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Wikipedia:Graph automorphism#0	In the mathematical field of graph theory, an automorphism of a graph is a form of symmetry in which the graph is mapped onto itself while preserving the edge–vertex connectivity. Formally, an automorphism of a graph G = (V, E) is a permutation σ of the vertex set V, such that the pair of vertices (u, v) form an edge if and only if the pair (σ(u), σ(v)) also form an edge. That is, it is a graph isomorphism from G to itself. Automorphisms may be defined in this way both for directed graphs and for undirected graphs. The composition of two automorphisms is another automorphism, and the set of automorphisms of a given graph, under the composition operation, forms a group, the automorphism group of the graph. In the opposite direction, by Frucht's theorem, all groups can be represented as the automorphism group of a connected graph – indeed, of a cubic graph. == Computational complexity == Constructing the automorphism group of a graph, in the form of a list of generators, is polynomial-time equivalent to the graph isomorphism problem, and therefore solvable in quasi-polynomial time, that is with running time 2 O ( ( log ⁡ n ) c ) {\displaystyle 2^{O((\log n)^{c})}} for some fixed c > 0 {\displaystyle c>0} . Consequently, like the graph isomorphism problem, the problem of finding a graph's automorphism group is known to belong to the complexity class NP, but not known to be in P nor to be NP-complete, and therefore may be NP-intermediate. The easier problem of testing whether a graph has any symmetries (nontrivial automorphisms), known as the graph automorphism problem, also has no known polynomial time solution. There is a polynomial time algorithm for solving the graph automorphism problem for graphs where vertex degrees are bounded by a constant. The graph automorphism problem is polynomial-time many-one reducible to the graph isomorphism problem, but the converse reduction is unknown. By contrast, hardness is known when the automorphisms are constrained in a certain fashion; for instance, determining the existence of a fixed-point-free automorphism (an automorphism that fixes no vertex) is NP-complete, and the problem of counting such automorphisms is ♯P-complete. == Algorithms, software and applications == While no worst-case polynomial-time algorithms are known for the general Graph Automorphism problem, finding the automorphism group (and printing out an irredundant set of generators) for many large graphs arising in applications is rather easy. Several open-source software tools are available for this task, including NAUTY, BLISS and SAUCY. SAUCY and BLISS are particularly efficient for sparse graphs, e.g., SAUCY processes some graphs with millions of vertices in mere seconds. However, BLISS and NAUTY can also produce Canonical Labeling, whereas SAUCY is currently optimized for solving Graph Automorphism. An important observation is that for a graph on n vertices, the automorphism group can be specified by no more than n − 1 {\displaystyle n-1} generators, and the above software packages are guaranteed to satisfy this bound as a side-effect of their algorithms (minimal sets of generators are harder to find and are not particularly useful in practice). It also appears that the total support (i.e., the number of vertices moved) of all generators is limited by a linear function of n, which is important in runtime analysis of these algorithms. However, this has not been established for a fact, as of March 2012. Practical applications of Graph Automorphism include graph drawing and other visualization tasks, solving structured instances of Boolean Satisfiability arising in the context of Formal verification and Logistics. Molecular symmetry can predict or explain chemical properties. == Symmetry display == Several graph drawing researchers have investigated algorithms for drawing graphs in such a way that the automorphisms of the graph become visible as symmetries of the drawing. This may be done either by using a method that is not designed around symmetries, but that automatically generates symmetric drawings when possible, or by explicitly identifying symmetries and using them to guide vertex placement in the drawing. It is not always possible to display all symmetries of the graph simultaneously, so it may be necessary to choose which symmetries to display and which to leave unvisualized. == Graph families defined by their automorphisms == Several families of graphs are defined by having certain types of automorphisms: An asymmetric graph is an undirected graph with only the trivial automorphism. A vertex-transitive graph is an undirected graph in which every vertex may be mapped by an automorphism into any other vertex. An edge-transitive graph is an undirected graph in which every edge may be mapped by an automorphism into any other edge. A symmetric graph is a graph such that every pair of adjacent vertices may be mapped by an automorphism into any other pair of adjacent vertices. A distance-transitive graph is a graph such that every pair of vertices may be mapped by an automorphism into any other pair of vertices that are the same distance apart. A semi-symmetric graph is a graph that is edge-transitive but not vertex-transitive. A half-transitive graph is a graph that is vertex-transitive and edge-transitive but not symmetric. A skew-symmetric graph is a directed graph together with a permutation σ on the vertices that maps edges to edges but reverses the direction of each edge. Additionally, σ is required to be an involution. Inclusion relationships between these families are indicated by the following table: == See also == Algebraic graph theory Distinguishing coloring == References == == External links == Weisstein, Eric W. "Graph automorphism". MathWorld.
Wikipedia:Graph energy#0	In mathematics, the energy of a graph is the sum of the absolute values of the eigenvalues of the adjacency matrix of the graph. This quantity is studied in the context of spectral graph theory. More precisely, let G be a graph with n vertices. It is assumed that G is a simple graph, that is, it does not contain loops or parallel edges. Let A be the adjacency matrix of G and let λ i {\displaystyle \lambda _{i}} , i = 1 , … , n {\displaystyle i=1,\ldots ,n} , be the eigenvalues of A. Then the energy of the graph is defined as: E ( G ) = ∑ i = 1 n \| λ i \| . {\displaystyle E(G)=\sum _{i=1}^{n}\|\lambda _{i}\|.} == References == Cvetković, Dragoš M.; Doob, Michael; Sachs, Horst (1980), Spectra of graphs, Pure and Applied Mathematics, vol. 87, New York: Academic Press Inc. [Harcourt Brace Jovanovich Publishers], ISBN 0-12-195150-2, MR 0572262. Gutman, Ivan (1978), "The energy of a graph", 10. Steiermärkisches Mathematisches Symposium (Stift Rein, Graz, 1978), Ber. Math.-Statist. Sekt. Forsch. Graz, vol. 103, pp. 1–22, MR 0525890. Gutman, Ivan (2001), "The energy of a graph: old and new results", Algebraic combinatorics and applications (Gößweinstein, 1999), Berlin: Springer, pp. 196–211, MR 1851951. Li, Xueliang; Shi, Yongtang; Gutman, Ivan (2012), Graph Energy, New York: Springer, ISBN 978-1-4614-4219-6.
