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c_wjjkj00jhk43 | In mathematics, the Picard group of a ringed space X, denoted by Pic(X), is the group of isomorphism classes of invertible sheaves (or line bundles) on X, with the group operation being tensor product. This construction is a global version of the construction of the divisor class group, or ideal class group, and is muc... | Picard scheme |
c_uf4qd5ggjwxx | {\displaystyle H^{1}(X,{\mathcal {O}}_{X}^{*}).\,} For integral schemes the Picard group is isomorphic to the class group of Cartier divisors. For complex manifolds the exponential sheaf sequence gives basic information on the Picard group. The name is in honour of Émile Picard's theories, in particular of divisors on ... | Picard scheme |
c_2aw6hf9wlr12 | In mathematics, the Picard–Fuchs equation, named after Émile Picard and Lazarus Fuchs, is a linear ordinary differential equation whose solutions describe the periods of elliptic curves. | Picard–Fuchs equation |
c_mh4jvt5smkvc | In mathematics, the Pidduck polynomials sn(x) are polynomials introduced by Pidduck (1910, 1912) given by the generating function ∑ n s n ( x ) n ! t n = ( 1 + t 1 − t ) x ( 1 − t ) − 1 {\displaystyle \displaystyle \sum _{n}{\frac {s_{n}(x)}{n! }}t^{n}=\left({\frac {1+t}{1-t}}\right)^{x}(1-t)^{-1}} (Roman 1984, 4.4.3),... | Pidduck polynomials |
c_s2dzqyrosrnt | In mathematics, the Pincherle derivative T ′ {\displaystyle T'} of a linear operator T: K → K {\displaystyle T:\mathbb {K} \to \mathbb {K} } on the vector space of polynomials in the variable x over a field K {\displaystyle \mathbb {K} } is the commutator of T {\displaystyle T} with the multiplication by x in the alg... | Pincherle derivative |
c_ug3u5euq6chv | In mathematics, the Pincherle polynomials Pn(x) are polynomials introduced by S. Pincherle (1891) given by the generating function ( 1 − 3 x t + t 3 ) − 1 / 2 = ∑ n = 0 ∞ P n ( x ) t n {\displaystyle \displaystyle (1-3xt+t^{3})^{-1/2}=\sum _{n=0}^{\infty }P_{n}(x)t^{n}} Humbert polynomials are a generalization of Pinch... | Pincherle polynomials |
c_g7gqqrltj2ex | In mathematics, the Pinsky phenomenon is a result in Fourier analysis. This phenomenon was discovered by Mark Pinsky of Northwestern University. It involves the spherical inversion of the Fourier transform. | Pinsky phenomenon |
c_v9pt2dbd3hc5 | The phenomenon involves a lack of convergence at a point due to a discontinuity at boundary. This lack of convergence in the Pinsky phenomenon happens far away from the boundary of the discontinuity, rather than at the discontinuity itself seen in the Gibbs phenomenon. This non-local phenomenon is caused by a lensing e... | Pinsky phenomenon |
c_op94zy39z65b | In mathematics, the Plancherel theorem (sometimes called the Parseval–Plancherel identity) is a result in harmonic analysis, proven by Michel Plancherel in 1910. It states that the integral of a function's squared modulus is equal to the integral of the squared modulus of its frequency spectrum. That is, if f ( x ) {\d... | Plancherel theorem |
c_oy4adgzjxyrf | This isometry is actually a unitary map. In effect, this makes it possible to speak of Fourier transforms of quadratically integrable functions. Plancherel's theorem remains valid as stated on n-dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . | Plancherel theorem |
c_s0gnqtghrmby | The theorem also holds more generally in locally compact abelian groups. There is also a version of the Plancherel theorem which makes sense for non-commutative locally compact groups satisfying certain technical assumptions. This is the subject of non-commutative harmonic analysis. | Plancherel theorem |
c_7l6awwdmae88 | The unitarity of the Fourier transform is often called Parseval's theorem in science and engineering fields, based on an earlier (but less general) result that was used to prove the unitarity of the Fourier series. Due to the polarization identity, one can also apply Plancherel's theorem to the L 2 ( R ) {\displaystyle... | Plancherel theorem |
