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Whitney's planarity criterion In mathematics, Whitney's planarity criterion is a matroid-theoretic characterization of planar graphs, named after Hassler Whitney.[1] It states that a graph G is planar if and only if its graphic matroid is also cographic (that is, it is the dual matroid of another graphic matroid). In purely graph-theoretic terms, this criterion can be stated as follows: There must be another (dual) graph G'=(V',E') and a bijective correspondence between the edges E' and the edges E of the original graph G, such that a subset T of E forms a spanning tree of G if and only if the edges corresponding to the complementary subset E-T form a spanning tree of G'. Algebraic duals An equivalent form of Whitney's criterion is that a graph G is planar if and only if it has a dual graph whose graphic matroid is dual to the graphic matroid of G. A graph whose graphic matroid is dual to the graphic matroid of G is known as an algebraic dual of G. Thus, Whitney's planarity criterion can be expressed succinctly as: a graph is planar if and only if it has an algebraic dual. Topological duals If a graph is embedded into a topological surface such as the plane, in such a way that every face of the embedding is a topological disk, then the dual graph of the embedding is defined as the graph (or in some cases multigraph) H that has a vertex for every face of the embedding, and an edge for every adjacency between a pair of faces. According to Whitney's criterion, the following conditions are equivalent: • The surface on which the embedding exists is topologically equivalent to the plane, sphere, or punctured plane • H is an algebraic dual of G • Every simple cycle in G corresponds to a minimal cut in H, and vice versa • Every simple cycle in H corresponds to a minimal cut in G, and vice versa • Every spanning tree in G corresponds to the complement of a spanning tree in H, and vice versa.[2] It is possible to define dual graphs of graphs embedded on nonplanar surfaces such as the torus, but these duals do not generally have the correspondence between cuts, cycles, and spanning trees required by Whitney's criterion. References 1. Whitney, Hassler (1932), "Non-separable and planar graphs", Transactions of the American Mathematical Society, 34 (2): 339–362, doi:10.1090/S0002-9947-1932-1501641-2. 2. Tutte, W. T. (1965), "Lectures on matroids", Journal of Research of the National Bureau of Standards, 69B: 1–47, doi:10.6028/jres.069b.001, MR 0179781. See in particular section 2.5, "Bon-matroid of a graph", pp. 5–6, section 5.6, "Graphic and co-graphic matroids", pp. 19–20, and section 9, "Graphic matroids", pp. 38–47.
Whitney umbrella In geometry, the Whitney umbrella (or Whitney's umbrella, named after American mathematician Hassler Whitney, and sometimes called a Cayley umbrella) is a specific self-intersecting ruled surface placed in three dimensions. It is the union of all straight lines that pass through points of a fixed parabola and are perpendicular to a fixed straight line which is parallel to the axis of the parabola and lies on its perpendicular bisecting plane. Formulas Whitney's umbrella can be given by the parametric equations in Cartesian coordinates $\left\{{\begin{aligned}x(u,v)&=uv,\\y(u,v)&=u,\\z(u,v)&=v^{2},\end{aligned}}\right.$ where the parameters u and v range over the real numbers. It is also given by the implicit equation $x^{2}-y^{2}z=0.$ This formula also includes the negative z axis (which is called the handle of the umbrella). Properties Whitney's umbrella is a ruled surface and a right conoid. It is important in the field of singularity theory, as a simple local model of a pinch point singularity. The pinch point and the fold singularity are the only stable local singularities of maps from R2 to R3. It is named after the American mathematician Hassler Whitney. In string theory, a Whitney brane is a D7-brane wrapping a variety whose singularities are locally modeled by the Whitney umbrella. Whitney branes appear naturally when taking Sen's weak coupling limit of F-theory. See also • Cross-cap • Right conoid • Ruled surface References • "Whitney's Umbrella". The Topological Zoo. The Geometry Center. Retrieved 2006-03-08. (Images and movies of the Whitney umbrella.)
Regular homotopy In the mathematical field of topology, a regular homotopy refers to a special kind of homotopy between immersions of one manifold in another. The homotopy must be a 1-parameter family of immersions. Similar to homotopy classes, one defines two immersions to be in the same regular homotopy class if there exists a regular homotopy between them. Regular homotopy for immersions is similar to isotopy of embeddings: they are both restricted types of homotopies. Stated another way, two continuous functions $f,g:M\to N$ are homotopic if they represent points in the same path-components of the mapping space $C(M,N)$, given the compact-open topology. The space of immersions is the subspace of $C(M,N)$ consisting of immersions, denoted by $\operatorname {Imm} (M,N)$. Two immersions $f,g:M\to N$ are regularly homotopic if they represent points in the same path-component of $\operatorname {Imm} (M,N)$. Examples Any two knots in 3-space are equivalent by regular homotopy, though not by isotopy. The Whitney–Graustein theorem classifies the regular homotopy classes of a circle into the plane; two immersions are regularly homotopic if and only if they have the same turning number – equivalently, total curvature; equivalently, if and only if their Gauss maps have the same degree/winding number. Stephen Smale classified the regular homotopy classes of a k-sphere immersed in $\mathbb {R} ^{n}$ – they are classified by homotopy groups of Stiefel manifolds, which is a generalization of the Gauss map, with here k partial derivatives not vanishing. More precisely, the set $I(n,k)$ of regular homotopy classes of embeddings of sphere $S^{k}$ in $\mathbb {R} ^{n}$ is in one-to-one correspondence with elements of group $\pi _{k}\left(V_{k}\left(\mathbb {R} ^{n}\right)\right)$. In case $k=n-1$ we have $V_{n-1}\left(\mathbb {R} ^{n}\right)\cong SO(n)$. Since $SO(1)$ is path connected, $\pi _{2}(SO(3))\cong \pi _{2}\left(\mathbb {R} P^{3}\right)\cong \pi _{2}\left(S^{3}\right)\cong 0$ and $\pi _{6}(SO(6))\to \pi _{6}(SO(7))\to \pi _{6}\left(S^{6}\right)\to \pi _{5}(SO(6)\to \pi _{5}(SO(7))$ and due to Bott periodicity theorem we have $\pi _{6}(SO(6))\cong \pi _{6}(\operatorname {Spin} (6))\cong \pi _{6}(SU(4))\cong \pi _{6}(U(4))\cong 0$ and since $\pi _{5}(SO(6))\cong \mathbb {Z} ,\ \pi _{5}(SO(7))\cong 0$ then we have $\pi _{6}(SO(7))\cong 0$. Therefore all immersions of spheres $S^{0},\ S^{2}$ and $S^{6}$ in euclidean spaces of one more dimension are regular homotopic. In particular, spheres $S^{n}$ embedded in $\mathbb {R} ^{n+1}$ admit eversion if $n=0,2,6$. A corollary of his work is that there is only one regular homotopy class of a 2-sphere immersed in $\mathbb {R} ^{3}$. In particular, this means that sphere eversions exist, i.e. one can turn the 2-sphere "inside-out". Both of these examples consist of reducing regular homotopy to homotopy; this has subsequently been substantially generalized in the homotopy principle (or h-principle) approach. Non-degenerate homotopy For locally convex, closed space curves, one can also define non-degenerate homotopy. Here, the 1-parameter family of immersions must be non-degenerate (i.e. the curvature may never vanish). There are 2 distinct non-degenerate homotopy classes.[1] Further restrictions of non-vanishing torsion lead to 4 distinct equivalence classes.[2] References 1. Feldman, E. A. (1968). "Deformations of closed space curves". Journal of Differential Geometry. 2 (1): 67–75. doi:10.4310/jdg/1214501138. 2. Little, John A. (1971). "Third order nondegenerate homotopies of space curves". Journal of Differential Geometry. 5 (3): 503–515. doi:10.4310/jdg/1214430012. • Whitney, Hassler (1937). "On regular closed curves in the plane". Compositio Mathematica. 4: 276–284. • Smale, Stephen (February 1959). "A classification of immersions of the two-sphere" (PDF). Transactions of the American Mathematical Society. 90 (2): 281–290. doi:10.2307/1993205. JSTOR 1993205. • Smale, Stephen (March 1959). "The classification of immersions of spheres in Euclidean spaces" (PDF). Annals of Mathematics. 69 (2): 327–344. doi:10.2307/1970186. JSTOR 1970186.
Clique complex Clique complexes, independence complexes, flag complexes, Whitney complexes and conformal hypergraphs are closely related mathematical objects in graph theory and geometric topology that each describe the cliques (complete subgraphs) of an undirected graph. Clique complex The clique complex X(G) of an undirected graph G is an abstract simplicial complex (that is, a family of finite sets closed under the operation of taking subsets), formed by the sets of vertices in the cliques of G. Any subset of a clique is itself a clique, so this family of sets meets the requirement of an abstract simplicial complex that every subset of a set in the family should also be in the family. The clique complex can also be viewed as a topological space in which each clique of k vertices is represented by a simplex of dimension k – 1. The 1-skeleton of X(G) (also known as the underlying graph of the complex) is an undirected graph with a vertex for every 1-element set in the family and an edge for every 2-element set in the family; it is isomorphic to G.[1] Negative example Every clique complex is an abstract simplicial complex, but the opposite is not true. For example, consider the abstract simplicial complex over {1,2,3,4} with maximal sets {1,2,3}, {2,3,4}, {4,1}. If it were the X(G) of some graph G, then G had to have the edges {1,2}, {1,3}, {2,3}, {2,4}, {3,4}, {4,1}, so X(G) should also contain the clique {1,2,3,4}. Independence complex The independence complex I(G) of an undirected graph G is an abstract simplicial complex formed by the sets of vertices in the independent sets of G. The clique complex of G is equivalent to the independence complex of the complement graph of G. Flag complex A flag complex is an abstract simplicial complex with an additional property called "2-determined": for every subset S of vertices, if every pair of vertices in S is in the complex, then S itself is in the complex too. Every clique complex is a flag complex: if every pair of vertices in S is a clique of size 2, then there is an edge between them, so S is a clique. Every flag complex is a clique complex: given a flag complex, define a graph G on the set of all vertices, where two vertices u,v are adjacent in G iff {u,v} is in the complex (this graph is called the 1-skeleton of the complex). By definition of a flag complex, every set of vertices that are pairwise-connected, is in the complex. Therefore, the flag complex is equal to the clique complex on G. Thus, flag complexes and clique complexes are essentially the same thing. However, in many cases it is convenient to define a flag complex directly from some data other than a graph, rather than indirectly as the clique complex of a graph derived from that data.[2] Mikhail Gromov defined the no-Δ condition to be the condition of being a flag complex. Whitney complex Clique complexes are also known as Whitney complexes, after Hassler Whitney. A Whitney triangulation or clean triangulation of a two-dimensional manifold is an embedding of a graph G onto the manifold in such a way that every face is a triangle and every triangle is a face. If a graph G has a Whitney triangulation, it must form a cell complex that is isomorphic to the Whitney complex of G. In this case, the complex (viewed as a topological space) is homeomorphic to the underlying manifold. A graph G has a 2-manifold clique complex, and can be embedded as a Whitney triangulation, if and only if G is locally cyclic; this means that, for every vertex v in the graph, the induced subgraph formed by the neighbors of v forms a single cycle.[3] Conformal hypergraph The primal graph G(H) of a hypergraph is the graph on the same vertex set that has as its edges the pairs of vertices appearing together in the same hyperedge. A hypergraph is said to be conformal if every maximal clique of its primal graph is a hyperedge, or equivalently, if every clique of its primal graph is contained in some hyperedge.[4] If the hypergraph is required to be downward-closed (so it contains all hyperedges that are contained in some hyperedge) then the hypergraph is conformal precisely when it is a flag complex. This relates the language of hypergraphs to the language of simplicial complexes. Examples and applications The barycentric subdivision of any cell complex C is a flag complex having one vertex per cell of C. A collection of vertices of the barycentric subdivision form a simplex if and only if the corresponding collection of cells of C form a flag (a chain in the inclusion ordering of the cells).[2] In particular, the barycentric subdivision of a cell complex on a 2-manifold gives rise to a Whitney triangulation of the manifold. The order complex of a partially ordered set consists of the chains (totally ordered subsets) of the partial order. If every pair of some subset is itself ordered, then the whole subset is a chain, so the order complex satisfies the no-Δ condition. It may be interpreted as the clique complex of the comparability graph of the partial order.[2] The matching complex of a graph consists of the sets of edges no two of which share an endpoint; again, this family of sets satisfies the no-Δ condition. It may be viewed as the clique complex of the complement graph of the line graph of the given graph. When the matching complex is referred to without any particular graph as context, it means the matching complex of a complete graph. The matching complex of a complete bipartite graph Km,n is known as a chessboard complex. It is the clique graph of the complement graph of a rook's graph,[5] and each of its simplices represents a placement of rooks on an m × n chess board such that no two of the rooks attack each other. When m = n ± 1, the chessboard complex forms a pseudo-manifold. The Vietoris–Rips complex of a set of points in a metric space is a special case of a clique complex, formed from the unit disk graph of the points; however, every clique complex X(G) may be interpreted as the Vietoris–Rips complex of the shortest path metric on the underlying graph G. Hodkinson & Otto (2003) describe an application of conformal hypergraphs in the logics of relational structures. In that context, the Gaifman graph of a relational structure is the same as the underlying graph of the hypergraph representing the structure, and a structure is guarded if it corresponds to a conformal hypergraph. Gromov showed that a cubical complex (that is, a family of hypercubes intersecting face-to-face) forms a CAT(0) space if and only if the complex is simply connected and the link of every vertex forms a flag complex. A cubical complex meeting these conditions is sometimes called a cubing or a space with walls.[1][6] Homology groups Meshulam[7] proves the following theorem on the homology of the clique complex. Given integers $l\geq 1,t\geq 0$, suppose a graph G satisfies a property called $P(l,t)$, which means that: • Every set of $l$ vertices in G has a common neighbor; • There exists a set A of vertices, that contains a common neighbor to every set of $l$ vertices, and in addition, the induced graph G[A] does not contain, as an induced subgraph, a copy of the 1-skeleton of the t-dimensional octahedral sphere. Then the j-th reduced homology of the clique complex X(G) is trivial for any j between 0 and $\max(l-t,\lfloor {l}/{2}\rfloor )-1$. See also • Simplex graph, a kind of graph having one node for every clique of the underlying graph • Partition matroid, a kind of matroid whose matroid intersections may form clique complexes Notes 1. Bandelt & Chepoi (2008). 2. Davis (2002). 3. Hartsfeld & Ringel (1991); Larrión, Neumann-Lara & Pizaña (2002); Malnič & Mohar (1992). 4. Berge (1989); Hodkinson & Otto (2003). 5. Dong & Wachs (2002). 6. Chatterji & Niblo (2005). 7. Meshulam, Roy (2001-01-01). "The Clique Complex and Hypergraph Matching". Combinatorica. 21 (1): 89–94. doi:10.1007/s004930170006. ISSN 1439-6912. S2CID 207006642. References • Bandelt, H.-J.; Chepoi, V. (2008), "Metric graph theory and geometry: a survey", in Goodman, J. E.; Pach, J.; Pollack, R. (eds.), Surveys on Discrete and Computational Geometry: Twenty Years Later (PDF), Contemporary Mathematics, vol. 453, Providence, RI: AMS, pp. 49–86. • Berge, C. (1989), Hypergraphs: Combinatorics of Finite Sets, North-Holland, ISBN 0-444-87489-5. • Chatterji, I.; Niblo, G. (2005), "From wall spaces to CAT(0) cube complexes", International Journal of Algebra and Computation, 15 (5–6): 875–885, arXiv:math.GT/0309036, doi:10.1142/S0218196705002669, S2CID 2786607. • Davis, M. W. (2002), "Nonpositive curvature and reflection groups", in Daverman, R. J.; Sher, R. B. (eds.), Handbook of Geometric Topology, Elsevier, pp. 373–422. • Dong, X.; Wachs, M. L. (2002), "Combinatorial Laplacian of the matching complex", Electronic Journal of Combinatorics, 9: R17, doi:10.37236/1634. • Hartsfeld, N.; Ringel, Gerhard (1991), "Clean triangulations", Combinatorica, 11 (2): 145–155, doi:10.1007/BF01206358, S2CID 28144260. • Hodkinson, I.; Otto, M. (2003), "Finite conformal hypergraph covers and Gaifman cliques in finite structures", The Bulletin of Symbolic Logic, 9 (3): 387–405, CiteSeerX 10.1.1.107.5000, doi:10.2178/bsl/1058448678. • Larrión, F.; Neumann-Lara, V.; Pizaña, M. A. (2002), "Whitney triangulations, local girth and iterated clique graphs", Discrete Mathematics, 258 (1–3): 123–135, doi:10.1016/S0012-365X(02)00266-2. • Malnič, A.; Mohar, B. (1992), "Generating locally cyclic triangulations of surfaces", Journal of Combinatorial Theory, Series B, 56 (2): 147–164, doi:10.1016/0095-8956(92)90015-P.
Whitney covering lemma In mathematical analysis, the Whitney covering lemma, or Whitney decomposition, asserts the existence of a certain type of partition of an open set in a Euclidean space. Originally it was employed in the proof of Hassler Whitney's extension theorem. The lemma was subsequently applied to prove generalizations of the Calderón–Zygmund decomposition. Roughly speaking, the lemma states that it is possible to decompose an open set by cubes each of whose diameters is proportional, within certain bounds, to its distance from the boundary of the open set. More precisely: Whitney Covering Lemma (Grafakos 2008, Appendix J) Let $\Omega $ be an open non-empty proper subset of $\mathbb {R} ^{n}$. Then there exists a family of closed cubes $\{Q_{j}\}_{j}$ such that • $\cup _{j}Q_{j}=\Omega $ and the $Q_{j}$'s have disjoint interiors. • ${\sqrt {n}}\ell (Q_{j})\leq \mathrm {dist} (Q_{j},\Omega ^{c})\leq 4{\sqrt {n}}\ell (Q_{j}).$ • If the boundaries of two cubes $Q_{j}$ and $Q_{k}$ touch then ${\frac {1}{4}}\leq {\frac {\ell (Q_{j})}{\ell (Q_{k})}}\leq 4.$ • For a given $Q_{j}$ there exist at most $12^{n}Q_{k}$'s that touch it. Where $\ell (Q)$ denotes the length of a cube $Q$. References • Grafakos, Loukas (2008). Classical Fourier Analysis. Springer. ISBN 978-0-387-09431-1. • DiBenedetto, Emmanuele (2002), Real analysis, Birkhäuser, ISBN 0-8176-4231-5. • Stein, Elias (1970), Singular Integrals and Differentiability Properties of Functions, Princeton University Press. • Whitney, Hassler (1934), "Analytic extensions of functions defined in closed sets", Transactions of the American Mathematical Society, American Mathematical Society, 36 (1): 63–89, doi:10.2307/1989708, JSTOR 1989708.
Whitney disk In mathematics, given two submanifolds A and B of a manifold X intersecting in two points p and q, a Whitney disc is a mapping from the two-dimensional disc D, with two marked points, to X, such that the two marked points go to p and q, one boundary arc of D goes to A and the other to B.[1] Their existence and embeddedness is crucial in proving the cobordism theorem, where it is used to cancel the intersection points; and its failure in low dimensions corresponds to not being able to embed a Whitney disc. Casson handles are an important technical tool for constructing the embedded Whitney disc relevant to many results on topological four-manifolds. Pseudoholomorphic Whitney discs are counted by the differential in Lagrangian intersection Floer homology. References 1. Scorpan, Alexandru (2005), The Wild World of 4-manifolds, American Mathematical Society, p. 560, ISBN 9780821837498.
Whitney inequality In mathematics, the Whitney inequality gives an upper bound for the error of best approximation of a function by polynomials in terms of the moduli of smoothness. It was first proved by Hassler Whitney in 1957,[1] and is an important tool in the field of approximation theory for obtaining upper estimates on the errors of best approximation. Statement of the theorem Denote the value of the best uniform approximation of a function $f\in C([a,b])$ by algebraic polynomials $P_{n}$ of degree $\leq n$ by $E_{n}(f)_{[a,b]}:=\inf _{P_{n}}{\\|f-P_{n}\\|_{C([a,b])}}$ The moduli of smoothness of order $k$ of a function $f\in C([a,b])$ are defined as: $\omega _{k}(t):=\omega _{k}(t;f;[a,b]):=\sup _{h\in [0,t]}\\|\Delta _{h}^{k}(f;\cdot )\\|_{C([a,b-kh])}\quad {\text{ for }}\quad t\in [0,(b-a)/k],$ $\omega _{k}(t):=\omega _{k}((b-a)/k)\quad {\text{ for}}\quad t>(b-a)/k,$ where $\Delta _{h}^{k}$ is the finite difference of order $k$. Theorem: [2] [Whitney, 1957] If $f\in C([a,b])$, then $E_{k-1}(f)_{[a,b]}\leq W_{k}\omega _{k}\left({\frac {b-a}{k}};f;[a,b]\right)$ where $W_{k}$ is a constant depending only on $k$. The Whitney constant $W(k)$ is the smallest value of $W_{k}$ for which the above inequality holds. The theorem is particularly useful when applied on intervals of small length, leading to good estimates on the error of spline approximation. Proof The original proof given by Whitney follows an analytic argument which utilizes the properties of moduli of smoothness. However, it can also be proved in a much shorter way using Peetre's K-functionals.[3] Let: $x_{0}:=a,\quad h:={\frac {b-a}{k}},\quad x_{j}:=x+0+jh,\quad F(x)=\int _{a}^{x}f(u)\,du,$ $G(x):=F(x)-L(x;F;x_{0},\ldots ,x_{k}),\quad g(x):=G'(x),$ $\omega _{k}(t):=\omega _{k}(t;f;[a,b])\equiv \omega _{k}(t;g;[a,b])$ where $L(x;F;x_{0},\ldots ,x_{k})$ is the Lagrange polynomial for $F$ at the nodes $\{x_{0},\ldots ,x_{k}\}$. Now fix some $x\in [a,b]$ and choose $\delta $ for which $(x+k\delta )\in [a,b]$. Then: $\int _{0}^{1}\Delta _{t\delta }^{k}(g;x)\,dt=(-1)^{k}g(x)+\sum _{j=1}^{k}(-1)^{k-j}{\binom {k}{j}}\int _{0}^{1}g(x+jt\delta )\,dt$ $=(-1)^{k}g(x)+\sum _{j=1}^{k}{(-1)^{k-j}{\binom {k}{j}}{\frac {1}{j\delta }}(G(x+j\delta )-G(x))},$ Therefore: $\|g(x)\|\leq \int _{0}^{1}\|\Delta _{t\delta }^{k}(g;x)\|\,dt+{\frac {2}{\|\delta \|}}\\|G\\|_{C([a,b])}\sum _{j=1}^{k}{\binom {k}{j}}{\frac {1}{j}}\leq \omega _{k}(\|\delta \|)+{\frac {1}{\|\delta \|}}2^{k+1}\\|G\\|_{C([a,b])}$ And since we have $\\|G\\|_{C([a,b])}\leq h\omega _{k}(h)$, (a property of moduli of smoothness) $E_{k-1}(f)_{[a,b]}\leq \\|g\\|_{C([a,b])}\leq \omega _{k}(\|\delta \|)+{\frac {1}{\|\delta \|}}h2^{k+1}\omega _{k}(h).$ Since $\delta $ can always be chosen in such a way that $h\geq \|\delta \|\geq h/2$, this completes the proof. Whitney constants and Sendov's conjecture It is important to have sharp estimates of the Whitney constants. It is easily shown that $W(1)=1/2$, and it was first proved by Burkill (1952) that $W(2)\leq 1$, who conjectured that $W(k)\leq 1$ for all $k$. Whitney was also able to prove that [2] $W(2)={\frac {1}{2}},\quad {\frac {8}{15}}\leq W(3)\leq 0.7\quad W(4)\leq 3.3\quad W(5)\leq 10.4$ and $W(k)\geq {\frac {1}{2}},\quad k\in \mathbb {N} $ In 1964, Brudnyi was able to obtain the estimate $W(k)=O(k^{2k})$, and in 1982, Sendov proved that $W(k)\leq (k+1)k^{k}$. Then, in 1985, Ivanov and Takev proved that $W(k)=O(k\ln k)$, and Binev proved that $W(k)=O(k)$. Sendov conjectured that $W(k)\leq 1$ for all $k$, and in 1985 was able to prove that the Whitney constants are bounded above by an absolute constant, that is, $W(k)\leq 6$ for all $k$. Kryakin, Gilewicz, and Shevchuk (2002)[4] were able to show that $W(k)\leq 2$ for $k\leq 82000$, and that $W(k)\leq 2+{\frac {1}{e^{2}}}$ for all $k$. References 1. Hassler, Whitney (1957). "On Functions with Bounded nth Differences". J. Math. Pures Appl. 36 (IX): 67–95. 2. Dzyadyk, Vladislav K.; Shevchuk, Igor A. (2008). "3.6". Theory of Uniform Approximation of Functions by Polynomials (1st ed.). Berlin, Germany: Walter de Gruyter. pp. 231–233. ISBN 978-3-11-020147-5. 3. Devore, R. A. K.; Lorentz, G. G. (4 November 1993). "6, Theorem 4.2". Constructive Approximation, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] (1st ed.). Berlin, Germany: Springer-Verlag. ISBN 978-3540506270. 4. Gilewicz, J.; Kryakin, Yu. V.; Shevchuk, I. A. (2002). "Boundedness by 3 of the Whitney Interpolation Constant". Journal of Approximation Theory. 119 (2): 271–290. doi:10.1006/jath.2002.3732.
Dual graph In the mathematical discipline of graph theory, the dual graph of a planar graph G is a graph that has a vertex for each face of G. The dual graph has an edge for each pair of faces in G that are separated from each other by an edge, and a self-loop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs (graphs that are already embedded in the plane) rather than planar graphs (graphs that may be embedded but for which the embedding is not yet known). For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph. Historically, the first form of graph duality to be recognized was the association of the Platonic solids into pairs of dual polyhedra. Graph duality is a topological generalization of the geometric concepts of dual polyhedra and dual tessellations, and is in turn generalized combinatorially by the concept of a dual matroid. Variations of planar graph duality include a version of duality for directed graphs, and duality for graphs embedded onto non-planar two-dimensional surfaces. These notions of dual graphs should not be confused with a different notion, the edge-to-vertex dual or line graph of a graph. The term dual is used because the property of being a dual graph is symmetric, meaning that if H is a dual of a connected graph G, then G is a dual of H. When discussing the dual of a graph G, the graph G itself may be referred to as the "primal graph". Many other graph properties and structures may be translated into other natural properties and structures of the dual. For instance, cycles are dual to cuts, spanning trees are dual to the complements of spanning trees, and simple graphs (without parallel edges or self-loops) are dual to 3-edge-connected graphs. Graph duality can help explain the structure of mazes and of drainage basins. Dual graphs have also been applied in computer vision, computational geometry, mesh generation, and the design of integrated circuits. Examples Cycles and dipoles A dipole graph A cycle graph The unique planar embedding of a cycle graph divides the plane into only two regions, the inside and outside of the cycle, by the Jordan curve theorem. However, in an n-cycle, these two regions are separated from each other by n different edges. Therefore, the dual graph of the n-cycle is a multigraph with two vertices (dual to the regions), connected to each other by n dual edges. Such a graph is called a multiple edge, linkage, or sometimes a dipole graph. Conversely, the dual to an n-edge dipole graph is an n-cycle.[1] Dual polyhedra Main article: Dual polyhedron According to Steinitz's theorem, every polyhedral graph (the graph formed by the vertices and edges of a three-dimensional convex polyhedron) must be planar and 3-vertex-connected, and every 3-vertex-connected planar graph comes from a convex polyhedron in this way. Every three-dimensional convex polyhedron has a dual polyhedron; the dual polyhedron has a vertex for every face of the original polyhedron, with two dual vertices adjacent whenever the corresponding two faces share an edge. Whenever two polyhedra are dual, their graphs are also dual. For instance the Platonic solids come in dual pairs, with the octahedron dual to the cube, the dodecahedron dual to the icosahedron, and the tetrahedron dual to itself.[2] Polyhedron duality can also be extended to duality of higher dimensional polytopes,[3] but this extension of geometric duality does not have clear connections to graph-theoretic duality. Self-dual graphs A plane graph is said to be self-dual if it is isomorphic to its dual graph. The wheel graphs provide an infinite family of self-dual graphs coming from self-dual polyhedra (the pyramids).[4][5] However, there also exist self-dual graphs that are not polyhedral, such as the one shown. Servatius & Christopher (1992) describe two operations, adhesion and explosion, that can be used to construct a self-dual graph containing a given planar graph; for instance, the self-dual graph shown can be constructed as the adhesion of a tetrahedron with its dual.[5] It follows from Euler's formula that every self-dual graph with n vertices has exactly 2n − 2 edges.[6] Every simple self-dual planar graph contains at least four vertices of degree three, and every self-dual embedding has at least four triangular faces.[7] Properties Many natural and important concepts in graph theory correspond to other equally natural but different concepts in the dual graph. Because the dual of the dual of a connected plane graph is isomorphic to the primal graph,[8] each of these pairings is bidirectional: if concept X in a planar graph corresponds to concept Y in the dual graph, then concept Y in a planar graph corresponds to concept X in the dual. Simple graphs versus multigraphs The dual of a simple graph need not be simple: it may have self-loops (an edge with both endpoints at the same vertex) or multiple edges connecting the same two vertices, as was already evident in the example of dipole multigraphs being dual to cycle graphs. As a special case of the cut-cycle duality discussed below, the bridges of a planar graph G are in one-to-one correspondence with the self-loops of the dual graph.[9] For the same reason, a pair of parallel edges in a dual multigraph (that is, a length-2 cycle) corresponds to a 2-edge cutset in the primal graph (a pair of edges whose deletion disconnects the graph). Therefore, a planar graph is simple if and only if its dual has no 1- or 2-edge cutsets; that is, if it is 3-edge-connected. The simple planar graphs whose duals are simple are exactly the 3-edge-connected simple planar graphs.[10] This class of graphs includes, but is not the same as, the class of 3-vertex-connected simple planar graphs. For instance, the figure showing a self-dual graph is 3-edge-connected (and therefore its dual is simple) but is not 3-vertex-connected. Uniqueness Because the dual graph depends on a particular embedding, the dual graph of a planar graph is not unique, in the sense that the same planar graph can have non-isomorphic dual graphs.[11] In the picture, the blue graphs are isomorphic but their dual red graphs are not. The upper red dual has a vertex with degree 6 (corresponding to the outer face of the blue graph) while in the lower red graph all degrees are less than 6. Hassler Whitney showed that if the graph is 3-connected then the embedding, and thus the dual graph, is unique.[12] By Steinitz's theorem, these graphs are exactly the polyhedral graphs, the graphs of convex polyhedra. A planar graph is 3-vertex-connected if and only if its dual graph is 3-vertex-connected. More generally, a planar graph has a unique embedding, and therefore also a unique dual, if and only if it is a subdivision of a 3-vertex-connected planar graph (a graph formed from a 3-vertex-connected planar graph by replacing some of its edges by paths). For some planar graphs that are not 3-vertex-connected, such as the complete bipartite graph K2,4, the embedding is not unique, but all embeddings are isomorphic. When this happens, correspondingly, all dual graphs are isomorphic. Because different embeddings may lead to different dual graphs, testing whether one graph is a dual of another (without already knowing their embeddings) is a nontrivial algorithmic problem. For biconnected graphs, it can be solved in polynomial time by using the SPQR trees of the graphs to construct a canonical form for the equivalence relation of having a shared mutual dual. For instance, the two red graphs in the illustration are equivalent according to this relation. However, for planar graphs that are not biconnected, this relation is not an equivalence relation and the problem of testing mutual duality is NP-complete.[13] Cuts and cycles A cutset in an arbitrary connected graph is a subset of edges defined from a partition of the vertices into two subsets, by including an edge in the subset when it has one endpoint on each side of the partition. Removing the edges of a cutset necessarily splits the graph into at least two connected components. A minimal cutset (also called a bond) is a cutset with the property that every proper subset of the cutset is not itself a cut. A minimal cutset of a connected graph necessarily separates its graph into exactly two components, and consists of the set of edges that have one endpoint in each component.[14] A simple cycle is a connected subgraph in which each vertex of the cycle is incident to exactly two edges of the cycle.[15] In a connected planar graph G, every simple cycle of G corresponds to a minimal cutset in the dual of G, and vice versa.[16] This can be seen as a form of the Jordan curve theorem: each simple cycle separates the faces of G into the faces in the interior of the cycle and the faces of the exterior of the cycle, and the duals of the cycle edges are exactly the edges that cross from the interior to the exterior.[17] The girth of any planar graph (the size of its smallest cycle) equals the edge connectivity of its dual graph (the size of its smallest cutset).[18] This duality extends from individual cutsets and cycles to vector spaces defined from them. The cycle space of a graph is defined as the family of all subgraphs that have even degree at each vertex; it can be viewed as a vector space over the two-element finite field, with the symmetric difference of two sets of edges acting as the vector addition operation in the vector space. Similarly, the cut space of a graph is defined as the family of all cutsets, with vector addition defined in the same way. Then the cycle space of any planar graph and the cut space of its dual graph are isomorphic as vector spaces.[19] Thus, the rank of a planar graph (the dimension of its cut space) equals the cyclotomic number of its dual (the dimension of its cycle space) and vice versa.[11] A cycle basis of a graph is a set of simple cycles that form a basis of the cycle space (every even-degree subgraph can be formed in exactly one way as a symmetric difference of some of these cycles). For edge-weighted planar graphs (with sufficiently general weights that no two cycles have the same weight) the minimum-weight cycle basis of the graph is dual to the Gomory–Hu tree of the dual graph, a collection of nested cuts that together include a minimum cut separating each pair of vertices in the graph. Each cycle in the minimum weight cycle basis has a set of edges that are dual to the edges of one of the cuts in the Gomory–Hu tree. When cycle weights may be tied, the minimum-weight cycle basis may not be unique, but in this case it is still true that the Gomory–Hu tree of the dual graph corresponds to one of the minimum weight cycle bases of the graph.[19] In directed planar graphs, simple directed cycles are dual to directed cuts (partitions of the vertices into two subsets such that all edges go in one direction, from one subset to the other). Strongly oriented planar graphs (graphs whose underlying undirected graph is connected, and in which every edge belongs to a cycle) are dual to directed acyclic graphs in which no edge belongs to a cycle. To put this another way, the strong orientations of a connected planar graph (assignments of directions to the edges of the graph that result in a strongly connected graph) are dual to acyclic orientations (assignments of directions that produce a directed acyclic graph).[20] In the same way, dijoins (sets of edges that include an edge from each directed cut) are dual to feedback arc sets (sets of edges that include an edge from each cycle).[21] Spanning trees A spanning tree may be defined as a set of edges that, together with all of the vertices of the graph, forms a connected and acyclic subgraph. But, by cut-cycle duality, if a set S of edges in a planar graph G is acyclic (has no cycles), then the set of edges dual to S has no cuts, from which it follows that the complementary set of dual edges (the duals of the edges that are not in S) forms a connected subgraph. Symmetrically, if S is connected, then the edges dual to the complement of S form an acyclic subgraph. Therefore, when S has both properties – it is connected and acyclic – the same is true for the complementary set in the dual graph. That is, each spanning tree of G is complementary to a spanning tree of the dual graph, and vice versa. Thus, the edges of any planar graph and its dual can together be partitioned (in multiple different ways) into two spanning trees, one in the primal and one in the dual, that together extend to all the vertices and faces of the graph but never cross each other. In particular, the minimum spanning tree of G is complementary to the maximum spanning tree of the dual graph.[22] However, this does not work for shortest path trees, even approximately: there exist planar graphs such that, for every pair of a spanning tree in the graph and a complementary spanning tree in the dual graph, at least one of the two trees has distances that are significantly longer than the distances in its graph.[23] An example of this type of decomposition into interdigitating trees can be seen in some simple types of mazes, with a single entrance and no disconnected components of its walls. In this case both the maze walls and the space between the walls take the form of a mathematical tree. If the free space of the maze is partitioned into simple cells (such as the squares of a grid) then this system of cells can be viewed as an embedding of a planar graph, in which the tree structure of the walls forms a spanning tree of the graph and the tree structure of the free space forms a spanning tree of the dual graph.[24] Similar pairs of interdigitating trees can also be seen in the tree-shaped pattern of streams and rivers within a drainage basin and the dual tree-shaped pattern of ridgelines separating the streams.[25] This partition of the edges and their duals into two trees leads to a simple proof of Euler’s formula V − E + F = 2 for planar graphs with V vertices, E edges, and F faces. Any spanning tree and its complementary dual spanning tree partition the edges into two subsets of V − 1 and F − 1 edges respectively, and adding the sizes of the two subsets gives the equation E = (V − 1) + (F − 1) which may be rearranged to form Euler's formula. According to Duncan Sommerville, this proof of Euler's formula is due to K. G. C. Von Staudt’s Geometrie der Lage (Nürnberg, 1847).[26] In nonplanar surface embeddings the set of dual edges complementary to a spanning tree is not a dual spanning tree. Instead this set of edges is the union of a dual spanning tree with a small set of extra edges whose number is determined by the genus of the surface on which the graph is embedded. The extra edges, in combination with paths in the spanning trees, can be used to generate the fundamental group of the surface.[27] Additional properties Any counting formula involving vertices and faces that is valid for all planar graphs may be transformed by planar duality into an equivalent formula in which the roles of the vertices and faces have been swapped. Euler's formula, which is self-dual, is one example. Another given by Harary involves the handshaking lemma, according to which the sum of the degrees of the vertices of any graph equals twice the number of edges. In its dual form, this lemma states that in a plane graph, the sum of the numbers of sides of the faces of the graph equals twice the number of edges.[28] The medial graph of a plane graph is isomorphic to the medial graph of its dual. Two planar graphs can have isomorphic medial graphs only if they are dual to each other.[29] A planar graph with four or more vertices is maximal (no more edges can be added while preserving planarity) if and only if its dual graph is both 3-vertex-connected and 3-regular.[30] A connected planar graph is Eulerian (has even degree at every vertex) if and only if its dual graph is bipartite.[31] A Hamiltonian cycle in a planar graph G corresponds to a partition of the vertices of the dual graph into two subsets (the interior and exterior of the cycle) whose induced subgraphs are both trees. In particular, Barnette's conjecture on the Hamiltonicity of cubic bipartite polyhedral graphs is equivalent to the conjecture that every Eulerian maximal planar graph can be partitioned into two induced trees.[32] If a planar graph G has Tutte polynomial TG(x,y), then the Tutte polynomial of its dual graph is obtained by swapping x and y. For this reason, if some particular value of the Tutte polynomial provides information about certain types of structures in G, then swapping the arguments to the Tutte polynomial will give the corresponding information for the dual structures. For instance, the number of strong orientations is TG(0,2) and the number of acyclic orientations is TG(2,0).[33] For bridgeless planar graphs, graph colorings with k colors correspond to nowhere-zero flows modulo k on the dual graph. For instance, the four color theorem (the existence of a 4-coloring for every planar graph) can be expressed equivalently as stating that the dual of every bridgeless planar graph has a nowhere-zero 4-flow. The number of k-colorings is counted (up to an easily computed factor) by the Tutte polynomial value TG(1 − k,0) and dually the number of nowhere-zero k-flows is counted by TG(0,1 − k).[34] An st-planar graph is a connected planar graph together with a bipolar orientation of that graph, an orientation that makes it acyclic with a single source and a single sink, both of which are required to be on the same face as each other. Such a graph may be made into a strongly connected graph by adding one more edge, from the sink back to the source, through the outer face. The dual of this augmented planar graph is itself the augmentation of another st-planar graph.[35] Variations Directed graphs In a directed plane graph, the dual graph may be made directed as well, by orienting each dual edge by a 90° clockwise turn from the corresponding primal edge.[35] Strictly speaking, this construction is not a duality of directed planar graphs, because starting from a graph G and taking the dual twice does not return to G itself, but instead constructs a graph isomorphic to the transpose graph of G, the graph formed from G by reversing all of its edges. Taking the dual four times returns to the original graph. Weak dual The weak dual of a plane graph is the subgraph of the dual graph whose vertices correspond to the bounded faces of the primal graph. A plane graph is outerplanar if and only if its weak dual is a forest. For any plane graph G, let G+ be the plane multigraph formed by adding a single new vertex v in the unbounded face of G, and connecting v to each vertex of the outer face (multiple times, if a vertex appears multiple times on the boundary of the outer face); then, G is the weak dual of the (plane) dual of G+.[36] Infinite graphs and tessellations The concept of duality applies as well to infinite graphs embedded in the plane as it does to finite graphs. However, care is needed to avoid topological complications such as points of the plane that are neither part of an open region disjoint from the graph nor part of an edge or vertex of the graph. When all faces are bounded regions surrounded by a cycle of the graph, an infinite planar graph embedding can also be viewed as a tessellation of the plane, a covering of the plane by closed disks (the tiles of the tessellation) whose interiors (the faces of the embedding) are disjoint open disks. Planar duality gives rise to the notion of a dual tessellation, a tessellation formed by placing a vertex at the center of each tile and connecting the centers of adjacent tiles.[37] The concept of a dual tessellation can also be applied to partitions of the plane into finitely many regions. It is closely related to but not quite the same as planar graph duality in this case. For instance, the Voronoi diagram of a finite set of point sites is a partition of the plane into polygons within which one site is closer than any other. The sites on the convex hull of the input give rise to unbounded Voronoi polygons, two of whose sides are infinite rays rather than finite line segments. The dual of this diagram is the Delaunay triangulation of the input, a planar graph that connects two sites by an edge whenever there exists a circle that contains those two sites and no other sites. The edges of the convex hull of the input are also edges of the Delaunay triangulation, but they correspond to rays rather than line segments of the Voronoi diagram. This duality between Voronoi diagrams and Delaunay triangulations can be turned into a duality between finite graphs in either of two ways: by adding an artificial vertex at infinity to the Voronoi diagram, to serve as the other endpoint for all of its rays,[38] or by treating the bounded part of the Voronoi diagram as the weak dual of the Delaunay triangulation. Although the Voronoi diagram and Delaunay triangulation are dual, their embedding in the plane may have additional crossings beyond the crossings of dual pairs of edges. Each vertex of the Delaunay triangle is positioned within its corresponding face of the Voronoi diagram. Each vertex of the Voronoi diagram is positioned at the circumcenter of the corresponding triangle of the Delaunay triangulation, but this point may lie outside its triangle. Nonplanar embeddings K7 is dual to the Heawood graph in the torus K6 is dual to the Petersen graph in the projective plane The concept of duality can be extended to graph embeddings on two-dimensional manifolds other than the plane. The definition is the same: there is a dual vertex for each connected component of the complement of the graph in the manifold, and a dual edge for each graph edge connecting the two dual vertices on either side of the edge. In most applications of this concept, it is restricted to embeddings with the property that each face is a topological disk; this constraint generalizes the requirement for planar graphs that the graph be connected. With this constraint, the dual of any surface-embedded graph has a natural embedding on the same surface, such that the dual of the dual is isomorphic to and isomorphically embedded to the original graph. For instance, the complete graph K7 is a toroidal graph: it is not planar but can be embedded in a torus, with each face of the embedding being a triangle. This embedding has the Heawood graph as its dual graph.[39] The same concept works equally well for non-orientable surfaces. For instance, K6 can be embedded in the projective plane with ten triangular faces as the hemi-icosahedron, whose dual is the Petersen graph embedded as the hemi-dodecahedron.[40] Even planar graphs may have nonplanar embeddings, with duals derived from those embeddings that differ from their planar duals. For instance, the four Petrie polygons of a cube (hexagons formed by removing two opposite vertices of the cube) form the hexagonal faces of an embedding of the cube in a torus. The dual graph of this embedding has four vertices forming a complete graph K4 with doubled edges. In the torus embedding of this dual graph, the six edges incident to each vertex, in cyclic order around that vertex, cycle twice through the three other vertices. In contrast to the situation in the plane, this embedding of the cube and its dual is not unique; the cube graph has several other torus embeddings, with different duals.[39] Many of the equivalences between primal and dual graph properties of planar graphs fail to generalize to nonplanar duals, or require additional care in their generalization. Another operation on surface-embedded graphs is the Petrie dual, which uses the Petrie polygons of the embedding as the faces of a new embedding. Unlike the usual dual graph, it has the same vertices as the original graph, but generally lies on a different surface.[41] Surface duality and Petrie duality are two of the six Wilson operations, and together generate the group of these operations.[42] Matroids and algebraic duals An algebraic dual of a connected graph G is a graph G* such that G and G* have the same set of edges, any cycle of G is a cut of G, and any cut of G is a cycle of G. Every planar graph has an algebraic dual, which is in general not unique (any dual defined by a plane embedding will do). The converse is actually true, as settled by Hassler Whitney in Whitney's planarity criterion:[43] A connected graph G is planar if and only if it has an algebraic dual. The same fact can be expressed in the theory of matroids. If M is the graphic matroid of a graph G, then a graph G* is an algebraic dual of G if and only if the graphic matroid of G* is the dual matroid of M. Then Whitney's planarity criterion can be rephrased as stating that the dual matroid of a graphic matroid M is itself a graphic matroid if and only if the underlying graph G of M is planar. If G is planar, the dual matroid is the graphic matroid of the dual graph of G. In particular, all dual graphs, for all the different planar embeddings of G, have isomorphic graphic matroids.[44] For nonplanar surface embeddings, unlike planar duals, the dual graph is not generally an algebraic dual of the primal graph. And for a non-planar graph G, the dual matroid of the graphic matroid of G is not itself a graphic matroid. However, it is still a matroid whose circuits correspond to the cuts in G, and in this sense can be thought of as a combinatorially generalized algebraic dual of G.[45] The duality between Eulerian and bipartite planar graphs can be extended to binary matroids (which include the graphic matroids derived from planar graphs): a binary matroid is Eulerian if and only if its dual matroid is bipartite.[31] The two dual concepts of girth and edge connectivity are unified in matroid theory by matroid girth: the girth of the graphic matroid of a planar graph is the same as the graph's girth, and the girth of the dual matroid (the graphic matroid of the dual graph) is the edge connectivity of the graph.[18] Applications Along with its use in graph theory, the duality of planar graphs has applications in several other areas of mathematical and computational study. In geographic information systems, flow networks (such as the networks showing how water flows in a system of streams and rivers) are dual to cellular networks describing drainage divides. This duality can be explained by modeling the flow network as a spanning tree on a grid graph of an appropriate scale, and modeling the drainage divide as the complementary spanning tree of ridgelines on the dual grid graph.[46] In computer vision, digital images are partitioned into small square pixels, each of which has its own color. The dual graph of this subdivision into squares has a vertex per pixel and an edge between pairs of pixels that share an edge; it is useful for applications including clustering of pixels into connected regions of similar colors.[47] In computational geometry, the duality between Voronoi diagrams and Delaunay triangulations implies that any algorithm for constructing a Voronoi diagram can be immediately converted into an algorithm for the Delaunay triangulation, and vice versa.[48] The same duality can also be used in finite element mesh generation. Lloyd's algorithm, a method based on Voronoi diagrams for moving a set of points on a surface to more evenly spaced positions, is commonly used as a way to smooth a finite element mesh described by the dual Delaunay triangulation. This method improves the mesh by making its triangles more uniformly sized and shaped.[49] In the synthesis of CMOS circuits, the function to be synthesized is represented as a formula in Boolean algebra. Then this formula is translated into two series–parallel multigraphs. These graphs can be interpreted as circuit diagrams in which the edges of the graphs represent transistors, gated by the inputs to the function. One circuit computes the function itself, and the other computes its complement. One of the two circuits is derived by converting the conjunctions and disjunctions of the formula into series and parallel compositions of graphs, respectively. The other circuit reverses this construction, converting the conjunctions and disjunctions of the formula into parallel and series compositions of graphs.[50] These two circuits, augmented by an additional edge connecting the input of each circuit to its output, are planar dual graphs.[51] History The duality of convex polyhedra was recognized by Johannes Kepler in his 1619 book Harmonices Mundi.[52] Recognizable planar dual graphs, outside the context of polyhedra, appeared as early as 1725, in Pierre Varignon's posthumously published work, Nouvelle Méchanique ou Statique. This was even before Leonhard Euler's 1736 work on the Seven Bridges of Königsberg that is often taken to be the first work on graph theory. Varignon analyzed the forces on static systems of struts by drawing a graph dual to the struts, with edge lengths proportional to the forces on the struts; this dual graph is a type of Cremona diagram.[53] In connection with the four color theorem, the dual graphs of maps (subdivisions of the plane into regions) were mentioned by Alfred Kempe in 1879, and extended to maps on non-planar surfaces by Lothar Heffter in 1891.[54] Duality as an operation on abstract planar graphs was introduced by Hassler Whitney in 1931.[55] Notes 1. van Lint, J. H.; Wilson, Richard Michael (1992), A Course in Combinatorics, Cambridge University Press, p. 411, ISBN 0-521-42260-4. 2. Bóna, Miklós (2006), A walk through combinatorics (2nd ed.), World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, p. 276, doi:10.1142/6177, ISBN 981-256-885-9, MR 2361255. 3. Ziegler, Günter M. (1995), "2.3 Polarity", Lectures on Polytopes, Graduate Texts in Mathematics, vol. 152, pp. 59–64. 4. 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(2012), Graph Theory Applications, Springer, pp. 66–67, ISBN 978-1-4612-0933-1. 12. Bondy, Adrian; Murty, U.S.R. (2008), "Planar Graphs", Graph Theory, Graduate Texts in Mathematics, vol. 244, Springer, Theorem 10.28, p. 267, doi:10.1007/978-1-84628-970-5, ISBN 978-1-84628-969-9, LCCN 2007923502 13. Angelini, Patrizio; Bläsius, Thomas; Rutter, Ignaz (2014), "Testing mutual duality of planar graphs", International Journal of Computational Geometry and Applications, 24 (4): 325–346, arXiv:1303.1640, doi:10.1142/S0218195914600103, MR 3349917. 14. Diestel, Reinhard (2006), Graph Theory, Graduate Texts in Mathematics, vol. 173, Springer, p. 25, ISBN 978-3-540-26183-4. 15. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001) [1990], Introduction to Algorithms (2nd ed.), MIT Press and McGraw-Hill, p. 1081, ISBN 0-262-03293-7 16. Godsil, Chris; Royle, Gordon F. 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Florek, Jan (2010), "On Barnette's conjecture", Discrete Mathematics, 310 (10–11): 1531–1535, doi:10.1016/j.disc.2010.01.018, MR 2601261. 33. Las Vergnas, Michel (1980), "Convexity in oriented matroids", Journal of Combinatorial Theory, Series B, 29 (2): 231–243, doi:10.1016/0095-8956(80)90082-9, MR 0586435. 34. Tutte, William Thomas (1953), A contribution to the theory of chromatic polynomials 35. di Battista, Giuseppe; Eades, Peter; Tamassia, Roberto; Tollis, Ioannis G. (1999), Graph Drawing: Algorithms for the Visualization of Graphs, Prentice Hall, p. 91, ISBN 978-0-13-301615-4. 36. Fleischner, Herbert J.; Geller, D. P.; Harary, Frank (1974), "Outerplanar graphs and weak duals", Journal of the Indian Mathematical Society, 38: 215–219, MR 0389672. 37. Weisstein, Eric W., "Dual Tessellation", MathWorld 38. Samet, Hanan (2006), Foundations of Multidimensional and Metric Data Structures, Morgan Kaufmann, p. 348, ISBN 978-0-12-369446-1. 39. 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(2006), "5.2 Duality in graphic matroids", Matroid Theory, Oxford graduate texts in mathematics, vol. 3, Oxford University Press, p. 143, ISBN 978-0-19-920250-8. 45. Tutte, W. T. (2012), Graph Theory As I Have Known It, Oxford Lecture Series in Mathematics and Its Applications, vol. 11, Oxford University Press, p. 87, ISBN 978-0-19-966055-1. 46. Chorley, Richard J.; Haggett, Peter (2013), Integrated Models in Geography, Routledge, p. 646, ISBN 978-1-135-12184-6. 47. Kandel, Abraham; Bunke, Horst; Last, Mark (2007), Applied Graph Theory in Computer Vision and Pattern Recognition, Studies in Computational Intelligence, vol. 52, Springer, p. 16, ISBN 978-3-540-68020-8. 48. Devadoss, Satyan L.; O'Rourke, Joseph (2011), Discrete and Computational Geometry, Princeton University Press, p. 111, ISBN 978-1-4008-3898-1. 49. Du, Qiang; Gunzburger, Max (2002), "Grid generation and optimization based on centroidal Voronoi tessellations", Applied Mathematics and Computation, 133 (2–3): 591–607, doi:10.1016/S0096-3003(01)00260-0. 50. Piguet, Christian (2004), "7.2.1 Static CMOS Logic", Low-Power Electronics Design, CRC Press, pp. 7-1 – 7-2, ISBN 978-1-4200-3955-9. 51. Kaeslin, Hubert (2008), "8.1.4 Composite or complex gates", Digital Integrated Circuit Design: From VLSI Architectures to CMOS Fabrication, Cambridge University Press, p. 399, ISBN 978-0-521-88267-5. 52. Richeson, David S. (2012), Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, pp. 58–61, ISBN 978-1-4008-3856-1. 53. Rippmann, Matthias (2016), Funicular Shell Design: Geometric Approaches to Form Finding and Fabrication of Discrete Funicular Structures, Habilitation thesis, Diss. ETH No. 23307, ETH Zurich, pp. 39–40, doi:10.3929/ethz-a-010656780, hdl:20.500.11850/116926. See also Erickson, Jeff (June 9, 2016), Reciprocal force diagrams from Nouvelle Méchanique ou Statique by Pierre de Varignon (1725), Is this the oldest illustration of duality between planar graphs?. 54. Biggs, Norman; Lloyd, E. Keith; Wilson, Robin J. (1998), Graph Theory, 1736–1936, Oxford University Press, p. 118, ISBN 978-0-19-853916-2. 55. Whitney, Hassler (1931), "A theorem on graphs", Annals of Mathematics, Second Series, 32 (2): 378–390, doi:10.2307/1968197, JSTOR 1968197, MR 1503003. External links • Weisstein, Eric W., "Dual graph", MathWorld
Whitney conditions In differential topology, a branch of mathematics, the Whitney conditions are conditions on a pair of submanifolds of a manifold introduced by Hassler Whitney in 1965. A stratification of a topological space is a finite filtration by closed subsets Fi , such that the difference between successive members Fi and F(i − 1) of the filtration is either empty or a smooth submanifold of dimension i. The connected components of the difference Fi − F(i − 1) are the strata of dimension i. A stratification is called a Whitney stratification if all pairs of strata satisfy the Whitney conditions A and B, as defined below. The Whitney conditions in Rn Let X and Y be two disjoint (locally closed) submanifolds of Rn, of dimensions i and j. • X and Y satisfy Whitney's condition A if whenever a sequence of points x1, x2, … in X converges to a point y in Y, and the sequence of tangent i-planes Tm to X at the points xm converges to an i-plane T as m tends to infinity, then T contains the tangent j-plane to Y at y. • X and Y satisfy Whitney's condition B if for each sequence x1, x2, … of points in X and each sequence y1, y2, … of points in Y, both converging to the same point y in Y, such that the sequence of secant lines Lm between xm and ym converges to a line L as m tends to infinity, and the sequence of tangent i-planes Tm to X at the points xm converges to an i-plane T as m tends to infinity, then L is contained in T. John Mather first pointed out that Whitney's condition B implies Whitney's condition A in the notes of his lectures at Harvard in 1970, which have been widely distributed. He also defined the notion of Thom–Mather stratified space, and proved that every Whitney stratification is a Thom–Mather stratified space and hence is a topologically stratified space. Another approach to this fundamental result was given earlier by René Thom in 1969. David Trotman showed in his 1977 Warwick thesis that a stratification of a closed subset in a smooth manifold M satisfies Whitney's condition A if and only if the subspace of the space of smooth mappings from a smooth manifold N into M consisting of all those maps which are transverse to all of the strata of the stratification, is open (using the Whitney, or strong, topology). The subspace of mappings transverse to any countable family of submanifolds of M is always dense by Thom's transversality theorem. The density of the set of transverse mappings is often interpreted by saying that transversality is a 'generic' property for smooth mappings, while the openness is often interpreted by saying that the property is 'stable'. The reason that Whitney conditions have become so widely used is because of Whitney's 1965 theorem that every algebraic variety, or indeed analytic variety, admits a Whitney stratification, i.e. admits a partition into smooth submanifolds satisfying the Whitney conditions. More general singular spaces can be given Whitney stratifications, such as semialgebraic sets (due to René Thom) and subanalytic sets (due to Heisuke Hironaka). This has led to their use in engineering, control theory and robotics. In a thesis under the direction of Wieslaw Pawlucki at the Jagellonian University in Kraków, Poland, the Vietnamese mathematician Ta Lê Loi proved further that every definable set in an o-minimal structure can be given a Whitney stratification. See also • Thom–Mather stratified space • Topologically stratified space • Thom's first isotopy lemma • Stratified space References • Mather, John Notes on topological stability, Harvard, 1970 (available on his webpage at Princeton University). • Thom, René Ensembles et morphismes stratifiés, Bulletin of the American Mathematical Society Vol. 75, pp. 240–284), 1969. • Trotman, David Stability of transversality to a stratification implies Whitney (a)-regularity, Inventiones Mathematicae 50(3), pp. 273–277, 1979. • Trotman, David Comparing regularity conditions on stratifications, Singularities, Part 2 (Arcata, Calif., 1981), volume 40 of Proc. Sympos. Pure Math., pp. 575–586. American Mathematical Society, Providence, R.I., 1983. • Whitney, Hassler Local properties of analytic varieties. Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) pp. 205–244 Princeton Univ. Press, Princeton, N. J., 1965. • Whitney, Hassler, Tangents to an analytic variety, Annals of Mathematics 81, no. 3 (1965), pp. 496–549.
Vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space $X$ (for example $X$ could be a topological space, a manifold, or an algebraic variety): to every point $x$ of the space $X$ we associate (or "attach") a vector space $V(x)$ in such a way that these vector spaces fit together to form another space of the same kind as $X$ (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over $X$. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space $V$ such that $V(x)=V$ for all $x$ in $X$: in this case there is a copy of $V$ for each $x$ in $X$ and these copies fit together to form the vector bundle $X\times V$ over $X$. Such vector bundles are said to be trivial. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a manifold we attach the tangent space to the manifold at that point. Tangent bundles are not, in general, trivial bundles. For example, the tangent bundle of the sphere is non-trivial by the hairy ball theorem. In general, a manifold is said to be parallelizable if, and only if, its tangent bundle is trivial. Vector bundles are almost always required to be locally trivial, which means they are examples of fiber bundles. Also, the vector spaces are usually required to be over the real or complex numbers, in which case the vector bundle is said to be a real or complex vector bundle (respectively). Complex vector bundles can be viewed as real vector bundles with additional structure. In the following, we focus on real vector bundles in the category of topological spaces. Definition and first consequences A real vector bundle consists of: 1. topological spaces $X$ (base space) and $E$ (total space) 2. a continuous surjection $\pi :E\to X$ (bundle projection) 3. for every $x$ in $X$, the structure of a finite-dimensional real vector space on the fiber $\pi ^{-1}(\{x\})$ where the following compatibility condition is satisfied: for every point $p$ in $X$, there is an open neighborhood $U\subseteq X$ of $p$, a natural number $k$, and a homeomorphism $\varphi \colon U\times \mathbb {R} ^{k}\to \pi ^{-1}(U)$ such that for all $x$ in $U$, • $(\pi \circ \varphi )(x,v)=x$ for all vectors $v$ in $\mathbb {R} ^{k}$, and • the map $v\mapsto \varphi (x,v)$ is a linear isomorphism between the vector spaces $\mathbb {R} ^{k}$ and $\pi ^{-1}(\{x\})$. The open neighborhood $U$ together with the homeomorphism $\varphi $ is called a local trivialization of the vector bundle. The local trivialization shows that locally the map $\pi $ "looks like" the projection of $U\times \mathbb {R} ^{k}$ on $U$. Every fiber $\pi ^{-1}(\{x\})$ is a finite-dimensional real vector space and hence has a dimension $k_{x}$. The local trivializations show that the function $x\to k_{x}$ is locally constant, and is therefore constant on each connected component of $X$. If $k_{x}$ is equal to a constant $k$ on all of $X$, then $k$ is called the rank of the vector bundle, and $E$ is said to be a vector bundle of rank $k$. Often the definition of a vector bundle includes that the rank is well defined, so that $k_{x}$ is constant. Vector bundles of rank 1 are called line bundles, while those of rank 2 are less commonly called plane bundles. The Cartesian product $X\times \mathbb {R} ^{k}$, equipped with the projection $X\times \mathbb {R} ^{k}\to X$, is called the trivial bundle of rank $k$ over $X$. Transition functions Given a vector bundle $E\to X$ of rank $k$, and a pair of neighborhoods $U$ and $V$ over which the bundle trivializes via ${\begin{aligned}\varphi _{U}\colon U\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(U),\\\varphi _{V}\colon V\times \mathbb {R} ^{k}&\mathrel {\xrightarrow {\cong } } \pi ^{-1}(V)\end{aligned}}$ the composite function $\varphi _{U}^{-1}\circ \varphi _{V}\colon (U\cap V)\times \mathbb {R} ^{k}\to (U\cap V)\times \mathbb {R} ^{k}$ is well-defined on the overlap, and satisfies $\varphi _{U}^{-1}\circ \varphi _{V}(x,v)=\left(x,g_{UV}(x)v\right)$ for some ${\text{GL}}(k)$-valued function $g_{UV}\colon U\cap V\to \operatorname {GL} (k).$ These are called the transition functions (or the coordinate transformations) of the vector bundle. The set of transition functions forms a Čech cocycle in the sense that $g_{UU}(x)=I,\quad g_{UV}(x)g_{VW}(x)g_{WU}(x)=I$ for all $U,V,W$ over which the bundle trivializes satisfying $U\cap V\cap W\neq \emptyset $. Thus the data $(E,X,\pi ,\mathbb {R} ^{k})$ defines a fiber bundle; the additional data of the $g_{UV}$ specifies a ${\text{GL}}(k)$ structure group in which the action on the fiber is the standard action of ${\text{GL}}(k)$. Conversely, given a fiber bundle $(E,X,\pi ,\mathbb {R} ^{k})$ with a ${\text{GL}}(k)$ cocycle acting in the standard way on the fiber $\mathbb {R} ^{k}$, there is associated a vector bundle. This is an example of the fibre bundle construction theorem for vector bundles, and can be taken as an alternative definition of a vector bundle. Subbundles Main article: Subbundle One simple method of constructing vector bundles is by taking subbundles of other vector bundles. Given a vector bundle $\pi :E\to X$ over a topological space, a subbundle is simply a subspace $F\subset E$ for which the restriction $\left.\pi \right\|_{F}$ of $\pi $ to $F$ gives $\left.\pi \right\|_{F}:F\to X$ the structure of a vector bundle also. In this case the fibre $F_{x}\subset E_{x}$ is a vector subspace for every $x\in X$. A subbundle of a trivial bundle need not be trivial, and indeed every real vector bundle over a compact space can be viewed as a subbundle of a trivial bundle of sufficiently high rank. For example, the Möbius band, a non-trivial line bundle over the circle, can be seen as a subbundle of the trivial rank 2 bundle over the circle. Vector bundle morphisms A morphism from the vector bundle π1: E1 → X1 to the vector bundle π2: E2 → X2 is given by a pair of continuous maps f: E1 → E2 and g: X1 → X2 such that g ∘ π1 = π2 ∘ f for every x in X1, the map π1−1({x}) → π2−1({g(x)}) induced by f is a linear map between vector spaces. Note that g is determined by f (because π1 is surjective), and f is then said to cover g. The class of all vector bundles together with bundle morphisms forms a category. Restricting to vector bundles for which the spaces are manifolds (and the bundle projections are smooth maps) and smooth bundle morphisms we obtain the category of smooth vector bundles. Vector bundle morphisms are a special case of the notion of a bundle map between fiber bundles, and are sometimes called (vector) bundle homomorphisms. A bundle homomorphism from E1 to E2 with an inverse which is also a bundle homomorphism (from E2 to E1) is called a (vector) bundle isomorphism, and then E1 and E2 are said to be isomorphic vector bundles. An isomorphism of a (rank k) vector bundle E over X with the trivial bundle (of rank k over X) is called a trivialization of E, and E is then said to be trivial (or trivializable). The definition of a vector bundle shows that any vector bundle is locally trivial. We can also consider the category of all vector bundles over a fixed base space X. As morphisms in this category we take those morphisms of vector bundles whose map on the base space is the identity map on X. That is, bundle morphisms for which the following diagram commutes: (Note that this category is not abelian; the kernel of a morphism of vector bundles is in general not a vector bundle in any natural way.) A vector bundle morphism between vector bundles π1: E1 → X1 and π2: E2 → X2 covering a map g from X1 to X2 can also be viewed as a vector bundle morphism over X1 from E1 to the pullback bundle gE2. Sections and locally free sheaves Given a vector bundle π: E → X and an open subset U of X, we can consider sections of π on U, i.e. continuous functions s: U → E where the composite π ∘ s is such that (π ∘ s)(u) = u for all u in U. Essentially, a section assigns to every point of U a vector from the attached vector space, in a continuous manner. As an example, sections of the tangent bundle of a differential manifold are nothing but vector fields on that manifold. Let F(U) be the set of all sections on U. F(U) always contains at least one element, namely the zero section: the function s that maps every element x of U to the zero element of the vector space π−1({x}). With the pointwise addition and scalar multiplication of sections, F(U) becomes itself a real vector space. The collection of these vector spaces is a sheaf of vector spaces on X. If s is an element of F(U) and α: U → R is a continuous map, then αs (pointwise scalar multiplication) is in F(U). We see that F(U) is a module over the ring of continuous real-valued functions on U. Furthermore, if OX denotes the structure sheaf of continuous real-valued functions on X, then F becomes a sheaf of OX-modules. Not every sheaf of OX-modules arises in this fashion from a vector bundle: only the locally free ones do. (The reason: locally we are looking for sections of a projection U × Rk → U; these are precisely the continuous functions U → Rk, and such a function is a k-tuple of continuous functions U → R.) Even more: the category of real vector bundles on X is equivalent to the category of locally free and finitely generated sheaves of OX-modules. So we can think of the category of real vector bundles on X as sitting inside the category of sheaves of OX-modules; this latter category is abelian, so this is where we can compute kernels and cokernels of morphisms of vector bundles. A rank n vector bundle is trivial if and only if it has n linearly independent global sections. Operations on vector bundles Most operations on vector spaces can be extended to vector bundles by performing the vector space operation fiberwise. For example, if E is a vector bundle over X, then there is a bundle E over X, called the dual bundle, whose fiber at x ∈ X is the dual vector space (Ex). Formally E can be defined as the set of pairs (x, φ), where x ∈ X and φ ∈ (Ex). The dual bundle is locally trivial because the dual space of the inverse of a local trivialization of E is a local trivialization of E: the key point here is that the operation of taking the dual vector space is functorial. There are many functorial operations which can be performed on pairs of vector spaces (over the same field), and these extend straightforwardly to pairs of vector bundles E, F on X (over the given field). A few examples follow. • The Whitney sum (named for Hassler Whitney) or direct sum bundle of E and F is a vector bundle E ⊕ F over X whose fiber over x is the direct sum Ex ⊕ Fx of the vector spaces Ex and Fx. • The tensor product bundle E ⊗ F is defined in a similar way, using fiberwise tensor product of vector spaces. • The Hom-bundle Hom(E, F) is a vector bundle whose fiber at x is the space of linear maps from Ex to Fx (which is often denoted Hom(Ex, Fx) or L(Ex, Fx)). The Hom-bundle is so-called (and useful) because there is a bijection between vector bundle homomorphisms from E to F over X and sections of Hom(E, F) over X. • Building on the previous example, given a section s of an endomorphism bundle Hom(E, E) and a function f: X → R, one can construct an eigenbundle by taking the fiber over a point x ∈ X to be the f(x)-eigenspace of the linear map s(x): Ex → Ex. Though this construction is natural, unless care is taken, the resulting object will not have local trivializations. Consider the case of s being the zero section and f having isolated zeroes. The fiber over these zeroes in the resulting "eigenbundle" will be isomorphic to the fiber over them in E, while everywhere else the fiber is the trivial 0-dimensional vector space. • The dual vector bundle E* is the Hom bundle Hom(E, R × X) of bundle homomorphisms of E and the trivial bundle R × X. There is a canonical vector bundle isomorphism Hom(E, F) = E* ⊗ F. Each of these operations is a particular example of a general feature of bundles: that many operations that can be performed on the category of vector spaces can also be performed on the category of vector bundles in a functorial manner. This is made precise in the language of smooth functors. An operation of a different nature is the pullback bundle construction. Given a vector bundle E → Y and a continuous map f: X → Y one can "pull back" E to a vector bundle fE over X. The fiber over a point x ∈ X is essentially just the fiber over f(x) ∈ Y. Hence, Whitney summing E ⊕ F can be defined as the pullback bundle of the diagonal map from X to X × X where the bundle over X × X is E × F. Remark: Let X be a compact space. Any vector bundle E over X is a direct summand of a trivial bundle; i.e., there exists a bundle E' such that E ⊕ E' is trivial. This fails if X is not compact: for example, the tautological line bundle over the infinite real projective space does not have this property.[1] Additional structures and generalizations Vector bundles are often given more structure. For instance, vector bundles may be equipped with a vector bundle metric. Usually this metric is required to be positive definite, in which case each fibre of E becomes a Euclidean space. A vector bundle with a complex structure corresponds to a complex vector bundle, which may also be obtained by replacing real vector spaces in the definition with complex ones and requiring that all mappings be complex-linear in the fibers. More generally, one can typically understand the additional structure imposed on a vector bundle in terms of the resulting reduction of the structure group of a bundle. Vector bundles over more general topological fields may also be used. If instead of a finite-dimensional vector space, if the fiber F is taken to be a Banach space then a Banach bundle is obtained.[2] Specifically, one must require that the local trivializations are Banach space isomorphisms (rather than just linear isomorphisms) on each of the fibers and that, furthermore, the transitions $g_{UV}\colon U\cap V\to \operatorname {GL} (F)$ are continuous mappings of Banach manifolds. In the corresponding theory for Cp bundles, all mappings are required to be Cp. Vector bundles are special fiber bundles, those whose fibers are vector spaces and whose cocycle respects the vector space structure. More general fiber bundles can be constructed in which the fiber may have other structures; for example sphere bundles are fibered by spheres. Smooth vector bundles A vector bundle (E, p, M) is smooth, if E and M are smooth manifolds, p: E → M is a smooth map, and the local trivializations are diffeomorphisms. Depending on the required degree of smoothness, there are different corresponding notions of Cp bundles, infinitely differentiable C∞-bundles and real analytic Cω-bundles. In this section we will concentrate on C∞-bundles. The most important example of a C∞-vector bundle is the tangent bundle (TM, πTM, M) of a C∞-manifold M. A smooth vector bundle can be characterized by the fact that it admits transition functions as described above which are smooth functions on overlaps of trivializing charts U and V. That is, a vector bundle E is smooth if it admits a covering by trivializing open sets such that for any two such sets U and V, the transition function $g_{UV}:U\cap V\to \operatorname {GL} (k,\mathbb {R} )$ is a smooth function into the matrix group GL(k,R), which is a Lie group. Similarly, if the transition functions are: • Cr then the vector bundle is a Cr vector bundle, • real analytic then the vector bundle is a real analytic vector bundle (this requires the matrix group to have a real analytic structure), • holomorphic then the vector bundle is a holomorphic vector bundle (this requires the matrix group to be a complex Lie group), • algebraic functions then the vector bundle is an algebraic vector bundle (this requires the matrix group to be an algebraic group). The C∞-vector bundles (E, p, M) have a very important property not shared by more general C∞-fibre bundles. Namely, the tangent space Tv(Ex) at any v ∈ Ex can be naturally identified with the fibre Ex itself. This identification is obtained through the vertical lift vlv: Ex → Tv(Ex), defined as $\operatorname {vl} _{v}w[f]:=\left.{\frac {d}{dt}}\right\|_{t=0}f(v+tw),\quad f\in C^{\infty }(E_{x}).$ The vertical lift can also be seen as a natural C∞-vector bundle isomorphism pE → VE, where (pE, pp, E) is the pull-back bundle of (E, p, M) over E through p: E → M, and VE := Ker(p) ⊂ TE is the vertical tangent bundle, a natural vector subbundle of the tangent bundle (TE, πTE, E) of the total space E. The total space E of any smooth vector bundle carries a natural vector field Vv := vlvv, known as the canonical vector field. More formally, V is a smooth section of (TE, πTE, E), and it can also be defined as the infinitesimal generator of the Lie-group action $(t,v)\mapsto e^{tv}$ given by the fibrewise scalar multiplication. The canonical vector field V characterizes completely the smooth vector bundle structure in the following manner. As a preparation, note that when X is a smooth vector field on a smooth manifold M and x ∈ M such that Xx = 0, the linear mapping $C_{x}(X):T_{x}M\to T_{x}M;\quad C_{x}(X)Y=(\nabla _{Y}X)_{x}$ does not depend on the choice of the linear covariant derivative ∇ on M. The canonical vector field V on E satisfies the axioms 1. The flow (t, v) → ΦtV(v) of V is globally defined. 2. For each v ∈ V there is a unique limt→∞ ΦtV(v) ∈ V. 3. Cv(V)∘Cv(V) = Cv(V) whenever Vv = 0. 4. The zero set of V is a smooth submanifold of E whose codimension is equal to the rank of Cv(V). Conversely, if E is any smooth manifold and V is a smooth vector field on E satisfying 1–4, then there is a unique vector bundle structure on E whose canonical vector field is V. For any smooth vector bundle (E, p, M) the total space TE of its tangent bundle (TE, πTE, E) has a natural secondary vector bundle structure (TE, p, TM), where p* is the push-forward of the canonical projection p: E → M. The vector bundle operations in this secondary vector bundle structure are the push-forwards +: T(E × E) → TE and λ: TE → TE of the original addition +: E × E → E and scalar multiplication λ: E → E. K-theory The K-theory group, K(X), of a compact Hausdorff topological space is defined as the abelian group generated by isomorphism classes [E] of complex vector bundles modulo the relation that, whenever we have an exact sequence $0\to A\to B\to C\to 0,$ then $[B]=[A]+[C]$ in topological K-theory. KO-theory is a version of this construction which considers real vector bundles. K-theory with compact supports can also be defined, as well as higher K-theory groups. The famous periodicity theorem of Raoul Bott asserts that the K-theory of any space X is isomorphic to that of the S2X, the double suspension of X. In algebraic geometry, one considers the K-theory groups consisting of coherent sheaves on a scheme X, as well as the K-theory groups of vector bundles on the scheme with the above equivalence relation. The two constructs are the same provided that the underlying scheme is smooth. See also General notions • Grassmannian: classifying spaces for vector bundle, among which projective spaces for line bundles • Characteristic class • Splitting principle • Stable bundle Topology and differential geometry • Gauge theory: the general study of connections on vector bundles and principal bundles and their relations to physics. • Connection: the notion needed to differentiate sections of vector bundles. Algebraic and analytic geometry • Algebraic vector bundle • Picard group • Holomorphic vector bundle Notes 1. Hatcher 2003, Example 3.6. 2. Lang 1995. Sources • Abraham, Ralph H.; Marsden, Jerrold E. (1978), Foundations of mechanics, London: Benjamin-Cummings, see section 1.5, ISBN 978-0-8053-0102-1. • Hatcher, Allen (2003), Vector Bundles & K-Theory (2.0 ed.). • Jost, Jürgen (2002), Riemannian Geometry and Geometric Analysis (3rd ed.), Berlin, New York: Springer-Verlag, ISBN 978-3-540-42627-1, see section 1.5. • Lang, Serge (1995), Differential and Riemannian manifolds, Berlin, New York: Springer-Verlag, ISBN 978-0-387-94338-1. • Lee, Jeffrey M. (2009), Manifolds and Differential Geometry, Graduate Studies in Mathematics, vol. 107, Providence: American Mathematical Society, ISBN 978-0-8218-4815-9. • Lee, John M. (2003), Introduction to Smooth Manifolds, New York: Springer, ISBN 0-387-95448-1 see Ch.5 • Rubei, Elena (2014), Algebraic Geometry, a concise dictionary, Berlin/Boston: Walter De Gruyter, ISBN 978-3-11-031622-3. External links • "Vector bundle", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Why is it useful to study vector bundles ? on MathOverflow • Why is it useful to classify the vector bundles of a space ? 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Whitney topologies In mathematics, and especially differential topology, functional analysis and singularity theory, the Whitney topologies are a countably infinite family of topologies defined on the set of smooth mappings between two smooth manifolds. They are named after the American mathematician Hassler Whitney. Construction Let M and N be two real, smooth manifolds. Furthermore, let C∞(M,N) denote the space of smooth mappings between M and N. The notation C∞ means that the mappings are infinitely differentiable, i.e. partial derivatives of all orders exist and are continuous.[1] Whitney Ck-topology For some integer k ≥ 0, let Jk(M,N) denote the k-jet space of mappings between M and N. The jet space can be endowed with a smooth structure (i.e. a structure as a C∞ manifold) which make it into a topological space. This topology is used to define a topology on C∞(M,N). For a fixed integer k ≥ 0 consider an open subset U ⊂ Jk(M,N), and denote by Sk(U) the following: $S^{k}(U)=\{f\in C^{\infty }(M,N):(J^{k}f)(M)\subseteq U\}.$ The sets Sk(U) form a basis for the Whitney Ck-topology on C∞(M,N).[2] Whitney C∞-topology For each choice of k ≥ 0, the Whitney Ck-topology gives a topology for C∞(M,N); in other words the Whitney Ck-topology tells us which subsets of C∞(M,N) are open sets. Let us denote by Wk the set of open subsets of C∞(M,N) with respect to the Whitney Ck-topology. Then the Whitney C∞-topology is defined to be the topology whose basis is given by W, where:[2] $W=\bigcup _{k=0}^{\infty }W^{k}.$ Dimensionality Notice that C∞(M,N) has infinite dimension, whereas Jk(M,N) has finite dimension. In fact, Jk(M,N) is a real, finite-dimensional manifold. To see this, let ℝk[x1,…,xm] denote the space of polynomials, with real coefficients, in m variables of order at most k and with zero as the constant term. This is a real vector space with dimension $\dim \left\{\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}]\right\}=\sum _{i=1}^{k}{\frac {(m+i-1)!}{(m-1)!\cdot i!}}=\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).$ Writing a = dim{ℝk[x1,…,xm]} then, by the standard theory of vector spaces ℝk[x1,…,xm] ≅ ℝa, and so is a real, finite-dimensional manifold. Next, define: $B_{m,n}^{k}=\bigoplus _{i=1}^{n}\mathbb {R} ^{k}[x_{1},\ldots ,x_{m}],\implies \dim \left\{B_{m,n}^{k}\right\}=n\dim \left\{A_{m}^{k}\right\}=n\left({\frac {(m+k)!}{m!\cdot k!}}-1\right).$ Using b to denote the dimension Bkm,n, we see that Bkm,n ≅ ℝb, and so is a real, finite-dimensional manifold. In fact, if M and N have dimension m and n respectively then:[3] $\dim \!\left\{J^{k}(M,N)\right\}=m+n+\dim \!\left\{B_{n,m}^{k}\right\}=m+n\left({\frac {(m+k)!}{m!\cdot k!}}\right).$ Topology Given the Whitney C∞-topology, the space C∞(M,N) is a Baire space, i.e. every residual set is dense.[4] References 1. Golubitsky, M.; Guillemin, V. (1974), Stable Mappings and Their Singularities, Springer, p. 1, ISBN 0-387-90072-1 2. Golubitsky & Guillemin (1974), p. 42. 3. Golubitsky & Guillemin (1974), p. 40. 4. Golubitsky & Guillemin (1974), p. 44.
A Course of Modern Analysis A Course of Modern Analysis: an introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions (colloquially known as Whittaker and Watson) is a landmark textbook on mathematical analysis written by Edmund T. Whittaker and George N. Watson, first published by Cambridge University Press in 1902.[1] The first edition was Whittaker's alone, but later editions were co-authored with Watson. A Course of Modern Analysis Cover of a 1996 reissue of the fourth edition of the book. AuthorEdmund T. Whittaker and George N. Watson LanguageEnglish SubjectMathematics PublisherCambridge University Press Publication date 1902 History Its first, second, third, and the fourth edition were published in 1902,[2] 1915,[3] 1920,[4] and 1927,[5] respectively. Since then, it has continuously been reprinted and is still in print today.[5][6] A revised, expanded and digitally reset fifth edition, edited by Victor H. Moll, was published in 2021.[7] The book is notable for being the standard reference and textbook for a generation of Cambridge mathematicians including Littlewood and Godfrey H. Hardy. Mary L. Cartwright studied it as preparation for her final honours on the advice of fellow student Vernon C. Morton, later Professor of Mathematics at Aberystwyth University.[8] But its reach was much further than just the Cambridge school; André Weil in his obituary of the French mathematician Jean Delsarte noted that Delsarte always had a copy on his desk.[9] In 1941 the book was included among a "selected list" of mathematical analysis books for use in universities in an article for that purpose published by American Mathematical Monthly.[10] Notable features Some idiosyncratic but interesting problems from an older era of the Cambridge Mathematical Tripos are in the exercises. The book was one of the earliest to use decimal numbering for its sections, an innovation the authors attribute to Giuseppe Peano.[11] Contents Below are the contents of the fourth edition: Part I. The Process of Analysis 1. Complex Numbers 2. The Theory of Convergence 3. Continuous Functions and Uniform Convergence 4. The Theory of Riemann Integration 5. The fundamental properties of Analytic Functions; Taylor's, Laurent's, and Liouville's Theorems 6. The Theory of Residues; application to the evaluation of Definite Integrals 7. The expansion of functions in Infinite Series 8. Asymptotic Expansions and Summable Series 9. Fourier Series and Trigonometrical Series 10. Linear Differential Equations 11. Integral Equations Part II. The Transcendental Functions 1. The Gamma Function 2. The Zeta Function of Riemann 3. The Hypergeometric Function 4. Legendre Functions 5. The Confluent Hypergeometric Function 6. Bessel Functions 7. The Equations of Mathematical Physics 8. Mathieu Functions 9. Elliptic Functions. General theorems and the Weierstrassian Functions 10. The Theta Functions 11. The Jacobian Elliptic Functions 12. Ellipsoidal Harmonics and Lamé's Equation Reception Reviews of the first edition George B. Mathews, in a 1903 review article published in The Mathematical Gazette opens by saying the book is "sure of a favorable reception" because of its "attractive account of some of the most valuable and interesting results of recent analysis".[12] He notes that Part I deals mainly with infinite series, focusing on power series and Fourier expansions while including the "elements of" complex integration and the theory of residues. Part II, in contrast, has chapters on the gamma function, Legendre functions, the hypergeometric series, Bessel functions, elliptic functions, and mathematical physics. Arthur S. Hathaway, in another 1903 review published in the Journal of the American Chemical Society, notes that the book centers around complex analysis, but that topics such as infinite series are "considered in all their phases" along with "all those important series and functions" developed by mathematicians such as Joseph Fourier, Friedrich Bessel, Joseph-Louis Lagrange, Adrien-Marie Legendre, Pierre-Simon Laplace, Carl Friedrich Gauss, Niels Henrik Abel, and others in their respective studies of "practice problems".[13] He goes on to say it "is a useful book for those who wish to make use of the most advanced developments of mathematical analysis in theoretical investigations of physical and chemical questions."[13] In a third review of the first edition, Maxime Bôcher, in a 1904 review published in the Bulletin of the American Mathematical Society notes that while the book falls short of the "rigor" of French, German, and Italian writers, it is a "gratifying sign of progress to find in an English book such an attempt at rigorous treatment as is here made".[1] He notes that important parts of the book were otherwise non-existent in the English language. See also • Bateman Manuscript Project References 1. Bôcher, Maxime (1904). "Review: A Course of Modern Analysis, by E. T. Whittaker". Bulletin of the American Mathematical Society (review). 10 (7): 351–354. doi:10.1090/s0002-9904-1904-01123-4. (4 pages) 2. Whittaker, Edmund Taylor (1902). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (1st ed.). Cambridge, UK: at the University Press. OCLC 1072208628. (xvi+378 pages) 3. Whittaker, Edmund Taylor; Watson, George Neville (1915). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (2nd ed.). Cambridge, UK: at the University Press. OCLC 474155529. (viii+560 pages) 4. Whittaker, Edmund Taylor; Watson, George Neville (1920). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (3rd ed.). Cambridge, UK: at the University Press. OCLC 1170617940. 5. Whittaker, Edmund Taylor; Watson, George Neville (1927-01-02). A Course Of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions (4th ed.). Cambridge, UK: at the University Press. ISBN 0-521-06794-4. ISBN 978-0-521-06794-2. (vi+608 pages) (reprinted: 1935, 1940, 1946, 1950, 1952, 1958, 1962, 1963, 1992) 6. Whittaker, Edmund Taylor; Watson, George Neville (1996) [1927]. A Course of Modern Analysis. Cambridge Mathematical Library (4th reissued ed.). Cambridge, UK: Cambridge University Press. doi:10.1017/cbo9780511608759. ISBN 978-0-521-58807-2. OCLC 802476524. ISBN 0-521-58807-3. (reprinted: 1999, 2000, 2002, 2010) 7. Whittaker, Edmund Taylor; Watson, George Neville (2021-08-26) [2021-08-07]. Moll, Victor Hugo (ed.). A Course of Modern Analysis (5th revised ed.). Cambridge, UK: Cambridge University Press. doi:10.1017/9781009004091. ISBN 978-1-31651893-9. ISBN 1-31651893-0. Archived from the original on 2021-08-10. Retrieved 2021-12-26. (700 pages) 8. O'Connor, John J.; Robertson, Edmund Frederick (October 2003). "Dame Mary Lucy Cartwright". MacTutor. St. Andrews, UK: St. Andrews University. Archived from the original on 2021-03-21. Retrieved 2021-03-21. 9. O'Connor, John J.; Robertson, Edmund Frederick (December 2005). "Jean Frédéric Auguste Delsarte". MacTutor. St. Andrews, UK: St. Andrews University. Archived from the original on 2021-03-21. Retrieved 2021-03-21. 10. "A Selected List of Mathematics Books for Colleges". The American Mathematical Monthly. 48 (9): 600–609. 1941. doi:10.1080/00029890.1941.11991146. ISSN 0002-9890. JSTOR 2303868. (10 pages) 11. Kowalski, Emmanuel [in German] (2008-06-03). "Peano paragraphing". E. Kowalski's blog - Comments on mathematics, mostly. Archived from the original on 2021-02-25. Retrieved 2021-03-21. 12. Mathews, George Ballard (1903). "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 2 (39): 290–292. doi:10.2307/3603560. ISSN 0025-5572. JSTOR 3603560. S2CID 221486387. (3 pages) 13. Hathaway, Arthur Stafford (February 1903). "A Course in Modern Analysis". Journal of the American Chemical Society (review). 25 (2): 220. doi:10.1021/ja02004a022. ISSN 0002-7863. Further reading • Jourdain, Philip E. B. (1916-01-01). "(1) A Course of Pure Mathematics. By G. H. Hardy. Cambridge University Press, 1908. Pp. xvi, 428. Cloth, 12s. net. (2) A Course of Pure Mathematics. By G. H. Hardy. Second edition. Cambridge University Press, 1914. Pp. xii, 443. Cloth, 12s. net. (3) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. By E. T. Whittaker. Cambridge University Press, 1902. Pp. xvi, 378. Cloth, 12s. 6d. net. (4) A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised. By E. T. Whittaker and G. N. Watson. Cambridge University Press, 1915. Pp. viii, 560. Cloth, 18s. net". VI. Critical Notices. Mind (review). XXV (4): 525–533. doi:10.1093/mind/XXV.4.525. ISSN 0026-4423. JSTOR 2248860. (9 pages) • Neville, Eric Harold (1921). "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 10 (152): 283. doi:10.2307/3604927. ISSN 0025-5572. JSTOR 3604927. (1 page) • Wrinch, Dorothy Maud (1921). "Review of A Course of Modern Analysis. Third Edition". Science Progress in the Twentieth Century (1919-1933) (review). Sage Publications, Inc. 15 (60): 658. ISSN 2059-4941. JSTOR 43769035. (1 page) • "Review of A Course of Modern Analysis". The Mathematical Gazette (review). 14 (196): 245. 1928. doi:10.2307/3606904. ISSN 0025-5572. JSTOR 3606904. (1 page) • "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". The American Mathematical Monthly (review). 28 (4): 176. 1921. doi:10.2307/2972291. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 0002-9890. JSTOR 2972291. • Φ (1916). "Review of A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions. Second edition, completely revised". The Monist (review). 26 (4): 639–640. ISSN 0026-9662. JSTOR 27900617. (2 pages) • "Review of A Course of Modern Analysis. An Introduction to the General Theory of Infinite Processes and of Analytical Functions, with an Account of the Principal Transcendental Functions. Second Edition". Science Progress (1916–1919) (review). Sage Publications, Inc. 11 (41): 160–161. 1916. ISSN 2059-495X. JSTOR 43426733. (2 pages) • "Review of A Course of Modern Analysis: An introduction to the General Theory of Infinite Processes and of Analytical Functions; With an Account of the Principal Transcendental Functions". The Mathematical Gazette (review). 8 (124): 306–307. 1916. doi:10.2307/3604810. ISSN 0025-5572. JSTOR 3604810. S2CID 40238008. (2 pages) • Schubert, A. (1963). "E. T. Whittaker and G. N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of the principal transcendental functions. Fourth Edition. 608 S. Cambridge 1962. Cambridge University Press. Preis brosch. 27/6 net". ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik (review). 43 (9): 435. Bibcode:1963ZaMM...43R.435S. doi:10.1002/zamm.19630430916. ISSN 1521-4001. (1 page) • "Modern Analysis. By E. T. Whittaker and G. N. Watson Pp. 608. 27s. 6d. 1962. (Cambridge University Press)". The Mathematical Gazette (review). 47 (359): 88. February 1963. doi:10.1017/S0025557200049032. ISSN 0025-5572. • "A Course of Modern Analysis". Nature (review). 97 (2432): 298–299. 1916-06-08. Bibcode:1916Natur..97..298.. doi:10.1038/097298a0. ISSN 1476-4687. S2CID 3980161. (1 page) • "A Course of Modern Analysis: An Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 106 (2669): 531. 1920-12-23. Bibcode:1920Natur.106R.531.. doi:10.1038/106531c0. hdl:2027/coo1.ark:/13960/t17m0tq6p. ISSN 1476-4687. S2CID 40238008. (1 page) • M.-T., L. M. (1928-03-17). "A Course of Modern Analysis: an Introduction to the General Theory of Infinite Processes and of Analytic Functions; with an Account of the Principal Transcendental Functions". Nature (review). 121 (3046): 417. Bibcode:1928Natur.121..417M. doi:10.1038/121417a0. ISSN 1476-4687. (1 page) • Stuart, S. N. (1981). "Table errata: A course of modern analysis [fourth edition, Cambridge Univ. Press, Cambridge, 1927; Jbuch 53, 180] by E. T. Whittaker and G. N. Watson". Mathematics of Computation (errata). American Mathematical Society. 36 (153): 315–320 [319]. doi:10.1090/S0025-5718-1981-0595076-1. ISSN 0025-5718. JSTOR 2007758. (1 of 6 pages) Sir Edmund Taylor Whittaker FRS FRSE LLD ScD Fields • Mathematics • Astronomy • Mathematical physics • History of science Notable works • A Course of Modern Analysis (1902) • Analytical Dynamics of Particles and Rigid Bodies (1904) • A History of the Theories of Aether and Electricity, from the age of Descartes to the Close of the Nineteenth Century (1910) • A History of the Theories of Aether and Electricity, the Classic Theories (1951) • A History of the Theories of Aether and Electricity, the Modern Theories (1900-1926) (1953) Eponym of • Whittaker function • Whittaker model • Whittaker–Nyquist–Kotelnikov–Shannon sampling theorem • Whittaker–Shannon interpolation formula • Sir Edmund Whittaker Memorial Prize Notable research • Rapidity • Special functions • Electromagnetism • General relativity • Harmonic functions • Automorphic functions • Confluent hypergeometric functions • Numerical analysis Notable family members • John Macnaghten Whittaker (son) • Edward Copson (son-in-law) Notable disputes • Lorentz-Poincaré-Einstein controversy • Fictitious Problems in Mathematics
Whittaker function In mathematics, a Whittaker function is a special solution of Whittaker's equation, a modified form of the confluent hypergeometric equation introduced by Whittaker (1903) to make the formulas involving the solutions more symmetric. More generally, Jacquet (1966, 1967) introduced Whittaker functions of reductive groups over local fields, where the functions studied by Whittaker are essentially the case where the local field is the real numbers and the group is SL2(R). Whittaker's equation is ${\frac {d^{2}w}{dz^{2}}}+\left(-{\frac {1}{4}}+{\frac {\kappa }{z}}+{\frac {1/4-\mu ^{2}}{z^{2}}}\right)w=0.$ It has a regular singular point at 0 and an irregular singular point at ∞. Two solutions are given by the Whittaker functions Mκ,μ(z), Wκ,μ(z), defined in terms of Kummer's confluent hypergeometric functions M and U by $M_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}M\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right)$ $W_{\kappa ,\mu }\left(z\right)=\exp \left(-z/2\right)z^{\mu +{\tfrac {1}{2}}}U\left(\mu -\kappa +{\tfrac {1}{2}},1+2\mu ,z\right).$ The Whittaker function $W_{\kappa ,\mu }(z)$ is the same as those with opposite values of μ, in other words considered as a function of μ at fixed κ and z it is even functions. When κ and z are real, the functions give real values for real and imaginary values of μ. These functions of μ play a role in so-called Kummer spaces.[1] Whittaker functions appear as coefficients of certain representations of the group SL2(R), called Whittaker models. References 1. Louis de Branges (1968). Hilbert spaces of entire functions. Prentice-Hall. ASIN B0006BUXNM. Sections 55-57. • Abramowitz, Milton; Stegun, Irene Ann, eds. (1983) [June 1964]. "Chapter 13". Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Applied Mathematics Series. Vol. 55 (Ninth reprint with additional corrections of tenth original printing with corrections (December 1972); first ed.). Washington D.C.; New York: United States Department of Commerce, National Bureau of Standards; Dover Publications. pp. 504, 537. ISBN 978-0-486-61272-0. LCCN 64-60036. MR 0167642. LCCN 65-12253. See also chapter 14. • Bateman, Harry (1953), Higher transcendental functions (PDF), vol. 1, McGraw-Hill. • Brychkov, Yu.A.; Prudnikov, A.P. (2001) [1994], "Whittaker function", Encyclopedia of Mathematics, EMS Press. • Daalhuis, Adri B. Olde (2010), "Whittaker function", in Olver, Frank W. J.; Lozier, Daniel M.; Boisvert, Ronald F.; Clark, Charles W. (eds.), NIST Handbook of Mathematical Functions, Cambridge University Press, ISBN 978-0-521-19225-5, MR 2723248. • Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus de l'Académie des Sciences, Série A et B, 262: A943–A945, ISSN 0151-0509, MR 0200390 • Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France, 95: 243–309, doi:10.24033/bsmf.1654, ISSN 0037-9484, MR 0271275 • Rozov, N.Kh. (2001) [1994], "Whittaker equation", Encyclopedia of Mathematics, EMS Press. • Slater, Lucy Joan (1960), Confluent hypergeometric functions, Cambridge University Press, MR 0107026. • Whittaker, Edmund T. (1903), "An expression of certain known functions as generalized hypergeometric functions", Bulletin of the A.M.S., Providence, R.I.: American Mathematical Society, 10 (3): 125–134, doi:10.1090/S0002-9904-1903-01077-5 Further reading • Hatamzadeh-Varmazyar, Saeed; Masouri, Zahra (2012-11-01). "A fast numerical method for analysis of one- and two-dimensional electromagnetic scattering using a set of cardinal functions". Engineering Analysis with Boundary Elements. 36 (11): 1631–1639. doi:10.1016/j.enganabound.2012.04.014. ISSN 0955-7997. • Gerasimov, A. A.; Lebedev, Dmitrii R.; Oblezin, Sergei V. (2012). "New integral representations of Whittaker functions for classical Lie groups". Russian Mathematical Surveys. 67 (1): 1–92. arXiv:0705.2886. Bibcode:2012RuMaS..67....1G. doi:10.1070/RM2012v067n01ABEH004776. ISSN 0036-0279. • Baudoin, Fabrice; O’Connell, Neil (2011). "Exponential functionals of brownian motion and class-one Whittaker functions". Annales de l'Institut Henri Poincaré, Probabilités et Statistiques. 47 (4): 1096–1120. Bibcode:2011AIHPB..47.1096B. doi:10.1214/10-AIHP401. S2CID 113388. • McKee, Mark (April 2009). "An Infinite Order Whittaker Function". Canadian Journal of Mathematics. 61 (2): 373–381. doi:10.4153/CJM-2009-019-x. ISSN 0008-414X. S2CID 55587239. • Mathai, A. M.; Pederzoli, Giorgio (1997-03-01). "Some properties of matrix-variate Laplace transforms and matrix-variate Whittaker functions". Linear Algebra and Its Applications. 253 (1): 209–226. doi:10.1016/0024-3795(95)00705-9. ISSN 0024-3795. • Whittaker, J. M. (May 1927). "On the Cardinal Function of Interpolation Theory". Proceedings of the Edinburgh Mathematical Society. 1 (1): 41–46. doi:10.1017/S0013091500007318. ISSN 1464-3839. • Cherednik, Ivan (2009). "Whittaker Limits of Difference Spherical Functions". International Mathematics Research Notices. 2009 (20): 3793–3842. arXiv:0807.2155. doi:10.1093/imrn/rnp065. ISSN 1687-0247. S2CID 6253357. • Slater, L. J. (October 1954). "Expansions of generalized Whittaker functions". Mathematical Proceedings of the Cambridge Philosophical Society. 50 (4): 628–631. Bibcode:1954PCPS...50..628S. doi:10.1017/S0305004100029765. ISSN 1469-8064. S2CID 122348447. • Etingof, Pavel (1999-01-12). "Whittaker functions on quantum groups and q-deformed Toda operators". arXiv:math/9901053. • McNamara, Peter J. (2011-01-15). "Metaplectic Whittaker functions and crystal bases". Duke Mathematical Journal. 156 (1): 1–31. arXiv:0907.2675. doi:10.1215/00127094-2010-064. ISSN 0012-7094. S2CID 979197. • Mathai, A. M.; Pederzoli, Giorgio (1998-01-15). "A whittaker function of matrix argument". Linear Algebra and Its Applications. 269 (1): 91–103. doi:10.1016/S0024-3795(97)00059-1. ISSN 0024-3795. • Frenkel, E.; Gaitsgory, D.; Kazhdan, D.; Vilonen, K. (1998). "Geometric realization of Whittaker functions and the Langlands conjecture". Journal of the American Mathematical Society. 11 (2): 451–484. doi:10.1090/S0894-0347-98-00260-4. ISSN 0894-0347. S2CID 13221400.