Wikipedia:Graph of a function#0	In mathematics, the graph of a function f {\displaystyle f} is the set of ordered pairs ( x , y ) {\displaystyle (x,y)} , where f ( x ) = y . {\displaystyle f(x)=y.} In the common case where x {\displaystyle x} and f ( x ) {\displaystyle f(x)} are real numbers, these pairs are Cartesian coordinates of points in a plane and often form a curve. The graphical representation of the graph of a function is also known as a plot. In the case of functions of two variables – that is, functions whose domain consists of pairs ( x , y ) {\displaystyle (x,y)} –, the graph usually refers to the set of ordered triples ( x , y , z ) {\displaystyle (x,y,z)} where f ( x , y ) = z {\displaystyle f(x,y)=z} . This is a subset of three-dimensional space; for a continuous real-valued function of two real variables, its graph forms a surface, which can be visualized as a surface plot. In science, engineering, technology, finance, and other areas, graphs are tools used for many purposes. In the simplest case one variable is plotted as a function of another, typically using rectangular axes; see Plot (graphics) for details. A graph of a function is a special case of a relation. In the modern foundations of mathematics, and, typically, in set theory, a function is actually equal to its graph. However, it is often useful to see functions as mappings, which consist not only of the relation between input and output, but also which set is the domain, and which set is the codomain. For example, to say that a function is onto (surjective) or not the codomain should be taken into account. The graph of a function on its own does not determine the codomain. It is common to use both terms function and graph of a function since even if considered the same object, they indicate viewing it from a different perspective. == Definition == Given a function f : X → Y {\displaystyle f:X\to Y} from a set X (the domain) to a set Y (the codomain), the graph of the function is the set G ( f ) = { ( x , f ( x ) ) : x ∈ X } , {\displaystyle G(f)=\{(x,f(x)):x\in X\},} which is a subset of the Cartesian product X × Y {\displaystyle X\times Y} . In the definition of a function in terms of set theory, it is common to identify a function with its graph, although, formally, a function is formed by the triple consisting of its domain, its codomain and its graph. == Examples == === Functions of one variable === The graph of the function f : { 1 , 2 , 3 } → { a , b , c , d } {\displaystyle f:\{1,2,3\}\to \{a,b,c,d\}} defined by f ( x ) = { a , if x = 1 , d , if x = 2 , c , if x = 3 , {\displaystyle f(x)={\begin{cases}a,&{\text{if }}x=1,\\d,&{\text{if }}x=2,\\c,&{\text{if }}x=3,\end{cases}}} is the subset of the set { 1 , 2 , 3 } × { a , b , c , d } {\displaystyle \{1,2,3\}\times \{a,b,c,d\}} G ( f ) = { ( 1 , a ) , ( 2 , d ) , ( 3 , c ) } . {\displaystyle G(f)=\{(1,a),(2,d),(3,c)\}.} From the graph, the domain { 1 , 2 , 3 } {\displaystyle \{1,2,3\}} is recovered as the set of first component of each pair in the graph { 1 , 2 , 3 } = { x : ∃ y , such that ( x , y ) ∈ G ( f ) } {\displaystyle \{1,2,3\}=\{x:\ \exists y,{\text{ such that }}(x,y)\in G(f)\}} . Similarly, the range can be recovered as { a , c , d } = { y : ∃ x , such that ( x , y ) ∈ G ( f ) } {\displaystyle \{a,c,d\}=\{y:\exists x,{\text{ such that }}(x,y)\in G(f)\}} . The codomain { a , b , c , d } {\displaystyle \{a,b,c,d\}} , however, cannot be determined from the graph alone. The graph of the cubic polynomial on the real line f ( x ) = x 3 − 9 x {\displaystyle f(x)=x^{3}-9x} is { ( x , x 3 − 9 x ) : x is a real number } . {\displaystyle \{(x,x^{3}-9x):x{\text{ is a real number}}\}.} If this set is plotted on a Cartesian plane, the result is a curve (see figure). === Functions of two variables === The graph of the trigonometric function f ( x , y ) = sin ⁡ ( x 2 ) cos ⁡ ( y 2 ) {\displaystyle f(x,y)=\sin(x^{2})\cos(y^{2})} is { ( x , y , sin ⁡ ( x 2 ) cos ⁡ ( y 2 ) ) : x and y are real numbers } . {\displaystyle \{(x,y,\sin(x^{2})\cos(y^{2})):x{\text{ and }}y{\text{ are real numbers}}\}.} If this set is plotted on a three dimensional Cartesian coordinate system, the result is a surface (see figure). Oftentimes it is helpful to show with the graph, the gradient of the function and several level curves. The level curves can be mapped on the function surface or can be projected on the bottom plane. The second figure shows such a drawing of the graph of the function: f ( x , y ) = − ( cos ⁡ ( x 2 ) + cos ⁡ ( y 2 ) ) 2 . {\displaystyle f(x,y)=-(\cos(x^{2})+\cos(y^{2}))^{2}.} == See also == == References == == Further reading == == External links == Weisstein, Eric W. "Function Graph." From MathWorld—A Wolfram Web Resource.
Wikipedia:Greedy algorithm for Egyptian fractions#0	In mathematics, the greedy algorithm for Egyptian fractions is a greedy algorithm, first described by Fibonacci, for transforming rational numbers into Egyptian fractions. An Egyptian fraction is a representation of an irreducible fraction as a sum of distinct unit fractions, such as ⁠5/6⁠ = ⁠1/2⁠ + ⁠1/3⁠. As the name indicates, these representations have been used as long ago as ancient Egypt, but the first published systematic method for constructing such expansions was described in 1202 in the Liber Abaci of Leonardo of Pisa (Fibonacci). It is called a greedy algorithm because at each step the algorithm chooses greedily the largest possible unit fraction that can be used in any representation of the remaining fraction. Fibonacci actually lists several different methods for constructing Egyptian fraction representations. He includes the greedy method as a last resort for situations when several simpler methods fail; see Egyptian fraction for a more detailed listing of these methods. The greedy method, and extensions of it for the approximation of irrational numbers, have been rediscovered several times by modern mathematicians, earliest and most notably by J. J. Sylvester (1880) A closely related expansion method that produces closer approximations at each step by allowing some unit fractions in the sum to be negative dates back to Lambert (1770). The expansion produced by this method for a number x {\displaystyle x} is called the greedy Egyptian expansion, Sylvester expansion, or Fibonacci–Sylvester expansion of x {\displaystyle x} . However, the term Fibonacci expansion usually refers, not to this method, but to representation of integers as sums of Fibonacci numbers. == Algorithm and examples == Fibonacci's algorithm expands the fraction x / y {\displaystyle x/y} to be represented, by repeatedly performing the replacement x y = 1 ⌈ y x ⌉ + ( − y ) mod x y ⌈ y x ⌉ {\displaystyle {\frac {x}{y}}={\frac {1}{\left\lceil {\frac {y}{x}}\right\rceil }}+{\frac {(-y){\bmod {x}}}{y\left\lceil {\frac {y}{x}}\right\rceil }}} (simplifying the second term in this replacement as necessary). For instance: 7 15 = 1 3 + 2 15 = 1 3 + 1 8 + 1 120 . {\displaystyle {\frac {7}{15}}={\frac {1}{3}}+{\frac {2}{15}}={\frac {1}{3}}+{\frac {1}{8}}+{\frac {1}{120}}.