c_dopsdks376kv | In mathematics, the Plancherel theorem for spherical functions is an important result in the representation theory of semisimple Lie groups, due in its final form to Harish-Chandra. It is a natural generalisation in non-commutative harmonic analysis of the Plancherel formula and Fourier inversion formula in the represe... | Plancherel theorem for spherical functions |
c_ozk53obbwo27 | The Plancherel theorem gives the eigenfunction expansion of radial functions for the Laplacian operator on the associated symmetric space X; it also gives the direct integral decomposition into irreducible representations of the regular representation on L2(X). In the case of hyperbolic space, these expansions were kno... | Plancherel theorem for spherical functions |
c_u32ebnvox571 | In mathematics, the Plücker map embeds the Grassmannian G r ( k , V ) {\displaystyle \mathbf {Gr} (k,V)} , whose elements are k-dimensional subspaces of an n-dimensional vector space V, either real or complex, in a projective space, thereby realizing it as an algebraic variety. More precisely, the Plücker map embeds G ... | Plucker relations |
c_tzv22w2m0cbb | In mathematics, the Pochhammer contour, introduced by Camille Jordan (1887) and Leo Pochhammer (1890), is a contour in the complex plane with two points removed, used for contour integration. If A and B are loops around the two points, both starting at some fixed point P, then the Pochhammer contour is the commutator A... | Pochhammer contour |
c_574rgkxeesns | In mathematics, the Pocklington–Lehmer primality test is a primality test devised by Henry Cabourn Pocklington and Derrick Henry Lehmer. The test uses a partial factorization of N − 1 {\displaystyle N-1} to prove that an integer N {\displaystyle N} is prime. It produces a primality certificate to be found with less eff... | Pocklington primality test |
c_wp2trxvh6a45 | In mathematics, the Poincaré conjecture is a theorem about the characterization of the 3-sphere, which is the hypersphere that bounds the unit ball in four-dimensional space. The conjecture states that: Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere. An equivalent form of the conjecture invol... | Mathematical conjecture |
c_67hidmrn0n9g | The Poincaré conjecture claims that if such a space has the additional property that each loop in the space can be continuously tightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higher dimensions for some time. After nearly a century of effort by mathematicia... | Mathematical conjecture |
c_9shbwoyqculx | The proof followed on from the program of Richard S. Hamilton to use the Ricci flow to attempt to solve the problem. Hamilton later introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way, but was unable to prove... | Mathematical conjecture |
c_cnnp8h3zz090 | In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology and cohomology groups of manifolds. It states that if M is an n-dimensional oriented closed manifold (compact and without boundary), then the kth cohomology group of M is isomorphic to the ( n − ... | Poincaré duality theorem |
c_0z5cpknzucco | In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the mode... | Poincaré inequality |
c_8xk547vix1hp | In mathematics, the Poincaré lemma gives a sufficient condition for a closed differential form to be exact (while an exact form is necessarily closed). Precisely, it states that every closed p-form on an open ball in Rn is exact for p with 1 ≤ p ≤ n. The lemma was introduced by Henri Poincaré in 1886.Especially in calc... | Poincare lemma |
c_ldu4u6p1eq2j | In mathematics, the Poincaré metric, named after Henri Poincaré, is the metric tensor describing a two-dimensional surface of constant negative curvature. It is the natural metric commonly used in a variety of calculations in hyperbolic geometry or Riemann surfaces. There are three equivalent representations commonly u... | Constant negative curvature |
c_0biwjymyf79g | The Poincaré disk model defines a model for hyperbolic space on the unit disk. The disk and the upper half plane are related by a conformal map, and isometries are given by Möbius transformations. A third representation is on the punctured disk, where relations for q-analogues are sometimes expressed. These various for... | Constant negative curvature |