Whittaker model In representation theory, a branch of mathematics, the Whittaker model is a realization of a representation of a reductive algebraic group such as GL2 over a finite or local or global field on a space of functions on the group. It is named after E. T. Whittaker even though he never worked in this area, because (Jacquet 1966, 1967) pointed out that for the group SL2(R) some of the functions involved in the representation are Whittaker functions. Irreducible representations without a Whittaker model are sometimes called "degenerate", and those with a Whittaker model are sometimes called "generic". The representation θ10 of the symplectic group Sp4 is the simplest example of a degenerate representation. Whittaker models for GL2 If G is the algebraic group GL2 and F is a local field, and τ is a fixed non-trivial character of the additive group of F and π is an irreducible representation of a general linear group G(F), then the Whittaker model for π is a representation π on a space of functions ƒ on G(F) satisfying $f\left({\begin{pmatrix}1&b\\0&1\end{pmatrix}}g\right)=\tau (b)f(g).$ Jacquet & Langlands (1970) used Whittaker models to assign L-functions to admissible representations of GL2. Whittaker models for GLn Let $G$ be the general linear group $\operatorname {GL} _{n}$, $\psi $ a smooth complex valued non-trivial additive character of $F$ and $U$ the subgroup of $\operatorname {GL} _{n}$ consisting of unipotent upper triangular matrices. A non-degenerate character on $U$ is of the form $\chi (u)=\psi (\alpha _{1}x_{12}+\alpha _{2}x_{23}+\cdots +\alpha _{n-1}x_{n-1n}),$ for $u=(x_{ij})$ ∈ $U$ and non-zero $\alpha _{1},\ldots ,\alpha _{n-1}$ ∈ $F$. If $(\pi ,V)$ is a smooth representation of $G(F)$, a Whittaker functional $\lambda $ is a continuous linear functional on $V$ such that $\lambda (\pi (u)v)=\chi (u)\lambda (v)$ for all $u$ ∈ $U$, $v$ ∈ $V$. Multiplicity one states that, for $\pi $ unitary irreducible, the space of Whittaker functionals has dimension at most equal to one. Whittaker models for reductive groups If G is a split reductive group and U is the unipotent radical of a Borel subgroup B, then a Whittaker model for a representation is an embedding of it into the induced (Gelfand–Graev) representation IndG U (χ), where χ is a non-degenerate character of U, such as the sum of the characters corresponding to simple roots. See also • Gelfand–Graev representation, roughly the sum of Whittaker models over a finite field. • Kirillov model References • Jacquet, Hervé (1966), "Une interprétation géométrique et une généralisation P-adique des fonctions de Whittaker en théorie des groupes semi-simples", Comptes Rendus de l'Académie des Sciences, Série A et B, 262: A943–A945, ISSN 0151-0509, MR 0200390 • Jacquet, Hervé (1967), "Fonctions de Whittaker associées aux groupes de Chevalley", Bulletin de la Société Mathématique de France, 95: 243–309, doi:10.24033/bsmf.1654, ISSN 0037-9484, MR 0271275 • Jacquet, H.; Langlands, Robert P. (1970), Automorphic forms on GL(2), Lecture Notes in Mathematics, Vol. 114, vol. 114, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0058988, ISBN 978-3-540-04903-6, MR 0401654 • J. A. Shalika, The multiplicity one theorem for $GL_{n}$, The Annals of Mathematics, 2nd. Ser., Vol. 100, No. 2 (1974), 171-193. Further reading • Jacquet, Hervé; Shalika, Joseph (1983). "The Whittaker models of induced representations". Pacific Journal of Mathematics. 109 (1): 107–120. doi:10.2140/pjm.1983.109.107. ISSN 0030-8730.
Wholeness axiom In mathematics, the wholeness axiom is a strong axiom of set theory introduced by Paul Corazza in 2000.[1] Statement The wholeness axiom states roughly that there is an elementary embedding j from the Von Neumann universe V to itself. This has to be stated carefully to avoid Kunen's inconsistency theorem stating (roughly) that no such embedding exists. More specifically, as Samuel Gomes da Silva states, "the inconsistency is avoided by omitting from the schema all instances of the Replacement Axiom for j-formulas".[2] Thus, the wholeness axiom differs from Reinhardt cardinals (another way of providing elementary embeddings from V to itself) by allowing the axiom of choice and instead modifying the axiom of replacement. However, Holmes, Forster & Libert (2012) write that Corrazza's theory should be "naturally viewed as a version of Zermelo set theory rather than ZFC".[3] If the wholeness axiom is consistent, then it is also consistent to add to the wholeness axiom the assertion that all sets are hereditarily ordinal definable.[4] The consistency of stratified versions of the wholeness axiom, introduced by Hamkins (2001),[4] was studied by Apter (2012).[5] References 1. Corazza, Paul (2000), "The Wholeness Axiom and Laver Sequences", Annals of Pure and Applied Logic, 105 (1–3): 157–260, doi:10.1016/s0168-0072(99)00052-4 2. Samuel Gomes da Silva, Review of "The wholeness axioms and the class of supercompact cardinals" by Arthur Apter. 3. Holmes, M. Randall; Forster, Thomas; Libert, Thierry (2012), "Alternative set theories", Sets and extensions in the twentieth century, Handb. Hist. Log., vol. 6, Elsevier/North-Holland, Amsterdam, pp. 559–632, doi:10.1016/B978-0-444-51621-3.50008-6, MR 3409865. 4. Hamkins, Joel David (2001), "The wholeness axioms and V = HOD", Archive for Mathematical Logic, 40 (1): 1–8, arXiv:math/9902079, doi:10.1007/s001530050169, MR 1816602, S2CID 15083392. 5. Apter, Arthur W. (2012), "The wholeness axioms and the class of supercompact cardinals", Bulletin of the Polish Academy of Sciences, Mathematics, 60 (2): 101–111, doi:10.4064/ba60-2-1, MR 2914539. External links • The Wholeness axiom in Cantor's attic
Why Johnny Can't Add Why Johnny Can't Add: The Failure of the New Math is a 1973 book by Morris Kline, in which the author severely criticized the teaching practices characteristic of the "New Math" fashion for school teaching, which were based on Bourbaki's approach to mathematical research, and were being pushed into schools in the United States.[1][2] Reactions were immediate, and the book became a best seller in its genre and was translated into many languages.[3] Why Johnny Can't Add: The Failure of the New Math AuthorMorris Kline SubjectCriticism of "New Math" education Published1973 References 1. Jürgen Maass; Wolfgang Schlöglmann (2006). New Mathematics Education Research and Practice. Sense Publishers. p. 1. ISBN 978-90-77874-74-5. 2. Joseph W. Dauben; Christoph J. Scriba (23 September 2002). Writing the History of Mathematics: Its Historical Development. Springer Science & Business Media. p. 458. ISBN 978-3-7643-6167-9. 3. Fey, James T.. 1978. “U.S.A.”. Educational Studies in Mathematics 9 (3). Springer: 339–353. https://www.jstor.org/stable/3481942. Further reading • "Review of Why Johnny Can't Add". Bulletin of the Orton Society. 24: 210. 1974-01-01. JSTOR 23769748. • Rising, Gerald R. (1974-01-01). "Review of Why Johnny Can't Add: The Failure of the New Math". The Arithmetic Teacher. 21 (5): 450. JSTOR 41190940. • Gillman, Leonard (1974-01-01). "Review of Why Johnny Can't Add: The Failure of the New Math". The American Mathematical Monthly. 81 (5): 531–532. doi:10.2307/2318615. JSTOR 2318615. • Niman, John (1973-01-01). "Review of Why Johnny Can't Add: The Failure of the New Math". Mathematics Magazine. 46 (4): 228–229. doi:10.2307/2688316. JSTOR 2688316. • McIntosh, Jerry (1973-01-01). Kline, Morris (ed.). "Kline's 'Gutsy Appraisal': New Math Needs Overhaul". The Phi Delta Kappan. 55 (1): 79–80. JSTOR 20297438. • Peak, Philip (1973-01-01). "Review of Why Johnny Can't Add: The Failure of the New Math (L, S, P)". The Mathematics Teacher. 66 (7): 641–642. JSTOR 27959458. • Moore, John W. (1973-01-01). "Why Johnny Can't Add". Journal of College Science Teaching. 3 (2): 167–168. JSTOR 42964980. External links • Text on-line, with permission of the current copyright holders
Wichmann–Hill Wichmann–Hill is a pseudorandom number generator proposed in 1982 by Brian Wichmann and David Hill.[1] It consists of three linear congruential generators with different prime moduli, each of which is used to produce a uniformly distributed number between 0 and 1. These are summed, modulo 1, to produce the result.[2] Summing three generators produces a pseudorandom sequence with cycle exceeding 6.95×1012.[3] Specifically, the moduli are 30269, 30307 and 30323, producing periods of 30268, 30306 and 30322. The overall period is the least common multiple of these: 30268×30306×30322/4 = 6953607871644. This has been confirmed by a brute-force search.[4][5] Implementation The following pseudocode is for implementation on machines capable of integer arithmetic up to 5,212,632: [r, s1, s2, s3] = function(s1, s2, s3) is // s1, s2, s3 should be random from 1 to 30,000. Use clock if available. s1 := mod(171 × s1, 30269) s2 := mod(172 × s2, 30307) s3 := mod(170 × s3, 30323) r := mod(s1/30269.0 + s2/30307.0 + s3/30323.0, 1) For machines limited to 16-bit signed integers, the following equivalent code only uses numbers up to 30,323: [r, s1, s2, s3] = function(s1, s2, s3) is // s1, s2, s3 should be random from 1 to 30,000. Use clock if available. s1 := 171 × mod(s1, 177) − 2 × floor(s1 / 177) s2 := 172 × mod(s2, 176) − 35 × floor(s2 / 176) s3 := 170 × mod(s3, 178) − 63 × floor(s3 / 178) r := mod(s1/30269 + s2/30307 + s3/30323, 1) The seed values s1, s2 and s3 must be initialized to non-zero values. References 1. Wichmann, Brian A.; Hill, I. David (1982). "Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator". Journal of the Royal Statistical Society. Series C (Applied Statistics). 31 (2): 188–190. doi:10.2307/2347988. JSTOR 2347988. 2. McLeod, A. Ian (1985). "Remark AS R58: A Remark on Algorithm AS 183. An Efficient and Portable Pseudo-Random Number Generator". Journal of the Royal Statistical Society. Series C (Applied Statistics). 34 (2): 198–200. doi:10.2307/2347378. JSTOR 2347378. Wichmann and Hill (1982) claim that their generator RANDOM will produce uniform pseudorandom numbers which are strictly greater than zero and less than one. However, depending on the precision of the machine, some zero values may be produced due to rounding error. The problem occurs with single-precision floating point when rounding to zero. 3. Wichmann, Brian; Hill, David (1984). "Correction: Algorithm AS 183: An Efficient and Portable Pseudo-Random Number Generator". Journal of the Royal Statistical Society. Series C (Applied Statistics). 33 (1): 123. doi:10.2307/2347676. JSTOR 2347676. 4. Rikitake, Kenji (16 March 2017). "AS183 PRNG algorithm internal state calculator in C". GitHub. 5. Zeisel, H. (1986). "Remark ASR 61: A Remark on Algorithm AS 183. An Efficient and Portable Pseudo-Random Number Generator". Journal of the Royal Statistical Society. Series C (Applied Statistics). 35 (1): 98. doi:10.1111/j.1467-9876.1986.tb01945.x. JSTOR 2347876. Wichmann and Hill (1982) suggested a pseudo-random generator based on summation of pseudo-random numbers based on summation of pseudo-random numbers generated by multiplicative congruential methods. This, however, is not more than an efficient method to implement a multiplicative congruential generator with a cycle length much greater than the maximal integer. Using the Chinese Remainder Theorem (see e.g. Knuth, 1981) you can prove that you will obtain the same results using only one multiplicative congruential generator Xn+1 = α⋅Xn modulo m with α = 1655 54252 64690, m = 2781 71856 04309. The original version, however, is still necessary to make a generator with such lengthy constants work on ordinary computer arithmetic.
Isserlis' theorem In probability theory, Isserlis' theorem or Wick's probability theorem is a formula that allows one to compute higher-order moments of the multivariate normal distribution in terms of its covariance matrix. It is named after Leon Isserlis. This theorem is also particularly important in particle physics, where it is known as Wick's theorem after the work of Wick (1950).[1] Other applications include the analysis of portfolio returns,[2] quantum field theory[3] and generation of colored noise.[4] Statement If $(X_{1},\dots ,X_{n})$ is a zero-mean multivariate normal random vector, then $\operatorname {E} [\,X_{1}X_{2}\cdots X_{n}\,]=\sum _{p\in P_{n}^{2}}\prod _{\{i,j\}\in p}\operatorname {E} [\,X_{i}X_{j}\,]=\sum _{p\in P_{n}^{2}}\prod _{\{i,j\}\in p}\operatorname {Cov} (\,X_{i},X_{j}\,),$ where the sum is over all the pairings of $\{1,\ldots ,n\}$, i.e. all distinct ways of partitioning $\{1,\ldots ,n\}$ into pairs $\{i,j\}$, and the product is over the pairs contained in $p$.[5][6] More generally, if $(Z_{1},\dots ,Z_{n})$ is a zero-mean complex-valued multivariate normal random vector, then the formula still holds. The expression on the right-hand side is also known as the hafnian of the covariance matrix of $(X_{1},\dots ,X_{n})$. Odd case If $n=2m+1$ is odd, there does not exist any pairing of $\{1,\ldots ,2m+1\}$. Under this hypothesis, Isserlis' theorem implies that $\operatorname {E} [\,X_{1}X_{2}\cdots X_{2m+1}\,]=0.$ This also follows from the fact that $-X=(-X_{1},\dots ,-X_{n})$ has the same distribution as $X$, which implies that $\operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=\operatorname {E} [\,(-X_{1})\cdots (-X_{2m+1})\,]=-\operatorname {E} [\,X_{1}\cdots X_{2m+1}\,]=0$. Even case In his original paper,[7] Leon Isserlis proves this theorem by mathematical induction, generalizing the formula for the $4^{\text{th}}$ order moments,[8] which takes the appearance $\operatorname {E} [\,X_{1}X_{2}X_{3}X_{4}\,]=\operatorname {E} [X_{1}X_{2}]\,\operatorname {E} [X_{3}X_{4}]+\operatorname {E} [X_{1}X_{3}]\,\operatorname {E} [X_{2}X_{4}]+\operatorname {E} [X_{1}X_{4}]\,\operatorname {E} [X_{2}X_{3}].$ If $n=2m$ is even, there exist $(2m)!/(2^{m}m!)=(2m-1)!!$ (see double factorial) pair partitions of $\{1,\ldots ,2m\}$: this yields $(2m)!/(2^{m}m!)=(2m-1)!!$ terms in the sum. For example, for $4^{\text{th}}$ order moments (i.e. $4$ random variables) there are three terms. For $6^{\text{th}}$-order moments there are $3\times 5=15$ terms, and for $8^{\text{th}}$-order moments there are $3\times 5\times 7=105$ terms. Proof Since the formula is linear on both sides, if we can prove the real case, we get the complex case for free. Let $\Sigma _{ij}=\operatorname {Cov} (X_{i},X_{j})$ be the covariance matrix, so that we have the zero-mean multivariate normal random vector $(X_{1},...,X_{n})\sim N(0,\Sigma )$. Since both sides of the formula are continuous with respect to $\Sigma $, it suffices to prove the case when $\Sigma $ is invertible. Using quadratic factorization $-x^{T}\Sigma ^{-1}x/2+v^{T}x-v^{T}\Sigma v/2=-(x-\Sigma v)^{T}\Sigma ^{-1}(x-\Sigma v)/2$, we get ${\frac {1}{\sqrt {(2\pi )^{n}\det \Sigma }}}\int e^{-x^{T}\Sigma ^{-1}x/2+v^{T}x}dx=e^{v^{T}\Sigma v/2}$ Differentiate under the integral sign with $\partial _{v_{1},...,v_{n}}\|_{v_{1},...,v_{n}=0}$ to obtain $E[X_{1}\cdots X_{n}]=\partial _{v_{1},...,v_{n}}\|_{v_{1},...,v_{n}=0}e^{v^{T}\Sigma v/2}$ . That is, we need only find the coefficient of term $v_{1}\cdots v_{n}$ in the Taylor expansion of $e^{v^{T}\Sigma v/2}$. If $n$ is odd, this is zero. So let $n=2m$, then we need only find the coefficient of term $v_{1}\cdots v_{n}$ in the polynomial ${\frac {1}{m!}}(v^{T}\Sigma v/2)^{m}$. Expand the polynomial and count, we obtain the formula. $\square $ Generalizations Gaussian integration by parts An equivalent formulation of the Wick's probability formula is the Gaussian integration by parts. If $(X_{1},\dots X_{n})$ is a zero-mean multivariate normal random vector, then $\operatorname {E} (X_{1}f(X_{1},\ldots ,X_{n}))=\sum _{i=1}^{n}\operatorname {Cov} (X_{1}X_{i})\operatorname {E} (\partial _{X_{i}}f(X_{1},\ldots ,X_{n})).$ This is a generalization of Stein's lemma. The Wick's probability formula can be recovered by induction, considering the function $f:\mathbb {R} ^{n}\to \mathbb {R} $ defined by $f(x_{1},\ldots ,x_{n})=x_{2}\ldots x_{n}$. Among other things, this formulation is important in Liouville conformal field theory to obtain conformal Ward identities, BPZ equations[9] and to prove the Fyodorov-Bouchaud formula.[10] Non-Gaussian random variables For non-Gaussian random variables, the moment-cumulants formula[11] replaces the Wick's probability formula. If $(X_{1},\dots X_{n})$ is a vector of random variables, then $\operatorname {E} (X_{1}\ldots X_{n})=\sum _{p\in P_{n}}\prod _{b\in p}\kappa {\big (}(X_{i})_{i\in b}{\big )},$ where the sum is over all the partitions of $\{1,\ldots ,n\}$, the product is over the blocks of $p$ and $\kappa {\big (}(X_{i})_{i\in b}{\big )}$ is the joint cumulant of $(X_{i})_{i\in b}$. See also • Wick's theorem • Cumulants • Normal distribution References 1. Wick, G.C. (1950). "The evaluation of the collision matrix". Physical Review. 80 (2): 268–272. Bibcode:1950PhRv...80..268W. doi:10.1103/PhysRev.80.268. 2. Repetowicz, Przemysław; Richmond, Peter (2005). "Statistical inference of multivariate distribution parameters for non-Gaussian distributed time series" (PDF). Acta Physica Polonica B. 36 (9): 2785–2796. Bibcode:2005AcPPB..36.2785R. 3. Perez-Martin, S.; Robledo, L.M. (2007). "Generalized Wick's theorem for multiquasiparticle overlaps as a limit of Gaudin's theorem". Physical Review C. 76 (6): 064314. arXiv:0707.3365. Bibcode:2007PhRvC..76f4314P. doi:10.1103/PhysRevC.76.064314. S2CID 119627477. 4. Bartosch, L. (2001). "Generation of colored noise". International Journal of Modern Physics C. 12 (6): 851–855. Bibcode:2001IJMPC..12..851B. doi:10.1142/S0129183101002012. S2CID 54500670. 5. Janson, Svante (June 1997). Gaussian Hilbert Spaces. doi:10.1017/CBO9780511526169. ISBN 9780521561280. Retrieved 2019-11-30. {{cite book}}: \|website= ignored (help) 6. Michalowicz, J.V.; Nichols, J.M.; Bucholtz, F.; Olson, C.C. (2009). "An Isserlis' theorem for mixed Gaussian variables: application to the auto-bispectral density". Journal of Statistical Physics. 136 (1): 89–102. Bibcode:2009JSP...136...89M. doi:10.1007/s10955-009-9768-3. S2CID 119702133. 7. Isserlis, L. (1918). "On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables". Biometrika. 12 (1–2): 134–139. doi:10.1093/biomet/12.1-2.134. JSTOR 2331932. 8. Isserlis, L. (1916). "On Certain Probable Errors and Correlation Coefficients of Multiple Frequency Distributions with Skew Regression". Biometrika. 11 (3): 185–190. doi:10.1093/biomet/11.3.185. JSTOR 2331846. 9. Kupiainen, Antti; Rhodes, Rémi; Vargas, Vincent (2019-11-01). "Local Conformal Structure of Liouville Quantum Gravity". Communications in Mathematical Physics. 371 (3): 1005–1069. arXiv:1512.01802. Bibcode:2019CMaPh.371.1005K. doi:10.1007/s00220-018-3260-3. ISSN 1432-0916. S2CID 55282482. 10. Remy, Guillaume (2020). "The Fyodorov–Bouchaud formula and Liouville conformal field theory". Duke Mathematical Journal. 169. arXiv:1710.06897. doi:10.1215/00127094-2019-0045. S2CID 54777103. 11. Leonov, V. P.; Shiryaev, A. N. (January 1959). "On a Method of Calculation of Semi-Invariants". Theory of Probability & Its Applications. 4 (3): 319–329. doi:10.1137/1104031. Further reading • Koopmans, Lambert G. (1974). The spectral analysis of time series. San Diego, CA: Academic Press.
Cyclostationary process A cyclostationary process is a signal having statistical properties that vary cyclically with time.[1] A cyclostationary process can be viewed as multiple interleaved stationary processes. For example, the maximum daily temperature in New York City can be modeled as a cyclostationary process: the maximum temperature on July 21 is statistically different from the temperature on December 20; however, it is a reasonable approximation that the temperature on December 20 of different years has identical statistics. Thus, we can view the random process composed of daily maximum temperatures as 365 interleaved stationary processes, each of which takes on a new value once per year. Definition There are two differing approaches to the treatment of cyclostationary processes.[2] The stochastic approach is to view measurements as an instance of an abstract stochastic process model. As an alternative, the more empirical approach is to view the measurements as a single time series of data--that which has actually been measured in practice and, for some parts of theory, conceptually extended from an observed finite time interval to an infinite interval. Both mathematical models lead to probabilistic theories: abstract stochastic probability for the stochastic process model and the more empirical Fraction Of Time (FOT) probability for the alternative model. The FOT probability of some event associated with the time series is defined to be the fraction of time that event occurs over the lifetime of the time series. In both approaches, the process or time series is said to be cyclostationary if and only if its associated probability distributions vary periodically with time. However, in the non-stochastic time-series approach, there is an alternative but equivalent definition: A time series that contains no finite-strength additive sine-wave components is said to exhibit cyclostationarity if and only if there exists some nonlinear time-invariant transformation of the time series that produces finite-strength (non-zero) additive sine-wave components. Wide-sense cyclostationarity An important special case of cyclostationary signals is one that exhibits cyclostationarity in second-order statistics (e.g., the autocorrelation function). These are called wide-sense cyclostationary signals, and are analogous to wide-sense stationary processes. The exact definition differs depending on whether the signal is treated as a stochastic process or as a deterministic time series. Cyclostationary stochastic process A stochastic process $x(t)$ of mean $\operatorname {E} [x(t)]$ and autocorrelation function: $R_{x}(t,\tau )=\operatorname {E} \{x(t+\tau )x^{}(t)\},\,$ where the star denotes complex conjugation, is said to be wide-sense cyclostationary with period $T_{0}$ if both $\operatorname {E} [x(t)]$ and $R_{x}(t,\tau )$ are cyclic in $t$ with period $T_{0},$ i.e.:[2] $\operatorname {E} [x(t)]=\operatorname {E} [x(t+T_{0})]{\text{ for all }}t$ $R_{x}(t,\tau )=R_{x}(t+T_{0};\tau ){\text{ for all }}t,\tau .$ The autocorrelation function is thus periodic in t and can be expanded in Fourier series: $R_{x}(t,\tau )=\sum _{n=-\infty }^{\infty }R_{x}^{n/T_{0}}(\tau )e^{j2\pi {\frac {n}{T_{0}}}t}$ where $R_{x}^{n/T_{0}}(\tau )$ is called cyclic autocorrelation function and equal to: $R_{x}^{n/T_{0}}(\tau )={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.$ The frequencies $n/T_{0},\,n\in \mathbb {Z} ,$ are called cycle frequencies. Wide-sense stationary processes are a special case of cyclostationary processes with only $R_{x}^{0}(\tau )\neq 0$. Cyclostationary time series A signal that is just a function of time and not a sample path of a stochastic process can exhibit cyclostationarity properties in the framework of the fraction-of-time point of view. This way, the cyclic autocorrelation function can be defined by:[2] ${\widehat {R}}_{x}^{n/T_{0}}(\tau )=\lim _{T\rightarrow +\infty }{\frac {1}{T}}\int _{-T/2}^{T/2}x(t+\tau )x^{}(t)e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t.$ If the time-series is a sample path of a stochastic process it is $R_{x}^{n/T_{0}}(\tau )=\operatorname {E} \left[{\widehat {R}}_{x}^{n/T_{0}}(\tau )\right]$. If the signal is further cycloergodic,[3] all sample paths exhibit the same cyclic time-averages with probability equal to 1 and thus $R_{x}^{n/T_{0}}(\tau )={\widehat {R}}_{x}^{n/T_{0}}(\tau )$ with probability 1. Frequency domain behavior The Fourier transform of the cyclic autocorrelation function at cyclic frequency α is called cyclic spectrum or spectral correlation density function and is equal to: $S_{x}^{\alpha }(f)=\int _{-\infty }^{+\infty }R_{x}^{\alpha }(\tau )e^{-j2\pi f\tau }\mathrm {d} \tau .$ The cyclic spectrum at zero cyclic frequency is also called average power spectral density. For a Gaussian cyclostationary process, its rate distortion function can be expressed in terms of its cyclic spectrum.[4] The reason $S_{x}^{\alpha }(f)$ is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the expected value of the product of the output of a one-sided bandpass filter with center frequency $f+\alpha /2$ and the conjugate of the output of another one-sided bandpass filter with center frequency $f-\alpha /2$, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.[5] For time series, the reason the cyclic spectral density function is called the spectral correlation density function is that it equals the limit, as filter bandwidth approaches zero, of the average over all time of the product of the output of a one-sided bandpass filter with center frequency $f+\alpha /2$ and the conjugate of the output of another one-sided bandpass filter with center frequency $f-\alpha /2$, with both filter outputs frequency shifted to a common center frequency, such as zero, as originally observed and proved in.[6] Example: linearly modulated digital signal An example of cyclostationary signal is the linearly modulated digital signal : $x(t)=\sum _{k=-\infty }^{\infty }a_{k}p(t-kT_{0})$ where $a_{k}\in \mathbb {C} $ are i.i.d. random variables. The waveform $p(t)$, with Fourier transform $P(f)$, is the supporting pulse of the modulation. By assuming $\operatorname {E} [a_{k}]=0$ and $\operatorname {E} [\|a_{k}\|^{2}]=\sigma _{a}^{2}$, the auto-correlation function is: ${\begin{aligned}R_{x}(t,\tau )&=\operatorname {E} [x(t+\tau )x^{}(t)]\\[6pt]&=\sum _{k,n}\operatorname {E} [a_{k}a_{n}^{}]p(t+\tau -kT_{0})p^{}(t-nT_{0})\\[6pt]&=\sigma _{a}^{2}\sum _{k}p(t+\tau -kT_{0})p^{}(t-kT_{0}).\end{aligned}}$ The last summation is a periodic summation, hence a signal periodic in t. This way, $x(t)$ is a cyclostationary signal with period $T_{0}$ and cyclic autocorrelation function: ${\begin{aligned}R_{x}^{n/T_{0}}(\tau )&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}R_{x}(t,\tau )e^{-j2\pi {\frac {n}{T_{0}}}t}\,\mathrm {d} t\\[6pt]&={\frac {1}{T_{0}}}\int _{-T_{0}/2}^{T_{0}/2}\sigma _{a}^{2}\sum _{k=-\infty }^{\infty }p(t+\tau -kT_{0})p^{}(t-kT_{0})e^{-j2\pi {\frac {n}{T_{0}}}t}\mathrm {d} t\\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}\sum _{k=-\infty }^{\infty }\int _{-T_{0}/2-kT_{0}}^{T_{0}/2-kT_{0}}p(\lambda +\tau )p^{}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}(\lambda +kT_{0})}\mathrm {d} \lambda \\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}\int _{-\infty }^{\infty }p(\lambda +\tau )p^{}(\lambda )e^{-j2\pi {\frac {n}{T_{0}}}\lambda }\mathrm {d} \lambda \\[6pt]&={\frac {\sigma _{a}^{2}}{T_{0}}}p(\tau )\left\{p^{}(-\tau )e^{j2\pi {\frac {n}{T_{0}}}\tau }\right\}.\end{aligned}}$ with $$ indicating convolution. The cyclic spectrum is: $S_{x}^{n/T_{0}}(f)={\frac {\sigma _{a}^{2}}{T_{0}}}P(f)P^{}\left(f-{\frac {n}{T_{0}}}\right).$ Typical raised-cosine pulses adopted in digital communications have thus only $n=-1,0,1$ non-zero cyclic frequencies. This same result can be obtained for the non-stochastic time series model of linearly modulated digital signals in which expectation is replaced with infinite time average, but this requires a somewhat modified mathematical method as originally observed and proved in.[7] Cyclostationary models It is possible to generalise the class of autoregressive moving average models to incorporate cyclostationary behaviour. For example, Troutman[8] treated autoregressions in which the autoregression coefficients and residual variance are no longer constant but vary cyclically with time. His work follows a number of other studies of cyclostationary processes within the field of time series analysis.[9][10] Polycyclostationarity In practice, signals exhibiting cyclicity with more than one incommensurate period arise and require a generalization of the theory of cyclostationarity. Such signals are called polycyclostationary if they exhibit a finite number of incommensurate periods and almost cyclostationary if they exhibit a countably infinite number. Such signals arise frequently in radio communications due to multiple transmissions with differing sine-wave carrier frequencies and digital symbol rates. The theory was introduced in [11] for stochastic processes and further developed in [12] for non-stochastic time series. Higher Order and Strict Sense Cyclostationarity The wide sense theory of time series exhibiting cyclostationarity, polycyclostationarity and almost cyclostationarity originated and developed by Gardner [13] was also generalized by Gardner to a theory of higher-order temporal and spectral moments and cumulants and a strict sense theory of cumulative probability distributions. The encyclopedic book [14] comprehensively teaches all of this and provides a scholarly treatment of the originating publications by Gardner and contributions thereafter by others. Applications • Cyclostationarity has extremely diverse applications in essentially all fields of engineering and science, as thoroughly documented in [15] and.[16] A few examples are: • Cyclostationarity is used in telecommunications for signal synchronization, transmitter and receiver optimization, and spectrum sensing for cognitive radio;[17] • In signals intelligence, cyclostationarity is used for signal interception;[18] • In econometrics, cyclostationarity is used to analyze the periodic behavior of financial-markets; • Queueing theory utilizes cyclostationary theory to analyze computer networks and car traffic; • Cyclostationarity is used to analyze mechanical signals produced by rotating and reciprocating machines. Angle-time cyclostationarity of mechanical signals Mechanical signals produced by rotating or reciprocating machines are remarkably well modelled as cyclostationary processes. The cyclostationary family accepts all signals with hidden periodicities, either of the additive type (presence of tonal components) or multiplicative type (presence of periodic modulations). This happens to be the case for noise and vibration produced by gear mechanisms, bearings, internal combustion engines, turbofans, pumps, propellers, etc. The explicit modelling of mechanical signals as cyclostationary processes has been found useful in several applications, such as in noise, vibration, and harshness (NVH) and in condition monitoring.[19] In the latter field, cyclostationarity has been found to generalize the envelope spectrum, a popular analysis technique used in the diagnostics of bearing faults. One peculiarity of rotating machine signals is that the period of the process is strictly linked to the angle of rotation of a specific component – the “cycle” of the machine. At the same time, a temporal description must be preserved to reflect the nature of dynamical phenomena that are governed by differential equations of time. Therefore, the angle-time autocorrelation function is used, $R_{x}(\theta ,\tau )=\operatorname {E} \{x(t(\theta )+\tau )x^{}(t(\theta ))\},\,$ where $\theta $ stands for angle, $t(\theta )$ for the time instant corresponding to angle $\theta $ and $\tau $ for time delay. Processes whose angle-time autocorrelation function exhibit a component periodic in angle, i.e. such that $R_{x}(\theta ;\tau )$ ;\tau )} has a non-zero Fourier-Bohr coefficient for some angular period $\Theta $, are called (wide-sense) angle-time cyclostationary. The double Fourier transform of the angle-time autocorrelation function defines the order-frequency spectral correlation, $S_{x}^{\alpha }(f)=\lim _{S\rightarrow +\infty }{\frac {1}{S}}\int _{-S/2}^{S/2}\int _{-\infty }^{+\infty }R_{x}(\theta ,\tau )e^{-j2\pi f\tau }e^{-j2\pi \alpha {\frac {\theta }{\Theta }}}\,\mathrm {d} \tau \,\mathrm {d} \theta $ where $\alpha $ is an order (unit in events per revolution) and $f$ a frequency (unit in Hz). For constant speed of rotation, $\omega $, angle is proportional to time, $\theta =\omega t$. Consequently, the angle-time autocorrelation is simply a cyclicity-scaled traditional autocorrelation; that is, the cycle frequencies are scaled by $\omega $. On the other hand, if the speed of rotation changes with time, then the signal is no longer cyclostationary (unless the speed varies periodically). Therefore, it is not a model for cyclostationary signals. It is not even a model for time-warped cyclostationarity, although it can be a useful approximation for sufficiently slow changes in speed of rotation. [20] References 1. Gardner, William A.; Antonio Napolitano; Luigi Paura (2006). "Cyclostationarity: Half a century of research". Signal Processing. Elsevier. 86 (4): 639–697. doi:10.1016/j.sigpro.2005.06.016. 2. Gardner, William A. (1991). "Two alternative philosophies for estimation of the parameters of time-series". IEEE Trans. Inf. Theory. 37 (1): 216–218. doi:10.1109/18.61145. 3. 1983 R. A. Boyles and W. A. Gardner. CYCLOERGODIC PROPERTIES OF DISCRETE-PARAMETER NONSTATIONARY STOCHASTIC PROCESSES. IEEE Transactions on Information Theory, Vol. IT-29, No. 1, pp. 105-114. 4. Kipnis, Alon; Goldsmith, Andrea; Eldar, Yonina (May 2018). "The Distortion Rate Function of Cyclostationary Gaussian Processes". IEEE Transactions on Information Theory. 65 (5): 3810–3824. arXiv:1505.05586. doi:10.1109/TIT.2017.2741978. S2CID 5014143. 5. W. A. Gardner. INTRODUCTION TO RANDOM PROCESSES WITH APPLICATIONS TO SIGNALS AND SYSTEMS. Macmillan, New York, 434 pages, 1985 6. W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. 7. W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. 8. Troutman, B.M. (1979) "Some results in periodic autoregression." Biometrika, 66 (2), 219–228 9. Jones, R.H., Brelsford, W.M. (1967) "Time series with periodic structure." Biometrika, 54, 403–410 10. Pagano, M. (1978) "On periodic and multiple autoregressions." Ann. Stat., 6, 1310–1317. 11. W. A. Gardner. STATIONARIZABLE RANDOM PROCESSES. IEEE Transactions on Information Theory, Vol. IT-24, No. 1, pp. 8-22. 1978 12. W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. 13. W. A. Gardner. STATISTICAL SPECTRAL ANALYSIS: A NONPROBABILISTIC THEORY. Prentice-Hall, Englewood Cliffs, NJ, 565 pages, 1987. 14. A. Napolitano, Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Academic Press, 2020. 15. W. A. Gardner. STATISTICALLY INFERRED TIME WARPING: EXTENDING THE CYCLOSTATIONARITY PARADIGM FROM REGULAR TO IRREGULAR STATISTICAL CYCLICITY IN SCIENTIFIC DATA. EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 59. doi: 10.1186/s13634-018-0564-6 16. A. Napolitano, Cyclostationary Processes and Time Series: Theory, Applications, and Generalizations. Academic Press, 2020. 17. W. A. Gardner. CYCLOSTATIONARITY IN COMMUNICATIONS AND SIGNAL PROCESSING. Piscataway, NJ: IEEE Press. 504 pages.1984. 18. W. A. Gardner. SIGNAL INTERCEPTION: A UNIFYING THEORETICAL FRAMEWORK FOR FEATURE DETECTION. IEEE Transactions on Communications, Vol. COM-36, No. 8, pp. 897-906. 1988 19. Antoni, Jérôme (2009). "Cyclostationarity by examples". Mechanical Systems and Signal Processing. Elsevier. 23 (4): 987–1036. doi:10.1016/j.ymssp.2008.10.010. 20. 2018 W. A. Gardner. STATISTICALLY INFERRED TIME WARPING: EXTENDING THE CYCLOSTATIONARITY PARADIGM FROM REGULAR TO IRREGULAR STATISTICAL CYCLICITY IN SCIENTIFIC DATA. EURASIP Journal on Advances in Signal Processing volume 2018, Article number: 59. doi: 10.1186/s13634-018-0564-6 External links • Noise in mixers, oscillators, samplers, and logic: an introduction to cyclostationary noise manuscript annotated presentation presentation
Widest path problem In graph algorithms, the widest path problem is the problem of finding a path between two designated vertices in a weighted graph, maximizing the weight of the minimum-weight edge in the path. The widest path problem is also known as the maximum capacity path problem. It is possible to adapt most shortest path algorithms to compute widest paths, by modifying them to use the bottleneck distance instead of path length.[1] However, in many cases even faster algorithms are possible. For instance, in a graph that represents connections between routers in the Internet, where the weight of an edge represents the bandwidth of a connection between two routers, the widest path problem is the problem of finding an end-to-end path between two Internet nodes that has the maximum possible bandwidth.[2] The smallest edge weight on this path is known as the capacity or bandwidth of the path. As well as its applications in network routing, the widest path problem is also an important component of the Schulze method for deciding the winner of a multiway election,[3] and has been applied to digital compositing,[4] metabolic pathway analysis,[5] and the computation of maximum flows.[6] A closely related problem, the minimax path problem or bottleneck shortest path problem asks for the path that minimizes the maximum weight of any of its edges. It has applications that include transportation planning.[7] Any algorithm for the widest path problem can be transformed into an algorithm for the minimax path problem, or vice versa, by reversing the sense of all the weight comparisons performed by the algorithm, or equivalently by replacing every edge weight by its negation. Undirected graphs In an undirected graph, a widest path may be found as the path between the two vertices in the maximum spanning tree of the graph, and a minimax path may be found as the path between the two vertices in the minimum spanning tree.[8][9][10] In any graph, directed or undirected, there is a straightforward algorithm for finding a widest path once the weight of its minimum-weight edge is known: simply delete all smaller edges and search for any path among the remaining edges using breadth-first search or depth-first search. Based on this test, there also exists a linear time algorithm for finding a widest s-t path in an undirected graph, that does not use the maximum spanning tree. The main idea of the algorithm is to apply the linear-time path-finding algorithm to the median edge weight in the graph, and then either to delete all smaller edges or contract all larger edges according to whether a path does or does not exist, and recurse in the resulting smaller graph.[9][11][12] Fernández, Garfinkel & Arbiol (1998) use undirected bottleneck shortest paths in order to form composite aerial photographs that combine multiple images of overlapping areas. In the subproblem to which the widest path problem applies, two images have already been transformed into a common coordinate system; the remaining task is to select a seam, a curve that passes through the region of overlap and divides one of the two images from the other. Pixels on one side of the seam will be copied from one of the images, and pixels on the other side of the seam will be copied from the other image. Unlike other compositing methods that average pixels from both images, this produces a valid photographic image of every part of the region being photographed. They weigh the edges of a grid graph by a numeric estimate of how visually apparent a seam across that edge would be, and find a bottleneck shortest path for these weights. Using this path as the seam, rather than a more conventional shortest path, causes their system to find a seam that is difficult to discern at all of its points, rather than allowing it to trade off greater visibility in one part of the image for lesser visibility elsewhere.[4] A solution to the minimax path problem between the two opposite corners of a grid graph can be used to find the weak Fréchet distance between two polygonal chains. Here, each grid graph vertex represents a pair of line segments, one from each chain, and the weight of an edge represents the Fréchet distance needed to pass from one pair of segments to another.[13] If all edge weights of an undirected graph are positive, then the minimax distances between pairs of points (the maximum edge weights of minimax paths) form an ultrametric; conversely every finite ultrametric space comes from minimax distances in this way.[14] A data structure constructed from the minimum spanning tree allows the minimax distance between any pair of vertices to be queried in constant time per query, using lowest common ancestor queries in a Cartesian tree. The root of the Cartesian tree represents the heaviest minimum spanning tree edge, and the children of the root are Cartesian trees recursively constructed from the subtrees of the minimum spanning tree formed by removing the heaviest edge. The leaves of the Cartesian tree represent the vertices of the input graph, and the minimax distance between two vertices equals the weight of the Cartesian tree node that is their lowest common ancestor. Once the minimum spanning tree edges have been sorted, this Cartesian tree can be constructed in linear time.[15] Directed graphs In directed graphs, the maximum spanning tree solution cannot be used. Instead, several different algorithms are known; the choice of which algorithm to use depends on whether a start or destination vertex for the path is fixed, or whether paths for many start or destination vertices must be found simultaneously. All pairs The all-pairs widest path problem has applications in the Schulze method for choosing a winner in multiway elections in which voters rank the candidates in preference order. The Schulze method constructs a complete directed graph in which the vertices represent the candidates and every two vertices are connected by an edge. Each edge is directed from the winner to the loser of a pairwise contest between the two candidates it connects, and is labeled with the margin of victory of that contest. Then the method computes widest paths between all pairs of vertices, and the winner is the candidate whose vertex has wider paths to each opponent than vice versa.[3] The results of an election using this method are consistent with the Condorcet method – a candidate who wins all pairwise contests automatically wins the whole election – but it generally allows a winner to be selected, even in situations where the Concorcet method itself fails.[16] The Schulze method has been used by several organizations including the Wikimedia Foundation.[17] To compute the widest path widths for all pairs of nodes in a dense directed graph, such as the ones that arise in the voting application, the asymptotically fastest known approach takes time O(n(3+ω)/2) where ω is the exponent for fast matrix multiplication. Using the best known algorithms for matrix multiplication, this time bound becomes O(n2.688).[18] Instead, the reference implementation for the Schulze method uses a modified version of the simpler Floyd–Warshall algorithm, which takes O(n3) time.[3] For sparse graphs, it may be more efficient to repeatedly apply a single-source widest path algorithm. Single source If the edges are sorted by their weights, then a modified version of Dijkstra's algorithm can compute the bottlenecks between a designated start vertex and every other vertex in the graph, in linear time. The key idea behind the speedup over a conventional version of Dijkstra's algorithm is that the sequence of bottleneck distances to each vertex, in the order that the vertices are considered by this algorithm, is a monotonic subsequence of the sorted sequence of edge weights; therefore, the priority queue of Dijkstra's algorithm can be implemented as a bucket queue: an array indexed by the numbers from 1 to m (the number of edges in the graph), where array cell i contains the vertices whose bottleneck distance is the weight of the edge with position i in the sorted order. This method allows the widest path problem to be solved as quickly as sorting; for instance, if the edge weights are represented as integers, then the time bounds for integer sorting a list of m integers would apply also to this problem.[12] Single source and single destination Berman & Handler (1987) suggest that service vehicles and emergency vehicles should use minimax paths when returning from a service call to their base. In this application, the time to return is less important than the response time if another service call occurs while the vehicle is in the process of returning. By using a minimax path, where the weight of an edge is the maximum travel time from a point on the edge to the farthest possible service call, one can plan a route that minimizes the maximum possible delay between receipt of a service call and arrival of a responding vehicle.[7] Ullah, Lee & Hassoun (2009) use maximin paths to model the dominant reaction chains in metabolic networks; in their model, the weight of an edge is the free energy of the metabolic reaction represented by the edge.[5] Another application of widest paths arises in the Ford–Fulkerson algorithm for the maximum flow problem. Repeatedly augmenting a flow along a maximum capacity path in the residual network of the flow leads to a small bound, O(m log U), on the number of augmentations needed to find a maximum flow; here, the edge capacities are assumed to be integers that are at most U. However, this analysis does not depend on finding a path that has the exact maximum of capacity; any path whose capacity is within a constant factor of the maximum suffices. Combining this approximation idea with the shortest path augmentation method of the Edmonds–Karp algorithm leads to a maximum flow algorithm with running time O(mn log U).[6] It is possible to find maximum-capacity paths and minimax paths with a single source and single destination very efficiently even in models of computation that allow only comparisons of the input graph's edge weights and not arithmetic on them.[12][19] The algorithm maintains a set S of edges that are known to contain the bottleneck edge of the optimal path; initially, S is just the set of all m edges of the graph. At each iteration of the algorithm, it splits S into an ordered sequence of subsets S1, S2, ... of approximately equal size; the number of subsets in this partition is chosen in such a way that all of the split points between subsets can be found by repeated median-finding in time O(m). The algorithm then reweights each edge of the graph by the index of the subset containing the edge, and uses the modified Dijkstra algorithm on the reweighted graph; based on the results of this computation, it can determine in linear time which of the subsets contains the bottleneck edge weight. It then replaces S by the subset Si that it has determined to contain the bottleneck weight, and starts the next iteration with this new set S. The number of subsets into which S can be split increases exponentially with each step, so the number of iterations is proportional to the iterated logarithm function, O(logn), and the total time is O(m logn).[19] In a model of computation where each edge weight is a machine integer, the use of repeated bisection in this algorithm can be replaced by a list-splitting technique of Han & Thorup (2002), allowing S to be split into O(√m) smaller sets Si in a single step and leading to a linear overall time bound.[20] Euclidean point sets A variant of the minimax path problem has also been considered for sets of points in the Euclidean plane. As in the undirected graph problem, this Euclidean minimax path problem can be solved efficiently by finding a Euclidean minimum spanning tree: every path in the tree is a minimax path. However, the problem becomes more complicated when a path is desired that not only minimizes the hop length but also, among paths with the same hop length, minimizes or approximately minimizes the total length of the path. The solution can be approximated using geometric spanners.[21] In number theory, the unsolved Gaussian moat problem asks whether or not minimax paths in the Gaussian prime numbers have bounded or unbounded minimax length. That is, does there exist a constant B such that, for every pair of points p and q in the infinite Euclidean point set defined by the Gaussian primes, the minimax path in the Gaussian primes between p and q has minimax edge length at most B?[22] References 1. Pollack, Maurice (1960), "The maximum capacity through a network", Operations Research, 8 (5): 733–736, doi:10.1287/opre.8.5.733, JSTOR 167387 2. Shacham, N. (1992), "Multicast routing of hierarchical data", IEEE International Conference on Communications (ICC '92), vol. 3, pp. 1217–1221, doi:10.1109/ICC.1992.268047, hdl:2060/19990017646, ISBN 978-0-7803-0599-1, S2CID 60475077; Wang, Zheng; Crowcroft, J. (1995), "Bandwidth-delay based routing algorithms", IEEE Global Telecommunications Conference (GLOBECOM '95), vol. 3, pp. 2129–2133, doi:10.1109/GLOCOM.1995.502780, ISBN 978-0-7803-2509-8, S2CID 9117583 3. Schulze, Markus (2011), "A new monotonic, clone-independent, reversal symmetric, and Condorcet-consistent single-winner election method", Social Choice and Welfare, 36 (2): 267–303, doi:10.1007/s00355-010-0475-4, S2CID 1927244 4. Fernández, Elena; Garfinkel, Robert; Arbiol, Roman (1998), "Mosaicking of aerial photographic maps via seams defined by bottleneck shortest paths", Operations Research, 46 (3): 293–304, doi:10.1287/opre.46.3.293, JSTOR 222823 5. Ullah, E.; Lee, Kyongbum; Hassoun, S. (2009), "An algorithm for identifying dominant-edge metabolic pathways", IEEE/ACM International Conference on Computer-Aided Design (ICCAD 2009), pp. 144–150 6. Ahuja, Ravindra K.; Magnanti, Thomas L.; Orlin, James B. (1993), "7.3 Capacity Scaling Algorithm", Network Flows: Theory, Algorithms and Applications, Prentice Hall, pp. 210–212, ISBN 978-0-13-617549-0 7. Berman, Oded; Handler, Gabriel Y. (1987), "Optimal Minimax Path of a Single Service Unit on a Network to Nonservice Destinations", Transportation Science, 21 (2): 115–122, doi:10.1287/trsc.21.2.115 8. Hu, T. C. (1961), "The maximum capacity route problem", Operations Research, 9 (6): 898–900, doi:10.1287/opre.9.6.898, JSTOR 167055 9. Punnen, Abraham P. (1991), "A linear time algorithm for the maximum capacity path problem", European Journal of Operational Research, 53 (3): 402–404, doi:10.1016/0377-2217(91)90073-5 10. Malpani, Navneet; Chen, Jianer (2002), "A note on practical construction of maximum bandwidth paths", Information Processing Letters, 83 (3): 175–180, doi:10.1016/S0020-0190(01)00323-4, MR 1904226 11. Camerini, P. M. (1978), "The min-max spanning tree problem and some extensions", Information Processing Letters, 7 (1): 10–14, doi:10.1016/0020-0190(78)90030-3 12. Kaibel, Volker; Peinhardt, Matthias A. F. (2006), On the bottleneck shortest path problem (PDF), ZIB-Report 06-22, Konrad-Zuse-Zentrum für Informationstechnik Berlin 13. Alt, Helmut; Godau, Michael (1995), "Computing the Fréchet distance between two polygonal curves" (PDF), International Journal of Computational Geometry and Applications, 5 (1–2): 75–91, doi:10.1142/S0218195995000064. 14. Leclerc, Bruno (1981), "Description combinatoire des ultramétriques", Centre de Mathématique Sociale. École Pratique des Hautes Études. Mathématiques et Sciences Humaines (in French) (73): 5–37, 127, MR 0623034 15. Demaine, Erik D.; Landau, Gad M.; Weimann, Oren (2009), "On Cartesian trees and range minimum queries", Automata, Languages and Programming, 36th International Colloquium, ICALP 2009, Rhodes, Greece, July 5-12, 2009, Lecture Notes in Computer Science, vol. 5555, pp. 341–353, doi:10.1007/978-3-642-02927-1_29, hdl:1721.1/61963, ISBN 978-3-642-02926-4 16. More specifically, the only kind of tie that the Schulze method fails to break is between two candidates who have equally wide paths to each other. 17. See Jesse Plamondon-Willard, Board election to use preference voting, May 2008; Mark Ryan, 2008 Wikimedia Board Election results, June 2008; 2008 Board Elections, June 2008; and 2009 Board Elections, August 2009. 18. Duan, Ran; Pettie, Seth (2009), "Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths", Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA '09), pp. 384–391. For an earlier algorithm that also used fast matrix multiplication to speed up all pairs widest paths, see Vassilevska, Virginia; Williams, Ryan; Yuster, Raphael (2007), "All-pairs bottleneck paths for general graphs in truly sub-cubic time", Proceedings of the 39th Annual ACM Symposium on Theory of Computing (STOC '07), New York: ACM, pp. 585–589, CiteSeerX 10.1.1.164.9808, doi:10.1145/1250790.1250876, ISBN 9781595936318, MR 2402484, S2CID 9353065 and Chapter 5 of Vassilevska, Virginia (2008), Efficient Algorithms for Path Problems in Weighted Graphs (PDF), Ph.D. thesis, Report CMU-CS-08-147, Carnegie Mellon University School of Computer Science 19. Gabow, Harold N.; Tarjan, Robert E. (1988), "Algorithms for two bottleneck optimization problems", Journal of Algorithms, 9 (3): 411–417, doi:10.1016/0196-6774(88)90031-4, MR 0955149 20. Han, Yijie; Thorup, M. (2002), "Integer sorting in O(n√log log n) expected time and linear space", Proc. 43rd Annual Symposium on Foundations of Computer Science (FOCS 2002), pp. 135–144, doi:10.1109/SFCS.2002.1181890, ISBN 978-0-7695-1822-0, S2CID 5245628. 21. Bose, Prosenjit; Maheshwari, Anil; Narasimhan, Giri; Smid, Michiel; Zeh, Norbert (2004), "Approximating geometric bottleneck shortest paths", Computational Geometry. Theory and Applications, 29 (3): 233–249, doi:10.1016/j.comgeo.2004.04.003, MR 2095376 22. Gethner, Ellen; Wagon, Stan; Wick, Brian (1998), "A stroll through the Gaussian primes", American Mathematical Monthly, 105 (4): 327–337, doi:10.2307/2589708, JSTOR 2589708, MR 1614871.
Antichain In mathematics, in the area of order theory, an antichain is a subset of a partially ordered set such that any two distinct elements in the subset are incomparable. The size of the largest antichain in a partially ordered set is known as its width. By Dilworth's theorem, this also equals the minimum number of chains (totally ordered subsets) into which the set can be partitioned. Dually, the height of the partially ordered set (the length of its longest chain) equals by Mirsky's theorem the minimum number of antichains into which the set can be partitioned. The family of all antichains in a finite partially ordered set can be given join and meet operations, making them into a distributive lattice. For the partially ordered system of all subsets of a finite set, ordered by set inclusion, the antichains are called Sperner families and their lattice is a free distributive lattice, with a Dedekind number of elements. More generally, counting the number of antichains of a finite partially ordered set is #P-complete. Definitions Let $S$ be a partially ordered set. Two elements $a$ and $b$ of a partially ordered set are called comparable if $a\leq b{\text{ or }}b\leq a.$ If two elements are not comparable, they are called incomparable; that is, $x$ and $y$ are incomparable if neither $x\leq y{\text{ nor }}y\leq x.$ A chain in $S$ is a subset $C\subseteq S$ in which each pair of elements is comparable; that is, $C$ is totally ordered. An antichain in $S$ is a subset $A$ of $S$ in which each pair of different elements is incomparable; that is, there is no order relation between any two different elements in $A.$ (However, some authors use the term "antichain" to mean strong antichain, a subset such that there is no element of the poset smaller than two distinct elements of the antichain.) Height and width A maximal antichain is an antichain that is not a proper subset of any other antichain. A maximum antichain is an antichain that has cardinality at least as large as every other antichain. The width of a partially ordered set is the cardinality of a maximum antichain. Any antichain can intersect any chain in at most one element, so, if we can partition the elements of an order into $k$ chains then the width of the order must be at most $k$ (if the antichain has more than $k$ elements, by the pigeonhole principle, there would be 2 of its elements belonging to the same chain, a contradiction). Dilworth's theorem states that this bound can always be reached: there always exists an antichain, and a partition of the elements into chains, such that the number of chains equals the number of elements in the antichain, which must therefore also equal the width.[1] Similarly, one can define the height of a partial order to be the maximum cardinality of a chain. Mirsky's theorem states that in any partial order of finite height, the height equals the smallest number of antichains into which the order may be partitioned.[2] Sperner families An antichain in the inclusion ordering of subsets of an $n$-element set is known as a Sperner family. The number of different Sperner families is counted by the Dedekind numbers,[3] the first few of which numbers are 2, 3, 6, 20, 168, 7581, 7828354, 2414682040998, 56130437228687557907788 (sequence A000372 in the OEIS). Even the empty set has two antichains in its power set: one containing a single set (the empty set itself) and one containing no sets. Join and meet operations Any antichain $A$ corresponds to a lower set $L_{A}=\{x:\exists y\in A{\mbox{ such that }}x\leq y\}.$ In a finite partial order (or more generally a partial order satisfying the ascending chain condition) all lower sets have this form. The union of any two lower sets is another lower set, and the union operation corresponds in this way to a join operation on antichains: $A\vee B=\{x\in A\cup B:\nexists y\in A\cup B{\mbox{ such that }}x<y\}.$ Similarly, we can define a meet operation on antichains, corresponding to the intersection of lower sets: $A\wedge B=\{x\in L_{A}\cap L_{B}:\nexists y\in L_{A}\cap L_{B}{\mbox{ such that }}x<y\}.$ The join and meet operations on all finite antichains of finite subsets of a set $X$ define a distributive lattice, the free distributive lattice generated by $X.$ Birkhoff's representation theorem for distributive lattices states that every finite distributive lattice can be represented via join and meet operations on antichains of a finite partial order, or equivalently as union and intersection operations on the lower sets of the partial order.[4] Computational complexity A maximum antichain (and its size, the width of a given partially ordered set) can be found in polynomial time.[5] Counting the number of antichains in a given partially ordered set is #P-complete.[6] References 1. Dilworth, Robert P. (1950), "A decomposition theorem for partially ordered sets", Annals of Mathematics, 51 (1): 161–166, doi:10.2307/1969503, JSTOR 1969503 2. Mirsky, Leon (1971), "A dual of Dilworth's decomposition theorem", American Mathematical Monthly, 78 (8): 876–877, doi:10.2307/2316481, JSTOR 2316481 3. Kahn, Jeff (2002), "Entropy, independent sets and antichains: a new approach to Dedekind's problem", Proceedings of the American Mathematical Society, 130 (2): 371–378, doi:10.1090/S0002-9939-01-06058-0, MR 1862115 4. Birkhoff, Garrett (1937), "Rings of sets", Duke Mathematical Journal, 3 (3): 443–454, doi:10.1215/S0012-7094-37-00334-X 5. Felsner, Stefan; Raghavan, Vijay; Spinrad, Jeremy (2003), "Recognition algorithms for orders of small width and graphs of small Dilworth number", Order, 20 (4): 351–364 (2004), doi:10.1023/B:ORDE.0000034609.99940.fb, MR 2079151, S2CID 1363140 6. Provan, J. Scott; Ball, Michael O. (1983), "The complexity of counting cuts and of computing the probability that a graph is connected", SIAM Journal on Computing, 12 (4): 777–788, doi:10.1137/0212053, MR 0721012 External links • Weisstein, Eric W. "Antichain". MathWorld. • "Antichain". PlanetMath. Order theory • Topics • Glossary • Category Key concepts • Binary relation • Boolean algebra • Cyclic order • Lattice • Partial order • Preorder • Total order • Weak ordering Results • Boolean prime ideal theorem • Cantor–Bernstein theorem • Cantor's isomorphism theorem • Dilworth's theorem • Dushnik–Miller theorem • Hausdorff maximal principle • Knaster–Tarski theorem • Kruskal's tree theorem • Laver's theorem • Mirsky's theorem • Szpilrajn extension theorem • Zorn's lemma Properties & Types (list) • Antisymmetric • Asymmetric • Boolean algebra • topics • Completeness • Connected • Covering • Dense • Directed • (Partial) Equivalence • Foundational • Heyting algebra • Homogeneous • Idempotent • Lattice • Bounded • Complemented • Complete • Distributive • Join and meet • Reflexive • Partial order • Chain-complete • Graded • Eulerian • Strict • Prefix order • Preorder • Total • Semilattice • Semiorder • Symmetric • Total • Tolerance • Transitive • Well-founded • Well-quasi-ordering (Better) • (Pre) Well-order Constructions • Composition • Converse/Transpose • Lexicographic order • Linear extension • Product order • Reflexive closure • Series-parallel partial order • Star product • Symmetric closure • Transitive closure Topology & Orders • Alexandrov topology & Specialization preorder • Ordered topological vector space • Normal cone • Order topology • Order topology • Topological vector lattice • Banach • Fréchet • Locally convex • Normed Related • Antichain • Cofinal • Cofinality • Comparability • Graph • Duality • Filter • Hasse diagram • Ideal • Net • Subnet • Order morphism • Embedding • Isomorphism • Order type • Ordered field • Ordered vector space • Partially ordered • Positive cone • Riesz space • Upper set • Young's lattice
Wiedersehen pair In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged. For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points. If every point of an oriented manifold (M, g) belongs to a Wiedersehen pair, then (M, g) is said to be a Wiedersehen manifold. The concept was introduced by the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n). See also • Cut locus (Riemannian manifold) References • Blaschke, Wilhelm (1921). Vorlesung über Differentialgeometrie I. Berlin: Springer-Verlag. • C. T. Yang (1980). "Odd-dimensional wiedersehen manifolds are spheres". J. Differential Geom. 15 (1): 91–96. doi:10.4310/jdg/1214435386. ISSN 0022-040X. • Chavel, Isaac (2006). Riemannian geometry: a modern introduction. New York: Cambridge University Press. pp. 328–329. ISBN 0-521-61954-8. External links • Weisstein, Eric W. "Wiedersehen pair". MathWorld. • Weisstein, Eric W. "Wiedersehen surface". MathWorld.