} in this expansion, the denominator 3 of the first unit fraction is the result of rounding ⁠15/7⁠ up to the next larger integer, and the remaining fraction ⁠2/15⁠ is the result of simplifying ⁠−15 mod 7/15 × 3⁠ = ⁠6/45⁠. The denominator of the second unit fraction, 8, is the result of rounding ⁠15/2⁠ up to the next larger integer, and the remaining fraction ⁠1/120⁠ is what is left from ⁠7/15⁠ after subtracting both ⁠1/3⁠ and ⁠1/8⁠. As each expansion step reduces the numerator of the remaining fraction to be expanded, this method always terminates with a finite expansion; however, compared to ancient Egyptian expansions or to more modern methods, this method may produce expansions that are quite long, with large denominators. For instance, this method expands 5 121 = 1 25 + 1 757 + 1 763 309 + 1 873 960 180 913 + 1 1 527 612 795 642 093 418 846 225 , {\displaystyle {\frac {5}{121}}={\frac {1}{25}}+{\frac {1}{757}}+{\frac {1}{763\,309}}+{\frac {1}{873\,960\,180\,913}}+{\frac {1}{1\,527\,612\,795\,642\,093\,418\,846\,225}},} while other methods lead to the much better expansion 5 121 = 1 33 + 1 121 + 1 363 . {\displaystyle {\frac {5}{121}}={\frac {1}{33}}+{\frac {1}{121}}+{\frac {1}{363}}.} Wagon (1991) suggests an even more badly-behaved example, ⁠31/311⁠. The greedy method leads to an expansion with ten terms, the last of which has over 500 digits in its denominator; however, ⁠31/311⁠ has a much shorter non-greedy representation, ⁠1/12⁠ + ⁠1/63⁠ + ⁠1/2799⁠ + ⁠1/8708⁠. == Sylvester's sequence and closest approximation == Sylvester's sequence 2, 3, 7, 43, 1807, ... (OEIS: A000058) can be viewed as generated by an infinite greedy expansion of this type for the number 1, where at each step we choose the denominator ⌊ ⁠y/x⁠ ⌋ + 1 instead of ⌈ ⁠y/x⁠ ⌉. Truncating this sequence to k terms and forming the corresponding Egyptian fraction, e.g. (for k = 4) 1 2 + 1 3 + 1 7 + 1 43 = 1805 1806 {\displaystyle {\frac {1}{2}}+{\frac {1}{3}}+{\frac {1}{7}}+{\frac {1}{43}}={\frac {1805}{1806}}} results in the closest possible underestimate of 1 by any k-term Egyptian fraction. That is, for example, any Egyptian fraction for a number in the open interval (⁠1805/1806⁠, 1) requires at least five terms. Curtiss (1922) describes an application of these closest-approximation results in lower-bounding the number of divisors of a perfect number, while Stong (1983) describes applications in group theory. == Maximum-length expansions and congruence conditions == Any fraction ⁠x/y⁠ requires at most x terms in its greedy expansion. Mays (1987) and Freitag & Phillips (1999) examine the conditions under which the greedy method produces an expansion of ⁠x/y⁠ with exactly x terms; these can be described in terms of congruence conditions on y. Every fraction ⁠1/y⁠ requires one term in its greedy expansion; the simplest such fraction is ⁠1/1⁠. Every fraction ⁠2/y⁠ requires two terms in its greedy expansion if and only if y ≡ 1 (mod 2); the simplest such fraction is ⁠2/3⁠. A fraction ⁠3/y⁠ requires three terms in its greedy expansion if and only if y ≡ 1 (mod 6), for then −y mod x = 2 and ⁠y(y + 2)/3⁠ is odd, so the fraction remaining after a single step of the greedy expansion, ( − y ) mod x y ⌈ y x ⌉ = 2 y ( y + 2 ) 3 {\displaystyle {\frac {(-y){\bmod {x}}}{y\left\lceil {\frac {y}{x}}\right\rceil }}={\frac {2}{\,{\frac {y(y+2)}{3}}\,}}} is in simplest terms. The simplest fraction ⁠3/y⁠ with a three-term expansion is ⁠3/7⁠. A fraction ⁠4/y⁠ requires four terms in its greedy expansion if and only if y ≡ 1 or 17 (mod 24), for then the numerator −y mod x of the remaining fraction is 3 and the denominator is 1 (mod 6). The simplest fraction ⁠4/y⁠ with a four-term expansion is ⁠4/17⁠. The Erdős–Straus conjecture states that all fractions ⁠4/y⁠ have an expansion with three or fewer terms, but when y ≡ 1 or 17 (mod 24) such expansions must be found by methods other than the greedy algorithm, with the 17 (mod 24) case being covered by the congruence relationship 2 (mod 3). More generally the sequence of fractions ⁠x/y⁠ that have x-term greedy expansions and that have the smallest possible denominator y for each x is == Approximation of polynomial roots == Stratemeyer (1930) and Salzer (1947) describe a method of finding an accurate approximation for the roots of a polynomial based on the greedy method. Their algorithm computes the greedy expansion of a root; at each step in this expansion it maintains an auxiliary polynomial that has as its root the remaining fraction to be expanded. Consider as an example applying this method to find the greedy expansion of the golden ratio, one of the two solutions of the polynomial equation P0(x) = x2 − x − 1 = 0. The algorithm of Stratemeyer and Salzer performs the following sequence of steps: Since P0(x) < 0 for x = 1, and P0(x) > 0 for all x ≥ 2, there must be a root of P0(x) between 1 and 2. That is, the first term of the greedy expansion of the golden ratio is ⁠1/1⁠. If x1 is the remaining fraction after the first step of the greedy expansion, it satisfies the equation P0(x1 + 1) = 0, which can be expanded as P1(x1) = x21 + x1 − 1 = 0. Since P1(x) < 0 for x = ⁠1/2⁠, and P1(x) > 0 for all x > 1, the root of P1 lies between ⁠1/2⁠ and 1, and the first term in its greedy expansion (the second term in the greedy expansion for the golden ratio) is ⁠1/2⁠. If x2 is the remaining fraction after this step of the greedy expansion, it satisfies the equation P1(x2 + ⁠1/2⁠) = 0, which can be expanded as P2(x2) = 4x22 + 8x2 − 1 = 0. Since P2(x) < 0 for x = ⁠1/9⁠, and P2(x) > 0 for all x > ⁠1/8⁠, the next term in the greedy expansion is ⁠1/9⁠. If x3 is the remaining fraction after this step of the greedy expansion, it satisfies the equation P2(x3 + ⁠1/9⁠) = 0, which can again be expanded as a polynomial equation with integer coefficients, P3(x3) = 324x23 + 720x3 − 5 = 0. Continuing this approximation process eventually produces the greedy expansion for the golden ratio, == Other integer sequences == The length, minimum denominator, and maximum denominator of the greedy expansion for all fractions with small numerators and denominators can be found in the On-Line Encyclopedia of Integer Sequences as sequences OEIS: A050205, OEIS: A050206, and OEIS: A050210, respectively. In addition, the greedy expansion of any irrational number leads to an infinite increasing sequence of integers, and the OEIS contains expansions of several well known constants. Some additional entries in the OEIS, though not labeled as being produced by the greedy algorithm, appear to be of the same type. == Related expansions == In general, if one wants an Egyptian fraction expansion in which the denominators are constrained in some way, it is possible to define a greedy algorithm in which at each step one chooses the expansion x y = 1 d + x d − y y d , {\displaystyle {\frac {x}{y}}={\frac {1}{d}}+{\frac {xd-y}{yd}},} where d {\displaystyle d} is chosen, among all possible values satisfying the constraints, as small as possible such that x d > y {\displaystyle xd>y} and such that d {\displaystyle d} is distinct from all previously chosen denominators. Examples of methods defined in this way include Engel expansion, in which each successive denominator must be a multiple of the previous one, and odd greedy expansion, in which all denominators are constrained to be odd numbers. However, it may be difficult to determine whether an algorithm of this type can always succeed in finding a finite expansion. In particular, it is unknown whether the odd greedy expansion terminates with a finite expansion for all fractions x / y {\displaystyle x/y} for which y {\displaystyle y} is odd, although it is possible to find finite odd expansions for these fractions by non-greedy methods. == Notes == == References ==