c_wvz1xesovbtp | In mathematics, the Poincaré residue is a generalization, to several complex variables and complex manifold theory, of the residue at a pole of complex function theory. It is just one of a number of such possible extensions. Given a hypersurface X ⊂ P n {\displaystyle X\subset \mathbb {P} ^{n}} defined by a degree d {\... | Poincare residue |
c_b04fxvdqk6f3 | In mathematics, the Poincaré separation theorem, also known as the Cauchy interlacing theorem, gives some upper and lower bounds of eigenvalues of a real symmetric matrix B'AB that can be considered as the orthogonal projection of a larger real symmetric matrix A onto a linear subspace spanned by the columns of B. The ... | Poincaré separation theorem |
c_t9ls7j0bgtjf | In mathematics, the Poincaré–Bendixson theorem is a statement about the long-term behaviour of orbits of continuous dynamical systems on the plane, cylinder, or two-sphere. | Poincaré–Bendixson theorem |
c_j481b5izhbw7 | In mathematics, the Poincaré–Hopf theorem (also known as the Poincaré–Hopf index formula, Poincaré–Hopf index theorem, or Hopf index theorem) is an important theorem that is used in differential topology. It is named after Henri Poincaré and Heinz Hopf. The Poincaré–Hopf theorem is often illustrated by the special case... | Poincaré–Hopf theorem |
c_81mlql69v9lz | In mathematics, the Poincaré–Lelong equation, studied by Lelong (1964), is the partial differential equation i ∂ ∂ ¯ u = ρ {\displaystyle i\partial {\overline {\partial }}u=\rho } on a Kähler manifold, where ρ is a positive (1,1)-form. | Poincaré–Lelong equation |
c_wl24hyk6kt3m | In mathematics, the Poincaré–Miranda theorem is a generalization of intermediate value theorem, from a single function in a single dimension, to n functions in n dimensions. It says as follows: Consider n {\displaystyle n} continuous functions of n {\displaystyle n} variables, f 1 , … , f n {\displaystyle f_{1},\ldots ... | Poincaré-Miranda theorem |
c_28dcxx4azgdz | In mathematics, the Poisson boundary is a measure space associated to a random walk. It is an object designed to encode the asymptotic behaviour of the random walk, i.e. how trajectories diverge when the number of steps goes to infinity. Despite being called a boundary it is in general a purely measure-theoretical obje... | Poisson boundary |
c_lumwhe2lpi6c | In mathematics, the Poisson summation formula is an equation that relates the Fourier series coefficients of the periodic summation of a function to values of the function's continuous Fourier transform. Consequently, the periodic summation of a function is completely defined by discrete samples of the original functio... | Poisson summation formula |
c_uwijeow0m8qz | In mathematics, the Pompeiu problem is a conjecture in integral geometry, named for Dimitrie Pompeiu, who posed the problem in 1929, as follows. Suppose f is a nonzero continuous function defined on a Euclidean space, and K is a simply connected Lipschitz domain, so that the integral of f vanishes on every congruent co... | Pompeiu problem |
c_slxix1itr92w | In mathematics, the Pontryagin classes, named after Lev Pontryagin, are certain characteristic classes of real vector bundles. The Pontryagin classes lie in cohomology groups with degrees a multiple of four. | Pontryagin number |
c_hjsa300rsa9o | In mathematics, the Pontryagin product, introduced by Lev Pontryagin (1939), is a product on the homology of a topological space induced by a product on the topological space. Special cases include the Pontryagin product on the homology of an abelian group, the Pontryagin product on an H-space, and the Pontryagin produ... | Pontryagin product |
c_mei08ecoanw1 | In mathematics, the Porteous formula, or Thom–Porteous formula, or Giambelli–Thom–Porteous formula, is an expression for the fundamental class of a degeneracy locus (or determinantal variety) of a morphism of vector bundles in terms of Chern classes. Giambelli's formula is roughly the special case when the vector bundl... | Degeneracy loci |
c_gpvve3hn7e36 | In mathematics, the Property P conjecture is a statement about 3-manifolds obtained by Dehn surgery on a knot in the 3-sphere. A knot in the 3-sphere is said to have Property P if every 3-manifold obtained by performing (non-trivial) Dehn surgery on the knot is not simply-connected. The conjecture states that all knots... | Property P conjecture |