Wieferich prime In number theory, a Wieferich prime is a prime number p such that p2 divides 2p − 1 − 1,[4] therefore connecting these primes with Fermat's little theorem, which states that every odd prime p divides 2p − 1 − 1. Wieferich primes were first described by Arthur Wieferich in 1909 in works pertaining to Fermat's Last Theorem, at which time both of Fermat's theorems were already well known to mathematicians.[5][6] Wieferich prime Named afterArthur Wieferich Publication year1909 Author of publicationWieferich, A. No. of known terms2 Conjectured no. of termsInfinite Subsequence of • Crandall numbers[1] • Wieferich numbers[2] • Lucas–Wieferich primes[3] • near-Wieferich primes First terms1093, 3511 Largest known term3511 OEIS indexA001220 Since then, connections between Wieferich primes and various other topics in mathematics have been discovered, including other types of numbers and primes, such as Mersenne and Fermat numbers, specific types of pseudoprimes and some types of numbers generalized from the original definition of a Wieferich prime. Over time, those connections discovered have extended to cover more properties of certain prime numbers as well as more general subjects such as number fields and the abc conjecture. As of April 2023, the only known Wieferich primes are 1093 and 3511 (sequence A001220 in the OEIS). Equivalent definitions The stronger version of Fermat's little theorem, which a Wieferich prime satisfies, is usually expressed as a congruence relation 2p -1 ≡ 1 (mod p2). From the definition of the congruence relation on integers, it follows that this property is equivalent to the definition given at the beginning. Thus if a prime p satisfies this congruence, this prime divides the Fermat quotient ${\tfrac {2^{p-1}-1}{p}}$. The following are two illustrative examples using the primes 11 and 1093: For p = 11, we get ${\tfrac {2^{10}-1}{11}}$ which is 93 and leaves a remainder of 5 after division by 11, hence 11 is not a Wieferich prime. For p = 1093, we get ${\tfrac {2^{1092}-1}{1093}}$ or 485439490310...852893958515 (302 intermediate digits omitted for clarity), which leaves a remainder of 0 after division by 1093 and thus 1093 is a Wieferich prime. Wieferich primes can be defined by other equivalent congruences. If p is a Wieferich prime, one can multiply both sides of the congruence 2p−1 ≡ 1 (mod p2) by 2 to get 2p ≡ 2 (mod p2). Raising both sides of the congruence to the power p shows that a Wieferich prime also satisfies 2p2 ≡2p ≡ 2 (mod p2), and hence 2pk ≡ 2 (mod p2) for all k ≥ 1. The converse is also true: 2pk ≡ 2 (mod p2) for some k ≥ 1 implies that the multiplicative order of 2 modulo p2 divides gcd(pk − 1, φ(p2)) = p − 1, that is, 2p−1 ≡ 1 (mod p2) and thus p is a Wieferich prime. This also implies that Wieferich primes can be defined as primes p such that the multiplicative orders of 2 modulo p and modulo p2 coincide: ordp2 2 = ordp 2, (By the way, ord10932 = 364, and ord35112 = 1755). H. S. Vandiver proved that 2p−1 ≡ 1 (mod p3) if and only if $1+{\tfrac {1}{3}}+\dots +{\tfrac {1}{p-2}}\equiv 0{\pmod {p^{2}}}$.[7]: 187 History and search status In 1902, Meyer proved a theorem about solutions of the congruence ap − 1 ≡ 1 (mod pr).[8]: 930 [9] Later in that decade Arthur Wieferich showed specifically that if the first case of Fermat's last theorem has solutions for an odd prime exponent, then that prime must satisfy that congruence for a = 2 and r = 2.[10] In other words, if there exist solutions to xp + yp + zp = 0 in integers x, y, z and p an odd prime with p ∤ xyz, then p satisfies 2p − 1 ≡ 1 (mod p2). In 1913, Bachmann examined the residues of ${\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$. He asked the question when this residue vanishes and tried to find expressions for answering this question.[11] The prime 1093 was found to be a Wieferich prime by W. Meissner in 1913 and confirmed to be the only such prime below 2000. He calculated the smallest residue of ${\tfrac {2^{t}-1}{p}}\,{\bmod {\,}}p$ for all primes p < 2000 and found this residue to be zero for t = 364 and p = 1093, thereby providing a counterexample to a conjecture by Grave about the impossibility of the Wieferich congruence.[12] E. Haentzschel later ordered verification of the correctness of Meissner's congruence via only elementary calculations.[13]: 664 Inspired by an earlier work of Euler, he simplified Meissner's proof by showing that 10932 \| (2182 + 1) and remarked that (2182 + 1) is a factor of (2364 − 1).[14] It was also shown that it is possible to prove that 1093 is a Wieferich prime without using complex numbers contrary to the method used by Meissner,[15] although Meissner himself hinted at that he was aware of a proof without complex values.[12]: 665 The prime 3511 was first found to be a Wieferich prime by N. G. W. H. Beeger in 1922[16] and another proof of it being a Wieferich prime was published in 1965 by Guy.[17] In 1960, Kravitz[18] doubled a previous record set by Fröberg[19] and in 1961 Riesel extended the search to 500000 with the aid of BESK.[20] Around 1980, Lehmer was able to reach the search limit of 6×109.[21] This limit was extended to over 2.5×1015 in 2006,[22] finally reaching 3×1015. It is now known that if any other Wieferich primes exist, they must be greater than 6.7×1015.[23] In 2007–2016, a search for Wieferich primes was performed by the distributed computing project Wieferich@Home.[24] In 2011–2017, another search was performed by the PrimeGrid project, although later the work done in this project was claimed wasted.[25] While these projects reached search bounds above 1×1017, neither of them reported any sustainable results. In 2020, PrimeGrid started another project that searches for Wieferich and Wall–Sun–Sun primes simultaneously. The new project uses checksums to enable independent double-checking of each subinterval, thus minimizing the risk of missing an instance because of faulty hardware.[26] The project ended in December 2022, definitely proving that a third Wieferich prime must exceed 264 (about 18×1018).[27] It has been conjectured (as for Wilson primes) that infinitely many Wieferich primes exist, and that the number of Wieferich primes below x is approximately log(log(x)), which is a heuristic result that follows from the plausible assumption that for a prime p, the (p − 1)-th degree roots of unity modulo p2 are uniformly distributed in the multiplicative group of integers modulo p2.[28] Properties Connection with Fermat's Last Theorem The following theorem connecting Wieferich primes and Fermat's Last Theorem was proven by Wieferich in 1909:[10] Let p be prime, and let x, y, z be integers such that xp + yp + zp = 0. Furthermore, assume that p does not divide the product xyz. Then p is a Wieferich prime. The above case (where p does not divide any of x, y or z) is commonly known as the first case of Fermat's Last Theorem (FLTI)[29][30] and FLTI is said to fail for a prime p, if solutions to the Fermat equation exist for that p, otherwise FLTI holds for p.[31] In 1910, Mirimanoff expanded[32] the theorem by showing that, if the preconditions of the theorem hold true for some prime p, then p2 must also divide 3p − 1 − 1. Granville and Monagan further proved that p2 must actually divide mp − 1 − 1 for every prime m ≤ 89.[33] Suzuki extended the proof to all primes m ≤ 113.[34] Let Hp be a set of pairs of integers with 1 as their greatest common divisor, p being prime to x, y and x + y, (x + y)p−1 ≡ 1 (mod p2), (x + ξy) being the pth power of an ideal of K with ξ defined as cos 2π/p + i sin 2π/p. K = Q(ξ) is the field extension obtained by adjoining all polynomials in the algebraic number ξ to the field of rational numbers (such an extension is known as a number field or in this particular case, where ξ is a root of unity, a cyclotomic number field).[33]: 332 From uniqueness of factorization of ideals in Q(ξ) it follows that if the first case of Fermat's last theorem has solutions x, y, z then p divides x+y+z and (x, y), (y, z) and (z, x) are elements of Hp.[33]: 333 Granville and Monagan showed that (1, 1) ∈ Hp if and only if p is a Wieferich prime.[33]: 333 Connection with the abc conjecture and non-Wieferich primes A non-Wieferich prime is a prime p satisfying 2p − 1 ≢ 1 (mod p2). J. H. Silverman showed in 1988 that if the abc conjecture holds, then there exist infinitely many non-Wieferich primes.[35] More precisely he showed that the abc conjecture implies the existence of a constant only depending on α such that the number of non-Wieferich primes to base α with p less than or equal to a variable X is greater than log(X) as X goes to infinity.[36]: 227 Numerical evidence suggests that very few of the prime numbers in a given interval are Wieferich primes. The set of Wieferich primes and the set of non-Wieferich primes, sometimes denoted by W2 and W2c respectively,[37] are complementary sets, so if one of them is shown to be finite, the other one would necessarily have to be infinite, because both are proper subsets of the set of prime numbers. It was later shown that the existence of infinitely many non-Wieferich primes already follows from a weaker version of the abc conjecture, called the ABC-(k, ε) conjecture.[38] Additionally, the existence of infinitely many non-Wieferich primes would also follow if there exist infinitely many square-free Mersenne numbers[39] as well as if there exists a real number ξ such that the set {n ∈ N : λ(2n − 1) < 2 − ξ} is of density one, where the index of composition λ(n) of an integer n is defined as ${\tfrac {\log n}{\log \gamma (n)}}$ and $\gamma (n)=\prod _{p\mid n}p$, meaning $\gamma (n)$ gives the product of all prime factors of n.[37]: 4 Connection with Mersenne and Fermat primes It is known that the nth Mersenne number Mn = 2n − 1 is prime only if n is prime. Fermat's little theorem implies that if p > 2 is prime, then Mp−1 (= 2p − 1 − 1) is always divisible by p. Since Mersenne numbers of prime indices Mp and Mq are co-prime, A prime divisor p of Mq, where q is prime, is a Wieferich prime if and only if p2 divides Mq.[40] Thus, a Mersenne prime cannot also be a Wieferich prime. A notable open problem is to determine whether or not all Mersenne numbers of prime index are square-free. If q is prime and the Mersenne number Mq is not square-free, that is, there exists a prime p for which p2 divides Mq, then p is a Wieferich prime. Therefore, if there are only finitely many Wieferich primes, then there will be at most finitely many Mersenne numbers with prime index that are not square-free. Rotkiewicz showed a related result: if there are infinitely many square-free Mersenne numbers, then there are infinitely many non-Wieferich primes.[41] Similarly, if p is prime and p2 divides some Fermat number Fn = 22n + 1, then p must be a Wieferich prime.[42] In fact, there exists a natural number n and a prime p that p2 divides $\Phi _{n}(2)$ (where $\Phi _{n}(x)$ is the n-th cyclotomic polynomial) if and only if p is a Wieferich prime. For example, 10932 divides $\Phi _{364}(2)$, 35112 divides $\Phi _{1755}(2)$. Mersenne and Fermat numbers are just special situations of $\Phi _{n}(2)$. Thus, if 1093 and 3511 are only two Wieferich primes, then all $\Phi _{n}(2)$ are square-free except $\Phi _{364}(2)$ and $\Phi _{1755}(2)$ (In fact, when there exists a prime p which p2 divides some $\Phi _{n}(2)$, then it is a Wieferich prime); and clearly, if $\Phi _{n}(2)$ is a prime, then it cannot be Wieferich prime. (Any odd prime p divides only one $\Phi _{n}(2)$ and n divides p − 1, and if and only if the period length of 1/p in binary is n, then p divides $\Phi _{n}(2)$. Besides, if and only if p is a Wieferich prime, then the period length of 1/p and 1/p2 are the same (in binary). Otherwise, this is p times than that.) For the primes 1093 and 3511, it was shown that neither of them is a divisor of any Mersenne number with prime index nor a divisor of any Fermat number, because 364 and 1755 are neither prime nor powers of 2.[43] Connection with other equations Scott and Styer showed that the equation px – 2y = d has at most one solution in positive integers (x, y), unless when p4 \| 2ordp 2 – 1 if p ≢ 65 (mod 192) or unconditionally when p2 \| 2ordp 2 – 1, where ordp 2 denotes the multiplicative order of 2 modulo p.[44]: 215, 217–218 They also showed that a solution to the equation ±ax1 ± 2y1 = ±ax2 ± 2y2 = c must be from a specific set of equations but that this does not hold, if a is a Wieferich prime greater than 1.25 x 1015.[45]: 258 Binary periodicity of p − 1 Johnson observed[46] that the two known Wieferich primes are one greater than numbers with periodic binary expansions (1092 = 0100010001002=44416; 3510 = 1101101101102=66668). The Wieferich@Home project searched for Wieferich primes by testing numbers that are one greater than a number with a periodic binary expansion, but up to a "bit pseudo-length" of 3500 of the tested binary numbers generated by combination of bit strings with a bit length of up to 24 it has not found a new Wieferich prime.[47] Abundancy of p − 1 It has been noted (sequence A239875 in the OEIS) that the known Wieferich primes are one greater than mutually friendly numbers (the shared abundancy index being 112/39). Connection with pseudoprimes It was observed that the two known Wieferich primes are the square factors of all non-square free base-2 Fermat pseudoprimes up to 25×109.[48] Later computations showed that the only repeated factors of the pseudoprimes up to 1012 are 1093 and 3511.[49] In addition, the following connection exists: Let n be a base 2 pseudoprime and p be a prime divisor of n. If ${\tfrac {2^{n-1}-1}{n}}\not \equiv 0{\pmod {p}}$, then also ${\tfrac {2^{p-1}-1}{p}}\not \equiv 0{\pmod {p}}$.[31]: 378 Furthermore, if p is a Wieferich prime, then p2 is a Catalan pseudoprime. Connection with directed graphs For all primes p up to 100000, L(pn+1) = L(pn) only in two cases: L(10932) = L(1093) = 364 and L(35112) = L(3511) = 1755, where L(m) is the number of vertices in the cycle of 1 in the doubling diagram modulo m. Here the doubling diagram represents the directed graph with the non-negative integers less than m as vertices and with directed edges going from each vertex x to vertex 2x reduced modulo m.[50]: 74 It was shown, that for all odd prime numbers either L(pn+1) = p · L(pn) or L(pn+1) = L(pn).[50]: 75 Properties related to number fields It was shown that $\chi _{D_{0}}{\big (}p{\big )}=1$ and $\lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1$ if and only if 2p − 1 ≢ 1 (mod p2) where p is an odd prime and $D_{0}<0$ is the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {1-p^{2}}}{\big )}$. Furthermore, the following was shown: Let p be a Wieferich prime. If p ≡ 3 (mod 4), let $D_{0}<0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {1-p}}{\big )}$ and if p ≡ 1 (mod 4), let $D_{0}<0$ be the fundamental discriminant of the imaginary quadratic field $\mathbb {Q} {\big (}{\sqrt {4-p}}{\big )}$. Then $\chi _{D_{0}}{\big (}p{\big )}=1$ and $\lambda \,\!_{p}{\big (}\mathbb {Q} {\big (}{\sqrt {D_{0}}}{\big )}{\big )}=1$ (χ and λ in this context denote Iwasawa invariants).[51]: 27 Furthermore, the following result was obtained: Let q be an odd prime number, k and p are primes such that p = 2k + 1, k ≡ 3 (mod 4), p ≡ −1 (mod q), p ≢ −1 (mod q3) and the order of q modulo k is ${\tfrac {k-1}{2}}$. Assume that q divides h+, the class number of the real cyclotomic field $\mathbb {Q} {\big (}\zeta \,\!_{p}+\zeta \,\!_{p}^{-1}{\big )}$, the cyclotomic field obtained by adjoining the sum of a p-th root of unity and its reciprocal to the field of rational numbers. Then q is a Wieferich prime.[52]: 55 This also holds if the conditions p ≡ −1 (mod q) and p ≢ −1 (mod q3) are replaced by p ≡ −3 (mod q) and p ≢ −3 (mod q3) as well as when the condition p ≡ −1 (mod q) is replaced by p ≡ −5 (mod q) (in which case q is a Wall–Sun–Sun prime) and the incongruence condition replaced by p ≢ −5 (mod q3).[53]: 376 Generalizations Near-Wieferich primes A prime p satisfying the congruence 2(p−1)/2 ≡ ±1 + Ap (mod p2) with small \|A\| is commonly called a near-Wieferich prime (sequence A195988 in the OEIS).[28][54] Near-Wieferich primes with A = 0 represent Wieferich primes. Recent searches, in addition to their primary search for Wieferich primes, also tried to find near-Wieferich primes.[23][55] The following table lists all near-Wieferich primes with \|A\| ≤ 10 in the interval [1×109, 3×1015].[56] This search bound was reached in 2006 in a search effort by P. Carlisle, R. Crandall and M. Rodenkirch.[22][57] Bigger entries are by PrimeGrid. p1 or −1A 3520624567+1−6 46262476201+1+5 47004625957−1+1 58481216789−1+5 76843523891−1+1 1180032105761+1−6 12456646902457+1+2 134257821895921+1+10 339258218134349−1+2 2276306935816523−1−3 82687771042557349-1-10 3156824277937156367+1+7 The sign +1 or -1 above can be easily predicted by Euler's criterion (and the second supplement to the law of quadratic reciprocity). Dorais and Klyve[23] used a different definition of a near-Wieferich prime, defining it as a prime p with small value of $\left\|{\tfrac {\omega (p)}{p}}\right\|$ where $\omega (p)={\tfrac {2^{p-1}-1}{p}}\,{\bmod {\,}}p$ is the Fermat quotient of 2 with respect to p modulo p (the modulo operation here gives the residue with the smallest absolute value). The following table lists all primes p ≤ 6.7 × 1015 with $\left\|{\tfrac {\omega (p)}{p}}\right\|\leq 10^{-14}$. p$\omega (p)$$\left\|{\tfrac {\omega (p)}{p}}\right\|\times 10^{14}$ 109300 351100 2276306935816523+60.264 3167939147662997−170.537 3723113065138349−360.967 5131427559624857−360.702 5294488110626977−310.586 6517506365514181+580.890 The two notions of nearness are related as follows. If $2^{(p-1)/2}\equiv \pm 1+Ap{\pmod {p^{2}}}$, then by squaring, clearly $2^{p-1}\equiv 1\pm 2Ap{\pmod {p^{2}}}$. So if A had been chosen with $\|A\|$ small, then clearly $\|\!\pm 2A\|=2\|A\|$ is also (quite) small, and an even number. However, when $\omega (p)$ is odd above, the related A from before the last squaring was not "small". For example, with $p=3167939147662997$, we have $2^{(p-1)/2}\equiv -1-1583969573831490p{\pmod {p^{2}}}$ which reads extremely non-near, but after squaring this is $2^{p-1}\equiv 1-17p{\pmod {p^{2}}}$ which is a near-Wieferich by the second definition. Base-a Wieferich primes Main article: Fermat quotient A Wieferich prime base a is a prime p that satisfies ap − 1 ≡ 1 (mod p2),[8] with a less than p but greater than 1. Such a prime cannot divide a, since then it would also divide 1. It's a conjecture that for every natural number a, there are infinitely many Wieferich primes in base a. Bolyai showed that if p and q are primes, a is a positive integer not divisible by p and q such that ap−1 ≡ 1 (mod q), aq−1 ≡ 1 (mod p), then apq−1 ≡ 1 (mod pq). Setting p = q leads to ap2−1 ≡ 1 (mod p2).[58]: 284 It was shown that ap2−1 ≡ 1 (mod p2) if and only if ap−1 ≡ 1 (mod p2).[58]: 285–286 Known solutions of ap−1 ≡ 1 (mod p2) for small values of a are:[59] (checked up to 5 × 1013) a primes p such that ap − 1 = 1 (mod p2) OEIS sequence 12, 3, 5, 7, 11, 13, 17, 19, 23, 29, ... (All primes) A000040 21093, 3511, ... A001220 311, 1006003, ... A014127 41093, 3511, ... 52, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... A123692 666161, 534851, 3152573, ... A212583 75, 491531, ... A123693 83, 1093, 3511, ... 92, 11, 1006003, ... 103, 487, 56598313, ... A045616 1171, ... 122693, 123653, ... A111027 132, 863, 1747591, ... A128667 1429, 353, 7596952219, ... A234810 1529131, 119327070011, ... A242741 161093, 3511, ... 172, 3, 46021, 48947, 478225523351, ... A128668 185, 7, 37, 331, 33923, 1284043, ... A244260 193, 7, 13, 43, 137, 63061489, ... A090968 20281, 46457, 9377747, 122959073, ... A242982 212, ... 2213, 673, 1595813, 492366587, 9809862296159, ... A298951 2313, 2481757, 13703077, 15546404183, 2549536629329, ... A128669 245, 25633, ... 252, 20771, 40487, 53471161, 1645333507, 6692367337, 188748146801, ... 263, 5, 71, 486999673, 6695256707, ... A306255 2711, 1006003, ... 283, 19, 23, ... 292, ... 307, 160541, 94727075783, ... A306256 317, 79, 6451, 2806861, ... A331424 325, 1093, 3511, ... 332, 233, 47441, 9639595369, ... 3446145917691, ... 353, 1613, 3571, ... 3666161, 534851, 3152573, ... 372, 3, 77867, 76407520781, ... A331426 3817, 127, ... 398039, ... 4011, 17, 307, 66431, 7036306088681, ... 412, 29, 1025273, 138200401, ... A331427 4223, 719867822369, ... 435, 103, 13368932516573, ... 443, 229, 5851, ... 452, 1283, 131759, 157635607, ... 463, 829, ... 47... 487, 257, ... 492, 5, 491531, ... 507, ... For more information, see[60][61][62] and.[63] (Note that the solutions to a = bk is the union of the prime divisors of k which does not divide b and the solutions to a = b) The smallest solutions of np−1 ≡ 1 (mod p2) are 2, 1093, 11, 1093, 2, 66161, 5, 3, 2, 3, 71, 2693, 2, 29, 29131, 1093, 2, 5, 3, 281, 2, 13, 13, 5, 2, 3, 11, 3, 2, 7, 7, 5, 2, 46145917691, 3, 66161, 2, 17, 8039, 11, 2, 23, 5, 3, 2, 3, ... (The next term > 4.9×1013) (sequence A039951 in the OEIS) There are no known solutions of np−1 ≡ 1 (mod p2) for n = 47, 72, 186, 187, 200, 203, 222, 231, 304, 311, 335, 355, 435, 454, 546, 554, 610, 639, 662, 760, 772, 798, 808, 812, 858, 860, 871, 983, 986, 1002, 1023, 1130, 1136, 1138, .... It is a conjecture that there are infinitely many solutions of ap−1 ≡ 1 (mod p2) for every natural number a. The bases b < p2 which p is a Wieferich prime are (for b > p2, the solutions are just shifted by k·p2 for k > 0), and there are p − 1 solutions < p2 of p and the set of the solutions congruent to p are {1, 2, 3, ..., p − 1}) (sequence A143548 in the OEIS) p values of b < p2 2 1 3 1, 8 5 1, 7, 18, 24 7 1, 18, 19, 30, 31, 48 11 1, 3, 9, 27, 40, 81, 94, 112, 118, 120 13 1, 19, 22, 23, 70, 80, 89, 99, 146, 147, 150, 168 17 1, 38, 40, 65, 75, 110, 131, 134, 155, 158, 179, 214, 224, 249, 251, 288 19 1, 28, 54, 62, 68, 69, 99, 116, 127, 234, 245, 262, 292, 293, 299, 307, 333, 360 23 1, 28, 42, 63, 118, 130, 170, 177, 195, 255, 263, 266, 274, 334, 352, 359, 399, 411, 466, 487, 501, 528 29 1, 14, 41, 60, 63, 137, 190, 196, 221, 236, 267, 270, 374, 416, 425, 467, 571, 574, 605, 620, 645, 651, 704, 778, 781, 800, 827, 840 The least base b > 1 which prime(n) is a Wieferich prime are 5, 8, 7, 18, 3, 19, 38, 28, 28, 14, 115, 18, 51, 19, 53, 338, 53, 264, 143, 11, 306, 31, 99, 184, 53, 181, 43, 164, 96, 68, 38, 58, 19, 328, 313, 78, 226, 65, 253, 259, 532, 78, 176, 276, 143, 174, 165, 69, 330, 44, 33, 332, 94, 263, 48, 79, 171, 747, 731, 20, ... (sequence A039678 in the OEIS) We can also consider the formula $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$, (because of the generalized Fermat little theorem, $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$ is true for all prime p and all natural number a such that both a and a + 1 are not divisible by p). It's a conjecture that for every natural number a, there are infinitely many primes such that $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$. Known solutions for small a are: (checked up to 4 × 1011) [64] $a$ primes $p$ such that $(a+1)^{p-1}-a^{p-1}\equiv 0{\pmod {p^{2}}}$ 1 1093, 3511, ... 2 23, 3842760169, 41975417117, ... 3 5, 250829, ... 4 3, 67, ... 5 3457, 893122907, ... 6 72673, 1108905403, 2375385997, ... 7 13, 819381943, ... 8 67, 139, 499, 26325777341, ... 9 67, 887, 9257, 83449, 111539, 31832131, ... 10 ... 11 107, 4637, 239357, ... 12 5, 11, 51563, 363901, 224189011, ... 13 3, ... 14 11, 5749, 17733170113, 140328785783, ... 15 292381, ... 16 4157, ... 17 751, 46070159, ... 18 7, 142671309349, ... 19 17, 269, ... 20 29, 162703, ... 21 5, 2711, 104651, 112922981, 331325567, 13315963127, ... 22 3, 7, 13, 94447, 1198427, 23536243, ... 23 43, 179, 1637, 69073, ... 24 7, 353, 402153391, ... 25 43, 5399, 21107, 35879, ... 26 7, 131, 653, 5237, 97003, ... 27 2437, 1704732131, ... 28 5, 617, 677, 2273, 16243697, ... 29 73, 101, 6217, ... 30 7, 11, 23, 3301, 48589, 549667, ... 31 3, 41, 416797, ... 32 95989, 2276682269, ... 33 139, 1341678275933, ... 34 83, 139, ... 35 ... 36 107, 137, 613, 2423, 74304856177, ... 37 5, ... 38 167, 2039, ... 39 659, 9413, ... 40 3, 23, 21029249, ... 41 31, 71, 1934399021, 474528373843, ... 42 4639, 1672609, ... 43 31, 4962186419, ... 44 36677, 17786501, ... 45 241, 26120375473, ... 46 5, 13877, ... 47 13, 311, 797, 906165497, ... 48 ... 49 3, 13, 2141, 281833, 1703287, 4805298913, ... 50 2953, 22409, 99241, 5427425917, ... Wieferich pairs Main article: Wieferich pair A Wieferich pair is a pair of primes p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) so that a Wieferich prime p ≡ 1 (mod 4) will form such a pair (p, 2): the only known instance in this case is p = 1093. There are only 7 known Wieferich pairs.[65] (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787) (sequence OEIS: A282293 in OEIS) Wieferich sequence Start with a(1) any natural number (>1), a(n) = the smallest prime p such that (a(n − 1))p − 1 = 1 (mod p2) but p2 does not divide a(n − 1) − 1 or a(n − 1) + 1. (If p2 divides a(n − 1) − 1 or a(n − 1) + 1, then the solution is a trivial solution) It is a conjecture that every natural number k = a(1) > 1 makes this sequence become periodic, for example, let a(1) = 2: 2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (sequence A359952 in the OEIS) Let a(1) = 83: 83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. Let a(1) = 59 (a longer sequence): 59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ..., it also gets 5. However, there are many values of a(1) with unknown status, for example, let a(1) = 3: 3, 11, 71, 47, ? (There are no known Wieferich primes in base 47). Let a(1) = 14: 14, 29, ? (There are no known Wieferich prime in base 29 except 2, but 22 = 4 divides 29 − 1 = 28) Let a(1) = 39 (a longer sequence): 39, 8039, 617, 101, 1050139, 29, ? (It also gets 29) It is unknown that values for a(1) > 1 exist such that the resulting sequence does not eventually become periodic. When a(n − 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown) Wieferich numbers A Wieferich number is an odd natural number n satisfying the congruence 2φ(n) ≡ 1 (mod n2), where φ denotes the Euler's totient function (according to Euler's theorem, 2φ(n) ≡ 1 (mod n) for every odd natural number n). If Wieferich number n is prime, then it is a Wieferich prime. The first few Wieferich numbers are: 1, 1093, 3279, 3511, 7651, 10533, 14209, 17555, 22953, 31599, 42627, 45643, 52665, 68859, 94797, 99463, ... (sequence A077816 in the OEIS) It can be shown that if there are only finitely many Wieferich primes, then there are only finitely many Wieferich numbers. In particular, if the only Wieferich primes are 1093 and 3511, then there exist exactly 104 Wieferich numbers, which matches the number of Wieferich numbers currently known.[2] More generally, a natural number n is a Wieferich number to base a, if aφ(n) ≡ 1 (mod n2).[66]: 31 Another definition specifies a Wieferich number as odd natural number n such that n and ${\tfrac {2^{m}-1}{n}}$ are not coprime, where m is the multiplicative order of 2 modulo n. The first of these numbers are:[67] 21, 39, 55, 57, 105, 111, 147, 155, 165, 171, 183, 195, 201, 203, 205, 219, 231, 237, 253, 273, 285, 291, 301, 305, 309, 327, 333, 355, 357, 385, 399, ... (sequence A182297 in the OEIS) As above, if Wieferich number q is prime, then it is a Wieferich prime. Weak Wieferich prime A weak Wieferich prime to base a is a prime p satisfies the condition ap ≡ a (mod p2) Every Wieferich prime to base a is also a weak Wieferich prime to base a. If the base a is squarefree, then a prime p is a weak Wieferich prime to base a if and only if p is a Wieferich prime to base a. Smallest weak Wieferich prime to base n are (start with n = 0) 2, 2, 1093, 11, 2, 2, 66161, 5, 2, 2, 3, 71, 2, 2, 29, 29131, 2, 2, 3, 3, 2, 2, 13, 13, 2, 2, 3, 3, 2, 2, 7, 7, 2, 2, 46145917691, 3, 2, 2, 17, 8039, 2, 2, 23, 5, 2, 2, 3, ... Wieferich prime with order n For integer n ≥2, a Wieferich prime to base a with order n is a prime p satisfies the condition ap−1 ≡ 1 (mod pn) Clearly, a Wieferich prime to base a with order n is also a Wieferich prime to base a with order m for all 2 ≤ m ≤ n, and Wieferich prime to base a with order 2 is equivalent to Wieferich prime to base a, so we can only consider the n ≥ 3 case. However, there are no known Wieferich prime to base 2 with order 3. The first base with known Wieferich prime with order 3 is 9, where 2 is a Wieferich prime to base 9 with order 3. Besides, both 5 and 113 are Wieferich prime to base 68 with order 3. Lucas–Wieferich primes Let P and Q be integers. The Lucas sequence of the first kind associated with the pair (P, Q) is defined by ${\begin{aligned}U_{0}(P,Q)&=0,\\U_{1}(P,Q)&=1,\\U_{n}(P,Q)&=P\cdot U_{n-1}(P,Q)-Q\cdot U_{n-2}(P,Q)\end{aligned}}$ for all $n\geq 2$. A Lucas–Wieferich prime associated with (P, Q) is a prime p such that Up−ε(P, Q) ≡ 0 (mod p2), where ε equals the Legendre symbol $\left({\tfrac {P^{2}-4Q}{p}}\right)$. All Wieferich primes are Lucas–Wieferich primes associated with the pair (3, 2).[3]: 2088 Fibonacci–Wieferich primes Let Q = −1. For every natural number P, the Lucas–Wieferich primes associated with (P, −1) are called P-Fibonacci–Wieferich primes or P-Wall–Sun–Sun primes. If P = 1, they are called Fibonacci–Wieferich primes. If P = 2, they are called Pell–Wieferich primes. For example, 241 is a Lucas–Wieferich prime associated with (3, −1), so it is a 3-Fibonacci–Wieferich prime or 3-Wall–Sun–Sun prime. In fact, 3 is a P-Fibonacci–Wieferich prime if and only if P congruent to 0, 4, or 5 (mod 9), which is analogous to the statement for traditional Wieferich primes that 3 is a base-n Wieferich prime if and only if n congruent to 1 or 8 (mod 9). Wieferich places Let K be a global field, i.e. a number field or a function field in one variable over a finite field and let E be an elliptic curve. If v is a non-archimedean place of norm qv of K and a ∈ K, with v(a) = 0 then v(aqv − 1 − 1) ≥ 1. v is called a Wieferich place for base a, if v(aqv − 1 − 1) > 1, an elliptic Wieferich place for base P ∈ E, if NvP ∈ E2 and a strong elliptic Wieferich place for base P ∈ E if nvP ∈ E2, where nv is the order of P modulo v and Nv gives the number of rational points (over the residue field of v) of the reduction of E at v.[68]: 206 See also • Wall–Sun–Sun prime – another type of prime number which in the broadest sense also resulted from the study of FLT • Wolstenholme prime – another type of prime number which in the broadest sense also resulted from the study of FLT • Wilson prime • Table of congruences – lists other congruences satisfied by prime numbers • PrimeGrid – primes search project • BOINC • Distributed computing References 1. Franco, Z.; Pomerance, C. 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(2006), "Relaxations of the ABC Conjecture using integer k'th roots" (PDF), New Zealand J. Math., 35 (2): 121–136 39. Ribenboim, P. (1979). 13 Lectures on Fermat's Last Theorem. New York: Springer. p. 154. ISBN 978-0-387-90432-0. 40. Mersenne Primes: Conjectures and Unsolved Problems 41. Rotkiewicz, A. (1965). "Sur les nombres de Mersenne dépourvus de diviseurs carrés et sur les nombres naturels n, tels que n2\|2n − 2". Mat. Vesnik (in French). 2 (17): 78–80. 42. Ribenboim, Paulo (1991), The little book of big primes, New York: Springer, p. 64, ISBN 978-0-387-97508-5 43. Bray, H. G.; Warren, L. J. (1967), "On the square-freeness of Fermat and Mersenne numbers", Pacific J. Math., 22 (3): 563–564, doi:10.2140/pjm.1967.22.563, MR 0220666, Zbl 0149.28204 44. Scott, R.; Styer, R. (April 2004). "On px − qy = c and related three term exponential Diophantine equations with prime bases". Journal of Number Theory. 105 (2): 212–234. doi:10.1016/j.jnt.2003.11.008. 45. Scott, R.; Styer, R. (2006). "On the generalized Pillai equation ±ax±by = c". Journal of Number Theory. 118 (2): 236–265. doi:10.1016/j.jnt.2005.09.001. 46. Wells Johnson (1977), "On the nonvanishing of Fermat quotients (mod p)", J. Reine Angew. Math., 292: 196–200 47. Dobeš, Jan; Kureš, Miroslav (2010). "Search for Wieferich primes through the use of periodic binary strings". Serdica Journal of Computing. 4: 293–300. Zbl 1246.11019. 48. Ribenboim, P. (2004). "Chapter 2. How to Recognize Whether a Natural Number is a Prime" (PDF). The Little Book of Bigger Primes. New York: Springer-Verlag. p. 99. ISBN 978-0-387-20169-6. 49. Pinch, R. G. E. (2000). The Pseudoprimes up to 1013. Lecture Notes in Computer Science. Vol. 1838. pp. 459–473. doi:10.1007/10722028_30. ISBN 978-3-540-67695-9. 50. Ehrlich, A. (1994), "Cycles in Doubling Diagrams mod m" (PDF), The Fibonacci Quarterly, 32 (1): 74–78. 51. Byeon, D. (2006), "Class numbers, Iwasawa invariants and modular forms" (PDF), Trends in Mathematics, 9 (1): 25–29, archived from the original (PDF) on 2012-04-26, retrieved 2012-09-05 52. Jakubec, S. (1995), "Connection between the Wieferich congruence and divisibility of h+" (PDF), Acta Arithmetica, 71 (1): 55–64, doi:10.4064/aa-71-1-55-64 53. Jakubec, S. (1998), "On divisibility of the class number h+ of the real cyclotomic fields of prime degree l" (PDF), Mathematics of Computation, 67 (221): 369–398, doi:10.1090/s0025-5718-98-00916-8 54. Joshua Knauer; Jörg Richstein (2005), "The continuing search for Wieferich primes" (PDF), Mathematics of Computation, 74 (251): 1559–1563, Bibcode:2005MaCom..74.1559K, doi:10.1090/S0025-5718-05-01723-0. 55. About project Wieferich@Home 56. PrimeGrid, Wieferich & near Wieferich primes p < 11e15 Archived 2012-10-18 at the Wayback Machine 57. Ribenboim, Paulo (2000), My numbers, my friends: popular lectures on number theory, New York: Springer, pp. 213–229, ISBN 978-0-387-98911-2 58. Kiss, E.; Sándor, J. (2004). "On a congruence by János Bolyai, connected with pseudoprimes" (PDF). Mathematica Pannonica. 15 (2): 283–288. 59. Fermat Quotient at The Prime Glossary 60. "Wieferich primes to base 1052". 61. "Wieferich primes to base 10125". 62. "Fermat quotients qp(a) that are divisible by p". www1.uni-hamburg.de. 2014-08-09. Archived from the original on 2014-08-09. Retrieved 2019-09-18. 63. "Wieferich primes with level ≥ 3". 64. "Solution of (a + 1)p−1 − ap−1 ≡ 0 (mod p2)". 65. Weisstein, Eric W. "Double Wieferich Prime Pair". MathWorld. 66. Agoh, T.; Dilcher, K.; Skula, L. (1997), "Fermat Quotients for Composite Moduli", Journal of Number Theory, 66 (1): 29–50, doi:10.1006/jnth.1997.2162 67. Müller, H. (2009). "Über Periodenlängen und die Vermutungen von Collatz und Crandall". Mitteilungen der Mathematischen Gesellschaft in Hamburg (in German). 28: 121–130. 68. Voloch, J. F. (2000), "Elliptic Wieferich Primes", Journal of Number Theory, 81 (2): 205–209, doi:10.1006/jnth.1999.2471 Further reading • Haussner, R. (1926), "Über die Kongruenzen 2p−1 − 1 ≡ 0 (mod p2) für die Primzahlen p=1093 und 3511", Archiv for Mathematik og Naturvidenskab (in German), 39 (5): 7, JFM 52.0141.06, DNB 363953469 • Haussner, R. (1927), "Über numerische Lösungen der Kongruenz up−1 − 1 ≡ 0 (mod p2)", Journal für die Reine und Angewandte Mathematik (in German), 1927 (156): 223–226, doi:10.1515/crll.1927.156.223, S2CID 117969297 • Ribenboim, P. (1979), Thirteen lectures on Fermat's Last Theorem, Springer-Verlag, pp. 139, 151, ISBN 978-0-387-90432-0 • Guy, Richard K. (2004), Unsolved Problems in Number Theory (3rd ed.), Springer Verlag, p. 14, ISBN 978-0-387-20860-2 • Crandall, R. E.; Pomerance, C. (2005), Prime numbers: a computational perspective (PDF), Springer Science+Business Media, pp. 31–32, ISBN 978-0-387-25282-7 • Ribenboim, P. (1996), The new book of prime number records, New York: Springer-Verlag, pp. 333–346, ISBN 978-0-387-94457-9 External links • Weisstein, Eric W. "Wieferich prime". MathWorld. • Fermat-/Euler-quotients (ap−1 − 1)/pk with arbitrary k • A note on the two known Wieferich primes • PrimeGrid's Wieferich Prime Search project page Prime number classes By formula • Fermat (22n + 1) • Mersenne (2p − 1) • Double Mersenne (22p−1 − 1) • Wagstaff (2p + 1)/3 • Proth (k·2n + 1) • Factorial (n! ± 1) • Primorial (pn# ± 1) • Euclid (pn# + 1) • Pythagorean (4n + 1) • Pierpont (2m·3n + 1) • Quartan (x4 + y4) • Solinas (2m ± 2n ± 1) • Cullen (n·2n + 1) • Woodall (n·2n − 1) • Cuban (x3 − y3)/(x − y) • Leyland (xy + yx) • Thabit (3·2n − 1) • Williams ((b−1)·bn − 1) • Mills (⌊A3n⌋) By integer sequence • Fibonacci • Lucas • Pell • Newman–Shanks–Williams • Perrin • Partitions • Bell • Motzkin By property • Wieferich (pair) • Wall–Sun–Sun • Wolstenholme • Wilson • Lucky • Fortunate • Ramanujan • Pillai • Regular • Strong • Stern • Supersingular (elliptic curve) • Supersingular (moonshine theory) • Good • Super • Higgs • Highly cototient • Unique Base-dependent • Palindromic • Emirp • Repunit (10n − 1)/9 • Permutable • Circular • Truncatable • Minimal • Delicate • Primeval • Full reptend • Unique • Happy • Self • Smarandache–Wellin • Strobogrammatic • Dihedral • Tetradic Patterns • Twin (p, p + 2) • Bi-twin chain (n ± 1, 2n ± 1, 4n ± 1, …) • Triplet (p, p + 2 or p + 4, p + 6) • Quadruplet (p, p + 2, p + 6, p + 8) • k-tuple • Cousin (p, p + 4) • Sexy (p, p + 6) • Chen • Sophie Germain/Safe (p, 2p + 1) • Cunningham (p, 2p ± 1, 4p ± 3, 8p ± 7, ...) • Arithmetic progression (p + a·n, n = 0, 1, 2, 3, ...) • Balanced (consecutive p − n, p, p + n) By size • Mega (1,000,000+ digits) • Largest known • list Complex numbers • Eisenstein prime • Gaussian prime Composite numbers • Pseudoprime • Catalan • Elliptic • Euler • Euler–Jacobi • Fermat • Frobenius • Lucas • Somer–Lucas • Strong • Carmichael number • Almost prime • Semiprime • Sphenic number • Interprime • Pernicious Related topics • Probable prime • Industrial-grade prime • Illegal prime • Formula for primes • Prime gap First 60 primes • 2 • 3 • 5 • 7 • 11 • 13 • 17 • 19 • 23 • 29 • 31 • 37 • 41 • 43 • 47 • 53 • 59 • 61 • 67 • 71 • 73 • 79 • 83 • 89 • 97 • 101 • 103 • 107 • 109 • 113 • 127 • 131 • 137 • 139 • 149 • 151 • 157 • 163 • 167 • 173 • 179 • 181 • 191 • 193 • 197 • 199 • 211 • 223 • 227 • 229 • 233 • 239 • 241 • 251 • 257 • 263 • 269 • 271 • 277 • 281 List of prime numbers
Wieferich pair In mathematics, a Wieferich pair is a pair of prime numbers p and q that satisfy pq − 1 ≡ 1 (mod q2) and qp − 1 ≡ 1 (mod p2) Wieferich pairs are named after German mathematician Arthur Wieferich. Wieferich pairs play an important role in Preda Mihăilescu's 2002 proof[1] of Mihăilescu's theorem (formerly known as Catalan's conjecture).[2] Known Wieferich pairs There are only 7 Wieferich pairs known:[3][4] (2, 1093), (3, 1006003), (5, 1645333507), (5, 188748146801), (83, 4871), (911, 318917), and (2903, 18787). (sequence OEIS: A124121 and OEIS: A124122 in OEIS) Wieferich triple A Wieferich triple is a triple of prime numbers p, q and r that satisfy pq − 1 ≡ 1 (mod q2), qr − 1 ≡ 1 (mod r2), and rp − 1 ≡ 1 (mod p2). There are 17 known Wieferich triples: (2, 1093, 5), (2, 3511, 73), (3, 11, 71), (3, 1006003, 3188089), (5, 20771, 18043), (5, 20771, 950507), (5, 53471161, 193), (5, 6692367337, 1601), (5, 6692367337, 1699), (5, 188748146801, 8807), (13, 863, 23), (17, 478225523351, 2311), (41, 138200401, 2953), (83, 13691, 821), (199, 1843757, 2251), (431, 2393, 54787), and (1657, 2281, 1667). (sequences OEIS: A253683, OEIS: A253684 and OEIS: A253685 in OEIS) Barker sequence Barker sequence or Wieferich n-tuple is a generalization of Wieferich pair and Wieferich triple. It is primes (p1, p2, p3, ..., pn) such that p1p2 − 1 ≡ 1 (mod p22), p2p3 − 1 ≡ 1 (mod p32), p3p4 − 1 ≡ 1 (mod p42), ..., pn−1pn − 1 ≡ 1 (mod pn2), pnp1 − 1 ≡ 1 (mod p12).[5] For example, (3, 11, 71, 331, 359) is a Barker sequence, or a Wieferich 5-tuple; (5, 188748146801, 453029, 53, 97, 76704103313, 4794006457, 12197, 3049, 41) is a Barker sequence, or a Wieferich 10-tuple. For the smallest Wieferich n-tuple, see OEIS: A271100, for the ordered set of all Wieferich tuples, see OEIS: A317721. Wieferich sequence Wieferich sequence is a special type of Barker sequence. Every integer k>1 has its own Wieferich sequence. To make a Wieferich sequence of an integer k>1, start with a(1)=k, a(n) = the smallest prime p such that a(n-1)p-1 = 1 (mod p) but a(n-1) ≠ 1 or -1 (mod p). It is a conjecture that every integer k>1 has a periodic Wieferich sequence. For example, the Wieferich sequence of 2: 2, 1093, 5, 20771, 18043, 5, 20771, 18043, 5, ..., it gets a cycle: {5, 20771, 18043}. (a Wieferich triple) The Wieferich sequence of 83: 83, 4871, 83, 4871, 83, 4871, 83, ..., it gets a cycle: {83, 4871}. (a Wieferich pair) The Wieferich sequence of 59: (this sequence needs more terms to be periodic) 59, 2777, 133287067, 13, 863, 7, 5, 20771, 18043, 5, ... it also gets 5. However, there are many values of a(1) with unknown status. For example, the Wieferich sequence of 3: 3, 11, 71, 47, ? (There are no known Wieferich primes in base 47). The Wieferich sequence of 14: 14, 29, ? (There are no known Wieferich primes in base 29 except 2, but 22 = 4 divides 29 - 1 = 28) The Wieferich sequence of 39: 39, 8039, 617, 101, 1050139, 29, ? (It also gets 29) It is unknown that values for k exist such that the Wieferich sequence of k does not become periodic. Eventually, it is unknown that values for k exist such that the Wieferich sequence of k is finite. When a(n - 1)=k, a(n) will be (start with k = 2): 1093, 11, 1093, 20771, 66161, 5, 1093, 11, 487, 71, 2693, 863, 29, 29131, 1093, 46021, 5, 7, 281, ?, 13, 13, 25633, 20771, 71, 11, 19, ?, 7, 7, 5, 233, 46145917691, 1613, 66161, 77867, 17, 8039, 11, 29, 23, 5, 229, 1283, 829, ?, 257, 491531, ?, ... (For k = 21, 29, 47, 50, even the next value is unknown) See also • Wieferich prime • Fermat quotient References 1. Preda Mihăilescu (2004). "Primary Cyclotomic Units and a Proof of Catalan's Conjecture". J. Reine Angew. Math. 2004 (572): 167–195. doi:10.1515/crll.2004.048. MR 2076124. 2. Jeanine Daems A Cyclotomic Proof of Catalan's Conjecture. 3. Weisstein, Eric W. "Double Wieferich Prime Pair". MathWorld. 4. OEIS: A124121, For example, currently there are two known double Wieferich prime pairs (p, q) with q = 5: (1645333507, 5) and (188748146801, 5). 5. List of all known Barker sequence Further reading • Bilu, Yuri F. (2004). "Catalan's conjecture (after Mihăilescu)". Astérisque. 294: vii, 1–26. Zbl 1094.11014. • Ernvall, Reijo; Metsänkylä, Tauno (1997). "On the p-divisibility of Fermat quotients". Math. Comp. 66 (219): 1353–1365. Bibcode:1997MaCom..66.1353E. doi:10.1090/S0025-5718-97-00843-0. MR 1408373. Zbl 0903.11002. • Steiner, Ray (1998). "Class number bounds and Catalan's equation". Math. Comp. 67 (223): 1317–1322. Bibcode:1998MaCom..67.1317S. doi:10.1090/S0025-5718-98-00966-1. MR 1468945. Zbl 0897.11009.
Wiener's Tauberian theorem In mathematical analysis, Wiener's tauberian theorem is any of several related results proved by Norbert Wiener in 1932.[1] They provide a necessary and sufficient condition under which any function in $L^{1}$ or $L^{2}$ can be approximated by linear combinations of translations of a given function.[2] Informally, if the Fourier transform of a function $f$ vanishes on a certain set $Z$, the Fourier transform of any linear combination of translations of $f$ also vanishes on $Z$. Therefore, the linear combinations of translations of $f$ cannot approximate a function whose Fourier transform does not vanish on $Z$. Wiener's theorems make this precise, stating that linear combinations of translations of $f$ are dense if and only if the zero set of the Fourier transform of $f$ is empty (in the case of $L^{1}$) or of Lebesgue measure zero (in the case of $L^{2}$). Gelfand reformulated Wiener's theorem in terms of commutative C-algebras, when it states that the spectrum of the $L^{1}$ group ring $L^{1}(\mathbb {R} )$ of the group $\mathbb {R} $ of real numbers is the dual group of $\mathbb {R} $. A similar result is true when $\mathbb {R} $ is replaced by any locally compact abelian group. The condition in $L^{1}$ Let $f\in L^{1}(\mathbb {R} $ be an integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{1}(\mathbb {R} )$ if and only if the Fourier transform of $f$ has no real zeros. Tauberian reformulation The following statement is equivalent to the previous result, and explains why Wiener's result is a Tauberian theorem: Suppose the Fourier transform of $f\in L^{1}$ has no real zeros, and suppose the convolution $fh$ tends to zero at infinity for some $h\in L^{\infty }$. Then the convolution $gh$ tends to zero at infinity for any $g\in L^{1}$. More generally, if $\lim _{x\to \infty }(fh)(x)=A\int f(x)\,dx$ for some $f\in L^{1}$ the Fourier transform of which has no real zeros, then also $\lim _{x\to \infty }(gh)(x)=A\int g(x)\,dx$ for any $g\in L^{1}$. Discrete version Wiener's theorem has a counterpart in $l^{1}(\mathbb {Z} )$: the span of the translations of $f\in l^{1}(\mathbb {Z} )$ is dense if and only if the Fourier transform $\varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }\,$ has no real zeros. The following statements are equivalent version of this result: • Suppose the Fourier transform of $f\in l^{1}(\mathbb {Z} )$ has no real zeros, and for some bounded sequence $h$ the convolution $fh$ tends to zero at infinity. Then $g*h$ also tends to zero at infinity for any $g\in l^{1}(\mathbb {Z} )$. • Let $\varphi $ be a function on the unit circle with absolutely convergent Fourier series. Then $1/\varphi $ has absolutely convergent Fourier series if and only if $\varphi $ has no zeros. Gelfand (1941a, 1941b) showed that this is equivalent to the following property of the Wiener algebra $A(\mathbb {T} )$, which he proved using the theory of Banach algebras, thereby giving a new proof of Wiener's result: • The maximal ideals of $A(\mathbb {T} )$ are all of the form $M_{x}=\left\{f\in A(\mathbb {T} )\mid f(x)=0\right\},\quad x\in \mathbb {T} .$ The condition in L2 Let $f\in L^{2}(\mathbb {R} )$ be a square-integrable function. The span of translations $f_{a}(x)=f(x+a)$ is dense in $L^{2}(\mathbb {R} )$ if and only if the real zeros of the Fourier transform of $f$ form a set of zero Lebesgue measure. The parallel statement in $l^{2}(\mathbb {Z} )$ is as follows: the span of translations of a sequence $f\in l^{2}(\mathbb {Z} )$ is dense if and only if the zero set of the Fourier transform $\varphi (\theta )=\sum _{n\in \mathbb {Z} }f(n)e^{-in\theta }$ has zero Lebesgue measure. Notes 1. See Wiener (1932). 2. see Rudin (1991). References • Gelfand, I. (1941a), "Normierte Ringe", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 3–24, MR 0004726 • Gelfand, I. (1941b), "Über absolut konvergente trigonometrische Reihen und Integrale", Rec. Math. (Mat. Sbornik), Nouvelle Série, 9 (51): 51–66, MR 0004727 • Rudin, W. (1991), Functional analysis, International Series in Pure and Applied Mathematics, New York: McGraw-Hill, Inc., ISBN 0-07-054236-8, MR 1157815 • Wiener, N. (1932), "Tauberian Theorems", Annals of Mathematics, 33 (1): 1–100, doi:10.2307/1968102, JSTOR 1968102 External links • Shtern, A.I. (2001) [1994], "Wiener Tauberian theorem", Encyclopedia of Mathematics, EMS Press
Wiener–Hopf method The Wiener–Hopf method is a mathematical technique widely used in applied mathematics. It was initially developed by Norbert Wiener and Eberhard Hopf as a method to solve systems of integral equations, but has found wider use in solving two-dimensional partial differential equations with mixed boundary conditions on the same boundary. In general, the method works by exploiting the complex-analytical properties of transformed functions. Typically, the standard Fourier transform is used, but examples exist using other transforms, such as the Mellin transform. In general, the governing equations and boundary conditions are transformed and these transforms are used to define a pair of complex functions (typically denoted with '+' and '−' subscripts) which are respectively analytic in the upper and lower halves of the complex plane, and have growth no faster than polynomials in these regions. These two functions will also coincide on some region of the complex plane, typically, a thin strip containing the real line. Analytic continuation guarantees that these two functions define a single function analytic in the entire complex plane, and Liouville's theorem implies that this function is an unknown polynomial, which is often zero or constant. Analysis of the conditions at the edges and corners of the boundary allows one to determine the degree of this polynomial. Wiener–Hopf decomposition The key step in many Wiener–Hopf problems is to decompose an arbitrary function $\Phi $ into two functions $\Phi _{\pm }$ with the desired properties outlined above. In general, this can be done by writing $\Phi _{+}(\alpha )={\frac {1}{2\pi i}}\int _{C_{1}}\Phi (z){\frac {dz}{z-\alpha }}$ and $\Phi _{-}(\alpha )=-{\frac {1}{2\pi i}}\int _{C_{2}}\Phi (z){\frac {dz}{z-\alpha }},$ where the contours $C_{1}$ and $C_{2}$ are parallel to the real line, but pass above and below the point $z=\alpha $, respectively. Similarly, arbitrary scalar functions may be decomposed into a product of +/− functions, i.e. $K(\alpha )=K_{+}(\alpha )K_{-}(\alpha )$, by first taking the logarithm, and then performing a sum decomposition. Product decompositions of matrix functions (which occur in coupled multi-modal systems such as elastic waves) are considerably more problematic since the logarithm is not well defined, and any decomposition might be expected to be non-commutative. A small subclass of commutative decompositions were obtained by Khrapkov, and various approximate methods have also been developed. Example Consider the linear partial differential equation ${\boldsymbol {L}}_{xy}f(x,y)=0,$ where ${\boldsymbol {L}}_{xy}$ is a linear operator which contains derivatives with respect to x and y, subject to the mixed conditions on y = 0, for some prescribed function g(x), $f=g(x){\text{ for }}x\leq 0,\quad f_{y}=0{\text{ when }}x>0$ and decay at infinity i.e. f → 0 as ${\boldsymbol {x}}\rightarrow \infty $. Taking a Fourier transform with respect to x results in the following ordinary differential equation ${\boldsymbol {L}}_{y}{\widehat {f}}(k,y)-P(k,y){\widehat {f}}(k,y)=0,$ where ${\boldsymbol {L}}_{y}$ is a linear operator containing y derivatives only, P(k,y) is a known function of y and k and ${\widehat {f}}(k,y)=\int _{-\infty }^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x.$ If a particular solution of this ordinary differential equation which satisfies the necessary decay at infinity is denoted F(k,y), a general solution can be written as ${\widehat {f}}(k,y)=C(k)F(k,y),$ where C(k) is an unknown function to be determined by the boundary conditions on y=0. The key idea is to split ${\widehat {f}}$ into two separate functions, ${\widehat {f}}_{+}$ and ${\widehat {f}}_{-}$ which are analytic in the lower- and upper-halves of the complex plane, respectively, ${\widehat {f}}_{+}(k,y)=\int _{0}^{\infty }f(x,y)e^{-ikx}\,{\textrm {d}}x,$ ${\widehat {f}}_{-}(k,y)=\int _{-\infty }^{0}f(x,y)e^{-ikx}\,{\textrm {d}}x.$ The boundary conditions then give ${\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}_{-}(k,0)+{\widehat {f}}_{+}(k,0)={\widehat {f}}(k,0)=C(k)F(k,0)$ and, on taking derivatives with respect to $y$, ${\widehat {f}}'_{-}(k,0)={\widehat {f}}'_{-}(k,0)+{\widehat {f}}'_{+}(k,0)={\widehat {f}}'(k,0)=C(k)F'(k,0).$ Eliminating $C(k)$ yields ${\widehat {g\,}}(k)+{\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K(k),$ where $K(k)={\frac {F'(k,0)}{F(k,0)}}.$ Now $K(k)$ can be decomposed into the product of functions $K^{-}$ and $K^{+}$ which are analytic in the upper and lower half-planes respectively. To be precise, $K(k)=K^{+}(k)K^{-}(k),$ where $\log K^{-}={\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k>0,$ $\log K^{+}=-{\frac {1}{2\pi i}}\int _{-\infty }^{\infty }{\frac {\log(K(z))}{z-k}}\,{\textrm {d}}z,\quad \operatorname {Im} k<0.$ (Note that this sometimes involves scaling $K$ so that it tends to $1$ as $k\rightarrow \infty $.) We also decompose $K^{+}{\widehat {g\,}}$ into the sum of two functions $G^{+}$ and $G^{-}$ which are analytic in the lower and upper half-planes respectively, i.e., $K^{+}(k){\widehat {g\,}}(k)=G^{+}(k)+G^{-}(k).$ This can be done in the same way that we factorised $K(k).$ Consequently, $G^{+}(k)+K_{+}(k){\widehat {f}}_{+}(k,0)={\widehat {f}}'_{-}(k,0)/K_{-}(k)-G^{-}(k).$ Now, as the left-hand side of the above equation is analytic in the lower half-plane, whilst the right-hand side is analytic in the upper half-plane, analytic continuation guarantees existence of an entire function which coincides with the left- or right-hand sides in their respective half-planes. Furthermore, since it can be shown that the functions on either side of the above equation decay at large k, an application of Liouville's theorem shows that this entire function is identically zero, therefore ${\widehat {f}}_{+}(k,0)=-{\frac {G^{+}(k)}{K^{+}(k)}},$ and so $C(k)={\frac {K^{+}(k){\widehat {g\,}}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}}.$ See also • Wiener filter • Riemann–Hilbert problem References • "Category:Wiener-Hopf - WikiWaves". wikiwaves.org. Retrieved 2020-05-19. • "Wiener-Hopf method", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Fornberg, Bengt. Complex variables and analytic functions : an illustrated introduction. Piret, Cécile. Philadelphia. ISBN 978-1-61197-597-0. OCLC 1124781689.
Wiener–Lévy theorem Wiener–Lévy theorem is a theorem in Fourier analysis, which states that a function of an absolutely convergent Fourier series has an absolutely convergent Fourier series under some conditions. The theorem was named after Norbert Wiener and Paul Lévy. Norbert Wiener first proved Wiener's 1/f theorem,[1] see Wiener's theorem. It states that if f has absolutely convergent Fourier series and is never zero, then its inverse 1/f also has an absolutely convergent Fourier series. Wiener–Levy theorem Paul Levy generalized Wiener's result,[2] showing that Let $F(\theta )=\sum \limits _{k=-\infty }^{\infty }c_{k}e^{ik\theta },\quad \theta \in [0,2\pi ]$ be an absolutely convergent Fourier series with $\\|F\\|=\sum \limits _{k=-\infty }^{\infty }\|c_{k}\|<\infty .$ The values of $F(\theta )$ lie on a curve $C$, and $H(t)$ is an analytic (not necessarily single-valued) function of a complex variable which is regular at every point of $C$. Then $H[F(\theta )]$ has an absolutely convergent Fourier series. The proof can be found in the Zygmund's classic book Trigonometric Series.[3] Example Let $H(\theta )=\ln(\theta )$ and $F(\theta )=\sum \limits _{k=0}^{\infty }p_{k}e^{ik\theta },(\sum \limits _{k=0}^{\infty }p_{k}=1$) is characteristic function of discrete probability distribution. So $F(\theta )$ is an absolutely convergent Fourier series. If $F(\theta )$ has no zeros, then we have $H[F(\theta )]=\ln \left(\sum \limits _{k=0}^{\infty }p_{k}e^{ik\theta }\right)=\sum _{k=0}^{\infty }c_{k}e^{ik\theta },$ where $\\|H\\|=\sum \limits _{k=0}^{\infty }\|c_{k}\|<\infty .$ The statistical application of this example can be found in discrete pseudo compound Poisson distribution[4] and zero-inflated model. If a discrete r.v. $X$ with $\Pr(X=i)=P_{i}$, $i\in \mathbb {N} $, has the probability generating function of the form $P(z)=\sum \limits _{i=0}^{\infty }P_{i}z^{i}=\exp \left\{\sum \limits _{i=1}^{\infty }\alpha _{i}\lambda (z^{i}-1)\right\},z=e^{ik\theta }$ where $\sum \limits _{i=1}^{\infty }\alpha _{i}=1$, $\sum \limits _{i=1}^{\infty }\left\|\alpha _{i}\right\|<\infty $, $\alpha _{i}\in \mathbb {R} $, and $\lambda >0$. Then $X$ is said to have the discrete pseudo compound Poisson distribution, abbreviated DPCP. We denote it as $X\sim DPCP({\alpha _{1}}\lambda ,{\alpha _{2}}\lambda ,\cdots )$. See also • Wiener's theorem (disambiguation) References 1. Wiener, N. (1932). "Tauberian Theorems". Annals of Mathematics. 33 (1): 1–100. doi:10.2307/1968102. JSTOR 1968102. 2. Lévy, P. (1935). "Sur la convergence absolue des séries de Fourier". Compositio Mathematica. 1: 1–14. 3. Zygmund, A. (2002). Trigonometric Series. Cambridge: Cambridge University Press. p. 245. 4. Huiming, Zhang; Li, Bo; G. Jay Kerns (2017). "A characterization of signed discrete infinitely divisible distributions". Studia Scientiarum Mathematicarum Hungarica. 54: 446–470. arXiv:1701.03892. doi:10.1556/012.2017.54.4.1377.
Wiener amalgam space In mathematics, amalgam spaces categorize functions with regard to their local and global behavior. While the concept of function spaces treating local and global behavior separately was already known earlier, Wiener amalgams, as the term is used today, were introduced by Hans Georg Feichtinger in 1980. The concept is named after Norbert Wiener. Let $X$ be a normed space with norm $\\|\cdot \\|_{X}$. Then the Wiener amalgam space[1] with local component $X$ and global component $L_{m}^{p}$, a weighted $L^{p}$ space with non-negative weight $m$, is defined by $W(X,L^{p})=\left\{f\ :\ \left(\int _{\mathbb {R} ^{d}}\\|f(\cdot ){\bar {g}}(\cdot -x)\\|_{X}^{p}m(x)^{p}\,dx\right)^{1/p}<\infty \right\},$ :\ \left(\int _{\mathbb {R} ^{d}}\\|f(\cdot ){\bar {g}}(\cdot -x)\\|_{X}^{p}m(x)^{p}\,dx\right)^{1/p}<\infty \right\},} where $g$ is a continuously differentiable, compactly supported function, such that $\sum _{x\in \mathbb {Z^{d}} }g(z-x)=1$, for all $z\in \mathbb {R} ^{d}$. Again, the space defined is independent of $g$. As the definition suggests, Wiener amalgams are useful to describe functions showing characteristic local and global behavior.[2] References 1. Wiener amalgam spaces for the Fundamental Identity of Gabor Analysis by Hans Georg Feichtinger and Franz Luef 2. Foundations of Time-Frequency Analysis by Karlheinz Gröchenig Functional analysis (topics – glossary) Spaces • Banach • Besov • Fréchet • Hilbert • Hölder • Nuclear • Orlicz • Schwartz • Sobolev • Topological vector Properties • Barrelled • Complete • Dual (Algebraic/Topological) • Locally convex • Reflexive • Reparable Theorems • Hahn–Banach • Riesz representation • Closed graph • Uniform boundedness principle • Kakutani fixed-point • Krein–Milman • Min–max • Gelfand–Naimark • Banach–Alaoglu Operators • Adjoint • Bounded • Compact • Hilbert–Schmidt • Normal • Nuclear • Trace class • Transpose • Unbounded • Unitary Algebras • Banach algebra • C-algebra • Spectrum of a C-algebra • Operator algebra • Group algebra of a locally compact group • Von Neumann algebra Open problems • Invariant subspace problem • Mahler's conjecture Applications • Hardy space • Spectral theory of ordinary differential equations • Heat kernel • Index theorem • Calculus of variations • Functional calculus • Integral operator • Jones polynomial • Topological quantum field theory • Noncommutative geometry • Riemann hypothesis • Distribution (or Generalized functions) Advanced topics • Approximation property • Balanced set • Choquet theory • Weak topology • Banach–Mazur distance • Tomita–Takesaki theory • Mathematics portal • Category • Commons
Wiener connector In network theory, the Wiener connector is a means of maximizing efficiency in connecting specified "query vertices" in a network. Given a connected, undirected graph and a set of query vertices in a graph, the minimum Wiener connector is an induced subgraph that connects the query vertices and minimizes the sum of shortest path distances among all pairs of vertices in the subgraph. In combinatorial optimization, the minimum Wiener connector problem is the problem of finding the minimum Wiener connector. It can be thought of as a version of the classic Steiner tree problem (one of Karp's 21 NP-complete problems), where instead of minimizing the size of the tree, the objective is to minimize the distances in the subgraph.[1][2] The minimum Wiener connector was first presented by Ruchansky et al. in 2015.[3] The minimum Wiener connector has applications in many domains where there is a graph structure and an interest in learning about connections between sets of individuals. For example, given a set of patients infected with a viral disease, which other patients should be checked to find the culprit? Or given a set of proteins of interest, which other proteins participate in pathways with them? The Wiener connector was named in honor of chemist Harry Wiener who first introduced the Wiener Index. Problem definition The Wiener index is the sum of shortest path distances in a (sub)graph. Using $d(u,v)$ to denote the shortest path between $u$ and $v$, the Wiener index of a (sub)graph $S$, denoted $W(S)$, is defined as $W(S)=\sum _{(u,v)\in S}d(u,v)$. The minimum Wiener connector problem is defined as follows. Given an undirected and unweighted graph with vertex set $V$ and edge set $E$ and a set of query vertices $Q\subseteq V$, find a connector $H\subseteq V$ of minimum Wiener index. More formally, the problem is to compute $\operatorname {*} {arg\,min}_{H}W(H\cup Q)$, that is, find a connector $H$ that minimizes the sum of shortest paths in $H$. Relationship to Steiner tree The minimum Wiener connector problem is related to the Steiner tree problem. In the former, the objective function in the minimization is the Wiener index of the connector, whereas in the latter, the objective function is the sum of the weights of the edges in the connector. The optimum solutions to these problems may differ, given the same graph and set of query vertices. In fact, a solution for the Steiner tree problem may be arbitrarily bad for the minimum Wiener connector problem; the graph on the right provides an example. Computational complexity Hardness The problem is NP-hard, and does not admit a polynomial-time approximation scheme unless P = NP.[3] This can be proven using the inapproximability of vertex cover in bounded degree graphs.[4] Although there is no polynomial-time approximation scheme, there is a polynomial-time constant-factor approximation—an algorithm that finds a connector whose Wiener index is within a constant multiplicative factor of the Wiener index of the optimum connector. In terms of complexity classes, the minimum Wiener connector problem is in APX but is not in PTAS unless P = NP. Exact algorithms An exhaustive search over all possible subsets of vertices to find the one that induces the connector of minimum Wiener index yields an algorithm that finds the optimum solution in $2^{O(n)}$ time (that is, exponential time) on graphs with n vertices. In the special case that there are exactly two query vertices, the optimum solution is the shortest path joining the two vertices, so the problem can be solved in polynomial time by computing the shortest path. In fact, for any fixed constant number of query vertices, an optimum solution can be found in polynomial time. Approximation algorithms There is a constant-factor approximation algorithm for the minimum Wiener connector problem that runs in time $O(q(m\log n+n\log ^{2}n))$ on a graph with n vertices, m edges, and q query vertices, roughly the same time it takes to compute shortest-path distances from the query vertices to every other vertex in the graph.[3] The central approach of this algorithm is to reduce the problem to the vertex-weighted Steiner tree problem, which admits a constant-factor approximation in particular instances related to the minimum Wiener connector problem. Behavior The minimum Wiener connector behaves like betweenness centrality. When the query vertices belong to the same community, the non-query vertices that form the minimum Wiener connector tend to belong to the same community and have high centrality within the community. Such vertices are likely to be influential vertices playing leadership roles in the community. In a social network, these influential vertices might be good users for spreading information or to target in a viral marketing campaign.[5] When the query vertices belong to different communities, the non-query vertices that form the minimum Wiener connector contain vertices adjacent to edges that bridge the different communities. These vertices span a structural hole in the graph and are important.[6] Applications The minimum Wiener connector is useful in applications in which one wishes to learn about the relationship between a set of vertices in a graph. For example, • in biology, it provides insight into how a set of proteins in a protein–protein interaction network are related, • in social networks (like Twitter), it demonstrates the communities to which a set of users belong and how these communities are related, • in computer networks, it may be useful in identifying an efficient way to route a multicast message to a set of destinations. References 1. Hwang, Frank; Richards, Dana; Winter, Dana; Winter, Pawel (1992). "The Steiner Tree Problem". Annals of Discrete Mathematics. 2. DIMACS Steiner Tree Challenge 3. Ruchansky, Natali; Bonchi, Francesco; Garcia-Soriano, David; Gullo, Francesco; Kourtellis, Nicolas (2015). "The Minimum Wiener Connector". SIGMOD. arXiv:1504.00513. Bibcode:2015arXiv150400513R. doi:10.1145/2723372.2749449. S2CID 2856346. 4. Dinur, Irit; Safra, Samuel (2005). "On the hardness of approximating minimum vertex cover". Annals of Mathematics. 162: 439–485. doi:10.4007/annals.2005.162.439. 5. Hinz, Oliver; Skiera, Bernd; Barrot, Christian; Becker, Jan U. (2011). "Seeding Strategies for Viral Marketing: An Empirical Comparison". Journal of Marketing. 75 (6): 55–71. doi:10.1509/jm.10.0088. S2CID 53972310. 6. Lou, Tiancheng; Tang, Jie (2013). "Mining Structural Hole Spanners Through Information Diffusion in Social Networks". Proceedings of the 22nd International Conference on World Wide Web. Rio de Janeiro, Brazil: International World Wide Web Conferences Steering Committee. pp. 825–836. ISBN 9781450320351.
Wiener deconvolution In mathematics, Wiener deconvolution is an application of the Wiener filter to the noise problems inherent in deconvolution. It works in the frequency domain, attempting to minimize the impact of deconvolved noise at frequencies which have a poor signal-to-noise ratio. The Wiener deconvolution method has widespread use in image deconvolution applications, as the frequency spectrum of most visual images is fairly well behaved and may be estimated easily. Wiener deconvolution is named after Norbert Wiener. Definition Given a system: $\ y(t)=(hx)(t)+n(t)$ where $$ denotes convolution and: • $\ x(t)$ is some original signal (unknown) at time $\ t$. • $\ h(t)$ is the known impulse response of a linear time-invariant system • $\ n(t)$ is some unknown additive noise, independent of $\ x(t)$ • $\ y(t)$ is our observed signal Our goal is to find some $\ g(t)$ so that we can estimate $\ x(t)$ as follows: $\ {\hat {x}}(t)=(gy)(t)$ where $\ {\hat {x}}(t)$ is an estimate of $\ x(t)$ that minimizes the mean square error $\ \epsilon (t)=\mathbb {E} \left\|x(t)-{\hat {x}}(t)\right\|^{2}$, with $\ \mathbb {E} $ denoting the expectation. The Wiener deconvolution filter provides such a $\ g(t)$. The filter is most easily described in the frequency domain: $\ G(f)={\frac {H^{}(f)S(f)}{\|H(f)\|^{2}S(f)+N(f)}}$ where: • $\ G(f)$ and $\ H(f)$ are the Fourier transforms of $\ g(t)$ and $\ h(t)$, • $\ S(f)=\mathbb {E} \|X(f)\|^{2}$ is the mean power spectral density of the original signal $\ x(t)$, • $\ N(f)=\mathbb {E} \|V(f)\|^{2}$ is the mean power spectral density of the noise $\ n(t)$, • $X(f)$, $Y(f)$, and $V(f)$ are the Fourier transforms of $x(t)$, and $y(t)$, and $n(t)$, respectively, • the superscript ${}^{}$ denotes complex conjugation. The filtering operation may either be carried out in the time-domain, as above, or in the frequency domain: $\ {\hat {X}}(f)=G(f)Y(f)$ and then performing an inverse Fourier transform on $\ {\hat {X}}(f)$ to obtain $\ {\hat {x}}(t)$. Note that in the case of images, the arguments $\ t$ and $\ f$ above become two-dimensional; however the result is the same. Interpretation The operation of the Wiener filter becomes apparent when the filter equation above is rewritten: ${\begin{aligned}G(f)&={\frac {1}{H(f)}}\left[{\frac {1}{1+1/(\|H(f)\|^{2}\mathrm {SNR} (f))}}\right]\end{aligned}}$ Here, $\ 1/H(f)$ is the inverse of the original system, $\ \mathrm {SNR} (f)=S(f)/N(f)$ is the signal-to-noise ratio, and $\ \|H(f)\|^{2}\mathrm {SNR} (f)$ is the ratio of the pure filtered signal to noise spectral density. When there is zero noise (i.e. infinite signal-to-noise), the term inside the square brackets equals 1, which means that the Wiener filter is simply the inverse of the system, as we might expect. However, as the noise at certain frequencies increases, the signal-to-noise ratio drops, so the term inside the square brackets also drops. This means that the Wiener filter attenuates frequencies according to their filtered signal-to-noise ratio. The Wiener filter equation above requires us to know the spectral content of a typical image, and also that of the noise. Often, we do not have access to these exact quantities, but we may be in a situation where good estimates can be made. For instance, in the case of photographic images, the signal (the original image) typically has strong low frequencies and weak high frequencies, while in many cases the noise content will be relatively flat with frequency. Derivation As mentioned above, we want to produce an estimate of the original signal that minimizes the mean square error, which may be expressed: $\ \epsilon (f)=\mathbb {E} \left\|X(f)-{\hat {X}}(f)\right\|^{2}$ . The equivalence to the previous definition of $\epsilon $, can be derived using Plancherel theorem or Parseval's theorem for the Fourier transform. If we substitute in the expression for $\ {\hat {X}}(f)$, the above can be rearranged to ${\begin{aligned}\epsilon (f)&=\mathbb {E} \left\|X(f)-G(f)Y(f)\right\|^{2}\\&=\mathbb {E} \left\|X(f)-G(f)\left[H(f)X(f)+V(f)\right]\right\|^{2}\\&=\mathbb {E} {\big \|}\left[1-G(f)H(f)\right]X(f)-G(f)V(f){\big \|}^{2}\end{aligned}}$ If we expand the quadratic, we get the following: ${\begin{aligned}\epsilon (f)&={\Big [}1-G(f)H(f){\Big ]}{\Big [}1-G(f)H(f){\Big ]}^{}\,\mathbb {E} \|X(f)\|^{2}\\&{}-{\Big [}1-G(f)H(f){\Big ]}G^{}(f)\,\mathbb {E} {\Big \{}X(f)V^{}(f){\Big \}}\\&{}-G(f){\Big [}1-G(f)H(f){\Big ]}^{}\,\mathbb {E} {\Big \{}V(f)X^{}(f){\Big \}}\\&{}+G(f)G^{}(f)\,\mathbb {E} \|V(f)\|^{2}\end{aligned}}$ However, we are assuming that the noise is independent of the signal, therefore: $\ \mathbb {E} {\Big \{}X(f)V^{}(f){\Big \}}=\mathbb {E} {\Big \{}V(f)X^{}(f){\Big \}}=0$ Substituting the power spectral densities $\ S(f)$ and $\ N(f)$, we have: $\epsilon (f)={\Big [}1-G(f)H(f){\Big ]}{\Big [}1-G(f)H(f){\Big ]}^{}S(f)+G(f)G^{}(f)N(f)$ To find the minimum error value, we calculate the Wirtinger derivative with respect to $\ G(f)$ and set it equal to zero. $\ {\frac {d\epsilon (f)}{dG(f)}}=0\Rightarrow G^{}(f)N(f)-H(f){\Big [}1-G(f)H(f){\Big ]}^{*}S(f)=0$ This final equality can be rearranged to give the Wiener filter. See also • Information field theory • Deconvolution • Wiener filter • Point spread function • Blind deconvolution • Fourier transform Wikimedia Commons has media related to An example of Wiener deconvolution on motion blured image (and source codes in MATLAB/GNU Octave).. References • Rafael Gonzalez, Richard Woods, and Steven Eddins. Digital Image Processing Using Matlab. Prentice Hall, 2003. External links • Comparison of different deconvolution methods.
Wiener equation A simple mathematical representation of Brownian motion, the Wiener equation, named after Norbert Wiener,[1] assumes the current velocity of a fluid particle fluctuates randomly: $\mathbf {v} ={\frac {d\mathbf {x} }{dt}}=g(t),$ where v is velocity, x is position, d/dt is the time derivative, and g(t) may for instance be white noise. Since velocity changes instantly in this formalism, the Wiener equation is not suitable for short time scales. In those cases, the Langevin equation, which looks at particle acceleration, must be used. References 1. Pesi R. Masani (6 December 2012). Norbert Wiener 1894–1964. Birkhäuser. pp. 134–. ISBN 978-3-0348-9252-0.
Wiener sausage In the mathematical field of probability, the Wiener sausage is a neighborhood of the trace of a Brownian motion up to a time t, given by taking all points within a fixed distance of Brownian motion. It can be visualized as a sausage of fixed radius whose centerline is Brownian motion. The Wiener sausage was named after Norbert Wiener by M. D. Donsker and S. R. Srinivasa Varadhan (1975) because of its relation to the Wiener process; the name is also a pun on Vienna sausage, as "Wiener" is German for "Viennese". The Wiener sausage is one of the simplest non-Markovian functionals of Brownian motion. Its applications include stochastic phenomena including heat conduction. It was first described by Frank Spitzer (1964), and it was used by Mark Kac and Joaquin Mazdak Luttinger (1973, 1974) to explain results of a Bose–Einstein condensate, with proofs published by M. D. Donsker and S. R. Srinivasa Varadhan (1975). Definitions The Wiener sausage Wδ(t) of radius δ and length t is the set-valued random variable on Brownian paths b (in some Euclidean space) defined by $W_{\delta }(t)({b})$ is the set of points within a distance δ of some point b(x) of the path b with 0≤x≤t. Volume of the Wiener sausage There has been a lot of work on the behavior of the volume (Lebesgue measure) \|Wδ(t)\| of the Wiener sausage as it becomes thin (δ→0); by rescaling, this is essentially equivalent to studying the volume as the sausage becomes long (t→∞). Spitzer (1964) showed that in 3 dimensions the expected value of the volume of the sausage is $E(\|W_{\delta }(t)\|)=2\pi \delta t+4\delta ^{2}{\sqrt {2\pi t}}+4\pi \delta ^{3}/3.$ In dimension d at least 3 the volume of the Wiener sausage is asymptotic to $\delta ^{d-2}\pi ^{d/2}2t/\Gamma ((d-2)/2)$ as t tends to infinity. In dimensions 1 and 2 this formula gets replaced by ${\sqrt {8t/\pi }}$ and $2{\pi }t/\log(t)$ respectively. Whitman (1964), a student of Spitzer, proved similar results for generalizations of Wiener sausages with cross sections given by more general compact sets than balls. References • Donsker, M. D.; Varadhan, S. R. S. (1975), "Asymptotics for the Wiener sausage", Communications on Pure and Applied Mathematics, 28 (4): 525–565, doi:10.1002/cpa.3160280406 • Hollander, F. den (2001) [1994], "Wiener sausage", Encyclopedia of Mathematics, EMS Press • Kac, M.; Luttinger, J. M. (1973), "Bose-Einstein condensation in the presence of impurities", J. Math. Phys., 14 (11): 1626–1628, Bibcode:1973JMP....14.1626K, doi:10.1063/1.1666234, MR 0342114 • Kac, M.; Luttinger, J. M. (1974), "Bose-Einstein condensation in the presence of impurities. II", J. Math. Phys., 15 (2): 183–186, Bibcode:1974JMP....15..183K, doi:10.1063/1.1666617, MR 0342115 • Simon, Barry (2005), Functional integration and quantum physics, Providence, RI: AMS Chelsea Publishing, ISBN 0-8218-3582-3, MR 2105995 Especially chapter 22. • Spitzer, F. (1964), "Electrostatic capacity, heat flow and Brownian motion", Probability Theory and Related Fields, 3 (2): 110–121, doi:10.1007/BF00535970, S2CID 198179345 • Spitzer, Frank (1976), Principles of random walks, Graduate Texts in Mathematics, vol. 34, New York-Heidelberg: Springer-Verlag, p. 40, MR 0171290 (Reprint of 1964 edition) • Sznitman, Alain-Sol (1998), Brownian motion, obstacles and random media, Springer Monographs in Mathematics, Berlin: Springer-Verlag, doi:10.1007/978-3-662-11281-6, ISBN 3-540-64554-3, MR 1717054 An advanced monograph covering the Wiener sausage. • Whitman, Walter William (1964), Some Strong Laws for Random Walks and Brownian Motion, PhD Thesis, Cornell U. 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Wiener series In mathematics, the Wiener series, or Wiener G-functional expansion, originates from the 1958 book of Norbert Wiener. It is an orthogonal expansion for nonlinear functionals closely related to the Volterra series and having the same relation to it as an orthogonal Hermite polynomial expansion has to a power series. For this reason it is also known as the Wiener–Hermite expansion. The analogue of the coefficients are referred to as Wiener kernels. The terms of the series are orthogonal (uncorrelated) with respect to a statistical input of white noise. This property allows the terms to be identified in applications by the Lee–Schetzen method. The Wiener series is important in nonlinear system identification. In this context, the series approximates the functional relation of the output to the entire history of system input at any time. The Wiener series has been applied mostly to the identification of biological systems, especially in neuroscience. The name Wiener series is almost exclusively used in system theory. In the mathematical literature it occurs as the Itô expansion (1951) which has a different form but is entirely equivalent to it. The Wiener series should not be confused with the Wiener filter, which is another algorithm developed by Norbert Wiener used in signal processing. Wiener G-functional expressions Given a system with an input/output pair $(x(t),y(t))$ where the input is white noise with zero mean value and power A, we can write the output of the system as sum of a series of Wiener G-functionals $y(n)=\sum _{p}(G_{p}x)(n)$ In the following the expressions of the G-functionals up to the fifth order will be given: $(G_{0}x)(n)=k_{0}=E\left\{y(n)\right\};$ $(G_{1}x)(n)=\sum _{\tau _{1}=0}^{N_{1}-1}k_{1}(\tau _{1})x(n-\tau _{1});$ $(G_{2}x)(n)=\sum _{\tau _{1},\tau _{2}=0}^{N_{2}-1}k_{2}(\tau _{1},\tau _{2})x(n-\tau _{1})x(n-\tau _{2})-A\sum _{\tau _{1}=0}^{N_{2}-1}k_{2}(\tau _{1},\tau _{1});$ $(G_{3}x)(n)=\sum _{\tau _{1},\ldots ,\tau _{3}=0}^{N_{3}-1}k_{3}(\tau _{1},\tau _{2},\tau _{3})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})-3A\sum _{\tau _{1}=0}^{N_{3}-1}\sum _{\tau _{2}=0}^{N_{3}-1}k_{3}(\tau _{1},\tau _{2},\tau _{2})x(n-\tau _{1});$ ${\begin{aligned}(G_{4}x)(n)={}&\sum _{\tau _{1},\ldots ,\tau _{4}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{2},\tau _{3},\tau _{4})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})x(n-\tau _{4})+{}\\[6pt]&{}-6A\sum _{\tau _{1},\tau _{2}=0}^{N_{4}-1}\sum _{\tau _{3}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{2},\tau _{3},\tau _{3})x(n-\tau _{1})x(n-\tau _{2})+3A^{2}\sum _{\tau _{1},\tau _{2}=0}^{N_{4}-1}k_{4}(\tau _{1},\tau _{1},\tau _{2},\tau _{2});\end{aligned}}$ ${\begin{aligned}(G_{5}x)(n)={}&\sum _{\tau _{1},\ldots ,\tau _{5}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{5})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})x(n-\tau _{4})x(n-\tau _{5})+{}\\[6pt]&{}-10A\sum _{\tau _{1},\ldots ,\tau _{3}=0}^{N_{5}-1}\sum _{\tau _{4}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{3},\tau _{4},\tau _{4})x(n-\tau _{1})x(n-\tau _{2})x(n-\tau _{3})\\[6pt]&{}+15A^{2}\sum _{\tau _{1}=0}^{N_{5}-1}\sum _{\tau _{2},\tau _{3}=0}^{N_{5}-1}k_{5}(\tau _{1},\tau _{2},\tau _{2},\tau _{3},\tau _{3})x(n-\tau _{1}).\end{aligned}}$ See also • Volterra series • System identification • Spike-triggered average References • Wiener, Norbert (1958). Nonlinear Problems in Random Theory. Wiley and MIT Press. • Lee and Schetzen; Schetzen‡, M. (1965). "Measurement of the Wiener kernels of a non-linear system by cross-correlation". International Journal of Control. First. 2 (3): 237–254. doi:10.1080/00207176508905543. • Itô K "A multiple Wiener integral" J. Math. Soc. Jpn. 3 1951 157–169 • Marmarelis, P.Z.; Naka, K. (1972). "White-noise analysis of a neuron chain: an application of the Wiener theory". Science. 175 (4027): 1276–1278. doi:10.1126/science.175.4027.1276. PMID 5061252. • Schetzen, Martin (1980). The Volterra and Wiener Theories of Nonlinear Systems. John Wiley and Sons. ISBN 978-0-471-04455-0. • Marmarelis, P.Z. (1991). "Wiener Analysis of Nonlinear Feedback". Sensory Systems Annals of Biomedical Engineering. 19 (4): 345–382. doi:10.1007/BF02584316. • Franz, M; Schölkopf, B. (2006). "A unifying view of Wiener and Volterra theory and polynomial kernel regression". Neural Computation. 18 (12): 3097–3118. doi:10.1162/neco.2006.18.12.3097. • L.A. Zadeh On the representation of nonlinear operators. IRE Westcon Conv. Record pt.2 1957 105-113.