Wikipedia:Greek numerals#0	Greek numerals, also known as Ionic, Ionian, Milesian, or Alexandrian numerals, is a system of writing numbers using the letters of the Greek alphabet. In modern Greece, they are still used for ordinal numbers and in contexts similar to those in which Roman numerals are still used in the Western world. For ordinary cardinal numbers, however, modern Greece uses Arabic numerals. == History == The Minoan and Mycenaean civilizations' Linear A and Linear B alphabets used a different system, called Aegean numerals, which included number-only symbols for powers of ten: 𐄇 = 1, 𐄐 = 10, 𐄙 = 100, 𐄢 = 1000, and 𐄫 = 10000. Attic numerals composed another system that came into use perhaps in the 7th century BC. They were acrophonic, derived (after the initial one) from the first letters of the names of the numbers represented. They ran = 1, = 5, = 10, = 100, = 1,000, and = 10,000. The numbers 50, 500, 5,000, and 50,000 were represented by the letter with minuscule powers of ten written in the top-right corner: , , , and . One-half was represented by 𐅁 (left half of a full circle) and one-quarter by ɔ (right side of a full circle). The same system was used outside of Attica, but the symbols varied with the local alphabets; for example, 1,000 was in Boeotia. The present system probably developed around Miletus in Ionia. 19th century classicists placed its development in the 3rd century BC, the occasion of its first widespread use. More thorough modern archaeology has caused the date to be pushed back at least to the 5th century BC, a little before Athens abandoned its pre-Eucleidean alphabet in favour of Miletus's in 402 BC, and it may predate that by a century or two. The present system uses the 24 letters adopted under Eucleides, as well as three Phoenician and Ionic ones that had not been dropped from the Athenian alphabet (although kept for numbers): digamma, koppa, and sampi. The position of those characters within the numbering system imply that the first two were still in use (or at least remembered as letters) while the third was not. The exact dating, particularly for sampi, is problematic since its uncommon value means the first attested representative near Miletus does not appear until the 2nd century BC, and its use is unattested in Athens until the 2nd century CE. (In general, Athenians resisted using the new numerals for the longest of any Greek state, but had fully adopted them by c. 50 CE.) == Description == Greek numerals are decimal, based on powers of 10. The units from 1 to 9 are assigned to the first nine letters of the old Ionic alphabet from alpha to theta. Instead of reusing these numbers to form multiples of the higher powers of ten, however, each multiple of ten from 10 to 90 was assigned its own separate letter from the next nine letters of the Ionic alphabet from iota to koppa. Each multiple of one hundred from 100 to 900 was then assigned its own separate letter as well, from rho to sampi. (That this was not the traditional location of sampi in the Ionic alphabetical order has led classicists to conclude that sampi had fallen into disuse as a letter by the time the system was created.) This alphabetic system operates on the additive principle in which the numeric values of the letters are added together to obtain the total. For example, 241 was represented as (200 + 40 + 1). (It was not always the case that the numbers ran from highest to lowest: a 4th-century BC inscription at Athens placed the units to the left of the tens. This practice continued in Asia Minor well into the Roman period.) In ancient and medieval manuscripts, these numerals were eventually distinguished from letters using overbars: α, β, γ, etc. In medieval manuscripts of the Book of Revelation, the number of the Beast 666 is written as χξϛ (600 + 60 + 6). (Numbers larger than 1,000 reused the same letters but included various marks to note the change.) Fractions were indicated as the denominator followed by a keraia (ʹ); γʹ indicated one third, δʹ one fourth and so on. As an exception, special symbol ∠ʹ indicated one half, and γ°ʹ or γoʹ was two-thirds. These fractions were additive (also known as Egyptian fractions); for example δʹ ϛʹ indicated 1⁄4 + 1⁄6 = 5⁄12. Although the Greek alphabet began with only majuscule forms, surviving papyrus manuscripts from Egypt show that uncial and cursive minuscule forms began early. These new letter forms sometimes replaced the former ones, especially in the case of the obscure numerals. The old Q-shaped koppa (Ϙ) began to be broken up ( and ) and simplified ( and ). The numeral for 6 changed several times. During antiquity, the original letter form of digamma (Ϝ) came to be avoided in favour of a special numerical one (). By the Byzantine era, the letter was known as episemon and written as or . This eventually merged with the sigma-tau ligature stigma ϛ ( or ). In modern Greek, a number of other changes have been made. Instead of extending an over bar over an entire number, the keraia (κεραία, lit. "hornlike projection") is marked to its upper right, a development of the short marks formerly used for single numbers and fractions. The modern keraia (´) is a symbol similar to the acute accent (´), the tonos (U+0384,΄) and the prime symbol (U+02B9, ʹ), but has its own Unicode character as U+0374. Alexander the Great's father Philip II of Macedon is thus known as Φίλιππος Βʹ in modern Greek. A lower left keraia (Unicode: U+0375, "Greek Lower Numeral Sign") is now standard for distinguishing thousands: 2019 is represented as ͵ΒΙΘʹ (2 × 1,000 + 10 + 9). The declining use of ligatures in the 20th century also means that stigma is frequently written as the separate letters ΣΤʹ, although a single keraia is used for the group. == Isopsephy == The practice of adding up the number values of Greek letters of words, names and phrases, thus connecting the meaning of words, names and phrases with others with equivalent numeric sums, is called isopsephy. Similar practices for the Hebrew and English are called gematria and English Qaballa, respectively. == Table == Alternatively, sub-sections of manuscripts are sometimes numbered by lowercase characters (αʹ. βʹ. γʹ. δʹ. εʹ. ϛʹ. ζʹ. ηʹ. θʹ.). In Ancient Greek, myriad notation is used for multiples of 10,000, for example βΜ for 20,000 or ρκγΜ͵δφξζ (also written on the line as ρκγΜ ͵δφξζ) for 1,234,567. == Higher numbers == In his text The Sand Reckoner, the natural philosopher Archimedes gives an upper bound of the number of grains of sand required to fill the entire universe, using a contemporary estimation of its size. This would defy the then-held notion that it is impossible to name a number greater than that of the sand on a beach or on the entire world. In order to do that, he had to devise a new numeral scheme with much greater range. Pappus of Alexandria reports that Apollonius of Perga developed a simpler system based on powers of the myriad; αΜ was 10,000, βΜ was 10,0002 = 100,000,000, γΜ was 10,0003 = 1012 and so on. == Zero == Hellenistic astronomers extended alphabetic Greek numerals into a sexagesimal positional numbering system by limiting each position to a maximum value of 50 + 9 and including a special symbol for zero, which was only used alone for a whole table cell, rather than combined with other digits, like today's modern zero, which is a placeholder in positional numeric notation. This system was probably adapted from Babylonian numerals by Hipparchus c. 140 BC. It was then used by Ptolemy (c. 140 BC), Theon (c. 380 AD) and Theon's daughter Hypatia (d. 415 AD). The symbol for zero is clearly different from that of the value for 70, omicron or "ο". In the 2nd-century papyrus shown here, one can see the symbol for zero in the lower right, and a number of larger omicrons elsewhere in the same papyrus. In Ptolemy's table of chords, the first fairly extensive trigonometric table, there were 360 rows, portions of which looked