasi-polynomial can be written as q ( k ) = c d ( k ) k d + c d − 1 ( k ) k d − 1 + ⋯ + c 0 ( k ) {\displaystyle q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots +c_{0}(k)} , where c i ( k ) {\displaystyle c_{i}(k)} is a periodic function with integral period.
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https://en.wikipedia.org/wiki/Quasi-polynomial
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If c d ( k ) {\displaystyle c_{d}(k)} is not identically zero, then the degree of q {\displaystyle q} is d {\displaystyle d} . Equivalently, a function f: N → N {\displaystyle f\colon \mathbb {N} \to \mathbb {N} } is a quasi-polynomial if there exist polynomials p 0 , … , p s − 1 {\displaystyle p_{0},\dots ,p_{s-1}} such that f ( n ) = p i ( n ) {\displaystyle f(n)=p_{i}(n)} when i ≡ n mod s {\displaystyle i\equiv n{\bmod {s}}} . The polynomials p i {\displaystyle p_{i}} are called the constituents of f {\displaystyle f} .
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https://en.wikipedia.org/wiki/Quasi-polynomial
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In mathematics, a quasi-projective variety in algebraic geometry is a locally closed subset of a projective variety, i.e., the intersection inside some projective space of a Zariski-open and a Zariski-closed subset. A similar definition is used in scheme theory, where a quasi-projective scheme is a locally closed subscheme of some projective space.
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https://en.wikipedia.org/wiki/Quasi-projective_variety
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In mathematics, a quasi-split group over a field is a reductive group with a Borel subgroup defined over the field. Simply connected quasi-split groups over a field correspond to actions of the absolute Galois group on a Dynkin diagram.
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https://en.wikipedia.org/wiki/Quasi-split_group
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In mathematics, a quasi-topology on a set X is a function that associates to every compact Hausdorff space C a collection of mappings from C to X satisfying certain natural conditions. A set with a quasi-topology is called a quasitopological space. They were introduced by Spanier, who showed that there is a natural quasi-topology on the space of continuous maps from one space to another.
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https://en.wikipedia.org/wiki/Quasitopological_space
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In mathematics, a quasicircle is a Jordan curve in the complex plane that is the image of a circle under a quasiconformal mapping of the plane onto itself. Originally introduced independently by Pfluger (1961) and Tienari (1962), in the older literature (in German) they were referred to as quasiconformal curves, a terminology which also applied to arcs. In complex analysis and geometric function theory, quasicircles play a fundamental role in the description of the universal Teichmüller space, through quasisymmetric homeomorphisms of the circle. Quasicircles also play an important role in complex dynamical systems.
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https://en.wikipedia.org/wiki/Quasicircle
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In mathematics, a quasiconvex function is a real-valued function defined on an interval or on a convex subset of a real vector space such that the inverse image of any set of the form ( − ∞ , a ) {\displaystyle (-\infty ,a)} is a convex set. For a function of a single variable, along any stretch of the curve the highest point is one of the endpoints. The negative of a quasiconvex function is said to be quasiconcave. All convex functions are also quasiconvex, but not all quasiconvex functions are convex, so quasiconvexity is a generalization of convexity. Univariate unimodal functions are quasiconvex or quasiconcave, however this is not necessarily the case for functions with multiple arguments. For example, the 2-dimensional Rosenbrock function is unimodal but not quasiconvex and functions with star-convex sublevel sets can be unimodal without being quasiconvex.
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https://en.wikipedia.org/wiki/Quasiconcave_function
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