Wiener–Araya graph The Wiener–Araya graph is, in graph theory, a graph on 42 vertices with 67 edges. It is hypohamiltonian, which means that it does not itself have a Hamiltonian cycle but every graph formed by removing a single vertex from it is Hamiltonian. It is also planar. Wiener-Araya graph Vertices42 Edges67 Radius5 Diameter7 Girth4 Automorphisms2 Chromatic number3 Chromatic index4 PropertiesHypohamiltonian Planar Table of graphs and parameters History Hypohamiltonian graphs were first studied by Sousselier in Problèmes plaisants et délectables (1963).[1] In 1967, Lindgren built an infinite sequence of hypohamiltonian graphs.[2] He first cited Gaudin, Herz and Rossi,[3] then Busacker and Saaty[4] as pioneers on this topic. From the start, the smallest hypohamiltonian graph is known: the Petersen graph. However, the hunt for the smallest planar hypohamiltonian graph continues. This question was first raised by Václav Chvátal in 1973.[5] The first candidate answer was provided in 1976 by Carsten Thomassen, who exhibited a 105-vertices construction, the 105-Thomassen graph.[6] In 1979, Hatzel improved this result with a planar hypohamiltonian graph on 57 vertices : the Hatzel graph.[7] This bound was lowered in 2007 by the 48-Zamfirescu graph.[8] In 2009, a graph built by Gábor Wiener and Makoto Araya became (with its 42 vertices) the smallest planar hypohamiltonian graph known.[9][10] In their paper, Wiener and Araya conjectured their graph to be optimal arguing that its order (42) appears to be the answer to The Ultimate Question of Life, the Universe, and Everything from The Hitchhiker's Guide to the Galaxy, a Douglas Adams novel. However, subsequently, smaller planar hypohamiltonian graphs have been discovered.[11] References 1. Sousselier, R. (1963), Problème no. 29: Le cercle des irascibles, vol. 7, Rev. Franç. Rech. Opérationnelle, pp. 405–406 2. Lindgren, W. F. (1967), "An infinite class of hypohamiltonian graphs", American Mathematical Monthly, 74 (9): 1087–1089, doi:10.2307/2313617, JSTOR 2313617, MR 0224501 3. Gaudin, T.; Herz, P.; Rossi (1964), "Solution du problème No. 29", Rev. Franç. Rech. Opérationnelle (in French), 8: 214–218 4. Busacker, R. G.; Saaty, T. L. (1965), Finite Graphs and Networks 5. Chvátal, V. (1973), "Flip-flops in hypo-Hamiltonian graphs", Canadian Mathematical Bulletin, 16: 33–41, doi:10.4153/cmb-1973-008-9, MR 0371722 6. Thomassen, Carsten (1976), "Planar and infinite hypohamiltonian and hypotraceable graphs", Discrete Mathematics, 14 (4): 377–389, doi:10.1016/0012-365x(76)90071-6, MR 0422086 7. Hatzel, Wolfgang (1979), "Ein planarer hypohamiltonscher Graph mit 57 Knoten", Mathematische Annalen (in German), 243 (3): 213–216, doi:10.1007/BF01424541, MR 0548801 8. Zamfirescu, Carol T.; Zamfirescu, Tudor I. (2007), "A planar hypohamiltonian graph with 48 vertices", Journal of Graph Theory, 55 (4): 338–342, doi:10.1002/jgt.20241, MR 2336805 9. Wiener, Gábor; Araya, Makoto (April 20, 2009), The ultimate question, arXiv:0904.3012, Bibcode:2009arXiv0904.3012W. 10. Wiener, Gábor; Araya, Makoto (2011), "On planar hypohamiltonian graphs", Journal of Graph Theory, 67 (1): 55–68, doi:10.1002/jgt.20513, MR 2809563, S2CID 5340663. 11. Jooyandeh, Mohammadreza; McKay, Brendan D.; Östergård, Patric R. J.; Pettersson, Ville H.; Zamfirescu, Carol T. (2017), "Planar hypohamiltonian graphs on 40 vertices", Journal of Graph Theory, 84 (2): 121–133, arXiv:1302.2698, doi:10.1002/jgt.22015, MR 3601121, S2CID 5535167 External links • Weisstein, Eric W., "Wiener-Araya Graph", MathWorld
Wiener–Wintner theorem In mathematics, the Wiener–Wintner theorem, named after Norbert Wiener and Aurel Wintner, is a strengthening of the ergodic theorem, proved by Wiener and Wintner (1941). Statement Suppose that τ is a measure-preserving transformation of a measure space S with finite measure. If f is a real-valued integrable function on S then the Wiener–Wintner theorem states that there is a measure 0 set E such that the average $\lim _{\ell \rightarrow \infty }{\frac {1}{2\ell +1}}\sum _{j=-\ell }^{\ell }e^{ij\lambda }f(\tau ^{j}P)$ exists for all real λ and for all P not in E. The special case for λ = 0 is essentially the Birkhoff ergodic theorem, from which the existence of a suitable measure 0 set E for any fixed λ, or any countable set of values λ, immediately follows. The point of the Wiener–Wintner theorem is that one can choose the measure 0 exceptional set E to be independent of λ. This theorem was even much more generalized by the Return Times Theorem. References • Assani, I. (2001) [1994], "Wiener–Wintner theorem", Encyclopedia of Mathematics, EMS Press • Wiener, Norbert; Wintner, Aurel (1941), "Harmonic analysis and ergodic theory", American Journal of Mathematics, 63: 415–426, doi:10.2307/2371534, ISSN 0002-9327, JSTOR 2371534, MR 0004098
Karl Weierstrass Karl Theodor Wilhelm Weierstrass (German: Weierstraß [ˈvaɪɐʃtʁaːs];[1] 31 October 1815 – 19 February 1897) was a German mathematician often cited as the "father of modern analysis". Despite leaving university without a degree, he studied mathematics and trained as a school teacher, eventually teaching mathematics, physics, botany and gymnastics.[2] He later received an honorary doctorate and became professor of mathematics in Berlin. Karl Weierstrass Karl Weierstraß Born(1815-10-31)31 October 1815 Ennigerloh, Kingdom of Prussia Died19 February 1897(1897-02-19) (aged 81) Berlin, Kingdom of Prussia NationalityGerman Alma mater • University of Bonn • Münster Academy Known for • Weierstrass function • Weierstrass product inequality • (ε, δ)-definition of limit • Weierstrass–Erdmann condition • Weierstrass theorems • Bolzano–Weierstrass theorem Awards • PhD (Hon): University of Königsberg (1854) • Copley Medal (1895) Scientific career FieldsMathematics InstitutionsGewerbeinstitut, Friedrich Wilhelm University Academic advisorsChristoph Gudermann Doctoral students • Nikolai Bugaev • Georg Cantor • Georg Frobenius • Lazarus Fuchs • Wilhelm Killing • Johannes Knoblauch • Leo Königsberger • Ernst Kötter • Sofia Kovalevskaya • Mathias Lerch • Hans von Mangoldt • Eugen Netto • Adolf Piltz • Carl Runge • Arthur Schoenflies • Friedrich Schottky • Hermann Schwarz • Ludwig Stickelberger Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis, proved the intermediate value theorem and the Bolzano–Weierstrass theorem, and used the latter to study the properties of continuous functions on closed bounded intervals. Biography Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh, in the Province of Westphalia.[3] Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were Catholic Rhinelanders. His interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn. He was sent to the University of Bonn upon graduation to prepare for a government position. Because his studies were to be in the fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study but continuing private study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the Münster Academy (which was even then famous for mathematics) and his father was able to obtain a place for him in a teacher training school in Münster. Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions. In 1843 he taught in Deutsch Krone in West Prussia and since 1848 he taught at the Lyceum Hosianum in Braunsberg.[4] Besides mathematics he also taught physics, botany, and gymnastics.[3] Weierstrass may have had an illegitimate child named Franz with the widow of his friend Carl Wilhelm Borchardt.[5] After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut in Berlin (an institute to educate technical workers which would later merge with the Bauakademie to form the Technical University of Berlin). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the Humboldt Universität zu Berlin. In 1870, at the age of fifty-five, Weierstrass met Sofia Kovalevsky whom he tutored privately after failing to secure her admission to the university. They had a fruitful intellectual, and kindly personal relationship that "far transcended the usual teacher-student relationship". He mentored her for four years, and regarded her as his best student, helping to secure a doctorate for her from Heidelberg University without the need for an oral thesis defense. He was immobile for the last three years of his life, and died in Berlin from pneumonia.[6] Mathematical contributions Soundness of calculus Weierstrass was interested in the soundness of calculus, and at the time there were somewhat ambiguous definitions of the foundations of calculus so that important theorems could not be proven with sufficient rigour. Although Bolzano had developed a reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions. The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of Cauchy in the 1820s.[7][8] Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that the (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is false in general. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence, which was first observed by Weierstrass's advisor, Christoph Gudermann, in an 1838 paper, where Gudermann noted the phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus. The formal definition of continuity of a function, as formulated by Weierstrass, is as follows: $\displaystyle f(x)$ is continuous at $\displaystyle x=x_{0}$ if $\displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0$ such that for every $x$ in the domain of $f$, $\displaystyle \ \|x-x_{0}\|<\delta \Rightarrow \|f(x)-f(x_{0})\|<\varepsilon .$ In simple English, $\displaystyle f(x)$ is continuous at a point $\displaystyle x=x_{0}$ if for each $x$ close enough to $x_{0}$, the function value $f(x)$ is very close to $f(x_{0})$, where the "close enough" restriction typically depends on the desired closeness of $f(x_{0})$ to $f(x).$ Using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals. Calculus of variations Weierstrass also made advances in the field of calculus of variations. Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory that paved the way for the modern study of the calculus of variations. Among several axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition, which gives sufficient conditions for an extremal to have a corner along a given extremum and allows one to find a minimizing curve for a given integral. Other analytical theorems See also: List of things named after Karl Weierstrass • Stone–Weierstrass theorem • Casorati–Weierstrass theorem • Weierstrass elliptic function • Weierstrass function • Weierstrass M-test • Weierstrass preparation theorem • Lindemann–Weierstrass theorem • Weierstrass factorization theorem • Weierstrass–Enneper parameterization Students • Edmund Husserl Honours and awards The lunar crater Weierstrass and the asteroid 14100 Weierstrass are named after him. Also, there is the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. Selected works • Zur Theorie der Abelschen Funktionen (1854) • Theorie der Abelschen Funktionen (1856) • Abhandlungen-1, Math. Werke. Bd. 1. Berlin, 1894 • Abhandlungen-2, Math. Werke. Bd. 2. Berlin, 1895 • Abhandlungen-3, Math. Werke. Bd. 3. Berlin, 1903 • Vorl. ueber die Theorie der Abelschen Transcendenten, Math. Werke. Bd. 4. Berlin, 1902 • Vorl. ueber Variationsrechnung, Math. Werke. Bd. 7. Leipzig, 1927 See also • List of things named after Karl Weierstrass References 1. Duden. Das Aussprachewörterbuch. 7. Auflage. Bibliographisches Institut, Berlin 2015, ISBN 978-3-411-04067-4 2. Weierstrass, Karl Theodor Wilhelm. (2018). In Helicon (Ed.), The Hutchinson unabridged encyclopedia with atlas and weather guide. [Online]. Abington: Helicon. Available from: http://libezproxy.open.ac.uk/login?url=https://search.credoreference.com/content/entry/heliconhe/weierstrass_karl_theodor_wilhelm/0?institutionId=292 [Accessed 8 July 2018]. 3. O'Connor, J. J.; Robertson, E. F. (October 1998). "Karl Theodor Wilhelm Weierstrass". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved 7 September 2014. 4. Elstrodt, Jürgen (2016), König, Wolfgang; Sprekels, Jürgen (eds.), "Die prägenden Jahre im Leben von Karl Weierstraß", Karl Weierstraß (1815–1897) (in German), Wiesbaden: Springer Fachmedien Wiesbaden, pp. 11–51, doi:10.1007/978-3-658-10619-5_2, ISBN 978-3-658-10618-8, retrieved 2023-08-12 5. Biermann, Kurt-R.; Schubring, Gert (1996). "Einige Nachträge zur Biographie von Karl Weierstraß. (German) [Some postscripts to the biography of Karl Weierstrass]". History of mathematics. San Diego, CA: Academic Press. pp. 65–91. 6. Dictionary of scientific biography. Gillispie, Charles Coulston,, American Council of Learned Societies. New York. 1970. p. 223. ISBN 978-0-684-12926-6. OCLC 89822.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link) 7. Grabiner, Judith V. (March 1983), "Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus" (PDF), The American Mathematical Monthly, 90 (3): 185–194, doi:10.2307/2975545, JSTOR 2975545, archived (PDF) from the original on 2014-11-29 8. Cauchy, A.-L. (1823), "Septième Leçon – Valeurs de quelques expressions qui se présentent sous les formes indéterminées ${\frac {\infty }{\infty }},\infty ^{0},\ldots $ Relation qui existe entre le rapport aux différences finies et la fonction dérivée", Résumé des leçons données à l'école royale polytechnique sur le calcul infinitésimal, Paris, archived from the original on 2009-05-04, retrieved 2009-05-01, p. 44. {{citation}}: External link in \|postscript= (help)CS1 maint: postscript (link) External links Wikimedia Commons has media related to Karl Weierstrass. Wikiquote has quotations related to Karl Weierstrass. • O'Connor, John J.; Robertson, Edmund F., "Karl Weierstrass", MacTutor History of Mathematics Archive, University of St Andrews • Digitalized versions of Weierstrass's original publications are freely available online from the library of the Berlin Brandenburgische Akademie der Wissenschaften. • Works by Karl Weierstrass at Project Gutenberg • Works by or about Karl Weierstrass at Internet Archive Copley Medallists (1851–1900) • Richard Owen (1851) • Alexander von Humboldt (1852) • Heinrich Wilhelm Dove (1853) • Johannes Peter Müller (1854) • Léon Foucault (1855) • Henri Milne-Edwards (1856) • Michel Eugène Chevreul (1857) • Charles Lyell (1858) • Wilhelm Eduard Weber (1859) • Robert Bunsen (1860) • Louis Agassiz (1861) • Thomas Graham (1862) • Adam Sedgwick (1863) • Charles Darwin (1864) • Michel Chasles (1865) • Julius Plücker (1866) • Karl Ernst von Baer (1867) • Charles Wheatstone (1868) • Henri Victor Regnault (1869) • James Prescott Joule (1870) • Julius Robert von Mayer (1871) • Friedrich Wöhler (1872) • Hermann von Helmholtz (1873) • Louis Pasteur (1874) • August Wilhelm von Hofmann (1875) • Claude Bernard (1876) • James Dwight Dana (1877) • Jean-Baptiste Boussingault (1878) • Rudolf Clausius (1879) • James Joseph Sylvester (1880) • Charles Adolphe Wurtz (1881) • Arthur Cayley (1882) • William Thomson (1883) • Carl Ludwig (1884) • Friedrich August Kekulé von Stradonitz (1885) • Franz Ernst Neumann (1886) • Joseph Dalton Hooker (1887) • Thomas Henry Huxley (1888) • George Salmon (1889) • Simon Newcomb (1890) • Stanislao Cannizzaro (1891) • Rudolf Virchow (1892) • George Gabriel Stokes (1893) • Edward Frankland (1894) • Karl Weierstrass (1895) • Karl Gegenbaur (1896) • Albert von Kölliker (1897) • William Huggins (1898) • John William Strutt (1899) • Marcellin Berthelot (1900) Authority control International • FAST • ISNI • VIAF • WorldCat National • Norway • France • BnF data • Catalonia • Germany • Israel • United States • Sweden • Latvia • Czech Republic • Australia • Croatia • Netherlands • Poland Academics • CiNii • Leopoldina • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • IdRef
Borchers algebra In mathematics, a Borchers algebra or Borchers–Uhlmann algebra or BU-algebra is the tensor algebra of a vector space, often a space of smooth test functions. They were studied by H. J. Borchers (1962), who showed that the Wightman distributions of a quantum field could be interpreted as a state, called a Wightman functional, on a Borchers algebra. A Borchers algebra with a state can often be used to construct an O*-algebra. The Borchers algebra of a quantum field theory has an ideal called the locality ideal, generated by elements of the form ab−ba for a and b having spacelike-separated support. The Wightman functional of a quantum field theory vanishes on the locality ideal, which is equivalent to the locality axiom for quantum field theory. References • Borchers, H.-J. (1962), "On structure of the algebra of field operators", Nuovo Cimento, 24 (2): 214–236, doi:10.1007/BF02745645, MR 0142320, S2CID 122439590 External links • Yngvason, Jakob (2009), The Borchers-Uhlmann Algebra and its Descendants (PDF)
Eugene Wigner Eugene Paul "E. P." Wigner (Hungarian: Wigner Jenő Pál, pronounced [ˈviɡnɛr ˈjɛnøː ˈpaːl]; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".[1] Eugene Wigner Wigner in 1963 Born Wigner Jenő Pál (1902-11-17)November 17, 1902 Budapest, Kingdom of Hungary, Austria-Hungary DiedJanuary 1, 1995(1995-01-01) (aged 92) Princeton, New Jersey, U.S. Citizenship • Hungary (by birth) • United States (naturalized 1937) Alma materBudapest University of Technology and Economics Technical University of Berlin Known for • Bargmann–Wigner equations • Law of conservation of parity • Wigner D-matrix • Wigner–Eckart theorem • Wigner's friend • Wigner semicircle distribution • Wigner's classification • Wigner distribution function • Wigner quasiprobability distribution • Wigner crystal • Wigner effect • Wigner energy • Relativistic Breit–Wigner distribution • Modified Wigner distribution function • Wigner–d'Espagnat inequality • Gabor–Wigner transform • Wigner's theorem • Jordan–Wigner transformation • Newton–Wigner localization • Wigner–Inonu contraction • Wigner–Seitz cell • Wigner–Seitz radius • Thomas–Wigner rotation • Wigner–Weyl transform • Wigner–Wilkins spectrum • 6-j symbol • 9-j symbol Spouses Amelia Frank (m. 1936; died 1937) Mary Annette Wheeler (m. 1941; died 1977) Eileen Clare-Patton Hamilton (m. 1979) Children3 Awards • Medal for Merit (1946) • Franklin Medal (1950) • Enrico Fermi Award (1958) • Atoms for Peace Award (1959) • Max Planck Medal (1961) • Nobel Prize in Physics (1963) • National Medal of Science (1969) • Albert Einstein Award (1972) • Wigner Medal (1978) Scientific career Fields • Theoretical physics • Atomic physics • Nuclear physics • Solid-state physics Institutions • University of Göttingen • University of Wisconsin–Madison • Princeton University • Manhattan Project ThesisBildung und Zerfall von Molekülen (1925) Doctoral advisorMichael Polanyi Other academic advisors • László Rátz • Richard Becker Doctoral students • John Bardeen • Victor Frederick Weisskopf • Marcos Moshinsky • Abner Shimony • Edwin Thompson Jaynes • Frederick Seitz • Conyers Herring • Frederick Tappert • J O Hirschfelder Signature A graduate of the Technical University of Berlin, Wigner worked as an assistant to Karl Weissenberg and Richard Becker at the Kaiser Wilhelm Institute in Berlin, and David Hilbert at the University of Göttingen. Wigner and Hermann Weyl were responsible for introducing group theory into physics, particularly the theory of symmetry in physics. Along the way he performed ground-breaking work in pure mathematics, in which he authored a number of mathematical theorems. In particular, Wigner's theorem is a cornerstone in the mathematical formulation of quantum mechanics. He is also known for his research into the structure of the atomic nucleus. In 1930, Princeton University recruited Wigner, along with John von Neumann, and he moved to the United States, where he obtained citizenship in 1937. Wigner participated in a meeting with Leo Szilard and Albert Einstein that resulted in the Einstein–Szilard letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs. Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first. During the Manhattan Project, he led a team whose task was to design nuclear reactors to convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He became director of research and development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in early 1946, but became frustrated with bureaucratic interference by the Atomic Energy Commission, and returned to Princeton. In the postwar period, he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964. In later life, he became more philosophical, and published The Unreasonable Effectiveness of Mathematics in the Natural Sciences, his best-known work outside technical mathematics and physics. Early life and education Wigner Jenő Pál was born in Budapest, Austria-Hungary on November 17, 1902, to middle class Jewish parents, Elisabeth Elsa Einhorn and Antal Anton Wigner, a leather tanner. He had an older sister, Berta, known as Biri, and a younger sister Margit, known as Manci,[2] who later married British theoretical physicist Paul Dirac.[3] He was home schooled by a professional teacher until the age of 9, when he started school at the third grade. During this period, Wigner developed an interest in mathematical problems.[4] At the age of 11, Wigner contracted what his doctors believed to be tuberculosis. His parents sent him to live for six weeks in a sanatorium in the Austrian mountains, before the doctors concluded that the diagnosis was mistaken.[5] Wigner's family was Jewish, but not religiously observant, and his Bar Mitzvah was a secular one. From 1915 through 1919, he studied at the secondary grammar school called Fasori Evangélikus Gimnázium, the school his father had attended. Religious education was compulsory, and he attended classes in Judaism taught by a rabbi.[6] A fellow student was János von Neumann, who was a year behind Wigner. They both benefited from the instruction of the noted mathematics teacher László Rátz.[7] In 1919, to escape the Béla Kun communist regime, the Wigner family briefly fled to Austria, returning to Hungary after Kun's downfall.[8] Partly as a reaction to the prominence of Jews in the Kun regime, the family converted to Lutheranism.[9] Wigner explained later in his life that his family decision to convert to Lutheranism "was not at heart a religious decision but an anti-communist one".[9] After graduating from the secondary school in 1920, Wigner enrolled at the Budapest University of Technical Sciences, known as the Műegyetem. He was not happy with the courses on offer,[10] and in 1921 enrolled at the Technische Hochschule Berlin (now Technical University of Berlin), where he studied chemical engineering.[11] He also attended the Wednesday afternoon colloquia of the German Physical Society. These colloquia featured leading researchers including Max Planck, Max von Laue, Rudolf Ladenburg, Werner Heisenberg, Walther Nernst, Wolfgang Pauli, and Albert Einstein.[12] Wigner also met the physicist Leó Szilárd, who at once became Wigner's closest friend.[13] A third experience in Berlin was formative. Wigner worked at the Kaiser Wilhelm Institute for Physical Chemistry and Electrochemistry (now the Fritz Haber Institute), and there he met Michael Polanyi, who became, after László Rátz, Wigner's greatest teacher. Polanyi supervised Wigner's DSc thesis, Bildung und Zerfall von Molekülen ("Formation and Decay of Molecules").[14] Middle years Wigner returned to Budapest, where he went to work at his father's tannery, but in 1926, he accepted an offer from Karl Weissenberg at the Kaiser Wilhelm Institute in Berlin. Weissenberg wanted someone to assist him with his work on x-ray crystallography, and Polanyi had recommended Wigner. After six months as Weissenberg's assistant, Wigner went to work for Richard Becker for two semesters. Wigner explored quantum mechanics, studying the work of Erwin Schrödinger. He also delved into the group theory of Ferdinand Frobenius and Eduard Ritter von Weber.[15] Wigner received a request from Arnold Sommerfeld to work at the University of Göttingen as an assistant to the great mathematician David Hilbert. This proved a disappointment, as the aged Hilbert's abilities were failing, and his interests had shifted to logic. Wigner nonetheless studied independently.[16] He laid the foundation for the theory of symmetries in quantum mechanics and in 1927 introduced what is now known as the Wigner D-matrix.[17] Wigner and Hermann Weyl were responsible for introducing group theory into quantum mechanics. The latter had written a standard text, Group Theory and Quantum Mechanics (1928), but it was not easy to understand, especially for younger physicists. Wigner's Group Theory and Its Application to the Quantum Mechanics of Atomic Spectra (1931) made group theory accessible to a wider audience.[18] In these works, Wigner laid the foundation for the theory of symmetries in quantum mechanics.[19] Wigner's theorem proved by Wigner in 1931, is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT symmetry are represented on the Hilbert space of states. According to the theorem, any symmetry transformation is represented by a linear and unitary or antilinear and antiunitary transformation of Hilbert space. The representation of a symmetry group on a Hilbert space is either an ordinary representation or a projective representation.[20][21] In the late 1930s, Wigner extended his research into atomic nuclei. By 1929, his papers were drawing notice in the world of physics. In 1930, Princeton University recruited Wigner for a one-year lectureship, at 7 times the salary that he had been drawing in Europe. Princeton recruited von Neumann at the same time. Jenő Pál Wigner and János von Neumann had collaborated on three papers together in 1928 and two in 1929. They anglicized their first names to "Eugene" and "John", respectively.[22] When their year was up, Princeton offered a five-year contract as visiting professors for half the year. The Technische Hochschule responded with a teaching assignment for the other half of the year. This was very timely, since the Nazis soon rose to power in Germany.[23] At Princeton in 1934, Wigner introduced his sister Margit "Manci" Wigner to the physicist Paul Dirac, with whom she remarried.[24] Princeton did not rehire Wigner when his contract ran out in 1936.[25] Through Gregory Breit, Wigner found new employment at the University of Wisconsin. There, he met his first wife, Amelia Frank, who was a physics student there. However, she died unexpectedly in 1937, leaving Wigner distraught. He therefore accepted a 1938 offer from Princeton to return there.[26] Wigner became a naturalized citizen of the United States on January 8, 1937, and he brought his parents to the United States.[27] Manhattan Project Although he was a professed political amateur, on August 2, 1939, he participated in a meeting with Leó Szilárd and Albert Einstein that resulted in the Einstein–Szilárd letter, which prompted President Franklin D. Roosevelt to initiate the Manhattan Project to develop atomic bombs.[28] Wigner was afraid that the German nuclear weapon project would develop an atomic bomb first, and even refused to have his fingerprints taken because they could be used to track him down if Germany won.[29] "Thoughts of being murdered," he later recalled, "focus your mind wonderfully."[29] On June 4, 1941, Wigner married his second wife, Mary Annette Wheeler, a professor of physics at Vassar College, who had completed her Ph.D. at Yale University in 1932. After the war she taught physics on the faculty of Rutgers University's Douglass College in New Jersey until her retirement in 1964. They remained married until her death in November 1977.[30][31] They had two children, David Wigner and Martha Wigner Upton.[32] During the Manhattan Project, Wigner led a team that included J. Ernest Wilkins Jr., Alvin M. Weinberg, Katharine Way, Gale Young and Edward Creutz. The group's task was to design the production nuclear reactors that would convert uranium into weapons grade plutonium. At the time, reactors existed only on paper, and no reactor had yet gone critical. In July 1942, Wigner chose a conservative 100 MW design, with a graphite neutron moderator and water cooling.[33] Wigner was present at a converted rackets court under the stands at the University of Chicago's abandoned Stagg Field on December 2, 1942, when the world's first atomic reactor, Chicago Pile One (CP-1) achieved a controlled nuclear chain reaction.[34] Wigner was disappointed that DuPont was given responsibility for the detailed design of the reactors, not just their construction. He threatened to resign in February 1943, but was talked out of it by the head of the Metallurgical Laboratory, Arthur Compton, who sent him on vacation instead. As it turned out, a design decision by DuPont to give the reactor additional load tubes for more uranium saved the project when neutron poisoning became a problem.[35] Without the additional tubes, the reactor could have been run at 35% power until the boron impurities in the graphite were burned up and enough plutonium produced to run the reactor at full power; but this would have set the project back a year.[36] During the 1950s, he would even work for DuPont on the Savannah River Site.[35] Wigner did not regret working on the Manhattan Project, and sometimes wished the atomic bomb had been ready a year earlier.[37] An important discovery Wigner made during the project was the Wigner effect. This is a swelling of the graphite moderator caused by the displacement of atoms by neutron radiation.[38] The Wigner effect was a serious problem for the reactors at the Hanford Site in the immediate post-war period, and resulted in production cutbacks and a reactor being shut down entirely.[39] It was eventually discovered that it could be overcome by controlled heating and annealing.[40] Through Manhattan project funding, Wigner and Leonard Eisenbud also developed an important general approach to nuclear reactions, the Wigner–Eisenbud R-matrix theory, which was published in 1947.[41] Later years Wigner was elected to the American Philosophical Society in 1944 and the United States National Academy of Sciences in 1945.[42][43] He accepted a position as the director of research and development at the Clinton Laboratory (now the Oak Ridge National Laboratory) in Oak Ridge, Tennessee in early 1946. Because he did not want to be involved in administrative duties, he became co-director of the laboratory, with James Lum handling the administrative chores as executive director.[44] When the newly created Atomic Energy Commission (AEC) took charge of the laboratory's operations at the start of 1947, Wigner feared that many of the technical decisions would be made in Washington.[45] He also saw the Army's continuation of wartime security policies at the laboratory as a "meddlesome oversight", interfering with research.[46] One such incident occurred in March 1947, when the AEC discovered that Wigner's scientists were conducting experiments with a critical mass of uranium-235 when the director of the Manhattan Project, Major General Leslie R. Groves, Jr., had forbidden such experiments in August 1946 after the death of Louis Slotin at the Los Alamos Laboratory. Wigner argued that Groves's order had been superseded, but was forced to terminate the experiments, which were completely different from the one that killed Slotin.[47] Feeling unsuited to a managerial role in such an environment, he left Oak Ridge in 1947 and returned to Princeton University,[48] although he maintained a consulting role with the facility for many years.[45] In the postwar period, he served on a number of government bodies, including the National Bureau of Standards from 1947 to 1951, the mathematics panel of the National Research Council from 1951 to 1954, the physics panel of the National Science Foundation, and the influential General Advisory Committee of the Atomic Energy Commission from 1952 to 1957 and again from 1959 to 1964.[49] He also contributed to civil defense.[50] Wigner was elected to the American Academy of Arts and Sciences in 1950.[51] Near the end of his life, Wigner's thoughts turned more philosophical. In 1960, he published a now classic article on the philosophy of mathematics and of physics, which has become his best-known work outside technical mathematics and physics, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences".[52] He argued that biology and cognition could be the origin of physical concepts, as we humans perceive them, and that the happy coincidence that mathematics and physics were so well matched, seemed to be "unreasonable" and hard to explain.[52] His original paper has provoked and inspired many responses across a wide range of disciplines. These included Richard Hamming in Computer Science,[53] Arthur Lesk in Molecular Biology,[54] Peter Norvig in data mining,[55] Max Tegmark in Physics,[56] Ivor Grattan-Guinness in Mathematics,[57] and Vela Velupillai in Economics.[58] Turning to philosophical questions about the theory of quantum mechanics, Wigner developed a thought experiment (later called Wigner's Friend paradox) to illustrate his belief that consciousness is foundational to the quantum mechanical measurement process. He thereby followed an ontological approach that sets human's consciousness at the center: "All that quantum mechanics purports to provide are probability connections between subsequent impressions (also called 'apperceptions') of the consciousness".[59] Measurements are understood as the interactions which create the impressions in our consciousness (and as a result modify the wave function of the "measured" physical system), an idea which has been called the "consciousness causes collapse" interpretation. Interestingly, Hugh Everett III (a student of Wigner) discussed Wigner's thought experiment in the introductory part of his 1957 dissertation as an "amusing, but extremely hypothetical drama".[60] In an early draft of Everett's work, one also finds a drawing of the Wigner's Friend situation,[61] which must be seen as the first evidence on paper of the thought experiment that was later assigned to be Wigner's. This suggests that Everett must at least have discussed the problem together with Wigner. In November 1963, Wigner called for the allocation of 10% of the national defense budget to be spent on nuclear blast shelters and survival resources, arguing that such an expenditure would be less costly than disarmament. Wigner considered a recent Woods Hole study's conclusion that a nuclear strike would kill 20% of Americans to be a very modest projection and that the country could recover from such an attack more quickly than Germany had recovered from the devastation of World War II.[62] Wigner was awarded the Nobel Prize in Physics in 1963 "for his contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles".[1] The prize was shared that year, with the other half of the award divided between Maria Goeppert-Mayer and J. Hans D. Jensen.[1] Wigner professed that he had never considered the possibility that this might occur, and added: "I never expected to get my name in the newspapers without doing something wicked."[63] He also won the Franklin Medal in 1950,[64] the Enrico Fermi award in 1958,[65] the Atoms for Peace Award in 1959,[66] the Max Planck Medal in 1961,[67] the National Medal of Science in 1969,[68] the Albert Einstein Award in 1972,[69] the Golden Plate Award of the American Academy of Achievement in 1974,[70] and the eponymous Wigner Medal in 1978.[71] In 1968 he gave the Josiah Willard Gibbs lecture.[72][73] After his retirement from Princeton in 1971, Wigner prepared the first edition of Symmetries and Reflections, a collection of philosophical essays, and became more involved in international and political meetings; around this time he became a leader[74] and vocal defender[75] of the Unification Church's annual International Conference on the Unity of the Sciences. Mary died in November 1977. In 1979, Wigner married his third wife, Eileen Clare-Patton (Pat) Hamilton, the widow of physicist Donald Ross Hamilton, the dean of the graduate school at Princeton University, who had died in 1972.[76] In 1992, at the age of 90, he published his memoirs, The Recollections of Eugene P. Wigner with Andrew Szanton. In it, Wigner said: "The full meaning of life, the collective meaning of all human desires, is fundamentally a mystery beyond our grasp. As a young man, I chafed at this state of affairs. But by now I have made peace with it. I even feel a certain honor to be associated with such a mystery."[77] In his collection of essays 'Philosophical Reflections and Syntheses' (1995), he commented: "It was not possible to formulate the laws of quantum mechanics in a fully consistent way without reference to consciousness."[78] Wigner was credited as a member of the advisory board for the Western Goals Foundation, a private domestic intelligence agency created in the US in 1979 to "fill the critical gap caused by the crippling of the FBI, the disabling of the House Un-American Activities Committee and the destruction of crucial government files".[79] Wigner died of pneumonia at the University Medical Center in Princeton, New Jersey on 1 January 1995.[80] He was survived by his wife Eileen (died 2010) and children Erika, David and Martha, and his sisters Bertha and Margit.[69] Publications • 1958 (with Alvin M. Weinberg). Physical Theory of Neutron Chain Reactors University of Chicago Press. ISBN 0-226-88517-8 • 1959. Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. New York: Academic Press. Translation by J. J. Griffin of 1931, Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren, Vieweg Verlag, Braunschweig. • 1970 Symmetries and Reflections: Scientific Essays. Indiana University Press, Bloomington ISBN 0-262-73021-9 • 1992 (as told to Andrew Szanton). The Recollections of Eugene P. Wigner. Plenum. ISBN 0-306-44326-0 • 1995 (with Jagdish Mehra and Arthur Wightman, eds.). Philosophical Reflections and Syntheses. Springer, Berlin ISBN 3-540-63372-3 Selected contributions Theoretical physics • Bargmann–Wigner equations • Jordan–Wigner transformation • Newton–Wigner localization • Polynomial Wigner–Ville distribution • Relativistic Breit–Wigner distribution • Thomas–Wigner rotation • Wigner–Eckart theorem • Wigner–Inonu contraction • Wigner–Seitz cell • Wigner–Seitz radius • Wigner–Weyl transform • Wigner–Wilkins spectrum • Wigner's classification • Wigner quasi-probability distribution • Wigner's friend • Wigner's theorem • Wigner crystal • Wigner D-matrix • Wigner effect • Wigner energy • Wigner lattice • Wigner's disease • Thomas–Wigner rotation • Von Neumann–Wigner interpretation • Wigner–Witmer correlation rules Mathematics • Gabor–Wigner transform • Modified Wigner distribution function • Wigner distribution function • Wigner semicircle distribution • Wigner rotation • Wigner quasi-probability distribution • Wigner semicircle distribution • 6-j symbol • 9-j symbol • Wigner 3-j symbols • Wigner–İnönü group contraction • Wigner surmise See also • List of things named after Eugene Wigner • The Martians (scientists) • List of Jewish Nobel laureates Notes 1. "The Nobel Prize in Physics 1963". Nobel Foundation. Retrieved May 19, 2015. 2. Szanton 1992, pp. 9–12. 3. Szanton 1992, pp. 164–166. 4. Szanton 1992, pp. 14–15. 5. Szanton 1992, pp. 22–24. 6. Szanton 1992, pp. 33–34, 47. 7. Szanton 1992, pp. 49–53. 8. Szanton 1992, pp. 40–43. 9. Szanton 1992, p. 38. 10. Szanton 1992, p. 59. 11. Szanton 1992, pp. 64–65. 12. Szanton 1992, pp. 68–75. 13. Szanton 1992, pp. 93–94. 14. Szanton 1992, pp. 76–84. 15. Szanton 1992, pp. 101–106. 16. Szanton 1992, pp. 109–112. 17. Wigner, E. (1927). "Einige Folgerungen aus der Schrödingerschen Theorie für die Termstrukturen". Zeitschrift für Physik (in German). 43 (9–10): 624–652. Bibcode:1927ZPhy...43..624W. doi:10.1007/BF01397327. S2CID 124334051. 18. Szanton 1992, pp. 116–119. 19. Wightman, A.S. (1995). "Eugene Paul Wigner 1902–1995" (PDF). Notices of the American Mathematical Society. 42 (7): 769–771. 20. Wigner 1931, pp. 251–254. 21. Wigner 1959, pp. 233–236. 22. Szanton 1992, pp. 127–132. 23. Szanton 1992, pp. 136, 153–155. 24. Szanton 1992, pp. 163–166. 25. Szanton 1992, pp. 171–172. 26. Szanton 1992, pp. 173–178. 27. Szanton 1992, pp. 184–185. 28. Szanton 1992, pp. 197–202. 29. Szanton 1992, p. 215. 30. Szanton 1992, pp. 205–207. 31. "Obituary: Mary Wigner". Physics Today. 31 (7): 58. July 1978. Bibcode:1978PhT....31g..58.. doi:10.1063/1.2995119. Archived from the original on 2013-09-27. 32. "Wigner Biography". St Andrews University. Retrieved August 10, 2013. 33. Szanton 1992, pp. 217–218. 34. "Chicago Pile 1 Pioneers". Los Alamos National Laboratory. Archived from the original on February 4, 2012. Retrieved August 10, 2013. 35. Szanton 1992, pp. 233–235. 36. Wigner & Weinberg 1992, p. 8. 37. Szanton 1992, p. 249. 38. Wigner, E. P. (1946). "Theoretical Physics in the Metallurgical Laboratory of Chicago". Journal of Applied Physics. 17 (11): 857–863. Bibcode:1946JAP....17..857W. doi:10.1063/1.1707653. 39. Rhodes 1995, p. 277. 40. Wilson, Richard (November 8, 2002). "A young Scientist's Meetings with Wigner in America". Budapest: Wigner Symposium, Hungarian Academy of Sciences. Retrieved May 16, 2015. 41. Leal, L. C. "Brief Review of R-Matrix Theory" (PDF). Retrieved August 12, 2013. • The original paper is Wigner, E. P.; Eisenbud, L. (1 July 1947). "Higher Angular Momenta and Long Range Interaction in Resonance Reactions". Physical Review. 72 (1): 29–41. Bibcode:1947PhRv...72...29W. doi:10.1103/PhysRev.72.29. 42. "APS Member History". search.amphilsoc.org. Retrieved 2023-04-03. 43. "Eugene P. Wigner". www.nasonline.org. Retrieved 2023-04-03. 44. Johnson & Schaffer 1994, p. 31. 45. Seitz, Frederick; Vogt, Erich; Weinberg, Alvin M. "Eugene Paul Wigner". Biographical Memoirs. National Academies Press. Retrieved 20 August 2013. 46. "ORNL History. Chapter 2: High-Flux Years. Section: Research and Regulations". ORNL Review. Oak Ridge National Laboratory's Communications and Community Outreach. Archived from the original on 16 March 2013. Retrieved 20 August 2013. Oak Ridge at that time was so terribly bureaucratized that I am sorry to say I could not stand it. 47. Hewlett & Duncan 1969, pp. 38–39. 48. Johnson & Schaffer 1994, p. 49. 49. Szanton 1992, p. 270. 50. Szanton 1992, pp. 288–290. 51. "Eugene Paul Wigner". American Academy of Arts & Sciences. 9 February 2023. Retrieved 2023-04-03. 52. Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. 13 (1): 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. S2CID 6112252. Archived from the original on February 28, 2011. Retrieved December 24, 2008. 53. Hamming, R. W. (1980). "The Unreasonable Effectiveness of Mathematics". The American Mathematical Monthly. 87 (2): 81–90. doi:10.2307/2321982. hdl:10945/55827. JSTOR 2321982. Archived from the original on 2007-02-03. Retrieved 2015-08-28. 54. Lesk, A. M. (2000). "The unreasonable effectiveness of mathematics in molecular biology". The Mathematical Intelligencer. 22 (2): 28–37. doi:10.1007/BF03025372. S2CID 120102813. 55. Halevy, A.; Norvig, P.; Pereira, F. (2009). "The Unreasonable Effectiveness of Data" (PDF). IEEE Intelligent Systems. 24 (2): 8–12. doi:10.1109/MIS.2009.36. S2CID 14300215. 56. Tegmark, Max (2008). "The Mathematical Universe". Foundations of Physics. 38 (2): 101–150. arXiv:0704.0646. Bibcode:2008FoPh...38..101T. doi:10.1007/s10701-007-9186-9. S2CID 9890455. 57. Grattan-Guinness, I. (2008). "Solving Wigner's mystery: The reasonable (though perhaps limited) effectiveness of mathematics in the natural sciences". The Mathematical Intelligencer. 30 (3): 7–17. doi:10.1007/BF02985373. S2CID 123174309. 58. Velupillai, K. V. (2005). "The unreasonable ineffectiveness of mathematics in economics" (PDF). Cambridge Journal of Economics. 29 (6): 849–872. CiteSeerX 10.1.1.194.6586. doi:10.1093/cje/bei084. Archived from the original (PDF) on 2005-03-11. Retrieved 2017-10-24. 59. Wigner, E. P. (1995), "Remarks on the Mind-Body Question", Philosophical Reflections and Syntheses, Berlin, Heidelberg: Springer Berlin Heidelberg, pp. 247–260, doi:10.1007/978-3-642-78374-6_20, ISBN 978-3-540-63372-3, retrieved 2021-12-01 60. Everett, Hugh (1957-07-01). ""Relative State" Formulation of Quantum Mechanics". Reviews of Modern Physics. 29 (3): 454–462. Bibcode:1957RvMP...29..454E. doi:10.1103/RevModPhys.29.454. ISSN 0034-6861. 61. Barrett, Jeffrey A.; Byrne, Peter, eds. (2012-05-20). The Everett Interpretation of Quantum Mechanics. doi:10.1515/9781400842742. ISBN 9781400842742. 62. Lyons, R. (1963, November 22). Asks Better Civil Defense for Atomic Victory. New York Daily News, p. 6. 63. Szanton 1992, p. 147. 64. "Eugene P. Wigner". The Franklin Institute. 2014-01-15. Retrieved May 19, 2015. 65. "Eugene P. Wigner, 1958". United States Department of Energy Office of Science. Retrieved May 19, 2015. 66. "Guide to Atoms for Peace Awards Records MC.0010". Massachusetts Institute of Technology. Archived from the original on August 5, 2015. Retrieved May 19, 2015. 67. "Preisträger Max Planck nach Jahren" (in German). Deutschen Physikalischen Gesellschaft. Archived from the original on September 23, 2015. Retrieved May 19, 2015. 68. "The President's National Medal of Science: Recipient Details". United States National Science Foundation. Retrieved May 19, 2015. 69. "Eugene P. Wigner". Princeton University. 70. "Golden Plate Awardees of the American Academy of Achievement". www.achievement.org. American Academy of Achievement. 71. "The Wigner Medal". University of Texas. Retrieved May 19, 2015. 72. "Josiah Willard Gibbs Lectures". American Mathematical Society. Retrieved May 15, 2015. 73. Wigner, Eugene P (1968). "Problems of symmetry in old and new physics". Bulletin of the American Mathematical Society. 75 (5): 793–815. doi:10.1090/S0002-9904-1968-12047-6. MR 1566474. 74. Seitz, Frederick; Vogt, Erich; Weinberg, Alvin. "Eugene Paul Wigner 1902-1995: A biographical memoir" (PDF). National Academy of Sciences. National Academies Press. Retrieved 9 May 2023. 75. Johnson, Thomas (9 November 1975). "'Unification' Science Parley Is Defended". The New York Times. Retrieved 9 May 2023. 76. Szanton 1992, p. 305. 77. Szanton 1992, p. 318. 78. Wigner, Mehra & Wightman 1995, p. 14. 79. Staff writer (Jan. 2, 1989). "Western Goals Foundation." Interhemispheric Resource Center/International Relations Center. Archived from the original. 80. Broad, William J. (January 4, 1995). "Eugene Wigner, 92, Quantum Theorist Who Helped Usher In Atomic Age, Dies". The New York Times. Retrieved May 19, 2015. References • Hewlett, Richard G.; Duncan, Francis (1969). Atomic Shield, 1947–1952 (PDF). A History of the United States Atomic Energy Commission. University Park, Pennsylvania: Pennsylvania State University Press. ISBN 978-0-520-07187-2. OCLC 3717478. Retrieved 7 March 2015. • Johnson, Leland; Schaffer, Daniel (1994). Oak Ridge National Laboratory: the first fifty years. Knoxville: University of Tennessee Press. ISBN 978-0-87049-853-4. • Rhodes, Richard (1995). Dark Sun: The Making of the Hydrogen Bomb. New York: Simon & Schuster. ISBN 978-0-684-80400-2. • N. Mukunda (1995) "Eugene Paul Wigner – A tribute", Current Science 69(4): 375–85 MR1347799 • Szanton, Andrew (1992). The Recollections of Eugene P. Wigner. Plenum. ISBN 978-0-306-44326-8. • Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren (in German). Braunschweig, Germany: Friedrich Vieweg und Sohn. ASIN B000K1MPEI. • Wigner, E. P. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. translation from German by J. J. Griffin. New York: Academic Press. ISBN 978-0-12-750550-3. • Wigner, E. P.; Weinberg, Alvin M. (1992). The collected works of Eugene Paul Wigner, Volume 5, Part A, Nuclear energy. Berlin: Springer. ISBN 978-0-387-55343-6. • Wigner, Eugene Paul; Mehra, Jagdish; Wightman, A. S. (1995). Volume 7, Part B, Philosophical Reflections and Syntheses. Berlin: Springer. ISBN 978-3-540-63372-3. External links Wikimedia Commons has media related to Eugene Wigner. Wikiquote has quotations related to Eugene Wigner. • 1964 Audio Interview with Eugene Wigner by Stephane Groueff Voices of the Manhattan Project • O'Connor, John J.; Robertson, Edmund F., "Eugene Wigner", MacTutor History of Mathematics Archive, University of St Andrews • Eugene Wigner at the Mathematics Genealogy Project • EPW contributions to the theory of the atomic nucleus and the elementary particles, particularly through the discovery and application of fundamental symmetry principles at the Wayback Machine (archived July 9, 2011) • 1984 interview with Wigner, in: The Princeton University Mathematics Community in the 1930s. at the Wayback Machine (archived October 5, 2012) • Oral history interview transcript with Eugene Wigner 21 November 1963, American Institute of Physics, Niels Bohr Library & Archives at the Wayback Machine (archived October 1, 2013) • Archived March 26, 2015, at the Wayback Machine • Wigner Jenö Iskolás Évei by Radnai Gyula, ELTE, Fizikai Szemle 2007/2 – 62.o. (Hungarian). Description of the childhood and especially of the school-years in Budapest, with some interesting photos too. • Interview with Eugene P. Wigner on John von Neumann at the Charles Babbage Institute, University of Minnesota, Minneapolis – Wigner talks about his association with John von Neumann during their school years in Hungary, their graduate studies in Berlin, and their appointments to Princeton in 1930. Wigner discusses von Neumann's contributions to the theory of quantum mechanics, Wigner's own work in this area, and von Neumann's interest in the application of theory to the atomic bomb project. • Works by or about Eugene Wigner at Internet Archive • Eugene Wigner on Nobelprize.org including the Nobel Lecture, December 12, 1963 Events, Laws of Nature, and Invariance Principles Laureates of the Nobel Prize in Physics 1901–1925 • 1901: Röntgen • 1902: Lorentz / Zeeman • 1903: Becquerel / P. Curie / M. Curie • 1904: Rayleigh • 1905: Lenard • 1906: J. J. Thomson • 1907: Michelson • 1908: Lippmann • 1909: Marconi / Braun • 1910: Van der Waals • 1911: Wien • 1912: Dalén • 1913: Kamerlingh Onnes • 1914: Laue • 1915: W. L. Bragg / W. H. Bragg • 1916 • 1917: Barkla • 1918: Planck • 1919: Stark • 1920: Guillaume • 1921: Einstein • 1922: N. Bohr • 1923: Millikan • 1924: M. Siegbahn • 1925: Franck / Hertz 1926–1950 • 1926: Perrin • 1927: Compton / C. Wilson • 1928: O. Richardson • 1929: De Broglie • 1930: Raman • 1931 • 1932: Heisenberg • 1933: Schrödinger / Dirac • 1934 • 1935: Chadwick • 1936: Hess / C. D. Anderson • 1937: Davisson / G. P. Thomson • 1938: Fermi • 1939: Lawrence • 1940 • 1941 • 1942 • 1943: Stern • 1944: Rabi • 1945: Pauli • 1946: Bridgman • 1947: Appleton • 1948: Blackett • 1949: Yukawa • 1950: Powell 1951–1975 • 1951: Cockcroft / Walton • 1952: Bloch / Purcell • 1953: Zernike • 1954: Born / Bothe • 1955: Lamb / Kusch • 1956: Shockley / Bardeen / Brattain • 1957: C. N. Yang / T. D. Lee • 1958: Cherenkov / Frank / Tamm • 1959: Segrè / Chamberlain • 1960: Glaser • 1961: Hofstadter / Mössbauer • 1962: Landau • 1963: Wigner / Goeppert Mayer / Jensen • 1964: Townes / Basov / Prokhorov • 1965: Tomonaga / Schwinger / Feynman • 1966: Kastler • 1967: Bethe • 1968: Alvarez • 1969: Gell-Mann • 1970: Alfvén / Néel • 1971: Gabor • 1972: Bardeen / Cooper / Schrieffer • 1973: Esaki / Giaever / Josephson • 1974: Ryle / Hewish • 1975: A. Bohr / Mottelson / Rainwater 1976–2000 • 1976: Richter / Ting • 1977: P. W. Anderson / Mott / Van Vleck • 1978: Kapitsa / Penzias / R. Wilson • 1979: Glashow / Salam / Weinberg • 1980: Cronin / Fitch • 1981: Bloembergen / Schawlow / K. Siegbahn • 1982: K. Wilson • 1983: Chandrasekhar / Fowler • 1984: Rubbia / Van der Meer • 1985: von Klitzing • 1986: Ruska / Binnig / Rohrer • 1987: Bednorz / Müller • 1988: Lederman / Schwartz / Steinberger • 1989: Ramsey / Dehmelt / Paul • 1990: Friedman / Kendall / R. Taylor • 1991: de Gennes • 1992: Charpak • 1993: Hulse / J. Taylor • 1994: Brockhouse / Shull • 1995: Perl / Reines • 1996: D. Lee / Osheroff / R. Richardson • 1997: Chu / Cohen-Tannoudji / Phillips • 1998: Laughlin / Störmer / Tsui • 1999: 't Hooft / Veltman • 2000: Alferov / Kroemer / Kilby 2001– present • 2001: Cornell / Ketterle / Wieman • 2002: Davis / Koshiba / Giacconi • 2003: Abrikosov / Ginzburg / Leggett • 2004: Gross / Politzer / Wilczek • 2005: Glauber / Hall / Hänsch • 2006: Mather / Smoot • 2007: Fert / Grünberg • 2008: Nambu / Kobayashi / Maskawa • 2009: Kao / Boyle / Smith • 2010: Geim / Novoselov • 2011: Perlmutter / Schmidt / Riess • 2012: Wineland / Haroche • 2013: Englert / Higgs • 2014: Akasaki / Amano / Nakamura • 2015: Kajita / McDonald • 2016: Thouless / Haldane / Kosterlitz • 2017: Weiss / Barish / Thorne • 2018: Ashkin / Mourou / Strickland • 2019: Peebles / Mayor / Queloz • 2020: Penrose / Genzel / Ghez • 2021: Parisi / Hasselmann / Manabe • 2022: Aspect / Clauser / Zeilinger 1963 Nobel Prize laureates Chemistry • Karl Ziegler (Germany) • Giulio Natta (Italy) Literature (1963) • Giorgos Seferis (Greece) Peace • International Committee of the Red Cross (Switzerland) • International Federation of Red Cross and Red Crescent Societies (Switzerland) Physics • Eugene Wigner (United States) • Maria Goeppert Mayer (United States) • J. Hans D. Jensen (Germany) Physiology or Medicine • John Eccles (Great Britain) • Alan Lloyd Hodgkin (Great Britain) • Andrew Huxley (Australia) Nobel Prize recipients 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 Hungarian or Hungarian-American Nobel Laureates Chemistry • Zsigmondy Richárd (1925) • Hevesy György (1943) • Polányi János (1986) • Oláh György (1994) • Avram Hershko (2004) Literature • Kertész Imre (2002) Physics • Lénárd Fülöp (1905) • Wigner Jenő (1963) • Gábor Dénes (1971) Physiology or Medicine • Bárány Róbert (1914) • Szent-Györgyi Albert (1937) • Békésy György (1961) Economic Sciences • Harsányi János (1994) United States National Medal of Science laureates Behavioral and social science 1960s 1964 Neal Elgar Miller 1980s 1986 Herbert A. Simon 1987 Anne Anastasi George J. Stigler 1988 Milton Friedman 1990s 1990 Leonid Hurwicz Patrick Suppes 1991 George A. Miller 1992 Eleanor J. Gibson 1994 Robert K. Merton 1995 Roger N. Shepard 1996 Paul Samuelson 1997 William K. Estes 1998 William Julius Wilson 1999 Robert M. Solow 2000s 2000 Gary Becker 2003 R. Duncan Luce 2004 Kenneth Arrow 2005 Gordon H. Bower 2008 Michael I. Posner 2009 Mortimer Mishkin 2010s 2011 Anne Treisman 2014 Robert Axelrod 2015 Albert Bandura Biological sciences 1960s 1963 C. B. van Niel 1964 Theodosius Dobzhansky Marshall W. Nirenberg 1965 Francis P. Rous George G. Simpson Donald D. Van Slyke 1966 Edward F. Knipling Fritz Albert Lipmann William C. Rose Sewall Wright 1967 Kenneth S. Cole Harry F. Harlow Michael Heidelberger Alfred H. Sturtevant 1968 Horace Barker Bernard B. Brodie Detlev W. Bronk Jay Lush Burrhus Frederic Skinner 1969 Robert Huebner Ernst Mayr 1970s 1970 Barbara McClintock Albert B. Sabin 1973 Daniel I. Arnon Earl W. Sutherland Jr. 1974 Britton Chance Erwin Chargaff James V. Neel James Augustine Shannon 1975 Hallowell Davis Paul Gyorgy Sterling B. Hendricks Orville Alvin Vogel 1976 Roger Guillemin Keith Roberts Porter Efraim Racker E. O. Wilson 1979 Robert H. Burris Elizabeth C. Crosby Arthur Kornberg Severo Ochoa Earl Reece Stadtman George Ledyard Stebbins Paul Alfred Weiss 1980s 1981 Philip Handler 1982 Seymour Benzer Glenn W. Burton Mildred Cohn 1983 Howard L. Bachrach Paul Berg Wendell L. Roelofs Berta Scharrer 1986 Stanley Cohen Donald A. Henderson Vernon B. Mountcastle George Emil Palade Joan A. Steitz 1987 Michael E. DeBakey Theodor O. Diener Harry Eagle Har Gobind Khorana Rita Levi-Montalcini 1988 Michael S. Brown Stanley Norman Cohen Joseph L. Goldstein Maurice R. Hilleman Eric R. Kandel Rosalyn Sussman Yalow 1989 Katherine Esau Viktor Hamburger Philip Leder Joshua Lederberg Roger W. Sperry Harland G. Wood 1990s 1990 Baruj Benacerraf Herbert W. Boyer Daniel E. Koshland Jr. Edward B. Lewis David G. Nathan E. Donnall Thomas 1991 Mary Ellen Avery G. Evelyn Hutchinson Elvin A. Kabat Robert W. Kates Salvador Luria Paul A. Marks Folke K. Skoog Paul C. Zamecnik 1992 Maxine Singer Howard Martin Temin 1993 Daniel Nathans Salome G. Waelsch 1994 Thomas Eisner Elizabeth F. Neufeld 1995 Alexander Rich 1996 Ruth Patrick 1997 James Watson Robert A. Weinberg 1998 Bruce Ames Janet Rowley 1999 David Baltimore Jared Diamond Lynn Margulis 2000s 2000 Nancy C. Andreasen Peter H. Raven Carl Woese 2001 Francisco J. Ayala George F. Bass Mario R. Capecchi Ann Graybiel Gene E. Likens Victor A. McKusick Harold Varmus 2002 James E. Darnell Evelyn M. Witkin 2003 J. Michael Bishop Solomon H. Snyder Charles Yanofsky 2004 Norman E. Borlaug Phillip A. Sharp Thomas E. Starzl 2005 Anthony Fauci Torsten N. Wiesel 2006 Rita R. Colwell Nina Fedoroff Lubert Stryer 2007 Robert J. Lefkowitz Bert W. O'Malley 2008 Francis S. Collins Elaine Fuchs J. Craig Venter 2009 Susan L. Lindquist Stanley B. Prusiner 2010s 2010 Ralph L. Brinster Rudolf Jaenisch 2011 Lucy Shapiro Leroy Hood Sallie Chisholm 2012 May Berenbaum Bruce Alberts 2013 Rakesh K. Jain 2014 Stanley Falkow Mary-Claire King Simon Levin Chemistry 1960s 1964 Roger Adams 1980s 1982 F. Albert Cotton Gilbert Stork 1983 Roald Hoffmann George C. Pimentel Richard N. Zare 1986 Harry B. Gray Yuan Tseh Lee Carl S. Marvel Frank H. Westheimer 1987 William S. Johnson Walter H. Stockmayer Max Tishler 1988 William O. Baker Konrad E. Bloch Elias J. Corey 1989 Richard B. Bernstein Melvin Calvin Rudolph A. Marcus Harden M. McConnell 1990s 1990 Elkan Blout Karl Folkers John D. Roberts 1991 Ronald Breslow Gertrude B. Elion Dudley R. Herschbach Glenn T. Seaborg 1992 Howard E. Simmons Jr. 1993 Donald J. Cram Norman Hackerman 1994 George S. Hammond 1995 Thomas Cech Isabella L. Karle 1996 Norman Davidson 1997 Darleane C. Hoffman Harold S. Johnston 1998 John W. Cahn George M. Whitesides 1999 Stuart A. Rice John Ross Susan Solomon 2000s 2000 John D. Baldeschwieler Ralph F. Hirschmann 2001 Ernest R. Davidson Gábor A. Somorjai 2002 John I. Brauman 2004 Stephen J. Lippard 2005 Tobin J. Marks 2006 Marvin H. Caruthers Peter B. Dervan 2007 Mostafa A. El-Sayed 2008 Joanna Fowler JoAnne Stubbe 2009 Stephen J. Benkovic Marye Anne Fox 2010s 2010 Jacqueline K. Barton Peter J. Stang 2011 Allen J. Bard M. Frederick Hawthorne 2012 Judith P. Klinman Jerrold Meinwald 2013 Geraldine L. Richmond 2014 A. Paul Alivisatos Engineering sciences 1960s 1962 Theodore von Kármán 1963 Vannevar Bush John Robinson Pierce 1964 Charles S. Draper Othmar H. Ammann 1965 Hugh L. Dryden Clarence L. Johnson Warren K. Lewis 1966 Claude E. Shannon 1967 Edwin H. Land Igor I. Sikorsky 1968 J. Presper Eckert Nathan M. Newmark 1969 Jack St. Clair Kilby 1970s 1970 George E. Mueller 1973 Harold E. Edgerton Richard T. Whitcomb 1974 Rudolf Kompfner Ralph Brazelton Peck Abel Wolman 1975 Manson Benedict William Hayward Pickering Frederick E. Terman Wernher von Braun 1976 Morris Cohen Peter C. Goldmark Erwin Wilhelm Müller 1979 Emmett N. Leith Raymond D. Mindlin Robert N. Noyce Earl R. Parker Simon Ramo 1980s 1982 Edward H. Heinemann Donald L. Katz 1983 Bill Hewlett George Low John G. Trump 1986 Hans Wolfgang Liepmann Tung-Yen Lin Bernard M. Oliver 1987 Robert Byron Bird H. Bolton Seed Ernst Weber 1988 Daniel C. Drucker Willis M. Hawkins George W. Housner 1989 Harry George Drickamer Herbert E. Grier 1990s 1990 Mildred Dresselhaus Nick Holonyak Jr. 1991 George H. Heilmeier Luna B. Leopold H. Guyford Stever 1992 Calvin F. Quate John Roy Whinnery 1993 Alfred Y. Cho 1994 Ray W. Clough 1995 Hermann A. Haus 1996 James L. Flanagan C. Kumar N. Patel 1998 Eli Ruckenstein 1999 Kenneth N. Stevens 2000s 2000 Yuan-Cheng B. Fung 2001 Andreas Acrivos 2002 Leo Beranek 2003 John M. Prausnitz 2004 Edwin N. Lightfoot 2005 Jan D. Achenbach 2006 Robert S. Langer 2007 David J. Wineland 2008 Rudolf E. Kálmán 2009 Amnon Yariv 2010s 2010 Shu Chien 2011 John B. Goodenough 2012 Thomas Kailath Mathematical, statistical, and computer sciences 1960s 1963 Norbert Wiener 1964 Solomon Lefschetz H. Marston Morse 1965 Oscar Zariski 1966 John Milnor 1967 Paul Cohen 1968 Jerzy Neyman 1969 William Feller 1970s 1970 Richard Brauer 1973 John Tukey 1974 Kurt Gödel 1975 John W. Backus Shiing-Shen Chern George Dantzig 1976 Kurt Otto Friedrichs Hassler Whitney 1979 Joseph L. Doob Donald E. Knuth 1980s 1982 Marshall H. Stone 1983 Herman Goldstine Isadore Singer 1986 Peter Lax Antoni Zygmund 1987 Raoul Bott Michael Freedman 1988 Ralph E. Gomory Joseph B. Keller 1989 Samuel Karlin Saunders Mac Lane Donald C. Spencer 1990s 1990 George F. Carrier Stephen Cole Kleene John McCarthy 1991 Alberto Calderón 1992 Allen Newell 1993 Martin David Kruskal 1994 John Cocke 1995 Louis Nirenberg 1996 Richard Karp Stephen Smale 1997 Shing-Tung Yau 1998 Cathleen Synge Morawetz 1999 Felix Browder Ronald R. Coifman 2000s 2000 John Griggs Thompson Karen Uhlenbeck 2001 Calyampudi R. Rao Elias M. Stein 2002 James G. Glimm 2003 Carl R. de Boor 2004 Dennis P. Sullivan 2005 Bradley Efron 2006 Hyman Bass 2007 Leonard Kleinrock Andrew J. Viterbi 2009 David B. Mumford 2010s 2010 Richard A. Tapia S. R. Srinivasa Varadhan 2011 Solomon W. Golomb Barry Mazur 2012 Alexandre Chorin David Blackwell 2013 Michael Artin Physical sciences 1960s 1963 Luis W. Alvarez 1964 Julian Schwinger Harold Urey Robert Burns Woodward 1965 John Bardeen Peter Debye Leon M. Lederman William Rubey 1966 Jacob Bjerknes Subrahmanyan Chandrasekhar Henry Eyring John H. Van Vleck Vladimir K. Zworykin 1967 Jesse Beams Francis Birch Gregory Breit Louis Hammett George Kistiakowsky 1968 Paul Bartlett Herbert Friedman Lars Onsager Eugene Wigner 1969 Herbert C. Brown Wolfgang Panofsky 1970s 1970 Robert H. Dicke Allan R. Sandage John C. Slater John A. Wheeler Saul Winstein 1973 Carl Djerassi Maurice Ewing Arie Jan Haagen-Smit Vladimir Haensel Frederick Seitz Robert Rathbun Wilson 1974 Nicolaas Bloembergen Paul Flory William Alfred Fowler Linus Carl Pauling Kenneth Sanborn Pitzer 1975 Hans A. Bethe Joseph O. Hirschfelder Lewis Sarett Edgar Bright Wilson Chien-Shiung Wu 1976 Samuel Goudsmit Herbert S. Gutowsky Frederick Rossini Verner Suomi Henry Taube George Uhlenbeck 1979 Richard P. Feynman Herman Mark Edward M. Purcell John Sinfelt Lyman Spitzer Victor F. Weisskopf 1980s 1982 Philip W. Anderson Yoichiro Nambu Edward Teller Charles H. Townes 1983 E. Margaret Burbidge Maurice Goldhaber Helmut Landsberg Walter Munk Frederick Reines Bruno B. Rossi J. Robert Schrieffer 1986 Solomon J. Buchsbaum H. Richard Crane Herman Feshbach Robert Hofstadter Chen-Ning Yang 1987 Philip Abelson Walter Elsasser Paul C. Lauterbur George Pake James A. Van Allen 1988 D. Allan Bromley Paul Ching-Wu Chu Walter Kohn Norman Foster Ramsey Jr. Jack Steinberger 1989 Arnold O. Beckman Eugene Parker Robert Sharp Henry Stommel 1990s 1990 Allan M. Cormack Edwin M. McMillan Robert Pound Roger Revelle 1991 Arthur L. Schawlow Ed Stone Steven Weinberg 1992 Eugene M. Shoemaker 1993 Val Fitch Vera Rubin 1994 Albert Overhauser Frank Press 1995 Hans Dehmelt Peter Goldreich 1996 Wallace S. Broecker 1997 Marshall Rosenbluth Martin Schwarzschild George Wetherill 1998 Don L. Anderson John N. Bahcall 1999 James Cronin Leo Kadanoff 2000s 2000 Willis E. Lamb Jeremiah P. Ostriker Gilbert F. White 2001 Marvin L. Cohen Raymond Davis Jr. Charles Keeling 2002 Richard Garwin W. Jason Morgan Edward Witten 2003 G. Brent Dalrymple Riccardo Giacconi 2004 Robert N. Clayton 2005 Ralph A. Alpher Lonnie Thompson 2006 Daniel Kleppner 2007 Fay Ajzenberg-Selove Charles P. Slichter 2008 Berni Alder James E. Gunn 2009 Yakir Aharonov Esther M. Conwell Warren M. 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Wigner's theorem Wigner's theorem, proved by Eugene Wigner in 1931,[2] is a cornerstone of the mathematical formulation of quantum mechanics. The theorem specifies how physical symmetries such as rotations, translations, and CPT are represented on the Hilbert space of states. The physical states in a quantum theory are represented by unit vectors in Hilbert space up to a phase factor, i.e. by the complex line or ray the vector spans. In addition, by the Born rule the absolute value of the unit vectors inner product, or equivalently the cosine squared of the angle between the lines the vectors span, corresponds to the transition probability. Ray space, in mathematics known as projective Hilbert space, is the space of all unit vectors in Hilbert space up to the equivalence relation of differing by a phase factor. By Wigner's theorem, any transformation of ray space that preserves the absolute value of the inner products can be represented by a unitary or antiunitary transformation of Hilbert space, which is unique up to a phase factor. As a consequence, the representation of a symmetry group on ray space can be lifted to a projective representation or sometimes even an ordinary representation on Hilbert space. Rays and ray space It is a postulate of quantum mechanics that vectors in Hilbert space that are scalar nonzero multiples of each other represent the same pure state. A ray belonging to the vector $\Psi \in H\setminus \{0\}$ is the complex line through the origin [3][4] ${\underline {\Psi }}=\left\{\lambda \Psi :\lambda \in \mathbb {C} \right\}$ :\lambda \in \mathbb {C} \right\}} . Two nonzero vectors $\Psi _{1},\Psi _{2}$ define the same ray, if and only if they differ by some nonzero complex number: $\Psi _{1}=\lambda \Psi _{2}$, $\lambda \in \mathbb {C} ^{}=\mathbb {C} \setminus \{0\}$. Alternatively, we can consider a ray ${\underline {\Psi }}$ as a set of vectors with norm 1 that span the same line, a unit ray, by intersecting the line ${\underline {\Psi }}$ with the unit sphere [5] $SH=\{\Phi \in H\mid \\|\Phi \\|^{2}=1\}$. Two unit vectors $\Psi _{1},\Psi _{2}$ then define the same unit ray ${\underline {\Psi _{1}}}={\underline {\Psi _{2}}}$ if they differ by a phase factor: $\Psi _{1}=e^{i\alpha }\Psi _{2}$. This is the more usual picture in physics. The set of rays is in one to one correspondence with the set of unit rays and we can identify them. There is also a one-to-one correspondence between physical pure states $\rho $ and (unit) rays ${\underline {\Phi }}$ given by $\rho =P_{\Phi }={\frac {\|\Phi \rangle \langle \Phi \|}{\langle \Phi \|\Phi \rangle }}$ where $P_{\Phi }$ is the orthogonal projection on the line ${\underline {\Phi }}$. In either interpretation, if $\Phi \in {\underline {\Psi }}$ or $P_{\Phi }=P_{\Psi }$ then $\Phi $ is a representative of ${\underline {\Psi }}$.[nb 1] The space of all rays is called ray space. It can be defined in several ways. One may define an equivalence relation $\approx $ on $H\setminus \{0\}$ by $\Psi \approx \Phi \Leftrightarrow \Psi =\lambda \Phi ,\quad \lambda \in \mathbb {C} \setminus \{0\},$ and define ray space as $\mathbb {P} H=H\setminus \{0\}/{\approx }.$ Alternatively define a relation ≅ as an equivalence relation on the sphere $SH$. The unit ray space $\mathbb {P} H$, is an incarnation of ray space defined (making no notational distinction with ray space) as the set of equivalence classes $\mathbb {P} H=SH/{\cong }.$ A third equivalent definition of ray space is as pure state ray space i.e. as density matrices that are orthogonal projections of rank 1 $\mathbb {P} H=\{P\in B(H)\mid P^{2}=P=P^{\dagger },\mathbb {tr} (P)=1\}.$ Each of these definitions make it clear that ray space is nothing but another name for projective Hilbert space.[6] If $\dim(H)=N$ is finite, $\mathbb {P} H$ has real dimension $2N-2$. In fact, $\mathbb {P} H$ is a compact complex manifold of dimension $N-1$ which (by choosing a basis) is isomorphic to the projective space $\mathbb {C} \mathbb {P} ^{N-1}=\mathbb {P} (\mathbb {C} ^{N})$. For example, the Bloch sphere $\mathbb {P} (\lambda _{1}\|+\rangle +\lambda _{2}\|-\rangle ,\ (\lambda _{1},\lambda _{2})\in \mathbb {C} ^{2}\setminus \{0\})$ is isomorphic to the Riemann sphere $\mathbb {C} \mathbb {P} ^{1}$. Ray space (i.e. projective space) takes a little getting used to, but is a very well studied object that predates quantum mechanics going back to the study of perspective by renaissance artists. It is not a vector space with well-defined linear combinations of rays. However, for every two vectors $\Psi _{1},\Psi _{2}$ and ratio of complex numbers $(\lambda _{1}:\lambda _{2})$ (i.e. element of $\mathbb {C} \mathbb {P} ^{1}$) there is a well defined ray ${\underline {\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}}}$. Moreover, for distinct rays ${\underline {\Psi }}_{1},{\underline {\Psi }}_{2}$ (i.e. linearly independent lines) there is a projective line of rays of the form ${\underline {\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}}}$ in $\mathbb {P} H$: all 1 dimensional complex lines in the 2 complex dimensional plane spanned by $\Psi _{1}$ and $\Psi _{2}$ in $H$). The Hilbert space structure on $H$ defines additional structure on ray space. Define the ray correlation (or ray product) ${\underline {\Psi }}\cdot {\underline {\Phi }}={\frac {\left\|\left\langle \Psi ,\Phi \right\rangle \right\|}{\\|\Phi \\|\\|\Psi \\|}}={\sqrt {\mathrm {tr} (P_{\Psi }P_{\Phi })}},$ where $\langle \,,\,\rangle $ is the Hilbert space inner product, and $\Psi ,\Phi $ are representatives of ${\underline {\Phi }}$ and ${\underline {\Psi }}$. Note that the righthand side is independent of the choice of representatives. The physical significance of this definition is that according to the Born rule, another postulate of quantum mechanics, the transition probabilities between normalised states $\Psi $ and $\Phi $ in Hilbert space is given by $P(\Psi \rightarrow \Phi )=\|\langle \Psi ,\Phi \rangle \|^{2}=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}$ i.e. we can define Born's rule on ray space by. $P({\underline {\Psi }}\to {\underline {\Phi }}):=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}.$ Geometrically, we can define an angle $\theta $ with $0\leq \theta \leq \pi /2$ between the lines ${\underline {\Phi }}$ and ${\underline {\Psi }}$ by $\cos(\theta )=({\underline {\Psi }}\cdot {\underline {\Phi }})$. The angle then turns out to satisfy the triangle inequality and defines a metric structure on ray space which comes from a Riemannian metric, the Fubini-Study metric. Symmetry transformations Loosely speaking, a symmetry transformation is a change in which "nothing happens"[7] or a "change in our point of view"[8] that does not change the outcomes of possible experiments. For example, translating a system in a homogeneous environment should have no qualitative effect on the outcomes of experiments made on the system. Likewise for rotating a system in an isotropic environment. This becomes even clearer when one considers the mathematically equivalent passive transformations, i.e. simply changes of coordinates and let the system be. Usually, the domain and range Hilbert spaces are the same. An exception would be (in a non-relativistic theory) the Hilbert space of electron states that is subjected to a charge conjugation transformation. In this case the electron states are mapped to the Hilbert space of positron states and vice versa. However this means that the symmetry acts on the direct sum of the Hilbert spaces. A transformation of a physical system is a transformation of states, hence mathematically a transformation, not of the Hilbert space, but of its ray space. Hence, in quantum mechanics, a transformation of a physical system gives rise to a bijective ray transformation $T$ ${\begin{aligned}T:\mathbb {P} H&\to \mathbb {P} H\\{\underline {\Psi }}&\mapsto T{\underline {\Psi }}.\\\end{aligned}}$ Since the composition of two physical transformations and the reversal of a physical transformation are also physical transformations, the set of all ray transformations so obtained is a group acting on $\mathbb {P} H$. Not all bijections of $\mathbb {P} H$ are permissible as symmetry transformations, however. Physical transformations must preserve Born's rule. For a physical transformation, the transition probabilities in the transformed and untransformed systems should be preserved: $P({\underline {\Psi }}\rightarrow {\underline {\Phi }})=\left({\underline {\Psi }}\cdot {\underline {\Phi }}\right)^{2}=\left(T{\underline {\Psi }}\cdot T{\underline {\Phi }}\right)^{2}=P\left(T\Psi \rightarrow T\Phi \right)$ A bijective ray transformation $\mathbb {P} H\to \mathbb {P} H$ is called a symmetry transformation iff[9] $T{\underline {\Psi }}\cdot T{\underline {\Phi }}={\underline {\Psi }}\cdot {\underline {\Phi }},\quad \forall {\underline {\Psi }},{\underline {\Phi }}\in \mathbb {P} H.$ A geometric interpretation, is that a symmetry transformation is an isometry of ray space. Some facts about symmetry transformations that can be verified using the definition: • The product of two symmetry transformations, i.e. two symmetry transformations applied in succession, is a symmetry transformation. • Any symmetry transformation has an inverse. • The identity transformation is a symmetry transformation. • Multiplication of symmetry transformations is associative. The set of symmetry transformations thus forms a group, the symmetry group of the system. Some important frequently occurring subgroups in the symmetry group of a system are realizations of • The symmetric group with its subgroups. This is important on the exchange of particle labels. • The Poincaré group. It encodes the fundamental symmetries of spacetime [NB: a symmetry is defined above as a map on the ray space describing a given system, the notion of symmetry of spacetime has not been defined and is not clear]. • Internal symmetry groups like SU(2) and SU(3). They describe so called internal symmetries, like isospin and color charge peculiar to quantum mechanical systems. These groups are also referred to as symmetry groups of the system. Statement of Wigner's theorem Preliminaries Some preliminary definitions are needed to state the theorem. A transformation $U:H\to K$ of Hilbert spaces is unitary if it is bijective and $\langle U\Psi ,U\Phi \rangle =\langle \Psi ,\Phi \rangle .$ Since $\langle U(\lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2}),\Phi '\rangle =\langle \lambda _{1}\Psi _{1}+\lambda _{2}\Psi _{2},U^{-1}\Phi '\rangle =\lambda _{1}\langle \Psi _{1},U^{-1}\Phi '\rangle +\lambda _{2}\langle \Psi _{2},U^{-1}\Phi '\rangle =$ $=\lambda _{1}\langle U\Psi _{1},\Phi '\rangle +\lambda _{2}\langle U\Psi _{2},\Phi '\rangle =\langle \lambda _{1}U\Psi _{1}+\lambda _{2}U\Psi _{2},\Phi '\rangle $ for all $\Phi '\in K$, a unitary transformation is automatically linear and $U^{\dagger }=U^{-1}$. Likewise, a transformation $A:H\to K$ is antiunitary if it is bijective and $\langle A\Psi ,A\Phi \rangle =\langle \Psi ,\Phi \rangle ^{}=\langle \Phi ,\Psi \rangle .$ As above, an antiunitary transformation is necessarily antilinear.[nb 2] Both variants are real linear and additive. Given a unitary transformation $U:H\to K$ of Hilbert spaces, define ${\begin{aligned}T_{U}:\mathbb {P} H&\to \mathbb {P} K\\{\underline {\Psi }}&\mapsto {\underline {U\Psi }}\\\end{aligned}}$ This is a symmetry transformation since $T{\underline {\Psi }}\cdot T{\underline {\Phi }}={\frac {\left\|\langle U\Psi ,U\Phi \rangle \right\|}{\\|U\Psi \\|\\|U\Phi \\|}}={\frac {\left\|\langle \Psi ,\Phi \rangle \right\|}{\\|\Psi \\|\\|\Phi \\|}}={\underline {\Psi }}\cdot {\underline {\Phi }}.$ In the same way an antiunitary transformation of Hilbert space induces a symmetry transformation. One says that a transformation $U:H\to K$ of Hilbert spaces is compatible with the transformation $T:\mathbb {P} H\to \mathbb {P} K$ of ray spaces if $T=T_{U}$ or equivalently $U\Psi \in T{\underline {\Psi }}$ for all $\Psi \in H\setminus \{0\}$.[10] Transformations of Hilbert space induced by either a unitary linear transformation or an antiunitary antilinear operator are obviously compatible with the transformations or ray space they induce as described. Statement Wigner's theorem states a converse of the above:[11] Wigner's theorem (1931) — If $H$ and $K$ are Hilbert spaces and if $T:\mathbb {P} H\to \mathbb {P} K$ is a symmetry transformation, then there exists a unitary or antiunitary transformation $V:H\to K$ which is compatible with $T$. If $\dim(H)\geq 2$ , $V$ is either unitary or antiunitary. If $\dim(H)=1$ (and $\mathbb {P} H$ and $\mathbb {P} K$ consist of a single point), all unitary transformations $U:H\to K$ and all antiunitary transformations $A:H\to K$ are compatible with $T$. If $V_{1}$ and $V_{2}$ are both compatible with $T$ then $V_{1}=e^{i\alpha }V_{2}$ for some $\alpha \in \mathbb {R} $ Proofs can be found in Wigner (1931, 1959), Bargmann (1964) and Weinberg (2002). Antiunitary and antilinear transformations are less prominent in physics. They are all related to a reversal of the direction of the flow of time.[12] Remark 1: The significance of the uniqueness part of the theorem is that it specifies the degree of uniqueness of the representation on $H$. For example, one might be tempted to believe that $V\Psi =Ue^{i\alpha (\Psi )}\Psi ,\alpha (\Psi )\in \mathbb {R} ,\Psi \in H\quad ({\text{wrong unless }}\alpha (\Psi ){\text{ is const.}})$ would be admissible, with $\alpha (\Psi )\neq \alpha (\Phi )$ for $\langle \Psi ,\Phi \rangle =0$ but this is not the case according to the theorem.[nb 3][13] In fact such a $V$ would not be additive. Remark 2: Whether $T$ must be represented by a unitary or antiunitary operator is determined by topology. If $\dim _{\mathbb {C} }(\mathbb {P} H)=\dim _{\mathbb {C} }(\mathbb {P} K)\geq 1$, the second cohomology $H^{2}(\mathbb {P} H)$ has a unique generator $c_{\mathbb {P} H}$ such that for a (equivalently for every) complex projective line $L\subset \mathbb {P} H$, one has $c_{\mathbb {P} H}\cap [L]=\deg _{L}(c_{\mathbb {P} H}\|_{L})=1$. Since $T$ is a homeomorphism, $T^{}c_{\mathbb {P} K}$ also generates $H^{2}(\mathbb {P} H)$ and so we have $T^{}c_{\mathbb {P} K}=\pm c_{\mathbb {P} H}$. If $U:H\to K$ is unitary, then $T_{U}^{}c_{\mathbb {P} K}=c_{\mathbb {P} H}$ while if $A:H\to K$ is anti linear then $T_{A}^{}c_{\mathbb {P} K}=-c_{\mathbb {P} H}$. Remark 3: Wigner's theorem is in close connection with the fundamental theorem of projective geometry[14] Representations and projective representations If G is a symmetry group (in this latter sense of being embedded as a subgroup of the symmetry group of the system acting on ray space), and if f, g, h ∈ G with fg = h, then $T(f)T(g)=T(h),$ where the T are ray transformations. From the uniqueness part of Wigner's theorem, one has for the compatible representatives U, $U(f)U(g)=\omega (f,g)U(fg)=e^{i\xi (f,g)}U(fg),$ where ω(f, g) is a phase factor.[nb 4] The function ω is called a 2-cocycle or Schur multiplier. A map U:G → GL(V) satisfying the above relation for some vector space V is called a projective representation or a ray representation. If ω(f, g) = 1, then it is called a representation. One should note that the terminology differs between mathematics and physics. In the linked article, term projective representation has a slightly different meaning, but the term as presented here enters as an ingredient and the mathematics per se is of course the same. If the realization of the symmetry group, g → T(g), is given in terms of action on the space of unit rays S = PH, then it is a projective representation G → PGL(H) in the mathematical sense, while its representative on Hilbert space is a projective representation G → GL(H) in the physical sense. Applying the last relation (several times) to the product fgh and appealing to the known associativity of multiplication of operators on H, one finds ${\begin{aligned}\omega (f,g)\omega (fg,h)&=\omega (g,h)\omega (f,gh),\\\xi (f,g)+\xi (fg,h)&=\xi (g,h)+\xi (f,gh)\quad (\operatorname {mod} 2\pi ).\end{aligned}}$ They also satisfy ${\begin{aligned}\omega (g,e)&=\omega (e,g)=1,\\\xi (g,e)&=\xi (e,g)=0\quad (\operatorname {mod} 2\pi ),\\\omega \left(g,g^{-1}\right)&=\omega (g^{-1},g),\\\xi \left(g,g^{-1}\right)&=\xi (g^{-1},g).\\\end{aligned}}$ Upon redefinition of the phases, $U(g)\mapsto {\hat {U}}(g)=\eta (g)U(g)=e^{i\zeta (g)}U(g),$ which is allowed by last theorem, one finds[15][16] ${\begin{aligned}{\hat {\omega }}(g,h)&=\omega (g,h)\eta (g)\eta (h)\eta (gh)^{-1},\\{\hat {\xi }}(g,h)&=\xi (g,h)+\zeta (g)+\zeta (h)-\zeta (gh)\quad (\operatorname {mod} 2\pi ),\end{aligned}}$ where the hatted quantities are defined by ${\hat {U}}(f){\hat {U}}(g)={\hat {\omega }}(f,g){\hat {U}}(fg)=e^{i{\hat {\xi }}(f,g)}{\hat {U}}(fg).$ Utility of phase freedom The following rather technical theorems and many more can be found, with accessible proofs, in Bargmann (1954). The freedom of choice of phases can be used to simplify the phase factors. For some groups the phase can be eliminated altogether. Theorem — If G is semisimple and simply connected, then ω(g, h) = 1 is possible.[17] In the case of the Lorentz group and its subgroup the rotation group SO(3), phases can, for projective representations, be chosen such that ω(g, h) = ± 1. For their respective universal covering groups, SL(2,C) and Spin(3), it is according to the theorem possible to have ω(g, h) = 1, i.e. they are proper representations. The study of redefinition of phases involves group cohomology. Two functions related as the hatted and non-hatted versions of ω above are said to be cohomologous. They belong to the same second cohomology class, i.e. they are represented by the same element in H2(G), the second cohomology group of G. If an element of H2(G) contains the trivial function ω = 0, then it is said to be trivial.[16] The topic can be studied at the level of Lie algebras and Lie algebra cohomology as well.[18][19] Assuming the projective representation g → T(g) is weakly continuous, two relevant theorems can be stated. An immediate consequence of (weak) continuity is that the identity component is represented by unitary operators.[nb 5] Theorem: (Wigner 1939) — The phase freedom can be used such that in a some neighborhood of the identity the map g → U(g) is strongly continuous.[20] Theorem (Bargmann) — In a sufficiently small neighborhood of e, the choice ω(g1, g2) ≡ 1 is possible for semisimple Lie groups (such as SO(n), SO(3,1) and affine linear groups, (in particular the Poincaré group). More precisely, this is exactly the case when the second cohomology group H2(g, R) of the Lie algebra g of G is trivial.[20] See also • Particle physics and representation theory Remarks 1. Here the possibility of superselection rules is ignored. It may be the case that a system cannot be prepared in specific states. For instance, superposition of states with different spin is generally believed impossible. Likewise, states being superpositions of states with different charge are considered impossible. Minor complications due to those issues are treated in Bogoliubov, Logunov & Todorov (1975) 2. Bäurle & de Kerf (1999, p. 342) This is stated but not proved. 3. There is an exception to this. If a superselection rule is in effect, then the phase may depend on in which sector of $H$ the element $\Psi $ resides, see Weinberg 2002, p. 53 4. Again there is an exception. If a superselection rule is in effect, then the phase may depend on in which sector of H h resides on which the operators act, see Weinberg 2002, p. 53 5. This is made plausible as follows. In an open neighborhood in the vicinity of the identity all operators can be expressed as squares. Whether an operator is unitary or antiunitary its square is unitary. Hence they are all unitary in a sufficiently small neighborhood. Such a neighborhood generates the identity. Notes 1. Seitz, Vogt & Weinberg 2000 2. Wigner 1931, pp. 251–254 (in German), Wigner 1959, pp. 233–236 (English translation). 3. Weinberg 2002, p. 49 4. Bäuerle & de Kerf 1999, p. 341 harvnb error: no target: CITEREFBäuerlede_Kerf1999 (help) 5. Simon et al. 2008 6. This approach is used in Bargmann 1964, which serves as a basis reference for the proof outline to be given below. 7. de Kerf & Bäuerle 1999 harvnb error: no target: CITEREFde_KerfBäuerle1999 (help) 8. Weinberg 2002, p. 50 9. de Kerf & Van Groesen 1999, p. 342 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) 10. Bargmann 1964 11. de Kerf & Van Groesen 1999, p. 343 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) 12. Weinberg 2002, p. 51 13. de Kerf & Van Groesen 1999, p. 344 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) This is stated but not proved. 14. Faure 2002 15. de Kerf & Van Groesen 1999, p. 346 harvnb error: no target: CITEREFde_KerfVan_Groesen1999 (help) There is an error in this formula in the book. 16. Weinberg 2002, p. 82 17. Weinberg 2002, Appendix B, Chapter 2 18. Bäurle & de Kerf 1999, pp. 347–349 19. Weinberg 2002, Section 2.7. 20. Straumann 2014 References • Bargmann, V. (1954). "On unitary ray representations of continuous groups". Ann. of Math. 59 (1): 1–46. doi:10.2307/1969831. JSTOR 1969831. • Bargmann, V. (1964). "Note on Wigner's Theorem on Symmetry Operations". Journal of Mathematical Physics. 5 (7): 862–868. Bibcode:1964JMP.....5..862B. doi:10.1063/1.1704188. • Bogoliubov, N. N.; Logunov, A.A.; Todorov, I. T. (1975). Introduction to axiomatic quantum field theory. Mathematical Physics Monograph Series. Vol. 18. Translated to English by Stephan A. Fulling and Ludmila G. Popova. New York: Benjamin. ASIN B000IM4HLS. • Bäurle, C. G. A.; de Kerf, E.A. (1999). E.A. Van Groesen; E. M. De Jager (eds.). Lie algebras Part 1:Finite and infinite dimensional Lie algebras and their applications in physics. Studies in mathematical physics. Vol. 1 (2nd ed.). Amsterdam: North-Holland. ISBN 0-444-88776-8. • Faure, Claude-Alain (2002). "An Elementary Proof of the Fundamental Theorem of Projective Geometry". Geometriae Dedicata. 90: 145–151. doi:10.1023/A:1014933313332. S2CID 115770315. • Seitz, F.; Vogt, E.; Weinberg, A. M. (2000). "Eugene Paul Wigner. 17 November 1902 -- 1 January 1995". Biogr. Mem. Fellows R. Soc. 46: 577–592. doi:10.1098/rsbm.1999.0102. • Simon, R.; Mukunda, N.; Chaturvedi, S.; Srinivasan, V.; Hamhalter, J. (2008). "Two elementary proofs of the Wigner theorem on symmetry in quantum mechanics". Phys. Lett. A. 372 (46): 6847–6852. arXiv:0808.0779. Bibcode:2008PhLA..372.6847S. doi:10.1016/j.physleta.2008.09.052. S2CID 53858196. • Straumann, N. (2014). "Unitary Representations of the inhomogeneous Lorentz Group and their Significance in Quantum Physics". In A. Ashtekar; V. Petkov (eds.). Springer Handbook of Spacetime. Springer Handbooks. pp. 265–278. arXiv:0809.4942. Bibcode:2014shst.book..265S. CiteSeerX 10.1.1.312.401. doi:10.1007/978-3-642-41992-8_14. ISBN 978-3-642-41991-1. S2CID 18493194. • Weinberg, S. (2002), The Quantum Theory of Fields, vol. I, Cambridge University Press, ISBN 978-0-521-55001-7 • Wigner, E. P. (1931). Gruppentheorie und ihre Anwendung auf die Quanten mechanik der Atomspektren (in German). Braunschweig, Germany: Friedrich Vieweg und Sohn. pp. 251–254. ASIN B000K1MPEI. • Wigner, E. P. (1959). Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. translation from German by J. J. Griffin. New York: Academic Press. pp. 233–236. ISBN 978-0-1275-0550-3. Further reading • Mouchet, Amaury (2013). "An alternative proof of Wigner theorem on quantum transformations based on elementary complex analysis". Physics Letters A. 377 (39): 2709–2711. arXiv:1304.1376. Bibcode:2013PhLA..377.2709M. doi:10.1016/j.physleta.2013.08.017. S2CID 42994708. • Molnar, Lajos (1999). "An Algebraic Approach to Wigner's Unitary-Antiunitary Theorem" (PDF). J. Austral. Math. Soc. Ser. A. 65 (3): 354–369. arXiv:math/9808033. Bibcode:1998math......8033M. doi:10.1017/s144678870003593x. S2CID 119593689.
Group contraction In theoretical physics, Eugene Wigner and Erdal İnönü have discussed[1] the possibility to obtain from a given Lie group a different (non-isomorphic) Lie group by a group contraction with respect to a continuous subgroup of it. That amounts to a limiting operation on a parameter of the Lie algebra, altering the structure constants of this Lie algebra in a nontrivial singular manner, under suitable circumstances.[2][3] For example, the Lie algebra of the 3D rotation group SO(3), [X1, X2] = X3, etc., may be rewritten by a change of variables Y1 = εX1, Y2 = εX2, Y3 = X3, as [Y1, Y2] = ε2 Y3, [Y2, Y3] = Y1, [Y3, Y1] = Y2. The contraction limit ε → 0 trivializes the first commutator and thus yields the non-isomorphic algebra of the plane Euclidean group, E2 ~ ISO(2). (This is isomorphic to the cylindrical group, describing motions of a point on the surface of a cylinder. It is the little group, or stabilizer subgroup, of null four-vectors in Minkowski space.) Specifically, the translation generators Y1, Y2, now generate the Abelian normal subgroup of E2 (cf. Group extension), the parabolic Lorentz transformations. Similar limits, of considerable application in physics (cf. correspondence principles), contract • the de Sitter group SO(4, 1) ~ Sp(2, 2) to the Poincaré group ISO(3, 1), as the de Sitter radius diverges: R → ∞; or • the super-anti-de Sitter algebra to the super-Poincaré algebra as the AdS radius diverges R → ∞; or • the Poincaré group to the Galilei group, as the speed of light diverges: c → ∞;[4] or • the Moyal bracket Lie algebra (equivalent to quantum commutators) to the Poisson bracket Lie algebra, in the classical limit as the Planck constant vanishes: ħ → 0. Notes 1. Inönü & Wigner 1953 2. Segal 1951, p. 221 3. Saletan 1961, p. 1 4. Gilmore 2006 References • Dooley, A. H.; Rice, J. W. (1985). "On contractions of semisimple Lie groups" (PDF). Transactions of the American Mathematical Society. 289 (1): 185–202. doi:10.2307/1999695. ISSN 0002-9947. JSTOR 1999695. MR 0779059. • Gilmore, Robert (2006). Lie Groups, Lie Algebras, and Some of Their Applications. Dover Books on Mathematics. Dover Publications. ISBN 0486445291. MR 1275599. • Inönü, E.; Wigner, E. P. (1953). "On the Contraction of Groups and Their Representations". Proc. Natl. Acad. Sci. 39 (6): 510–24. Bibcode:1953PNAS...39..510I. doi:10.1073/pnas.39.6.510. PMC 1063815. PMID 16589298. • Saletan, E. J. (1961). "Contraction of Lie Groups". Journal of Mathematical Physics. 2 (1): 1–21. Bibcode:1961JMP.....2....1S. doi:10.1063/1.1724208. • Segal, I. E. (1951). "A class of operator algebras which are determined by groups". Duke Mathematical Journal. 18: 221. doi:10.1215/S0012-7094-51-01817-0.
Wigner D-matrix The Wigner D-matrix is a unitary matrix in an irreducible representation of the groups SU(2) and SO(3). It was introduced in 1927 by Eugene Wigner, and plays a fundamental role in the quantum mechanical theory of angular momentum. The complex conjugate of the D-matrix is an eigenfunction of the Hamiltonian of spherical and symmetric rigid rotors. The letter D stands for Darstellung, which means "representation" in German. Definition of the Wigner D-matrix Let Jx, Jy, Jz be generators of the Lie algebra of SU(2) and SO(3). In quantum mechanics, these three operators are the components of a vector operator known as angular momentum. Examples are the angular momentum of an electron in an atom, electronic spin, and the angular momentum of a rigid rotor. In all cases, the three operators satisfy the following commutation relations, $[J_{x},J_{y}]=iJ_{z},\quad [J_{z},J_{x}]=iJ_{y},\quad [J_{y},J_{z}]=iJ_{x},$ where i is the purely imaginary number and Planck's constant ħ has been set equal to one. The Casimir operator $J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}$ commutes with all generators of the Lie algebra. Hence, it may be diagonalized together with Jz. This defines the spherical basis used here. That is, there is a complete set of kets (i.e. orthonormal basis of joint eigenvectors labelled by quantum numbers that define the eigenvalues) with $J^{2}\|jm\rangle =j(j+1)\|jm\rangle ,\quad J_{z}\|jm\rangle =m\|jm\rangle ,$ where j = 0, 1/2, 1, 3/2, 2, ... for SU(2), and j = 0, 1, 2, ... for SO(3). In both cases, m = −j, −j + 1, ..., j. A 3-dimensional rotation operator can be written as ${\mathcal {R}}(\alpha ,\beta ,\gamma )=e^{-i\alpha J_{z}}e^{-i\beta J_{y}}e^{-i\gamma J_{z}},$ where α, β, γ are Euler angles (characterized by the keywords: z-y-z convention, right-handed frame, right-hand screw rule, active interpretation). The Wigner D-matrix is a unitary square matrix of dimension 2j + 1 in this spherical basis with elements $D_{m'm}^{j}(\alpha ,\beta ,\gamma )\equiv \langle jm'\|{\mathcal {R}}(\alpha ,\beta ,\gamma )\|jm\rangle =e^{-im'\alpha }d_{m'm}^{j}(\beta )e^{-im\gamma },$ where $d_{m'm}^{j}(\beta )=\langle jm'\|e^{-i\beta J_{y}}\|jm\rangle =D_{m'm}^{j}(0,\beta ,0)$ is an element of the orthogonal Wigner's (small) d-matrix. That is, in this basis, $D_{m'm}^{j}(\alpha ,0,0)=e^{-im'\alpha }\delta _{m'm}$ is diagonal, like the γ matrix factor, but unlike the above β factor. Wigner (small) d-matrix Wigner gave the following expression:[1] $d_{m'm}^{j}(\beta )=[(j+m')!(j-m')!(j+m)!(j-m)!]^{\frac {1}{2}}\sum _{s=s_{\mathrm {min} }}^{s_{\mathrm {max} }}\left[{\frac {(-1)^{m'-m+s}\left(\cos {\frac {\beta }{2}}\right)^{2j+m-m'-2s}\left(\sin {\frac {\beta }{2}}\right)^{m'-m+2s}}{(j+m-s)!s!(m'-m+s)!(j-m'-s)!}}\right].$ The sum over s is over such values that the factorials are nonnegative, i.e. $s_{\mathrm {min} }=\mathrm {max} (0,m-m')$, $s_{\mathrm {max} }=\mathrm {min} (j+m,j-m')$. Note: The d-matrix elements defined here are real. In the often-used z-x-z convention of Euler angles, the factor $(-1)^{m'-m+s}$ in this formula is replaced by $(-1)^{s}i^{m-m'},$ causing half of the functions to be purely imaginary. The realness of the d-matrix elements is one of the reasons that the z-y-z convention, used in this article, is usually preferred in quantum mechanical applications. The d-matrix elements are related to Jacobi polynomials $P_{k}^{(a,b)}(\cos \beta )$ with nonnegative $a$ and $b.$[2] Let $k=\min(j+m,j-m,j+m',j-m').$ If $k={\begin{cases}j+m:&a=m'-m;\quad \lambda =m'-m\\j-m:&a=m-m';\quad \lambda =0\\j+m':&a=m-m';\quad \lambda =0\\j-m':&a=m'-m;\quad \lambda =m'-m\\\end{cases}}$ Then, with $b=2j-2k-a,$ the relation is $d_{m'm}^{j}(\beta )=(-1)^{\lambda }{\binom {2j-k}{k+a}}^{\frac {1}{2}}{\binom {k+b}{b}}^{-{\frac {1}{2}}}\left(\sin {\frac {\beta }{2}}\right)^{a}\left(\cos {\frac {\beta }{2}}\right)^{b}P_{k}^{(a,b)}(\cos \beta ),$ where $a,b\geq 0.$ Properties of the Wigner D-matrix The complex conjugate of the D-matrix satisfies a number of differential properties that can be formulated concisely by introducing the following operators with $(x,y,z)=(1,2,3),$ ${\begin{aligned}{\hat {\mathcal {J}}}_{1}&=i\left(\cos \alpha \cot \beta {\frac {\partial }{\partial \alpha }}+\sin \alpha {\partial \over \partial \beta }-{\cos \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{2}&=i\left(\sin \alpha \cot \beta {\partial \over \partial \alpha }-\cos \alpha {\partial \over \partial \beta }-{\sin \alpha \over \sin \beta }{\partial \over \partial \gamma }\right)\\{\hat {\mathcal {J}}}_{3}&=-i{\partial \over \partial \alpha }\end{aligned}}$ which have quantum mechanical meaning: they are space-fixed rigid rotor angular momentum operators. Further, ${\begin{aligned}{\hat {\mathcal {P}}}_{1}&=i\left({\cos \gamma \over \sin \beta }{\partial \over \partial \alpha }-\sin \gamma {\partial \over \partial \beta }-\cot \beta \cos \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{2}&=i\left(-{\sin \gamma \over \sin \beta }{\partial \over \partial \alpha }-\cos \gamma {\partial \over \partial \beta }+\cot \beta \sin \gamma {\partial \over \partial \gamma }\right)\\{\hat {\mathcal {P}}}_{3}&=-i{\partial \over \partial \gamma },\\\end{aligned}}$ which have quantum mechanical meaning: they are body-fixed rigid rotor angular momentum operators. The operators satisfy the commutation relations $\left[{\mathcal {J}}_{1},{\mathcal {J}}_{2}\right]=i{\mathcal {J}}_{3},\qquad {\hbox{and}}\qquad \left[{\mathcal {P}}_{1},{\mathcal {P}}_{2}\right]=-i{\mathcal {P}}_{3},$ and the corresponding relations with the indices permuted cyclically. The ${\mathcal {P}}_{i}$ satisfy anomalous commutation relations (have a minus sign on the right hand side). The two sets mutually commute, $\left[{\mathcal {P}}_{i},{\mathcal {J}}_{j}\right]=0,\quad i,j=1,2,3,$ and the total operators squared are equal, ${\mathcal {J}}^{2}\equiv {\mathcal {J}}_{1}^{2}+{\mathcal {J}}_{2}^{2}+{\mathcal {J}}_{3}^{2}={\mathcal {P}}^{2}\equiv {\mathcal {P}}_{1}^{2}+{\mathcal {P}}_{2}^{2}+{\mathcal {P}}_{3}^{2}.$ Their explicit form is, ${\mathcal {J}}^{2}={\mathcal {P}}^{2}=-{\frac {1}{\sin ^{2}\beta }}\left({\frac {\partial ^{2}}{\partial \alpha ^{2}}}+{\frac {\partial ^{2}}{\partial \gamma ^{2}}}-2\cos \beta {\frac {\partial ^{2}}{\partial \alpha \partial \gamma }}\right)-{\frac {\partial ^{2}}{\partial \beta ^{2}}}-\cot \beta {\frac {\partial }{\partial \beta }}.$ The operators ${\mathcal {J}}_{i}$ act on the first (row) index of the D-matrix, ${\begin{aligned}{\mathcal {J}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}&=m'D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}\\({\mathcal {J}}_{1}\pm i{\mathcal {J}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}&={\sqrt {j(j+1)-m'(m'\pm 1)}}D_{m'\pm 1,m}^{j}(\alpha ,\beta ,\gamma )^{}\end{aligned}}$ The operators ${\mathcal {P}}_{i}$ act on the second (column) index of the D-matrix, ${\mathcal {P}}_{3}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}=mD_{m'm}^{j}(\alpha ,\beta ,\gamma )^{},$ and, because of the anomalous commutation relation the raising/lowering operators are defined with reversed signs, $({\mathcal {P}}_{1}\mp i{\mathcal {P}}_{2})D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}={\sqrt {j(j+1)-m(m\pm 1)}}D_{m',m\pm 1}^{j}(\alpha ,\beta ,\gamma )^{}.$ Finally, ${\mathcal {J}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}={\mathcal {P}}^{2}D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}=j(j+1)D_{m'm}^{j}(\alpha ,\beta ,\gamma )^{}.$ In other words, the rows and columns of the (complex conjugate) Wigner D-matrix span irreducible representations of the isomorphic Lie algebras generated by $\{{\mathcal {J}}_{i}\}$ and $\{-{\mathcal {P}}_{i}\}$. An important property of the Wigner D-matrix follows from the commutation of ${\mathcal {R}}(\alpha ,\beta ,\gamma )$ with the time reversal operator T, $\langle jm'\|{\mathcal {R}}(\alpha ,\beta ,\gamma )\|jm\rangle =\langle jm'\|T^{\dagger }{\mathcal {R}}(\alpha ,\beta ,\gamma )T\|jm\rangle =(-1)^{m'-m}\langle j,-m'\|{\mathcal {R}}(\alpha ,\beta ,\gamma )\|j,-m\rangle ^{},$ or $D_{m'm}^{j}(\alpha ,\beta ,\gamma )=(-1)^{m'-m}D_{-m',-m}^{j}(\alpha ,\beta ,\gamma )^{}.$ Here, we used that $T$ is anti-unitary (hence the complex conjugation after moving $T^{\dagger }$ from ket to bra), $T\|jm\rangle =(-1)^{j-m}\|j,-m\rangle $ and $(-1)^{2j-m'-m}=(-1)^{m'-m}$. A further symmetry implies $(-1)^{m'-m}D_{mm'}^{j}(\alpha ,\beta ,\gamma )=D_{m'm}^{j}(\gamma ,\beta ,\alpha )~.$ Orthogonality relations The Wigner D-matrix elements $D_{mk}^{j}(\alpha ,\beta ,\gamma )$ form a set of orthogonal functions of the Euler angles $\alpha ,\beta ,$ and $\gamma $: $\int _{0}^{2\pi }d\alpha \int _{0}^{\pi }d\beta \sin \beta \int _{0}^{2\pi }d\gamma \,\,D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )^{\ast }D_{mk}^{j}(\alpha ,\beta ,\gamma )={\frac {8\pi ^{2}}{2j+1}}\delta _{m'm}\delta _{k'k}\delta _{j'j}.$ This is a special case of the Schur orthogonality relations. Crucially, by the Peter–Weyl theorem, they further form a complete set. The fact that $D_{mk}^{j}(\alpha ,\beta ,\gamma )$ are matrix elements of a unitary transformation from one spherical basis $\|lm\rangle $ to another ${\mathcal {R}}(\alpha ,\beta ,\gamma )\|lm\rangle $ is represented by the relations:[3] $\sum _{k}D_{m'k}^{j}(\alpha ,\beta ,\gamma )^{}D_{mk}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'},$ $\sum _{k}D_{km'}^{j}(\alpha ,\beta ,\gamma )^{}D_{km}^{j}(\alpha ,\beta ,\gamma )=\delta _{m,m'}.$ The group characters for SU(2) only depend on the rotation angle β, being class functions, so, then, independent of the axes of rotation, $\chi ^{j}(\beta )\equiv \sum _{m}D_{mm}^{j}(\beta )=\sum _{m}d_{mm}^{j}(\beta )={\frac {\sin \left({\frac {(2j+1)\beta }{2}}\right)}{\sin \left({\frac {\beta }{2}}\right)}},$ and consequently satisfy simpler orthogonality relations, through the Haar measure of the group,[4] ${\frac {1}{\pi }}\int _{0}^{2\pi }d\beta \sin ^{2}\left({\frac {\beta }{2}}\right)\chi ^{j}(\beta )\chi ^{j'}(\beta )=\delta _{j'j}.$ The completeness relation (worked out in the same reference, (3.95)) is $\sum _{j}\chi ^{j}(\beta )\chi ^{j}(\beta ')=\delta (\beta -\beta '),$ whence, for $\beta '=0,$ $\sum _{j}\chi ^{j}(\beta )(2j+1)=\delta (\beta ).$ Kronecker product of Wigner D-matrices, Clebsch-Gordan series The set of Kronecker product matrices $\mathbf {D} ^{j}(\alpha ,\beta ,\gamma )\otimes \mathbf {D} ^{j'}(\alpha ,\beta ,\gamma )$ forms a reducible matrix representation of the groups SO(3) and SU(2). Reduction into irreducible components is by the following equation:[3] $D_{mk}^{j}(\alpha ,\beta ,\gamma )D_{m'k'}^{j'}(\alpha ,\beta ,\gamma )=\sum _{J=\|j-j'\|}^{j+j'}\langle jmj'm'\|J\left(m+m'\right)\rangle \langle jkj'k'\|J\left(k+k'\right)\rangle D_{\left(m+m'\right)\left(k+k'\right)}^{J}(\alpha ,\beta ,\gamma )$ The symbol $\langle j_{1}m_{1}j_{2}m_{2}\|j_{3}m_{3}\rangle $ is a Clebsch–Gordan coefficient. Relation to spherical harmonics and Legendre polynomials For integer values of $l$, the D-matrix elements with second index equal to zero are proportional to spherical harmonics and associated Legendre polynomials, normalized to unity and with Condon and Shortley phase convention: $D_{m0}^{\ell }(\alpha ,\beta ,\gamma )={\sqrt {\frac {4\pi }{2\ell +1}}}Y_{\ell }^{m}(\beta ,\alpha )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta })\,e^{-im\alpha }.$ This implies the following relationship for the d-matrix: $d_{m0}^{\ell }(\beta )={\sqrt {\frac {(\ell -m)!}{(\ell +m)!}}}\,P_{\ell }^{m}(\cos {\beta }).$ A rotation of spherical harmonics $\langle \theta ,\phi \|\ell m'\rangle $ then is effectively a composition of two rotations, $\sum _{m'=-\ell }^{\ell }Y_{\ell }^{m'}(\theta ,\phi )~D_{m'~m}^{\ell }(\alpha ,\beta ,\gamma ).$ When both indices are set to zero, the Wigner D-matrix elements are given by ordinary Legendre polynomials: $D_{0,0}^{\ell }(\alpha ,\beta ,\gamma )=d_{0,0}^{\ell }(\beta )=P_{\ell }(\cos \beta ).$ In the present convention of Euler angles, $\alpha $ is a longitudinal angle and $\beta $ is a colatitudinal angle (spherical polar angles in the physical definition of such angles). This is one of the reasons that the z-y-z convention is used frequently in molecular physics. From the time-reversal property of the Wigner D-matrix follows immediately $\left(Y_{\ell }^{m}\right)^{*}=(-1)^{m}Y_{\ell }^{-m}.$ There exists a more general relationship to the spin-weighted spherical harmonics: $D_{ms}^{\ell }(\alpha ,\beta ,-\gamma )=(-1)^{s}{\sqrt {\frac {4\pi }{2{\ell }+1}}}{}_{s}Y_{\ell }^{m}(\beta ,\alpha )e^{is\gamma }.$[5] Connection with transition probability under rotations The absolute square of an element of the D-matrix, $F_{mm'}(\beta )=\|D_{mm'}^{j}(\alpha ,\beta ,\gamma )\|^{2},$ gives the probability that a system with spin $j$ prepared in a state with spin projection $m$ along some direction will be measured to have a spin projection $m'$ along a second direction at an angle $\beta $ to the first direction. The set of quantities $F_{mm'}$ itself forms a real symmetric matrix, that depends only on the Euler angle $\beta $, as indicated. Remarkably, the eigenvalue problem for the $F$ matrix can be solved completely:[6][7] $\sum _{m'=-j}^{j}F_{mm'}(\beta )f_{\ell }^{j}(m')=P_{\ell }(\cos \beta )f_{\ell }^{j}(m)\qquad (\ell =0,1,\ldots ,2j).$ Here, the eigenvector, $f_{\ell }^{j}(m)$, is a scaled and shifted discrete Chebyshev polynomial, and the corresponding eigenvalue, $P_{\ell }(\cos \beta )$, is the Legendre polynomial. Relation to Bessel functions In the limit when $\ell \gg m,m^{\prime }$ we have $D_{mm'}^{\ell }(\alpha ,\beta ,\gamma )\approx e^{-im\alpha -im'\gamma }J_{m-m'}(\ell \beta )$ where $J_{m-m'}(\ell \beta )$ is the Bessel function and $\ell \beta $ is finite. List of d-matrix elements Using sign convention of Wigner, et al. the d-matrix elements $d_{m'm}^{j}(\theta )$ for j = 1/2, 1, 3/2, and 2 are given below. for j = 1/2 ${\begin{aligned}d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {1}{2}}&=\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {1}{2}}&=-\sin {\frac {\theta }{2}}\end{aligned}}$ for j = 1 ${\begin{aligned}d_{1,1}^{1}&={\frac {1}{2}}(1+\cos \theta )\\[6pt]d_{1,0}^{1}&=-{\frac {1}{\sqrt {2}}}\sin \theta \\[6pt]d_{1,-1}^{1}&={\frac {1}{2}}(1-\cos \theta )\\[6pt]d_{0,0}^{1}&=\cos \theta \end{aligned}}$ for j = 3/2 ${\begin{aligned}d_{{\frac {3}{2}},{\frac {3}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(1+\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {\sqrt {3}}{2}}(1+\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {\sqrt {3}}{2}}(1-\cos \theta )\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {3}{2}},-{\frac {3}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(1-\cos \theta )\sin {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},{\frac {1}{2}}}^{\frac {3}{2}}&={\frac {1}{2}}(3\cos \theta -1)\cos {\frac {\theta }{2}}\\[6pt]d_{{\frac {1}{2}},-{\frac {1}{2}}}^{\frac {3}{2}}&=-{\frac {1}{2}}(3\cos \theta +1)\sin {\frac {\theta }{2}}\end{aligned}}$ for j = 2[8] ${\begin{aligned}d_{2,2}^{2}&={\frac {1}{4}}\left(1+\cos \theta \right)^{2}\\[6pt]d_{2,1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1+\cos \theta \right)\\[6pt]d_{2,0}^{2}&={\sqrt {\frac {3}{8}}}\sin ^{2}\theta \\[6pt]d_{2,-1}^{2}&=-{\frac {1}{2}}\sin \theta \left(1-\cos \theta \right)\\[6pt]d_{2,-2}^{2}&={\frac {1}{4}}\left(1-\cos \theta \right)^{2}\\[6pt]d_{1,1}^{2}&={\frac {1}{2}}\left(2\cos ^{2}\theta +\cos \theta -1\right)\\[6pt]d_{1,0}^{2}&=-{\sqrt {\frac {3}{8}}}\sin 2\theta \\[6pt]d_{1,-1}^{2}&={\frac {1}{2}}\left(-2\cos ^{2}\theta +\cos \theta +1\right)\\[6pt]d_{0,0}^{2}&={\frac {1}{2}}\left(3\cos ^{2}\theta -1\right)\end{aligned}}$ Wigner d-matrix elements with swapped lower indices are found with the relation: $d_{m',m}^{j}=(-1)^{m-m'}d_{m,m'}^{j}=d_{-m,-m'}^{j}.$ Symmetries and special cases ${\begin{aligned}d_{m',m}^{j}(\pi )&=(-1)^{j-m}\delta _{m',-m}\\[6pt]d_{m',m}^{j}(\pi -\beta )&=(-1)^{j+m'}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(\pi +\beta )&=(-1)^{j-m}d_{m',-m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(2\pi +\beta )&=(-1)^{2j}d_{m',m}^{j}(\beta )\\[6pt]d_{m',m}^{j}(-\beta )&=d_{m,m'}^{j}(\beta )=(-1)^{m'-m}d_{m',m}^{j}(\beta )\end{aligned}}$ See also • Clebsch–Gordan coefficients • Tensor operator • Symmetries in quantum mechanics References 1. Wigner, E. P. (1951) [1931]. Gruppentheorie und ihre Anwendungen auf die Quantenmechanik der Atomspektren. Braunschweig: Vieweg Verlag. OCLC 602430512. Translated into English by Group Theory and its Application to the Quantum Mechanics of Atomic Spectra. Translated by Griffin, J.J. Elsevier. 2013 [1959]. ISBN 978-1-4832-7576-5. 2. Biedenharn, L. C.; Louck, J. D. (1981). Angular Momentum in Quantum Physics. Reading: Addison-Wesley. ISBN 0-201-13507-8. 3. Rose, Morris Edgar (1995) [1957]. Elementary theory of angular momentum. Dover. ISBN 0-486-68480-6. OCLC 31374243. 4. Schwinger, J. (January 26, 1952). On Angular Momentum (Technical report). Harvard University, Nuclear Development Associates. doi:10.2172/4389568. NYO-3071, TRN: US200506%%295. 5. Shiraishi, M. (2013). "Appendix A: Spin-Weighted Spherical Harmonic Function" (PDF). Probing the Early Universe with the CMB Scalar, Vector and Tensor Bispectrum (PhD). Nagoya University. pp. 153–4. ISBN 978-4-431-54180-6. 6. Meckler, A. (1958). "Majorana formula". Physical Review. 111 (6): 1447. doi:10.1103/PhysRev.111.1447. 7. Mermin, N.D.; Schwarz, G.M. (1982). "Joint distributions and local realism in the higher-spin Einstein-Podolsky-Rosen experiment". Foundations of Physics. 12 (2): 101. doi:10.1007/BF00736844. S2CID 121648820. 8. Edén, M. (2003). "Computer simulations in solid-state NMR. I. Spin dynamics theory". Concepts in Magnetic Resonance Part A. 17A (1): 117–154. doi:10.1002/cmr.a.10061. External links • Amsler, C.; et al. (Particle Data Group) (2008). "PDG Table of Clebsch-Gordan Coefficients, Spherical Harmonics, and d-Functions" (PDF). Physics Letters B667.