as follows: π ε ϱ ι φ ε ϱ ε ι ω ~ ν ε υ ' ϑ ε ι ω ~ ν ε ‘ ξ η κ o σ τ ω ~ ν π δ ∠ ′ π ε π ε ∠ ′ π ϛ π ϛ ∠ ′ π ζ π μ α γ π α δ ι ε π α κ ζ κ β π α ν κ δ π β ι γ ι ϑ π β λ ϛ ϑ ∘ ∘ μ ϛ κ ε ∘ ∘ μ ϛ ι δ ∘ ∘ μ ϛ γ ∘ ∘ μ ε ν β ∘ ∘ μ ε μ ∘ ∘ μ ε κ ϑ {\displaystyle {\begin{array}{ccc}\pi \varepsilon \varrho \iota \varphi \varepsilon \varrho \varepsilon \iota {\tilde {\omega }}\nu &\varepsilon {\overset {\text{'}}{\upsilon }}\vartheta \varepsilon \iota {\tilde {\omega }}\nu &{\overset {\text{‘}}{\varepsilon }}\xi \eta \kappa o\sigma \tau {\tilde {\omega }}\nu \\{\begin{array}{\|l\|}\hline \pi \delta \angle '\\\pi \varepsilon \\\pi \varepsilon \angle '\\\hline \pi \mathrm {\stigma} \\\pi \mathrm {\stigma} \angle '\\\pi \zeta \\\hline \end{array}}&{\begin{array}{\|r\|r\|r\|}\hline \pi &\mu \alpha &\gamma \\\pi \alpha &\delta &\iota \varepsilon \\\pi \alpha &\kappa \zeta &\kappa \beta \\\hline \pi \alpha &\nu &\kappa \delta \\\pi \beta &\iota \gamma &\iota \vartheta \\\pi \beta &\lambda \mathrm {\stigma} &\vartheta \\\hline \end{array}}&{\begin{array}{\|r\|r\|r\|r\|}\hline \circ &\circ &\mu \mathrm {\stigma} &\kappa \varepsilon \\\circ &\circ &\mu \mathrm {\stigma} &\iota \delta \\\circ &\circ &\mu \mathrm {\stigma} &\gamma \\\hline \circ &\circ &\mu \varepsilon &\nu \beta \\\circ &\circ &\mu \varepsilon &\mu \\\circ &\circ &\mu \varepsilon &\kappa \vartheta \\\hline \end{array}}\end{array}}} Each number in the first column, labeled περιφερειῶν, ["regions"] is the number of degrees of arc on a circle. Each number in the second column, labeled εὐθειῶν, ["straight lines" or "segments"] is the length of the corresponding chord of the circle, when the diameter is 120. Thus πδ represents an 84° arc, and the ∠′ after it means one-half, so that πδ∠′ means 84+1⁄2°. In the next column we see π μα γ , meaning 80 + ⁠41/60⁠ + ⁠3/60²⁠. That is the length of the chord corresponding to an arc of 84+1⁄2° when the diameter of the circle is 120. The next column, labeled ἑξηκοστῶν, for "sixtieths", is the number to be added to the chord length for each 1' increase in the arc, over the span of the next 1°. Thus that last column was used for linear interpolation. The Greek sexagesimal placeholder or zero symbol changed over time: The symbol used on papyri during the second century was a very small circle with an overbar several diameters long, terminated or not at both ends in various ways. Later, the overbar shortened to only one diameter, similar to the modern o-macron (ō) which was still being used in late medieval Arabic manuscripts whenever alphabetic numerals were used, later the overbar was omitted in Byzantine manuscripts, leaving a bare ο (omicron). This gradual change from an invented symbol to ο does not support the hypothesis that the latter was the initial of οὐδέν meaning "nothing". Note that the letter ο was still used with its original numerical value of 70; however, there was no ambiguity, as 70 could not appear in the fractional part of a sexagesimal number, and zero was usually omitted when it was the integer. Some of Ptolemy's true zeros appeared in the first line of each of his eclipse tables, where they were a measure of the angular separation between the center of the Moon and either the center of the Sun (for solar eclipses) or the center of Earth's shadow (for lunar eclipses). All of these zeros took the form ο \| ο ο, where Ptolemy actually used three of the symbols described in the previous paragraph. The vertical bar (\|) indicates that the integral part on the left was in a separate column labeled in the headings of his tables as digits (of five arc-minutes each), whereas the fractional part was in the next column labeled minute of immersion, meaning sixtieths (and thirty-six-hundredths) of a digit. The Greek zero was added to Unicode in Version 4.1.0 at U+1018A 𐆊 GREEK ZERO SIGN. == See also == Alphabetic numeral system – Type of numeral system Attic numerals – Symbolic number notation used by the ancient Greeks Cyrillic numerals – Numeral system derived from the Cyrillic script Greek mathematics – Mathematics of Ancient GreecePages displaying short descriptions of redirect targets Greek numerals in Unicode – Graphemes for various number systemsPages displaying short descriptions of redirect targets (acrophonic, not alphabetic, numerals) Hebrew numerals – Numeral system using letters of the Hebrew alphabet, based on the Greek system History of ancient numeral systems – Symbols representing numbers History of arithmetic – Branch of elementary mathematicsPages displaying short descriptions of redirect targets History of communication Isopsephy – Numerological connection between words whose letters' number values have equal sums List of numeral system topics List of numeral systems Number of the beast – Number associated with the Beast of Revelation Roman numerals – Numbers in the Roman numeral system == References == == External links == The Greek Number Converter
Wikipedia:Green's identities#0	In mathematics, Green's identities are a set of three identities in vector calculus relating the bulk with the boundary of a region on which differential operators act. They are named after the mathematician George Green, who discovered Green's theorem. == Green's first identity == This identity is derived from the divergence theorem applied to the vector field F = ψ ∇φ while using an extension of the product rule that ∇ ⋅ (ψ X ) = ∇ψ ⋅X + ψ ∇⋅X: Let φ and ψ be scalar functions defined on some region U ⊂ Rd, and suppose that φ is twice continuously differentiable, and ψ is once continuously differentiable. Using the product rule above, but letting X = ∇φ, integrate ∇⋅(ψ∇φ) over U. Then ∫ U ( ψ Δ φ + ∇ ψ ⋅ ∇ φ ) d V = ∮ ∂ U ψ ( ∇ φ ⋅ n ) d S = ∮ ∂ U ψ ∇ φ ⋅ d S {\displaystyle \int _{U}\left(\psi \,\Delta \varphi +\nabla \psi \cdot \nabla \varphi \right)\,dV=\oint _{\partial U}\psi \left(\nabla \varphi \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \,\nabla \varphi \cdot d\mathbf {S} } where ∆ ≡ ∇2 is the Laplace operator, ∂U is the boundary of region U, n is the outward pointing unit normal to the surface element dS and dS = ndS is the oriented surface element. This theorem is a special case of the divergence theorem, and is essentially the higher dimensional equivalent of integration by parts with ψ and the gradient of φ replacing u and v. Note that Green's first identity above is a special case of the more general identity derived from the divergence theorem by substituting F = ψΓ, ∫ U ( ψ ∇ ⋅ Γ + Γ ⋅ ∇ ψ ) d V = ∮ ∂ U ψ ( Γ ⋅ n ) d S = ∮ ∂ U ψ Γ ⋅ d S . {\displaystyle \int _{U}\left(\psi \,\nabla \cdot \mathbf {\Gamma } +\mathbf {\Gamma } \cdot \nabla \psi \right)\,dV=\oint _{\partial U}\psi \left(\mathbf {\Gamma } \cdot \mathbf {n} \right)\,dS=\oint _{\partial U}\psi \mathbf {\Gamma } \cdot d\mathbf {S} ~.} == Green's second identity == If φ and ψ are both twice continuously differentiable on U ⊂ R3, and ε is once continuously differentiable, one may choose F = ψε ∇φ − φε ∇ψ to obtain ∫ U [ ψ ∇ ⋅ ( ε ∇ φ ) − φ ∇ ⋅ ( ε ∇ ψ ) ] d V = ∮ ∂ U ε ( ψ ∂ φ ∂ n − φ ∂ ψ ∂ n ) d S . {\displaystyle \int _{U}\left[\psi \,\nabla \cdot \left(\varepsilon \,\nabla \varphi \right)-\varphi \,\nabla \cdot \left(\varepsilon \,\nabla \psi \right)\right]\,dV=\oint _{\partial U}\varepsilon \left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.} For the special case of ε = 1 all across U ⊂ R3, then, ∫ U ( ψ ∇ 2 φ − φ ∇ 2 ψ ) d V = ∮ ∂ U ( ψ ∂ φ ∂ n − φ ∂ ψ ∂ n ) d S . {\displaystyle \int _{U}\left(\psi \,\nabla ^{2}\varphi -\varphi \,\nabla ^{2}\psi \right)\,dV=\oint _{\partial U}\left(\psi {\partial \varphi \over \partial \mathbf {n} }-\varphi {\partial \psi \over \partial \mathbf {n} }\right)\,dS.