c_omyzy0zdhnvp | This conjecture can be thought of as a first step to resolving the Poincaré conjecture, since the Lickorish–Wallace theorem says any closed, orientable 3-manifold results from Dehn surgery on a link. If a knot K ⊂ S 3 {\displaystyle K\subset \mathbb {S} ^{3}} has Property P, then one cannot construct a counterexample t... | Property P conjecture |
c_u3lm7e6w91nw | In mathematics, the Prouhet–Tarry–Escott problem asks for two disjoint multisets A and B of n integers each, whose first k power sum symmetric polynomials are all equal. That is, the two multisets should satisfy the equations ∑ a ∈ A a i = ∑ b ∈ B b i {\displaystyle \sum _{a\in A}a^{i}=\sum _{b\in B}b^{i}} for each int... | Prouhet–Tarry–Escott problem |
c_snge2z5iftuc | In mathematics, the Prouhet–Thue–Morse constant, named for Eugène Prouhet, Axel Thue, and Marston Morse, is the number—denoted by τ—whose binary expansion 0.01101001100101101001011001101001... is given by the Prouhet–Thue–Morse sequence. That is, τ = ∑ n = 0 ∞ t n 2 n + 1 = 0.412454033640 … {\displaystyle \tau =\sum _{... | Prouhet–Thue–Morse constant |
c_6ilzqykc1itq | In mathematics, the Prym variety construction (named for Friedrich Prym) is a method in algebraic geometry of making an abelian variety from a morphism of algebraic curves. In its original form, it was applied to an unramified double covering of a Riemann surface, and was used by F. Schottky and H. W. E. Jung in relati... | Prym variety |
c_dqoby9nvh30c | Then from φ construct the corresponding morphism ψ: J1 → J2,which can be defined on a divisor class D of degree zero by applying φ to each point of the divisor. This is a well-defined morphism, often called the norm homomorphism. Then the Prym variety of φ is the kernel of ψ. To qualify that somewhat, to get an abelian... | Prym variety |
c_lo89vv1thjro | Or in other words take the largest abelian subvariety of J1 on which ψ is trivial. The theory of Prym varieties was dormant for a long time, until revived by David Mumford around 1970. It now plays a substantial role in some contemporary theories, for example of the Kadomtsev–Petviashvili equation. | Prym variety |
c_86dui6ri4pgm | One advantage of the method is that it allows one to apply the theory of curves to the study of a wider class of abelian varieties than Jacobians. For example, principally polarized abelian varieties (p.p.a.v. 's) of dimension > 3 are not generally Jacobians, but all p.p.a.v. | Prym variety |
c_g237lrijgwiw | 's of dimension 5 or less are Prym varieties. It is for this reason that p.p.a.v. 's are fairly well understood up to dimension 5. | Prym variety |
c_x6cpom6kv0qc | In mathematics, the Prékopa–Leindler inequality is an integral inequality closely related to the reverse Young's inequality, the Brunn–Minkowski inequality and a number of other important and classical inequalities in analysis. The result is named after the Hungarian mathematicians András Prékopa and László Leindler. | Prékopa–Leindler inequality |
c_emzb0jngxg9v | In mathematics, the Prüfer manifold or Prüfer surface is a 2-dimensional Hausdorff real analytic manifold that is not paracompact. It was introduced by Radó (1925) and named after Heinz Prüfer. | Prüfer surface |
c_rzhxkdzplctf | In mathematics, the Puppe sequence is a construction of homotopy theory, so named after Dieter Puppe. It comes in two forms: a long exact sequence, built from the mapping fibre (a fibration), and a long coexact sequence, built from the mapping cone (which is a cofibration). Intuitively, the Puppe sequence allows us to ... | Fiber sequence |
c_a5vnoyu9s9uw | In mathematics, the Pythagoras number or reduced height of a field describes the structure of the set of squares in the field. The Pythagoras number p(K) of a field K is the smallest positive integer p such that every sum of squares in K is a sum of p squares. A Pythagorean field is a field with Pythagoras number 1: th... | Pythagoras number |
c_0r2gfn2wpklm | In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite the right angle) is equal to the sum of the areas of the squares on the other ... | The Pythagorean theorem |