Wigner distribution function The Wigner distribution function (WDF) is used in signal processing as a transform in time-frequency analysis. The WDF was first proposed in physics to account for quantum corrections to classical statistical mechanics in 1932 by Eugene Wigner, and it is of importance in quantum mechanics in phase space (see, by way of comparison: Wigner quasi-probability distribution, also called the Wigner function or the Wigner–Ville distribution). Given the shared algebraic structure between position-momentum and time-frequency conjugate pairs, it also usefully serves in signal processing, as a transform in time-frequency analysis, the subject of this article. Compared to a short-time Fourier transform, such as the Gabor transform, the Wigner distribution function provides the highest possible temporal vs frequency resolution which is mathematically possible within the limitations of the uncertainty principle. The downside is the introduction of large cross terms between every pair of signal components and between positive and negative frequencies, which makes the original formulation of the function a poor fit for most analysis applications. Subsequent modifications have been proposed which preserve the sharpness of the Wigner distribution function but largely suppress cross terms. Mathematical definition There are several different definitions for the Wigner distribution function. The definition given here is specific to time-frequency analysis. Given the time series $x[t]$, its non-stationary auto-covariance function is given by $C_{x}(t_{1},t_{2})=\left\langle \left(x[t_{1}]-\mu [t_{1}]\right)\left(x[t_{2}]-\mu [t_{2}]\right)^{}\right\rangle ,$ where $\langle \cdots \rangle $ denotes the average over all possible realizations of the process and $\mu (t)$ is the mean, which may or may not be a function of time. The Wigner function $W_{x}(t,f)$ is then given by first expressing the autocorrelation function in terms of the average time $t=(t_{1}+t_{2})/2$ and time lag $\tau =t_{1}-t_{2}$, and then Fourier transforming the lag. $W_{x}(t,f)=\int _{-\infty }^{\infty }C_{x}\left(t+{\frac {\tau }{2}},t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .$ So for a single (mean-zero) time series, the Wigner function is simply given by $W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\frac {\tau }{2}}\right)\,x^{}\left(t-{\frac {\tau }{2}}\right)\,e^{-2\pi i\tau f}\,d\tau .$ The motivation for the Wigner function is that it reduces to the spectral density function at all times $t$ for stationary processes, yet it is fully equivalent to the non-stationary autocorrelation function. Therefore, the Wigner function tells us (roughly) how the spectral density changes in time. Time-frequency analysis example Here are some examples illustrating how the WDF is used in time-frequency analysis. Constant input signal When the input signal is constant, its time-frequency distribution is a horizontal line along the time axis. For example, if x(t) = 1, then $W_{x}(t,f)=\int _{-\infty }^{\infty }e^{-i2\pi \tau \,f}\,d\tau =\delta (f).$ Sinusoidal input signal When the input signal is a sinusoidal function, its time-frequency distribution is a horizontal line parallel to the time axis, displaced from it by the sinusoidal signal's frequency. For example, if x(t) = e i2πkt, then ${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }e^{i2\pi k\left(t+{\frac {\tau }{2}}\right)}e^{-i2\pi k\left(t-{\frac {\tau }{2}}\right)}e^{-i2\pi \tau \,f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{-i2\pi \tau \left(f-k\right)}\,d\tau \\&=\delta (f-k).\end{aligned}}$ Chirp input signal When the input signal is a linear chirp function, the instantaneous frequency is a linear function. This means that the time frequency distribution should be a straight line. For example, if $x(t)=e^{i2\pi kt^{2}}$ , then its instantaneous frequency is ${\frac {1}{2\pi }}{\frac {d(2\pi kt^{2})}{dt}}=2kt~,$ and its WDF ${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }e^{i2\pi k\left(t+{\frac {\tau }{2}}\right)^{2}}e^{-i2\pi k\left(t-{\frac {\tau }{2}}\right)^{2}}e^{-i2\pi \tau \,f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{i4\pi kt\tau }e^{-i2\pi \tau f}\,d\tau \\&=\int _{-\infty }^{\infty }e^{-i2\pi \tau (f-2kt)}\,d\tau \\&=\delta (f-2kt)~.\end{aligned}}$ Delta input signal When the input signal is a delta function, since it is only non-zero at t=0 and contains infinite frequency components, its time-frequency distribution should be a vertical line across the origin. This means that the time frequency distribution of the delta function should also be a delta function. By WDF ${\begin{aligned}W_{x}(t,f)&=\int _{-\infty }^{\infty }\delta \left(t+{\frac {\tau }{2}}\right)\delta \left(t-{\frac {\tau }{2}}\right)e^{-i2\pi \tau \,f}\,d\tau \\&=4\int _{-\infty }^{\infty }\delta (2t+\tau )\delta (2t-\tau )e^{-i2\pi \tau f}\,d\tau \\&=4\delta (4t)e^{i4\pi tf}\\&=\delta (t)e^{i4\pi tf}\\&=\delta (t).\end{aligned}}$ The Wigner distribution function is best suited for time-frequency analysis when the input signal's phase is 2nd order or lower. For those signals, WDF can exactly generate the time frequency distribution of the input signal. Boxcar function $x(t)={\begin{cases}1&\|t\|<1/2\\0&{\text{otherwise}}\end{cases}}\qquad $ , the rectangular function ⇒ $W_{x}(t,f)={\begin{cases}{\frac {1}{\pi f}}\sin(2\pi f\{1-2\|t\|\})&\|t\|<1/2\\0&{\mbox{otherwise}}\end{cases}}$ Cross term property The Wigner distribution function is not a linear transform. A cross term ("time beats") occurs when there is more than one component in the input signal, analogous in time to frequency beats.[1] In the ancestral physics Wigner quasi-probability distribution, this term has important and useful physics consequences, required for faithful expectation values. By contrast, the short-time Fourier transform does not have this feature. Negative features of the WDF are reflective of the Gabor limit of the classical signal and physically unrelated to any possible underlay of quantum structure. The following are some examples that exhibit the cross-term feature of the Wigner distribution function. • $x(t)={\begin{cases}\cos(2\pi t)&t\leq -2\\\cos(4\pi t)&-2<t\leq 2\\\cos(3\pi t)&t>2\end{cases}}$ • $x(t)=e^{it^{3}}$ In order to reduce the cross-term difficulty, several approaches have been proposed in the literature,[2][3] some of them leading to new transforms as the modified Wigner distribution function, the Gabor–Wigner transform, the Choi-Williams distribution function and Cohen's class distribution. Properties of the Wigner distribution function The Wigner distribution function has several evident properties listed in the following table. Projection property ${\begin{aligned}\|x(t)\|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,df\\\|X(f)\|^{2}&=\int _{-\infty }^{\infty }W_{x}(t,f)\,dt\end{aligned}}$ Energy property $\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }W_{x}(t,f)\,df\,dt=\int _{-\infty }^{\infty }\|x(t)\|^{2}\,dt=\int _{-\infty }^{\infty }\|X(f)\|^{2}\,df$ Recovery property ${\begin{aligned}\int _{-\infty }^{\infty }W_{x}\left({\frac {t}{2}},f\right)e^{i2\pi ft}\,df&=x(t)x^{}(0)\\\int _{-\infty }^{\infty }W_{x}\left(t,{\frac {f}{2}}\right)e^{i2\pi ft}\,dt&=X(f)X^{}(0)\end{aligned}}$ Mean condition frequency and mean condition time ${\begin{aligned}X(f)&=\|X(f)\|e^{i2\pi \psi (f)},\quad x(t)=\|x(t)\|e^{i2\pi \phi (t)},\\{\text{if }}\phi '(t)&=\|x(t)\|^{-2}\int _{-\infty }^{\infty }fW_{x}(t,f)\,df\\{\text{ and }}-\psi '(f)&=\|X(f)\|^{-2}\int _{-\infty }^{\infty }tW_{x}(t,f)\,dt\end{aligned}}$ Moment properties ${\begin{aligned}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }t^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }t^{n}\|x(t)\|^{2}\,dt\\\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }f^{n}W_{x}(t,f)\,dt\,df&=\int _{-\infty }^{\infty }f^{n}\|X(f)\|^{2}\,df\end{aligned}}$ Real properties $W_{x}^{}(t,f)=W_{x}(t,f)$ Region properties ${\begin{aligned}{\text{If }}x(t)&=0{\text{ for }}t>t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t>t_{0}\\{\text{If }}x(t)&=0{\text{ for }}t<t_{0}{\text{ then }}W_{x}(t,f)=0{\text{ for }}t<t_{0}\end{aligned}}$ Multiplication theorem ${\begin{aligned}{\text{If }}y(t)&=x(t)h(t)\\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(t,\rho )W_{h}(t,f-\rho )\,d\rho \end{aligned}}$ Convolution theorem ${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t-\tau )h(\tau )\,d\tau \\{\text{then }}W_{y}(t,f)&=\int _{-\infty }^{\infty }W_{x}(\rho ,f)W_{h}(t-\rho ,f)\,d\rho \end{aligned}}$ Correlation theorem ${\begin{aligned}{\text{If }}y(t)&=\int _{-\infty }^{\infty }x(t+\tau )h^{}(\tau )\,d\tau {\text{ then }}\\W_{y}(t,\omega )&=\int _{-\infty }^{\infty }W_{x}(\rho ,\omega )W_{h}(-t+\rho ,\omega )\,d\rho \end{aligned}}$ Time-shifting covariance ${\begin{aligned}{\text{If }}y(t)&=x(t-t_{0})\\{\text{then }}W_{y}(t,f)&=W_{x}(t-t_{0},f)\end{aligned}}$ Modulation covariance ${\begin{aligned}{\text{If }}y(t)&=e^{i2\pi f_{0}t}x(t)\\{\text{then }}W_{y}(t,f)&=W_{x}(t,f-f_{0})\end{aligned}}$ Scale covariance ${\begin{aligned}{\text{If }}y(t)&={\sqrt {a}}x(at){\text{ for some }}a>0{\text{ then }}\\{\text{then }}W_{y}(t,f)&=W_{x}(at,{\frac {f}{a}})\end{aligned}}$ Windowed Wigner Distribution Function When a signal is not time limited, its Wigner Distribution Function is hard to implement. Thus, we add a new function(mask) to its integration part, so that we only have to implement part of the original function instead of integrating all the way from negative infinity to positive infinity. Original function: $W_{x}(t,f)=\int _{-\infty }^{\infty }x\left(t+{\frac {\tau }{2}}\right)\cdot x^{}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau $ Function with mask: $W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau $ $w(\tau )$ is real and time-limited Implementation According to definition: ${\begin{aligned}W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau \\W_{x}(t,f)=2\int _{-\infty }^{\infty }w(2\tau ')x\left(t+\tau '\right)\cdot x^{}\left(t-\tau '\right)e^{-j4\pi \tau 'f}\cdot d\tau '\\W_{x}(n\Delta _{t},m\Delta _{f})=2\sum _{p=-\infty }^{\infty }w(2p\Delta _{t})x((n+p)\Delta _{t})x^{\ast }((n-p)\Delta _{t})e^{-j4\pi mp\Delta _{t}\Delta _{f}}\Delta _{t}\end{aligned}}$ Suppose that $w(t)=0$ for $\|t\|>B\rightarrow w(2p\Delta _{t})=0$ for $p<-Q$ and $p>Q$ ${\begin{aligned}W_{x}(n\Delta _{t},m\Delta _{f})=2\sum _{p=-Q}^{Q}w(2p\Delta _{t})x((n+p)\Delta _{t})x^{\ast }((n-p)\Delta _{t})e^{-j4\pi mp\Delta _{t}\Delta _{f}}\Delta _{t}\end{aligned}}$ We take $x(t)=\delta (t-t_{1})+\delta (t-t_{2})$ as example ${\begin{aligned}W_{x}(t,f)=\int _{-\infty }^{\infty }w(\tau )x\left(t+{\frac {\tau }{2}}\right)\cdot x^{}\left(t-{\frac {\tau }{2}}\right)e^{-j2\pi \tau f}\cdot d\tau \,,\end{aligned}}$ where $w(\tau )$ is a real function And then we compare the difference between two conditions. Ideal: $W_{x}(t,f)=0,{\text{ for }}t\neq t_{2},t_{1}$ When mask function $w(\tau )=1$, which means no mask function. $y(t,\tau )=x(t+{\frac {\tau }{2}})$ $y^{}(t,-\tau )=x^{}(t-{\frac {\tau }{2}})$ $W_{x}(t,f)=\int _{-\infty }^{\infty }x(t+{\frac {\tau }{2}})x^{}(t-{\frac {\tau }{2}}e^{-j2\pi \tau f}d\tau $ $=\int _{-\infty }^{\infty }[\delta (t+{\frac {\tau }{2}}-t_{1})+\delta (t+{\frac {\tau }{2}}-t_{2})][\delta (t-{\frac {\tau }{2}}-t_{1})+\delta (t-{\frac {\tau }{2}}-t_{2})]e^{-j2\pi \tau f}\cdot d\tau $ $=4\int _{-\infty }^{\infty }[\delta (2t+\tau -2t_{1})+\delta (2t+\tau -2t_{2})][\delta (2t-\tau -2t_{1})+\delta (2t-\tau -2t_{2})]e^{j2\pi \tau f}\cdot d\tau $ 3 Conditions Then we consider the condition with mask function: We can see that $w(\tau )$ have value only between –B to B, thus conducting with $w(\tau )$ can remove cross term of the function. But if x(t) is not a Delta function nor a narrow frequency function, instead, it is a function with wide frequency or ripple. The edge of the signal may still exist between –B and B, which still cause the cross term problem. for example: See also • Time-frequency representation • Short-time Fourier transform • Spectrogram • Gabor transform • Autocorrelation • Gabor–Wigner transform • Modified Wigner distribution function • Optical equivalence theorem • Polynomial Wigner–Ville distribution • Cohen's class distribution function • Wigner quasi-probability distribution • Transformation between distributions in time-frequency analysis • Bilinear time–frequency distribution References 1. F. Hlawatsch and P. Flandrin, "The interference structure of the Wigner distribution and related time-frequency signal representations", in W. Mecklenbräuker and F. Hlawatsch, The Wigner Distribution - Theory and Applications in Signal Processing 2. B. Boashah (Ed.), Time Frequency Signal Analysis and Processing, Elsevier, 2003 3. P. Flandrin, Time-Frequency/Time-Scale Analysis, Elsevier, 1998 Further reading • Wigner, E. (1932). "On the Quantum Correction for Thermodynamic Equilibrium" (PDF). Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466. • J. Ville, 1948. "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 . • T. A. C. M. Classen and W. F. G. Mecklenbrauker, 1980. "The Wigner distribution-a tool for time-frequency signal analysis; Part I," Philips J. Res., vol. 35, pp. 217–250. • L. Cohen (1989): Proceedings of the IEEE 77 pp. 941–981, Time-frequency distributions---a review • L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995. ISBN 978-0135945322 • S. Qian and D. Chen, Joint Time-Frequency Analysis: Methods and Applications, Chap. 5, Prentice Hall, N.J., 1996. • B. Boashash, "Note on the Use of the Wigner Distribution for Time Frequency Signal Analysis", IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 9, pp. 1518–1521, Sept. 1988. doi:10.1109/29.90380. B. Boashash, editor,Time-Frequency Signal Analysis and Processing – A Comprehensive Reference, Elsevier Science, Oxford, 2003, ISBN 0-08-044335-4. • F. Hlawatsch, G. F. Boudreaux-Bartels: "Linear and quadratic time-frequency signal representation," IEEE Signal Processing Magazine, pp. 21–67, Apr. 1992. • R. L. Allen and D. W. Mills, Signal Analysis: Time, Frequency, Scale, and Structure, Wiley- Interscience, NJ, 2004. • Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2015. • Kakofengitis, D., & Steuernagel, O. (2017). "Wigner's quantum phase space current in weakly anharmonic weakly excited two-state systems" European Physical Journal Plus 14.07.2017 External links Wikimedia Commons has media related to Wigner distribution function. Look up wigner distribution function in Wiktionary, the free dictionary. • Sonogram Visible Speech Under GPL Licensed Freeware for the visual extraction of the Wigner Distribution.
Wigner semicircle distribution The Wigner semicircle distribution, named after the physicist Eugene Wigner, is the probability distribution on [−R, R] whose probability density function f is a scaled semicircle (i.e., a semi-ellipse) centered at (0, 0): $f(x)={2 \over \pi R^{2}}{\sqrt {R^{2}-x^{2}\,}}\,$ Wigner semicircle Probability density function Cumulative distribution function Parameters $R>0\!$ radius (real) Support $x\in [-R;+R]\!$ PDF ${\frac {2}{\pi R^{2}}}\,{\sqrt {R^{2}-x^{2}}}\!$ CDF ${\frac {1}{2}}+{\frac {x{\sqrt {R^{2}-x^{2}}}}{\pi R^{2}}}+{\frac {\arcsin \!\left({\frac {x}{R}}\right)}{\pi }}\!$ for $-R\leq x\leq R$ Mean $0\,$ Median $0\,$ Mode $0\,$ Variance ${\frac {R^{2}}{4}}\!$ Skewness $0\,$ Ex. kurtosis $-1\,$ Entropy $\ln(\pi R)-{\frac {1}{2}}\,$ MGF $2\,{\frac {I_{1}(R\,t)}{R\,t}}$ CF $2\,{\frac {J_{1}(R\,t)}{R\,t}}$ for −R ≤ x ≤ R, and f(x) = 0 if \|x\| > R. The parameter R is commonly referred to as the "radius" parameter of the distribution. The Wigner distribution also coincides with a scaled beta distribution. That is, if Y is a beta-distributed random variable with parameters α = β = 3/2, then the random variable X = 2RY – R exhibits a Wigner semicircle distribution with radius R. The distribution arises as the limiting distribution of the eigenvalues of many random symmetric matrices, that is, as the dimensions of the random matrix approach infinity. The distribution of the spacing or gaps between eigenvalues is addressed by the similarly named Wigner surmise. General properties The Chebyshev polynomials of the third kind are orthogonal polynomials with respect to the Wigner semicircle distribution. For positive integers n, the 2n-th moment of this distribution is $E(X^{2n})=\left({R \over 2}\right)^{2n}C_{n}\,$ where X is any random variable with this distribution and Cn is the nth Catalan number $C_{n}={1 \over n+1}{2n \choose n},\,$ so that the moments are the Catalan numbers if R = 2. (Because of symmetry, all of the odd-order moments are zero.) Making the substitution $x=R\cos(\theta )$ into the defining equation for the moment generating function it can be seen that: $M(t)={\frac {2}{\pi }}\int _{0}^{\pi }e^{Rt\cos(\theta )}\sin ^{2}(\theta )\,d\theta $ which can be solved (see Abramowitz and Stegun §9.6.18) to yield: $M(t)=2\,{\frac {I_{1}(Rt)}{Rt}}$ where $I_{1}(z)$ is the modified Bessel function. Similarly, the characteristic function is given by:[1][2][3] $\varphi (t)=2\,{\frac {J_{1}(Rt)}{Rt}}$ where $J_{1}(z)$ is the Bessel function. (See Abramowitz and Stegun §9.1.20), noting that the corresponding integral involving $\sin(Rt\cos(\theta ))$ is zero.) In the limit of $R$ approaching zero, the Wigner semicircle distribution becomes a Dirac delta function. Relation to free probability In free probability theory, the role of Wigner's semicircle distribution is analogous to that of the normal distribution in classical probability theory. Namely, in free probability theory, the role of cumulants is occupied by "free cumulants", whose relation to ordinary cumulants is simply that the role of the set of all partitions of a finite set in the theory of ordinary cumulants is replaced by the set of all noncrossing partitions of a finite set. Just as the cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is normal, so also, the free cumulants of degree more than 2 of a probability distribution are all zero if and only if the distribution is Wigner's semicircle distribution. Related distributions Wigner (spherical) parabolic distribution Wigner parabolic Parameters $R>0\!$ radius (real) Support $x\in [-R;+R]\!$ PDF ${\frac {3}{4R^{3}}}\,(R^{2}-x^{2})$ CDF ${\frac {1}{4R^{3}}}\,(2R-x)\,(R+x)^{2}$ MGF $3\,{\frac {i_{1}(R\,t)}{R\,t}}$ CF $3\,{\frac {j_{1}(R\,t)}{R\,t}}$ The parabolic probability distribution supported on the interval [−R, R] of radius R centered at (0, 0): $f(x)={3 \over \ 4R^{3}}{(R^{2}-x^{2})}\,$ for −R ≤ x ≤ R, and f(x) = 0 if \|x\| > R. Example. The joint distribution is $\int _{0}^{\pi }\int _{0}^{+2\pi }\int _{0}^{R}f_{X,Y,Z}(x,y,z)R^{2}\,dr\sin(\theta )\,d\theta \,d\phi =1;$ $f_{X,Y,Z}(x,y,z)={\frac {3}{4\pi }}$ Hence, the marginal PDF of the spherical (parametric) distribution is:[4] $f_{X}(x)=\int _{-{\sqrt {1-y^{2}-x^{2}}}}^{+{\sqrt {1-y^{2}-x^{2}}}}\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}f_{X,Y,Z}(x,y,z)\,dy\,dz;$ $f_{X}(x)=\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}2{\sqrt {1-y^{2}-x^{2}}}\,dy\,;$ $f_{X}(x)={3 \over \ 4}{(1-x^{2})}\,;$ such that R=1 The characteristic function of a spherical distribution becomes the pattern multiplication of the expected values of the distributions in X, Y and Z. The parabolic Wigner distribution is also considered the monopole moment of the hydrogen like atomic orbitals. Wigner n-sphere distribution The normalized N-sphere probability density function supported on the interval [−1, 1] of radius 1 centered at (0, 0): $f_{n}(x;n)={(1-x^{2})^{(n-1)/2}\Gamma (1+n/2) \over {\sqrt {\pi }}\Gamma ((n+1)/2)}\,(n>=-1)$, for −1 ≤ x ≤ 1, and f(x) = 0 if \|x\| > 1. Example. The joint distribution is $\int _{-{\sqrt {1-y^{2}-x^{2}}}}^{+{\sqrt {1-y^{2}-x^{2}}}}\int _{-{\sqrt {1-x^{2}}}}^{+{\sqrt {1-x^{2}}}}\int _{0}^{1}f_{X,Y,Z}(x,y,z){{\sqrt {1-x^{2}-y^{2}-z^{2}}}^{(n)}}dxdydz=1;$ $f_{X,Y,Z}(x,y,z)={\frac {3}{4\pi }}$ Hence, the marginal PDF distribution is [4] $f_{X}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;$ such that R=1 The cumulative distribution function (CDF) is $F_{X}(x)={2x\Gamma (1+n/2)_{2}F_{1}(1/2,(1-n)/2;3/2;x^{2}) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\,;$ such that R=1 and n >= -1 The characteristic function (CF) of the PDF is related to the beta distribution as shown below $CF(t;n)={_{1}F_{1}(n/2,;n;jt/2)}\,\urcorner (\alpha =\beta =n/2);$ In terms of Bessel functions this is $CF(t;n)={\Gamma (n/2+1)J_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);$ Raw moments of the PDF are $\mu '_{N}(n)=\int _{-1}^{+1}x^{N}f_{X}(x;n)dx={(1+(-1)^{N})\Gamma (1+n/2) \over \ {2{\sqrt {\pi }}}\Gamma ((2+n+N)/2)};$ Central moments are $\mu _{0}(x)=1$ $\mu _{1}(n)=\mu _{1}'(n)$ $\mu _{2}(n)=\mu _{2}'(n)-\mu _{1}'^{2}(n)$ $\mu _{3}(n)=2\mu _{1}'^{3}(n)-3\mu _{1}'(n)\mu _{2}'(n)+\mu _{3}'(n)$ $\mu _{4}(n)=-3\mu _{1}'^{4}(n)+6\mu _{1}'^{2}(n)\mu _{2}'(n)-4\mu '_{1}(n)\mu '_{3}(n)+\mu '_{4}(n)$ The corresponding probability moments (mean, variance, skew, kurtosis and excess-kurtosis) are: $\mu (x)=\mu _{1}'(x)=0$ $\sigma ^{2}(n)=\mu _{2}'(n)-\mu ^{2}(n)=1/(2+n)$ $\gamma _{1}(n)=\mu _{3}/\mu _{2}^{3/2}=0$ $\beta _{2}(n)=\mu _{4}/\mu _{2}^{2}=3(2+n)/(4+n)$ $\gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3=-6/(4+n)$ Raw moments of the characteristic function are: $\mu '_{N}(n)=\mu '_{N;E}(n)+\mu '_{N;O}(n)=\int _{-1}^{+1}cos^{N}(xt)f_{X}(x;n)dx+\int _{-1}^{+1}sin^{N}(xt)f_{X}(x;n)dx;$ For an even distribution the moments are [5] $\mu _{1}'(t;n:E)=CF(t;n)$ $\mu _{1}'(t;n:O)=0$ $\mu _{1}'(t;n)=CF(t;n)$ $\mu _{2}'(t;n:E)=1/2(1+CF(2t;n))$ $\mu _{2}'(t;n:O)=1/2(1-CF(2t;n))$ $\mu '_{2}(t;n)=1$ $\mu _{3}'(t;n:E)=(CF(3t)+3CF(t;n))/4$ $\mu _{3}'(t;n:O)=0$ $\mu _{3}'(t;n)=(CF(3t;n)+3CF(t;n))/4$ $\mu _{4}'(t;n:E)=(3+4CF(2t;n)+CF(4t;n))/8$ $\mu _{4}'(t;n:O)=(3-4CF(2t;n)+CF(4t;n))/8$ $\mu _{4}'(t;n)=(3+CF(4t;n))/4$ Hence, the moments of the CF (provided N=1) are $\mu (t;n)=\mu _{1}'(t)=CF(t;n)=_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})$ $\sigma ^{2}(t;n)=1-\|CF(t;n)\|^{2}=1-\|_{0}F_{1}({2+n \over 2},-t^{2}/4)\|^{2}$ $\gamma _{1}(n)={\mu _{3} \over \mu _{2}^{3/2}}={_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4})-_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+8\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})\|^{3} \over 4(1-\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}\|^{(3/2)}}$ $\beta _{2}(n)={\mu _{4} \over \mu _{2}^{2}}={3+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))+3_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})(-1+\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}\|^{2})) \over 4(-1+\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}\|^{2}}$ $\gamma _{2}(n)=\mu _{4}/\mu _{2}^{2}-3={-9+_{0}F_{1}({2+n \over 2},-4t^{2})-(4_{0}F_{1}({2+n \over 2},-t^{2}/4)(_{0}F_{1}({2+n \over 2},-9{t^{2} \over 4}))-9_{0}F_{1}({2+n \over 2},-{t^{2} \over 4})+6\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}\|^{3}) \over 4(-1+\|_{0}F_{1}({2+n \over 2},-{t^{2} \over 4}))^{2}\|^{2}}$ Skew and Kurtosis can also be simplified in terms of Bessel functions. The entropy is calculated as $H_{N}(n)=\int _{-1}^{+1}f_{X}(x;n)\ln(f_{X}(x;n))dx$ The first 5 moments (n=-1 to 3), such that R=1 are $\ -\ln(2/\pi );n=-1$ $\ -\ln(2);n=0$ $\ -1/2+\ln(\pi );n=1$ $\ 5/3-\ln(3);n=2$ $\ -7/4-\ln(1/3\pi );n=3$ N-sphere Wigner distribution with odd symmetry applied The marginal PDF distribution with odd symmetry is [4] $f{_{X}}(x;n)={(1-x^{2})^{(n-1)/2)}\Gamma (1+n/2) \over \ {\sqrt {\pi }}\Gamma ((n+1)/2)}\operatorname {sgn}(x)\,;$ such that R=1 Hence, the CF is expressed in terms of Struve functions $CF(t;n)={\Gamma (n/2+1)H_{n/2}(t)/(t/2)^{(n/2)}}\,\urcorner (n>=-1);$ "The Struve function arises in the problem of the rigid-piston radiator mounted in an infinite baffle, which has radiation impedance given by" [6] $Z={\rho c\pi a^{2}[R_{1}(2ka)-iX_{1}(2ka)],}$ $R_{1}={1-{2J_{1}(x) \over 2x},}$ $X_{1}={{2H_{1}(x) \over x},}$ Example (Normalized Received Signal Strength): quadrature terms The normalized received signal strength is defined as $\|R\|={{1 \over N}\|}\sum _{k=1}^{N}\exp[ix_{n}t]\|$ and using standard quadrature terms $x={1 \over N}\sum _{k=1}^{N}\cos(x_{n}t)$ $y={1 \over N}\sum _{k=1}^{N}\sin(x_{n}t)$ Hence, for an even distribution we expand the NRSS, such that x = 1 and y = 0, obtaining ${\sqrt {x^{2}+y^{2}}}=x+{3 \over 2}y^{2}-{3 \over 2}xy^{2}+{1 \over 2}x^{2}y^{2}+O(y^{3})+O(y^{3})(x-1)+O(y^{3})(x-1)^{2}+O(x-1)^{3}$ The expanded form of the Characteristic function of the received signal strength becomes [7] $E[x]={1 \over N}CF(t;n)$ $E[y^{2}]={1 \over 2N}(1-CF(2t;n))$ $E[x^{2}]={1 \over 2N}(1+CF(2t;n))$ $E[xy^{2}]={t^{2} \over 3N^{2}}CF(t;n)^{3}+({N-1 \over 2N^{2}})(1-tCF(2t;n))CF(t;n)$ $E[x^{2}y^{2}]={1 \over 8N^{3}}(1-CF(4t;n))+({N-1 \over 4N^{3}})(1-CF(2t;n)^{2})+({N-1 \over 3N^{3}})t^{2}CF(t;n)^{4}+({(N-1)(N-2) \over N^{3}})CF(t;n)^{2}(1-CF(2t;n))$ See also • Wigner surmise • The Wigner semicircle distribution is the limit of the Kesten–McKay distributions, as the parameter d tends to infinity. • In number-theoretic literature, the Wigner distribution is sometimes called the Sato–Tate distribution. See Sato–Tate conjecture. • Marchenko–Pastur distribution or Free Poisson distribution References 1. Buchanan, Kristopher; Flores, Carlos; Wheeland, Sara; Jensen, Jeffrey; Grayson, David; Huff, Gregory (2017). "Transmit beamforming for radar applications using circularly tapered random arrays". 2017 IEEE Radar Conference (Radar Conf). pp. 0112–0117. doi:10.1109/RADAR.2017.7944181. ISBN 978-1-4673-8823-8. S2CID 38429370. 2. Ryan, Buchanan (29 May 2014). Theory and Applications of Aperiodic (Random) Phased Arrays (Thesis). hdl:1969.1/157918. 3. Overturf, Drew; Buchanan, Kristopher; Jensen, Jeffrey; Wheeland, Sara; Huff, Gregory (2017). "Investigation of beamforming patterns from volumetrically distributed phased arrays". MILCOM 2017 - 2017 IEEE Military Communications Conference (MILCOM). pp. 817–822. doi:10.1109/MILCOM.2017.8170756. ISBN 978-1-5386-0595-0. S2CID 11591305. https://ieeexplore.ieee.org/abstract/document/8170756/ 4. Buchanan, K.; Huff, G. H. (July 2011). "A comparison of geometrically bound random arrays in euclidean space". 2011 IEEE International Symposium on Antennas and Propagation (APSURSI). pp. 2008–2011. doi:10.1109/APS.2011.5996900. ISBN 978-1-4244-9563-4. S2CID 10446533. 5. Thomas M. Cover (1963). "Antenna pattern distribution from random array" (PDF) (MEMORANDUM RM-3502--PR). Santa Monica: The RAND Corporation. Archived (PDF) from the original on September 4, 2021. 6. W., Weisstein, Eric. "Struve Function". mathworld.wolfram.com. Retrieved 2017-07-28.{{cite web}}: CS1 maint: multiple names: authors list (link) 7. "Advanced Beamforming for Distributed and Multi-Beam Networks" (PDF). • Milton Abramowitz and Irene A. Stegun, eds. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. External links • Eric W. Weisstein et al., Wigner's semicircle Probability distributions (list) Discrete univariate with finite support • Benford • Bernoulli • beta-binomial • binomial • categorical • hypergeometric • negative • Poisson binomial • Rademacher • soliton • discrete uniform • Zipf • Zipf–Mandelbrot with infinite support • beta negative binomial • Borel • Conway–Maxwell–Poisson • discrete phase-type • Delaporte • extended negative binomial • Flory–Schulz • Gauss–Kuzmin • geometric • logarithmic • mixed Poisson • negative binomial • Panjer • parabolic fractal • Poisson • Skellam • Yule–Simon • zeta Continuous univariate supported on a bounded interval • arcsine • ARGUS • Balding–Nichols • Bates • beta • beta rectangular • continuous Bernoulli • Irwin–Hall • Kumaraswamy • logit-normal • noncentral beta • PERT • raised cosine • reciprocal • triangular • U-quadratic • uniform • Wigner semicircle supported on a semi-infinite interval • Benini • Benktander 1st kind • Benktander 2nd kind • beta prime • Burr • chi • chi-squared • noncentral • inverse • scaled • Dagum • Davis • Erlang • hyper • exponential • hyperexponential • hypoexponential • logarithmic • F • noncentral • folded normal • Fréchet • gamma • generalized • inverse • gamma/Gompertz • Gompertz • shifted • half-logistic • half-normal • Hotelling's T-squared • inverse Gaussian • generalized • Kolmogorov • Lévy • log-Cauchy • log-Laplace • log-logistic • log-normal • log-t • Lomax • matrix-exponential • Maxwell–Boltzmann • Maxwell–Jüttner • Mittag-Leffler • Nakagami • Pareto • phase-type • Poly-Weibull • Rayleigh • relativistic Breit–Wigner • Rice • truncated normal • type-2 Gumbel • Weibull • discrete • Wilks's lambda supported on the whole real line • Cauchy • exponential power • Fisher's z • Kaniadakis κ-Gaussian • Gaussian q • generalized normal • generalized hyperbolic • geometric stable • Gumbel • Holtsmark • hyperbolic secant • Johnson's SU • Landau • Laplace • asymmetric • logistic • noncentral t • normal (Gaussian) • normal-inverse Gaussian • skew normal • slash • stable • Student's t • Tracy–Widom • variance-gamma • Voigt with support whose type varies • generalized chi-squared • generalized extreme value • generalized Pareto • Marchenko–Pastur • Kaniadakis κ-exponential • Kaniadakis κ-Gamma • Kaniadakis κ-Weibull • Kaniadakis κ-Logistic • Kaniadakis κ-Erlang • q-exponential • q-Gaussian • q-Weibull • shifted log-logistic • Tukey lambda Mixed univariate continuous- discrete • Rectified Gaussian Multivariate (joint) • Discrete: • Ewens • multinomial • Dirichlet • negative • Continuous: • Dirichlet • generalized • multivariate Laplace • multivariate normal • multivariate stable • multivariate t • normal-gamma • inverse • Matrix-valued: • LKJ • matrix normal • matrix t • matrix gamma • inverse • Wishart • normal • inverse • normal-inverse • complex Directional Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham Degenerate and singular Degenerate Dirac delta function Singular Cantor Families • Circular • compound Poisson • elliptical • exponential • natural exponential • location–scale • maximum entropy • mixture • Pearson • Tweedie • wrapped • Category • Commons
Wigner surmise In mathematical physics, the Wigner surmise is a statement about the probability distribution of the spaces between points in the spectra of nuclei of heavy atoms, which have many degrees of freedom, or quantum systems with few degrees of freedom but chaotic classical dynamics. It was proposed by Eugene Wigner in probability theory.[1] The surmise was a result of Wigner's introduction of random matrices in the field of nuclear physics. The surmise consists of two postulates: • In a simple sequence (spin and parity are same), the probability density function for a spacing is given by, $p_{w}(s)={\frac {\pi s}{2}}e^{-\pi s^{2}/4}.$ Here, $s={\frac {S}{D}}$ where S is a particular spacing and D is the mean distance between neighboring intervals.[2] • In a mixed sequence (spin and parity are different), the probability density function can be obtained by randomly superimposing simple sequences. The above result is exact for $2\times 2$ real symmetric matrices $M$, with elements that are independent standard gaussian random variables, with joint distribution proportional to $e^{-{\frac {1}{2}}{\rm {Tr}}(M^{2})}=e^{-{\frac {1}{2}}{\rm {Tr}}\left({\begin{array}{cc}a&b\\b&c\\\end{array}}\right)^{2}}=e^{-{\frac {1}{2}}a^{2}-{\frac {1}{2}}c^{2}-b^{2}}.$ In practice, it is a good approximation for the actual distribution for real symmetric matrices of any dimension. The corresponding result for complex hermitian matrices (which is also exact in the $2\times 2$ case and a good approximation in general) with distribution proportional to $e^{-{\frac {1}{2}}{\rm {Tr}}(MM^{\dagger })}$, is given by $p_{w}(s)={\frac {32s^{2}}{\pi ^{2}}}e^{-4s^{2}/\pi }.$ History During the conference on Neutron Physics by Time-of-Flight, held at Gatlinburg, Tennessee, November 1 and 2, 1956, Wigner delivered a presentation on the theoretical arrangement of neighboring neutron resonances (with matching spin and parity) in heavy nuclei. In the presentation he gave the following guess:[3][4] Perhaps I am now too courageous when I try to guess the distribution of the distances between successive levels (of energies of heavy nuclei). Theoretically, the situation is quite simple if one attacks the problem in a simpleminded fashion. The question is simply what are the distances of the characteristic values of a symmetric matrix with random coefficients. — Eugene Wigner, Results and theory of resonance absorption [5] See also • Wigner semicircle distribution References 1. Mehta, Madan Lal (6 October 2004). Random Matrices By Madan Lal Mehta. p. 13. ISBN 9780080474113. 2. Benenti, Giuliano; Casati, Giulio; Strini, Giuliano (2004). Principles of Quantum Computation and Information. p. 406. ISBN 9789812563453. 3. Conference on Neutron Physics by Time-of-Flight (1957) [1956]. Conference on Neutron Physics by Time-of-Flight, held at Gatlinburg, Tennessee, November 1 and 2, 1956; Oak Ridge National Laboratory Report ORNL-2309. Oak Ridge National Laboratory. p. 67. 4. Porter, Charles E. (1965). Statistical Theories of Spectra: Fluctuations. Elsevier Science & Technology Books. p. 208. ISBN 978-0-12-562356-8. 5. Barrett, Owen; Firk, Frank W. K.; Miller, Steven J.; Turnage-Butterbaugh, Caroline (2016), "From Quantum Systems to L-Functions: Pair Correlation Statistics and Beyond", Open Problems in Mathematics, Cham: Springer International Publishing, pp. 123–171, ISBN 978-3-319-32160-8, retrieved 2023-05-13
Wigner quasiprobability distribution The Wigner quasiprobability distribution (also called the Wigner function or the Wigner–Ville distribution, after Eugene Wigner and Jean-André Ville) is a quasiprobability distribution. It was introduced by Eugene Wigner in 1932[1] to study quantum corrections to classical statistical mechanics. The goal was to link the wavefunction that appears in Schrödinger's equation to a probability distribution in phase space. It is a generating function for all spatial autocorrelation functions of a given quantum-mechanical wavefunction ψ(x). Thus, it maps[2] on the quantum density matrix in the map between real phase-space functions and Hermitian operators introduced by Hermann Weyl in 1927,[3] in a context related to representation theory in mathematics (see Weyl quantization). In effect, it is the Wigner–Weyl transform of the density matrix, so the realization of that operator in phase space. It was later rederived by Jean Ville in 1948 as a quadratic (in signal) representation of the local time-frequency energy of a signal,[4] effectively a spectrogram. In 1949, José Enrique Moyal, who had derived it independently, recognized it as the quantum moment-generating functional,[5] and thus as the basis of an elegant encoding of all quantum expectation values, and hence quantum mechanics, in phase space (see Phase-space formulation). It has applications in statistical mechanics, quantum chemistry, quantum optics, classical optics and signal analysis in diverse fields, such as electrical engineering, seismology, time–frequency analysis for music signals, spectrograms in biology and speech processing, and engine design. Relation to classical mechanics A classical particle has a definite position and momentum, and hence it is represented by a point in phase space. Given a collection (ensemble) of particles, the probability of finding a particle at a certain position in phase space is specified by a probability distribution, the Liouville density. This strict interpretation fails for a quantum particle, due to the uncertainty principle. Instead, the above quasiprobability Wigner distribution plays an analogous role, but does not satisfy all the properties of a conventional probability distribution; and, conversely, satisfies boundedness properties unavailable to classical distributions. For instance, the Wigner distribution can and normally does take on negative values for states which have no classical model—and is a convenient indicator of quantum-mechanical interference. (See below for a characterization of pure states whose Wigner functions are non-negative.) Smoothing the Wigner distribution through a filter of size larger than ħ (e.g., convolving with a phase-space Gaussian, a Weierstrass transform, to yield the Husimi representation, below), results in a positive-semidefinite function, i.e., it may be thought to have been coarsened to a semi-classical one.[lower-alpha 1] Regions of such negative value are provable (by convolving them with a small Gaussian) to be "small": they cannot extend to compact regions larger than a few ħ, and hence disappear in the classical limit. They are shielded by the uncertainty principle, which does not allow precise location within phase-space regions smaller than ħ, and thus renders such "negative probabilities" less paradoxical. Definition and meaning The Wigner distribution W(x,p) of a pure state is defined as $W(x,p)~{\stackrel {\text{def}}{=}}~{\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\psi ^{}(x+y)\psi (x-y)e^{2ipy/\hbar }\,dy,$ where ψ is the wavefunction, and x and p are position and momentum, but could be any conjugate variable pair (e.g. real and imaginary parts of the electric field or frequency and time of a signal). Note that it may have support in x even in regions where ψ has no support in x ("beats"). It is symmetric in x and p: $W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\varphi ^{}(p+q)\varphi (p-q)e^{-2ixq/\hbar }\,dq,$ where φ is the normalized momentum-space wave function, proportional to the Fourier transform of ψ. In 3D, $W({\vec {r}},{\vec {p}})={\frac {1}{(2\pi )^{3}}}\int \psi ^{}({\vec {r}}+\hbar {\vec {s}}/2)\psi ({\vec {r}}-\hbar {\vec {s}}/2)e^{i{\vec {p}}\cdot {\vec {s}}}\,d^{3}s.$ In the general case, which includes mixed states, it is the Wigner transform of the density matrix: $W(x,p)={\frac {1}{\pi \hbar }}\int _{-\infty }^{\infty }\langle x-y\|{\hat {\rho }}\|x+y\rangle e^{2ipy/\hbar }\,dy,$ where ⟨x\|ψ⟩ = ψ(x). This Wigner transformation (or map) is the inverse of the Weyl transform, which maps phase-space functions to Hilbert-space operators, in Weyl quantization. Thus, the Wigner function is the cornerstone of quantum mechanics in phase space. In 1949, José Enrique Moyal elucidated how the Wigner function provides the integration measure (analogous to a probability density function) in phase space, to yield expectation values from phase-space c-number functions g(x, p) uniquely associated to suitably ordered operators Ĝ through Weyl's transform (see Wigner–Weyl transform and property 7 below), in a manner evocative of classical probability theory. Specifically, an operator's Ĝ expectation value is a "phase-space average" of the Wigner transform of that operator: $\langle {\hat {G}}\rangle =\int dx\,dp\,W(x,p)g(x,p).$ Mathematical properties 1. W(x, p) is a real-valued function. 2. The x and p probability distributions are given by the marginals: $\int _{-\infty }^{\infty }dp\,W(x,p)=\langle x\|{\hat {\rho }}\|x\rangle .$ If the system can be described by a pure state, one gets $\int _{-\infty }^{\infty }dp\,W(x,p)=\|\psi (x)\|^{2}.$ $\int _{-\infty }^{\infty }dx\,W(x,p)=\langle p\|{\hat {\rho }}\|p\rangle .$ If the system can be described by a pure state, one has $\int _{-\infty }^{\infty }dx\,W(x,p)=\|\varphi (p)\|^{2}.$ $\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)=\operatorname {Tr} ({\hat {\rho }}).$ Typically the trace of the density matrix ${\hat {\rho }}$ is equal to 1. 3. W(x, p) has the following reflection symmetries: • Time symmetry: $\psi (x)\to \psi (x)^{}\Rightarrow W(x,p)\to W(x,-p).$ • Space symmetry: $\psi (x)\to \psi (-x)\Rightarrow W(x,p)\to W(-x,-p).$ 4. W(x, p) is Galilei-covariant: $\psi (x)\to \psi (x+y)\Rightarrow W(x,p)\to W(x+y,p).$ It is not Lorentz-covariant. 5. The equation of motion for each point in the phase space is classical in the absence of forces: ${\frac {\partial W(x,p)}{\partial t}}={\frac {-p}{m}}{\frac {\partial W(x,p)}{\partial x}}.$ In fact, it is classical even in the presence of harmonic forces. 6. State overlap is calculated as $\|\langle \psi \|\theta \rangle \|^{2}=2\pi \hbar \int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W_{\psi }(x,p)W_{\theta }(x,p).$ 7. Operator expectation values (averages) are calculated as phase-space averages of the respective Wigner transforms: $g(x,p)\equiv \int _{-\infty }^{\infty }dy\,\left\langle x-{\frac {y}{2}}\right\|{\hat {G}}\left\|x+{\frac {y}{2}}\right\rangle e^{ipy/\hbar },$ $\langle \psi \|{\hat {G}}\|\psi \rangle =\operatorname {Tr} ({\hat {\rho }}{\hat {G}})=\int _{-\infty }^{\infty }dx\int _{-\infty }^{\infty }dp\,W(x,p)g(x,p).$ 8. For W(x, p) to represent physical (positive) density matrices, it must satisfy $\int _{-\infty }^{\infty }dx\,\int _{-\infty }^{\infty }dp\,W(x,p)W_{\theta }(x,p)\geq 0$ for all pure states \|θ⟩. 9. By virtue of the Cauchy–Schwarz inequality, for a pure state, it is constrained to be bounded: $-{\frac {2}{h}}\leq W(x,p)\leq {\frac {2}{h}}.$ This bound disappears in the classical limit, ħ → 0. In this limit, W(x, p) reduces to the probability density in coordinate space x, usually highly localized, multiplied by δ-functions in momentum: the classical limit is "spiky". Thus, this quantum-mechanical bound precludes a Wigner function which is a perfectly localized δ-function in phase space, as a reflection of the uncertainty principle.[6] 10. The Wigner transformation is simply the Fourier transform of the antidiagonals of the density matrix, when that matrix is expressed in a position basis.[7] Examples See also: Phase-space formulation § Simple harmonic oscillator Let $\|m\rangle \equiv {\frac {a^{\dagger m}}{\sqrt {m!}}}\|0\rangle $ be the $m$-th Fock state of a quantum harmonic oscillator. Groenewold (1946) discovered its associated Wigner function, in dimensionless variables: $W_{\|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi }}e^{-(x^{2}+p^{2})}L_{m}{\big (}2(p^{2}+x^{2}){\big )},$ where $L_{m}(x)$ denotes the $m$-th Laguerre polynomial. This may follow from the expression for the static eigenstate wavefunctions, $u_{m}(x)=\pi ^{-1/4}H_{m}(x)e^{-x^{2}/2},$ where $H_{m}$ is the $m$-th Hermite polynomial. From the above definition of the Wigner function, upon a change of integration variables, $W_{\|m\rangle }(x,p)={\frac {(-1)^{m}}{\pi ^{3/2}2^{m}m!}}e^{-x^{2}-p^{2}}\int _{-\infty }^{\infty }d\zeta \,e^{-\zeta ^{2}}H_{m}(\zeta -ip+x)H_{m}(\zeta -ip-x).$ The expression then follows from the integral relation between Hermite and Laguerre polynomials.[8] Evolution equation for Wigner function Main articles: Wigner–Weyl transform and Phase space formulation The Wigner transformation is a general invertible transformation of an operator Ĝ on a Hilbert space to a function g(x, p) on phase space and is given by $g(x,p)=\int _{-\infty }^{\infty }ds\,e^{ips/\hbar }\left\langle x-{\frac {s}{2}}\right\|{\hat {G}}\left\|x+{\frac {s}{2}}\right\rangle .$ Hermitian operators map to real functions. The inverse of this transformation, from phase space to Hilbert space, is called the Weyl transformation: $\langle x\|{\hat {G}}\|y\rangle =\int _{-\infty }^{\infty }{\frac {dp}{h}}e^{ip(x-y)/\hbar }g\left({\frac {x+y}{2}},p\right)$ (not to be confused with the distinct Weyl transformation in differential geometry). The Wigner function W(x, p) discussed here is thus seen to be the Wigner transform of the density matrix operator ρ̂. Thus the trace of an operator with the density matrix Wigner-transforms to the equivalent phase-space integral overlap of g(x, p) with the Wigner function. The Wigner transform of the von Neumann evolution equation of the density matrix in the Schrödinger picture is Moyal's evolution equation for the Wigner function: ${\frac {\partial W(x,p,t)}{\partial t}}=-\{\{W(x,p,t),H(x,p)\}\},$ where H(x, p) is the Hamiltonian, and {{⋅, ⋅}} is the Moyal bracket. In the classical limit, ħ → 0, the Moyal bracket reduces to the Poisson bracket, while this evolution equation reduces to the Liouville equation of classical statistical mechanics. Formally, the classical Liouville equation can be solved in terms of the phase-space particle trajectories which are solutions of the classical Hamilton equations. This technique of solving partial differential equations is known as the method of characteristics. This method transfers to quantum systems, where the characteristics' "trajectories" now determine the evolution of Wigner functions. The solution of the Moyal evolution equation for the Wigner function is represented formally as $W(x,p,t)=W{\big (}\star {\big (}x_{-t}(x,p),p_{-t}(x,p){\big )},0{\big )},$ where $x_{t}(x,p)$ and $p_{t}(x,p)$ are the characteristic trajectories subject to the quantum Hamilton equations with initial conditions $x_{t=0}(x,p)=x$ and $p_{t=0}(x,p)=p$, and where $\star $-product composition is understood for all argument functions. Since $\star $-composition of functions is thoroughly nonlocal (the "quantum probability fluid" diffuses, as observed by Moyal), vestiges of local trajectories in quantum systems are barely discernible in the evolution of the Wigner distribution function.[lower-alpha 2] In the integral representation of $\star $-products, successive operations by them have been adapted to a phase-space path integral, to solve the evolution equation for the Wigner function[9] (see also [10][11][12]). This non-local feature of Moyal time evolution[13] is illustrated in the gallery below, for Hamiltonians more complex than the harmonic oscillator. In the classical limit, the trajectory nature of the time evolution of Wigner functions becomes more and more distinct. At ħ = 0, the characteristics' trajectories reduce to the classical trajectories of particles in phase space. Examples of Wigner-function time evolutions • Pure state in a Morse potential. The green dashed lines represent level set of the Hamiltonian. • Pure state in a quartic potential. The solid lines represent the level set of the Hamiltonian. • Tunnelling of a wave packet through a potential barrier. The solid lines represent the level set of the Hamiltonian. • Long-time evolution of a mixed state ρ in an anharmonic potential well. Marginals are plotted on the right (p) and top (x). • An equilibrium mixed state ρ (evolves to itself), in the same anharmonic potential. Harmonic-oscillator time evolution In the special case of the quantum harmonic oscillator, however, the evolution is simple and appears identical to the classical motion: a rigid rotation in phase space with a frequency given by the oscillator frequency. This is illustrated in the gallery below. This same time evolution occurs with quantum states of light modes, which are harmonic oscillators. Examples of Wigner-function time evolutions in a quantum harmonic oscillator • A coherent state.[14] • Combined ground state and 1st excited state.[14] • A cat state; the marginals are plotted on the right (p) and underneath (x). Classical limit The Wigner function allows one to study the classical limit, offering a comparison of the classical and quantum dynamics in phase space.[15][16] It has been suggested that the Wigner function approach can be viewed as a quantum analogy to the operatorial formulation of classical mechanics introduced in 1932 by Bernard Koopman and John von Neumann: the time evolution of the Wigner function approaches, in the limit ħ → 0, the time evolution of the Koopman–von Neumann wavefunction of a classical particle.[17] Positivity of the Wigner function As already noted, the Wigner function of quantum state typically takes some negative values. Indeed, for a pure state in one variable, if $W(x,p)\geq 0$ for all $x$ and $p$, then the wave function must have the form $\psi (x)=e^{-ax^{2}+bx+c}$ for some complex numbers $a,b,c$ with $\operatorname {Re} (a)>0$ (Hudson's theorem[18]). Note that $a$ is allowed to be complex, so that $\psi $ is not necessarily a Gaussian wave packet in the usual sense. Thus, pure states with non-negative Wigner functions are not necessarily minimum-uncertainty states in the sense of the Heisenberg uncertainty formula; rather, they give equality in the Schrödinger uncertainty formula, which includes an anticommutator term in addition to the commutator term. (With careful definition of the respective variances, all pure-state Wigner functions lead to Heisenberg's inequality all the same.) In higher dimensions, the characterization of pure states with non-negative Wigner functions is similar; the wave function must have the form $\psi (x)=e^{-(x,Ax)+b\cdot x+c},$ where $A$ is a symmetric complex matrix whose real part is positive-definite, $b$ is a complex vector, and c is a complex number.[19] The Wigner function of any such state is a Gaussian distribution on phase space. Soto and Claverie[19] give an elegant proof of this characterization, using the Segal–Bargmann transform. The reasoning is as follows. The Husimi Q function of $\psi $ may be computed as the squared magnitude of the Segal–Bargmann transform of $\psi $, multiplied by a Gaussian. Meanwhile, the Husimi Q function is the convolution of the Wigner function with a Gaussian. If the Wigner function of $\psi $ is non-negative everywhere on phase space, then the Husimi Q function will be strictly positive everywhere on phase space. Thus, the Segal–Bargmann transform $F(x+ip)$ of $\psi $ will be nowhere zero. Thus, by a standard result from complex analysis, we have $F(x+ip)=e^{g(x+ip)}$ for some holomorphic function $g$. But in order for $F$ to belong to the Segal–Bargmann space—that is, for $F$ to be square-integrable with respect to a Gaussian measure—$g$ must have at most quadratic growth at infinity. From this, elementary complex analysis can be used to show that $g$ must actually be a quadratic polynomial. Thus, we obtain an explicit form of the Segal–Bargmann transform of any pure state whose Wigner function is non-negative. We can then invert the Segal–Bargmann transform to obtain the claimed form of the position wave function. There does not appear to be any simple characterization of mixed states with non-negative Wigner functions. The Wigner function in relation to other interpretations of quantum mechanics It has been shown that the Wigner quasiprobability distribution function can be regarded as an ħ-deformation of another phase-space distribution function that describes an ensemble of de Broglie–Bohm causal trajectories.[20] Basil Hiley has shown that the quasi-probability distribution may be understood as the density matrix re-expressed in terms of a mean position and momentum of a "cell" in phase space, and the de Broglie–Bohm interpretation allows one to describe the dynamics of the centers of such "cells".[21][22] There is a close connection between the description of quantum states in terms of the Wigner function and a method of quantum states reconstruction in terms of mutually unbiased bases.[23] Uses of the Wigner function outside quantum mechanics • In the modelling of optical systems such as telescopes or fibre telecommunications devices, the Wigner function is used to bridge the gap between simple ray tracing and the full wave analysis of the system. Here p/ħ is replaced with k = \|k\| sin θ ≈ \|k\|θ in the small-angle (paraxial) approximation. In this context, the Wigner function is the closest one can get to describing the system in terms of rays at position x and angle θ while still including the effects of interference.[24] If it becomes negative at any point, then simple ray tracing will not suffice to model the system. That is to say, negative values of this function are a symptom of the Gabor limit of the classical light signal and not of quantum features of light associated with ħ. • In signal analysis, a time-varying electrical signal, mechanical vibration, or sound wave are represented by a Wigner function. Here, x is replaced with the time, and p/ħ is replaced with the angular frequency ω = 2πf, where f is the regular frequency. • In ultrafast optics, short laser pulses are characterized with the Wigner function using the same f and t substitutions as above. Pulse defects such as chirp (the change in frequency with time) can be visualized with the Wigner function. See adjacent figure. • In quantum optics, x and p/ħ are replaced with the X and P quadratures, the real and imaginary components of the electric field (see coherent state). Measurements of the Wigner function • Quantum tomography • Frequency-resolved optical gating Other related quasiprobability distributions Main article: Quasiprobability distribution The Wigner distribution was the first quasiprobability distribution to be formulated, but many more followed, formally equivalent and transformable to and from it (see Transformation between distributions in time–frequency analysis). As in the case of coordinate systems, on account of varying properties, several such have with various advantages for specific applications: • Glauber P representation • Husimi Q representation Nevertheless, in some sense, the Wigner distribution holds a privileged position among all these distributions, since it is the only one whose requisite star-product drops out (integrates out by parts to effective unity) in the evaluation of expectation values, as illustrated above, and so can be visualized as a quasiprobability measure analogous to the classical ones. Historical note As indicated, the formula for the Wigner function was independently derived several times in different contexts. In fact, apparently, Wigner was unaware that even within the context of quantum theory, it had been introduced previously by Heisenberg and Dirac,[25][26] albeit purely formally: these two missed its significance, and that of its negative values, as they merely considered it as an approximation to the full quantum description of a system such as the atom. (Incidentally, Dirac would later become Wigner's brother-in-law, marrying his sister Manci.) Symmetrically, in most of his legendary 18-month correspondence with Moyal in the mid-1940s, Dirac was unaware that Moyal's quantum-moment generating function was effectively the Wigner function, and it was Moyal who finally brought it to his attention.[27] See also • Heisenberg group • Wigner–Weyl transform • Phase space formulation • Moyal bracket • Negative probability • Optical equivalence theorem • Modified Wigner distribution function • Cohen's class distribution function • Wigner distribution function • Transformation between distributions in time–frequency analysis • Squeezed coherent state • Bilinear time–frequency distribution • Continuous-variable quantum information Footnotes 1. Specifically, since this convolution is invertible, in fact, no information has been sacrificed, and the full quantum entropy has not increased yet. However, if this resulting Husimi distribution is then used as a plain measure in a phase-space integral evaluation of expectation values without the requisite star product of the Husimi representation, then, at that stage, quantum information has been forfeited and the distribution is a semi-classical one, effectively. That is, depending on its usage in evaluating expectation values, the very same distribution may serve as a quantum or a classical distribution function. 2. Quantum characteristics should not be confused with trajectories of the Feynman path integral, or trajectories of the de Broglie–Bohm theory. This three-fold ambiguity allows better understanding of the position of Niels Bohr, who vigorously but counterproductively opposed the notion of trajectory in the atomic physics. At the 1948 Pocono Conference, e.g., he said to Richard Feynman: "... one could not talk about the trajectory of an electron in the atom, because it was something not observable". ("The Beat of a Different Drum: The Life and Science of Richard Feynman", by Jagdish Mehra (Oxford, 1994, pp. 245–248)). Arguments of this kind were widely used in the past by Ernst Mach in his criticism of an atomic theory of physics and later, in the 1960s, by Geoffrey Chew, Tullio Regge and others to motivate replacing the local quantum field theory by the S-matrix theory. Today, statistical physics entirely based on atomistic concepts is included in standard courses, the S-matrix theory went out of fashion, while the Feynman path-integral method has been recognized as the most efficient method in gauge theories. References 1. E. P. Wigner (1932). "On the quantum correction for thermodynamic equilibrium". Physical Review. 40 (5): 749–759. Bibcode:1932PhRv...40..749W. doi:10.1103/PhysRev.40.749. hdl:10338.dmlcz/141466. 2. H. J. Groenewold (1946). "On the principles of elementary quantum mechanics". Physica. 12 (7): 405–460. Bibcode:1946Phy....12..405G. doi:10.1016/S0031-8914(46)80059-4. 3. H. Weyl (1927). "Quantenmechanik und gruppentheorie". Zeitschrift für Physik. 46 (1–2): 1. Bibcode:1927ZPhy...46....1W. doi:10.1007/BF02055756. S2CID 121036548.; H. Weyl, Gruppentheorie und Quantenmechanik (Leipzig: Hirzel) (1928); H. Weyl, The Theory of Groups and Quantum Mechanics (Dover, New York, 1931). 4. J. Ville, "Théorie et Applications de la Notion de Signal Analytique", Câbles et Transmission, 2, 61–74 (1948). 5. Moyal, J. E. (1949). "Quantum mechanics as a statistical theory". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 45 (1): 99–124. doi:10.1017/s0305004100000487. ISSN 0305-0041. S2CID 124183640. 6. Curtright, T. L.; Zachos, C. K. (2012). "Quantum Mechanics in Phase Space". Asia Pacific Physics Newsletter. 1: 37. arXiv:1104.5269. doi:10.1142/S2251158X12000069. S2CID 119230734.; C. Zachos, D. Fairlie, and T. Curtright, Quantum Mechanics in Phase Space (World Scientific, Singapore, 2005). ISBN 978-981-238-384-6. 7. Hawkes, Peter W. (2018). Advances in Imaging and Electron Physics. Academic Press. p. 47. ISBN 9780128155424. 8. Schleich, Wolfgang P. (2001-02-09). Quantum Optics in Phase Space (1st ed.). Wiley. p. 105. doi:10.1002/3527602976. ISBN 978-3-527-29435-0. 9. B. Leaf (1968). "Weyl transform in nonrelativistic quantum dynamics". Journal of Mathematical Physics. 9 (5): 769–781. Bibcode:1968JMP.....9..769L. doi:10.1063/1.1664640. 10. P. Sharan (1979). "Star-product representation of path integrals". Physical Review D. 20 (2): 414–418. Bibcode:1979PhRvD..20..414S. doi:10.1103/PhysRevD.20.414. 11. M. S. Marinov (1991). "A new type of phase-space path integral". Physics Letters A. 153 (1): 5–11. Bibcode:1991PhLA..153....5M. doi:10.1016/0375-9601(91)90352-9. 12. B. Segev: Evolution kernels for phase space distributions. In: M. A. Olshanetsky; Arkady Vainshtein (2002). Multiple Facets of Quantization and Supersymmetry: Michael Marinov Memorial Volume. World Scientific. pp. 68–90. ISBN 978-981-238-072-2. Retrieved 26 October 2012. See especially section 5. "Path integral for the propagator" on pages 86–89. Also online. 13. M. Oliva, D. Kakofengitis, and O. Steuernagel (2018). "Anharmonic quantum mechanical systems do not feature phase space trajectories". Physica A. 502: 201–210. arXiv:1611.03303. Bibcode:2018PhyA..502..201O. doi:10.1016/j.physa.2017.10.047. S2CID 53691877.{{cite journal}}: CS1 maint: multiple names: authors list (link) 14. Curtright, T. L., Time-dependent Wigner Functions. 15. See, for example: Wojciech H. Zurek, Decoherence and the transition from quantum to classical – revisited, Los Alamos Science, 27, 2002, arXiv:quant-ph/0306072, pp. 15 ff. 16. See, for example: C. Zachos, D. Fairlie, T. Curtright, Quantum mechanics in phase space: an overview with selected papers, World Scientific, 2005. ISBN 978-981-4520-43-0. 17. Bondar, Denys I.; Cabrera, Renan; Zhdanov, Dmitry V.; Rabitz, Herschel A. (2013). "Wigner phase-space distribution as a wave function". Physical Review A. 88 (5): 052108. arXiv:1202.3628. doi:10.1103/PhysRevA.88.052108. ISSN 1050-2947. S2CID 119155284. 18. Hudson, Robin L. (1974). "When is the Wigner quasi-probability density non-negative?". Reports on Mathematical Physics. 6 (2): 249–252. Bibcode:1974RpMP....6..249H. doi:10.1016/0034-4877(74)90007-X. 19. F. Soto and P. Claverie, "When is the Wigner function of multidimensional systems nonnegative?", Journal of Mathematical Physics 24 (1983) 97–100. 20. Dias, Nuno Costa; Prata, João Nuno (2002). "Bohmian trajectories and quantum phase space distributions". Physics Letters A. 302 (5–6): 261–272. arXiv:quant-ph/0208156v1. doi:10.1016/s0375-9601(02)01175-1. ISSN 0375-9601. S2CID 39936409. 21. B. J. Hiley: Phase space descriptions of quantum phenomena, in: A. Khrennikov (ed.): Quantum Theory: Re-consideration of Foundations–2, pp. 267–286, Växjö University Press, Sweden, 2003 (PDF). 22. B. Hiley: Moyal's characteristic function, the density matrix and von Neumann's idempotent (preprint). 23. F. C. Khanna, P. A. Mello, M. Revzen, Classical and Quantum Mechanical State Reconstruction, arXiv:1112.3164v1 [quant-ph] (submitted December 14, 2011). 24. Bazarov, Ivan V. (2012-05-03). "Synchrotron radiation representation in phase space". Physical Review Special Topics - Accelerators and Beams. American Physical Society (APS). 15 (5): 050703. doi:10.1103/physrevstab.15.050703. ISSN 1098-4402. S2CID 53489256. 25. W. Heisenberg (1931). "Über die inkohärente Streuung von Röntgenstrahlen". Physikalische Zeitschrift. 32: 737–740. 26. Dirac, P. A. M. (1930). "Note on Exchange Phenomena in the Thomas Atom". Mathematical Proceedings of the Cambridge Philosophical Society. Cambridge University Press (CUP). 26 (3): 376–385. doi:10.1017/s0305004100016108. ISSN 0305-0041. S2CID 97185766. 27. Ann Moyal, (2006), "Maverick Mathematician: The Life and Science of J. E. Moyal", ANU E-press, 2006, ISBN 1-920942-59-9. Further reading • M. Levanda and V. Fleurov, "Wigner quasi-distribution function for charged particles in classical electromagnetic fields", Annals of Physics, 292, 199–231 (2001). arXiv:cond-mat/0105137. External links • wigner Wigner function implementation in QuTiP. • Quantum Optics Gallery. • Sonogram Visible Speech GPL-licensed freeware for the Wigner quasiprobability distribution of signal files.
Wijsman convergence Wijsman convergence is a variation of Hausdorff convergence suitable for work with unbounded sets. Intuitively, Wijsman convergence is to convergence in the Hausdorff metric as pointwise convergence is to uniform convergence. History The convergence was defined by Robert Wijsman.[1] The same definition was used earlier by Zdeněk Frolík.[2] Yet earlier, Hausdorff in his book Grundzüge der Mengenlehre defined so called closed limits; for proper metric spaces it is the same as Wijsman convergence. Definition Let (X, d) be a metric space and let Cl(X) denote the collection of all d-closed subsets of X. For a point x ∈ X and a set A ∈ Cl(X), set $d(x,A)=\inf _{a\in A}d(x,a).$ A sequence (or net) of sets Ai ∈ Cl(X) is said to be Wijsman convergent to A ∈ Cl(X) if, for each x ∈ X, $d(x,A_{i})\to d(x,A).$ Wijsman convergence induces a topology on Cl(X), known as the Wijsman topology. Properties • The Wijsman topology depends very strongly on the metric d. Even if two metrics are uniformly equivalent, they may generate different Wijsman topologies. • Beer's theorem: if (X, d) is a complete, separable metric space, then Cl(X) with the Wijsman topology is a Polish space, i.e. it is separable and metrizable with a complete metric. • Cl(X) with the Wijsman topology is always a Tychonoff space. Moreover, one has the Levi-Lechicki theorem: (X, d) is separable if and only if Cl(X) is either metrizable, first-countable or second-countable. • If the pointwise convergence of Wijsman convergence is replaced by uniform convergence (uniformly in x), then one obtains Hausdorff convergence, where the Hausdorff metric is given by $d_{\mathrm {H} }(A,B)=\sup _{x\in X}{\big \|}d(x,A)-d(x,B){\big \|}.$ The Hausdorff and Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space. See also • Hausdorff distance • Kuratowski convergence • Vietoris topology • Hemicontinuity References Notes 1. Wijsman, Robert A. (1966). "Convergence of sequences of convex sets, cones and functions. II". Trans. Amer. Math. Soc. American Mathematical Society. 123 (1): 32–45. doi:10.2307/1994611. JSTOR 1994611. MR0196599 2. Z. Frolík, Concerning topological convergence of sets, Czechoskovak Math. J. 10 (1960), 168–180 Bibliography • Beer, Gerald (1993). Topologies on closed and closed convex sets. Mathematics and its Applications 268. Dordrecht: Kluwer Academic Publishers Group. pp. xii+340. ISBN 0-7923-2531-1. MR1269778 • Beer, Gerald (1994). "Wijsman convergence: a survey". Set-Valued Anal. 2 (1–2): 77–94. doi:10.1007/BF01027094. MR1285822 External links • Som Naimpally (2001) [1994], "Wijsman convergence", Encyclopedia of Mathematics, EMS Press
Wiktor Eckhaus Wiktor Eckhaus (28 June 1930 – 1 October 2000) was a Polish–Dutch mathematician, known for his work on the field of differential equations. He was Professor Emeritus of Applied Mathematics at the Utrecht University. Wiktor Eckhaus Born(1930-06-28)28 June 1930[1] Stanisławów, Poland Died1 October 2000(2000-10-01) (aged 70) Amstelveen,[2] Netherlands NationalityNetherlands[3] Alma materMassachusetts Institute of Technology Known forEckhaus instability Eckhaus equation Scientific career Fieldsmathematics, aerodynamics InstitutionsUtrecht University, Delft University of Technology, National Aerospace Laboratory Doctoral advisorLeon Trilling Biography Eckhaus was born into a wealthy family, and raised in Warsaw where his father was managing a fur company. During the German occupation of Poland, he, his mother and sister had to hide because of their Jewish descent. His father, after being a prisoner of war, joined the Russian Army. After the war, in 1947, the re-united family came to Amsterdam – via a refugee camp in Austria. Wiktor passed the state exam of the Hogere Burgerschool in 1948, and started to study aeronautics at the Delft University of Technology. Following his graduation he worked with the National Aerospace Laboratory in Amsterdam, from 1953 till 1957. In the period 1957–1960 he worked at the Massachusetts Institute of Technology, where Eckhaus earned a PhD in 1959 under Leon Trilling on a dissertation entitled "Some problems of unsteady flow with discontinuities". In 1960, he became a "maître de recherches" (senior research fellow) at the Department of Mechanics of the Sorbonne. In 1964 he was a visiting professor at the University of Amsterdam and the Mathematical Centre. Thereafter, in 1965, he became professor at the Delft University of Technology, in pure and applied mathematics and mechanics. From 1972 until his retirement in 1994, Eckhaus was professor of applied mathematics at the Utrecht University. Initially he studied the flow around airfoils, leading to his research on the stability of solutions to (weakly nonlinear) differential equations. This resulted in what is now known as the Eckhaus instability criterion and Eckhaus instability, appearing for instance as a secondary instability in models of Rayleigh–Bénard convection. Later, Eckhaus worked on singular perturbation theory and soliton equations. In 1983 he treated strongly singular relaxation oscillations – called "canards" (French for "ducks") – resulting in his most-read paper "Relaxation oscillations including a standard chase on French ducks".[4] Eckhaus used standard methods of analysis, on a problem qualified before, by Marc Diener, as an example of a problem only treatable through the use of non-standard analysis.[5] He became a member of the Royal Netherlands Academy of Arts and Sciences in 1987.[2] Publications • Eckhaus, W. (1965), Studies in nonlinear stability theory, Springer Tracts in Natural Philosophy, vol. 6, Springer, ISBN 978-3-642-88319-4 • —— (1973), Matched asymptotic expansions and singular perturbations, Mathematics Studies, vol. 6, North Holland, ISBN 978-0-7204-2600-7 • —— (1979), Asymptotic analysis of singular perturbations, Studies in Mathematics and its Applications, vol. 9, North Holland, ISBN 978-0-444-85306-6 • ——; van Harten, A. (1981), The inverse scattering transformation and the theory of solitons – An introduction, Mathematics Studies, vol. 50, North Holland, ISBN 978-0-444-55731-5 • —— (1975), "New approach to the asymptotic theory of nonlinear oscillations and wave propagation", Journal of Mathematical Analysis and Applications, 49 (3): 575–611, doi:10.1016/0022-247X(75)90200-0 • —— (1983), "Relaxation oscillations including a standard chase on French ducks", in Verhulst, F. (ed.), Asymptotic Analysis II – Surveys and New Trends, Lecture Notes in Mathematics, vol. 985, Springer, pp. 449–494, doi:10.1007/BFb0062381, ISBN 978-3-540-12286-9 • —— (1993), "The Ginzburg–Landau manifold is an attractor", Journal of Nonlinear Science, 3 (1): 329–348, Bibcode:1993JNS.....3..329E, doi:10.1007/BF02429869, S2CID 122662589 • ——; de Jager, E.M. (1966), "Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type", Arch. Rat. Mech. Anal., 23 (1): 26–86, Bibcode:1966ArRMA..23...26E, doi:10.1007/BF00281135, S2CID 843282 • —— (1997), Witus en de jaren van angst – Een reconstructie [Witus and the years of fear – A reconstruction] (Autobiography) (in Dutch), Bas Lubberhuizen, ISBN 9789073978690 Notes 1. See Eckhaus (1997) and Doelman et al. (2001) for a discussion on his date and place of birth. 2. Wiktor Eckhaus (1929 – 2000), Koninklijke Nederlandse Akademie van Wetenschappen, retrieved 2014-09-06 3. Zitting 1964-1965-8184 – Naturalisatie van van den Berg, Lilli Elfriede en 28 anderen [Session 1964-1965-8184 – Naturalisation of van den Berg, Lilli Elfriede and 28 others] (pdf) (in Dutch), Staten Generaal, 14 August 1965, retrieved 2014-09-06 4. Eckhaus (1983) 5. Martin Wechselberger (ed.). "Canards". Scholarpedia. References • Doelman, A.; Duistermaat, H.; Grasman, J.; van Harten, A. (2001), "In memoriam Wiktor Eckhaus" (PDF), Nieuw Archief voor Wiskunde (in Dutch), Koninklijk Wiskundig Genootschap, 2 (1): 18–20. Also appeared as: Duistermaat, J.J.; Doelman, A.; Grasman, J.; van Harten, A. (2003), "Levensbericht W. Eckhaus" [Eulogy W. Eckhaus] (PDF), Levensberichten en Herdenkingen (in Dutch), Amsterdam: Koninklijke Nederlandse Akademie van Wetenschappen: 19–24. External links • Wiktor Eckhaus at the Mathematics Genealogy Project Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Netherlands Other • IdRef
Wilberd van der Kallen Wilberd Leo Johan van der Kallen (born 15 January 1947 in Nieuwer-Amstel)[1] is a Dutch mathematician. W. L. J. van der Kallen completed his undergraduate study of mathematics and physics at Utrecht University.[2] There he received his PhD in 1973 with thesis advisor T. A. Springer and thesis Infinitesimally central extensions of Chevalley groups.[3] In 1969 van der Kallen became a teaching assistant in Utrecht University's Mathematics Department and has spent his career there, eventually becoming a tenured professor. His research deals with algebraic K-theory and the representation theory of algebraic groups, among other topics. He has frequently been a visiting professor at Northwestern University in Evanston, Illinois and at the Tata Institute of Fundamental Research in Mumbai.[2] He is the author or coauthor of over 60 research articles.[2] In 1977 he published an analogue of a 1977 theorem of Andrei Suslin[4][5] and a generalization of a 1969 theorem of Hideya Matsumoto.[6] In 1978 van der Kallen was an invited speaker at the International Congress of Mathematicians in Helsinki.[7] His 1980 paper Homology stability for linear groups[8] has over 200 citations. His 1977 paper Rational and generic cohomology, written with 3 other mathematicians,[9] has over 240 citations. Books • van der Kallen, Wilberd (1993). Lectures on Frobenius Splittings and B-modules. ISBN 978-81-85198-60-6; 98 pages{{cite book}}: CS1 maint: postscript (link) • van der Kallen, Wilberd (15 November 2006). Infinitesimally Central Extensions of Chevalley Groups. Lecture Notes in Mathematics, vol. 356. Springer. ISBN 978-3-540-37857-0; pbk reprint of 1973 original, 154 pages{{cite book}}: CS1 maint: postscript (link) • Cohen, Arjeh M.; Hesselink, Wim H.; van der Kallen, Wilberd L. J.; Strooker, Jan R., eds. (15 November 2006). Algebraic Groups. Utrecht 1986: Proceedings of a Symposium in Honour of T.A. Springer. Lecture Notes in Mathematics, vol. 1271. Springer. ISBN 978-3-540-47834-8; reprint of 1987 original{{cite book}}: CS1 maint: postscript (link) References 1. "XI 5 Wilberd van der Kallen" (PDF). deel 10 : Albert (X 7) — wilberdk.home.xs4all.nl. 2. "Wilberd van der Kallen (with links to publication list)". Geometry and Quantum Theory (GQT) (a national Dutch mathematical research cluster). 3. Wilberd L. J. van der Kallen at the Mathematics Genealogy Project 4. van der Kallen, Wilberd (1977). "Another presentation for Steinberg groups" (PDF). Indagationes Mathematicae (Proceedings). 80 (4): 304–312. doi:10.1016/1385-7258(77)90026-9. 5. Suslin, A. A. (1977). "On the Structure of the Special Linear Group over Polynomial Rings". Mathematics of the USSR-Izvestiya. 11 (2): 221–238. Bibcode:1977IzMat..11..221S. doi:10.1070/IM1977v011n02ABEH001709. 6. van der Kallen, W. (1977). "The $K_{2}$ of rings with many units" (PDF). Annales scientifiques de l'École Normale Supérieure. 10 (4): 473–515. doi:10.24033/asens.1334. 7. van der Kallen, W. (1978). "Generators and relations in algebraic K-theory" (PDF). Proceedings of the International Conference of Mathematicians at Helsinki. Vol. 1. pp. 305–310. 8. van der Kallen, Wilberd (1980). "Homology stability for linear groups". Inventiones Mathematicae. 60 (3): 269–295. Bibcode:1980InMat..60..269K. doi:10.1007/BF01390018. S2CID 54671324. 9. Cline, Edward; Parshall, Brian; Scott, Leonard; van der Kallen, W. (197). "Rational and generic cohomology" (PDF). Inventiones Mathematicae. 39 (2): 143–163. doi:10.1007/BF01390106. S2CID 14358269. External links • "Wilberd van der Kallen". Universiteit Utrecht (webspace.science.uu.nl). • "Slides of some talks". • "Linear Algebraic Groups: an overview by Wilberd van der Kallen". YouTube. matsciencechannel. August 20, 2013. Authority control International • ISNI • VIAF National • United States Academics • Google Scholar • MathSciNet • Mathematics Genealogy Project • ORCID Other • IdRef
Gibbs phenomenon In mathematics, the Gibbs phenomenon is the oscillatory behavior of the Fourier series of a piecewise continuously differentiable periodic function around a jump discontinuity. The $ N$th partial Fourier series of the function (formed by summing the $ N$ lowest constituent sinusoids of the Fourier series of the function) produces large peaks around the jump which overshoot and undershoot the function values. As more sinusoids are used, this approximation error approaches a limit of about 9% of the jump, though the infinite Fourier series sum does eventually converge almost everywhere (pointwise convergence on continuous points) except points of discontinuity.[1] The Gibbs phenomenon was observed by experimental physicists and was believed to be due to imperfections in the measuring apparatus,[2] but it is in fact a mathematical result. It is one cause of ringing artifacts in signal processing. Description The Gibbs phenomenon is a behavior of the Fourier series of a function with a jump discontinuity and is described as the following: As more Fourier series constituents or components are taken, the Fourier series shows the first overshoot in the oscillatory behavior around the jump point approaching ~ 9% of the (full) jump and this oscillation does not disappear but gets closer to the point so that the integral of the oscillation approaches to zero (i.e., zero energy in the oscillation). At the jump point, the Fourier series gives the average of the function's both side limits toward the point. Square wave example The three pictures on the right demonstrate the Gibbs phenomenon for a square wave (with peak-to-peak amplitude of $ c$ from $ -c/2$ to $ c/2$ and the periodicity $ L$) whose $ N$th partial Fourier series is ${\frac {2c}{\pi }}\left(\sin(\omega x)+{\frac {1}{3}}\sin(3\omega x)+\cdots +{\frac {1}{N-1}}\sin((N-1)\omega x)\right)$ where $ \omega =2\pi /L$. More precisely, this square wave is the function $ f(x)$ which equals ${\tfrac {c}{2}}$ between $ 2n(L/2)$ and $ (2n+1)(L/2)$ and $ -{\tfrac {c}{2}}$ between $ (2n+1)(L/2)$ and $ (2n+2)(L/2)$ for every integer $ n$; thus, this square wave has a jump discontinuity of peak-to-peak height $ c$ at every integer multiple of $ L/2$. As more sinusoidal terms are added (i.e., increasing $ N$), the error of the partial Fourier series converges to a fixed height. But because the width of the error continues to narrow, the area of the error – and hence the energy of the error – converges to 0.[3] The square wave analysis reveals that the error exceeds the height (from zero) ${\tfrac {c}{2}}$ of the square wave by ${\frac {c}{\pi }}\int _{0}^{\pi }{\frac {\sin(t)}{t}}\ dt-{\frac {c}{2}}=c\cdot (0.089489872236\dots ).$ (OEIS: A243268) or about 9% of the full jump $ c$. More generally, at any discontinuity of a piecewise continuously differentiable function with a jump of $ c$, the $ N$th partial Fourier series of the function will (for a very large $ N$ value) overshoot this jump by an error approaching $ c\cdot (0.089489872236\dots )$ at one end and undershoot it by the same amount at the other end; thus the "full jump" in the partial Fourier series will be about 18% larger than the full jump in the original function. At the discontinuity, the partial Fourier series will converge to the midpoint of the jump (regardless of the actual value of the original function at the discontinuity) as a consequence of Dirichlet's theorem.[4] The quantity $\int _{0}^{\pi }{\frac {\sin t}{t}}\ dt=(1.851937051982\dots )={\frac {\pi }{2}}+\pi \cdot (0.089489872236\dots )$ (OEIS: A036792) is sometimes known as the Wilbraham–Gibbs constant. History The Gibbs phenomenon was first noticed and analyzed by Henry Wilbraham in an 1848 paper.[5] The paper attracted little attention until 1914 when it was mentioned in Heinrich Burkhardt's review of mathematical analysis in Klein's encyclopedia.[6] In 1898, Albert A. Michelson developed a device that could compute and re-synthesize the Fourier series.[7] A widespread myth says that when the Fourier coefficients for a square wave were input to the machine, the graph would oscillate at the discontinuities, and that because it was a physical device subject to manufacturing flaws, Michelson was convinced that the overshoot was caused by errors in the machine. In fact the graphs produced by the machine were not good enough to exhibit the Gibbs phenomenon clearly, and Michelson may not have noticed it as he made no mention of this effect in his paper (Michelson & Stratton 1898) about his machine or his later letters to Nature.[8] Inspired by correspondence in Nature between Michelson and A. E. H. Love about the convergence of the Fourier series of the square wave function, J. Willard Gibbs published a note in 1898 pointing out the important distinction between the limit of the graphs of the partial sums of the Fourier series of a sawtooth wave and the graph of the limit of those partial sums. In his first letter Gibbs failed to notice the Gibbs phenomenon, and the limit that he described for the graphs of the partial sums was inaccurate. In 1899 he published a correction in which he described the overshoot at the point of discontinuity (Nature, April 27, 1899, p. 606). In 1906, Maxime Bôcher gave a detailed mathematical analysis of that overshoot, coining the term "Gibbs phenomenon"[9] and bringing the term into widespread use.[8] After the existence of Henry Wilbraham's paper became widely known, in 1925 Horatio Scott Carslaw remarked, "We may still call this property of Fourier's series (and certain other series) Gibbs's phenomenon; but we must no longer claim that the property was first discovered by Gibbs."[10] Explanation Informally, the Gibbs phenomenon reflects the difficulty inherent in approximating a discontinuous function by a finite series of continuous sinusoidal waves. It is important to put emphasis on the word finite, because even though every partial sum of the Fourier series overshoots around each discontinuity it is approximating, the limit of summing an infinite number of sinusoidal waves does not. The overshoot peaks moves closer and closer to the discontinuity as more terms are summed, so convergence is possible. There is no contradiction (between the overshoot error converging to a non-zero height even though the infinite sum has no overshoot), because the overshoot peaks move toward the discontinuity. The Gibbs phenomenon thus exhibits pointwise convergence, but not uniform convergence. For a piecewise continuously differentiable (class C1) function, the Fourier series converges to the function at every point except at jump discontinuities. At jump discontinuities, the infinite sum will converge to the jump discontinuity's midpoint (i.e. the average of the values of the function on either side of the jump), as a consequence of Dirichlet's theorem.[4] The Gibbs phenomenon is closely related to the principle that the smoothness of a function controls the decay rate of its Fourier coefficients. Fourier coefficients of smoother functions will more rapidly decay (resulting in faster convergence), whereas Fourier coefficients of discontinuous functions will slowly decay (resulting in slower convergence). For example, the discontinuous square wave has Fourier coefficients $({\tfrac {1}{1}},{\scriptstyle {\text{0}}},{\tfrac {1}{3}},{\scriptstyle {\text{0}}},{\tfrac {1}{5}},{\scriptstyle {\text{0}}},{\tfrac {1}{7}},{\scriptstyle {\text{0}}},{\tfrac {1}{9}},{\scriptstyle {\text{0}}},\dots )$ that decay only at the rate of ${\tfrac {1}{n}}$, while the continuous triangle wave has Fourier coefficients $({\tfrac {1}{1^{2}}},{\scriptstyle {\text{0}}},{\tfrac {-1}{3^{2}}},{\scriptstyle {\text{0}}},{\tfrac {1}{5^{2}}},{\scriptstyle {\text{0}}},{\tfrac {-1}{7^{2}}},{\scriptstyle {\text{0}}},{\tfrac {1}{9^{2}}},{\scriptstyle {\text{0}}},\dots )$ that decay at a much faster rate of ${\tfrac {1}{n^{2}}}$. This only provides a partial explanation of the Gibbs phenomenon, since Fourier series with absolutely convergent Fourier coefficients would be uniformly convergent by the Weierstrass M-test and would thus be unable to exhibit the above oscillatory behavior. By the same token, it is impossible for a discontinuous function to have absolutely convergent Fourier coefficients, since the function would thus be the uniform limit of continuous functions and therefore be continuous, a contradiction. See Convergence of Fourier series § Absolute convergence. Solutions In practice, the difficulties associated with the Gibbs phenomenon can be ameliorated by using a smoother method of Fourier series summation, such as Fejér summation or Riesz summation, or by using sigma-approximation. Using a continuous wavelet transform, the wavelet Gibbs phenomenon never exceeds the Fourier Gibbs phenomenon.[11] Also, using the discrete wavelet transform with Haar basis functions, the Gibbs phenomenon does not occur at all in the case of continuous data at jump discontinuities,[12] and is minimal in the discrete case at large change points. In wavelet analysis, this is commonly referred to as the Longo phenomenon. In the polynomial interpolation setting, the Gibbs phenomenon can be mitigated using the S-Gibbs algorithm.[13] Formal mathematical description of the Gibbs phenomenon Let $ f:{\mathbb {R} }\to {\mathbb {R} }$ be a piecewise continuously differentiable function which is periodic with some period $ L>0$. Suppose that at some point $ x_{0}$, the left limit $ f(x_{0}^{-})$ and right limit $ f(x_{0}^{+})$ of the function $ f$ differ by a non-zero jump of $ c$: $f(x_{0}^{+})-f(x_{0}^{-})=c\neq 0.$ For each positive integer $ N$ ≥ 1, let $ S_{N}f(x)$ be the $ N$th partial Fourier series ($ S_{N}$ can be treated as a mathematical operator on functions.) $S_{N}f(x):=\sum _{-N\leq n\leq N}{\widehat {f}}(n)e^{\frac {i2\pi nx}{L}}={\frac {1}{2}}a_{0}+\sum _{n=1}^{N}\left(a_{n}\cos \left({\frac {2\pi nx}{L}}\right)+b_{n}\sin \left({\frac {2\pi nx}{L}}\right)\right),$ where the Fourier coefficients $ {\widehat {f}}(n),a_{n},b_{n}$ for integers $ n$ are given by the usual formulae ${\widehat {f}}(n):={\frac {1}{L}}\int _{0}^{L}f(x)e^{-{\frac {i2\pi nx}{L}}}\,dx$ $a_{0}:={\frac {1}{L}}\int _{0}^{L}f(x)\ dx$ $a_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\cos \left({\frac {2\pi nx}{L}}\right)\,dx$ $b_{n}:={\frac {2}{L}}\int _{0}^{L}f(x)\sin \left({\frac {2\pi nx}{L}}\right)\,dx.$ Then we have $\lim _{N\to \infty }S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=f(x_{0}^{+})+c\cdot (0.089489872236\dots )$ and $\lim _{N\to \infty }S_{N}f\left(x_{0}-{\frac {L}{2N}}\right)=f(x_{0}^{-})-c\cdot (0.089489872236\dots )$ but $\lim _{N\to \infty }S_{N}f(x_{0})={\frac {f(x_{0}^{-})+f(x_{0}^{+})}{2}}.$ More generally, if $ x_{N}$ is any sequence of real numbers which converges to $ x_{0}$ as $ N\to \infty $, and if the jump of $ a$ is positive then $\limsup _{N\to \infty }S_{N}f(x_{N})\leq f(x_{0}^{+})+c\cdot (0.089489872236\dots )$ and $\liminf _{N\to \infty }S_{N}f(x_{N})\geq f(x_{0}^{-})-c\cdot (0.089489872236\dots ).$ If instead the jump of $ c$ is negative, one needs to interchange limit superior ($ \limsup $) with limit inferior ($ \liminf $), and also interchange the $ \leq $ and $ \geq $ signs, in the above two inequalities. Proof of the Gibbs phenomenon in a general case Stated again, let $ f:{\mathbb {R} }\to {\mathbb {R} }$ be a piecewise continuously differentiable function which is periodic with some period $ L>0$, and this function has multiple jump discontinuity points denoted $ x_{i}$ where $ i=0,1,2,$ and so on. At each discontinuity, the amount of the vertical full jump is $ c_{i}$. Then, $ f$ can be expressed as the sum of a continuous function $ f_{c}$ and a multi-step function $ f_{s}$ which is the sum of step functions such as[14] $f=f_{c}+f_{s},$ $f_{s}=f_{s_{1}}+f_{s_{2}}+f_{s_{3}}+\cdots ,$ $f_{s_{i}}(x)={\begin{cases}0&{\text{if }}x\leq x_{i},\\c_{i},&{\text{if }}x>x_{i}.\end{cases}}$ $ S_{N}f(x)$ as the $ N$th partial Fourier series of $ f=f_{c}+f_{s}=f_{c}+\left(f_{s_{1}}+f_{s_{2}}+f_{s_{3}}+\ldots \right)$ will converge well at all $ x$ points except points near discontinuities $ x_{i}$. Around each discontinuity point $ x_{i}$, $ f_{s_{i}}$ will only have the Gibbs phenomenon of its own (the maximum oscillatory convergence error of ~ 9 % of the jump $c_{i}$, as shown in the square wave analysis) because other functions are continuous ($f_{c}$) or flat zero ($f_{s_{j}}$ where $j\neq i$) around that point. This proves how the Gibbs phenomenon occurs at every discontinuity. Signal processing explanation From a signal processing point of view, the Gibbs phenomenon is the step response of a low-pass filter, and the oscillations are called ringing or ringing artifacts. Truncating the Fourier transform of a signal on the real line, or the Fourier series of a periodic signal (equivalently, a signal on the circle), corresponds to filtering out the higher frequencies with an ideal (brick-wall) low-pass filter. This can be represented as convolution of the original signal with the impulse response of the filter (also known as the kernel), which is the sinc function. Thus, the Gibbs phenomenon can be seen as the result of convolving a Heaviside step function (if periodicity is not required) or a square wave (if periodic) with a sinc function: the oscillations in the sinc function cause the ripples in the output. In the case of convolving with a Heaviside step function, the resulting function is exactly the integral of the sinc function, the sine integral; for a square wave the description is not as simply stated. For the step function, the magnitude of the undershoot is thus exactly the integral of the left tail until the first negative zero: for the normalized sinc of unit sampling period, this is $ \int _{-\infty }^{-1}{\frac {\sin(\pi x)}{\pi x}}\,dx.$ The overshoot is accordingly of the same magnitude: the integral of the right tail or (equivalently) the difference between the integral from negative infinity to the first positive zero minus 1 (the non-overshooting value). The overshoot and undershoot can be understood thus: kernels are generally normalized to have integral 1, so they result in a mapping of constant functions to constant functions – otherwise they have gain. The value of a convolution at a point is a linear combination of the input signal, with coefficients (weights) the values of the kernel. If a kernel is non-negative, such as for a Gaussian kernel, then the value of the filtered signal will be a convex combination of the input values (the coefficients (the kernel) integrate to 1, and are non-negative), and will thus fall between the minimum and maximum of the input signal – it will not undershoot or overshoot. If, on the other hand, the kernel assumes negative values, such as the sinc function, then the value of the filtered signal will instead be an affine combination of the input values and may fall outside of the minimum and maximum of the input signal, resulting in undershoot and overshoot, as in the Gibbs phenomenon. Taking a longer expansion – cutting at a higher frequency – corresponds in the frequency domain to widening the brick-wall, which in the time domain corresponds to narrowing the sinc function and increasing its height by the same factor, leaving the integrals between corresponding points unchanged. This is a general feature of the Fourier transform: widening in one domain corresponds to narrowing and increasing height in the other. This results in the oscillations in sinc being narrower and taller, and (in the filtered function after convolution) yields oscillations that are narrower (and thus with smaller area) but which do not have reduced magnitude: cutting off at any finite frequency results in a sinc function, however narrow, with the same tail integrals. This explains the persistence of the overshoot and undershoot. • Oscillations can be interpreted as convolution with a sinc. • Higher cutoff makes the sinc narrower but taller, with the same magnitude tail integrals, yielding higher frequency oscillations, but whose magnitude does not vanish. Thus, the features of the Gibbs phenomenon are interpreted as follows: • the undershoot is due to the impulse response having a negative tail integral, which is possible because the function takes negative values; • the overshoot offsets this, by symmetry (the overall integral does not change under filtering); • the persistence of the oscillations is because increasing the cutoff narrows the impulse response but does not reduce its integral – the oscillations thus move towards the discontinuity, but do not decrease in magnitude. Square wave analysis We examine the $ N$th partial Fourier series $ S_{N}f(x)$ of a square wave $ f(x)$ with the periodicity $ L$ and a discontinuity of a vertical "full" jump $ c$ from $ y=y_{0}$ at $ x=x_{0}$. Because the case of odd $ N$ is very similar, let us just deal with the case when $ N$ is even: $S_{N}f(x)=\left(y_{0}+{\frac {c}{2}}\right)+{\frac {2c}{\pi }}\left(\sin(\omega (x-x_{0}))+{\frac {1}{3}}\sin(3\omega (x-x_{0}))+\cdots +{\frac {1}{N-1}}\sin((N-1)\omega (x-x_{0}))\right)$ with $ \omega ={\frac {2\pi }{L}}$. ($ N=2N'$ where $ N'$ is the number of non-zero sinusoidal Fourier series components so there are literatures using $ N'$ instead of $ N$.) Substituting $ x=x_{0}$ (a point of discontinuity), we obtain $S_{N}f(x_{0})=\left(y_{0}+{\frac {c}{2}}\right)={\frac {f(0^{-})+f(0^{+})}{2}}={\frac {y_{0}+(y_{0}+c)}{2}}$ as claimed above. (The first term that only survives is the average of the Fourier series.) Next, we find the first maximum of the oscillation around the discontinuity $ x=x_{0}$ by checking the first and second derivatives of $ S_{N}f(x)$. The first condition for the maximum is that the first derivative equals to zero as ${\frac {d}{dx}}S_{N}f(x)={\frac {2c}{\pi }}\left(\cos(\omega (x-x_{0}))+\cos(3\omega (x-x_{0}))+\cdots +\cos((N-1)\omega (x-x_{0}))\right)={\frac {c}{\pi }}{\frac {\sin(N\omega (x-x_{0}))}{\sin(\omega (x-x_{0}))}}=0$ where the 2nd equality is from one of Lagrange's trigonometric identities. Solving this condition gives $ x-x_{0}=k\pi /(N\omega )=kL/(2N)$ for integers $ k$ excluding multiples of $ N\omega $ to avoid the zero denominator, so $ k=1,2,\ldots ,N\omega -1,N\omega +1,\ldots $ and their negatives are allowed. The second derivative of $ S_{N}f(x)$ at $ x-x_{0}=kL/(2N)$ is ${\frac {d^{2}}{dx^{2}}}S_{N}f(x)={\frac {c\omega }{\pi }}\left({\frac {N\cos(N\omega (x-x_{0}))\sin(\omega (x-x_{0}))-\sin(N\omega (x-x_{0}))\cos(\omega (x-x_{0}))}{\sin ^{2}(\omega (x-x_{0}))}}\right),$ $\left.{\frac {d^{2}}{dx^{2}}}S_{N}f(x)\right\vert _{x_{0}+kL/(2N)}={\begin{cases}{\frac {2c}{L}}{\frac {N}{\sin(k\pi /N)}},&{\text{if }}k{\text{ is even,}}\\[4pt]{\frac {2c}{L}}{\frac {-N}{\sin(k\pi /N)}},&{\text{if }}k{\text{ is odd.}}\end{cases}}$ Thus, the first maximum occurs at $ x=x_{0}+L/(2N)$ ($ k=1$) and $ S_{N}f(x)$ at this $ x$ value is $S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=\left(y_{0}+{\frac {c}{2}}\right)+{\frac {2c}{\pi }}\left(\sin \left({\frac {\pi }{N}}\right)+{\frac {1}{3}}\sin \left({\frac {3\pi }{N}}\right)+\cdots +{\frac {1}{N-1}}\sin \left({\frac {(N-1)\pi }{N}}\right)\right)$ If we introduce the normalized sinc function $ \operatorname {sinc} (x)={\frac {\sin(\pi x)}{\pi x}}$ for $ x\neq 0$, we can rewrite this as $S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)=(y_{0}+{\frac {c}{2}})+c\left[{\frac {2}{N}}\operatorname {sinc} \left({\frac {1}{N}}\right)+{\frac {2}{N}}\operatorname {sinc} \left({\frac {3}{N}}\right)+\cdots +{\frac {2}{N}}\operatorname {sinc} \left({\frac {(N-1)}{N}}\right)\right].$ For a sufficiently large $ N$, the expression in the square brackets is a Riemann sum approximation to the integral $ \int _{0}^{1}\operatorname {sinc} (x)\ dx$ (more precisely, it is a midpoint rule approximation with spacing ${\tfrac {2}{N}}$). Since the sinc function is continuous, this approximation converges to the integral as $N\to \infty $. Thus, we have ${\begin{aligned}\lim _{N\to \infty }S_{N}f\left(x_{0}+{\frac {L}{2N}}\right)&=(y_{0}+{\frac {c}{2}})+c\int _{0}^{1}\operatorname {sinc} (x)\,dx\\[8pt]&=(y_{0}+{\frac {c}{2}})+{\frac {c}{\pi }}\int _{x=0}^{1}{\frac {\sin(\pi x)}{\pi x}}\,d(\pi x)\\[8pt]&=(y_{0}+{\frac {c}{2}})+{\frac {c}{\pi }}\int _{0}^{\pi }{\frac {\sin(t)}{t}}\ dt\quad =\quad (y_{0}+c)+c\cdot (0.089489872236\dots ),\end{aligned}}$ which was claimed in the previous section. A similar computation shows $\lim _{N\to \infty }S_{N}f\left(x_{0}-{\frac {L}{2N}}\right)=-c\int _{0}^{1}\operatorname {sinc} (x)\,dx=y_{0}-c\cdot (0.089489872236\dots ).$ Consequences The Gibbs phenomenon is undesirable because it causes artifacts, namely clipping from the overshoot and undershoot, and ringing artifacts from the oscillations. In the case of low-pass filtering, these can be reduced or eliminated by using different low-pass filters. In MRI, the Gibbs phenomenon causes artifacts in the presence of adjacent regions of markedly differing signal intensity. This is most commonly encountered in spinal MRIs where the Gibbs phenomenon may simulate the appearance of syringomyelia. The Gibbs phenomenon manifests as a cross pattern artifact in the discrete Fourier transform of an image,[15] where most images (e.g. micrographs or photographs) have a sharp discontinuity between boundaries at the top / bottom and left / right of an image. When periodic boundary conditions are imposed in the Fourier transform, this jump discontinuity is represented by continuum of frequencies along the axes in reciprocal space (i.e. a cross pattern of intensity in the Fourier transform). And although this article mainly focused on the difficulty with trying to construct discontinuities without artifacts in the time domain with only a partial Fourier series, it is also important to consider that because the inverse Fourier transform is extremely similar to the Fourier transform, there equivalently is difficulty with trying to construct discontinuities in the frequency domain using only a partial Fourier series. Thus for instance because idealized brick-wall and rectangular filters have discontinuities in the frequency domain, their exact representation in the time domain necessarily requires an infinitely-long sinc filter impulse response, since a finite impulse response will result in Gibbs rippling in the frequency response near cut-off frequencies, though this rippling can be reduced by windowing finite impulse response filters (at the expense of wider transition bands).[16] See also • Mach bands • Pinsky phenomenon • Runge's phenomenon (a similar phenomenon in polynomial approximations) • σ-approximation which adjusts a Fourier summation to eliminate the Gibbs phenomenon which would otherwise occur at discontinuities • Sine integral Notes 1. H. S. Carslaw (1930). "Chapter IX". Introduction to the theory of Fourier's series and integrals (Third ed.). New York: Dover Publications Inc. 2. Vretblad 2000 Section 4.7. 3. "6.7: Gibbs Phenomena". Engineering LibreTexts. 2020-05-24. Retrieved 2022-03-03. 4. M. Pinsky (2002). Introduction to Fourier Analysis and Wavelets. United states of America: Brooks/Cole. p. 27. 5. Wilbraham, Henry (1848) "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3 : 198–201. 6. Encyklopädie der Mathematischen Wissenschaften mit Einschluss ihrer Anwendungen (PDF). Vol. II T. 1 H 1. Wiesbaden: Vieweg+Teubner Verlag. 1914. p. 1049. Retrieved 14 September 2016. 7. Hammack, Bill; Kranz, Steve; Carpenter, Bruce (2014-10-29). Albert Michelson's Harmonic Analyzer: A Visual Tour of a Nineteenth Century Machine that Performs Fourier Analysis. Articulate Noise Books. ISBN 9780983966173. Retrieved 14 September 2016. 8. Hewitt, Edwin; Hewitt, Robert E. (1979). "The Gibbs-Wilbraham phenomenon: An episode in Fourier analysis". Archive for History of Exact Sciences. 21 (2): 129–160. doi:10.1007/BF00330404. S2CID 119355426. Available on-line at: National Chiao Tung University: Open Course Ware: Hewitt & Hewitt, 1979. Archived 2016-03-04 at the Wayback Machine 9. Bôcher, Maxime (April 1906) "Introduction to the theory of Fourier's series", Annals of Mathethematics, second series, 7 (3) : 81–152. The Gibbs phenomenon is discussed on pages 123–132; Gibbs's role is mentioned on page 129. 10. Carslaw, H. S. (1 October 1925). "A historical note on Gibbs' phenomenon in Fourier's series and integrals". Bulletin of the American Mathematical Society. 31 (8): 420–424. doi:10.1090/s0002-9904-1925-04081-1. ISSN 0002-9904. Retrieved 14 September 2016. 11. Rasmussen, Henrik O. "The Wavelet Gibbs Phenomenon". In Wavelets, Fractals and Fourier Transforms, Eds M. Farge et al., Clarendon Press, Oxford, 1993. 12. Susan E., Kelly (1995). "Gibbs Phenomenon for Wavelets" (PDF). Applied and Computational Harmonic Analysis (3). Archived from the original (PDF) on 2013-09-09. Retrieved 2012-03-31. 13. De Marchi, Stefano; Marchetti, Francesco; Perracchione, Emma; Poggiali, Davide (2020). "Polynomial interpolation via mapped bases without resampling". J. Comput. Appl. Math. 364: 112347. doi:10.1016/j.cam.2019.112347. ISSN 0377-0427. S2CID 199688130. 14. Fay, Temple H.; Kloppers, P. Hendrik (2001). "The Gibbs' phenomenon". International Journal of Mathematical Education in Science and Technology. 32 (1): 73–89. doi:10.1080/00207390117151. 15. R. Hovden, Y. Jiang, H.L. Xin, L.F. Kourkoutis (2015). "Periodic Artifact Reduction in Fourier Transforms of Full Field Atomic Resolution Images". Microscopy and Microanalysis. 21 (2): 436–441. arXiv:2210.09024. doi:10.1017/S1431927614014639. PMID 25597865. S2CID 22435248.{{cite journal}}: CS1 maint: multiple names: authors list (link) 16. "Gibbs phenomenon \| RecordingBlogs". www.recordingblogs.com. Retrieved 2022-03-05. References • Gibbs, J. Willard (1898), "Fourier's Series", Nature, 59 (1522): 200, doi:10.1038/059200b0, ISSN 0028-0836, S2CID 4004787 • Gibbs, J. Willard (1899), "Fourier's Series", Nature, 59 (1539): 606, doi:10.1038/059606a0, ISSN 0028-0836, S2CID 13420929 • Michelson, A. A.; Stratton, S. W. (1898), "A new harmonic analyser", Philosophical Magazine, 5 (45): 85–91 • Zygmund, Antoni (1959). Trigonometric Series (2nd ed.). Cambridge University Press. Volume 1, Volume 2. • Wilbraham, Henry (1848), "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3: 198–201 • Paul J. Nahin, Dr. Euler's Fabulous Formula, Princeton University Press, 2006. Ch. 4, Sect. 4. • Vretblad, Anders (2000), Fourier Analysis and its Applications, Graduate Texts in Mathematics, vol. 223, New York: Springer Publishing, p. 93, ISBN 978-0-387-00836-3 External links • Media related to Gibbs phenomenon at Wikimedia Commons • "Gibbs phenomenon", Encyclopedia of Mathematics, EMS Press, 2001 [1994] • Weisstein, Eric W., "Gibbs Phenomenon". From MathWorld—A Wolfram Web Resource. • Prandoni, Paolo, "Gibbs Phenomenon". • Radaelli-Sanchez, Ricardo, and Richard Baraniuk, "Gibbs Phenomenon". The Connexions Project. (Creative Commons Attribution License) • Horatio S Carslaw : Introduction to the theory of Fourier's series and integrals.pdf (introductiontot00unkngoog.pdf ) at archive.org • A Python implementation of the S-Gibbs algorithm mitigating the Gibbs Phenomenon https://github.com/pog87/FakeNodes.