} In the equation above, ∂φ/∂n is the directional derivative of φ in the direction of the outward pointing surface normal n of the surface element dS, ∂ φ ∂ n = ∇ φ ⋅ n = ∇ n φ . {\displaystyle {\partial \varphi \over \partial \mathbf {n} }=\nabla \varphi \cdot \mathbf {n} =\nabla _{\mathbf {n} }\varphi .} Explicitly incorporating this definition in the Green's second identity with ε = 1 results in ∫ U ( ψ ∇ 2 φ − φ ∇ 2 ψ ) d V = ∮ ∂ U ( ψ ∇ φ − φ ∇ ψ ) ⋅ d S . {\displaystyle \int _{U}\left(\psi \,\nabla ^{2}\varphi -\varphi \,\nabla ^{2}\psi \right)\,dV=\oint _{\partial U}\left(\psi \nabla \varphi -\varphi \nabla \psi \right)\cdot d\mathbf {S} .} In particular, this demonstrates that the Laplacian is a self-adjoint operator in the L2 inner product for functions vanishing on the boundary so that the right hand side of the above identity is zero. == Green's third identity == Green's third identity derives from the second identity by choosing φ = G, where the Green's function G is taken to be a fundamental solution of the Laplace operator, ∆. This means that: Δ G ( x , η ) = δ ( x − η ) . {\displaystyle \Delta G(\mathbf {x} ,{\boldsymbol {\eta }})=\delta (\mathbf {x} -{\boldsymbol {\eta }})~.} For example, in R3, a solution has the form G ( x , η ) = − 1 4 π ‖ x − η ‖ . {\displaystyle G(\mathbf {x} ,{\boldsymbol {\eta }})={\frac {-1}{4\pi \\|\mathbf {x} -{\boldsymbol {\eta }}\\|}}~.} Green's third identity states that if ψ is a function that is twice continuously differentiable on U, then ∫ U [ G ( y , η ) Δ ψ ( y ) ] d V y − ψ ( η ) = ∮ ∂ U [ G ( y , η ) ∂ ψ ∂ n ( y ) − ψ ( y ) ∂ G ( y , η ) ∂ n ] d S y . {\displaystyle \int _{U}\left[G(\mathbf {y} ,{\boldsymbol {\eta }})\,\Delta \psi (\mathbf {y} )\right]\,dV_{\mathbf {y} }-\psi ({\boldsymbol {\eta }})=\oint _{\partial U}\left[G(\mathbf {y} ,{\boldsymbol {\eta }}){\partial \psi \over \partial \mathbf {n} }(\mathbf {y} )-\psi (\mathbf {y} ){\partial G(\mathbf {y} ,{\boldsymbol {\eta }}) \over \partial \mathbf {n} }\right]\,dS_{\mathbf {y} }.} A simplification arises if ψ is itself a harmonic function, i.e. a solution to the Laplace equation. Then ∇2ψ = 0 and the identity simplifies to ψ ( η ) = ∮ ∂ U [ ψ ( y ) ∂ G ( y , η ) ∂ n − G ( y , η ) ∂ ψ ∂ n ( y ) ] d S y . {\displaystyle \psi ({\boldsymbol {\eta }})=\oint _{\partial U}\left[\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}-G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \psi }{\partial \mathbf {n} }}(\mathbf {y} )\right]\,dS_{\mathbf {y} }.} The second term in the integral above can be eliminated if G is chosen to be the Green's function that vanishes on the boundary of U (Dirichlet boundary condition), ψ ( η ) = ∮ ∂ U ψ ( y ) ∂ G ( y , η ) ∂ n d S y . {\displaystyle \psi ({\boldsymbol {\eta }})=\oint _{\partial U}\psi (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}\,dS_{\mathbf {y} }~.} This form is used to construct solutions to Dirichlet boundary condition problems. Solutions for Neumann boundary condition problems may also be simplified, though the Divergence theorem applied to the differential equation defining Green's functions shows that the Green's function cannot integrate to zero on the boundary, and hence cannot vanish on the boundary. See Green's functions for the Laplacian or for a detailed argument, with an alternative. For the Neumann boundary condition, an appropriate choice of Green's function can be made to simplify the integral. First note ∫ U Δ G ( y , η ) d V y = 1 = ∮ ∂ U ∂ G ( y , η ) ∂ n d S y {\displaystyle \int _{U}\Delta G(\mathbf {y} ,{\boldsymbol {\eta }})dV_{y}=1=\oint _{\partial U}{\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}dS_{y}} and so ∂ G ( y , η ) ∂ n {\displaystyle {\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}} cannot vanish on surface S {\displaystyle S} . A convenient choice is ∂ G ( y , η ) ∂ n = 1 / A {\displaystyle {\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}=1/{\mathcal {A}}} , where A {\displaystyle {\mathcal {A}}} is the area of the surface S {\displaystyle S} . The integral can be simplified to ψ ( η ) = ⟨ ψ ⟩ S − ∮ ∂ U G ( y , η ) ∂ ψ ∂ n ( y ) d S y . {\displaystyle \psi ({\boldsymbol {\eta }})=\langle \psi \rangle _{S}-\oint _{\partial U}G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \psi }{\partial \mathbf {n} }}(\mathbf {y} )\,dS_{\mathbf {y} }.} where ⟨ ψ ⟩ S = 1 A ∮ ∂ U ψ ( y ) d S y {\displaystyle \langle \psi \rangle _{S}={\frac {1}{\mathcal {A}}}\oint _{\partial U}\psi (\mathbf {y} )dS_{y}} is the average value of ψ {\displaystyle \psi } on surface S {\displaystyle S} . Furthermore, if ψ {\displaystyle \psi } is a solution to the Laplace's equation, divergence theorem implies it must satisfy ∮ ∂ U ∂ ψ ∂ n ( y ) d S y = ∫ U Δ ψ ( y ) d V y = 0 {\displaystyle \oint _{\partial U}{\frac {\partial \psi }{\partial \mathbf {n} }}(\mathbf {y} )dS_{y}=\int _{U}\Delta \psi (\mathbf {y} )dV_{y}=0} . This is a necessary condition for the Neumann boundary problem to have a solution. It can be further verified that the above identity also applies when ψ is a solution to the Helmholtz equation or wave equation and G is the appropriate Green's function. In such a context, this identity is the mathematical expression of the Huygens principle, and leads to Kirchhoff's diffraction formula and other approximations. == On manifolds == Green's identities hold on a Riemannian manifold. In this setting, the first two are ∫ M u Δ v d V + ∫ M ⟨ ∇ u , ∇ v ⟩ d V = ∫ ∂ M u N v d V ~ ∫ M ( u Δ v − v Δ u ) d V = ∫ ∂ M ( u N v − v N u ) d V ~ {\displaystyle {\begin{aligned}\int _{M}u\,\Delta v\,dV+\int _{M}\langle \nabla u,\nabla v\rangle \,dV&=\int _{\partial M}uNv\,d{\widetilde {V}}\\\int _{M}\left(u\,\Delta v-v\,\Delta u\right)\,dV&=\int _{\partial M}(uNv-vNu)\,d{\widetilde {V}}\end{aligned}}} where u and v are smooth real-valued functions on M, dV is the volume form compatible with the metric, d V ~ {\displaystyle d{\widetilde {V}}} is the induced volume form on the boundary of M, N is the outward oriented unit vector field normal to the boundary, and Δu = div(grad u) is the Laplacian. == Green's vector identities == === First vector identity === Using the vector Laplacian identity and the divergence identity, expand P ⋅ Δ Q {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} } P ⋅ Δ Q = ∇ ⋅ ( P × ∇ × Q ) − ( ∇ × P ) ⋅ ( ∇ × Q ) + P ⋅ [ ∇ ( ∇ ⋅ Q ) ] {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} =\nabla \cdot (\mathbf {P} \times \nabla \times \mathbf {Q} )-(\nabla \times \mathbf {P} )\cdot (\nabla \times \mathbf {Q} )+\mathbf {P} \cdot [\nabla (\nabla \cdot \mathbf {Q} )]} The last term can be simplified by expanding components P ⋅ [ ∇ ( ∇ ⋅ Q ) ] = P i [ ∇ i ( ∇ j Q j ) ] = ∇ i [ P i ( ∇ j Q j ) ] − ( ∇ i P i ) ( ∇ j Q j ) = ∇ ⋅ [ P ( ∇ ⋅ Q ) ] − ( ∇ ⋅ P ) ( ∇ ⋅ Q ) {\displaystyle {\begin{aligned}\mathbf {P} \cdot [\nabla (\nabla \cdot \mathbf {Q} )]&=P^{i}[\nabla _{i}(\nabla _{j}Q^{j})]\\&=\nabla _{i}[P^{i}(\nabla _{j}Q^{j})]-(\nabla _{i}P^{i})(\nabla _{j}Q^{j})\\&=\nabla \cdot [\mathbf {P} (\nabla \cdot \mathbf {Q} )]-(\nabla \cdot \mathbf {P} )(\nabla \cdot \mathbf {Q} )\end{aligned}}} The identity can be rewritten as P ⋅ Δ Q = ∇ ⋅ ( P × ∇ × Q ) − ( ∇ × P ) ⋅ ( ∇ × Q ) + ∇ ⋅ [ P ( ∇ ⋅ Q ) ] − ( ∇ ⋅ P ) ( ∇ ⋅ Q ) {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} =\nabla \cdot (\mathbf {P} \times \nabla \times \mathbf {Q} )-(\nabla \times \mathbf {P} )\cdot (\nabla \times \mathbf {Q} )+\nabla \cdot [\mathbf {P} (\nabla \cdot \mathbf {Q} )]-(\nabla \cdot \mathbf {P} )(\nabla \cdot \mathbf {Q} )} In integral form, this is ∮ ∂ U n ⋅ [ P × ∇ × Q + P ( ∇ ⋅ Q ) ] d S = ∫ U [ P ⋅ Δ Q + ( ∇ × P ) ⋅ ( ∇ × Q ) + ( ∇ ⋅ P ) ( ∇ ⋅ Q ) ] d V {\displaystyle \oint _{\partial U}\mathbf {n} \cdot [\mathbf {P} \times \nabla \times \mathbf {Q} +\mathbf {P} (\nabla \cdot \mathbf {Q} )]dS=\int _{U}[\mathbf {P} \cdot \Delta \mathbf {Q} +(\nabla \times \mathbf {P} )\cdot (\nabla \times \mathbf {Q} )+(\nabla \cdot \mathbf {P} )(\nabla \cdot \mathbf {Q} )]dV} === Second vector identity === Green's second identity establishes a relationship between second and (the divergence of) first order derivatives of two scalar functions. In differential form p m Δ q m − q m Δ p m = ∇ ⋅ ( p m ∇ q m − q m ∇ p m ) , {\displaystyle p_{m}\,\Delta q_{m}-q_{m}\,\Delta p_{m}=\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\,\nabla p_{m}\right),} where pm and qm are two arbitrary twice continuously differentiable scalar fields. This identity is of great importance in physics because continuity equations can thus be established for scalar fields such as mass or energy. In vector diffraction theory, two versions of Green's second identity are introduced. One variant invokes the divergence of a cross product and states a relationship in terms of the curl-curl of the field P ⋅ ( ∇ × ∇ × Q ) − Q ⋅ ( ∇ × ∇ × P ) = ∇ ⋅ ( Q × ( ∇ × P ) − P × ( ∇ × Q ) ) . {\displaystyle \mathbf {P} \cdot \left(\nabla \times \nabla \times \mathbf {Q} \right)-\mathbf {Q} \cdot \left(\nabla \times \nabla \times \mathbf {P} \right)=\nabla \cdot \left(\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)-\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)\right).} This equation can be written in terms of the Laplacians, P ⋅ Δ Q − Q ⋅ Δ P + Q ⋅ [ ∇ ( ∇ ⋅ P ) ] − P ⋅ [ ∇ ( ∇ ⋅ Q ) ] = ∇ ⋅ ( P × ( ∇ × Q ) − Q × ( ∇ × P ) ) . {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} +\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]=\nabla \cdot \left(\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right).} However, the terms Q ⋅ [ ∇ ( ∇ ⋅ P ) ] − P ⋅ [ ∇ ( ∇ ⋅ Q ) ] , {\displaystyle \mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]-\mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right],} could not be readily written in terms of a divergence. The other approach introduces bi-vectors, this formulation requires a dyadic Green function. The derivation presented here avoids these problems. Consider that the scalar fields in Green's second identity are the Cartesian components of vector fields, i.e., P = ∑ m p m e ^ m , Q = ∑ m q m e ^ m . {\displaystyle \mathbf {P} =\sum _{m}p_{m}{\hat {\mathbf {e} }}_{m},\qquad \mathbf {Q} =\sum _{m}q_{m}{\hat {\mathbf {e} }}_{m}.} Summing up the equation for each component, we obtain ∑ m [ p m Δ q m − q m Δ p m ] = ∑ m ∇ ⋅ ( p m ∇ q m − q m ∇ p m ) . {\displaystyle \sum _{m}\left[p_{m}\Delta q_{m}-q_{m}\Delta p_{m}\right]=\sum _{m}\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right).} The LHS according to the definition of the dot product may be written in vector form as ∑ m [ p m Δ q m − q m Δ p m ] = P ⋅ Δ Q − Q ⋅ Δ P . {\displaystyle \sum _{m}\left[p_{m}\,\Delta q_{m}-q_{m}\,\Delta p_{m}\right]=\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} .} The RHS is a bit more awkward to express in terms of vector operators. Due to the distributivity of the divergence operator over addition, the sum of the divergence is equal to the divergence of the sum, i.e., ∑ m ∇ ⋅ ( p m ∇ q m − q m ∇ p m ) = ∇ ⋅ ( ∑ m p m ∇ q m − ∑ m q m ∇ p m ) . {\displaystyle \sum _{m}\nabla \cdot \left(p_{m}\nabla q_{m}-q_{m}\nabla p_{m}\right)=\nabla \cdot \left(\sum _{m}p_{m}\nabla q_{m}-\sum _{m}q_{m}\nabla p_{m}\right).} Recall the vector identity for the gradient of a dot product, ∇ ( P ⋅ Q ) = ( P ⋅ ∇ ) Q + ( Q ⋅ ∇ ) P + P × ( ∇ × Q ) + Q × ( ∇ × P ) , {\displaystyle \nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)+\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right),} which, written out in vector components is given by ∇ ( P ⋅ Q ) = ∇ ∑ m p m q m = ∑ m p m ∇ q m + ∑ m q m ∇ p m . {\displaystyle \nabla \left(\mathbf {P} \cdot \mathbf {Q} \right)=\nabla \sum _{m}p_{m}q_{m}=\sum _{m}p_{m}\nabla q_{m}+\sum _{m}q_{m}\nabla p_{m}.} This result is similar to what we wish to evince in vector terms 'except' for the minus sign. Since the differential operators in each term act either over one vector (say p m {\displaystyle p_{m}} ’s) or the other ( q m {\displaystyle q_{m}} ’s), the contribution to each term must be ∑ m p m ∇ q m = ( P ⋅ ∇ ) Q + P × ( ∇ × Q ) , {\displaystyle \sum _{m}p_{m}\nabla q_{m}=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right),} ∑ m q m ∇ p m = ( Q ⋅ ∇ ) P + Q × ( ∇ × P ) . {\displaystyle \sum _{m}q_{m}\nabla p_{m}=\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right).} These results can be rigorously proven to be correct through evaluation of the vector components. Therefore, the RHS can be written in vector form as ∑ m p m ∇ q m − ∑ m q m ∇ p m = ( P ⋅ ∇ ) Q + P × ( ∇ × Q ) − ( Q ⋅ ∇ ) P − Q × ( ∇ × P ) . {\displaystyle \sum _{m}p_{m}\nabla q_{m}-\sum _{m}q_{m}\nabla p_{m}=\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right).} Putting together these two results, a result analogous to Green's theorem for scalar fields is obtained, Theorem for vector fields: P ⋅ Δ Q − Q ⋅ Δ P = [ ( P ⋅ ∇ ) Q + P × ( ∇ × Q ) − ( Q ⋅ ∇ ) P − Q × ( ∇ × P ) ] . {\displaystyle \color {OliveGreen}\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\left[\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].} The curl of a cross product can be written as ∇ × ( P × Q ) = ( Q ⋅ ∇ ) P − ( P ⋅ ∇ ) Q + P ( ∇ ⋅ Q ) − Q ( ∇ ⋅ P ) ; {\displaystyle \nabla \times \left(\mathbf {P} \times \mathbf {Q} \right)=\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} -\left(\mathbf {P} \cdot \nabla \right)\mathbf {Q} +\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right);} Green's vector identity can then be rewritten as P ⋅ Δ Q − Q ⋅ Δ P = ∇ ⋅ [ P ( ∇ ⋅ Q ) − Q ( ∇ ⋅ P ) − ∇ × ( P × Q ) + P × ( ∇ × Q ) − Q × ( ∇ × P ) ] . {\displaystyle \mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)-\nabla \times \left(\mathbf {P} \times \mathbf {Q} \right)+\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].} Since the divergence of a curl is zero, the third term vanishes to yield Green's second vector identity: P ⋅ Δ Q − Q ⋅ Δ P = ∇ ⋅ [ P ( ∇ ⋅ Q ) − Q ( ∇ ⋅ P ) + P × ( ∇ × Q ) − Q × ( ∇ × P ) ] . {\displaystyle \color {OliveGreen}\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} =\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)+\mathbf {P} \times \left(\nabla \times \mathbf {Q} \right)-\mathbf {Q} \times \left(\nabla \times \mathbf {P} \right)\right].} With a similar procedure, the Laplacian of the dot product can be expressed in terms of the Laplacians of the factors Δ ( P ⋅ Q ) = P ⋅ Δ Q − Q ⋅ Δ P + 2 ∇ ⋅ [ ( Q ⋅ ∇ ) P + Q × ∇ × P ] . {\displaystyle \Delta \left(\mathbf {P} \cdot \mathbf {Q} \right)=\mathbf {P} \cdot \Delta \mathbf {Q} -\mathbf {Q} \cdot \Delta \mathbf {P} +2\nabla \cdot \left[\left(\mathbf {Q} \cdot \nabla \right)\mathbf {P} +\mathbf {Q} \times \nabla \times \mathbf {P} \right].} As a corollary, the awkward terms can now be written in terms of a divergence by comparison with the vector Green equation, P ⋅ [ ∇ ( ∇ ⋅ Q ) ] − Q ⋅ [ ∇ ( ∇ ⋅ P ) ] = ∇ ⋅ [ P ( ∇ ⋅ Q ) − Q ( ∇ ⋅ P ) ] . {\displaystyle \mathbf {P} \cdot \left[\nabla \left(\nabla \cdot \mathbf {Q} \right)\right]-\mathbf {Q} \cdot \left[\nabla \left(\nabla \cdot \mathbf {P} \right)\right]=\nabla \cdot \left[\mathbf {P} \left(\nabla \cdot \mathbf {Q} \right)-\mathbf {Q} \left(\nabla \cdot \mathbf {P} \right)\right].