c_wnowt67o6ecx | The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proved numerous times by many different methods – possibly the most for any mathematical theorem. The proofs are diverse, including both geometric proofs and algebraic proofs, with some dating back thousands of years. | The Pythagorean theorem |
c_2svkm2omagq1 | When Euclidean space is represented by a Cartesian coordinate system in analytic geometry, Euclidean distance satisfies the Pythagorean relation: the squared distance between two points equals the sum of squares of the difference in each coordinate between the points. The theorem can be generalized in various ways: to ... | The Pythagorean theorem |
c_h9lncx6865hr | In mathematics, the Quillen determinant line bundle is a line bundle over the space of Cauchy–Riemann operators of a vector bundle over a Riemann surface, introduced by Quillen (1985). Quillen proved the existence of the Quillen metric on the determinant line bundle, a Hermitian metric defined using the analytic torsio... | Quillen determinant line bundle |
c_nhkpn6tgrixi | In mathematics, the Quillen–Lichtenbaum conjecture is a conjecture relating étale cohomology to algebraic K-theory introduced by Quillen (1975, p. 175), who was inspired by earlier conjectures of Lichtenbaum (1973). Kahn (1997) and Rognes & Weibel (2000) proved the Quillen–Lichtenbaum conjecture at the prime 2 for some... | Lichtenbaum conjecture |
c_x6d6kgdd28vm | In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in March 1991 to encourage research into computational number theory and th... | RSA number |
c_ve62pgsqjdq2 | Cash prizes of varying size, up to US$200,000 (and prizes up to $20,000 awarded), were offered for factorization of some of them. The smallest RSA number was factored in a few days. Most of the numbers have still not been factored and many of them are expected to remain unfactored for many years to come. | RSA number |
c_c3yb95uo0mf3 | As of February 2020, the smallest 23 of the 54 listed numbers have been factored. While the RSA challenge officially ended in 2007, people are still attempting to find the factorizations. According to RSA Laboratories, "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength o... | RSA number |
c_mi23q08a2xza | Some of the smaller prizes had been awarded at the time. The remaining prizes were retracted. | RSA number |
c_6d8vajuoa6iy | The first RSA numbers generated, from RSA-100 to RSA-500, were labeled according to their number of decimal digits. Later, beginning with RSA-576, binary digits are counted instead. An exception to this is RSA-617, which was created before the change in the numbering scheme. The numbers are listed in increasing order b... | RSA number |
c_sklqaajwq7ah | In mathematics, the Rabinowitsch trick, introduced by J.L. Rabinowitsch (1929), is a short way of proving the general case of the Hilbert Nullstellensatz from an easier special case (the so-called weak Nullstellensatz), by introducing an extra variable. The Rabinowitsch trick goes as follows. | Rabinowitsch trick |
c_adxutt34qeav | Let K be an algebraically closed field. Suppose the polynomial f in K vanishes whenever all polynomials f1,....,fm vanish. Then the polynomials f1,....,fm, 1 − x0f have no common zeros (where we have introduced a new variable x0), so by the weak Nullstellensatz for K they generate the unit ideal of K. Spelt out, this m... | Rabinowitsch trick |
c_648s687iauwa | In mathematics, the Radon transform is the integral transform which takes a function f defined on the plane to a function Rf defined on the (two-dimensional) space of lines in the plane, whose value at a particular line is equal to the line integral of the function over that line. The transform was introduced in 1917 b... | Filtered backprojection |
c_mg2gange7plf | It was later generalized to higher-dimensional Euclidean spaces and more broadly in the context of integral geometry. The complex analogue of the Radon transform is known as the Penrose transform. The Radon transform is widely applicable to tomography, the creation of an image from the projection data associated with c... | Filtered backprojection |