Wild arc In geometric topology, a wild arc is an embedding of the unit interval into 3-dimensional space not equivalent to the usual one in the sense that there does not exist an ambient isotopy taking the arc to a straight line segment. Antoine (1920) found the first example of a wild arc, and Fox & Artin (1948) found another example called the Fox-Artin arc whose complement is not simply connected. See also • Wild knot • Horned sphere Further reading • Antoine, L. (1920), "Sur la possibilité d'étendre l'homéomorphie de deux figures à leurs voisinages", C. R. Acad. Sci. Paris (in French), 171: 661 • Fox, Ralph H.; Harrold, O. G. (1962), "The Wilder arcs", Topology of 3-manifolds and related topics (Proc. The Univ. of Georgia Institute, 1961), Prentice Hall, pp. 184–187, MR 0140096 • Fox, Ralph H.; Artin, Emil (1948), "Some wild cells and spheres in three-dimensional space", Annals of Mathematics, Second Series, 49 (4): 979–990, doi:10.2307/1969408, ISSN 0003-486X, JSTOR 1969408, MR 0027512 • Hocking, John Gilbert; Young, Gail Sellers (1988) [1961]. Topology. Dover. pp. 176–177. ISBN 0-486-65676-4. • McPherson, James M. (1973), "Wild arcs in three-space. I. Families of Fox–Artin arcs", Pacific Journal of Mathematics, 45 (2): 585–598, doi:10.2140/pjm.1973.45.585, ISSN 0030-8730, MR 0343276 Topology Fields • General (point-set) • Algebraic • Combinatorial • Continuum • Differential • Geometric • low-dimensional • Homology • cohomology • Set-theoretic • Digital Key concepts • Open set / Closed set • Interior • Continuity • Space • compact • Connected • Hausdorff • metric • uniform • Homotopy • homotopy group • fundamental group • Simplicial complex • CW complex • Polyhedral complex • Manifold • Bundle (mathematics) • Second-countable space • Cobordism Metrics and properties • Euler characteristic • Betti number • Winding number • Chern number • Orientability Key results • Banach fixed-point theorem • De Rham cohomology • Invariance of domain • Poincaré conjecture • Tychonoff's theorem • Urysohn's lemma • Category • Mathematics portal • Wikibook • Wikiversity • Topics • general • algebraic • geometric • Publications
Wild number Originally, wild numbers are the numbers supposed to belong to a fictional sequence of numbers imagined to exist in the mathematical world of the mathematical fiction The Wild Numbers authored by Philibert Schogt, a Dutch philosopher and mathematician.[1] Even though Schogt has given a definition of the wild number sequence in his novel, it is couched in a deliberately imprecise language that the definition turns out to be no definition at all. However, the author claims that the first few members of the sequence are 11, 67, 2, 4769, 67. Later, inspired by this wild and erratic behaviour of the fictional wild numbers, American mathematician J. C. Lagarias used the terminology to describe a precisely defined sequence of integers which shows somewhat similar wild and erratic behaviour. Lagaria's wild numbers are connected with the Collatz conjecture and the concept of the 3x + 1 semigroup.[2][3] The original fictional sequence of wild numbers has found a place in the On-Line Encyclopedia of Integer Sequences.[4] The wild number problem In the novel The Wild Numbers, The Wild Number Problem is formulated as follows: • Beauregard had defined a number of deceptively simple operations, which, when applied to a whole number, at first resulted in fractions. But if the same steps were repeated often enough, the eventual outcome was once again a whole number. Or, as Beauregard cheerfully observed: “In all numbers lurks a wild number, guaranteed to emerge when you provoke them long enough” . 0 yielded the wild number 11, 1 brought forth 67, 2 itself, 3 suddenly manifested itself as 4769, 4, surprisingly, brought forth 67 again. Beauregard himself had found fifty different wild numbers. The money prize was now awarded to whoever found a new one.[5] But it has not been specified what those "deceptively simple operations" are. Consequently, there is no way of knowing how those numbers 11, 67, etc. were obtained and no way of finding what the next wild number would be. History of The Wild Number Problem The novel The Wild Numbers has constructed a fictitious history for The Wild Number Problem. The important milestones in this history can be summarised as follows. DateEvent 1823Anatole Millechamps de Beauregard poses the Wild Number Problem in its original form. 1830sThe problem is generalised: How many wild numbers are there? Are there infinitely many wild numbers? It was conjectured that all numbers are wild. 1907Heinrich Riedel disproves the conjecture by showing that 3 is not a wild number. Later he also proves that there are infinitely many non-wild numbers. Early 1960sDimitri Arkanov sparks renewed interest in the almost forgotten problem by discovering a fundamental relationship between wild numbers and prime numbers. The presentIsaac Swift finds a solution. Real wild numbers In mathematics, the multiplicative semigroup, denoted by W0, generated by the set $\left\{{\frac {3n+2}{2n+1}}:n\geq 0\right\}$ is called the Wooley semigroup in honour of the American mathematician Trevor D. Wooley. The multiplicative semigroup, denoted by W, generated by the set $\left\{{\frac {1}{2}}\right\}\cup \left\{{\frac {3n+2}{2n+1}}:n\geq 0\right\}$ is called the wild semigroup. The set of integers in W0 is itself a multiplicative semigroup. It is called the Wooley integer semigroup and members of this semigroup are called Wooley integers. Similarly, the set of integers in W is itself a multiplicative semigroup. It is called the wild integer semigroup and members of this semigroup are called wild numbers.[6] The wild numbers in OEIS The On-Line Encyclopedia of Integer Sequences (OEIS) has an entry with the identifying number A058883 relating to the wild numbers. According to OEIS, "apparently these are completely fictional and there is no mathematical explanation". However, the OEIS has some entries relating to pseudo-wild numbers carrying well-defined mathematical explanations.[4] Sequences of pseudo-wild numbers Even though the sequence of wild numbers is entirely fictional, several mathematicians have tried to find rules that would generate the sequence of the fictional wild numbers. All these attempts have resulted in failures. However, in the process, certain new sequences of integers were created having similar wild and erratic behavior. These well-defined sequences are referred to as sequences of pseudo-wild numbers. A good example of this is the one discovered by the Dutch mathematician Floor van Lamoen. This sequence is defined as follows:[7][8] For a rational number p/q let $f(p/q)={\frac {pq}{{\text{ sum of digits of }}p{\text{ and }}q}}$. For a positive integer n, the n-th pseudo-wild number is the number obtained by iterating f, starting at n/1, until an integer is reached, and if no integer is reached the pseudo-wild number is 0. For example, taking n=2, we have ${\frac {2}{1}},{\frac {2}{3}},{\frac {6}{5}},{\frac {30}{11}},66$ and so the second pseudo-wild number is 66. The first few pseudo-wild numbers are 66, 66, 462, 180, 66, 31395, 714, 72, 9, 5. References 1. Philibert Schogt (March 23, 2000). The Wild Numbers: A Novel (First ed.). Four Walls Eight Windows. ISBN 978-1568581668. 2. Emmer, Michele (2013). Imagine Math 2: Between Culture and Mathematics. Springer Science & Business Media. pp. 37–38. ISBN 9788847028890. 3. Applegate, David; Lagarias, Jeffrey C. (2006). "The 3x + 1 semigroup". Journal of Number Theory. 117 (1): 146–159. doi:10.1016/j.jnt.2005.06.010. MR 2204740. 4. "A058883 : The "Wild Numbers", from the novel of the same title (Version 1)". OEIS. The OEIS Foundation. Retrieved 19 March 2016. 5. Philibert Schogt (March 23, 2000). The Wild Numbers: A Novel (First ed.). Four Walls Eight Windows. p. 34. ISBN 978-1568581668. 6. Jeffrey C. Lagarias (February 2006). "Wild and Wooley Numbers" (PDF). American Mathematical Monthly. 113 (2): 98–108. doi:10.2307/27641862. JSTOR 27641862. Retrieved 28 March 2016. 7. Schogt, Philibert (2012). "The Wild Number Problem: Math or fiction?". arXiv:1211.6583 [math.HO]. 8. "A059175". OEIS. The OEIS Foundation. Retrieved 30 March 2016.
Wild problem In the mathematical areas of linear algebra and representation theory, a problem is wild if it contains the problem of classifying pairs of square matrices up to simultaneous similarity.[1][2][3] Examples of wild problems are classifying indecomposable representations of any quiver that is neither a Dynkin quiver (i.e. the underlying undirected graph of the quiver is a (finite) Dynkin diagram) nor a Euclidean quiver (i.e., the underlying undirected graph of the quiver is an affine Dynkin diagram).[4] Necessary and sufficient conditions have been proposed to check the simultaneously block triangularization and diagonalization of a finite set of matrices under the assumption that each matrix is diagonalizable over the field of the complex numbers.[5] See also • Semi-invariant of a quiver References 1. Nazarova, L. A. (1974), "Representations of partially ordered sets of infinite type", Funkcional'nyi Analiz i ego Priloženija, 8 (4): 93–94, MR 0354455 2. Gabriel, P.; Nazarova, L. A.; Roĭter, A. V.; Sergeĭchuk, V. V.; Vossieck, D. (1993), "Tame and wild subspace problems", Akademīya Nauk Ukraïni, 45 (3): 313–352, doi:10.1007/BF01061008, MR 1238674, S2CID 122603779 3. Shavarovskiĭ, B. Z. (2004), "On some "tame" and "wild" aspects of the problem of semiscalar equivalence of polynomial matrices", Matematicheskie Zametki, 76 (1): 119–132, doi:10.1023/B:MATN.0000036747.26055.cb, MR 2099848, S2CID 120324840 4. Drozd, Yuriy A.; Golovashchuk, Natalia S.; Zembyk, Vasyl V. (2017), "Representations of nodal algebras of type E", Algebra and Discrete Mathematics, 23 (1): 16–34, hdl:123456789/155928, MR 3634499 5. Mesbahi, Afshin; Haeri, Mohammad (2015), "Conditions on decomposing linear systems with more than one matrix to block triangular or diagonal form", IEEE Transactions on Automatic Control, 60 (1): 233–239, doi:10.1109/TAC.2014.2326292, MR 3299432, S2CID 27053281
Wiles's proof of Fermat's Last Theorem Wiles's proof of Fermat's Last Theorem is a proof by British mathematician Andrew Wiles of a special case of the modularity theorem for elliptic curves. Together with Ribet's theorem, it provides a proof for Fermat's Last Theorem. Both Fermat's Last Theorem and the modularity theorem were almost universally considered inaccessible to prove by contemporaneous mathematicians, meaning that they were believed to be impossible to prove using current knowledge.[1]: 203–205, 223, 226 Wiles first announced his proof on 23 June 1993 at a lecture in Cambridge entitled "Modular Forms, Elliptic Curves and Galois Representations".[2] However, in September 1993 the proof was found to contain an error. One year later on 19 September 1994, in what he would call "the most important moment of [his] working life", Wiles stumbled upon a revelation that allowed him to correct the proof to the satisfaction of the mathematical community. The corrected proof was published in 1995.[3] Wiles's proof uses many techniques from algebraic geometry and number theory, and has many ramifications in these branches of mathematics. It also uses standard constructions of modern algebraic geometry, such as the category of schemes and Iwasawa theory, and other 20th-century techniques which were not available to Fermat. The proof's method of identification of a deformation ring with a Hecke algebra (now referred to as an R=T theorem) to prove modularity lifting theorems has been an influential development in algebraic number theory. Together, the two papers which contain the proof are 129 pages long,[4][5] and consumed over seven years of Wiles's research time. John Coates described the proof as one of the highest achievements of number theory, and John Conway called it "the proof of the [20th] century."[6] Wiles's path to proving Fermat's Last Theorem, by way of proving the modularity theorem for the special case of semistable elliptic curves, established powerful modularity lifting techniques and opened up entire new approaches to numerous other problems. For proving Fermat's Last Theorem, he was knighted, and received other honours such as the 2016 Abel Prize. When announcing that Wiles had won the Abel Prize, the Norwegian Academy of Science and Letters described his achievement as a "stunning proof".[3] Precursors to Wiles's proof Fermat's Last Theorem and progress prior to 1980 Main article: Fermat's Last Theorem Fermat's Last Theorem, formulated in 1637, states that no three positive integers a, b, and c can satisfy the equation $a^{n}+b^{n}=c^{n}$ if n is an integer greater than two (n > 2). Over time, this simple assertion became one of the most famous unproved claims in mathematics. Between its publication and Andrew Wiles's eventual solution over 350 years later, many mathematicians and amateurs attempted to prove this statement, either for all values of n > 2, or for specific cases. It spurred the development of entire new areas within number theory. Proofs were eventually found for all values of n up to around 4 million, first by hand, and later by computer. However, no general proof was found that would be valid for all possible values of n, nor even a hint how such a proof could be undertaken. The Taniyama–Shimura–Weil conjecture Main article: Modularity theorem Separately from anything related to Fermat's Last Theorem, in the 1950s and 1960s Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between elliptic curves and modular forms. These were mathematical objects with no known connection between them. Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways. They conjectured that every rational elliptic curve is also modular. This became known as the Taniyama–Shimura conjecture. In the West, this conjecture became well known through a 1967 paper by André Weil, who gave conceptual evidence for it; thus, it is sometimes called the Taniyama–Shimura–Weil conjecture. By around 1980, much evidence had been accumulated to form conjectures about elliptic curves, and many papers had been written which examined the consequences if the conjecture were true, but the actual conjecture itself was unproven and generally considered inaccessible—meaning that mathematicians believed a proof of the conjecture was probably impossible using current knowledge. For decades, the conjecture remained an important but unsolved problem in mathematics. Around 50 years after first being proposed, the conjecture was finally proven and renamed the modularity theorem, largely as a result of Andrew Wiles's work described below. Frey's curve On yet another separate branch of development, in the late 1960s, Yves Hellegouarch came up with the idea of associating hypothetical solutions (a, b, c) of Fermat's equation with a completely different mathematical object: an elliptic curve.[7] The curve consists of all points in the plane whose coordinates (x, y) satisfy the relation $y^{2}=x(x-a^{n})(x+b^{n}).$ Such an elliptic curve would enjoy very special properties due to the appearance of high powers of integers in its equation and the fact that an + bn = cn would be an nth power as well. In 1982–1985, Gerhard Frey called attention to the unusual properties of this same curve, now called a Frey curve. He showed that it was likely that the curve could link Fermat and Taniyama, since any counterexample to Fermat's Last Theorem would probably also imply that an elliptic curve existed that was not modular. Frey showed that there were good reasons to believe that any set of numbers (a, b, c, n) capable of disproving Fermat's Last Theorem could also probably be used to disprove the Taniyama–Shimura–Weil conjecture. Therefore, if the Taniyama–Shimura–Weil conjecture were true, no set of numbers capable of disproving Fermat could exist, so Fermat's Last Theorem would have to be true as well. Mathematically, the conjecture says that each elliptic curve with rational coefficients can be constructed in an entirely different way, not by giving its equation but by using modular functions to parametrise coordinates x and y of the points on it. Thus, according to the conjecture, any elliptic curve over Q would have to be a modular elliptic curve, yet if a solution to Fermat's equation with non-zero a, b, c and n greater than 2 existed, the corresponding curve would not be modular, resulting in a contradiction. If the link identified by Frey could be proven, then in turn, it would mean that a disproof of Fermat's Last Theorem would disprove the Taniyama–Shimura–Weil conjecture, or by contraposition, a proof of the latter would prove the former as well.[8] Ribet's theorem Main article: Ribet's theorem To complete this link, it was necessary to show that Frey's intuition was correct: that a Frey curve, if it existed, could not be modular. In 1985, Jean-Pierre Serre provided a partial proof that a Frey curve could not be modular. Serre did not provide a complete proof of his proposal; the missing part (which Serre had noticed early on[9]: 1 ) became known as the epsilon conjecture (sometimes written ε-conjecture; now known as Ribet's theorem). Serre's main interest was in an even more ambitious conjecture, Serre's conjecture on modular Galois representations, which would imply the Taniyama–Shimura–Weil conjecture. However his partial proof came close to confirming the link between Fermat and Taniyama. In the summer of 1986, Ken Ribet succeeded in proving the epsilon conjecture, now known as Ribet's theorem. His article was published in 1990. In doing so, Ribet finally proved the link between the two theorems by confirming, as Frey had suggested, that a proof of the Taniyama–Shimura–Weil conjecture for the kinds of elliptic curves Frey had identified, together with Ribet's theorem, would also prove Fermat's Last Theorem. In mathematical terms, Ribet's theorem showed that if the Galois representation associated with an elliptic curve has certain properties (which Frey's curve has), then that curve cannot be modular, in the sense that there cannot exist a modular form which gives rise to the same Galois representation.[10] Situation prior to Wiles's proof Following the developments related to the Frey curve, and its link to both Fermat and Taniyama, a proof of Fermat's Last Theorem would follow from a proof of the Taniyama–Shimura–Weil conjecture—or at least a proof of the conjecture for the kinds of elliptic curves that included Frey's equation (known as semistable elliptic curves). • From Ribet's Theorem and the Frey curve, any 4 numbers able to be used to disprove Fermat's Last Theorem could also be used to make a semistable elliptic curve ("Frey's curve") that could never be modular; • But if the Taniyama–Shimura–Weil conjecture were also true for semistable elliptic curves, then by definition every Frey's curve that existed must be modular. • The contradiction could have only one answer: if Ribet's theorem and the Taniyama–Shimura–Weil conjecture for semistable curves were both true, then it would mean there could not be any solutions to Fermat's equation—because then there would be no Frey curves at all, meaning no contradictions would exist. This would finally prove Fermat's Last Theorem. However, despite the progress made by Serre and Ribet, this approach to Fermat was widely considered unusable as well, since almost all mathematicians saw the Taniyama–Shimura–Weil conjecture itself as completely inaccessible to proof with current knowledge.[1]: 203–205, 223, 226 For example, Wiles's ex-supervisor John Coates stated that it seemed "impossible to actually prove",[1]: 226 and Ken Ribet considered himself "one of the vast majority of people who believed [it] was completely inaccessible".[1]: 223 Andrew Wiles Hearing of Ribet's 1986 proof of the epsilon conjecture, English mathematician Andrew Wiles, who had studied elliptic curves and had a childhood fascination with Fermat, decided to begin working in secret towards a proof of the Taniyama–Shimura–Weil conjecture, since it was now professionally justifiable,[11] as well as because of the enticing goal of proving such a long-standing problem. Ribet later commented that "Andrew Wiles was probably one of the few people on earth who had the audacity to dream that you can actually go and prove [it]."[1]: 223 Announcement and subsequent developments Wiles initially presented his proof in 1993. It was finally accepted as correct, and published, in 1995, following the correction of a subtle error in one part of his original paper. His work was extended to a full proof of the modularity theorem over the following six years by others, who built on Wiles's work. Announcement and final proof (1993–1995) During 21–23 June 1993, Wiles announced and presented his proof of the Taniyama–Shimura conjecture for semistable elliptic curves, and hence of Fermat's Last Theorem, over the course of three lectures delivered at the Isaac Newton Institute for Mathematical Sciences in Cambridge, England.[2] There was a relatively large amount of press coverage afterwards.[12] After the announcement, Nick Katz was appointed as one of the referees to review Wiles's manuscript. In the course of his review, he asked Wiles a series of clarifying questions that led Wiles to recognise that the proof contained a gap. There was an error in one critical portion of the proof which gave a bound for the order of a particular group: the Euler system used to extend Kolyvagin and Flach's method was incomplete. The error would not have rendered his work worthless—each part of Wiles's work was highly significant and innovative by itself, as were the many developments and techniques he had created in the course of his work, and only one part was affected.[1]: 289, 296–297 Without this part proved, however, there was no actual proof of Fermat's Last Theorem. Wiles spent almost a year trying to repair his proof, initially by himself and then in collaboration with his former student Richard Taylor, without success.[13][14][15] By the end of 1993, rumours had spread that under scrutiny, Wiles's proof had failed, but how seriously was not known. Mathematicians were beginning to pressure Wiles to disclose his work whether or not complete, so that the wider community could explore and use whatever he had managed to accomplish. Instead of being fixed, the problem, which had originally seemed minor, now seemed very significant, far more serious, and less easy to resolve.[16] Wiles states that on the morning of 19 September 1994, he was on the verge of giving up and was almost resigned to accepting that he had failed, and to publishing his work so that others could build on it and find the error. He states that he was having a final look to try to understand the fundamental reasons why his approach could not be made to work, when he had a sudden insight that the specific reason why the Kolyvagin–Flach approach would not work directly also meant that his original attempt using Iwasawa theory could be made to work if he strengthened it using experience gained from the Kolyvagin–Flach approach since then. Each was inadequate by itself, but fixing one approach with tools from the other would resolve the issue and produce a class number formula (CNF) valid for all cases that were not already proven by his refereed paper:[13][17] I was sitting at my desk examining the Kolyvagin–Flach method. It wasn't that I believed I could make it work, but I thought that at least I could explain why it didn't work. Suddenly I had this incredible revelation. I realised that, the Kolyvagin–Flach method wasn't working, but it was all I needed to make my original Iwasawa theory work from three years earlier. So out of the ashes of Kolyvagin–Flach seemed to rise the true answer to the problem. It was so indescribably beautiful; it was so simple and so elegant. I couldn't understand how I'd missed it and I just stared at it in disbelief for twenty minutes. Then during the day I walked around the department, and I'd keep coming back to my desk looking to see if it was still there. It was still there. I couldn't contain myself, I was so excited. It was the most important moment of my working life. Nothing I ever do again will mean as much. — Andrew Wiles, quoted by Simon Singh[18] On 6 October Wiles asked three colleagues (including Faltings) to review his new proof,[19] and on 24 October 1994 Wiles submitted two manuscripts, "Modular elliptic curves and Fermat's Last Theorem"[4] and "Ring theoretic properties of certain Hecke algebras",[5] the second of which Wiles had written with Taylor and proved that certain conditions were met which were needed to justify the corrected step in the main paper. The two papers were vetted and finally published as the entirety of the May 1995 issue of the Annals of Mathematics. The new proof was widely analysed, and became accepted as likely correct in its major components.[6][10][11] These papers established the modularity theorem for semistable elliptic curves, the last step in proving Fermat's Last Theorem, 358 years after it was conjectured. Subsequent developments Fermat claimed to "... have discovered a truly marvelous proof of this, which this margin is too narrow to contain".[20][21] Wiles's proof is very complex, and incorporates the work of so many other specialists that it was suggested in 1994 that only a small number of people were capable of fully understanding at that time all the details of what he had done.[2][22] The complexity of Wiles's proof motivated a 10-day conference at Boston University; the resulting book of conference proceedings aimed to make the full range of required topics accessible to graduate students in number theory.[9] As noted above, Wiles proved the Taniyama–Shimura–Weil conjecture for the special case of semistable elliptic curves, rather than for all elliptic curves. Over the following years, Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor (sometimes abbreviated as "BCDT") carried the work further, ultimately proving the Taniyama–Shimura–Weil conjecture for all elliptic curves in a 2001 paper.[23] Now proved, the conjecture became known as the modularity theorem. In 2005, Dutch computer scientist Jan Bergstra posed the problem of formalizing Wiles's proof in such a way that it could be verified by computer.[24] Summary of Wiles's proof Wiles proved the modularity theorem for semistable elliptic curves, from which Fermat’s last theorem follows using proof by contradiction. In this proof method, one assumes the opposite of what is to be proved, and shows if that were true, it would create a contradiction. The contradiction shows that the assumption (that the conclusion is wrong) must have been incorrect, requiring the conclusion to hold. The proof falls roughly in two parts. In the first part, Wiles proves a general result about "lifts", known as the "modularity lifting theorem". This first part allows him to prove results about elliptic curves by converting them to problems about Galois representations of elliptic curves. He then uses this result to prove that all semistable curves are modular, by proving that the Galois representations of these curves are modular. Outline proof Comment Part 1: setting up the proof 1 We start by assuming (for the sake of contradiction) that Fermat's Last Theorem is incorrect. That would mean there is at least one non-zero solution (a, b, c, n) (with all numbers rational and n > 2 and prime) to an + bn = cn. 2 Ribet's theorem (using Frey and Serre's work) shows that we can create a semistable elliptic curve E using the numbers (a, b, c, and n), which is never modular. If we can prove that all such elliptic curves will be modular (meaning that they match a modular form), then we have our contradiction and have proved our assumption (that such a set of numbers exists) was wrong. If the assumption is wrong, that means no such numbers exist, which proves Fermat's Last Theorem is correct. 3 Suppose that Fermat's Last Theorem is incorrect. This means a set of numbers (a, b, c, n) must exist that is a solution of Fermat's equation, and we can use the solution to create a Frey curve which is semistable and elliptic. So we assume that (somehow) we have found a solution and created such a curve (which we will call "E"), and see what happens. Part 2: the modularity lifting theorem 4 Galois representations of elliptic curves ρ(E, p) for any prime p > 3 have been studied by many mathematicians. Wiles aims first of all to prove a result about these representations, that he will use later: that if a semistable elliptic curve E has a Galois representation ρ(E, p) that is modular, the elliptic curve itself must be modular. Proving this is helpful in two ways: it makes counting and matching easier, and, significantly, to prove the representation is modular, we would only have to prove it for one single prime number p, and we can do this using any prime that makes our work easy – it does not matter which prime we use. This is the most difficult part of the problem – technically it means proving that if the Galois representation ρ(E, p) is a modular form, so are all the other related Galois representations ρ(E, p∞) for all powers of p.[3] This is the so-called "modular lifting problem", and Wiles approached it using deformations. Any elliptic curve (or a representation of an elliptic curve) can be categorized as either reducible or irreducible. The proof will be slightly different depending whether or not the elliptic curve's representation is reducible. To compare elliptic curves and modular forms directly is difficult; past efforts to count and match elliptic curves and modular forms had all failed. But elliptic curves can be represented within Galois theory. Wiles realized that working with the representations of elliptic curves instead of the curves themselves would make counting and matching them to modular forms far easier. From this point on, the proof primarily aims to prove: (1) if the geometric Galois representation of a semistable elliptic curve is modular, so is the curve itself; and (2) the geometric Galois representations of all semistable elliptic curves are modular. Together, these allow us to work with representations of curves rather than directly with elliptic curves themselves. Our original goal will have been transformed into proving the modularity of geometric Galois representations of semistable elliptic curves, instead. Wiles described this realization as a "key breakthrough". A Galois representation of an elliptic curve is G → GL(Zp). To show that a geometric Galois representation of an elliptic curve is a modular form, we need to find a normalized eigenform whose eigenvalues (which are also its Fourier series coefficients) satisfy a congruence relationship for all but a finite number of primes. 5 Wiles's initial strategy is to count and match using proof by induction and a class number formula ("CNF"): an approach in which, once the hypothesis is proved for one elliptic curve, it can automatically be extended to be proven for all subsequent elliptic curves. It was in this area that Wiles found difficulties, first with horizontal Iwasawa theory and later with his extension of Kolyvagin–Flach. Wiles's work extending Kolyvagin–Flach was mainly related to making Kolyvagin–Flach strong enough to prove the full CNF he would use. It later turned out that neither of these approaches by itself could produce a CNF able to cover all types of semistable elliptic curves, and the final piece of his proof in 1995 was to realize that he could succeed by strengthening Iwasawa theory with the techniques from Kolyvagin–Flach. 6 At this point, the proof has shown a key point about Galois representations: If the geometric Galois representation ρ(E, p) of a semistable elliptic curve E is irreducible and modular (for some prime number p > 2), then subject to some technical conditions, E is modular. This is Wiles's lifting theorem (or modularity lifting theorem), a major and revolutionary accomplishment at the time. Crucially, this result does not just show that modular irreducible representations imply modular curves. It also means we can prove a representation is modular by using any prime number > 2 that we find easiest to use (because proving it for just one prime > 2 proves it for all primes > 2). So we can try to prove all of our elliptic curves are modular by using one prime number as p - but if we do not succeed in proving this for all elliptic curves, perhaps we can prove the rest by choosing different prime numbers as 'p' for the difficult cases. The proof must cover the Galois representations of all semistable elliptic curves E, but for each individual curve, we only need to prove it is modular using one prime number p.) Part 3: Proving that all semistable elliptic curves are modular 7 With the lifting theorem proved, we return to the original problem. We will categorize all semistable elliptic curves based on the reducibility of their Galois representations, and use the powerful lifting theorem on the results. From above, it does not matter which prime is chosen for the representations. We can use any one prime number that is easiest. 3 is the smallest prime number more than 2, and some work has already been done on representations of elliptic curves using ρ(E, 3), so choosing 3 as our prime number is a helpful starting point. Wiles found that it was easier to prove the representation was modular by choosing a prime p = 3 in the cases where the representation ρ(E, 3) is irreducible, but the proof when ρ(E, 3) is reducible was easier to prove by choosing p = 5. So the proof splits in two at this point. The proof's use of both p = 3 and p = 5 below, is the so-called "3/5 switch" referred to in some descriptions of the proof, which Wiles noticed in a paper of Mazur's in 1993, although the trick itself dates back to the 19th century. The switch between p = 3 and p = 5 has since opened a significant area of study in its own right (see Serre's modularity conjecture). 8 If the Galois representation ρ(E, 3) (i.e., using p = 3) is irreducible, then it was known from around 1980 that its Galois representation is also always modular. Wiles uses his modularity lifting theorem to make short work of this case: • If the representation ρ(E, 3) is irreducible, then we know the representation is also modular (Langlands and Tunnell), but... • ... if the representation is both irreducible and modular then E itself is modular (modularity lifting theorem). Langlands and Tunnell proved this in two papers in the early 1980s. The proof is based on the fact that ρ(E, 3) has the same symmetry group as the general quartic equation in one variable, which was one of the few general classes of diophantine equation known at that time to be modular. This existing result for p = 3 is crucial to Wiles's approach and is one reason for initially using p = 3. 9 So we now consider what happens if ρ(E, 3) is reducible. Wiles found that when the representation of an elliptic curve using p = 3 is reducible, it was easier to work with p = 5 and use his new lifting theorem to prove that ρ(E, 5) will always be modular, than to try and prove directly that ρ(E, 3) itself is modular (remembering that we only need to prove it for one prime). 5 is the next prime number after 3, and any prime number can be used, perhaps 5 will be an easier prime number to work with than 3? But it looks hopeless initially to prove that ρ(E, 5) is always modular, for much the same reason that the general quintic equation cannot be solved by radicals. So Wiles has to find a way around this. 9.1 If ρ(E, 3) and ρ(E, 5) are both reducible, Wiles proved directly that ρ(E, 5) must be modular. 9.2 The last case is if ρ(E, 3) is reducible and ρ(E, 5) is irreducible. Wiles showed that in this case, one could always find another semistable elliptic curve F such that the representation ρ(F, 3) is irreducible and also the representations ρ(E, 5) and ρ(F, 5) are isomorphic (they have identical structures). - The first of these properties shows that F must be modular (Langlands and Tunnell again: all irreducible representations with p = 3 are modular). - If F is modular then we know ρ(F, 5) must be modular as well. - But because the representations of E and F with p = 5 have exactly the same structure, and we know that ρ(F, 5) is modular, ρ(E, 5) must be modular as well. 9.3 Therefore, if ρ(E, 3) is reducible, we have proved that ρ(E, 5) will always be modular. But if ρ(E, 5) is modular, then the modularity lifting theorem shows that E itself is modular. This step shows the real power of the modularity lifting theorem. Results 10 We have now proved that whether or not ρ(E, 3) is irreducible, E (which could be any semistable elliptic curve) will always be modular. This means that all semistable elliptic curves must be modular. This proves: (a) The Taniyama–Shimura–Weil conjecture for semistable elliptic curves; and also (b) Because there cannot be a contradiction, it also proves that the kinds of elliptic curves described by Frey cannot actually exist. Therefore no solutions to Fermat's equation can exist either, so Fermat's Last Theorem is also true. We have our proof by contradiction, because we have proven that if Fermat's Last Theorem is incorrect, we could create a semistable elliptic curve that cannot be modular (Ribet's Theorem) and must be modular (Wiles). As it cannot be both, the only answer is that no such curve exists. Mathematical detail of Wiles's proof Overview Wiles opted to attempt to match elliptic curves to a countable set of modular forms. He found that this direct approach was not working, so he transformed the problem by instead matching the Galois representations of the elliptic curves to modular forms. Wiles denotes this matching (or mapping) that, more specifically, is a ring homomorphism: $R_{n}\rightarrow \mathbf {T} _{n}.$ $R$ is a deformation ring and $\mathbf {T} $ is a Hecke ring. Wiles had the insight that in many cases this ring homomorphism could be a ring isomorphism (Conjecture 2.16 in Chapter 2, §3 of the 1995 paper[4]). He realised that the map between $R$ and $\mathbf {T} $ is an isomorphism if and only if two abelian groups occurring in the theory are finite and have the same cardinality. This is sometimes referred to as the "numerical criterion". Given this result, Fermat's Last Theorem is reduced to the statement that two groups have the same order. Much of the text of the proof leads into topics and theorems related to ring theory and commutation theory. Wiles's goal was to verify that the map $R\rightarrow \mathbf {T} $ is an isomorphism and ultimately that $R=\mathbf {T} $. In treating deformations, Wiles defined four cases, with the flat deformation case requiring more effort to prove and treated in a separate article in the same volume entitled "Ring-theoretic properties of certain Hecke algebras". Gerd Faltings, in his bulletin, gives the following commutative diagram (p. 745): or ultimately that $R=\mathbf {T} $, indicating a complete intersection. Since Wiles could not show that $R=\mathbf {T} $ directly, he did so through $\mathbf {Z} _{3},\mathbf {F} _{3}$ and $\mathbf {T} /{\mathfrak {m}}$ via lifts. In order to perform this matching, Wiles had to create a class number formula (CNF). He first attempted to use horizontal Iwasawa theory but that part of his work had an unresolved issue such that he could not create a CNF. At the end of the summer of 1991, he learned about an Euler system recently developed by Victor Kolyvagin and Matthias Flach that seemed "tailor made" for the inductive part of his proof, which could be used to create a CNF, and so Wiles set his Iwasawa work aside and began working to extend Kolyvagin and Flach's work instead, in order to create the CNF his proof would require.[25] By the spring of 1993, his work had covered all but a few families of elliptic curves, and in early 1993, Wiles was confident enough of his nearing success to let one trusted colleague into his secret. Since his work relied extensively on using the Kolyvagin–Flach approach, which was new to mathematics and to Wiles, and which he had also extended, in January 1993 he asked his Princeton colleague, Nick Katz, to help him review his work for subtle errors. Their conclusion at the time was that the techniques Wiles used seemed to work correctly.[1]: 261–265 [26] Wiles's use of Kolyvagin–Flach would later be found to be the point of failure in the original proof submission, and he eventually had to revert to Iwasawa theory and a collaboration with Richard Taylor to fix it. In May 1993, while reading a paper by Mazur, Wiles had the insight that the 3/5 switch would resolve the final issues and would then cover all elliptic curves. General approach and strategy Given an elliptic curve E over the field Q of rational numbers $E({\bar {\mathbf {Q} }})$, for every prime power $\ell ^{n}$, there exists a homomorphism from the absolute Galois group $\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )$ to $\operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ),$ the group of invertible 2 by 2 matrices whose entries are integers modulo $\ell ^{n}$. This is because $E({\bar {\mathbf {Q} }})$, the points of E over ${\bar {\mathbf {Q} }}$, form an abelian group, on which $\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )$ acts; the subgroup of elements x such that $\ell ^{n}x=0$ is just $(\mathbf {Z} /\ell ^{n}\mathbf {Z} )^{2}$, and an automorphism of this group is a matrix of the type described. Less obvious is that given a modular form of a certain special type, a Hecke eigenform with eigenvalues in Q, one also gets a homomorphism from the absolute Galois group $\operatorname {Gal} ({\bar {\mathbf {Q} }}/\mathbf {Q} )\rightarrow \operatorname {GL} _{2}(\mathbf {Z} /l^{n}\mathbf {Z} ).$ This goes back to Eichler and Shimura. The idea is that the Galois group acts first on the modular curve on which the modular form is defined, thence on the Jacobian variety of the curve, and finally on the points of $\ell ^{n}$ power order on that Jacobian. The resulting representation is not usually 2-dimensional, but the Hecke operators cut out a 2-dimensional piece. It is easy to demonstrate that these representations come from some elliptic curve but the converse is the difficult part to prove. Instead of trying to go directly from the elliptic curve to the modular form, one can first pass to the $(\mathrm {mod} \,\ell ^{n})$ representation for some ℓ and n, and from that to the modular form. In the case ℓ = 3 and n= 1, results of the Langlands–Tunnell theorem show that the $(\mathrm {mod} \,3)$ representation of any elliptic curve over Q comes from a modular form. The basic strategy is to use induction on n to show that this is true for ℓ = 3 and any n, that ultimately there is a single modular form that works for all n. To do this, one uses a counting argument, comparing the number of ways in which one can lift a $(\mathrm {mod} \,\ell ^{n})$ Galois representation to $(\mathrm {mod} \,\ell ^{n+1})$ and the number of ways in which one can lift a $(\mathrm {mod} \,\ell ^{n})$ modular form. An essential point is to impose a sufficient set of conditions on the Galois representation; otherwise, there will be too many lifts and most will not be modular. These conditions should be satisfied for the representations coming from modular forms and those coming from elliptic curves. 3–5 trick If the original $(\mathrm {mod} \,3)$ representation has an image which is too small, one runs into trouble with the lifting argument, and in this case, there is a final trick which has since been studied in greater generality in the subsequent work on the Serre modularity conjecture. The idea involves the interplay between the $(\mathrm {mod} \,3)$ and $(\mathrm {mod} \,5)$ representations. In particular, if the mod-5 Galois representation ${\overline {\rho }}_{E,5}$ associated to an semistable elliptic curve E over Q is irreducible, then there is another semistable elliptic curve E' over Q such that its associated mod-5 Galois representation ${\overline {\rho }}_{E',5}$ is isomorphic to ${\overline {\rho }}_{E,5}$ and such that its associated mod-3 Galois representation ${\overline {\rho }}_{E,3}$ is irreducible (and therefore modular by Langlands–Tunnell).[27] Structure of Wiles's proof In his 108-page article published in 1995, Wiles divides the subject matter up into the following chapters (preceded here by page numbers): Introduction 443 Chapter 1 455 1. Deformations of Galois representations 472 2. Some computations of cohomology groups 475 3. Some results on subgroups of GL2(k) Chapter 2 479 1. The Gorenstein property 489 2. Congruences between Hecke rings 503 3. The main conjectures Chapter 3 517 Estimates for the Selmer group Chapter 4 525 1. The ordinary CM case 533 2. Calculation of η Chapter 5 541 Application to elliptic curves Appendix 545 Gorenstein rings and local complete intersections Gerd Faltings subsequently provided some simplifications to the 1995 proof, primarily in switching from geometric constructions to rather simpler algebraic ones.[19][28] The book of the Cornell conference also contained simplifications to the original proof.[9] Overviews available in the literature Wiles's paper is over 100 pages long and often uses the specialised symbols and notations of group theory, algebraic geometry, commutative algebra, and Galois theory. The mathematicians who helped to lay the groundwork for Wiles often created new specialised concepts and technical jargon. Among the introductory presentations are an email which Ribet sent in 1993;[29][30] Hesselink's quick review of top-level issues, which gives just the elementary algebra and avoids abstract algebra;[24] or Daney's web page, which provides a set of his own notes and lists the current books available on the subject. Weston attempts to provide a handy map of some of the relationships between the subjects.[31] F. Q. Gouvêa's 1994 article "A Marvelous Proof", which reviews some of the required topics, won a Lester R. Ford award from the Mathematical Association of America.[32][33] Faltings' 5-page technical bulletin on the matter is a quick and technical review of the proof for the non-specialist.[34] For those in search of a commercially available book to guide them, he recommended that those familiar with abstract algebra read Hellegouarch, then read the Cornell book,[9] which is claimed to be accessible to "a graduate student in number theory". The Cornell book does not cover the entirety of the Wiles proof.[12] See also • Abstract algebra • p-adic number • Semistable curves References 1. Fermat's Last Theorem, Simon Singh, 1997, ISBN 1-85702-521-0 2. Kolata, Gina (24 June 1993). "At Last, Shout of 'Eureka!' In Age-Old Math Mystery". The New York Times. Retrieved 21 January 2013. 3. "The Abel Prize 2016". Norwegian Academy of Science and Letters. 2016. Retrieved 29 June 2017. 4. Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. 5. Taylor R, Wiles A (1995). "Ring theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. JSTOR 2118560. OCLC 37032255. Archived from the original on 27 November 2001. 6. "NOVA – Transcripts – The Proof – PBS". PBS. September 2006. Retrieved 29 June 2017. 7. Hellegouarch, Yves (2001). Invitation to the Mathematics of Fermat–Wiles. Academic Press. ISBN 978-0-12-339251-0. 8. Singh, pp. 194–198; Aczel, pp. 109–114. 9. G. Cornell, J. H. Silverman and G. Stevens, Modular forms and Fermat's Last Theorem, ISBN 0-387-94609-8 10. Daney, Charles (13 March 1996). "The Proof of Fermat's Last Theorem". Archived from the original on 10 December 2008. Retrieved 29 June 2017. 11. "Andrew Wiles on Solving Fermat". PBS. 1 November 2000. Retrieved 29 June 2017. 12. Buzzard, Kevin (22 February 1999). "Review of Modular forms and Fermat's Last Theorem, by G. Cornell, J. H. Silverman, and G. Stevens" (PDF). Bulletin of the American Mathematical Society. 36 (2): 261–266. doi:10.1090/S0273-0979-99-00778-8. 13. Singh, pp. 269–277. 14. Kolata, Gina (28 June 1994). "A Year Later, Snag Persists In Math Proof". The New York Times. ISSN 0362-4331. Retrieved 29 June 2017. 15. Kolata, Gina (3 July 1994). "June 26-July 2; A Year Later Fermat's Puzzle Is Still Not Quite Q.E.D." The New York Times. ISSN 0362-4331. Retrieved 29 June 2017. 16. Singh, pp. 175–185. 17. Aczel, pp. 132–134. 18. Singh pp. 186–187 (text condensed). 19. "Fermat's last theorem". MacTutor History of Mathematics. February 1996. Retrieved 29 June 2017. 20. Cornell, Gary; Silverman, Joseph H.; Stevens, Glenn (2013). Modular Forms and Fermat's Last Theorem (illustrated ed.). Springer Science & Business Media. p. 549. ISBN 978-1-4612-1974-3. Extract of page 549 21. O'Carroll, Eoin (17 August 2011). "Why Pierre de Fermat is the patron saint of unfinished business". The Christian Science Monitor. ISSN 0882-7729. Retrieved 29 June 2017. 22. Granville, Andrew. "History of Fermat's Last Theorem". Retrieved 29 June 2017. 23. Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001). "On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises". Journal of the American Mathematical Society. 14 (4): 843–939. doi:10.1090/S0894-0347-01-00370-8. ISSN 0894-0347. 24. Hesselink, Wim H. (3 April 2008). "Computer verification of Wiles' proof of Fermat's Last Theorem". www.cs.rug.nl. Retrieved 29 June 2017. 25. Singh p.259-262 26. Singh, pp. 239–243; Aczel, pp. 122–125. 27. Chapter 5 of Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem" (PDF). Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. JSTOR 2118559. OCLC 37032255. 28. Malek, Massoud (6 January 1996). "Fermat's Last Theorem". Retrieved 29 June 2017. 29. "sci.math FAQ: Wiles attack". www.faqs.org. Retrieved 29 June 2017. 30. "Fermat's Last Theorem, a Theorem at Last" (PDF). FOCUS. August 1993. Retrieved 29 June 2017. 31. Weston, Tom. "Research Summary Topics". people.math.umass.edu. Retrieved 29 June 2017. 32. Gouvêa, Fernando (1994). "A Marvelous Proof". American Mathematical Monthly. 101 (3): 203–222. doi:10.2307/2975598. JSTOR 2975598. Retrieved 29 June 2017. 33. "The Mathematical Association of America's Lester R. Ford Award". Retrieved 29 June 2017. 34. Faltings, Gerd (July 1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the American Mathematical Society. 42 (7): 743–746. Bibliography • Aczel, Amir (1 January 1997). Fermat's Last Theorem: Unlocking the Secret of an Ancient Mathematical Problem. ISBN 978-1-56858-077-7. Zbl 0878.11003. • Coates, John (July 1996). "Wiles Receives NAS Award in Mathematics" (PDF). Notices of the AMS. 43 (7): 760–763. Zbl 1029.01513. • Cornell, Gary (1 January 1998). Modular Forms and Fermat's Last Theorem. ISBN 978-0-387-94609-2. Zbl 0878.11004. (Cornell, et al.) • Daney, Charles (2003). "The Mathematics of Fermat's Last Theorem". Archived from the original on 3 August 2004. Retrieved 5 August 2004. • Darmon, H. (9 September 2007). "Wiles' theorem and the arithmetic of elliptic curves" (PDF). • Faltings, Gerd (July 1995). "The Proof of Fermat's Last Theorem by R. Taylor and A. Wiles" (PDF). Notices of the AMS. 42 (7): 743–746. ISSN 0002-9920. Zbl 1047.11510. • Frey, Gerhard (1986). "Links between stable elliptic curves and certain diophantine equations". Ann. Univ. Sarav. Ser. Math. 1: 1–40. Zbl 0586.10010. • Hellegouarch, Yves (1 January 2001). Invitation to the Mathematics of Fermat–Wiles. ISBN 978-0-12-339251-0. Zbl 0887.11003. See review • Mozzochi, Charles (7 December 2000). The Fermat Diary. American Mathematical Society. ISBN 978-0-8218-2670-6. Zbl 0955.11002. See also Gouvêa, Fernando Q. (2001). "Review: Wiles's Proof, 1993–1995: The Fermat Diary by C. J. Mozzochi". American Scientist. 89 (3): 281–282. JSTOR 27857485. • Mozzochi, Charles (6 July 2006). The Fermat Proof. Trafford Publishing. ISBN 978-1-4120-2203-3. Zbl 1104.11001. • O'Connor, J. J.; Robertson, E. F. (1996). "Fermat's last theorem". Retrieved 5 August 2004. • van der Poorten, Alfred (1 January 1996). Notes on Fermat's Last Theorem. ISBN 978-0-471-06261-5. Zbl 0882.11001. • Ribenboim, Paulo (1 January 2000). Fermat's Last Theorem for Amateurs. ISBN 978-0-387-98508-4. Zbl 0920.11016. • Singh, Simon (October 1998). Fermat's Enigma. New York: Anchor Books. ISBN 978-0-385-49362-8. Zbl 0930.00002. • Simon Singh "The Whole Story". Archived from the original on 10 May 2011. Edited version of ~2,000-word essay published in Prometheus magazine, describing Andrew Wiles's successful journey. • Richard Taylor and Andrew Wiles (May 1995). "Ring-theoretic properties of certain Hecke algebras". Annals of Mathematics. 141 (3): 553–572. CiteSeerX 10.1.1.128.531. doi:10.2307/2118560. ISSN 0003-486X. JSTOR 2118560. OCLC 37032255. Zbl 0823.11030. • Wiles, Andrew (1995). "Modular elliptic curves and Fermat's Last Theorem". Annals of Mathematics. 141 (3): 443–551. CiteSeerX 10.1.1.169.9076. doi:10.2307/2118559. ISSN 0003-486X. JSTOR 2118559. OCLC 37032255. Zbl 0823.11029. External links • Weisstein, Eric W. "Fermat's Last Theorem". MathWorld. • "The Proof". PBS. The title of one edition of the PBS television series NOVA discusses Andrew Wiles's effort to prove Fermat's Last Theorem that broadcast on BBC Horizon and UTV/Documentary as Fermat's Last Theorem (Adobe Flash) (subscription required) • Wiles, Ribet, Shimura–Taniyama–Weil and Fermat's Last Theorem • Are mathematicians finally satisfied with Andrew Wiles's proof of Fermat's Last Theorem? Why has this theorem been so difficult to prove?, Scientific American, 21 October 1999 Explanations of the proof (varying levels) • Overview of Wiles proof, accessible to non-experts, by Henri Darmon • Very short summary of the proof by Charles Daney • 140 page students work-through of the proof, with exercises, by Nigel Boston Pierre de Fermat Work • Fermat's Last Theorem • Fermat number • Fermat's principle • Fermat's little theorem • Fermat polygonal number theorem • Fermat pseudoprime • Fermat point • Fermat's theorem (stationary points) • Fermat's theorem on sums of two squares • Fermat's spiral • Fermat's right triangle theorem Related • List of things named after Pierre de Fermat • Wiles's proof of Fermat's Last Theorem • Fermat's Last Theorem in fiction • Fermat Prize • Fermat's Last Tango (2000 musical) • Fermat's Last Theorem (popular science book)
Wilf–Zeilberger pair In mathematics, specifically combinatorics, a Wilf–Zeilberger pair, or WZ pair, is a pair of functions that can be used to certify certain combinatorial identities. WZ pairs are named after Herbert S. Wilf and Doron Zeilberger, and are instrumental in the evaluation of many sums involving binomial coefficients, factorials, and in general any hypergeometric series. A function's WZ counterpart may be used to find an equivalent and much simpler sum. Although finding WZ pairs by hand is impractical in most cases, Gosper's algorithm provides a sure method to find a function's WZ counterpart, and can be implemented in a symbolic manipulation program. Definition Two functions F and G form a WZ pair if and only if the following two conditions hold: $F(n+1,k)-F(n,k)=G(n,k+1)-G(n,k),$ $\lim _{M\to \pm \infty }G(n,M)=0.$ Together, these conditions ensure that $\sum _{k=-\infty }^{\infty }[F(n+1,k)-F(n,k)]=0$ because the function G telescopes: ${\begin{aligned}\sum _{k=-\infty }^{\infty }[F(n+1,k)-F(n,k)]&{}=\lim _{M\to \infty }\sum _{k=-M}^{M}[F(n+1,k)-F(n,k)]\\&{}=\lim _{M\to \infty }\sum _{k=-M}^{M}[G(n,k+1)-G(n,k)]\\&{}=\lim _{M\to \infty }[G(n,M+1)-G(n,-M)]\\&{}=0-0\\&{}=0.\end{aligned}}$ Therefore, $\sum _{k=-\infty }^{\infty }F(n+1,k)=\sum _{k=-\infty }^{\infty }F(n,k),$ that is $\sum _{k=-\infty }^{\infty }F(n,k)={\text{const}}.$ The constant does not depend on n. Its value can be found by substituting n = n0 for a particular n0. If F and G form a WZ pair, then they satisfy the relation $G(n,k)=R(n,k)F(n,k-1),$ where $R(n,k)$ is a rational function of n and k and is called the WZ proof certificate. Example A Wilf–Zeilberger pair can be used to verify the identity $\sum _{k=0}^{\infty }(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}={2n \choose n}.$ Divide the identity by its right-hand side: $\sum _{k=0}^{\infty }{\frac {(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}}{2n \choose n}}=1.$ Use the proof certificate $R(n,k)={\frac {2k-1}{2n+1}}$ to verify that the left-hand side does not depend on n, where ${\begin{aligned}F(n,k)&={\frac {(-1)^{k}{n \choose k}{2k \choose k}4^{n-k}}{2n \choose n}},\\G(n,k)&=R(n,k)F(n,k-1).\end{aligned}}$ Now F and G form a Wilf–Zeilberger pair. To prove that the constant in the right-hand side of the identity is 1, substitute n = 0, for instance. References • Marko Petkovsek; Herbert Wilf and Doron Zeilberger (1996). A=B. AK Peters. ISBN 1-56881-063-6. • Tefera, Akalu (2010), "What Is . . . a Wilf-Zeilberger Pair?" (PDF), AMS Notices, 57 (4): 508–509. See also • Almkvist–Zeilberger method, an analog of WZ method for evaluating definite integrals. • List of mathematical identities External links • Gosper's algorithm gives a method for generating WZ pairs when they exist. • Generatingfunctionology provides details on the WZ method of identity certification.
Wilf equivalence In the study of permutations and permutation patterns, Wilf equivalence is an equivalence relation on permutation classes. Two permutation classes are Wilf equivalent when they have the same numbers of permutations of each possible length, or equivalently if they have the same generating functions.[1] The equivalence classes for Wilf equivalence are called Wilf classes;[2] they are the combinatorial classes of permutation classes. The counting functions and Wilf equivalences among many specific permutation classes are known. Wilf equivalence may also be described for individual permutations rather than permutation classes. In this context, two permutations are said to be Wilf equivalent if the principal permutation classes formed by forbidding them are Wilf equivalent.[1] References 1. Bevan, David (2015), Permutation patterns: basic definitions and notation, arXiv:1506.06673, Bibcode:2015arXiv150606673B 2. Steingrímsson, Einar (2013), "Some open problems on permutation patterns", Surveys in combinatorics 2013, London Math. Soc. Lecture Note Ser., vol. 409, Cambridge Univ. Press, Cambridge, pp. 239–263, MR 3156932
Wilfred Kaplan Wilfred Kaplan (November 28, 1915 – December 26, 2007) was a professor of mathematics at the University of Michigan for 46 years, from 1940 through 1986. His research focused on dynamical systems, the topology of curve families, complex function theory, and differential equations. In total, he authored over 30 research papers and 11 textbooks. Wilfred Kaplan Wilfred Kaplan, circa 1960 Born(1915-11-28)November 28, 1915 Boston, Massachusetts, US DiedDecember 26, 2007(2007-12-26) (aged 92) Ann Arbor, Michigan, US NationalityAmerican SpouseIda Roetting (married 1938) Children • Roland Kaplan • Muriel Kaplan Parents • Jacob Kaplan • Anne Kaplan Academic background Alma materHarvard University ThesisRegular Curve-Families Filling the Plane (1939) Doctoral advisorHassler Whitney Academic work DisciplineMathematics Doctoral students • William M. Boothby • Helen F. Cullen • George R. Sell For over thirty years Kaplan was an active member of the American Association of University Professors (AAUP) and served as president of the University of Michigan chapter from 1978 to 1985. Early life Education Wilfred Kaplan was born in Boston, Massachusetts to Jacob and Anne Kaplan.[1] He attended Boston Latin School[1] and furthered his education at Harvard University, where he was granted his A.B. in mathematics in 1936 and graduated summa cum laude.[2] Later that same year he received his master's degree at Harvard.[3] Kaplan received a Rogers Fellow scholarship to study in Europe from 1936-1937, during his second year of graduate school.[3] He was based out of Zürich, Switzerland where many of the mathematicians working on the applications of topology to differential equations were located.[3] He also spent a month in Rome to work with famous mathematician Tullio Levi-Civita.[3] Upon returning to the United States, Kaplan accepted a yearlong teaching fellowship at Rice Institute for the 1938-1939 school year, thus completing his graduate program.[2] He received his Ph.D. from Harvard in 1939 under the advisement of Hassler Whitney. His dissertation covered regular curve families filling the plane.[4] Personal life While attending lectures at the Eidgenössische Technische Hochschule (ETH) Zürich he met a fellow mathematician, Ida Roetting, whom he nicknamed Heidi and would eventually marry in 1938.[3] The couple lived in Houston for a year after their wedding while Kaplan taught at the Rice Institute. The Kaplans had two children, Roland and Muriel.[1] Wilfred Kaplan died at the age of 92 after a short illness.[2] Work Teaching and research After Kaplan's short teaching position at Rice Institute, he went on to teach at the College of William and Mary in Virginia for one year.[3] In 1940 he was invited by T. H. Hildebrandt to join the faculty at the University of Michigan, after he had previously attended the Topology Congress.[5] The mathematics department at this time was diminishing due to the effects of World War II. Enrollment was down and some of the faculty had been granted leaves to do military research.[5] When asked to record his contribution to the war effort, Kaplan mentioned teaching math exclusively to Air Force pre-meteorology students in the spring and summer of 1943, as well as teaching Navy V-12 and Army ASTP students for the majority of the academic year 1943-44. In June 1944, Kaplan worked at Brown University as a researcher in an Applied Mathematics Group for the Taylor Model Basin, the Watertown Arsenal and the Bureau of Ordnance of the Navy Department. He continued his research at Brown for 17 months. In May 1947 he outlined a curriculum for a new Lectures on Mathematics Project sponsored by the Office of Naval Research.[3] His early research focused primarily on dynamical systems, and the topology of curve families. In 1955, he became especially interested in complex function theory and made a significant contribution to mathematics in his study of a special class of Schlicht functions, for which he showed that the Bieberbach Conjecture held.[2][6] His later research took on a more applied approach as he returned once more to differential equations, this time engaging in a more global analysis. In total, Wilfred Kaplan authored about 30 research papers.[2][3] Kaplan was named assistant professor in 1944, associate professor in 1950 and full professor in 1956.[3] His lectures were characterized by clarity and directness, a skill which allowed him to write several popular mathematics textbooks.[2] An updated version of his Advanced Calculus textbook is still used widely today. A selection of Kaplan's books can be found in the bibliography. Kaplan taught a variety of undergraduate and graduate courses during his time at Michigan, and further advised eight doctoral students.[4] Kaplan's skillful teaching won him respect among students and coworkers. Donald Lewis, chair of the mathematics department and co-author said,[2] "First and foremost, Wilfred Kaplan was a teacher. He enjoyed conveying the beauty and usefulness of mathematics, and his students responded enthusiastically. He was a superb expositor, and his ability to elegantly convey mathematical ideas explains the enormous impact of his textbooks. When we were writing our joint texts, he never came to a meeting without a new idea to be incorporated." One of Kaplan's primary goals as an educator was bridging the gap between pure and applied mathematics. He sympathized with the plight of engineers who felt the pressure to master more and more math concepts and then master the additional skill of applying it to their field. Kaplan wrote math textbooks specifically for engineers, such as Advanced Mathematics for Engineers (1981), because he believed teachers needed to work on presenting mathematical knowledge more efficiently to this group.[3] Furthermore, he argued that science students in general, but specifically engineers, needed to be given other resources such as textbooks and articles to further their study outside of lectures along with the tools to employ those resources appropriately. He urged other textbook authors to use clear and simple language whenever possible, in order to “make the more advanced material accessible to those with limited background.[3]” He also taught a class called "Mathematical Ideas in Science and the Humanities," which focused on the use of math as an instrument to organize thinking about complex problems.[3] More than just learning specific math content, Kaplan believed math was a medium through which to teach conciseness and how to recognize analogies, determine logical consequences of assumptions, and learn what questions need to be asked to tackle a given problem in any field.[3] AAUP Wilfred Kaplan became a member of the national American Association of University Professors (AAUP) in 1946.[6] He served as the vice president of the Michigan Conference AAUP from 1966 to 1968 and as president from 1969 to 1970.[7] From 1973 to 1978, Kaplan served on the executive committee of the University of Michigan Chapter of the AAUP and took over as president in the years from 1978 through 1985. He continued being an active member even in his retirement, serving as executive secretary from 1985 to 2002.[3] Kaplan received grievances from faculty members and supported collective bargaining (although the University faculty was never unionized). One of his primary concerns was retired professors on fixed incomes who were suffering under rising inflation. He sought to obtain a grant to provide the required financial aid. He also argued that retirees should receive more information about the health care options available to them, and he secured increases in maximum coverage and the annual reinstatement amounts allowed to retirees under the university health plan.[3] In the 1980s, Kaplan wrote an extensive proposal for a study of higher education in the United States. He argued that there should be more research of the inevitable challenges that would arise and that the University of Michigan would be a great case study whose results would be relevant for many public universities. In the proposal he outlined a variety of topics to be explored in the study: the historical background of higher education including tuition rates, enrollment rates, and changes in social customs; a study of changing demographics; the economic need for college-trained people for the betterment of society; and a thorough account of the present resources available to higher education and how these could be modified for greater efficiency. In the 1990s, Kaplan's correspondence and reports focused heavily on the "grave difficulties" between faculty government leaders (members of the Senate Assembly) and the top administration offices, specifically President Duderstadt and Provost Whitaker. Many faculty grievances were concerned with the many decisions being made without faculty input and included complaints that President Duderstadt had only “modest interest in the views of others within the faculty.” [3] Additional clubs/Memberships Kaplan was involved in a wide variety of other clubs as well. His interest in art lead him to become the president of the Washtenaw County Council for Arts.[3] Kaplan also made a significant financial contribution to the ONCE Group, a group of artists, musicians, and film-makers known for their annual ONCE music festival in Ann Arbor; the group also spearheaded a film festival and a theatre ensemble in the 1960s. He was also a member of the American Mathematical Society (AMS), the American Physical Society (APS), and the Mathematical Association of America (MAA). He was on the State's Higher Education Capital Investment Advisory Committee although little is known about his specific role there.[3] He was also Vice Chair of the University's Senate Advisory Committee on Academic Affairs, a subcommittee of the Senate Assembly. Additionally, Kaplan served as the treasurer of the Ann Arbor Unitarian Fellowship from 1972 to 2002.[3] In his later years he was president of the University of Michigan Retirees Association.[3] Awards Kaplan was named collegiate professor by the Board of Regents of the university from 1973-1975 for his accomplishments as a teacher.[3] In 1984, he received the Good Teaching Award, from what was then called the AMOCO foundation, now BP Amoco.[8] The award recognized excellence in undergraduate instruction and sought to incentive great teaching. While serving on the Senate Advisory Committee on University Affairs, Kaplan received their Distinguished Faculty Governance Award in 1986.[9] Retirement Wilfred Kaplan retired in May 1986 after 46 years of service to the university. It was then that he received the emeritus distinction. Despite his retirement, Kaplan was still involved with the university. In the years to come he would receive several requests from the University Regents Commission to return to active duty to teach specific classes in the mathematics department.[3] In 1990 he helped establish the Academic Freedom Lecture Fund (AFLF), allowing professors that were suspended or fired during the McCarthy era to hold lectures on campus.[2][6] The film Keeping in Mind, an account of the mistreatment of the three professors who were suspended for their unpopular views during the McCarthy era, was played in the spring of 1989. After an audience member suggested the University make amends for its mistreatment of the three professors, the AAUP pursued this goal. First, university officials were contacted and a proposal was sent to the Senate Advisory Committee on University Affairs in October 1989. The Senate Assembly established the Academic Lecture Freedom Fund which was funded, in part, by the national AAUP.[10] Kaplan was on the AFLF's board of directors until his death.[6] After his wife's death in 2005, Kaplan wrote a book titled Bill and Heidi: Beginning of our Lives Together, which was a translated composition of all their early correspondence before their wedding.[3] Kaplan died on December 26, 2007, after a short illness.[2] After his death, Walter Dublin, Professor Emeritus of Mechanical Engineering, wrote about Kaplan in a letter to the editor of the Ann Arbor News that Kaplan "worked tirelessly to improve the faculty--and, de facto, the university--by his work on many committees, work that spanned multiple decades. While he was a leader, he was never domineering, but always logical. Often, he would quietly remind his associates when they had strayed from their stated purpose or point out a legal or historical obstacle to what was being considered. He was always up to date and on the mark until he died, many years after he had retired."[9] Selected bibliography • Kaplan, Wilfred. Advanced Calculus. Reading, Mass.: Addison-Wesley, 1952. Print. • Kaplan, Wilfred. Ordinary Differential Equations. Reading, Mass.: Addison-Wesley Pub., 1958. Print. • Kaplan, Wilfred, and Donald J. Lewis. Calculus and Linear Algebra. New York: Wiley, 1970. Print. • Kaplan, Wilfred. Advanced Mathematics for Engineers. Reading, Mass.: Addison-Wesley Pub., 1981. Print. • Kaplan, Wilfred. Operational Methods for Linear Systems. Reading, Mass.: Addison-Wesley Pub., 1962. Print. • Kaplan, Wilfred. "Regular curve-families filling the plane, I." Duke Mathematical Journal 7.1 (1940): 154–185. Web. • Kaplan, Wilfred. "Regular Curve-families Filling the Plane, II." Duke Mathematical Journal 8.1 (1941): 11-46. Web. • Kaplan, Wilfred. "Topology of Level Curves of Harmonic Functions." Transactions of the American Mathematical Society 63.3 (1948): 514. Web. • Kaplan, Wilfred. "Dynamics of Linear Systems (Vaclav Dolez Al)." SIAM Review 8.2 (1966): 246-2. ProQuest. Web. 21 Mar. 2015. • Kaplan, Wilfred. "Applications of Undergraduate Mathematics in Engineering (Ben Noble)." SIAM Review 10.3 (1968): 383-2. ProQuest. Web. 21 Mar. 2015. • Kaplan, Wilfred. "Topics in Mathematical System Theory (Rudolf E. Kalman, Peter L. Falb and Michael A. Arbib)." SIAM Review 12.1 (1970): 157-2. ProQuest. Web. 21 Mar. 2015. • Kaplan, Wilfred. "Topics in Ordinary Differential Equations: A Potpourri (William D. Lakin and David A. Sanchez)." SIAM Review 14.3 (1972): 508-2. ProQuest. Web. 21 Mar. 2015. • Kaplan, Wilfred. "Ordinary Differential Equations in the Complex Domain (Einar Hills)." SIAM Review 19.4 (1977): 749-1. ProQuest. Web. 21 Mar. 2015. • Kaplan, Wilfred. "Green's Functions and Boundary Value Problems (Ivar Stackgold)." SIAM Review 23.1 (1981): 117-2. ProQuest. Web. 21 Mar. 2015. References 1. Kalte, Pamela M.; Nemeh, Katherine H.; editors, Noah Schusterbauer, project (2005). American men & women of science a biographical directory of today's leaders in physical, biological, and related sciences (22nd ed.). Detroit: Thomson Gale. p. 211. ISBN 9781414404622. Retrieved 21 March 2015. {{cite book}}: \|last3= has generic name (help)CS1 maint: multiple names: authors list (link) 2. Duren, Peter. "Obituaries". The University Record. University of Michigan. Retrieved 21 March 2015. 3. "Finding Aid". Bentley Historical Library. Retrieved 30 January 2015. 4. Wilfred Kaplan at the Mathematics Genealogy Project 5. Kaplan, Wilfred. "Mathematics at the University of Michigan" (PDF). University of Michigan. Retrieved 21 March 2015. 6. "Faculty History Project: Wilfred Kaplan". um2017. University of Michigan. Retrieved 24 March 2015. 7. "Past Officers/Boards". Michigan Conference American Association of University Professors. Retrieved 24 March 2015. 8. "GOOD TEACHING AWARD (formerly AMOCO)" (PDF). University of Oklahoma Health Sciences Center. 9. Maloney, Wendi (May 2008). "Wilfred Kaplan, 1915-2007". Academe. 94 (3): 9. ProQuest 232312313. 10. "Academic Freedom Lecture Fund". University of Michigan. Retrieved 21 March 2015. Authority control International • FAST • ISNI • VIAF National • Catalonia • Germany • Israel • United States • Czech Republic • Korea • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Wilfrid Dixon Wilfrid Joseph Dixon (December 13, 1915 – September 20, 2008) was an American mathematician and statistician. He made notable contributions to nonparametric statistics, statistical education and experimental design. Wilfrid Dixon Born(1915-12-13)December 13, 1915 Portland, Oregon DiedSeptember 20, 2008(2008-09-20) (aged 92) NationalityAmerican Alma materPrinceton University University of Wisconsin–Madison Known forBMDP Scientific career FieldsMathematical statistics InstitutionsUniversity of California, Los Angeles University of Oregon Doctoral advisorSamuel S. Wilks Doctoral studentsPaula Diehr A native of Portland, Oregon, Dixon received a bachelor's degree in mathematics from Oregon State College in 1938. He continued his graduate studies at the University of Wisconsin–Madison, where he earned a master's degree in 1939. Under supervision of Samuel S. Wilks, he then earned a Ph.D. in mathematical statistics from Princeton in 1944.[1] During World War II, he was an operations analyst on Guam. Dixon was on the faculties at Oklahoma (1942–1943), Oregon (1946–1955), and UCLA (1955–1986, then emeritus). While at Oregon, Dixon (together with A.M. Mood) described and provided theory and estimation methods for the adaptive Up-and-Down experimental design, which was new and poorly documented at the time.[2] This article became the cornerstone publication for up-and-down, a family of designs used in many scientific, engineering and medical fields, and to which Dixon continued to contribute in later years. In 1951 Dixon, together with Frank Massey, published a statistics textbook - the first such textbook intended to a non-mathematical audience.[3] In 1955 he was elected as a Fellow of the American Statistical Association.[4] In the 1960s at UCLA, Dixon developed BMDP, a statistical software package for biomedical analyses.[5] His daughter, Janet D. Elashoff, is also a statistician who became a UCLA faculty member, and an ASA fellow in 1978.[6] In December 2008 she funded the W. J. Dixon Award for Excellence in Statistical Consulting of the American Statistical Association in his honor.[7] References 1. "Wilfrid J. Dixon *44". Princeton Alumni Weekly. September 23, 2009. 2. Dixon, WJ; Mood, AM (1948). "A method for obtaining and analyzing sensitivity data". Journal of the American Statistical Association. 43 (241): 109–126. doi:10.1080/01621459.1948.10483254. 3. Dixon, WJ; Massey, FJ (1951). Introduction to statistical analysis. McGraw-Hill. 4. View/Search Fellows of the ASA, accessed 2016-07-23. 5. "Oral History of Wilfrid J. (Wil) Dixon and Linda Glassner: Interviewed by Luanne Johnson" (PDF). March 27, 1986. 6. ASA Fellows, Caucus for Women in Statistics, 29 March 2016, retrieved 2017-10-24 7. W. J. Dixon Award for Excellence in Statistical Consulting, American Statistical Association, retrieved 2017-10-24 External links • Wilfrid Dixon at the Mathematics Genealogy Project Authority control International • FAST • ISNI • VIAF National • Norway • France • BnF data • Israel • United States • Czech Republic • Australia • Croatia • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Wilfrid Hodges Wilfrid Augustine Hodges, FBA (born 27 May 1941) is a British mathematician and logician known for his work in model theory. Wilfrid Hodges Wilfrid Hodges, 1988 at the MFO Born (1941-05-27) 27 May 1941 Alma materNew College, Oxford Parent(s)H. A. Hodges, Vera Joan Willis Scientific career FieldsModel theory Doctoral advisorJohn Crossley Doctoral studentsAlex Wilkie President of the DLMPST/IUHPST In office 2008–2011 Preceded byAdolf Grünbaum Succeeded byElliott Sober Life Hodges attended New College, Oxford (1959–65), where he received degrees in both Literae Humaniores and (Christianic) Theology. In 1970 he was awarded a doctorate for a thesis in Logic. He lectured in both Philosophy and Mathematics at Bedford College, University of London. He has held visiting appointments in the department of philosophy at the University of California and in the department of mathematics at University of Colorado. Hodges was Professor of Mathematics at Queen Mary College, University of London from 1987 to 2006 and is the author of books on logic. Honors and awards Hodges was President of the British Logic Colloquium, of the European Association for Logic, Language and Information and of the Division of Logic, Methodology, and Philosophy of Science. In 2009 he was elected a Fellow of the British Academy. Writing style Hodges' books are written in an informal style. The "Notes on Notation" in his book "Model theory" end with the following characteristic sentence: 'I' means I, 'we' means we. When this 780-page book appeared in 1993, it became one of the standard textbooks on model theory. Due to its success an abbreviated version (but with a new chapter on stability theory) was published as a paperback. Bibliography Only first editions are listed. • Hodges, Wilfrid (1977). Logic – An Introduction to Elementary Logic. Penguin Books.[1] • Hodges, Wilfrid (1985). Building Models by Games. London Mathematical Society Student Texts. Cambridge University Press. ISBN 9780521268974. • Hodges, Wilfrid (1993). Model Theory. Encyclopedia of Mathematics. Cambridge University Press. ISBN 0-521-30442-3.[2] • Hodges, Wilfrid (1997). A Shorter Model Theory. Cambridge University Press. ISBN 0-521-58713-1. • Chiswell, Ian; Hodges, Wilfrid (2007). Mathematical Logic. Oxford University Press. ISBN 978-0-19-921562-1. References 1. Leeds, Stephen (1980). "Review of Logic by Wilfrid Hodges". Journal of Symbolic Logic. 45 (2): 382–383. doi:10.2307/2273212. JSTOR 2273212. S2CID 117796294. 2. Baldwin, John T. (1995). "Review: Model Theory by Wilfrid Hodges" (PDF). Bull. Amer. Math. Soc. (N.S.). 32 (2): 280–285. doi:10.1090/s0273-0979-1995-00578-1. External links • Home page of Wilfrid Hodges • Wilfrid Hodges at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Latvia • Czech Republic • Netherlands • Poland Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Wilhelm Ahrens Wilhelm Ahrens (3 March 1872 – 23 May 1927) was a German mathematician and writer on recreational mathematics.[1] Biography Ahrens was born in Lübz at the Elde in Mecklenburg and studied from 1890 to 1897 at the University of Rostock, Humboldt University of Berlin, and the University of Freiburg. In 1895 at the University of Rostock he received his Promotion (Ph.D.), summa cum laude, under the supervision of Otto Staude[2] with dissertation entitled Über eine Gattung n-fach periodischer Functionen von n reellen Veränderlichen.[3] From 1895 to 1896 he taught at the German school in Antwerp and then studied another semester under Sophus Lie in Leipzig. In 1897 Ahrens was a teacher in Magdeburg at the Baugewerkeschule, from 1901 at the engineering school. Inspired by Sophus Lie, he wrote "On transformation groups, all of whose subgroups are invariant" (Hamburger Math Society Vol 4, 1902). He worked a lot on the history of mathematics and mathematical games (recreational mathematics), about which he wrote a great work and also contributed to the Encyclopedia of mathematical sciences His predecessors were the great Jacques Ozanam in France, where the number theorist Édouard Lucas (1842–1891) in the 19th century wrote similar books, and Walter William Rouse Ball (1850–1925) in England (Mathematical recreations and essays 1892), Sam Loyd (1841–1901) in the U.S. and Henry Dudeney (1857–1930) in England. In this sense Martin Gardner (1914-2010) and Ian Stewart, the editor of the math column in Scientific American, might be regarded as his successors. He also wrote a book of quotations and anecdotes about mathematicians. He was the author of numerous journal articles. Scherz und Ernst in der Mathematik According to R. C. Archibald: Ahrens's Scherz und Ernst in der Mathematik ... is strictly a book of quotations; secondly, each quotation is invariably given in the original language, spoken or written; thirdly, exact bibliographical data are provided for all quotations; fourthly, the quotations follow one another consecutively from pages 1 to 495 without grouping under subject headings. A 24-page detailed index of subjects and authors provides the means for rapid orientation. Names of living mathematicians are rarely met with, but references to the "old masters" such as Abel, Euclid, Euler, Gauss, Helmholtz, Lagrange, Laplace, Steiner, and Weierstrass, are very numerous. The whole constitutes a most admirable piece of work and must long serve as a desirable model for works of like nature.[4] Bibliography • Mathematische Unterhaltungen und Spiele [Mathematical Recreations and Games], 1901 • Mathematische Spiele [Mathematical Games], 1902 • Scherz und Ernst in der Mathematik; geflügelte und ungeflügelte Worte [Fun and seriousness in mathematics: well-known and less well-known words], 1904. Scherz und Ernst in der Mathematik is available for free viewing and download at the Internet Archive; 2002 Auflage • Gelehrten-Anekdoten [Scholarly anecdotes], 1911 • Mathematiker-Anekdoten [Anecdotes of Mathematicians], 1916;[5] Zweite, stark veränderte Auflage (2nd revised edition) 1920[6] References 1. britannica.com 2. See entries of Wilhelm Ahrens in Rostock Matrikelportal 3. Wilhelm Ahrens at the Mathematics Genealogy Project 4. Archibald, R. C. (1916). "Review of Memorabilia Mathematica or the Philomath's Quotation-Book by Robert Edouard Moritz". Bulletin of the American Mathematical Society. 22 (4): 188–192. doi:10.1090/S0002-9904-1916-02751-0. MR 1559751. 5. Smith, David Eugene (1916). "Book Review: Mathematiker-Anekdoten". Bulletin of the American Mathematical Society. 23: 44–47. doi:10.1090/S0002-9904-1916-02873-4. MR 1559860. 6. Carmichael, R. D. (1921). "Book Review: Mathematiker Anekdoten. Zweite, stark veränderte Auflage". Bulletin of the American Mathematical Society. 27 (5): 230–232. doi:10.1090/S0002-9904-1921-03410-0. MR 1560405. External links • O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ernst Martin Georg Ahrens", MacTutor History of Mathematics Archive, University of St Andrews • Wilhelm Ernst Martin Georg Ahrens at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Italy • Israel • United States • Latvia • Czech Republic • Netherlands • Poland Academics • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Karl Wilhelm Feuerbach Karl Wilhelm Feuerbach (30 May 1800 – 12 March 1834) was a German geometer and the son of legal scholar Paul Johann Anselm Ritter von Feuerbach, and the brother of philosopher Ludwig Feuerbach. After receiving his doctorate at age 22, he became a professor of mathematics at the Gymnasium at Erlangen. In 1822 he wrote a small book on mathematics noted mainly for a theorem on the nine-point circle, which is now known as Feuerbach's theorem. In 1827 he introduced homogeneous coordinates, independently of Möbius.[1] Karl Wilhelm Feuerbach Born30 May 1800 (1800-05-30) Jena, Saxe-Weimar, Holy Roman Empire Died12 March 1834 (1834-03-13) (aged 33) Erlangen, Germany NationalityGerman Alma materAlbert Ludwigs University of Freiburg Known forFeuerbach's theorem Scientific career FieldsMathematician InstitutionsUniversity of Basel Notes Brother of Ludwig Andreas Feuerbach Works • Eigenschaften einiger merkwürdigen Punkte des geradlinigen Dreiecks und mehrerer durch sie bestimmten Linien und Figuren. Eine analytisch-trigonometrische Abhandlung (Monograph ed.), Nürnberg: Wiessner, 1822. online book at Google Books ("Properties of some special points in the plane of a triangle, and various lines and figures determined by these points: an analytic-trigonometric treatment") • Grundriss zu analytischen Untersuchungen der dreyeckigen Pyramide ("Foundations of the analytic theory of the triangular pyramid") References 1. "Feuerbach". sfabel.tripod.com. Retrieved 2020-07-21. External links • Works by or about Karl Wilhelm Feuerbach at Internet Archive • O'Connor, John J.; Robertson, Edmund F., "Karl Wilhelm Feuerbach", MacTutor History of Mathematics Archive, University of St Andrews • Feuerbach's Theorem: a Proof • Karl Wilhelm Feuerbach: Geometer • Media related to Karl Wilhelm Feuerbach (geometer) at Wikimedia Commons Authority control International • ISNI • VIAF National • Germany • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Wilhelm Fuhrmann Wilhelm Ferdinand Fuhrmann (28 February 1833 - 11 June 1904) was a German mathematician. The Fuhrmann circle and the Fuhrmann triangle are named after him.[1] Biography Fuhrmann was born on 28 February 1833 in Burg bei Magdeburg. Fuhrmann had shortly worked as sailor before he returned to school and attended the Altstadt Gymnasium in Königsberg, where his teachers noticed his interest and talent in mathematics and geography. He graduated in 1853 and went on to study mathematics and physics at the University of Königsberg. One of his peers later remembered him as the most talented and diligent student of his class. Fuhrmann however despite his talent did not pursue a career at the university, instead he became a math and science teacher at the Burgschule in Königsberg after his graduation. He joined the school in 1860 and remained there until his death in 1904.[2] Fuhrmann authored several books and a number of papers on different mathematical subjects. Today he is best remembered for his interest in and contribution to elementary geometry. With Synthetische Beweise planimetrischer Sätze he wrote an influential book on the subject and in 1890 he published an article entitled Sur un nouveau cercle associé à un triangle in the Belgian math journal Mathesis. In this article Fuhrmann described the circle and the triangle that now carry his name.[2][3] Publications Papers • Transformationen der Theta-Funktionen (1864) • Einige Untersuchungen über die Abhängigkeit geometrischer Gebilde (1869) • Einige Anmerkungen der projektiven Eigenschaften der Figuren (1875) • Aufgaben über Kegelschnitte (1879) • Aufgaben aus der niederen Analysis (1886) • Der Brocardsche Winkel (1889) • "Sur un nouveau cercle associé à un triangle". In: Mathesis, 1890 (English translation) • Sätze und Aufgaben aus der sphärischen Trigonometrie (1894) • Beiträge zur Transformation algebraisch-trigonometrischer Figuren Teil 1 (1898) • Beiträge zur Transformation algebraisch-trigonometrischer Figuren Teil 2 (1899) • Kollineare und orthologische Dreiecke (1902) • Aufgaben aus der analytischen Geometrie (1904, post mortem) Books • Synthetische Beweise planimetrischer Sätze. Berlin: L. Simion, 1890. Heute: Wentworth Press, 2018, ISBN 9780270116830 (online copy in the Internet Archive) • Kollineare und orthologische Dreiecke. Königsberg: Hartung, 1902. • Wegweiser in der Arithmetik, Algebra und niedern Analysis. Leipzig: Teubner, 1886. References 1. Roger A. Johnson: Advanced Euclidean Geometry. Dover 2007, ISBN 978-0-486-46237-0, pp. 228–229, 300 (originally published 1929 with Houghton Mifflin Company (Boston) as Modern Geometry). 2. L. Saalschütz: "Zur Erinnerung an Wilhelm Fuhrmann". In: Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 14, 1905, pp. 56–60. (online copy) 3. Jan Vonk, J. Chris Fisher: "Translation of Fuhrmann’s “Sur un nouveau cercle associe´ a un triangle”. In: Forum Geometricorum, Volume 11 (2011), pp. 13–26 Authority control International • VIAF National • Germany • Netherlands Academics • zbMATH People • Deutsche Biographie
Wilhelm Grunwald Wilhelm Grunwald (15 July 1909 – 7 June 1989) was a German mathematician who introduced the Grunwald–Wang theorem, though his original statement and proof of this contained a small error that was corrected by Shianghao Wang. He later left mathematics to become a science librarian, and was director of the Göttingen university library (Roquette 2005, p.29). References • Grunwald, W. (1933), "Ein allgemeiner Existenzsatz für algebraische Zahlkörper", Journal für die reine und angewandte Mathematik, 169: 103–107 • Roquette, Peter (2005), The Brauer–Hasse–Noether theorem in historical perspective (PDF), Schriften der Mathematisch-Naturwissenschaftlichen Klasse der Heidelberger Akademie der Wissenschaften [Publications of the Mathematics and Natural Sciences Section of Heidelberg Academy of Sciences], vol. 15, Berlin, New York: Springer-Verlag, ISBN 978-3-540-23005-2 • Wilhelm Grunwald at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Germany Academics • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie
Wilhelm Lexis Wilhelm Lexis (17 July 1837, Eschweiler, Germany – 24 August 1914, Göttingen, Germany), full name Wilhelm Hector Richard Albrecht Lexis,[1] was a German statistician, economist, and social scientist. The Oxford Dictionary of Statistics cites him as a "pioneer of the analysis of demographic time series".[2] Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram. Wilhelm Lexis Born Wilhelm Hector Richard Albrecht Lexis (1837-07-17)17 July 1837 Eschweiler Died24 August 1914(1914-08-24) (aged 77) Göttingen CitizenshipGerman Scientific career FieldsSocial scientist Doctoral advisorAugust Beer[1] Doctoral studentsLadislaus Bortkiewicz[1] Life Lexis graduated in 1859 from the University of Bonn, where he studied science and mathematics. He spent some time afterwards in various occupations and, in 1861, went to Paris to study social science. It was there that Lexis became acquainted with the work of Adolphe Quetelet, whose quantitative approach to the social sciences was to guide much of Lexis' work. He spent about ten years in Paris, after which he took a teaching position in Strasbourg (France). At some point during this period, Lexis wrote his first book (Introduction to the Theory of Population Statistics) and had it published in 1875, by which time he was teaching at the Imperial University of Dorpat (today the University of Tartu) in what is today Tartu, Estonia. Starting in 1876, Lexis was the chair of the Economics Department at the University of Freiburg. The various papers written by him during his eight-year tenure at Freiburg were, in the eyes of statistics historian Stephen Stigler, "his most important statistical work". Foremost among them was the 1879 paper "On the Theory of the Stability of Statistical Series", which introduced the quantity now often called the Lexis ratio. Lexis moved on from Freiburg to the University of Breslau but stayed there only a few years (from 1884 to 1887). He then settled in Göttingen, taking a position at that city's University. In 1895, he established a course in actuarial science at the university, the first ever in Germany. In 1901, Lexis became a member of the Insurance Advisory Council for Germany's Federal Insurance Supervisory Office. He remained a member of the Council until his death in 1914. During this final period of his life, Lexis published two more books: Treatises on Population and Social Statistics (Jena: Gustav Fischer, 1903) and General Economics (Leipzig: Teubner, 1910). He was also the editor of a book on the German education system.[3][4] Work Throughout his professional career, Lexis published books and articles on a wide variety of topics, including demography, economics and mathematical statistics. However, little of that work proved to have lasting significance. Today, Lexis is largely remembered for two items that bear his name—the Lexis ratio and the Lexis diagram. His theory of mortality has also enjoyed a recent revival of interest. Lexis ratio To Lexis, a time series was "stable" if the underlying probability giving rise to the observed rates remained constant from year to year (or, more generally, from one measurement period to the next). Using modern terminology, such a time series would be called a zero-order moving-average series (also known as a white noise process). Lexis was aware that many series were not stable. For non-stable series, he imagined that the underlying probabilities varied over time, being affected by what he called "physical" forces (as opposed to the random "non-essential" forces that would cause an observed rate to be different than the underlying probability). In his 1879 paper "On the Theory of the Stability of Statistical Series",[5] Lexis set himself the task of devising a method for distinguishing between stable and non-stable time series. To this end, Lexis created a test statistic equal to the ratio between (i) the probable error of the observed rates and (ii) the probable error that would be expected if the underlying probabilities for each of the observed rates were all equal to the average rate observed across all of the observations. He called this ratio Q. Lexis then reasoned that if Q was sufficiently close to 1, then the time series was exhibiting what he called "normal dispersion" and one could assume that it was stable. If Q was substantially greater than 1, then the series was exhibiting "supernomal dispersion" and one must conclude that physical forces were having a discernible effect on the variability of the observations. Lexis used a Q value of 1.41 (i.e., the square root of 2) as the dividing line between "normal" and "supernormal" dispersion. "Stability of Statistical Series" is the only one of Lexis' works cited in his entry in the Oxford Dictionary of Statistics. It is also the only one that receives an extended discussion in Stigler's A History of Statistics. And yet, Stigler ends his discussion by labeling the work a failure. To Stigler, its chief value was the discussion that it generated from other researchers in the field. It was those other researchers, and not Lexis, who created the modern science of time-series analysis.[6] Lexis diagram Although it can take various forms, the typical Lexis diagram is a graphical illustration of the lifetime of either an individual or a cohort of same-aged individuals. On the diagram, each such lifetime appears as a straight line in a two-dimensional plane, with one dimension representing time and the other representing age. The use of Lexis diagrams is very common amongst demographers, so much so that they often are used without being identified as Lexis diagrams.[7] Lexis introduced his diagram in his first book, Introduction to the Theory of Population Statistics (Strasbourg: Trubner, 1875). However, the notion of using a time vs. age diagram appears to have been developed more or less simultaneously by other authors. See the paper by Vandeschrick (2001) for more detail. Theory of mortality In his 1877 book On the Theory of Mass Phenomena in Human Society (Freiburg: Wagnersche Buchhandlung), Lexis proposed that all human deaths could be classified into one of three types: (i) normal deaths, (ii) infant deaths and (iii) premature adult deaths. He also proposed that the normal deaths were subject to random forces such that, if all infant and other premature deaths were eliminated, the ages at which people died would exhibit a normal (i.e., Gaussian) distribution. Furthermore, the average of those ages would be equal to the age at which most adults are actually observed to die (i.e., the modal age at death), even though the actual observations are taking place in the presence of infant and other premature deaths.[8] In the adjacent diagram, the normal deaths are represented by the vertically-shaded bell-shaped area centered slightly above age 70; the infant deaths are represented by the unshaded area starting at age 0; the premature deaths are represented by the horizontally-shaded area bridging the infant and normal deaths. Although Lexis' theory did generate some contemporaneous discussion, it never supplanted the traditional demographic measures of life expectancy and age-adjusted mortality rates. However, recent research suggests that the modal age at death might be a useful statistic for tracking changes in the lifespans of the elderly. For a survey of the contemporaneous response to Lexis' theory, see section IV ("Reception of Lexis' hypothesis in the late 19th century") of Véron and Rohrbasser (2003). For a discussion of the modern-day use of the modal age at death, see Horiuchi et al. (2013). Further reading • Horiuchi, Shiro; Ouelette, Nadine; Cheung, Siu Lan Karen; Robine, Jean-Marie (2013). "Modal Age at Death: Lifespan Indicator in the Era of Longevity Extension" (PDF). Vienna Yearbook of Population Research. 11: 37–69. doi:10.1553/populationyearbook2013s37. • Vandeschrick, Christophe (2001). "The Lexis Diagram, a Misnomer" (PDF). Demographic Research. 4: 97–124. doi:10.4054/DemRes.2001.4.3. • Véron, Jacques; Rohrbasser, Jean-Marc (2003). "Wilhelm Lexis: The Normal Length of Life as an Expression of the 'Nature of Things'". Population. 53 (3): 303–322. Two biographies of Lexis are: • Heiss, Klaus-Peter (1978) "Wilhelm Lexis", in Kruskal, William H. and Tanur, Judith M. (eds.) International Encyclopedia of Statistics (New York: Free Press), Volume 1, pages 507-512 • Klein, Felix (1914) "Wilhelm Lexis" in Jahresbericht der Deutschen Mathematiker-Vereinigung, Volume 23, pages 314-317 (obituary, in German) References • Koch, Peter (1985) "Wilhem Lexis" in Neue Deutsche Biographie (Berlin: Duncker & Humblot) Volume 14, pages 421-422 (in German) • Stigler, Stephen M. (1986) The History of Statistics: The Measurement of Uncertainty before 1900 (Cambridge, Massachusetts: Belknap Press, ISBN 0-674-40340-1), chapter 6 ("Attempts to Revive the Binomial"), pages 221-238 • Upton, Graham and Cook, Ian (2006) A Dictionary of Statistics, Second edition (Oxford: Oxford University Press, ISBN 0-19-861431-4), pages 237-238 1. Lexis' page at the Mathematics Genealogy Project Note that the date of death given in the MacTutor biography does not agree with the German sources, including the 1914 obituary by Felix Klein. These other sources give the date as shown above. 2. Upton and Cook (2006), page 238 3. A General Overview of the History and Organisation of Public Education in the German Empire (Berlin: A. Asher, 1904) 4. Factual details in this section are taken from the Lexis entry in the Neue Deutsche Biographie. The relationship of Lexis' work to that of Quetelet is from page 223 of Stigler (1986), as is the direct quote concerning Lexis' Freiburg-period work. 5. Jahrbücher für National Ökonomie und Statistik, Volume 32, 1879, pages 60-98. The distinction between "physical" and "non-essential" forces is made on page 66. 6. Stigler's discussion of the Lexis ratio is at pages 229 through 234 of Stigler (1986) (i.e., the sections titled "The Dispersion of Series" and "Lexis's Analysis and Interpretation"). His finding that Lexis' work was a failure is at pages 234 through 236 ("Why Lexis Failed"). The effect on other researchers is at pages 237-238 ("Lexian Dispersion after Lexis"). 7. For example, see Dick London's discussion of U.S. Census Bureau techniques in chapter 9 of his Survival Models and their Estimation (Winsted, Connecticut: Actex, 1988 ISBN 0-936031-01-8). Also see Kenneth P. Veit's "Stationary Population Methods" in the Transactions of the Society of Actuaries, Volume XVI (1964), page 233 ff. (available here). 8. The discussion of normal vs. premature deaths starts at page 45 of Mass Phenomena. Note that Lexis uses the word jugendlichen to describe the infant deaths. Although Lexis' word might equally well be translated as "youth", his calculations later in the text indicate that no jugendlichen deaths are assumed to take place after age 15. Véron and Rohrbasser (2003) and Horiuchi et al. (2013) both translate Lexis' word as "infant". External links Wikisource has the text of a 1905 New International Encyclopedia article about "Wilhelm Lexis". Wikimedia Commons has media related to Wilhelm Lexis. Works by Lexis • Lexis' Ph.D. dissertation (in Latin) • Introduction to the Theory of Population Statistics (1875) (in German) • On the Theory of Mass Phenomena in Human Society (1877) (in German) • On the Theory of the Stability of Statistical Series (1879) (in German) • Treatises on Population and Social Statistics (1903) (in German) • 1904 text on the German education system (in English translation) Biographies of Lexis • Entry in the Neue Deutsche Biographie (in German) • Entry in the Oxford Dictionary of Statistics • Obituary by Felix Klein (in German) • O'Connor, John J.; Robertson, Edmund F., "Wilhelm Lexis", MacTutor History of Mathematics Archive, University of St Andrews Authority control International • FAST • ISNI • VIAF National • Norway • 2 • France • BnF data • Germany • Israel • Belgium • United States • Latvia • Japan • Czech Republic • Australia • Greece • Netherlands • Poland • Portugal • Vatican Academics • Mathematics Genealogy Project People • Deutsche Biographie • Trove Other • IdRef
Wilhelm Ackermann Wilhelm Friedrich Ackermann (/ˈækərmən/; German: [ˈakɐˌman]; 29 March 1896 – 24 December 1962) was a German mathematician and logician best known for his work in mathematical logic[1] and the Ackermann function, an important example in the theory of computation. Wilhelm Ackermann Wilhelm Ackermann in c. 1935 Born(1896-03-29)29 March 1896 Herscheid, German Empire Died24 December 1962(1962-12-24) (aged 66) Lüdenscheid, West Germany NationalityGerman Alma materUniversity of Göttingen Known for • Ackermann coding Ackermann function Ackermann set theory Scientific career FieldsMathematics Doctoral advisorDavid Hilbert Biography Ackermann was born in Herscheid, Germany, and was awarded a Ph.D. by the University of Göttingen in 1925 for his thesis Begründung des "tertium non datur" mittels der Hilbertschen Theorie der Widerspruchsfreiheit, which was a consistency proof of arithmetic apparently without Peano induction (although it did use e.g. induction over the length of proofs). This was one of two major works in proof theory in the 1920s and the only one following Hilbert's school of thought.[1] From 1929 until 1948, he taught at the Arnoldinum Gymnasium in Burgsteinfurt, and then at Lüdenscheid until 1961. He was also a corresponding member of the Akademie der Wissenschaften (Academy of Sciences) in Göttingen, and was an honorary professor at the University of Münster. In 1928, Ackermann helped David Hilbert turn his 1917 – 22 lectures on introductory mathematical logic into a text, Principles of Mathematical Logic. This text contained the first exposition ever of first-order logic, and posed the problem of its completeness and decidability (Entscheidungsproblem). Ackermann went on to construct consistency proofs for set theory (1937), full arithmetic (1940), type-free logic (1952), and a new axiomatization of set theory (1956). Later in life, Ackermann continued working as a high school teacher. Still, he kept continually engaged in the field of research and published many contributions to the foundations of mathematics until the end of his life. He died in Lüdenscheid, West Germany in December 1962. See also • Ackermann's bijection • Ackermann coding • Ackermann function • Ackermann ordinal • Ackermann set theory • Hilbert–Ackermann system • Entscheidungsproblem • Ordinal notation • Inverse Ackermann function Bibliography • 1928. "On Hilbert's construction of the real numbers" in Jean van Heijenoort, ed., 1967. From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press: 493–507. • 1940. "Zur Widerspruchsfreiheit der Zahlentheorie", Mathematische Annalen, vol. 117, pp 162–194. • 1950 (1928). (with David Hilbert) Principles of Mathematical Logic. Chelsea. Translation of 1938 German edition. • 1954. Solvable cases of the decision problem. North Holland. References 1. O'Connor, J J; Robertson, E F; Felscher, Walter. "Wilhelm Ackermann". MacTutor History of Mathematics. Retrieved 18 August 2021. External links • O'Connor, John J.; Robertson, Edmund F., "Wilhelm Ackermann", MacTutor History of Mathematics Archive, University of St Andrews • Wilhelm Ackermann at the Mathematics Genealogy Project • Erich Friedman's page on Ackermann at Stetson University • Hermes, In memoriam WILHELM ACKERMANN 1896-1962 (PDF, 945 KB) • Author profile in the database zbMATH Authority control International • FAST • ISNI • 2 • VIAF National • Norway • Spain • France • BnF data • Catalonia • Germany • Italy • Israel • United States • Sweden • Latvia • Japan • Czech Republic • Australia • Croatia • Netherlands • Poland Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • IdRef
Wilhelm Blaschke Wilhelm Johann Eugen Blaschke (13 September 1885 – 17 March 1962) was an Austrian mathematician working in the fields of differential and integral geometry. Wilhelm Blaschke Born(1885-09-13)13 September 1885 Graz Died17 March 1962(1962-03-17) (aged 76) Hamburg NationalityAustrian Alma materUniversity of Vienna Known forBlaschke product Blaschke selection theorem Blaschke–Santaló inequality Scientific career FieldsMathematics InstitutionsUniversity of Hamburg Doctoral advisorWilhelm Wirtinger Doctoral studentsShiing-Shen Chern Luis Santaló Emanuel Sperner Other notable studentsAlberto Dou Mas de Xaxàs Education and career Blaschke was the son of mathematician Josef Blaschke, who taught geometry at the Landes Oberrealschule in Graz. After studying for two years at the Technische Hochschule in Graz, he went to the University of Vienna, and completed a doctorate in 1908 under the supervision of Wilhelm Wirtinger.[1] His dissertation was Über eine besondere Art von Kurven vierter Klasse.[2] After completing his doctorate he spent several years visiting mathematicians at the major universities in Italy and Germany. He spent two years each in positions in Prague, Leipzig, Göttingen, and Tübingen until, in 1919, he took the professorship at the University of Hamburg that he would keep for the rest of his career.[1] His students at Hamburg included Shiing-Shen Chern, Luis Santaló, and Emanuel Sperner.[2] In 1933 Blaschke signed the Vow of allegiance of the Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State.[3][4] However, he defended Kurt Reidemeister against the Nazis and, in the early 1930s, campaigned against Ludwig Bieberbach for leadership of the German Mathematical Society, arguing that the society should remain international and apolitical in opposition to Bieberbach's wish to "enforce Nazi policies on German mathematics and race". However, by 1936 he was supporting Nazi policies, called himself "a Nazi at Heart", and was described by colleagues as "Mussolinetto" for his fascist beliefs.[1] He officially joined the Nazi Party in 1937.[3] After the war, Blaschke was removed from his position at the University of Hamburg for his Nazi affiliation, but after an appeal his professorship was restored in 1946.[1] He remained at the university until his retirement in 1953.[1] Publications In 1916 Blaschke published one of the first books devoted to convex sets: Circle and Sphere (Kreis und Kugel). Drawing on dozens of sources, Blaschke made a thorough review of the subject with citations within the text to attribute credit in a classical area of mathematics. • Kreis und Kugel, Leipzig, Veit 1916; 3rd edn. Berlin, de Gruyter 1956 • Vorlesungen über Differentialgeometrie, 3 vols., Springer, Grundlehren der mathematischen Wissenschaften 1921-1929 (vol. 1, Elementare Differentialgeometrie;[5] vol. 2, Affine Differentialgeometrie; vol. 3, Differentialgeometrie der Kreise und Kugeln, 1929) • with G. Bol: Geometrie der Gewebe. Berlin: Springer 1938[6] • Ebene Kinematik. Leipzig: B.G. Teubner 1938,[7] 2nd expanded edn. with Hans Robert Müller, Oldenbourg, München 1956 • Nicht-Euklidische Geometrie und Mechanik I, II, III. Leipzig: B.G.Teubner (1942) • Zur Bewegungsgeometrie auf der Kugel. In: Sitzungsberichte der Heidelberger Akademie der Wissenschaften (1948) • Einführung in die Differentialgeometrie. Springer 1950,[8] 2nd expanded edn. with H. Reichardt 1960 • with Kurt Leichtweiß: Elementare Differentialgeometrie. Berlin: Springer (5th edn. 1973) • Reden und Reisen eines Geometers. Berlin : VEB Deutscher Verlag der Wissenschaften (1961; 2nd expanded edn.) • Mathematik und Leben, Wiesbaden, Steiner 1951 • Griechische und anschauliche Geometrie, Oldenbourg 1953 • Projektive Geometrie, 3rd edn, Birkhäuser 1954 • Analytische Geometrie, 2nd edn., Birkhäuser 1954 • Einführung in die Geometrie der Waben, Birkhäuser 1955[9] • Vorlesungen über Integralgeometrie, VEB, Berlin 1955 • Reden und Reisen eines Geometers, 1957 • Kinematik und Quaternionen. Berlin: VEB Deutscher Verlag der Wissenschaften (1960) • Gesammelte Werke, Thales, Essen 1985 Namesake A number of mathematical theorems and concepts is associated with the name of Blaschke. • Blaschke selection theorem • Blaschke–Lebesgue theorem • Blaschke product • Blaschke sum • Blaschke condition • Blaschke–Busemann measure • Blaschke–Santaló inequality • Blaschke conjecture: "The only Wiedersehen manifolds in any dimension are the standard Euclidean spheres." See also • Affine differential geometry • Affine geometry of curves • Body of constant brightness • Web (differential geometry) • Pestov–Ionin theorem References 1. O'Connor, John J.; Robertson, Edmund F. "Wilhelm Blaschke". MacTutor History of Mathematics Archive. University of St Andrews. 2. Wilhelm Blaschke at the Mathematics Genealogy Project 3. Heinrich Behnke (1898-1979): zwischen Mathematik und deren Didaktik, Uta Hartmann, 2009 ISBN 9783631588604 4. Ernst Klee: Das Personenlexikon zum Dritten Reich. Wer war was vor und nach 1945. Fischer Taschenbuch Verlag, Zweite aktualisierte Auflage, Frankfurt am Main 2005, S. 52. 5. Bliss, G. A. (1923). "Blaschke on Differential Geometry". Bull. Amer. Math. Soc. 29 (7): 322–325. doi:10.1090/S0002-9904-1923-03737-3. 6. Walker, R. J. (1939). "Review: W. Blaschke and G. Bol, Geometrie der Gewebe". Bull. Amer. Math. Soc. 45 (9): 652–653. doi:10.1090/s0002-9904-1939-07052-3. 7. Synge, J. L. (1939). "Review: W. Blaschke, Eben Kinematik". Bull. Amer. Math. Soc. 45 (11): 814–815. doi:10.1090/s0002-9904-1939-07071-7. 8. Allendoerfer, Carl B. (1951). "Review, W. Blaschke, Einführung in die Differentialgeometrie". Bull. Amer. Math. Soc. 57 (1, Part 1): 84–85. doi:10.1090/s0002-9904-1951-09454-9. 9. Hsiung, C. C. (1957). "Review: W. Blaschke, Einführung in die Geometrie der Waben". Bull. Amer. Math. Soc. 63 (3): 203–204. doi:10.1090/s0002-9904-1957-10104-9. External links • Newspaper clippings about Wilhelm Blaschke in the 20th Century Press Archives of the ZBW Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • Belgium • United States • Latvia • Czech Republic • Netherlands • Vatican Academics • CiNii • Leopoldina • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
Wilkie's theorem In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. Formulations In terms of model theory, Wilkie's theorem deals with the language Lexp = (+, −, ·, <, 0, 1, ex), the language of ordered rings with an exponential function ex. Suppose φ(x1, ..., xm) is a formula in this language. Then Wilkie's theorem states that there is an integer n ≥ m and polynomials f1, ..., fr ∈ Z[x1, ..., xn, ex1, ..., exn] such that φ(x1, ..., xm) is equivalent to the existential formula $\exists x_{m+1}\ldots \exists x_{n}\,f_{1}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=\cdots =f_{r}(x_{1},\ldots ,x_{n},e^{x_{1}},\ldots ,e^{x_{n}})=0.$ Thus, while this theory does not have full quantifier elimination, formulae can be put in a particularly simple form. This result proves that the theory of the structure Rexp, that is the real ordered field with the exponential function, is model complete.[1] In terms of analytic geometry, the theorem states that any definable set in the above language — in particular the complement of an exponential variety — is in fact a projection of an exponential variety. An exponential variety over a field K is the set of points in Kn where a finite collection of exponential polynomials simultaneously vanish. Wilkie's theorem states that if we have any definable set in an Lexp structure K = (K, +, −, ·, 0, 1, ex), say X ⊂ Km, then there will be an exponential variety in some higher dimension Kn such that the projection of this variety down onto Km will be precisely X. Gabrielov's theorem The result can be considered as a variation of Gabrielov's theorem. This earlier theorem of Andrei Gabrielov dealt with sub-analytic sets, or the language Lan of ordered rings with a function symbol for each proper analytic function on Rm restricted to the closed unit cube [0, 1]m. Gabrielov's theorem states that any formula in this language is equivalent to an existential one, as above.[2] Hence the theory of the real ordered field with restricted analytic functions is model complete. Intermediate results Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. As an intermediate result Wilkie asked when the complement of a sub-analytic set could be defined using the same analytic functions that described the original set. It turns out the required functions are the Pfaffian functions.[1] In particular the theory of the real ordered field with restricted, totally defined Pfaffian functions is model complete.[3] Wilkie's approach for this latter result is somewhat different from his proof of Wilkie's theorem, and the result that allowed him to show that the Pfaffian structure is model complete is sometimes known as Wilkie's theorem of the complement. See also.[4] References 1. A.J. Wilkie, Model completeness results for expansions of the ordered field of real numbers by restricted pfaffian functions and the exponential functions, J. Amer. Math. Soc. 9 (1996), pp. 1051–1094. 2. A. Gabrielov, Projections of semi-analytic sets, Functional Anal. Appl. 2 (1968), pp.282–291. 3. A.J. Wilkie, A theorem of the complement and some new o-minimal structures, Sel. Math. 5 (1999), pp.397–421. 4. M. Karpinski and A. Macintyre, A generalization of Wilkie's theorem of the complement, and an application to Pfaffian closure, Sel. math., New ser. 5 (1999), pp.507-516
Wilkie investment model The Wilkie investment model, often just called Wilkie model, is a stochastic asset model developed by A. D. Wilkie that describes the behavior of various economics factors as stochastic time series. These time series are generated by autoregressive models. The main factor of the model which influences all asset prices is the consumer price index. The model is mainly in use for actuarial work and asset liability management. Because of the stochastic properties of that model it is mainly combined with Monte Carlo methods. Wilkie first proposed the model in 1986, in a paper published in the Transactions of the Faculty of Actuaries.[1] It has since been the subject of extensive study and debate.[2][3] Wilkie himself updated and expanded the model in a second paper published in 1995.[4] He advises to use that model to determine the "funnel of doubt", which can be seen as an interval of minimum and maximum development of a corresponding economic factor. Components • price inflation • wage inflation • share yield • share dividend • consols yield (long-term interest rate) • bank rate (short-term interest rate) References 1. Wilkie, A.D. (1986). "A stochastic investment model for Actuarial Use" (PDF). Transactions of the Faculty of Actuaries. 39: 341–403. doi:10.1017/S0071368600009009. 2. Geoghegan, T J; Clarkson, R S; Feldman, K S; Green, S J; Kitts, A; Lavecky, J P; Ross, F J M; Smith, W J; Toutounchi, A (27 January 1992). "Report on the Wilkie investment model". Journal of the Institute of Actuaries. 119: 173–228. doi:10.1017/S0020268100019879. 3. Şahin, Şule; Cairns, Andrew; Kleinow, Torsten; Wilkie, A. D. (12 June 2008). Revisiting the Wilkie Investment Model (PDF). International Actuarial Association, AFIR/ERM Sectional Colloquium, Rome, 2008. 4. Wilkie, A.D. (1995). "More on a stochastic asset model for actuarial use". British Actuarial Journal. 1 (5): 777–964. doi:10.1017/S1357321700001331. S2CID 153338215.
Wilkinson matrix In linear algebra, Wilkinson matrices are symmetric, tridiagonal, order-N matrices with pairs of nearly, but not exactly, equal eigenvalues.[1] It is named after the British mathematician James H. Wilkinson. For N = 7, the Wilkinson matrix is given by ${\begin{bmatrix}3&1&0&0&0&0&0\\1&2&1&0&0&0&0\\0&1&1&1&0&0&0\\0&0&1&0&1&0&0\\0&0&0&1&1&1&0\\0&0&0&0&1&2&1\\0&0&0&0&0&1&3\\\end{bmatrix}}.$ Wilkinson matrices have applications in many fields, including scientific computing, numerical linear algebra, and signal processing. References 1. Wilkinson (1965). The Algebraic Eigenvalue Problem. Oxford University Press. ISBN 0-19-853418-3. Numerical linear algebra Key concepts • Floating point • Numerical stability Problems • System of linear equations • Matrix decompositions • Matrix multiplication (algorithms) • Matrix splitting • Sparse problems Hardware • CPU cache • TLB • Cache-oblivious algorithm • SIMD • Multiprocessing Software • MATLAB • Basic Linear Algebra Subprograms (BLAS) • LAPACK • Specialized libraries • General purpose software
Wilks' theorem In statistics Wilks' theorem offers an asymptotic distribution of the log-likelihood ratio statistic, which can be used to produce confidence intervals for maximum-likelihood estimates or as a test statistic for performing the likelihood-ratio test. Statistical tests (such as hypothesis testing) generally require knowledge of the probability distribution of the test statistic. This is often a problem for likelihood ratios, where the probability distribution can be very difficult to determine. A convenient result by Samuel S. Wilks says that as the sample size approaches $\infty $, the distribution of the test statistic $-2\log(\Lambda )$ asymptotically approaches the chi-squared ($\chi ^{2}$) distribution under the null hypothesis $H_{0}$.[1] Here, $\Lambda $ denotes the likelihood ratio, and the $\chi ^{2}$ distribution has degrees of freedom equal to the difference in dimensionality of $\Theta $ and $\Theta _{0}$, where $\Theta $ is the full parameter space and $\Theta _{0}$ is the subset of the parameter space associated with $H_{0}$. This result means that for large samples and a great variety of hypotheses, a practitioner can compute the likelihood ratio $\Lambda $ for the data and compare $-2\log(\Lambda )$ to the $\chi ^{2}$ value corresponding to a desired statistical significance as an approximate statistical test. The theorem no longer applies when the true value of the parameter is on the boundary of the parameter space: Wilks’ theorem assumes that the ‘true’ but unknown values of the estimated parameters lie within the interior of the supported parameter space. In practice, one will notice the problem if the estimate lies on that boundary. In that event, the likelihood test is still a sensible test statistic and even possess some asymptotic optimality properties, but the significance (the p-value) can not be reliably estimated using the chi-squared distribution with the number of degrees of freedom prescribed by Wilks. In some cases, the asymptotic null-hypothesis distribution of the statistic is a mixture of chi-square distributions with different numbers of degrees of freedom. Use Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by D) is twice the log of the likelihoods ratio, i.e., it is twice the difference in the log-likelihoods: ${\begin{aligned}D&=-2\ln \left({\frac {\text{likelihood for null model}}{\text{likelihood for alternative model}}}\right)\\[5pt]&=2\ln \left({\frac {\text{likelihood for alternative model}}{\text{likelihood for null model}}}\right)\\[5pt]&=2\times [\ln({\text{likelihood for alternative model}})-\ln({\text{likelihood for null model}})]\\[5pt]\end{aligned}}$ The model with more parameters (here alternative) will always fit at least as well — i.e., have the same or greater log-likelihood — than the model with fewer parameters (here null). Whether the fit is significantly better and should thus be preferred is determined by deriving how likely (p-value) it is to observe such a difference D by chance alone, if the model with fewer parameters were true. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to $\,df_{\text{alt}}-df_{\text{null}}\,$,[2] respectively the number of free parameters of models alternative and null. For example: If the null model has 1 parameter and a log-likelihood of −8024 and the alternative model has 3 parameters and a log-likelihood of −8012, then the probability of this difference is that of chi-squared value of $2\times (-8012-(-8024))=24$ with $3-1=2$ degrees of freedom, and is equal to $6\times 10^{-6}$. Certain assumptions[1] must be met for the statistic to follow a chi-squared distribution, but empirical p-values may also be computed if those conditions are not met. Examples Coin tossing An example of Pearson's test is a comparison of two coins to determine whether they have the same probability of coming up heads. The observations can be put into a contingency table with rows corresponding to the coin and columns corresponding to heads or tails. The elements of the contingency table will be the number of times each coin came up heads or tails. The contents of this table are our observations X. ${\begin{array}{c\|cc}X&{\text{Heads}}&{\text{Tails}}\\\hline {\text{Coin 1}}&k_{\mathrm {1H} }&k_{\mathrm {1T} }\\{\text{Coin 2}}&k_{\mathrm {2H} }&k_{\mathrm {2T} }\end{array}}$ Here Θ consists of the possible combinations of values of the parameters $p_{\mathrm {1H} }$, $p_{\mathrm {1T} }$, $p_{\mathrm {2H} }$, and $p_{\mathrm {2T} }$, which are the probability that coins 1 and 2 come up heads or tails. In what follows, $i=1,2$ and $j=\mathrm {H,T} $. The hypothesis space H is constrained by the usual constraints on a probability distribution, $0\leq p_{ij}\leq 1$, and $p_{i\mathrm {H} }+p_{i\mathrm {T} }=1$. The space of the null hypothesis $H_{0}$ is the subspace where $p_{1j}=p_{2j}$. The dimensionality of the full parameter space Θ is 2 (either of the $p_{1j}$ and either of the $p_{2j}$ may be treated as free parameters under the hypothesis $H$), and the dimensionality of $\Theta _{0}$ is 1 (only one of the $p_{ij}$ may be considered a free parameter under the null hypothesis $H_{0}$). Writing $n_{ij}$ for the best estimates of $p_{ij}$ under the hypothesis H, the maximum likelihood estimate is given by $n_{ij}={\frac {k_{ij}}{k_{i\mathrm {H} }+k_{i\mathrm {T} }}}\,.$ Similarly, the maximum likelihood estimates of $p_{ij}$ under the null hypothesis $H_{0}$ are given by $m_{ij}={\frac {k_{1j}+k_{2j}}{k_{\mathrm {1H} }+k_{\mathrm {2H} }+k_{\mathrm {1T} }+k_{\mathrm {2T} }}}\,,$ which does not depend on the coin i. The hypothesis and null hypothesis can be rewritten slightly so that they satisfy the constraints for the logarithm of the likelihood ratio to have the desired distribution. Since the constraint causes the two-dimensional H to be reduced to the one-dimensional $H_{0}$, the asymptotic distribution for the test will be $\chi ^{2}(1)$, the $\chi ^{2}$ distribution with one degree of freedom. For the general contingency table, we can write the log-likelihood ratio statistic as $-2\log \Lambda =2\sum _{i,j}k_{ij}\log {\frac {n_{ij}}{m_{ij}}}\,.$ Invalidity for random or mixed effects models Wilks’ theorem assumes that the true but unknown values of the estimated parameters are in the interior of the parameter space. This is commonly violated in random or mixed effects models, for example, when one of the variance components is negligible relative to the others. In some such cases, one variance component can be effectively zero, relative to the others, or in other cases the models can be improperly nested. To be clear: These limitations on Wilks’ theorem do not negate any power properties of a particular likelihood ratio test.[3] The only issue is that a $\chi ^{2}$ distribution is sometimes a poor choice for estimating the statistical significance of the result. Bad examples Pinheiro and Bates (2000) showed that the true distribution of this likelihood ratio chi-square statistic could be substantially different from the naïve $\chi ^{2}$ – often dramatically so.[4] The naïve assumptions could give significance probabilities (p-values) that are, on average, far too large in some cases and far too small in others. In general, to test random effects, they recommend using Restricted maximum likelihood (REML). For fixed-effects testing, they say, “a likelihood ratio test for REML fits is not feasible”, because changing the fixed effects specification changes the meaning of the mixed effects, and the restricted model is therefore not nested within the larger model.[4] As a demonstration, they set either one or two random effects variances to zero in simulated tests. In those particular examples, the simulated p-values with k restrictions most closely matched a 50–50 mixture of $\chi ^{2}(k)$ and $\chi ^{2}(k-1)$. (With k = 1 , $\chi ^{2}(0)$ is 0 with probability 1. This means that a good approximation was $\,0.5\,\chi ^{2}(1)\,.$)[4] Pinheiro and Bates also simulated tests of different fixed effects. In one test of a factor with 4 levels (degrees of freedom = 3), they found that a 50–50 mixture of $\chi ^{2}(3)$ and $\chi ^{2}(4)$ was a good match for actual p-values obtained by simulation – and the error in using the naïve $\chi ^{2}(3)$ “may not be too alarming.”[4] However, in another test of a factor with 15 levels, they found a reasonable match to $\chi ^{2}(18)$ – 4 more degrees of freedom than the 14 that one would get from a naïve (inappropriate) application of Wilks’ theorem, and the simulated p-value was several times the naïve $\chi ^{2}(14)$. They conclude that for testing fixed effects, “it's wise to use simulation.”[lower-alpha 1] See also • Bayes factor • Model selection • Sup-LR test Notes 1. Pinheiro and Bates (2000)[4] provided a simulate.lme function in their nlme package for S-PLUS and R to support REML simulation; see ref.[5] References 1. Wilks, Samuel S. (1938). "The large-sample distribution of the likelihood ratio for testing composite hypotheses". The Annals of Mathematical Statistics. 9 (1): 60–62. doi:10.1214/aoms/1177732360. 2. Huelsenbeck, J.P.; Crandall, K.A. (1997). "Phylogeny Estimation and Hypothesis Testing Using Maximum Likelihood". Annual Review of Ecology and Systematics. 28: 437–466. doi:10.1146/annurev.ecolsys.28.1.437. 3. Neyman, Jerzy; Pearson, Egon S. (1933). "On the problem of the most efficient tests of statistical hypotheses" (PDF). Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences. 231 (694–706): 289–337. Bibcode:1933RSPTA.231..289N. doi:10.1098/rsta.1933.0009. JSTOR 91247. 4. Pinheiro, José C.; Bates, Douglas M. (2000). Mixed-Effects Models in S and S-PLUS. Springer-Verlag. pp. 82–93. ISBN 0-387-98957-9. 5. "Simulate results from lme models" (PDF). R-project.org (software documentation). Package nlme. 12 May 2019. pp. 281–282. Retrieved 8 June 2019. Other sources • Casella, George; Berger, Roger L. (2001). Statistical Inference (Second ed.). ISBN 0-534-24312-6. • Mood, A.M.; Graybill, F.A. (1963). Introduction to the Theory of Statistics (2nd ed.). McGraw-Hill. ISBN 978-0070428638. • Cox, D.R.; Hinkley, D.V. (1974). Theoretical Statistics. Chapman and Hall. ISBN 0-412-12420-3. • Stuart, A.; Ord, K.; Arnold, S. (1999). Kendall's Advanced Theory of Statistics. Vol. 2A. London: Arnold. ISBN 978-0-340-66230-4. External links • "Likelihood Ratio: Wilks's Theorem". Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model specification • Lp space • Parameter • location • scale • shape • Parametric family • Likelihood (monotone) • Location–scale family • Exponential family • Completeness • Sufficiency • Statistical functional • Bootstrap • U • V • Optimal decision • loss function • Efficiency • Statistical distance • divergence • Asymptotics • Robustness Frequentist inference Point estimation • Estimating equations • Maximum likelihood • Method of moments • M-estimator • Minimum distance • Unbiased estimators • Mean-unbiased minimum-variance • Rao–Blackwellization • Lehmann–Scheffé theorem • Median unbiased • Plug-in Interval estimation • Confidence interval • Pivot • Likelihood interval • Prediction interval • Tolerance interval • Resampling • Bootstrap • Jackknife Testing hypotheses • 1- & 2-tails • Power • Uniformly most powerful test • Permutation test • Randomization test • Multiple comparisons Parametric tests • Likelihood-ratio • Score/Lagrange multiplier • Wald Specific tests • Z-test (normal) • Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • Ordinary least squares • General linear model • Bayesian regression Non-standard predictors • Nonlinear regression • Nonparametric • Semiparametric • Isotonic • Robust • Heteroscedasticity • Homoscedasticity Generalized linear model • Exponential families • Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance • Analysis of variance (ANOVA, anova) • Analysis of covariance • Multivariate ANOVA • Degrees of freedom Categorical / Multivariate / Time-series / Survival analysis Categorical • Cohen's kappa • Contingency table • Graphical model • Log-linear model • McNemar's test • Cochran–Mantel–Haenszel statistics Multivariate • Regression • Manova • Principal components • Canonical correlation • Discriminant analysis • Cluster analysis • Classification • Structural equation model • Factor analysis • Multivariate distributions • Elliptical distributions • Normal Time-series General • Decomposition • Trend • Stationarity • Seasonal adjustment • Exponential smoothing • Cointegration • Structural break • Granger causality Specific tests • Dickey–Fuller • Johansen • Q-statistic (Ljung–Box) • Durbin–Watson • Breusch–Godfrey Time domain • Autocorrelation (ACF) • partial (PACF) • Cross-correlation (XCF) • ARMA model • ARIMA model (Box–Jenkins) • Autoregressive conditional heteroskedasticity (ARCH) • Vector autoregression (VAR) Frequency domain • Spectral density estimation • Fourier analysis • Least-squares spectral analysis • Wavelet • Whittle likelihood Survival Survival function • Kaplan–Meier estimator (product limit) • Proportional hazards models • Accelerated failure time (AFT) model • First hitting time Hazard function • Nelson–Aalen estimator Test • Log-rank test Applications Biostatistics • Bioinformatics • Clinical trials / studies • Epidemiology • Medical statistics Engineering statistics • Chemometrics • Methods engineering • Probabilistic design • Process / quality control • Reliability • System identification Social statistics • Actuarial science • Census • Crime statistics • Demography • Econometrics • Jurimetrics • National accounts • Official statistics • Population statistics • Psychometrics Spatial statistics • Cartography • Environmental statistics • Geographic information system • Geostatistics • Kriging • Category • Mathematics portal • Commons • WikiProject
Wilks Memorial Award The Wilks Memorial Award is awarded by the American Statistical Association to recognize outstanding contributions to statistics. It was established in 1964 and is awarded yearly. It is named in memory of the statistician Samuel S. Wilks. The award consists of a medal, a citation and a cash honorarium of US$1500 (as of 2008).[1] Wilks Memorial Award Awarded forOutstanding contributions to statistics CountryUSA Presented byAmerican Statistical Association First awarded1964 Websitehttps://www.amstat.org/ASA/Your-Career/Awards/Samuel-S-Wilks-Memorial-Award.aspx Recipients • 1964 Frank E. Grubbs • 1965 John W. Tukey • 1966 Leslie E. Simon • 1967 William G. Cochran • 1968 Jerzy Neyman • 1969 W. J. Youden • 1970 George W. Snedecor • 1971 Harold F. Dodge • 1972 George E.P. Box • 1973 Herman Otto Hartley • 1974 Cuthbert Daniel • 1975 Herbert Solomon • 1976 Solomon Kullback • 1977 Churchill Eisenhart • 1978 William Kruskal • 1979 Alexander M. Mood • 1980 W. Allen Wallis • 1981 Holbrook Working • 1982 Frank Proschan • 1983 W. Edwards Deming • 1984 Z. W. Birnbaum • 1985 Leo A. Goodman • 1986 Frederick Mosteller • 1987 Herman Chernoff • 1988 Theodore W. Anderson • 1989 C. R. Rao • 1990 Bradley Efron • 1991 Ingram Olkin • 1992 Wilfrid Dixon • 1993 Norman L. Johnson • 1994 Emanuel Parzen • 1995 Donald Rubin • 1996 Erich L. Lehmann • 1997 Leslie Kish • 1998 David O. Siegmund • 1999 Lynne Billard • 2000 Stephen Fienberg • 2001 George C. Tiao • 2002 Lawrence D. Brown • 2003 David L. Wallace • 2004 Paul Meier • 2005 Roderick J. A. Little • 2006 Marvin Zelen • 2007 Colin L. Mallows • 2008 Scott Zeger • 2009 Lee-Jen Wei • 2010 Pranab K. Sen • 2011 Nan Laird • 2012 Peter Gavin Hall • 2013 Kanti Mardia • 2014 Madan L. Puri • 2015 James O. Berger • 2016 David Donoho • 2017 Wayne Fuller • 2018 Peter J. Bickel • 2019 Alan E. Gelfand • 2020 Malay Ghosh • 2021 Sallie Ann Keller • 2022 Jessica Utts[2] References 1. "Wilks Memorial Award". American Statistical Association. 2. "Professors Utts and Stern Honored with American Statistical Association Awards". www.stat.uci.edu. Retrieved 2023-05-05.
Shapiro–Wilk test The Shapiro–Wilk test is a test of normality. It was published in 1965 by Samuel Sanford Shapiro and Martin Wilk.[1] Not to be confused with the likelihood-ratio test, which is sometimes referred to as Wilks test. Theory The Shapiro–Wilk test tests the null hypothesis that a sample x1, ..., xn came from a normally distributed population. The test statistic is $W={\left(\sum _{i=1}^{n}a_{i}x_{(i)}\right)^{2} \over \sum _{i=1}^{n}(x_{i}-{\overline {x}})^{2}},$ where • $x_{(i)}$ with parentheses enclosing the subscript index i is the ith order statistic, i.e., the ith-smallest number in the sample (not to be confused with $x_{i}$). • ${\overline {x}}=\left(x_{1}+\cdots +x_{n}\right)/n$ is the sample mean. The coefficients $a_{i}$ are given by:[1] $(a_{1},\dots ,a_{n})={m^{\mathsf {T}}V^{-1} \over C},$ where C is a vector norm:[2] $C=\\|V^{-1}m\\|=(m^{\mathsf {T}}V^{-1}V^{-1}m)^{1/2}$ and the vector m, $m=(m_{1},\dots ,m_{n})^{\mathsf {T}}\,$ is made of the expected values of the order statistics of independent and identically distributed random variables sampled from the standard normal distribution; finally, $V$ is the covariance matrix of those normal order statistics.[3] There is no name for the distribution of $W$. The cutoff values for the statistics are calculated through Monte Carlo simulations.[2] Interpretation The null-hypothesis of this test is that the population is normally distributed. Thus, if the p value is less than the chosen alpha level, then the null hypothesis is rejected and there is evidence that the data tested are not normally distributed. On the other hand, if the p value is greater than the chosen alpha level, then the null hypothesis (that the data came from a normally distributed population) can not be rejected (e.g., for an alpha level of .05, a data set with a p value of less than .05 rejects the null hypothesis that the data are from a normally distributed population – consequently, a data set with a p value more than the .05 alpha value fails to reject the null hypothesis that the data is from a normally distributed population).[4] Like most statistical significance tests, if the sample size is sufficiently large this test may detect even trivial departures from the null hypothesis (i.e., although there may be some statistically significant effect, it may be too small to be of any practical significance); thus, additional investigation of the effect size is typically advisable, e.g., a Q–Q plot in this case.[5] Power analysis Monte Carlo simulation has found that Shapiro–Wilk has the best power for a given significance, followed closely by Anderson–Darling when comparing the Shapiro–Wilk, Kolmogorov–Smirnov, and Lilliefors.[6] Approximation Royston proposed an alternative method of calculating the coefficients vector by providing an algorithm for calculating values that extended the sample size from 50 to 2,000.[7] This technique is used in several software packages including GraphPad Prism, Stata,[8][9] SPSS and SAS.[10] Rahman and Govidarajulu extended the sample size further up to 5,000.[11] See also • Anderson–Darling test • Cramér–von Mises criterion • D'Agostino's K-squared test • Kolmogorov–Smirnov test • Lilliefors test • Normal probability plot • Shapiro–Francia test References 1. Shapiro, S. S.; Wilk, M. B. (1965). "An analysis of variance test for normality (complete samples)". Biometrika. 52 (3–4): 591–611. doi:10.1093/biomet/52.3-4.591. JSTOR 2333709. MR 0205384. p. 593 2. RMD (2022). "The Shapiro-Wilk and related tests for normality" (PDF). Retrieved 2022-06-16. 3. Davis, C. S.; Stephens, M. A. (1978). The covariance matrix of normal order statistics (PDF) (Technical report). Department of Statistics, Stanford University, Stanford, California. Technical Report No. 14. Retrieved 2022-06-17. 4. "How do I interpret the Shapiro–Wilk test for normality?". JMP. 2004. Retrieved March 24, 2012. 5. Field, Andy (2009). Discovering statistics using SPSS (3rd ed.). Los Angeles [i.e. Thousand Oaks, Calif.]: SAGE Publications. p. 143. ISBN 978-1-84787-906-6. 6. Razali, Nornadiah; Wah, Yap Bee (2011). "Power comparisons of Shapiro–Wilk, Kolmogorov–Smirnov, Lilliefors and Anderson–Darling tests". Journal of Statistical Modeling and Analytics. 2 (1): 21–33. Retrieved 30 March 2017. 7. Royston, Patrick (September 1992). "Approximating the Shapiro–Wilk W-test for non-normality". Statistics and Computing. 2 (3): 117–119. doi:10.1007/BF01891203. S2CID 122446146. 8. Royston, Patrick. "Shapiro–Wilk and Shapiro–Francia Tests". Stata Technical Bulletin, StataCorp LP. 1 (3). 9. Shapiro–Wilk and Shapiro–Francia tests for normality 10. Park, Hun Myoung (2002–2008). "Univariate Analysis and Normality Test Using SAS, Stata, and SPSS". [working paper]. Retrieved 29 July 2023. 11. Rahman und Govidarajulu (1997). "A modification of the test of Shapiro and Wilk for normality". Journal of Applied Statistics. 24 (2): 219–236. doi:10.1080/02664769723828. External links • Worked example using Excel • Algorithm AS R94 (Shapiro Wilk) FORTRAN code • Exploratory analysis using the Shapiro–Wilk normality test in R • Real Statistics Using Excel: the Shapiro-Wilk Expanded Test Statistics • Outline • Index Descriptive statistics Continuous data Center • Mean • Arithmetic • Arithmetic-Geometric • Cubic • Generalized/power • Geometric • Harmonic • Heronian • Heinz • Lehmer • Median • Mode Dispersion • Average absolute deviation • Coefficient of variation • Interquartile range • Percentile • Range • Standard deviation • Variance Shape • Central limit theorem • Moments • Kurtosis • L-moments • Skewness Count data • Index of dispersion Summary tables • Contingency table • Frequency distribution • Grouped data Dependence • Partial correlation • Pearson product-moment correlation • Rank correlation • Kendall's τ • Spearman's ρ • Scatter plot Graphics • Bar chart • Biplot • Box plot • Control chart • Correlogram • Fan chart • Forest plot • Histogram • Pie chart • Q–Q plot • Radar chart • Run chart • Scatter plot • Stem-and-leaf display • Violin plot Data collection Study design • Effect size • Missing data • Optimal design • Population • Replication • Sample size determination • Statistic • Statistical power Survey methodology • Sampling • Cluster • Stratified • Opinion poll • Questionnaire • Standard error Controlled experiments • Blocking • Factorial experiment • Interaction • Random assignment • Randomized controlled trial • Randomized experiment • Scientific control Adaptive designs • Adaptive clinical trial • Stochastic approximation • Up-and-down designs Observational studies • Cohort study • Cross-sectional study • Natural experiment • Quasi-experiment Statistical inference Statistical theory • Population • Statistic • Probability distribution • Sampling distribution • Order statistic • Empirical distribution • Density estimation • Statistical model • Model specification • Lp space • Parameter • location • scale • shape • Parametric family • Likelihood (monotone) • Location–scale family • Exponential family • Completeness • Sufficiency • Statistical functional • Bootstrap • U • V • Optimal decision • loss function • Efficiency • Statistical distance • divergence • Asymptotics • Robustness Frequentist inference Point estimation • Estimating equations • Maximum likelihood • Method of moments • M-estimator • Minimum distance • Unbiased estimators • Mean-unbiased minimum-variance • Rao–Blackwellization • Lehmann–Scheffé theorem • Median unbiased • Plug-in Interval estimation • Confidence interval • Pivot • Likelihood interval • Prediction interval • Tolerance interval • Resampling • Bootstrap • Jackknife Testing hypotheses • 1- & 2-tails • Power • Uniformly most powerful test • Permutation test • Randomization test • Multiple comparisons Parametric tests • Likelihood-ratio • Score/Lagrange multiplier • Wald Specific tests • Z-test (normal) • Student's t-test • F-test Goodness of fit • Chi-squared • G-test • Kolmogorov–Smirnov • Anderson–Darling • Lilliefors • Jarque–Bera • Normality (Shapiro–Wilk) • Likelihood-ratio test • Model selection • Cross validation • AIC • BIC Rank statistics • Sign • Sample median • Signed rank (Wilcoxon) • Hodges–Lehmann estimator • Rank sum (Mann–Whitney) • Nonparametric anova • 1-way (Kruskal–Wallis) • 2-way (Friedman) • Ordered alternative (Jonckheere–Terpstra) • Van der Waerden test Bayesian inference • Bayesian probability • prior • posterior • Credible interval • Bayes factor • Bayesian estimator • Maximum posterior estimator • Correlation • Regression analysis Correlation • Pearson product-moment • Partial correlation • Confounding variable • Coefficient of determination Regression analysis • Errors and residuals • Regression validation • Mixed effects models • Simultaneous equations models • Multivariate adaptive regression splines (MARS) Linear regression • Simple linear regression • Ordinary least squares • General linear model • Bayesian regression Non-standard predictors • Nonlinear regression • Nonparametric • Semiparametric • Isotonic • Robust • Heteroscedasticity • Homoscedasticity Generalized linear model • Exponential families • Logistic (Bernoulli) / Binomial / Poisson regressions Partition of variance • Analysis of variance (ANOVA, anova) • Analysis of covariance • Multivariate ANOVA • Degrees of freedom Categorical / Multivariate / Time-series / Survival analysis Categorical • Cohen's kappa • Contingency table • Graphical model • Log-linear model • McNemar's test • Cochran–Mantel–Haenszel statistics Multivariate • Regression • Manova • Principal components • Canonical correlation • Discriminant analysis • Cluster analysis • Classification • Structural equation model • Factor analysis • Multivariate distributions • Elliptical distributions • Normal Time-series General • Decomposition • Trend • Stationarity • Seasonal adjustment • Exponential smoothing • Cointegration • Structural break • Granger causality Specific tests • Dickey–Fuller • Johansen • Q-statistic (Ljung–Box) • Durbin–Watson • Breusch–Godfrey Time domain • Autocorrelation (ACF) • partial (PACF) • Cross-correlation (XCF) • ARMA model • ARIMA model (Box–Jenkins) • Autoregressive conditional heteroskedasticity (ARCH) • Vector autoregression (VAR) Frequency domain • Spectral density estimation • Fourier analysis • Least-squares spectral analysis • Wavelet • Whittle likelihood Survival Survival function • Kaplan–Meier estimator (product limit) • Proportional hazards models • Accelerated failure time (AFT) model • First hitting time Hazard function • Nelson–Aalen estimator Test • Log-rank test Applications Biostatistics • Bioinformatics • Clinical trials / studies • Epidemiology • Medical statistics Engineering statistics • Chemometrics • Methods engineering • Probabilistic design • Process / quality control • Reliability • System identification Social statistics • Actuarial science • Census • Crime statistics • Demography • Econometrics • Jurimetrics • National accounts • Official statistics • Population statistics • Psychometrics Spatial statistics • Cartography • Environmental statistics • Geographic information system • Geostatistics • Kriging • Category • Mathematics portal • Commons • WikiProject
Willard L. Miranker Willard L. Miranker (March 8, 1932 – April 28, 2011) was an American mathematician and computer scientist, known for his contributions to applied mathematics and numerical mathematics.[1] Raised in Brooklyn, New York, he earned B.A. (1952), M.S. (1953) and Ph.D. (1956) from the Courant Institute at New York University, the latter on the thesis The Asymptotic Theory of Solutions of U + (K2)U = 0 advised by Joseph Keller. He then worked for the mathematics department at Bell Labs (1956–1958) before joining IBM Research (1961). After retirement from IBM, he joined the computer science faculty at Yale University (1989) as research faculty. He also held professor affiliations at California Institute of Technology (1963), Hebrew University of Jerusalem (1968), Yale University (1973), University of Paris-Sud (1974), City University of New York (1966–) and New York University (1970–1973). Miranker's work[2] includes articles and books on stiff differential equations,[3] interval arithmetic,[4] analog computing, and neural networks and the modeling of consciousness. Miranker was also an accomplished and prolific painter. Over the course of his life, Willard Miranker painted ~4000 watercolors/aquarelles and ~200 oil paintings, many of which are displayed online. He exhibited internationally in New York City, Paris and Bonn.[5] Awards • Fellow of the American Association for the Advancement of Science References 1. "Willard L. Miranker". Findagrave. Retrieved 2015-09-21. 2. Willard L. Miranker, 60 years, Computing 48:1-3, 1992 fulltext 3. Miranker, Willard L., Numerical Methods for Stiff Equations And Singular Perturbation Problems, D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981. ISBN 90-277-1107-0 4. Kulisch, Ulrich W.; Miranker, Willard L. (1981). Rheinboldt, Werner (ed.). Computer arithmetic in theory and practice. Computer Science and Applied Mathematics (1 ed.). New York, USA: Academic Press, Inc. ISBN 978-0-12-428650-4. 5. The Guide from New York Times (August 29, 1993). External links • Works by Willard L. Miranker at Open Library • Paintings by Will Miranker • Willard L. Miranker at the Mathematics Genealogy Project • short page at Yale University • FindaGrave entry Authority control International • ISNI • VIAF National • Norway • Germany • Israel • United States • Netherlands Academics • CiNii • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Willem van Zwet Willem Rutger van Zwet (31 March 1934 – 2 July 2020) was a Dutch mathematical statistician.[2] He was a professor at Leiden University between 1968 and 1999. Willem van Zwet Born31 March 1934 Leiden, Netherlands Died2 July 2020(2020-07-02) (aged 86) Oegstgeest, Netherlands NationalityDutch OccupationMathematician Scientific career ThesisConvex Transformations of Random Variables[1] (1964) Doctoral advisorJan Hemelrijk Doctoral studentsSara van de Geer Aad van der Vaart Biography Van Zwet was born on 31 March 1934 in Leiden.[3] Van Zwet obtained his doctoral degree in 1964 under the supervision of Jan Hemelrijk at the University of Amsterdam with a thesis titled "Convex Transformations of Random Variables".[4] After that, he worked at the Centrum Wiskunde & Informatica in Amsterdam, and became a lector of statistics at Leiden University in 1964 and was named professor in 1968. He retired in 1999.[3] From 1992 to 1999, van Zwet was the Director of the Thomas Stieltjes Institute of Mathematics. He co-founded Eurandom in 1997, and served as its director until 2000. From 1997 to 1999, he was also the President of the International Statistical Institute. Van Zwet was a Fellow of the Institute of Mathematical Statistics and a member of the Academia Europaea since 1990.[5] He received the Humboldt Prize in 2006. He won the Adolphe Quetelet Medal in 1993, and had been a Fellow of the Royal Statistical Society since 1978. In 1979, he became a member of the Royal Netherlands Academy of Arts and Sciences.[6] In 1996, he was made Knight of the Order of the Netherlands Lion, and was named Doctoris Honoris causa of Charles University the following year. He died on 2 July 2020 in Oegstgeest.[3] References 1. Willem van Zwet at the Mathematics Genealogy Project 2. "In Memoriarium: William van Zwet". International Statistical Institute. 2 July 2020. 3. "Willem Rutger van Zwet". Leiden University. Archived from the original on 9 July 2020. 4. "Willem Rutger van Zwet". North Dakota State University. 5. "Willem van Zwet". Academia Europaea. Archived from the original on 28 March 2019. 6. "Prof. dr. w.r. Willem van Zwet" (in Dutch). KNAW genootschap. Archived from the original on 8 July 2020. Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Willerton's fish In knot theory, Willerton's fish is an unexplained relationship between the first two Vassiliev invariants of a knot. These invariants are c2, the quadratic coefficient of the Alexander–Conway polynomial, and j3, an order-three invariant derived from the Jones polynomial.[1][2] When the values of c2 and j3, for knots of a given fixed crossing number, are used as the x and y coordinates of a scatter plot, the points of the plot appear to fill a fish-shaped region of the plane, with a lobed body and two sharp tail fins. The region appears to be bounded by cubic curves,[2] suggesting that the crossing number, c2, and j3 may be related to each other by not-yet-proven inequalities.[1] This shape is named after Simon Willerton,[1] who first observed this phenomenon and described the shape of the scatterplots as "fish-like".[3] References 1. Chmutov, S.; Duzhin, S.; Mostovoy, J. (2012), "14.3 Willerton's fish and bounds for c2 and j3", Introduction to Vassiliev knot invariants (PDF), Cambridge University Press, Cambridge, pp. 419–420, arXiv:1103.5628, doi:10.1017/CBO9781139107846, ISBN 978-1-107-02083-2, MR 2962302. 2. Dunin-Barkowski, P.; Sleptsov, A.; Smirnov, A. (2013), "Kontsevich integral for knots and Vassiliev invariants", International Journal of Modern Physics A, 28 (17): 1330025, arXiv:1112.5406, Bibcode:2013IJMPA..2830025D, doi:10.1142/S0217751X13300251, MR 3081407. See in particular Section 4.2.1, "Willerton's fish and families of knots". 3. Willerton, Simon (2002), "On the first two Vassiliev invariants", Experimental Mathematics, 11 (2): 289–296, MR 1959269.