c_1uihz4pbie84 | In mathematics, the Radon–Nikodym theorem is a result in measure theory that expresses the relationship between two measures defined on the same measurable space. A measure is a set function that assigns a consistent magnitude to the measurable subsets of a measurable space. Examples of a measure include area and volum... | Density function (measure theory) |
c_mylvn0tfzh8k | This can be expressed as ν ( A ) = ∫ A f d μ , {\displaystyle \nu (A)=\int _{A}f\,d\mu ,} where ν is the new measure being defined for any measurable subset A and the function f is the density at a given point. The integral is with respect to an existing measure μ, which may often be the canonical Lebesgue measure on t... | Density function (measure theory) |
c_neow1ah0qc0r | The function f is then called the Radon–Nikodym derivative and is denoted by d ν d μ {\displaystyle {\tfrac {d\nu }{d\mu }}} . An important application is in probability theory, leading to the probability density function of a random variable. The theorem is named after Johann Radon, who proved the theorem for the spec... | Density function (measure theory) |
c_75k01ezyclfb | In mathematics, the Radó–Kneser–Choquet theorem, named after Tibor Radó, Hellmuth Kneser and Gustave Choquet, states that the Poisson integral of a homeomorphism of the unit circle is a harmonic diffeomorphism of the open unit disk. The result was stated as a problem by Radó and solved shortly afterwards by Kneser in 1... | Radó–Kneser–Choquet theorem |
c_7xwjwyz2rsmh | In mathematics, the Rainville polynomials pn(z) are polynomials introduced by Rainville (1945) given by the generating function e w I 0 ( z w ) = ∑ n p n ( z ) w n {\displaystyle \displaystyle e^{w}I_{0}(zw)=\sum _{n}p_{n}(z)w^{n}} Boas & Buck (1958, p.46). | Rainville polynomials |
c_mex46foeuczw | In mathematics, the Ramanujan conjecture, due to Srinivasa Ramanujan (1916, p. 176), states that Ramanujan's tau function given by the Fourier coefficients τ(n) of the cusp form Δ(z) of weight 12 Δ ( z ) = ∑ n > 0 τ ( n ) q n = q ∏ n > 0 ( 1 − q n ) 24 = q − 24 q 2 + 252 q 3 − 1472 q 4 + 4830 q 5 − ⋯ , {\displaystyle \... | Ramanujan conjecture |
c_1uwbah3gxvzb | In mathematics, the Ramanujan–Soldner constant (also called the Soldner constant) is a mathematical constant defined as the unique positive zero of the logarithmic integral function. It is named after Srinivasa Ramanujan and Johann Georg von Soldner. Its value is approximately μ ≈ 1.451369234883381050283968485892027449... | Ramanujan–Soldner constant |
c_6klocifix4p6 | In mathematics, the Ran space (or Ran's space) of a topological space X is a topological space Ran ( X ) {\displaystyle \operatorname {Ran} (X)} whose underlying set is the set of all nonempty finite subsets of X: for a metric space X the topology is induced by the Hausdorff distance. The notion is named after Ziv Ra... | Ran space |
c_3hx5u1zuzoyk | In mathematics, the Rankin–Cohen bracket of two modular forms is another modular form, generalizing the product of two modular forms. Rankin (1956, 1957) gave some general conditions for polynomials in derivatives of modular forms to be modular forms, and Cohen (1975) found the explicit examples of such polynomials tha... | Rankin–Cohen bracket |
c_9m6vfs3a89wl | In mathematics, the Rauzy fractal is a fractal set associated with the Tribonacci substitution s ( 1 ) = 12 , s ( 2 ) = 13 , s ( 3 ) = 1 . {\displaystyle s(1)=12,\ s(2)=13,\ s(3)=1\,.} It was studied in 1981 by Gérard Rauzy, with the idea of generalizing the dynamic properties of the Fibonacci morphism. That fractal se... | Rauzy fractal |
c_zb408ixfoa65 | In mathematics, the Ravenel conjectures are a set of mathematical conjectures in the field of stable homotopy theory posed by Douglas Ravenel at the end of a paper published in 1984. It was earlier circulated in preprint. The problems involved have largely been resolved, with all but the "telescope conjecture" being pr... | Ravenel's conjectures |
c_gfhznzb86mgi | The telescope conjecture is now generally believed not to be true, though there are some conflicting claims concerning it in the published literature, and is taken to be an open problem. Ravenel's conjectures exerted influence on the field through the founding of the approach of chromatic homotopy theory. The first of ... | Ravenel's conjectures |