Willi Rinow Willi Ludwig August Rinow (February 28th, 1907 in Berlin – March 29th, 1979 in Greifswald) was a German mathematician who specialized in differential geometry and topology. Rinow was the son of a schoolteacher. In 1926, he attended the Humboldt University of Berlin, studying mathematics and physics under professors such as Max Planck, Ludwig Bieberbach, and Heinz Hopf. There, he received his doctorate in 1931 (Über Zusammenhänge zwischen der Differentialgeometrie im Großen und im Kleinen, Math. Zeitschrift volume 35, 1932, page 512). In 1933, he worked at the Jahrbuch über die Fortschritte der Mathematik in Berlin. In 1937, he joined the Nazi Party.[1] During 1937—1940, he was an editor of the journal Deutsche Mathematik. In 1937, he became a professor in Berlin and lectured there until 1950. His lecturing was interrupted because of his work as a mathematician at the Oberspreewerk in Berlin (a producer of radio and telecommunications technology) from 1946 to 1949. In 1950, he became a professor at the University of Greifswald. He retired in 1972. The Hopf–Rinow theorem is named after Hopf and Rinow. In 1959, he became the director of the Institute for Pure Mathematics at the German Academy of Sciences at Berlin and president of the German Mathematical Society. War work During World War II, Rinow worked as a cryptanalyst in Subsection F of Referat I of Group IV of the Inspectorate 7/VI, that was later called the General der Nachrichtenaufklärung (GdNA), achieving the rank of Colonel. Rinow worked on researching methods to solve foreign ciphers. Rinow was subordinated to Herbert von Denffer who was Director of the section and subordinated to Hans Pietsch who was Director of the section. Major Rudolf Hentze was head of the group. Otto Buggisch who was a cryptanalyst who working in the GdNA and earlier the OKW/Chi and was interrogated by TICOM agents after the war, stated that Rinow was one of the most capable people in the unit.[2] Publications • Die innere Geometrie der metrischen Räume, Springer 1961[3] • Lehrbuch der Topologie, Berlin, Deutscher Verlag der Wissenschaften 1975 • Rinow Über Zusammenhänge der Differentialgeometrie im Großen und Kleinen, Mathematische Zeitschrift, volume 32, 1932, pages 512-528, Dissertation Further reading • Renate Tobies: Biographisches Lexikon in Mathematik promovierter Personen, 2006 References 1. Harry Waibel: Diener vieler Herren : Ehemalige NS-Funktionäre in der SBZ/DDR. Peter Lang, Frankfurt 2011 ISBN 978-3-631-63542-1 page 270 2. "Volume 4 – Signal Intelligence Service of the Army High Command" (PDF). NSA. p. 182. Retrieved 12 November 2016. This article incorporates text from this source, which is in the public domain. 3. Green, Leon W. (1963). "Review: Die innere Geometrie der metrischen Räume by Willi Rinow" (PDF). Bull. Amer. Math. Soc. 69 (2): 210–212. doi:10.1090/s0002-9904-1963-10916-7. German Signals intelligence organisations before 1945 • The Type of organisation • Name of organisation • People Military (?) Wehrmacht High Command Cipher Bureau • Erich Fellgiebel • Albert Praun • Hugo Kettler • Wilhelm Fenner • Erich Hüttenhain • Peter Novopashenny • Walter Fricke • Karl Stein • Wolfgang Franz • Gisbert Hasenjaeger • Heinrich Scholz • Werner Liebknecht • Gottfried Köthe • Ernst Witt • Helmut Grunsky • Georg Hamel • Georg Aumann • Oswald Teichmüller • Alexander Aigner • Werner Weber • Otto Leiberich • Otto Buggisch • Fritz Menzer General der Nachrichtenaufklärung • Erich Fellgiebel • Fritz Thiele • Wilhelm Gimmler • Hugo Kettler • Fritz Boetzel • Otto Buggisch • Fritz Menzer • Herbert von Denffer • Ludwig Föppl • Horst Schubert • Friedrich Böhm • Bruno von Freytag-Löringhoff • Johannes Marquart • Willi Rinow • Rudolf Kochendörffer • Hans Pietsch • Guido Hoheisel • Hans-Peter Luzius • Wilhelm Vauck • Rudolf Bailovic • Alfred Kneschke Luftnachrichten Abteilung 350 • Wolfgang Martini • Ferdinand Voegele B-Dienst • Kurt Fricke • Ludwig Stummel • Heinz Bonatz • Wilhelm Tranow • Erhard Maertens • Fritz Krauss Abwehr (?) • Wilhelm Canaris Civilian (?) Pers Z S • Kurt Selchow • Horst Hauthal • Rudolf Schauffler • Johannes Benzing • Otfried Deubner • Hans Rohrbach • Helmut Grunsky • Erika Pannwitz • Karl Schröter Research Office of the Reich Air Ministry • Hermann Göring • Gottfried Schapper • Hans Schimpf • Prince Christoph of Hesse Training (?) GdNA Training Referat Heer and Luftwaffe Signals School • German Radio Intelligence Operations during World War II Authority control International • ISNI • VIAF • WorldCat National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • Leopoldina • MathSciNet • Mathematics Genealogy Project • Scopus • zbMATH People • Deutsche Biographie Other • IdRef
William A. Massey (mathematician) William Alfred Massey is an American mathematician and operations researcher, the Edwin S. Wilsey Professor of Operations Research and Financial Engineering at Princeton University. He is an expert in queueing theory. William A Massey Born1956 (age 66–67) Jefferson City, Missouri OccupationEdwin S Wilsey Professor of Princeton University Parent(s)Juliete and Richard Massey Biography Massey was born in Jefferson City, Missouri in 1956,[1] the son of Juliette and Richard Massey Sr., both educators. His family moved to St. Louis, Missouri when he was four.[2] He went to college at Princeton University, graduating in 1977.[2] Massey obtained his Ph.D. from Stanford University in 1981, with a thesis on queueing theory supervised by Joseph Keller.[3] His first research publication was developed during a summer program at Bell Laboratories while he was a graduate student and was published in 1978. After earning his doctorate, he became a permanent staff member at Bell Labs.[2] In 2001, Massey moved to his current position at Princeton, becoming the first African-American Princeton undergraduate alumnus to return as a faculty member.[2] Contributions Massey has made many original contributions as a mathematician by developing a theory of "dynamical queueing systems". Classical queueing models assumed that calling rates were constant so they could use the static, equilibrium analysis of time homogeneous Markov chains. However, real communication systems call for the large scale analysis of queueing models with time-varying rates. His thesis at Stanford University created a dynamic, asymptotic method for time inhomogeneous Markov chains called "uniform acceleration" to deal with such problems. Moreover, his research on queueing networks led to new methods of comparing multi-dimensional, Markov processes by viewing them as "stochastic orderings" on "partially ordered spaces". Finally, one of his most cited papers develops an algorithm to find a dynamic, optimal server staffing schedule for telephone call centers with time varying demand, which led to a patent. Another highly cited paper creates a temporally and spatially dynamic model for the offered load traffic of wireless communication networks. Awards and honors In 2006, Massey won the Blackwell–Tapia Prize of the Institute for Mathematics and its Applications for his "outstanding record of achievement in mathematical research and his mentoring of minorities and women in the field of mathematics".[2] He was elected to the 2006 class of Fellows of the Institute for Operations Research and the Management Sciences.[4] In 2012 he became a fellow of the American Mathematical Society.[5] Massey's accomplishments have earned him recognition by Mathematically Gifted & Black as a Black History Month 2018 Honoree.[6] References 1. "William A. Massey". Mathematicians of the African diaspora. State University of New York at Buffalo. Retrieved 16 April 2017. 2. "Massey's mentorship creates network of mathematicians", Princeton Weekly Bulletin, October 23, 2006. 3. William Alfred Massey at the Mathematics Genealogy Project. 4. Fellows: Alphabetical List, Institute for Operations Research and the Management Sciences, retrieved 2019-10-09 5. List of Fellows of the American Mathematical Society, retrieved 2013-02-02. 6. "William A. Massey". Mathematically Gifted & Black. External links • http://www.princeton.edu/~wmassey/ William A Massey Profile Authority control: Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William A. Stein William Arthur Stein (born February 21, 1974 in Santa Barbara, California) is a software developer and previously a professor of mathematics at the University of Washington. William A. Stein Born William Arthur Stein (1974-02-21) 21 February 1974 Santa Barbara, California Occupation(s)Software Developer, Professor of Mathematics Known forLead developer of SageMath and founder of CoCalc. Websitewww.wstein.org He is the lead developer of SageMath and founder of CoCalc. Stein does computational and theoretical research into the problem of computing with modular forms and the Birch and Swinnerton-Dyer conjecture.[1] He is considered "a leading expert in the field of computational arithmetic".[2] References 1. "NSF Award Search: Award # 0555776 - Explicit Approaches to Modular Forms and Modular Abelian Varieties". 2. Kleinert, Werner. "Zbl 1110.11015". Zentralblatt MATH. Retrieved 18 April 2012. External links • Official website • William A. Stein at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Korea • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Allen Whitworth William Allen Whitworth (1 February 1840 – 12 March 1905) was an English mathematician and a priest in the Church of England.[1][2] Education and mathematical career Whitworth was born in Runcorn; his father, William Whitworth, was a school headmaster, and he was the oldest of six siblings. He was schooled at the Sandicroft School in Northwich and then at St John's College, Cambridge, earning a B.A. in 1862 as 16th Wrangler. He taught mathematics at the Portarlington School and the Rossall School, and was a professor of mathematics at Queen's College in Liverpool from 1862 to 1864. He returned to Cambridge to earn a master's degree in 1865, and was a fellow there from 1867 to 1882.[1] Mathematical contributions As an undergraduate, Whitworth became the founding editor in chief of the Messenger of Mathematics, and he continued as its editor until 1880.[1] He published works about the logarithmic spiral and about trilinear coordinates, but his most famous mathematical publication is the book Choice and Chance: An Elementary Treatise on Permutations, Combinations, and Probability (first published in 1867 and extended over several later editions).[1] The first edition of the book treated the subject primarily from the point of view of arithmetic calculations, but had an appendix on algebra, and was based on lectures he had given at Queen's College.[2] Later editions added material on enumerative combinatorics (the numbers of ways of arranging items into groups with various constraints), derangements, frequentist probability, life expectancy, and the fairness of bets, among other topics.[2] Among the other contributions in this book, Whitworth was the first to use ordered Bell numbers to count the number of weak orderings of a set, in the 1886 edition. These numbers had been studied earlier by Arthur Cayley, but for a different problem.[3] He was the first to publish Bertrand's ballot theorem, in 1878; the theorem is misnamed after Joseph Louis François Bertrand, who rediscovered the same result in 1887.[4] He is the inventor of the E[X] notation for the expected value of a random variable X, still commonly in use,[5] and he coined the name "subfactorial" for the number of derangements of n items.[6] Another of Whitworth's contributions, in geometry, concerns equable shapes, shapes whose area has the same numerical value (with a different set of units) as their perimeter. As Whitworth showed with D. Biddle in 1904, there are exactly five equable triangles with integer sides: the two right triangles with side lengths (5,12,13) and (6,8,10), and the three triangles with side lengths (6,25,29), (7,15,20), and (9,10,17).[7] Religious career Whitworth was ordained as a deacon in 1865, and became a priest in 1866. He served as the curate of St Anne's Church in Birkenhead in 1865, of the Church of St Luke, Liverpool from 1866 to 1870 and of Christ Church in Liverpool from 1870 to 1875. He was then a vicar in London at St John the Evangelist's in Hammersmith. From 1886 to 1905 he was vicar of All Saints, Margaret Street.[1] He was the Hulsean Lecturer in 1903.[1] References 1. Lee, Sidney, ed. (1912). "Whitworth, William Allen" . Dictionary of National Biography (2nd supplement). Vol. 2. London: Smith, Elder & Co. 2. Irwin, J. O. (1967). "William Allen Whitworth and a Hundred Years of Probability". Journal of the Royal Statistical Society. Series A. 130 (2): 147–176. doi:10.2307/2343399. JSTOR 2343399.. 3. Pippenger, Nicholas (2010), "The hypercube of resistors, asymptotic expansions, and preferential arrangements", Mathematics Magazine, 83 (5): 331–346, arXiv:0904.1757, doi:10.4169/002557010X529752, MR 2762645, S2CID 17260512. 4. Feller, William (1968). An Introduction to Probability Theory and its Applications, Volume I (3rd ed.). Wiley. p. 69.. 5. Aldrich, John (2007). "Earliest Uses of Symbols in Probability and Statistics". Retrieved 13 March 2013.. 6. Cajori, Florian (2011), A History of Mathematical Notations: Two Volumes in One, Cosimo, Inc., p. 77, ISBN 9781616405717. 7. Dickson, Leonard Eugene (2005), History of the Theory of Numbers, Volume Il: Diophantine Analysis, Courier Dover Publications, p. 199, ISBN 9780486442334. External links • Works by or about William Allen Whitworth at Internet Archive Authority control International • FAST • ISNI • VIAF National • Germany • Israel • United States • Australia • Netherlands Academics • zbMATH People • Trove Other • IdRef
W. B. R. Lickorish William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow of Pembroke College, Cambridge. His research interests include topology and knot theory. He was one of the discoverers of the HOMFLY polynomial invariant of links, and proved the Lickorish-Wallace theorem which states that all closed orientable 3-manifolds can be obtained by Dehn surgery on a link. W. B. R. Lickorish Lickorish in 1974 Born19 February 1938 (1938-02-19) (age 85) NationalityBritish Alma materUniversity of Cambridge Known forTopology AwardsChauvenet Prize (1991) Senior Whitehead Prize (1991) Scientific career FieldsMathematician Doctoral advisorErik Christopher Zeeman Doctoral studentsMarc Lackenby Education Lickorish received his Ph.D from Cambridge in 1964; his thesis was written under the supervision of Christopher Zeeman.[1] Recognition and awards In 1991, Lickorish received the Senior Whitehead Prize from the London Mathematical Society.[2] Lickorish and Kenneth Millett won the 1991 Chauvenet Prize for their paper "The New Polynomial Invariants of Knots and Links".[3] Lickorish was included in the 2019 class of fellows of the American Mathematical Society "for contributions to knot theory and low-dimensional topology".[4] Selected publications • Lickorish, W. B. R. (November 1962). "A Representation of Orientable Combinatorial 3-Manifolds". Annals of Mathematics. 76 (3): 531–540. doi:10.2307/1970373. JSTOR 1970373. • Freyd, Peter; Yetter, David; Hoste, Jim; Lickorish, W.B.R.; Millett, Kenneth; Ocneanu, Adrian (1985). "A New Polynomial Invariant of Knots and Links". Bulletin of the American Mathematical Society. 12 (2): 239–246. doi:10.1090/S0273-0979-1985-15361-3. • Lickorish, W. B. R. (1997). An Introduction to Knot Theory. Graduate Texts in Mathematics 175. Springer. ISBN 0-387-98254-X. See also • Lickorish twist theorem • Lickorish–Wallace theorem References 1. W. B. R. Lickorish at the Mathematics Genealogy Project 2. London Mathematical Society. "List of Prizewinners". Retrieved 1 April 2015. 3. Lickorish, W. B. R.; Millett, K. C. (1988). "The New Polynomial Invariants of Knots and Links". Mathematics Magazine. Taylor & Francis. 61 (1): 3–23. doi:10.1080/0025570x.1988.11977338. ISSN 0025-570X. 4. "2019 Class of the Fellows of the AMS". American Mathematical Society. Retrieved 7 November 2018. Chauvenet Prize recipients • 1925 G. A. Bliss • 1929 T. H. Hildebrandt • 1932 G. H. Hardy • 1935 Dunham Jackson • 1938 G. T. Whyburn • 1941 Saunders Mac Lane • 1944 R. H. Cameron • 1947 Paul Halmos • 1950 Mark Kac • 1953 E. J. McShane • 1956 Richard H. Bruck • 1960 Cornelius Lanczos • 1963 Philip J. Davis • 1964 Leon Henkin • 1965 Jack K. Hale and Joseph P. LaSalle • 1967 Guido Weiss • 1968 Mark Kac • 1970 Shiing-Shen Chern • 1971 Norman Levinson • 1972 François Trèves • 1973 Carl D. Olds • 1974 Peter D. Lax • 1975 Martin Davis and Reuben Hersh • 1976 Lawrence Zalcman • 1977 W. Gilbert Strang • 1978 Shreeram S. Abhyankar • 1979 Neil J. A. Sloane • 1980 Heinz Bauer • 1981 Kenneth I. Gross • 1982 No award given. • 1983 No award given. • 1984 R. Arthur Knoebel • 1985 Carl Pomerance • 1986 George Miel • 1987 James H. Wilkinson • 1988 Stephen Smale • 1989 Jacob Korevaar • 1990 David Allen Hoffman • 1991 W. B. Raymond Lickorish and Kenneth C. Millett • 1992 Steven G. Krantz • 1993 David H. Bailey, Jonathan M. Borwein and Peter B. Borwein • 1994 Barry Mazur • 1995 Donald G. Saari • 1996 Joan Birman • 1997 Tom Hawkins • 1998 Alan Edelman and Eric Kostlan • 1999 Michael I. Rosen • 2000 Don Zagier • 2001 Carolyn S. Gordon and David L. Webb • 2002 Ellen Gethner, Stan Wagon, and Brian Wick • 2003 Thomas C. Hales • 2004 Edward B. Burger • 2005 John Stillwell • 2006 Florian Pfender & Günter M. Ziegler • 2007 Andrew J. Simoson • 2008 Andrew Granville • 2009 Harold P. Boas • 2010 Brian J. McCartin • 2011 Bjorn Poonen • 2012 Dennis DeTurck, Herman Gluck, Daniel Pomerleano & David Shea Vela-Vick • 2013 Robert Ghrist • 2014 Ravi Vakil • 2015 Dana Mackenzie • 2016 Susan H. Marshall & Donald R. Smith • 2017 Mark Schilling • 2018 Daniel J. Velleman • 2019 Tom Leinster • 2020 Vladimir Pozdnyakov & J. Michael Steele • 2021 Travis Kowalski • 2022 William Dunham, Ezra Brown & Matthew Crawford Authority control International • ISNI • VIAF National • Norway • France • BnF data • Germany • Israel • United States • Japan • Netherlands Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Bigelow Easton William Bigelow Easton was an American mathematician who proved Easton's theorem about the possible values of the continuum function. His advisor at Princeton was the famed mathematician and computer scientist Alonzo Church.[1] Publications • Easton, W. (1970), "Powers of regular cardinals", Ann. Math. Logic, 1 (2): 139–178, doi:10.1016/0003-4843(70)90012-4 References 1. "William Bigelow Easton". Mathematics Genealogy Project. Retrieved 2 November 2022. Authority control International • ISNI • VIAF National • Netherlands Academics • MathSciNet • Mathematics Genealogy Project
William Binney (intelligence official) William "Bill" Edward Binney (born September 1943) [4] is a former intelligence official with the United States National Security Agency (NSA)[5] and whistleblower. He retired on October 31, 2001, after more than 30 years with the agency. William Binney Binney at the Congress on Privacy & Surveillance (2013) of the École polytechnique fédérale de Lausanne (EPFL) Born William Edward Binney September 1943 (age 79) Pennsylvania, U.S. EducationPennsylvania State University (B.S., 1970) OccupationCryptanalyst-mathematician EmployerNational Security Agency (NSA) Known forCryptography, SIGINT analysis, whistleblowing Awards • Joe A. Callaway Award for Civic Courage (2012)[1] • Sam Adams Award (2015)[2] • Allard Prize for International Integrity (2019)[3] Signature He was a critic of his former employers during the George W. Bush administration, and later criticized the NSA's data-collection policies during the Barack Obama administration. He dissented from the view that Russia interfered with the 2016 US election. More specifically, he was critical of the view that Russia hacked the DNC server.[6] Biography Binney grew up in rural Pennsylvania and graduated with a Bachelor of Science degree in mathematics from the Pennsylvania State University in 1970. He said that he volunteered for the Army during the Vietnam era in order to select work that would interest him rather than be drafted and have no input. He was found to have strong aptitudes for mathematics, analysis, and code breaking,[7] and served from 1965 to 1969 in the Army Security Agency before going to the NSA in 1970. Binney was a Russia specialist and worked in the operations side of intelligence, starting as an analyst and ending as a Technical Director prior to becoming a geopolitical world Technical Director. In the 1990s, he co-founded a unit on automating signals intelligence with NSA research chief John Taggart.[8] Binney's NSA career culminated as Technical Leader for intelligence in 2001. He has expertise in intelligence analysis, traffic analysis, systems analysis, knowledge management, and mathematics (including set theory, number theory, and probability).[9][10] After retiring from the NSA, he founded, together with fellow NSA whistleblower J. Kirk Wiebe, Entity Mapping, LLC, a private intelligence agency to market their analysis program to government agencies.[11] Whistleblowing National Security Agency surveillance Map of global NSA data collection as of 2007, with countries subject to the most data collection shown in red Programs Pre-1978 • ECHELON • MINARET • SHAMROCK • PROMIS Since 1978 • Upstream collection • BLARNEY • FAIRVIEW • Main Core • ThinThread • Genoa Since 1990 • RAMPART-A Since 1998 • Tailored Access Operations Since 2001 • OAKSTAR • STORMBREW • Trailblazer • Turbulence • Genoa II • Total Information Awareness • President's Surveillance Program • Terrorist Surveillance Program Since 2007 • PRISM • Dropmire • Stateroom • Bullrun • MYSTIC Databases, tools etc. • PINWALE • MARINA • Main Core • MAINWAY • TRAFFICTHIEF • DISHFIRE • XKeyscore • ICREACH • BOUNDLESSINFORMANT GCHQ collaboration • MUSCULAR • Tempora Legislation • Safe Streets Act • Privacy Act of 1974 • FISA • ECPA • Patriot Act • Homeland Security Act • Protect America Act of 2007 • FISA Amendments Act of 2008 Institutions • FISC • Senate Intelligence Committee • National Security Council Lawsuits • ACLU v. NSA • Hepting v. AT&T • Jewel v. NSA • Clapper v. Amnesty • Klayman v. Obama • ACLU v. Clapper • Wikimedia v. NSA • US v. Moalin Whistleblowers • William Binney • Thomas Drake • Mark Klein • Thomas Tamm • Russ Tice Publication • 2005 warrantless surveillance scandal • Global surveillance disclosures (2013–present) Related • Cablegate • Surveillance of reporters • Mail tracking • UN diplomatic spying • Insider Threat Program • Mass surveillance in the United States • Mass surveillance in the United Kingdom Concepts • SIGINT • Metadata Collaboration United States • CSS • CYBERCOM • DOJ • FBI • CIA • DHS • IAO Five Eyes • CSEC • GCHQ • ASD • GCSB Other • DGSE • BND In September 2002, he, along with J. Kirk Wiebe and Edward Loomis, asked the U.S. Defense Department Inspector General (DoD IG) to investigate the NSA for allegedly wasting "millions and millions of dollars" on Trailblazer, a system intended to analyze mass collection of data carried on communications networks such as the Internet. Binney had been one of the inventors of an alternative system, ThinThread, which was shelved when Trailblazer was chosen instead. Binney has also been publicly critical of the NSA for spying on U.S. citizens, saying of its expanded surveillance after the September 11, 2001 attacks that "it's better than anything that the KGB, the Stasi, or the Gestapo and SS ever had"[12] as well as noting Trailblazer's ineffectiveness and unjustified high cost compared to the far less intrusive ThinThread.[13] He was furious that the NSA hadn't uncovered the 9/11 plot and stated that intercepts it had collected but not analyzed likely would have garnered timely attention with his leaner more focused system.[10] Post-NSA career After he left the NSA in 2001, Binney was one of several people investigated as part of an inquiry into a 2005 exposé by The New York Times on the agency's warrantless eavesdropping program.[14] Binney was cleared of wrongdoing after three interviews with FBI agents beginning in March 2007, but in early July 2007, in an unannounced early morning raid, a dozen agents armed with rifles appeared at his house, one of whom entered the bathroom and pointed his gun at Binney, who was taking a shower. The FBI confiscated a desktop computer, disks, and personal and business records.[15] The NSA revoked his security clearance, forcing him to close a business he ran with former colleagues at a loss of a reported $300,000 in annual income. The FBI raided the homes of Wiebe and Loomis, as well as House Intelligence Committee staffer Diane Roark, the same morning. Several months later the FBI raided the home of then still active NSA executive Thomas Andrews Drake who had also contacted DoD IG, but anonymously with confidentiality assured. The Assistant Inspector General, John Crane, in charge of the Whistleblower Program, suspecting his superiors provided confidential information to the United States Department of Justice (DOJ), challenged them, was eventually forced from his position, and subsequently himself became a public whistleblower. The punitive treatment of Binney, Drake, and the other whistleblowers also led Edward Snowden to go public with his revelations rather than report through the internal whistleblower program.[16] In 2012, Binney and his co-plaintiffs went to federal court to retrieve the confiscated items.[17] Allegations on intercepts Binney is known for making the claim that the NSA collects and stores information about every U.S. communication.[18] Binney was invited as a witness by the NSA commission of the German Bundestag. On July 3, 2014 Der Spiegel wrote, he said that the NSA wanted to have information about everything. In Binney's view this is a totalitarian approach, which had previously been seen only in dictatorships.[19] Binney stated that the goal was to control people. Meanwhile, he said that it is possible in principle to monitor the whole population, abroad and in the U.S., which in his view contradicts the United States Constitution.[19] In August 2014, Binney was among the signatories of an open letter by the group Veteran Intelligence Professionals for Sanity to German chancellor Angela Merkel in which they urged the Chancellor to be suspicious of U.S. intelligence regarding the alleged invasion by Russia in Eastern Ukraine.[20][21][22] In the open letter, the group said: [A]ccusations of a major Russian "invasion" of Ukraine appear not to be supported by reliable intelligence. Rather, the "intelligence" seems to be of the same dubious, politically "fixed" kind used 12 years ago to "justify" the U.S.-led attack on Iraq.[21] Russian Interference in the 2016 election Binney has said he voted for Trump in the 2016 presidential election, calling Hillary Clinton a "war monger".[23] Binney has asserted that the U.S. intelligence community's assessment that Russia interfered in the 2016 presidential election is false, and that the Democratic National Committee e-mails were leaked by an insider instead.[24][25][26] An investigation by Duncan Campbell later detailed how Binney had been persuaded by a pro-Kremlin disinformant that the theft of the DNC emails was an inside job, and not the work of Russian agents (contrary to the findings of the US intelligence community).[27] The disinformation agent altered metadata in the files released by Guccifer 2.0 (whom the US intelligence community identifies as a Russian military intelligence operation) to make it appear as if the documents came from a computer in the Eastern United States, not Russia. (Specifically, the local time zone of the computer's system clock was changed to UTC−05:00.)[27] Binney appeared on Fox News at least ten times between September 2016 and November 2017 to promote this theory.[18][24][25] Binney said that the "intelligence community wasn't being honest here".[24] He has been a frequent guest on RT and Fox News and has been frequently cited on Breitbart News.[18] In October 2017, Binney met with CIA Director Mike Pompeo at the behest of President Trump to discuss his theory.[24] However, on meeting Campbell and analysing the material again, Binney changed his position: he said there was “no evidence to prove where the download/copy was done”, and that the files he had based his previous assessment were “manipulated” and a “fabrication”.[27] Role in apparent release of the Nunes Memo On January 23, 2018, Binney made an appearance on InfoWars[28] in connection with the Nunes memo, a Congressional document alleging irregularities in the application of the FISA Act, which at that time was not publicly available although its potential release was a topic of public debate.[29] During the show, host Alex Jones announced that Binney had been able to provide him with the actual memo, and the purported leaked document was shown on air.[30] However, this was in fact a public document that had been available on the website of the Office of the Director of National Intelligence since at least May 2017.[31][32] The actual Nunes memo was released February 2, 2018.[33] Claims of fraud in the 2020 election After Joe Biden won the 2020 United States presidential election and Donald Trump refused to concede, Binney doubted the official results and claimed that there had been large-scale voter fraud. One of Binney's tweets alleging missing votes was based on a mistaken conflation between eligible voters and an outdated number of registered voters; this was cited in an article by The Gateway Pundit, which in turn was promoted by Trump.[34][35][36][37] Documentary film Binney's story is recounted in A Good American, a documentary film.[38] See also • MAINWAY • PRISM (surveillance program) • Mark Klein • Thomas Tamm • Russ Tice • Perry Fellwock • Targeted surveillance • Citizenfour – a 2014 documentary • A Good American – a 2015 documentary References 1. "For Immediate Release: Callaway Awards Tuesday November 13, 2012". The Joe A. Callaway Award for Civic Justice. November 13, 2012. Retrieved July 1, 2013. 2. Hannah Borno (January 23, 2015). "NSA whistleblower William Binney wins 2015 Sam Adams award". International Business Times. Retrieved August 31, 2015. 3. "Past Winners and Honourees of the Allard Prize". Allard Prize For International Integrity. Allard Prize Foundation. Retrieved October 24, 2020. 4. Video-Interview by Thomas Drake (October 26, 2011). "William Edward Binney Collection" (Video; 25 Min). Veterans History Project. American Folklife Center of the Library of Congress. Retrieved June 29, 2013. 5. "Three NSA Whistleblowers Back EFF's Lawsuit Over Government's Massive Spying Program". Electronic Frontier Foundation. July 2, 2012. Retrieved May 11, 2013. 6. Kopan, Tal (June 21, 2016). "DNC hack: What you need to know". CNN. Retrieved March 16, 2023. 7. "Keynote Address: William Binney". Schedule – HOPE Number Nine. 2600 Enterprises. 2012. Retrieved May 11, 2013. 8. "The Government Is Profiling You". MIT Center for Internet and Society. November 12, 2012. Retrieved June 8, 2013. 9. "Sworn Declaration of Whistleblower William Binney on NSA Domestic Surveillance Capabilities". Public Intelligence. July 16, 2012. Retrieved May 11, 2013. 10. Mayer, Jane (May 23, 2011). "The Secret Sharer: Is Thomas Drake an enemy of the state?". The New Yorker. 11. "NSA Whistleblowers William (Bill) Binney and J. Kirk Wiebe". Government Accountability Project website. Archived from the original on December 12, 2013. Retrieved June 9, 2013. 12. Shorrock, Tim (April 15, 2013). "The Untold Story: Obama's Crackdown on Whistleblowers: The NSA Four reveal how a toxic mix of cronyism and fraud blinded the agency before 9/11". The Nation. 13. "NSA Whistleblowers William (Bill) Binney and J. Kirk Wiebe". Government Accountability Project. Archived from the original on December 12, 2013. Retrieved May 11, 2013. 14. "The FRONTLINE Interview: William Binney – United States of Secrets". FRONTLINE. Retrieved November 10, 2020. 15. "Exclusive: National Security Agency Whistleblower William Binney on Growing State Surveillance". Democracy Now!. Retrieved September 23, 2016. 16. Hertsgaard, Mark; Kasten, Felix; Rosenbach, Marcel; Stark, Holger (May 22, 2016). "Blowing the Whistle: Former US Official Reveals Risks Faced by Internal Critics". Der Spiegel. Retrieved June 16, 2016. 17. Bronner, Ethan; Charlie Savage; Scott Shane (May 25, 2013). "Leak Inquiries Show How Wide A Net U.S. Cast". The New York Times. 18. "Was Donald Trump behind meeting of CIA chief and conspiracy theorist?". NBC News. Retrieved November 9, 2017. 19. "Untersuchungsausschuss im Bundestag: US-Informant vergleicht NSA mit einer Diktatur". Spiegel. Spiegelonline GmbH. July 3, 2014. Retrieved July 4, 2014. 20. "US-Geheimdienst-Pensionäre warnen Merkel vor Fehlinformationen" (in German). Der Tagesspiegel. September 4, 2014. 21. "The State Department Says Russia Is Invading Ukraine—Should We Believe It?". The Nation. September 2, 2014. Retrieved September 28, 2015. 22. "US-Geheimdienst-Pensionäre warnen Merkel vor Fehlinformationen". Der Tagesspiegel Online. September 4, 2014. 23. "Was Donald Trump behind meeting of CIA chief and conspiracy theorist?". NBC News. Retrieved August 4, 2019. 24. Campbell, Duncan; Risen, James (November 7, 2017). "CIA Director Met Advocate of Disputed DNC Hack Theory — at Trump's Request". The Intercept. Retrieved November 7, 2017. 25. "Conservative media figures are embracing a wild WikiLeaks conspiracy theory that the CIA hacked the DNC, and then framed Russia". Business Insider. Retrieved November 7, 2017. 26. "Why Some U.S. Ex-Spies Don't Buy the Russia Story". Bloomberg.com. August 10, 2017. Retrieved November 7, 2017. 27. "Briton ran pro-Kremlin disinformation campaign that helped Trump deny Russian links". ComputerWeekly.com. Retrieved August 1, 2018. 28. "Alex Jones tries and fails to pass off a publicly available document as the House GOP's secret Russia memo". Media Matters for America. 29. Mckew, Molly K. "How Twitter Bots and Trump Fans Made #ReleaseTheMemo Go Viral". POLITICO Magazine. 30. "Read the disputed memo here". CNN. February 2, 2018. 31. "Fact Check: Did Infowars Release the 'Secret FISA Memo'?". The Weekly Standard. January 23, 2018. 32. Matsakis, Louise (January 24, 2018). "The Cynical Misdirection Behind #ReleaseTheMemo". Wired – via wired.com. 33. McCarthy, Tom; Yuhas, Alan (February 2, 2018). "'Nunes memo' published after Trump declassifies controversial document". The Guardian – via theguardian.com. 34. "PolitiFact - Bogus analysis leads to ridiculous claim about Biden votes". PolitiFact. 2020. Retrieved December 28, 2020.{{cite web}}: CS1 maint: url-status (link) 35. "Fact check: Claim that turnout numbers prove U.S. election fraud uses wrong figures". Reuters. January 4, 2021. Retrieved February 19, 2021. 36. Bump, Philip (December 21, 2020). "Analysis - Another good example of how laughably flimsy Trump's electoral fraud claims are". Washington Post. Retrieved February 19, 2021. 37. "Flawed Calculation Behind False Claim of Fraudulent Votes". FactCheck.org. December 21, 2020. Retrieved February 19, 2021. 38. Kenigsberg, Ben (February 2, 2017). "Review: In 'A Good American,' Examining Sept. 11 and Data Collection". The New York Times. ISSN 0362-4331. Retrieved February 7, 2017. External links Wikiquote has quotations related to William Binney. Wikimedia Commons has media related to William Binney. • The Future of Freedom: A Feature Interview with NSA Whistleblower William Binney, February 2015 • William.Binney.HOPE.9.KEYNOTE.Part1, related to ThinThread development • William.Binney.HOPE.9.KEYNOTE.Part2, related to ThinThread development • "Who's Watching the N.S.A. Watchers?: Giving In to the Surveillance State", Shane Harris op-ed in The New York Times, 22 August 2012 • "The National Security Agency's Domestic Spying Program", Laura Poitras opinion piece in The New York Times, 22 August 2012 • 'The Program' – a video by Laura Poitras for The New York Times, 22 August 2012. • William Binney at IMDb • O'Brien, Alexa. "Retired NSA Technical Director Explains Snowden Docs". Archived from the original on October 15, 2014. Retrieved October 7, 2014. • Web site for documentary A Good American Laureates of the Sam Adams Award • 2002: Coleen Rowley • 2003: Katharine Gun • 2004: Sibel Edmonds • 2005: Craig Murray • 2006: Samuel Provance • 2007: Andrew Wilkie • 2008: Frank Grevil • 2009: Larry Wilkerson • 2010: Julian Assange • 2011: Thomas A. Drake and Jesselyn Radack • 2012: Thomas Fingar • 2013: Edward Snowden • 2014: Chelsea Manning • 2015: William Binney • 2016: John Kiriakou • 2017: Seymour Hersh • 2018: Karen Kwiatkowski • 2019: Jeffrey Sterling • 2020: Annie Machon • 2021: Daniel Hale Authority control International • VIAF National • Norway • Germany • Israel • United States Other • IdRef
William Brouncker, 2nd Viscount Brouncker William Brouncker, 2nd Viscount Brouncker FRS (c. 1620 – 5 April 1684) was an Anglo-Irish peer and mathematician who served as the president of the Royal Society from 1662 to 1677. Best known for introducing Brouncker's formula, he also worked as a civil servant, serving as a commissioner in the Royal Navy. Brouncker was a friend and colleague of Samuel Pepys, and features prominently in the Pepys' diary. The Right Honourable The Viscount Brouncker President of the Royal Society In office 1662–1677 Preceded byOffice established Succeeded byJoseph Williamson Personal details Bornc. 1620 Castlelyons, Ireland Died5 April 1684(1684-04-05) (aged 64) Westminster, London ResidenceEngland Alma materUniversity of Oxford Known forBrouncker's formula, leadership of Royal Society Scientific career FieldsMathematician, civil servant InstitutionsSaint Catherine's Hospital Academic advisorsJohn Wallis Biography Brouncker was born c. 1620 in Castlelyons, County Cork, the elder son of William Brouncker (1585–1649), 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from Melksham in Wiltshire. His grandfather Sir Henry Brouncker (died 1607) had been Lord President of Munster 1603–1607, and settled his family in Ireland. His father was created a viscount in the Peerage of Ireland in 1645 for his services to the Crown. Although the first viscount had fought for the Crown in the Anglo-Scots war of 1639, malicious gossip said that he paid the then enormous sum of £1200 for the title and was almost ruined as a result. He died only a few months afterwards. William obtained a DM at the University of Oxford in 1647. Until 1660 he played no part in public life: being a staunch Royalist, he felt it best to live quietly and devote himself to his mathematical studies. He was one of the founders and the first president of the Royal Society. In 1662, he became chancellor to Queen Catherine, then head of the Saint Catherine's Hospital. He was appointed one of the commissioners of the Royal Navy in 1664, and his career thereafter can be traced in the Diary of Samuel Pepys; despite their frequent disagreements, Samuel Pepys on the whole respected Brouncker more than most of his other colleagues, writing in 1668 that "in truth he is the best of them". Although his attendance at the Royal Society had become infrequent, and he had quarrelled with some of his fellow members, he was nonetheless greatly displeased to be deprived of the presidency in 1677. He was commissioner for executing the office of Lord High Admiral of England from 1679.[1] Abigail Williams Brouncker never married, but lived for many years with the actress Abigail Williams (much to Pepys' disgust) and left most of his property to her. She was the daughter of Sir Henry Clere (died 1622), first and last of the Clere Baronets, and the estranged wife of John Williams, otherwise Cromwell, second son of Sir Oliver Cromwell, and first cousin to the renowned Oliver Cromwell. She and John had a son and a daughter. The fire of 1673 which destroyed the Royal Navy Office started in her private closet: this is unlikely to have improved her relations with Samuel Pepys, whose private apartments were also destroyed in the blaze. On Brouncker's death in 1684, his title passed to his brother Henry, one of the most detested men of the era. William left him almost nothing in his will "for reasons I think not fit to mention". Mathematical works His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola,[2] which requires approximation of the natural logarithm function by infinite series.[3] He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi. Brouncker's formula This formula provides a development of π/4 in a generalized continued fraction: ${\frac {\pi }{4}}={\cfrac {1}{1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+{\cfrac {9^{2}}{2+\ddots }}}}}}}}}}}}$ The convergents are related to the Leibniz formula for pi: for instance ${\frac {1}{1+{\frac {1^{2}}{2}}}}={\frac {2}{3}}=1-{\frac {1}{3}}$ and ${\frac {1}{1+{\frac {1^{2}}{2+{\frac {3^{2}}{2}}}}}}={\frac {13}{15}}=1-{\frac {1}{3}}+{\frac {1}{5}}.$ Because of its slow convergence, Brouncker's formula is not useful for practical computations of π. Brouncker's formula can also be expressed as[4] ${\frac {4}{\pi }}=1+{\cfrac {1^{2}}{2+{\cfrac {3^{2}}{2+{\cfrac {5^{2}}{2+{\cfrac {7^{2}}{2+{\cfrac {9^{2}}{2+\ddots }}}}}}}}}}$ See also • List of presidents of the Royal Society References 1. "No. 1485". The London Gazette. 9 February 1679. p. 2. 2. W. Brouncker (1667) The Squaring of the Hyperbola, Philosophical Transactions of the Royal Society of London, abridged edition 1809, v. i, pp 233–6, link form Biodiversity Heritage Library 3. Julian Coolidge Mathematics of Great Amateurs, chapter 11, pp. 136–46 4. John Wallis, Arithmetica Infinitorum, ... (Oxford, England: Leon Lichfield, 1656), page 182. Brouncker expressed, as a continued fraction, the ratio of the area of a circle to the area of the circumscribed square (i.e., 4/π). The continued fraction appears at the top of page 182 (roughly) as: ☐ = 1 1/2 9/2 25/2 49/2 81/2 &c , where the square denotes the ratio that is sought. (Note: On the preceding page, Wallis names Brouncker as: "Dom. Guliel. Vicecon, & Barone Brouncher" (Lord William Viscount and Baron Brouncker).) External links • O'Connor, John J.; Robertson, Edmund F., "William Brouncker, 2nd Viscount Brouncker", MacTutor History of Mathematics Archive, University of St Andrews • "Brouncker, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. Presidents of the Royal Society 17th century • Viscount Brouncker (1662) • Joseph Williamson (1677) • Christopher Wren (1680) • John Hoskyns (1682) • Cyril Wyche (1683) • Samuel Pepys (1684) • Earl of Carbery (1686) • Earl of Pembroke (1689) • Robert Southwell (1690) • Charles Montagu (1695) • Lord Somers (1698) 18th century • Isaac Newton (1703) • Hans Sloane (1727) • Martin Folkes (1741) • Earl of Macclesfield (1752) • Earl of Morton (1764) • James Burrow (1768) • James West (1768) • James Burrow (1772) • John Pringle (1772) • Joseph Banks (1778) 19th century • William Hyde Wollaston (1820) • Humphry Davy (1820) • Davies Gilbert (1827) • Duke of Sussex (1830) • Marquess of Northampton (1838) • Earl of Rosse (1848) • Lord Wrottesley (1854) • Benjamin Collins Brodie (1858) • Edward Sabine (1861) • George Biddell Airy (1871) • Joseph Dalton Hooker (1873) • William Spottiswoode (1878) • Thomas Henry Huxley (1883) • George Gabriel Stokes (1885) • Lord Kelvin (1890) • Joseph Lister (1895) 20th century • William Huggins (1900) • Lord Rayleigh (1905) • Archibald Geikie (1908) • William Crookes (1913) • J. J. Thomson (1915) • Charles Scott Sherrington (1920) • Ernest Rutherford (1925) • Frederick Gowland Hopkins (1930) • William Henry Bragg (1935) • Henry Hallett Dale (1940) • Robert Robinson (1945) • Lord Adrian (1950) • Cyril Norman Hinshelwood (1955) • Howard Florey (1960) • Patrick Blackett (1965) • Alan Lloyd Hodgkin (1970) • Lord Todd (1975) • Andrew Huxley (1980) • George Porter (1985) • Sir Michael Atiyah (1990) • Sir Aaron Klug (1995) 21st century • Lord May (2000) • Lord Rees (2005) • Sir Paul Nurse (2010) • Venki Ramakrishnan (2015) • Adrian Smith (2020) Authority control International • FAST • ISNI • VIAF National • Germany • Italy • Belgium • United States • Australia • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie • Trove Other • RISM • SNAC • IdRef
William B. Johnson (mathematician) William Buhmann Johnson (born December 5, 1944) is an American mathematician, one of the namesakes of the Johnson–Lindenstrauss lemma. He is Distinguished Professor and A.G. & M.E. Owen Chair of Mathematics at Texas A&M University. His research specialties include the theory of Banach spaces, nonlinear functional analysis, and probability theory.[1] He was born in Palo Alto, California and raised from an early age in Dallas, Texas. Johnson graduated from Southern Methodist University in 1966,[2] and earned a doctorate from Iowa State University in 1969 under the supervision of James A. Dyer.[3] After faculty positions at the University of Houston, and Ohio State University, he joined the Texas A&M faculty in 1984. In 2007, Johnson was awarded the Stefan Banach Medal of the Polish Academy of Sciences.[4][5] In 2012 he became a fellow of the American Mathematical Society.[6] In 2018 he was an Invited Speaker at the International Congress of Mathematicians in Rio de Janeiro.[7] His doctoral students include Edward Odell. References 1. Faculty directory listing, Texas A&M Mathematics, retrieved 2013-01-26. 2. Faculty web page, retrieved 2013-01-26. 3. William Buhmann Johnson at the Mathematics Genealogy Project 4. Stefan Banach Medal, Polish Academy of Sciences, retrieved 2013-01-26. 5. 2007 Personal News, Texas A&M Mathematics, retrieved 2013-01-26. 6. List of Fellows of the American Mathematical Society, retrieved 2013-01-26. 7. Johnson, William B. "Some 20+ year old problems about Banach spaces and operators on them" (PDF). Proceedings of the ICM – 2018 Rio de Janeiro. Vol. 2. pp. 1669–1686. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • DBLP • Google Scholar • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • IdRef
William C. Waterhouse William Charles Waterhouse (December 31, 1941 – June 26, 2016) was an American mathematician. He was a professor emeritus of Mathematics at Pennsylvania State University,[1] after having taught there for over 35 years.[2] The early part of his career was at Cornell University.[2] His research interests included abstract algebra, number theory, group schemes, and the history of mathematics.[3] William C. Waterhouse Born(1941-12-31)December 31, 1941 Galveston, Texas, U.S. DiedJune 26, 2016(2016-06-26) (aged 74) State College, Pennsylvania, U.S. NationalityAmerican Alma materHarvard University AwardsLester R. Ford Award (1984, 1995) Scientific career FieldsMathematics InstitutionsPennsylvania State University Cornell University ThesisAbelian Varieties over Finite Fields (1968) Doctoral advisorJohn Tate Early life and education Waterhouse was born in Galveston, Texas, on December 31, 1941, the son of William T. Waterhouse and Grace D. Waterhouse,[4] but grew up in Denver, Colorado.[5] His father was an engineer who was employed with the United States Bureau of Reclamation.[6] He attended East High School in Denver.[7] In a high school mathematics competition spanning the states of Colorado, South Dakota, and Wyoming, he received the highest score in the competition's history and helped his school gain the top mark.[7] As a senior, he took the Scholastic Aptitude Test and received a near-perfect 797 on the verbal portion and a perfect 800 on the math portion.[6] He then received perfect 800 scores on three different college board Achievement Tests, those for English Composition, for Chemistry, and for advanced Mathematics,[6] a feat that the Associated Press filed a story about and that ran in a number of newspapers around the country.[8] Time magazine ran a profile of him as well.[6] Waterhouse received the National Merit Scholarship and the General Motors Scholarship;[6] he graduated from East High in 1959.[7] Waterhouse attended Harvard College. There he was a standout in the Putnam Competition: As a sophomore in the 1960 competition, he was not part of Harvard's three-person team that finished second overall, but he did achieve a top-ten individual mark;[9] as a junior in the 1961 competitition, he attained the highest individual level – a top-five score – while helping his Harvard team to a fourth-place finish overall;[10] and as a senior in the 1962 competition, he again was a Putnam Fellow with a top-five score and helped his Harvard team to a third-place finish overall.[11] After graduating from Harvard College with a bachelor's degree summa cum laude and being elected to Phi Beta Kappa, Waterhouse continued at Harvard Graduate School of Arts and Sciences where he received a master's degree.[12] While in the graduate school, he was awarded a National Science Foundation Fellowship.[12] He then received his Ph.D. in 1968 from Harvard for his thesis Abelian Varieties over Finite Fields under the supervision of John Tate.[13] Career Waterhouse began teaching at Cornell University in 1968.[4] He had a career-long interest in the history of mathematics,[3] and while at Cornell wrote a history of the early years of that university's Oliver Club, a discussion forum begun by pioneering Cornell mathematican James Edward Oliver in the 1890s.[14] Waterhouse remained an assistant professor at Cornell until 1975, at which point he was appointed an associate professor at Penn State.[12] At Penn State, he subsequently became a full professor.[5] In 1980 he married Betty Ann Senk, a doctoral student and teacher in comparative literature at Penn State.[15] They lived in State College, Pennsylvania.[15] According to his obituary published in the Centre Daily Times, during his career Waterhouse published over 250 articles in scholarly journals and other publications.[4] He was the author of the 1979 textbook Introduction to Affine Group Schemes for Springer-Verlag.[16] Telegraphic Reviews characterized the work as a "fairly intuitive and accessible" development of the topic, suitable for second-year graduate students.[17] In a 1986 volume for Springer-Verlag, he edited the 1966 translation by Arthur A. Clarke of Gauss's Disquisitiones Arithmeticae.[18] Waterhouse and his wife collaborated on several translations of works by German mathematicians.[19] He was a member of the Mathematical Association of America and the American Mathematical Society. [12] Waterhouse long had an interest in classical studies;[4] as such, he was a member of the Classical Association of the Middle West and South.[20] He published a number of entries about language- and classics-focused matters in the journal Notes and Queries, [21] as well as, in the journal Classical World, an exegesis of an unusual word form found in Ovid's Amores.[22] Waterhouse also had an interest in investigating quotations, whether via a Usenet newsgroup or publishing with The Skeptics Society.[23] By 2012, Waterhouse had moved to emeritus status.[1] Waterhouse died on June 26, 2016, in State College, Pennsylvania.[4] Awards and honors Waterhouse twice won the Lester R. Ford Award of the Mathematical Association of America, given to authors of articles of expository excellence. The first was in 1984 for his paper "Do Symmetric Problems Have Symmetric Solutions?"[13] and the second was in 1995 for his paper "A Counterexample for Germain".[24] The latter has been characterized as "a historical and mathematical detective story" that investigated an aspect of the correspondence between Carl Friedrich Gauss and Sophie Germain, a French mathematician who used a pseudonym to disguise the fact that she was a woman.[25] According to his obituary, Waterhouse had a special pride in having won the two Lester R. Ford Awards.[4] References 1. "Mathematics Department: Emeriti". Pennsylvania State University. Archived from the original on October 30, 2012. Retrieved March 15, 2023. 2. "News from the AMS: William C. 'Bill' Waterhouse (1941–2016)". American Mathematical Society. September 6, 2016. Retrieved March 15, 2023. 3. PSU Mathematics Department - Faculty, retrieved 2010-02-06. 4. "William C. Waterhouse Obituary". Centre Daily Times. June 29, 2016. Retrieved April 21, 2022.. Also available at this CAMWS page. 5. Waterhouse, William C. (1989). "Two Elementary Proofs of an Inequality (and 1½ Better Ones)". The College Mathematics Journal. 20 (3): 201–205. doi:10.1080/07468342.1989.11973231. 6. "Education: The Good Student". Time. April 27, 1959. 7. "uncertain". The Denver Post. March 25, 1959. 8. See for example "Flawless Performance". The Tampa Times. Associated Press. June 13, 1959. p. 2-C – via Newspapers.com. Newspapers.com shows over fifty papers running the AP report. 9. Bush, L. E. (1961). "The 1960 William Lowell Putnam Mathematical Competition". The American Mathematical Monthly. 68 (7): 629–637. doi:10.2307/2311508. 10. Bush, L. E. (1962). "The 1961 William Lowell Putnam Mathematical Competition". The American Mathematical Monthly. 69 (8): 759–767. doi:10.2307/2310772. 11. "Three Math Students Win Third in Contest". The Harvard Crimson. March 16, 1963. 12. "2 Named in Mathematics". Centre Daily Times. State College, Pennsylvania. September 10, 1975. p. 20 – via Newspapers.com. 13. "MAA Writing Awards: Do Symmetric Problems Have Symmetric Solutions?". Mathematical Association of America. Retrieved February 7, 2010. 14. "About the Oliver Club". Department of Mathematics, Cornell University. Retrieved February 8, 2023. 15. "Senk–Waterhouse Marriage". Centre Daily Times. State College, Pennsylvania. October 17, 1980. p. 10 – via Newspapers.com. 16. Graduate Texts in Mathematics 66, Springer-Verlag, 1979, ISBN 978-0-387-90421-4. 17. "Telegraphic Reviews". The American Mathematical Monthly. 87 (3): 233–238. 1980 – via JSTOR. 18. Reprinted in 1986 by Springer-Verlag, ISBN 978-0-387-96254-2. 19. Wynn, James; Reyes, G. Mitchell (2021). "From Division to Multiplication: Uncovering the Relationship Between Mathematics and Rhetoric Through Transdisciplinary Scholarship". In Wynn, James; Reyes, G. Mitchell (eds.). Arguing with Numbers: The Intersections of Rhetoric and Mathematics. Pennsylvania State University Press. p. 23. also Poonen, Bjorn (2017). Rational Points on Varieties. Providence, Rhode Island: American Mathematical Society. p. 317. 20. "CAMWS Necrology". Program: 115th Annual CAMWS Meeting (PDF). Lincoln, Nebraska: Classical Association of the Middle West and South. 2019. p. 69. 21. See search results at Notes and Queries archive, accessed March 10, 2023. 22. Waterhouse, William C. (2008). "Emodulanda in Ovid's Amores 1.1". Classical World. 101 (4): 533–534. doi:10.1353/clw.0.0027 – via Project Muse. 23. Waterhouse, William C. (January 5, 2006). "Did Einstein Praise the Church?". eSkeptic. The Skeptics Society. Retrieved March 15, 2023. 24. "MAA Writing Awards: A Counterexample for Germain". Mathematical Association of America. Retrieved February 7, 2010. 25. "Penn State Briefs: Math Professor Wins Award". Centre Daily Times. August 21, 1995. p. 3 – via Newspapers.com. External links • William Charles Waterhouse at the Mathematics Genealogy Project • Abelian Varieties over Finite Fields dissertation, at French Numdam service • Early History of the Oliver Club at Cornell University Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • Belgium • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
Bill Casselman William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a Professor Emeritus at the University of British Columbia.[1] He is closely connected to the Langlands program and has been involved in posting all of the work of Robert Langlands on the internet.[2] Bill Casselman Casselman in 1969 Born William Allen Casselman (1941-11-27) November 27, 1941 Glen Ridge, New Jersey, U.S. CitizenshipCanadian Alma materPrinceton University Scientific career FieldsRepresentation theory Automorphic forms Geometric combinatorics Structure of algebraic groups InstitutionsUniversity of British Columbia Doctoral advisorGoro Shimura InfluencesRobert Langlands Career Casselman did his undergraduate work at Harvard College where his advisor was Raoul Bott and received his Ph.D from Princeton University in 1966 where his advisor was Goro Shimura. He was a visiting scholar at the Institute for Advanced Study in 1974, 1983, and 2001.[3] He emigrated to Canada in 1971 and is a Professor Emeritus in mathematics at the University of British Columbia.[1] Research Casselman specializes in representation theory, automorphic forms, geometric combinatorics, and the structure of algebraic groups. He has an interest in mathematical graphics[4] and has been the graphics editor of the Notices of the American Mathematical Society since January, 2001.[5] Awards In 2012, he became one of the inaugural fellows of the American Mathematical Society.[6] Selected publications • Casselman, Bill (1973). "On some results of Atkin and Lehner". Mathematische Annalen. 201 (4): 301–314. doi:10.1007/BF01428197. S2CID 121867474. • Casselman, Bill (1977). "Characters and Jacquet modules". Mathematische Annalen. 230 (2): 101–105. doi:10.1007/BF01370657. ISSN 0025-5831. S2CID 121574262. • Casselman, Bill (1980). "The unramified principal series of p-adic groups. I. The Spherical function". Compositio Mathematica. 40 (3): 387–406. • Casselman, Bill; Shalika, Joseph (1980). "The unramified principal series of p-adic groups. II. The Whittaker function". Compositio Mathematica. 41 (2): 207–231. • Casselman, Bill; Milicic, Dragan (1982). "representations". Duke Mathematical Journal. 49 (4): 869–930. doi:10.1215/S0012-7094-82-04943-2. ISSN 0012-7094. • Borel, Armand; Casselman, Bill (1983). "L2-cohomology of locally symmetric manifolds of finite volume". Duke Mathematical Journal. 50 (3): 625–647. doi:10.1215/S0012-7094-83-05029-9. S2CID 122723214. • Casselman, Bill; Shahidi, Freydoon (1998). "On irreducibility of standard modules for generic representations". Annales Scientifiques de l'École Normale Supérieure. 31 (4): 561–589. doi:10.1016/S0012-9593(98)80107-9. • Casselman, Bill (2005). Mathematical Illustrations: A Manual of Geometry and PostScript. Cambridge University Press. ISBN 0521839211. References 1. "Emeriti and Retirees". University of British Columbia Mathematics Department. Retrieved September 27, 2020. 2. Institute for Advanced Study: The Work of Robert Langlands 3. Institute for Advanced Study: A Community of Scholars: Casselman, William 4. Mathematical Illustrations: A Manual of Geometry and PostScript reviewed by Denis Roegel in Notices of the AMS 5. Notices of the American Mathematical Society: Editors and Staff 6. List of Fellows of the American Mathematical Society, retrieved 2015-11-18. External links • Publications of Bill Casselman • Bill Casselman's Home Page • William Allen Casselman at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Chapple (surveyor) William Chapple (1718–1781) was an English surveyor and mathematician. His mathematical discoveries were mostly in plane geometry and include: • the first proof of the existence of the orthocentre of a triangle, • a formula for the distance between the incentre and circumcentre of a triangle, • the discovery of Poncelet's porism on triangles with a common incircle and circumcircle. He was also one of the earliest mathematicians to calculate the values of annuities. Life Chapple was born in Witheridge on 25 January 1719 [O.S. 14 January 1718], the son of a poor farmer and parish clerk.[1] He was a devoted bibliophile,[2] and gained much of his knowledge of mathematics from Ward's The Young Mathematician's Guide: Being a Plain and Easie Introduction to the Mathematicks, in Five Parts.[3] He became an assistant to the parish priest, and a regular contributor to The Ladies' Diary, especially concerning mathematical problems. He also later contributed work on West Country English to The Gentleman's Magazine.[1] His correspondence led him to become, in 1738, the clerk for a surveyor in Exeter. He married the surveyor's niece, supervised the construction of a new hospital in Exeter, and became secretary of the hospital.[1] He also worked as the estate steward for William Courtenay, 1st Viscount Courtenay.[4] In 1772 he began work on an update to Tristram Risdon's Survey of the County of Devon, and spent much of the rest of his life working on it; it was published in part throughout his life, and in complete form posthumously in 1785.[1] He died in early September 1781.[1] A tablet in his memory could be found in the west end of the nave of the Church of St Mary Major, Exeter, prior to that church's demolition in 1971.[5] Chapple Road in Witheridge is named after him.[2] Contributions to mathematics Andrea del Centina writes that: "To illustrate the work of Chapple, whose arguments are often confused and whose logic is very poor, even for the standard of his time, is not easy especially when trying to keep as faithful as possible to his thought."[3] Nevertheless, Chapple made several significant discoveries in mathematics. Plane geometry Euler's theorem in geometry gives a formula for the distance $d$ between the incentre and circumcentre of a circle, as a function of the inradius $r$ and circumradius $R$: $d={\sqrt {R(R-2r)}}.$ An immediate consequence is the related inequality $R\geq 2r$. Although these results are named for Leonhard Euler, who published them in 1765, they were found earlier by Chapple, in a 1746 essay in The Gentleman's Magazine.[6][7] In the same work he stated that, when two circles are the incircle and circumcircle of a triangle, then there is an infinite family of triangles for which they are the incircle and circumcircle. This is the triangular case of Poncelet's closure theorem, which applies more generally to polygons of any number of sides and to conics other than circles. It is the first known mathematical publication on pairs of inscribed and circumscribed circles of polygons, and significantly predates Poncelet's own 1822 work in this area.[3] In 1749, Chapple published the first known proof of the existence of the orthocentre of a triangle, the point where the three perpendiculars from the vertices to the sides meet. The orthocentre itself was known previously, but Chapple writes that its existence was "often taken for granted, but no where demonstrated".[8] Finance Chapple learned of the problem of valuation of annuities through his correspondence with John Rowe and Thomas Simpson, and carried out this valuation for Courtenay. In this, he became one of the first mathematicians to work on this problem, along with Simpson, Abraham de Moivre, James Dodson, and William Jones.[4] References 1. Pengelly, W. (1887), "Prince's "Worthies of Devon" and the "Dictionary of National Biography", part III", Report & Transactions of the Devonshire Association, Devonshire Association for the Advancement of Science Literature & the Arts, 19: 217–348. See in particular "Chapple, William", pp. 316–318. 2. "William Chapple", Witheridge Historical Archive, retrieved 18 November 2019 3. Del Centina, Andrea (2016), "Poncelet's porism: a long story of renewed discoveries, I", Archive for History of Exact Sciences, 70 (1): 1–122, doi:10.1007/s00407-015-0163-y, MR 3437893, S2CID 253898210 4. Bellhouse, David R. (2017), Leases for Lives: Life Contingent Contracts and the Emergence of Actuarial Science in Eighteenth-Century England, Cambridge University Press, p. 79, ISBN 9781108509121 5. Lysons, Daniel (1822), Magna Brittanica; being a concise topographical account of the several counties of Great Britain, Vol. VI: Devonshire, Thomas Cadell, p. 215 6. Milne, Antony (2015), "The Euler and Grace-Danielsson inequalities for nested triangles and tetrahedra: a derivation and generalisation using quantum information theory", Journal of Geometry, 106 (3): 455–463, doi:10.1007/s00022-014-0257-8, MR 3420559 7. Chapple, William (1749), "An essay on the properties of triangles inscribed in, and circumscribed about two given circles", Miscellanea Curiosa Mathematica, The Gentleman's Magazine, vol. 4, pp. 117–124 8. Bogomolny, Alexander, "A Possibly First Proof of the Concurrence of Altitudes", Cut The Knot, retrieved 17 November 2019. See also Chapple's letter with the proof. Authority control International • FAST • VIAF National • United States
William Charles Brenke William Charles Brenke (April 12, 1874, Berlin – 1964)[1] was an American mathematician who introduced Brenke polynomials and wrote several undergraduate textbooks. He received his PhD in mathematics at Harvard under Maxime Bôcher. Brenke taught at the University of Nebraska-Lincoln mathematics department from 1908 to 1944 and was chair of the department from 1934 to 1944. He retired in 1943 but his successor, Ralph Hull, was put on official leave to do war work and returned from leave in 1945.[2] Publications • Brenke, W. C. (1930). "On polynomial solutions of a class of differential equations of the second order". Bull. Amer. Math. Soc. 36 (2): 77–84. doi:10.1090/s0002-9904-1930-04888-0. MR 1561893. • Brenke, W. C. (1933). "On the summability and general sum of a series of Legendre polynomials". Bull. Amer. Math. Soc. 39 (10): 821–824. doi:10.1090/s0002-9904-1933-05753-1. MR 1562735. References 1. According to ancestry.com Brenke married Kate Read in 1898, had three children, and died in 1964. 2. General History - Department of Mathematics - University of Nebraska-Lincoln External links • William Charles Brenke at the Mathematics Genealogy Project • NEGenWeb Project - Lancaster County Who's Who in Nebraska, 1940 Authority control International • ISNI • VIAF National • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William Christopher Swinbank William Christopher Swinbank (8 May 1913 – 28 December 1973) was a British-born meteorological physicist who worked at the UK Meteorological Office, the CSIRO Australia and the NCAR Colorado. His main areas of research were fog prediction, upper atmosphere analysis, wind predictions, hail storms and turbulent fluxes. Early life William Swinbank was born on 8 May 1913 in the small coal mining village of Easington in County Durham,[1] UK where his father worked above ground at a local colliery as a mechanical engineer. Swinbank was the oldest of three boys. He won a scholarship to Henry Smith Grammar School in Hartlepool and went up to Durham University (Hatfield College) where he took a Double Honours in Mathematics and Physics, graduating 1934.[2] Career After leaving university, Swinbank worked briefly as a schoolteacher and also in industrial research. In 1938 he commenced working as a meteorologist at the UK Met Office. The Met Office 1938–1948 In 1938, Swinbank began work as a Met Office Technical Officer for the Air Ministry and was attached to various RAF bases to carry out weather forecasting duties. In 1940, he was given the task of investigating the issue of fog, since this was critical to the operations of an airbase. He initially worked with C.S. Durst in the areas of cloud physics and turbulence. In 1942 he moved to the forecasting headquarters at RAF Dunstable in Bedfordshire and continued his research on fog.[2] During this time, he worked with C.H.B. Priestley, P. A Sheppard and Sverre Petterssen and came to the realisation that fog needed to be studied in relation to larger weather systems.[2] He joined the Upper Air Unit at Dunstable which was headed by Petterssen and as the war progressed, Swinbank became more involved in the use of upper air analysis forecasting techniques and further developed the use of isobaric analysis for producing the synoptic charts used for weather forecasting.[2] His research into predicting fog was valuable for guiding returning RAF bombers back to base after bombing raids in the early morning over Germany as fog was most prevalent in these hours.[3] In 1944, RAF Dunstable played an important role in forecasting the weather patterns leading up to the D-Day landings[4] and Swinbank, along with Priestly and Andrezej Berson were involved in this forecasting.[5] After the war, Swinbank returned to the study of fog as well as agricultural physics and worked closely with the meteorologist H. L. Penman.[2] In 1947 Swinbank and Priestley wrote a paper titled The Vertical Transfer of Heat by Turbulence in the Atmosphere, a paper that was acknowledged as being a landmark in micrometeorology[6][7] and laid the foundation for the development of eddy covariance.[8] CSIRO 1948–1969 In 1948, Swinbank moved to Australia to take up a position in the newly formed section of Meteorological Physics (later renamed Atmospheric Physics) at the Commonwealth Scientific and Industrial Research Organisation in Aspendale, a suburb of Melboune. He worked with C. H. B. Priestley who had joined the organisation two years earlier.[9] His early work at the organisation was in micrometeorolgy where he developed the use of hot wire anemometry for the direct measurement of the turbulent fluxes of heat, water vapour and momentum.[6] This research was presented at the International Symposium on Atmospheric Turbulence in the Boundary Layer at MIT in 1951. In the discussion following Swinbank's presentation, one participant commented thus: "We have had to rely too much on hypothesis and conjecture in the processing and analysis of micrometeorological observations due to the fact that most observers did not measure the totality of atmospheric elements. I want to complement the group in Australia for the collection of the complete, detailed and accurate data on eddy fluxes."[10] Swinbank was an advocate for large scale atmospheric field experiments and between 1962 and 1964 established research projects in Kerang (Vic) and later on in Hay (NSW) for the recording of accurate data for later analysis.[2] These experiments recorded wind, temperature and moisture in the atmosphere, initially to a height of 16 metres, and later to 32 metres. In the later experiments at Hay, measurements were taken up to 1000 metres.[5] Swinbank was instrumental in the establishment of ozone monitoring in Australia.[6] NCAR 1969–1973 In 1969, Swinbank took leave of absence from the CSIRO to work at the NCAR in Colorado. He was free to pursue his work on turbulence and the boundary layer. In 1971, he was appointed Director of the National Hail Research Experiment (NHRE), and consequently resigned from the CSIRO. The NHRE developed in response to the impacts of hailstorms on crops in the Mid-West. By taking detailed atmospheric measurements within the storm cell, it was hoped that a storm could be seeded with silver oxide to break up the moisture into rain or small hail and so prevent the development of large destructive hailstones.[11] The project ran for five years until 1976. Personal life In 1939 Swinbank married Ivy Hook[12] and they had a daughter, Susan. Both Ivy and Susan died in 1940 within a day of each other.[13] In 1942 he met Angela Pinney while they were both working at RAF Dunstable in Bedfordshire and they married in December of that year. In 1948, Swinbank, his wife and two children moved to Australia and lived in Mt Eliza near Melbourne. He came to Australia to work in the newly founded CSIRO. They had four more children. In 1969, Swinbank took up a position at NCAR in Colorado and he died in Boulder on 28 December 1973. In May 1974, the Boundary-Layer Meteorology journal published a memorial edition to which colleagues from around the world contributed articles.[14] Awards • 1941 Elected Fellow of the Royal Meteorological Society[15] • 1968 Winner of the Buchan Prize by the Royal Meteorological Society[6] (in conjunction with A. J. Dyer) • 1970 Elected Fellow of the Australian Academy of Science[2] • Fellow of the Australian Institute of Physics[16] Selected publications • Priestley, C. H. B and Swinbank, W. C. (1947) Vertical transport of heat by turbulence in the atmosphere Proc. R. Soc. Lond. A189: 543–56 • Swinbank, W. C. (1948) Note on the formation of fog over a snow surface. Quarterly Journal of the Royal Meteorological Society. 1948; Vol 74(Issue 321-322):406-407. • Swinbank, W. C. (1949) Prediction diagrams for radiation fog. Durham theses, Durham University. Available at Durham E-Theses Online: http://etheses.dur.ac.uk/10469/ • Swinbank, W. C. (1951) The measurement of vertical transfer of heat and water vapor by eddies in the lower atmosphere. Journal of Meteorology. 1951; 8(3):135-145 • Swinbank, W. C. (1956) The physics of natural evaporation. Australian Meteorological Magazine. 1956; (14):58-59. • Swinbank, W. C. (1960) Wind profile in thermally stratified flow. Nature. 1960; 186(4723):463-464. • Swinbank, W. C. (1963) Long‐wave radiation from clear skies. Quarterly Journal of the Royal Meteorological Society, 89(381):339-348. doi: 10.1002/QJ.49708938105 • Swinbank, W. C and Dyer A.J. (1967) An experimental study in micro-meteorology. Quarterly Journal of the Royal Meteorological Society. Vol 93 (Issue 398):494-500. • Swinbank, W. C. & Dyer, A. J. (1968). Micrometeorological expeditions 1962-1964. Melbourne: C.S.I.R.O • Swinbank, W. C. (1970). Structure of wind and the shearing stress in the planetary boundary layer. Archiv Für Meteorologie, Geophysik Und Bioklimatologie, Serie A, 19, 1-12. Notes 1. "FreeBMD Entry Info". www.freebmd.org.uk. Retrieved 21 December 2022. 2. "William Christopher Swinbank 1913-1973". Australian Academy of Science. Retrieved 20 December 2022. 3. Petterssen, S (2001). Weathering the Storm: Sverre Petterssen, the D-Day Forecast and the Rise of Modern Meteorology. Ed J R Fleming. p. 128. 4. "The Role of the Met Office in the D-Day Landings" (PDF). 5. Garratt, John; Angus, David; Holper, Paul (1998). Winds of Change:Fifty Years of Achievements in the CSIRO Division of Atmospheric Research 1946-1996. CSIRO Publishing. 6. Dyer, A. J. (1974). "Obituary — William Christopher Swinbank". Boundary-Layer Meteorology. 6 (3–4): I. doi:10.1007/BF02137671. ISSN 0006-8314. S2CID 189836328. 7. Ward, Colin (13 January 2015). "Charles Henry Brian (Bill) Priestley (1915-1998)". CSIROpedia. Retrieved 23 December 2022. 8. Berg, Peter; Huettel, Markus; Glud, Ronnie N.; Reimers, Clare E.; Attard, Karl M. (January 2022). "Aquatic Eddy Covariance: The Method and Its Contributions to Defining Oxygen and Carbon Fluxes in Marine Environments". Annual Review of Marine Science. 14: 431–455. doi:10.1146/annurev-marine-042121-012329. PMID 34587456. S2CID 238230170. 9. Centre for Transformative Innovation, Swinburne University of Technology. "CSIRO Division of Meteorological Physics - Corporate Body - Encyclopedia of Australian Science and Innovation". www.eoas.info. Retrieved 23 December 2022. 10. Hewson, Edgar Wendell (1952). "International Symposium of Turbulence in The Boundary Layer". 11. "NHRE Information release 1971". 12. "FreeBMD District Info". www.freebmd.org.uk. Retrieved 21 December 2022. 13. "Well-known sportswoman's death". Biggleswade Chronicle. 20 March 1940. Retrieved 31 December 2022 – via britishnewspaperarchive.co.uk. 14. "Boundary-Layer Meteorology journal volume 6 issue3". 15. "Proceedings of the Royal Meteorological Society". Quarterly Journal of the Royal Meteorological Society. 68 (293): 76–83. December 1941. doi:10.1002/qj.49706829310.{{cite journal}}: CS1 maint: url-status (link) 16. "Australian Institute of Physics" (PDF). External links • List of publications by W C Swinbank while he was working at the CSIRO • National Hail Research Experiment Film (1974) 18 mins Swinbank was the Technical Advisor. • National Hail Research Experiment Summary reports (1970 -1976) Authority control National • Australia People • Trove
William Davidson Niven Sir William Davidson Niven KCB FRS (24 March 1842 – 29 May 1917) was a Scottish mathematician and electrical engineer. Sir William Davidson Niven Born William Davidson Niven (1842-03-24)March 24, 1842 Peterhead, Scotland DiedMay 29, 1917(1917-05-29) (aged 75) Sidcup; England Resting placePeterhead Old Churchyard 57.504068°N 1.790279°W / 57.504068; -1.790279 Alma materUniversity of Aberdeen Trinity College, Cambridge Known forEditor of James Clerk Maxwell's papers Scientific career FieldsMathematics InstitutionsRoyal Naval College, Greenwich InfluencesJames Clerk Maxwell InfluencedAlfred North Whitehead After an early teaching career at Cambridge, Niven was Director of Studies at the Royal Naval College, Greenwich, for thirty years. Life Niven was born at Peterhead in Aberdeenshire, one of five notable mathematician brothers: Charles and James the best known. He graduated first from the University of Aberdeen,[1] then from Trinity College, Cambridge, where he was a Wrangler and was elected a Fellow of his college. In 1882 Niven became Director of Studies at the Royal Naval College, Greenwich, succeeding Thomas Archer Hirst.[2] He was appointed a Companion of the Order of the Bath (Civil division) in Queen Victoria's Diamond Jubilee Honours of 1897. He retired in 1903, when he was knighted by being appointed a Knight Commander of the Order of the Bath.[2] Niven was a colleague of James Clerk Maxwell (1831–1879), whose scientific papers he edited after his death. Among Niven's students was Alfred North Whitehead, to whom he taught mathematics, by instructing him in the physics of Maxwell.[3] In retirement Niven lived at Eastburn, Sidcup, Kent,[2] where he died in 1917.[1] Major publications Niven edited works by J. C. Maxwell: • 1881: A Treatise on Electricity and Magnetism, 2nd edition • 1890: The Scientific Papers of James Clerk Maxwell from Biodiversity Heritage Library Notes 1. Ronny Desmet, Michel Weber, Whitehead. The Algebra of Metaphysics (2010), p. 116 2. 'NIVEN, Sir William Davidson', in Who Was Who 1916–1928 (London: A. & C. Black, 1992 reprint, ISBN 0-7136-3143-0) 3. Frank Northen Magill, Alison Aves (1999) Dictionary of World Biography, p. 3,965 Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Czech Republic • Netherlands Academics • CiNii • MathSciNet • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
William Duke (mathematician) William Drexel Duke (born 1958) is an American mathematician specializing in number theory. Duke studied at the University of New Mexico and then at New York University (Courant Institute), from which he received his Ph.D. in 1986 under the direction of Peter Sarnak. After a postdoctoral stint at the University of California, San Diego he joined the faculty of Rutgers University, where he stayed until becoming a Professor of Mathematics at the University of California, Los Angeles. Since 2015, he has been Chair of the mathematics department at UCLA.[1] Honors Duke gave an Invited Address at the 1998 International Congress of Mathematicians in Berlin.[2][3][4] Duke gave an AMS Invited Address at a 2001 Fall sectional meeting of the American Mathematical Society in Irvine, California.[5] He was selected as a fellow of the American Mathematical Society in 2016 "for contributions to analytic number theory and the theory of automorphic forms".[6] Duke is an Editorial Board Member for the book series "Monographs in Number Theory" published by World Scientific.[7] Students • Amanda Folsom Selected publications • Duke, W. (1988) Hyperbolic distribution problems and half-integral weight Maass forms, Inventiones Mathematicae, 92, 73–90. • Duke, W., Schulze-Pillot, R. (1993) Representation of integers by positive ternary quadratic forms and equidistribution of lattice points on ellipsoids, Duke Mathematical Journal, 71, 143–179. • Duke, W., Friedlander, J., Iwaniec, H. (1993) Bounds for automorphic L-functions, Inventiones Mathematicae, 112, 1–8. • Duke, W., Friedlander, J., Iwaniec, H. (1994) Bounds for automorphic L-functions II, Inventiones Mathematicae, 115, 219–239. • Duke, W., Friedlander, J., Iwaniec, H. (1995), Equidistribution of roots of a quadratic congruence to prime moduli, Annals of Mathematics, 141, 423–441. • Duke, W. (1995) The critical order of vanishing of automorphic L-functions with large level, Inventiones Mathematicae, 119, 165–174. • Duke, W., Kowalski, E. (2000), A problem of Linnik for elliptic curves and mean-value estimates for automorphic representations. With an appendix by Dinakar Ramakrishnan, Inventiones Mathematicae, 139, 1–39. • Duke, W., Friedlander, J., Iwaniec, H. (2002), The subconvexity problem for Artin L-functions, Inventiones Mathematicae, 149, 489–577. References 1. Faculty profile, Department of Mathematics, University of California, Los Angeles. Accessed May 15, 2016. 2. ICM Plenary and Invited Speakers since 1897, International Mathematical Union. Accessed May 15, 2016. 3. Invited speakers for ICM-98, Notices of the American Mathematical Society, 45 (1998), no. 5, p. 621 4. Duke, William (1998). "Bounds for arithmetic multiplicities". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 163–172. 5. AMS Sectional Meeting Invited Addresses, American Mathematical Society. Accessed May 15, 2016 6. 2016 Class of the Fellows of the AMS, American Mathematical Society. Accessed May 15, 2016. 7. Monographs in Number Theory, (series info), World Scientific. Accessed May 15, 2016 External links • William Duke's webpage at UCLA • William Drexel Duke at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Norway • France • BnF data • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Elwood Byerly William Elwood Byerly (13 December 1849 – 20 December 1935) was an American mathematician at Harvard University where he was the "Perkins Professor of Mathematics". He was noted for his excellent teaching and textbooks.[1] Byerly was the first to receive a Ph.D. from Harvard, and Harvard's chair "William Elwood Byerly Professor in Mathematics" is named after him. Byerly Hall in Radcliffe Yard, Radcliffe Institute for Advanced Study, Harvard University is also named for him. William Elwood Byerly Born1849 Died1935 (aged 85–86) Textbooks Among the textbooks he wrote are: • Elements of the Differential Calculus (1879) • Harmonic Functions (1906) • Problems in Differential Calculus • Introduction to the Calculus of Variations (1917) • Elements of the Integral Calculus (1881) • An Elementary Treatise on Fourier's Series (1893) • An Introduction to the Use of Generalized Coordinates in Mechanics and Physics (1916) References • J. L. Coolidge, "William Elwood Byerly—In memoriam", Bull. Amer. Math. Soc. Volume 42, Number 5 (1936), pp. 295–298. • Edwin H. Hall, "William Elwood Byerly (1849-1935)", Proceedings of the American Academy of Arts and Sciences, Vol. 71, No. 10 (Mar., 1937), pp. 492–494. Notes 1. National Academy of Sciences (2002). Biographical Memoirs. National Academies Press. p. 255. ISBN 978-0-309-08476-5. Authority control International • FAST • ISNI • VIAF National • France • BnF data • Germany • Israel • United States • Netherlands Academics • CiNii • Mathematics Genealogy Project Other • IdRef
William Esson William Esson, FRS (17 May 1838 – 28 August 1916) was a British mathematician. William Esson Born1838 Dundee, Scotland Died28 August 1916(1916-08-28) (aged 78) Abingdon, England NationalityBritish Alma materSt John's College, Oxford Known forMathematics of the rate of chemical change Scientific career InstitutionsUniversity of Oxford Early life He was born in Carnoustie, Scotland.[1] Esson attended St John's College, Oxford. Career He then became a Fellow of Merton College.[2] In 1892, he became the Savilian Professor of Geometry at the University of Oxford, based at New College. He worked on problems in chemistry with Augustus George Vernon Harcourt. In 1869 he was elected a Fellow of the Royal Society and in 1895 delivered, jointly with Harcourt, their Bakerian Lecture on the Laws of Connexion between the Conditions of a Chemical Change and its Amount. III. Further Researches on the Reaction of Hydrogen Dioxide and Hydrogen Iodide.[3] He was on the governing body of Abingdon School until 1900.[4] Personal life In 1874, Esson leased 13 Bradmore Road in North Oxford.[5] He died in Abingdon, England.[6] References 1. Obituary notice, Fellow: Esson, William, Monthly Notices of the Royal Astronomical Society, Vol. 77, p.299, 1917MNRAS..77..299., The SAO/NASA Astrophysics Data System 2. Obituary, Royal Society of Chemistry 3. "Fellow Details". Royal Society. Retrieved 20 January 2017. 4. "School Notes" (PDF). The Abingdonian. 5. Hinchcliffe, Tanis (1992). North Oxford. New Haven & London: Yale University Press. p. 220. ISBN 0-14-071045-0. 6. GRO Register of Deaths: Deaths SEP 1916 2c 348 ABINGDON — Willian Esson, aged 78 Savilian Professors Chairs established by Sir Henry Savile Savilian Professors of Astronomy • John Bainbridge (1620) • John Greaves (1642) • Seth Ward (1649) • Christopher Wren (1661) • Edward Bernard (1673) • David Gregory (1691) • John Caswell (1709) • John Keill (1712) • James Bradley (1721) • Thomas Hornsby (1763) • Abraham Robertson (1810) • Stephen Rigaud (1827) • George Johnson (1839) • William Donkin (1842) • Charles Pritchard (1870) • Herbert Turner (1893) • Harry Plaskett (1932) • Donald Blackwell (1960) • George Efstathiou (1994) • Joseph Silk (1999) • Steven Balbus (2012) Savilian Professors of Geometry • Henry Briggs (1619) • Peter Turner (1631) • John Wallis (1649) • Edmond Halley (1704) • Nathaniel Bliss (1742) • Joseph Betts (1765) • John Smith (1766) • Abraham Robertson (1797) • Stephen Rigaud (1810) • Baden Powell (1827) • Henry John Stephen Smith (1861) • James Joseph Sylvester (1883) • William Esson (1897) • Godfrey Harold Hardy (1919) • Edward Charles Titchmarsh (1931) • Michael Atiyah (1963) • Ioan James (1969) • Richard Taylor (1995) • Nigel Hitchin (1997) • Frances Kirwan (2017) University of Oxford portal Authority control International • ISNI • VIAF National • United States Other • SNAC
William Beckner (mathematician) William Beckner (born September 15, 1941) is an American mathematician, known for his work in harmonic analysis, especially geometric inequalities. He is the Paul V. Montgomery Centennial Memorial Professor in Mathematics at The University of Texas at Austin. William Beckner Born (1941-09-15) September 15, 1941 Kirksville, Missouri, USA NationalityAmerican Alma materPrinceton University University of Missouri (Columbia) Known forSharp Inequalities Babenko–Beckner inequality AwardsSalem Prize (1975) Scientific career FieldsMathematics InstitutionsUniversity of Texas, Austin Princeton University University of Chicago Doctoral advisorElias Stein Doctoral studentsEmanuel Carneiro Education Beckner earned his Bachelor of Science in physics from the University of Missouri in Columbia, Missouri in 1963, where he became a member of the Phi Beta Kappa Society. He later earned his Ph.D. in mathematics at Princeton University in Princeton, New Jersey, where his doctoral adviser was Elias Stein. He also completed some postgraduate work in mathematics under adviser A.P. Calderon at the University of Chicago. Awards and honors • Salem Prize • Sloan Fellow • Fellow of the American Mathematical Society.[1] Selected publications • Beckner, William (1975). "Inequalities in Fourier Analysis". Annals of Mathematics. 102 (1): 159–182. doi:10.2307/1970980. JSTOR 1970980. • Beckner, William (1993). "Sharp Sobolev inequalities on the sphere and the Moser–Trudinger inequality". Annals of Mathematics. 138 (1): 213–242. doi:10.2307/2946638. JSTOR 2946638. • Beckner, William (1995). "Geometric inequalities in Fourier analysis". Essays on Fourier Analysis in Honor of Elias M. Stein. Princeton University Press. pp. 36–68. ISBN 9780691603650. JSTOR j.ctt7ztk2g.5. See also • Babenko–Beckner inequality • Hirschman uncertainty References 1. List of Fellows of the American Mathematical Society, retrieved 2012-11-10. External links • Beckner's home page • William Beckner at the Mathematics Genealogy Project Authority control International • ISNI • VIAF National • Israel • United States • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • IdRef
William Floyd (mathematician) William J. Floyd is an American mathematician specializing in topology. He is currently a professor at Virginia Polytechnic Institute and State University. Floyd received a PhD in mathematics from Princeton University 1978 under the direction of William Thurston.[1] Mathematical contributions Most of Floyd's research is in the areas of geometric topology and geometric group theory. Floyd and Allen Hatcher classified all the incompressible surfaces in punctured-torus bundles over the circle.[2] In a 1980 paper[3] Floyd introduced a way to compactify a finitely generated group by adding to it a boundary which came to be called the Floyd boundary.[4][5] Floyd also wrote a number of joint papers with James W. Cannon and Walter R. Parry exploring a combinatorial approach to the Cannon conjecture[6][7][8] using finite subdivision rules. This represents one of the few plausible lines of attack of the conjecture.[9] References 1. William J. Floyd. Mathematics Genealogy Project. Accessed February 6, 2010 2. Floyd, W.; Hatcher, A. Incompressible surfaces in punctured-torus bundles. Topology and its Applications, vol. 13 (1982), no. 3, pp. 263–282 3. Floyd, William J., Group completions and limit sets of Kleinian groups. Inventiones Mathematicae, vol. 57 (1980), no. 3, pp. 205–218 4. Karlsson, Anders, Free subgroups of groups with nontrivial Floyd boundary. Communications in Algebra, vol. 31 (2003), no. 11, pp. 5361–5376. 5. Buckley, Stephen M.; Kokkendorff, Simon L., Comparing the Floyd and ideal boundaries of a metric space. Transactions of the American Mathematical Society, vol. 361 (2009), no. 2, pp. 715–734 6. J. W. Cannon, W. J. Floyd, W. R. Parry. Sufficiently rich families of planar rings. Annales Academiæ Scientiarium Fennicæ. Mathematica. vol. 24 (1999), no. 2, pp. 265–304. 7. J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196. 8. J. W. Cannon, W. J. Floyd, W. R. Parry. Expansion complexes for finite subdivision rules. I. Conformal Geometry and Dynamics, vol. 10 (2006), pp. 63–99. 9. Ilya Kapovich, and Nadia Benakli, in Boundaries of hyperbolic groups, Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), pp. 39–93, Contemporary Mathematics, 296, American Mathematical Society, Providence, RI, 2002, ISBN 0-8218-2822-3 MR1921706; pp. 63–64 External links • William Floyd at the Mathematics Genealogy Project • William Floyd's webpage, Department of Mathematics, Virginia Polytechnic Institute and State University Authority control: Academics • DBLP • MathSciNet • Mathematics Genealogy Project • zbMATH
William Francis Pohl William Francis Pohl (16 September 1937 – 9 December 1988)[1] was an American mathematician, specializing in differential geometry and known for the Clifton–Pohl torus. Pohl received from the University of Chicago his B.A in 1957 and his M.A.1958. He completed his Ph.D. at Berkeley in 1961 under the direction of Shiing-Shen Chern with dissertation Differential Geometry of Higher Order.[2] His dissertation was published in 1962 in the journal Topology[3] and has received over 120 citations in the mathematical literature. He was a member of the mathematics faculty at the University of Minnesota from September 1964 until his untimely death. Pohl engaged in a famous controversy arguing against Francis Crick[4] but, in view of additional empirical evidence, conceded about 1979 or 1980 that Crick was correct.[5] Pohl sang liturgical music in Catholic religious services and wrote an article in 1966 from which the journal Sacred Music published an excerpt in 2011.[6] In the early 1970s, Dr. William F. Pohl, a professor of mathematics at the University of Minnesota, sang the Gregorian chant, mostly solo, while developing a small schola of Chorale volunteers to assist him. Dr. Pohl guided the chant during the aftermath of the Second Vatican Council when all the liturgical books were being revised ? no small task, but as some may recall, he was no small man. By 1975, in cooperation with Monsignor Richard J. Schuler, pastor of Saint Agnes, and Harold Hughesdon, its master of ceremonies, Dr. Pohl, joined by a number of dedicated volunteers, had begun the custom of singing Sunday vespers weekly and the full office of Tenebrae during Holy Week. Organist David Bevan arrived from England in 1976 to accompany the Chorale, and he assumed directorship of the Gregorian chant after Dr. Pohl's retirement in 1977.[7] William Pohl later married Hildegard Bastian (now Hildegard Pohl), and fathered 5 children, Annetta Pohl, Agatha Pohl, Agnes Pohl, Lawrence Pohl, and John Pohl. Selected publications • Pohl, William F. (1966). "Connexions in differential geometry of higher order". Transactions of the American Mathematical Society. 125 (2): 310–325. doi:10.1090/s0002-9947-1966-0203628-1. JSTOR 1994357. • "The self-linking number of a closed space curve (Gauss integral formula treated for disjoint closed space curves linking number)" (PDF). Journal of Mathematics and Mechanics. 17: 975–985. 1968. • Pohl, William F. (1968). "Some integral formulas for space curves and their generalization". American Journal of Mathematics. 90 (4): 1321–1345. doi:10.2307/2373302. JSTOR 2373302. • with T. F. Banchoff: Banchoff, Thomas F.; Pohl, William F. (1971). "A generalization of the isoperimetric inequality". Journal of Differential Geometry. 6 (2): 175–192. doi:10.4310/jdg/1214430403. MR 0305319. • with John Alvord Little: Little, John A.; Pohl, William F. (1971). "On tight immersions of maximal codimension" (PDF). Inventiones Mathematicae. 13 (3): 179–204. Bibcode:1971InMat..13..179L. doi:10.1007/BF01404629. hdl:2027.42/46589. S2CID 54785966. • with Nicolaas H. Kuiper: Kuiper, Nicolaas H.; Pohl, William F. (1977). "Tight topological embeddings of the real projective plane in E5 ". Inventiones Mathematicae. 42 (1): 177–199. Bibcode:1977InMat..42..177K. doi:10.1007/BF01389787. S2CID 120800935. • Pohl, William F. (1981). "The probability of linking of random closed curves". Geometry Symposium Utrecht 1980. Lecture Notes in Mathematics. Vol. 894. Springer Berlin Heidelberg. pp. 113–126. doi:10.1007/BFb0096227. ISBN 978-3-540-11167-2. References 1. Minnesota Historical Society Death Certificate Search, 1904–2001 2. William Francis Pohl at the Mathematics Genealogy Project 3. Pohl, W. F. (1962). "Differential geometry of higher order". Topology. 1 (3): 169–211. doi:10.1016/0040-9383(62)90103-9. hdl:10338.dmlcz/101530. 4. Pohl, W. F.; Roberts, George W. (October 1978). "Topological consideration in the theory of replication of DNA". Journal of Mathematical Biology. 6 (4): 383–402. doi:10.1007/BF02463003. PMID 750633. S2CID 29082243. 5. Pohl, W. F. (March 1980). "DNA and differential geometry". The Mathematical Intelligencer. 3 (1): 20–27. doi:10.1007/BF03023391. S2CID 119798941. 6. Pohl, W. F. (2011). "Liturgical Music and the Liturgical Movement (1966)". Sacred Music. 136 (3): 37. 7. ""Sacred Choral Music" at Saint Agnes, Minneapolis, 2) The Schola Cantorum at Saint Agnes". catholicforum.com. 19 July 2006. Authority control International • VIAF National • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH
William Frederick Eberlein William Frederick Eberlein (June 25, 1917, Shawano, Wisconsin – 1986, Rochester, New York) was an American mathematician, specializing in mathematical analysis and mathematical physics. Life Eberlein studied from 1936 to 1942 at the University of Wisconsin and at Harvard University, where he received in 1942 a PhD for the thesis Closure, Convexity, and Linearity in Banach Spaces under the direction of Marshall Stone.[1] He was married twice—to Mary Bernarda Barry and Patricia Ramsay James. He had four children with Mary Barry, including Patrick Barry Eberlein, another renowned mathematician. Patricia Ramsay James was a mathematician who moved into computer science as the field opened up; their one child is Kristen James Eberlein, the chair of the OASIS Darwin Information Typing Architecture Technical Committee. Work Eberlein had academic positions at the Institute for Advanced Study (1947–1948), at the University of Wisconsin (1948–1955), at Wayne State University (1955–1956), and from 1957 at the University of Rochester, where he remained for the rest of his career.[2] His doctoral students include William F. Donoghue, Jr.[3] and A. Wayne Wymore. Contributions He worked on functional analysis, harmonic analysis, ergodic theory, mean value theorems, and numerical integration. Eberlein also worked on spacetime models, internal symmetries in gauge theory, and spinors.[2] His name is attached to the Eberlein–Šmulian theorem in functional analysis[4] and the Eberlein compacta in topology.[5] References 1. William Frederick Eberlein at the Mathematics Genealogy Project 2. A Guide to the W. F. Eberlein Papers, 1936–1986, Briscoe Center for American History, University of Texas at Austin, retrieved 2014-06-19. 3. Gelbaum, Bernard Russell. "In Memoriam: William F. Donoghue, Jr". University of California. 4. Conway, John B. (1990), A Course in Functional Analysis, Graduate Texts in Mathematics, vol. 96, Springer, p. 163, ISBN 9780387972459. 5. Arhangel'skii, A. V. (2003), "Eberlein compacta", in Hart, K. P.; Nagata, Jun-iti; Vaughan, J. E. (eds.), Encyclopedia of General Topology, Elsevier, pp. 145–146, ISBN 9780080530864. Authority control International • FAST • ISNI • VIAF • 2 National • Germany • United States Academics • MathSciNet • Mathematics Genealogy Project • zbMATH People • Deutsche Biographie Other • SNAC • IdRef
William Fulton (mathematician) William Edgar Fulton (born August 29, 1939) is an American mathematician, specializing in algebraic geometry. William Fulton William Fulton at Oberwolfach in 2006 Born (1939-08-29) August 29, 1939 Naugatuck, Connecticut, US NationalityAmerican Alma materPrinceton University AwardsLeroy P. Steele Prize (2010) Scientific career FieldsMathematics InstitutionsUniversity of Michigan University of Chicago Brown University Brandeis University Doctoral advisorGerard Washnitzer Other academic advisorsJohn Milnor John Coleman Moore Goro Shimura Doctoral studentsRobert Lazarsfeld Education and career He received his undergraduate degree from Brown University in 1961 and his doctorate from Princeton University in 1966. His Ph.D. thesis, written under the supervision of Gerard Washnitzer, was on The fundamental group of an algebraic curve. Fulton worked at Princeton and Brandeis University from 1965 until 1970, when he began teaching at Brown. In 1987 he moved to the University of Chicago.[1] He is, as of 2011, a professor at the University of Michigan.[2] Fulton is known as the author or coauthor of a number of popular texts, including Algebraic Curves and Representation Theory. Awards and honors In 1996 he received the Steele Prize for mathematical exposition for his text Intersection Theory.[1] Fulton is a member of the United States National Academy of Sciences since 1997; a fellow of the American Academy of Arts and Sciences from 1998, and was elected a foreign member of the Royal Swedish Academy of Sciences in 2000.[3] In 2010, he was awarded the Steele Prize for Lifetime Achievement.[4] In 2012 he became a fellow of the American Mathematical Society.[5] Selected works • Algebraic Curves: An Introduction To Algebraic Geometry, with Richard Weiss. New York: Benjamin, 1969. Reprint ed.: Redwood City, CA, USA: Addison-Wesley, Advanced Book Classics, 1989. ISBN 0-201-51010-3. Full text online. • William Fulton (1998), Intersection theory, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 2 (2nd ed.), Berlin, New York: Springer-Verlag, doi:10.1007/978-1-4612-1700-8, ISBN 978-3-540-62046-4, MR 1644323 1st edn. 1984.[6] • Fulton, William; Harris, Joe (1991). Representation Theory, A First Course. Graduate Texts in Mathematics. Vol. 129. Berlin, New York: Springer-Verlag. ISBN 978-0-387-97495-8. MR 1153249. See also • Fulton–Hansen connectedness theorem References 1. Announcement of the 1996 Steele Prizes at the American Mathematical Society web site, accessed July 15, 2009. 2. University of Michigan mathematics department, alphabetical faculty listing, accessed November 13, 2011. 3. "View from the Chair's Office - Fulton Named Distinguished Mel Hochster University Professor". University of Michigan. Retrieved 3 April 2022. 4. http://www.ams.org/ams/press/steele-lifetime-2010.html AMS announcement of 2010 Steele Prize for Lifetime Achievement 5. List of Fellows of the American Mathematical Society, retrieved 2012-12-29. 6. Kleiman, Steven L. (1985). "Review: Intersection theory, by W. Fulton and Introduction to intersection theory in algebraic geometry, by W. Fulton" (PDF). Bull. Amer. Math. Soc. (N.S.). 12 (1): 137–143. doi:10.1090/s0273-0979-1985-15319-4. External links • Fulton's home page at the University of Michigan Authority control International • FAST • ISNI • VIAF National • France • BnF data • Catalonia • Germany • Israel • Belgium • United States • Japan • Czech Republic • Australia • Netherlands Academics • MathSciNet • Mathematics Genealogy Project • zbMATH Other • SNAC • IdRef
William George Horner William George Horner (9 June 1786 – 22 September 1837) was a British mathematician. Proficient in classics and mathematics, he was a schoolmaster, headmaster and schoolkeeper who wrote extensively on functional equations, number theory and approximation theory, but also on optics. His contribution to approximation theory is honoured in the designation Horner's method, in particular respect of a paper in Philosophical Transactions of the Royal Society of London for 1819. The modern invention of the zoetrope, under the name Daedaleum in 1834, has been attributed to him.[1][2][3] Horner died comparatively young, before the establishment of specialist, regular scientific periodicals. So, the way others have written about him has tended to diverge, sometimes markedly, from his own prolific, if dispersed, record of publications and the contemporary reception of them. Family life The eldest son of the Rev. William Horner, a Wesleyan minister, Horner was born in Bristol. He was educated at Kingswood School, a Wesleyan foundation near Bristol, and at the age of sixteen became an assistant master there. In four years he rose to be headmaster (1806), but left in 1809, setting up his own school, The Classical Seminary, at Grosvenor Place, Bath, which he kept until he died there 22 September 1837. He and his wife Sarah (1787?–1864) had six daughters and two sons. Physical sciences, optics Although Horner's article on the Dædalum (zoetrope) appeared in Philosophical Magazine only in January, 1834, he had published on Camera lucida as early as August, 1815. Mathematics Horner's name first appears in the list of solvers of the mathematical problems in The Ladies' Diary: or, Woman's Almanack for 1811, continuing in the successive annual issues until that for 1817. Up until the issue for 1816, he is listed as solving all but a few of the fifteen problems each year; several of his answers were printed, along with two problems he proposed. He also contributed to other departments of the Diary, not without distinction, reflecting the fact that he was known to be an all-rounder, competent in the classics as well as in mathematics. Horner was ever vigilant in his reading, as shown by his characteristic return to the Diary for 1821 in a discussion of the Prize Problem, where he reminds readers of an item in (Thomson's) Annals of Philosophy for 1817; several other problems in the Diary that year were solved by his youngest brother, Joseph. His record in The Gentleman's Diary: or, Mathematical Repository for this period is similar, including one of two published modes of proof in the volume for 1815 of a problem posed the previous year by Thomas Scurr (d. 1836), now dubbed the Butterfly theorem. Leaving the headmastership of Kingswood School would have given him more time for this work, while the appearance of his name in these publications, which were favoured by a network of mathematics teachers, would have helped publicize his own school. At this stage, Horner's efforts turned more to The Mathematical Repository, edited by Thomas Leybourn, but to contributing occasional articles, rather than the problem section, as well as to Annals of Philosophy, where Horner begins by responding to other contributors and works up to independent articles of his own; he has a careful style with acknowledgements and, more often than not, cannot resist adding further detail. Several contributions pave the way for, or are otherwise related to, his most celebrated mathematical paper, in Philosophical Transactions of the Royal Society of London in 1819, which was read by title at the closing meeting for the session on 1 July 1819, with Davies Gilbert in the Chair. The article, with significant editorial notes by Thomas Stephens Davies, was reprinted as a commemorative tribute in The Ladies' Diary for 1838. The issue of The Gentleman's Diary for that year contains a short obituary notice. A careful analysis of this paper has appeared recently in Craig Smoryński's History of Mathematics: A Supplement.[4] While a sequel was read before the Royal Society, publication was declined for Philosophical Transactions, having to await appearance in a sequence of parts in the first two volumes of The Mathematician in the mid-1840s, again largely at the instigation of T. S. Davies. However, Horner published on diverse topics in The Philosophical Magazine well into the 1830s. Davies mooted an edition of Horner's collected papers, but this project never came to fruition, partly on account of Davies' own early death. Contemporary reception Some idea of Horner's standing with his contemporaries is provided by exchanges in the issues of Annals of Philosophy for July and August, 1817. Thomas Thomson, in commending to an enquirer Euler's work on algebra, is under some impression that the English translation is by Horner.[5]: 86 Horner writes promptly to correct this,[5]: 170 supposing the translation to be the work of Peter Barlow. Thomson, a professor in Glasgow, might not have known that the translation, originally published as far back as 1789, was the work of Francis Horner MP, an Edinburgh native, who had died only that February. Peter Barlow and continued fractions When Peter Barlow wrote, in 1845, he remembered Davies, but not Horner, asking to borrow a book by Budan (both Davies and Horner were living in Bath at the time). Barlow also had a vague recollection that the material on approximations Horner sent him related to continued fractions, rather than what appeared in the Philosophical Transactions. Horner clearly held Barlow in high regard and it would have been natural for Horner to approach him to request both books and critical advice as Horner draws attention to Barlow's article in New Series of the Mathematical Repository[6] and in his survey of approximation methods in the following volume of the Repository (bound up in 1819). The anonymous reviewer for The Monthly Review in the issue for December, 1820 writes that he has seen Horner's letter to Barlow and that the letter confirms that Horner already had his method of approximation at that date (1818). The methods of both Barlow and Horner use a nesting of expressions akin to continued fractions. Horner was aware of Lagrange's use of continued fractions at least through his reading of Bonnycastle's Algebra which is also mentioned in the survey article in the Repository. Horner may have rewritten his paper either under guidance or of his own volition, with an eye to publication in Philosophical Transactions. Horner goes on to write on the use of continued fractions in the summation of series in Annals of Philosophy in 1826 and on their use in improvements they yield in the solution of equations in Quarterly Journal of Science, Literature and the Arts running over into 1827; he explicitly cites work of Lagrange. Barlow's memory of events may have been confused by the appearance of this later work. Publications • New and important combinations with the Camera Lucida, dated Bath, 15 August 1815, Annals of Philosophy, 6 (Oct. 1815), 281–283. • I. On Annuities. - II. Imaginary cube roots. - III. Roots of Binomials, dated Bath, 9 September 1816, Annals of Philosophy, 8 (Oct. 1816), 279–284. • Corrections of the paper inserted in the last number of the Annals, p. 279, dated Bath, 3 October 1816 Annals of Philosophy, 8 (Nov. 1816), 388–389. • Formulas for estimating the height of mountains, dated Bath, 13 February 1817 Annals of Philosophy, 9 (March, 1817), 251–252. • On cubic equations, dated Bath, 17 January 1817, Annals of Philosophy, 9 (May, 1817), 378–381. • Solution of the equation ψnx=x, Annals of Philosophy, 10 (Nov, 1817), 341–346. • On reversion of series, especially in connection with the equation ψα−1ψαx=x, dated Bath, 10 November 1817, Annals of Philosophy, 11 (Feb, 1818), 108–112. • On popular methods of approximation, dated Bath, 1819, Math. Rep. New Series, 4 (1819), Part II, 131–136. • 'A Tribute of Friendship,’ a poem addressed to his friend Thomas Fussell, appended to a 'Funeral Sermon on Mrs. Fussell,’ Bristol, 1820. • On algebraic transformation, as deducible from first principles, and connected with continuous approximations, and the theory of finite and fluxional differences, including some new modes of numerical solution, one of ten papers read at the table at the meeting of the Royal on 19 June 1823, immediately before the long vacation adjournment until 20 November 1823; one of the three papers of the set not published in Phil. Trans. that year; published in issues in the first two volumes of The Mathematician bound up in 1845 and 1847. • Extension of Theorem of Fermat, dated 26 December, Annals of Philosophy New Series, 11 (Feb, 1826), 81–83. • On the solutions of the Function ψzx and their limitations, Art 1-8, dated Bath, 11 February 1826, Annals of Philosophy New Series, 11 (March, 1826), 168–183; Art 9-17, ibid, 11 (April, 1826), 241–246. • Reply to Mr. Herapath, dated Bath, 2 April 1826, Annals of Philosophy New Series, 11 (May, 1826), 363 • On the use of continued fractions with unrestricted numerators in summation of series, Art 1-4, dated Bath, 24 April 1826, Annals of Philosophy New Series, 11 (June, 1826), 416–421; Art 5-6, ibid, 12 (July, 1826), 48–51. • 'Natural Magic,’ a pamphlet on optics dealing with virtual images, London, 1832. • On the properties of the Dædaleum, a new instrument of optical illusion, Phil. Mag., Ser. 3, 4 (Jan, 1834), 36-41. • On the autoptic spectrum of certain vessels within the eye, as delineated in shadow on the retina, Phil. Mag., Ser. 3, 4 (April, 1834), 262-271. • Considerations relative to an interesting case in equations, Phil. Mag., Ser. 3, 5 (Sept, 1834), 188-191. • On the signs of the trigonometrical lines, Phil. Mag., Ser. 3, 6 (Feb, 1836), 86-90. • On the theory of congeneric surd equations, Communicated by T. S. Davies, Phil. Mag., Ser. 3, 8 (Jan, 1836), 43-50. • New demonstration of an original proposition in the theory of numbers, Communicated by T. R. Phillips, Phil. Mag., Ser. 3, 11 (Nov, 1837), 456-459. • 'Questions for the Examination of Pupils on … General History,’ Bath, 1843, 12mo. A complete edition of Horner's works was promised by Thomas Stephens Davies, but never appeared. Other contemporary literature • P. Barlow, On the resolution of the irreducible case in cubic equations, Math. Rep., NS IV (1814), 46-57 [includes Table for the solution of the irreducible case in cubic equations (6pp.)]. • P. Barlow, A new method of approximating towards the roots of equations of all dimensions, Math. Rep., NS IV (1814), No. 12, 67–71. • T. Holdred, A New Method of Solving Equations with Ease and Expedition; by which the True Value of the Unknown Quantity is Found Without Previous Reduction. With a Supplement, Containing Two Other Methods of Solving Equations, Derived from the Same Principle(Richard Watts. Sold by Davis and Dickson, mathematical and philosophical booksellers, 17, St. Martin's-le-Grand; and by the author, 2, Denzel Street, Clare-Market, 1820), 56pp.. Notes 1. Zoetrope. EarlyCinema.com. Retrieved on 2011-10-11. 2. Glossary – Z. Wernernekes.de. Retrieved on 2011-10-11. 3. Philosophical magazine. Taylor & Francis. 1834. p. 36. 4. History of Mathematics: A Supplement. New York, NY: Springer. 2008. ISBN 9780387754802. esp. Chap. 7 5. Thomas Thomson, ed. (1817). "Annals of Philosophy". X. London. hdl:2027/mdp.39015066710818. {{cite journal}}: Cite journal requires \|journal= (help) 6. Thomas Leybourn, ed. (1814). "New Series of the Mathematical Repository". III. {{cite journal}}: Cite journal requires \|journal= (help) References • Register of Kingswood School, 1748-1922 (1923), p. 89. • 1861 Census External links • Bath: Births, Marriages and Deaths • Prof. Neville Fletcher, Research School of Physics and Engineering, ANU • Australian Academy of Science: Interview with Neville Fletcher Attribution This article incorporates text from a publication now in the public domain: "Horner, William George". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. Authority control: Academics • zbMATH
William Galbraith (mathematician) Rev William Galbraith (1786 – 27 October 1850) was a Scottish mathematician. He taught mathematics and nautical astronomy in Edinburgh, and took an interest in surveying work, becoming an advocate of the extension of the work of triangulating Great Britain.[1] Early life He was born at Greenlaw, Berwickshire.[2] Initially he was a schoolmaster. His pupil William Rutherford walked long distances to attend his school at Eccles. Subsequently, he moved to Edinburgh, and graduated A.M. at the University of Edinburgh in 1821.[3] Surveyor During the 1830s Galbraith became interested in the surveying problems of Scotland. In 1831 he pointed out that Arthur's Seat had a strongly magnetic peak.[4] In 1837 he pointed out the impact of anomalies in measurement, work that received recognition;[5] it was topical because of the 1836 geological map of Scotland by John MacCulloch, with which critics had found fault on topographical as well as geological grounds.[6] A paper on the locations of places on the River Clyde was recognised in 1837 by a gold medal, from the Society for the Encouragement of the Useful Arts for Scotland.[7] Galbraith followed with detailed Remarks on the Geographical Position of some Points on the West Coast of Scotland (1838).[8] Having made some accurate surveys of his own, he lobbied for further attention from the national survey.[1] Later life About 1832 Galbraith was licensed a minister by the presbytery of Dunse. He married Eleanor Gale in 1833.[3] Galbraith was buried with his wife in the north-east section of the Grange Cemetery in Edinburgh.[9] Works Galbraith's major works combined textbook material with mathematical tables: • Mathematical and Astronomical Tables (1827):[10] review.[11] • Trigonometrical Surveying, Levelling, and Railway Engineering (1842)[12] He edited John Ainslie's 1812 treatise on land surveying (1849),[13] and with William Rutherford revised John Bonnycastle's Algebra.[14] Notes 1. "National Museums of Scotland - Rule (Detail)". 2. "Biographical notices of Mr. William Galbraith". Monthly Notices of the Royal Astronomical Society. 11 (4): 86. 1851. Bibcode:1851MNRAS..11...86.. doi:10.1093/mnras/11.4.67a. 3. Royal Astronomical Society (1851). Memoirs. Society. pp. 194–. Retrieved 6 November 2012. 4. "The topographical, statistical, and historical gazetteer of Scotland". 1848. 5. Kaiserl. Akademie der Wissenschaften in Wien (1851). Almanach der Kaiserlichen Akademie der Wissenschaften für das Jahr . Harvard University. Wien : K.K. Hof- und Staatsdruckerei. 6. Cumming, David A. "MacCulloch, John". Oxford Dictionary of National Biography (online ed.). Oxford University Press. doi:10.1093/ref:odnb/17412. (Subscription or UK public library membership required.) 7. The Edinburgh Philosophical Journal. Constable. 1837. pp. 1–. Retrieved 6 November 2012. 8. The Edinburgh New Philosophical Journal. A. and C. Black. 1838. pp. 300–. Retrieved 4 May 2012. 9. "Monuments and monumental inscriptions in Scotland". 10. William Galbraith (1827). Mathematical and Astronomical Tables. Retrieved 4 May 2012. 11. Robert Jameson; Sir William Jardine; Henry Darwin Rogers (1827). The Edinburgh New Philosophical Journal: exhibiting a view of the progressive discoveries and improvements in the sciences and the arts. A. and C. Black. pp. 404–. Retrieved 4 May 2012. 12. "Trigonometrical surveying, levelling, and railway engineering". 1842. 13. "A treatise on land surveying [ed.] by W. Galbraith". 1849. 14. "Rutherford, William" . Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900. External links • Online Books page Authority control International • ISNI • VIAF National • Germany • Israel • United States
William Gardiner (mathematician) William Gardiner (died 1752) was an English mathematician.[1] His logarithmic tables of sines and tangents (Tables of logarithms, 1742) had various reprints and saw use by scientists and other mathematicians. Works • A literal exposition of two prophecies cited by St. Matthew out of the Old Testament (1726), reprinted as A literal exposition of two remarkable prophecies in the Old Testament : relating to Jesus Christ the Messiah (1728) • Practical surveying improved : or, land-measuring, according to the present most correct methods.: 1737 (erroneously given as 1773 in Worldcat), many reprints • Tables of logarithms, for all numbers from 1 to 102100, and for the sines and tangents to every ten seconds of each degree in the quadrant; as also, for the sines of the first 72 minutes to every single second; with other useful and necessary tables: 1742, reprinted many times, translated in French and Italian[2] References 1. "Gardiner, William, -1752". National Library of the Czech Republic. 2. Gardiner, William (1796). Tavole logaritmiche (in Italian). Florence: Pietro Allegrini & Gioacchino Pagani. Authority control International • ISNI • VIAF National • France • BnF data • Germany • Italy • Czech Republic Other • IdRef

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