c_oeew08fwd1o3 | The telescope conjecture, which was #4 on the original list, remains of substantial interest because of its connection with the convergence of an Adams–Novikov spectral sequence. While opinion has been generally against the truth of the original statement, investigations of associated phenomena (for a triangulated cate... | Ravenel's conjectures |
c_dcp74u8uban8 | In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix M {\displaystyle M} and nonzero vector x {\displaystyle x} is defined as:For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the conjugate transpose x ∗ {\displaystyle x^{*}} to the usual tr... | Rayleigh quotient |
c_2k6vaplrrmli | It can be shown that, for a given matrix, the Rayleigh quotient reaches its minimum value λ min {\displaystyle \lambda _{\min }} (the smallest eigenvalue of M {\displaystyle M} ) when x {\displaystyle x} is v min {\displaystyle v_{\min }} (the corresponding eigenvector). Similarly, R ( M , x ) ≤ λ max {\displaystyle R(... | Rayleigh quotient |
c_4q1bjjf7kc6s | It is also used in eigenvalue algorithms (such as Rayleigh quotient iteration) to obtain an eigenvalue approximation from an eigenvector approximation. The range of the Rayleigh quotient (for any matrix, not necessarily Hermitian) is called a numerical range and contains its spectrum. When the matrix is Hermitian, the ... | Rayleigh quotient |
c_m1ht8m0ky18f | Still in functional analysis, λ max {\displaystyle \lambda _{\max }} is known as the spectral radius. In the context of C ⋆ {\displaystyle C^{\star }} -algebras or algebraic quantum mechanics, the function that to M {\displaystyle M} associates the Rayleigh–Ritz quotient R ( M , x ) {\displaystyle R(M,x)} for a fixed x... | Rayleigh quotient |
c_hxdkg5g4v2fl | In mathematics, the Rayleigh theorem for eigenvalues pertains to the behavior of the solutions of an eigenvalue equation as the number of basis functions employed in its resolution increases. Rayleigh, Lord Rayleigh, and 3rd Baron Rayleigh are the titles of John William Strutt, after the death of his father, the 2nd Ba... | Rayleigh theorem for eigenvalues |
c_o653k6bf7g4e | Except for metals, most of these other materials have an energy or a band gap, i.e., the difference between the lowest, unoccupied energy and the highest, occupied energy. For crystals, the energy spectrum is in bands and there is a band gap, if any, as opposed to energy gap. Given the diverse contributions of Lord Ray... | Rayleigh theorem for eigenvalues |
c_y9nps4m1m5z4 | In mathematics, the Redheffer star product is a binary operation on linear operators that arises in connection to solving coupled systems of linear equations. It was introduced by Raymond Redheffer in 1959, and has subsequently been widely adopted in computational methods for scattering matrices. Given two scattering m... | Redheffer star product |
c_m70pz1y2bc0a | In mathematics, the Reeb foliation is a particular foliation of the 3-sphere, introduced by the French mathematician Georges Reeb (1920–1993). It is based on dividing the sphere into two solid tori, along a 2-torus: see Clifford torus. Each of the solid tori is then foliated internally, in codimension 1, and the dividi... | Reeb foliation |
c_rl24vqpvu34c | In mathematics, the Reeb vector field, named after the French mathematician Georges Reeb, is a notion that appears in various domains of contact geometry including: in a contact manifold, given a contact 1-form α {\displaystyle \alpha } , the Reeb vector field satisfies R ∈ k e r d α , α ( R ) = 1 {\displaystyle R\in \... | Reeb vector field |
c_jpwq89jb4qnq | In mathematics, the Rees matrix semigroups are a special class of semigroups introduced by David Rees in 1940. They are of fundamental importance in semigroup theory because they are used to classify certain classes of simple semigroups. | Rees matrix semigroup |
c_mirikg4q98fw | In mathematics, the Regiomontanus's angle maximization problem, is a famous optimization problem posed by the 15th-century German mathematician Johannes Müller (also known as Regiomontanus). The problem is as follows: A painting hangs from a wall. Given the heights of the top and bottom of the painting above the viewer... | Regiomontanus' angle maximization problem |
c_kh42g4b4p7b1 | In mathematics, the Rellich–Kondrachov theorem is a compact embedding theorem concerning Sobolev spaces. It is named after the Austrian-German mathematician Franz Rellich and the Russian mathematician Vladimir Iosifovich Kondrashov. Rellich proved the L2 theorem and Kondrashov the Lp theorem. | Rellich–Kondrachov theorem |
c_78emm59liw41 | In mathematics, the Remez inequality, discovered by the Soviet mathematician Evgeny Yakovlevich Remez (Remez 1936), gives a bound on the sup norms of certain polynomials, the bound being attained by the Chebyshev polynomials. | Remez inequality |
c_4ws06m9xmull | In mathematics, the Riemann Xi function is a variant of the Riemann zeta function, and is defined so as to have a particularly simple functional equation. The function is named in honour of Bernhard Riemann. | Riemann Ξ function |
c_w3rje8a8jfh4 | In mathematics, the Riemann hypothesis is the conjecture that the Riemann zeta function has its zeros only at the negative even integers and complex numbers with real part 1/2. Many consider it to be the most important unsolved problem in pure mathematics. It is of great interest in number theory because it implies res... | Critical line (mathematics) |
c_nzardr3xtbrl | The Riemann hypothesis and some of its generalizations, along with Goldbach's conjecture and the twin prime conjecture, make up Hilbert's eighth problem in David Hilbert's list of twenty-three unsolved problems; it is also one of the Clay Mathematics Institute's Millennium Prize Problems, which offers a million dollars... | Critical line (mathematics) |
c_fvgw396orn0k | It has zeros at the negative even integers; that is, ζ(s) = 0 when s is one of −2, −4, −6, .... These are called its trivial zeros. The zeta function is also zero for other values of s, which are called nontrivial zeros. The Riemann hypothesis is concerned with the locations of these nontrivial zeros, and states that: ... | Critical line (mathematics) |
c_rtwjib1qg3km | In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture that the non-trivial zeros of the Riemann zeta function all have real part 1/2. The name is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields. The Riemann hypothesis impl... | Conjecture |
c_ga40qwzoarc7 | In mathematics, the Riemann series theorem, also called the Riemann rearrangement theorem, named after 19th-century German mathematician Bernhard Riemann, says that if an infinite series of real numbers is conditionally convergent, then its terms can be arranged in a permutation so that the new series converges to an a... | Riemann rearrangement theorem |
c_37m9kygwavt9 | Thus the original series is conditionally convergent, and can be rearranged (by taking the first two positive terms followed by the first negative term, followed by the next two positive terms and then the next negative term, etc.) to give a series that converges to a different sum: 1 + 1/2 − 1 + 1/3 + 1/4 − 1/2 + ⋯ = ... | Riemann rearrangement theorem |
c_8wg0f4spafj2 | In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity. This extended plane represents the extended complex numbers, that is, the complex numbers plus a value ∞ {\displaystyle \infty } for infinity. With the Riemann model,... | Extended complex numbers |
c_zrudeabi80fn | For example, any rational function on the complex plane can be extended to a holomorphic function on the Riemann sphere, with the poles of the rational function mapping to infinity. More generally, any meromorphic function can be thought of as a holomorphic function whose codomain is the Riemann sphere. In geometry, th... | Extended complex numbers |
c_a91qo8un8lw7 | In projective geometry, the sphere can be thought of as the complex projective line P 1 ( C ) {\displaystyle \mathbf {P} ^{1}(\mathbf {C} )} , the projective space of all complex lines in C 2 {\displaystyle \mathbf {C} ^{2}} . As with any compact Riemann surface, the sphere may also be viewed as a projective algebraic ... | Extended complex numbers |
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