} This result can be verified by expanding the divergence of a scalar times a vector on the RHS. === Third vector identity === The third vector identity can be derived using the free space scalar Green's function. Take the scalar Green's function definition Δ G ( x , η ) = δ ( x − η ) {\displaystyle \Delta G(\mathbf {x} ,{\boldsymbol {\eta }})=\delta (\mathbf {x} -{\boldsymbol {\eta }})} , multiply by P {\displaystyle \mathbf {P} } and subtract G ∇ i ∇ i P {\displaystyle G\nabla _{i}\nabla ^{i}\mathbf {P} } . ∇ i ( P ∇ i G − G ∇ i P ) = P δ ( x − η ) − G Δ P {\displaystyle \nabla _{i}(\mathbf {P} \nabla ^{i}G-G\nabla ^{i}\mathbf {P} )=\mathbf {P} \delta (\mathbf {x} -{\boldsymbol {\eta }})-G\Delta \mathbf {P} } Integrate over volume U {\displaystyle U} and use divergence theorem. ∮ ∂ U ( P ( y ) ∂ G ( y , η ) ∂ n − G ( y , η ) ∂ P ( y ) ∂ n ) d S y + ∫ U G ( y , η ) Δ P ( y ) d V y = { P ( η ) for η ∈ U 0 for η ∉ U {\displaystyle \oint _{\partial U}{\bigg (}\mathbf {P} (\mathbf {y} ){\frac {\partial G(\mathbf {y} ,{\boldsymbol {\eta }})}{\partial \mathbf {n} }}-G(\mathbf {y} ,{\boldsymbol {\eta }}){\frac {\partial \mathbf {P} (\mathbf {y} )}{\partial \mathbf {n} }}{\bigg )}dS_{y}+\int _{U}G(\mathbf {y} ,{\boldsymbol {\eta }})\Delta \mathbf {P} (\mathbf {y} )dV_{y}=\left\{{\begin{matrix}\mathbf {P} ({\boldsymbol {\eta }})&{\text{for }}~{\boldsymbol {\eta }}\in U\\0&{\text{for }}~{\boldsymbol {\eta }}\not \in U\\\end{matrix}}\right.} == See also == Green's function Kirchhoff integral theorem Lagrange's identity (boundary value problem) == References == == External links == "Green formulas", Encyclopedia of Mathematics, EMS Press, 2001 [1994] [1] Green's Identities at Wolfram MathWorld
Wikipedia:Greg Kuperberg#0	Greg Kuperberg (born July 4, 1967) is a Polish-born American mathematician known for his contributions to geometric topology, quantum algebra, and combinatorics. Kuperberg is a professor of mathematics at the University of California, Davis. == Biography == Kuperberg is the son of two mathematicians, Krystyna Kuperberg and Włodzimierz Kuperberg. He was born in Poland in 1967, but his family emigrated to Sweden in 1969 due to the 1968 Polish political crisis. In 1972, Kuperberg's family moved to the United States, eventually settling in Auburn, Alabama. Kuperberg wrote three computer games for the IBM Personal Computer in 1982 and 1983 (which were published by Orion Software): Paratrooper, PC-Man and J-Bird. (video game clones of Sabotage, Pac-Man and Q*bert, respectively) He enrolled at Harvard University in 1983 and received a bachelor's degree in 1987. He was ranked Top 10 in the 1986 William Lowell Putnam Mathematical Competition. Upon leaving Harvard, Kuperberg studied at the University of California, Berkeley under Andrew Casson, receiving a Ph.D. in geometric topology and quantum algebra in 1991. From 1991 until 1992, Kuperberg was a NSF postdoctoral fellow and adjunct assistant professor at Berkeley, and from 1992 to 1995 held a Dickson Instructorship at the University of Chicago. From 1995 through 1996, Kuperberg was Gibbs Assistant Professor at Yale University after which he joined the mathematics faculty at the University of California, Davis. In 2012 he became a fellow of the American Mathematical Society. Kuperberg is married to physicist Rena Zieve, who is a professor of physics at UC Davis. == Selected publications == Kuperberg has over fifty publications, including two in the Annals of Mathematics. Kuperberg, Greg (1994). "The quantum G2 link invariant". International Journal of Mathematics. 5 (1): 61–85. arXiv:math/9201302. doi:10.1142/S0129167X94000048. Kuperberg, Greg (1996). "Non-involutory Hopf algebras and 3-manifold invariants". Duke Mathematical Journal. 84: 83–129. arXiv:q-alg/9712047. doi:10.1215/S0012-7094-96-08403-3. S2CID 15086645. with Krystyna Kuperberg: Kuperberg, Greg; Kuperberg, Krystyna (1996). "Generalized counterexamples to the Seifert conjecture". Annals of Mathematics. Second Series. 144 (3): 547–576. arXiv:math/9802040. doi:10.2307/2118536. JSTOR 2118536. S2CID 16309410. Kuperberg, Greg (2002). "Symmetry classes of alternating-sign matrices under one roof". Annals of Mathematics. Second Series. 156 (3): 835–866. arXiv:math/0008184. doi:10.2307/3597283. JSTOR 3597283. S2CID 7965653. Kuperberg, Greg (2005). "A subexponential-time quantum algorithm for the dihedral hidden subgroup problem". SIAM Journal on Computing. 35 (1): 170–188. arXiv:quant-ph/0302112. doi:10.1137/S0097539703436345. S2CID 15965140. Kuperberg, Greg (2006). "Numerical cubature using error-correcting codes". SIAM Journal on Numerical Analysis. 44 (3): 897–907. arXiv:math/0402047. doi:10.1137/040615572. S2CID 18689951. Kuperberg, Greg (1 May 2014), "Knottedness is in NP, modulo GRH", Advances in Mathematics, 256: 493–506, doi:10.1016/j.aim.2014.01.007 == References == == External links == Greg Kuperberg, faculty page at UC-Davis Greg Kuperberg at the Mathematics Genealogy Project Bits from my personal collection - the original IBM PC and Orion Software
Wikipedia:Gregory Chaitin#0	Gregory John Chaitin ( CHY-tin; born 25 June 1947) is an Argentine-American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, in particular a computer-theoretic result equivalent to Gödel's incompleteness theorem. He is considered to be one of the founders of what is today known as algorithmic (Solomonoff–Kolmogorov–Chaitin, Kolmogorov or program-size) complexity together with Andrei Kolmogorov and Ray Solomonoff. Along with the works of e.g. Solomonoff, Kolmogorov, Martin-Löf, and Leonid Levin, algorithmic information theory became a foundational part of theoretical computer science, information theory, and mathematical logic. It is a common subject in several computer science curricula. Besides computer scientists, Chaitin's work draws attention of many philosophers and mathematicians to fundamental problems in mathematical creativity and digital philosophy. == Mathematics and computer science == Gregory Chaitin is Jewish. He attended the Bronx High School of Science and the City College of New York, where he (still in his teens) developed the theory that led to his independent discovery of algorithmic complexity. Chaitin has defined Chaitin's constant Ω, a real number whose digits are equidistributed and which is sometimes informally described as an expression of the probability that a random program will halt. Ω has the mathematical property that it is definable, with asymptotic approximations from below (but not from above), but not computable. Chaitin is also the originator of using graph coloring to do register allocation in compiling, a process known as Chaitin's algorithm. He was formerly a researcher at IBM's Thomas J. Watson Research Center in New York. He has written more than 10 books that have been translated to about 15 languages. He is today interested in questions of metabiology and information-theoretic formalizations of the theory of evolution, and is a member of the Institute for Advanced Studies at Mohammed VI Polytechnic University. == Other scholarly contributions == Chaitin also writes about philosophy, especially metaphysics and philosophy of mathematics (particularly about epistemological matters in mathematics). In metaphysics, Chaitin claims that algorithmic information theory is the key to solving problems in the field of biology (obtaining a formal definition of 'life', its origin and evolution) and neuroscience (the problem of consciousness and the study of the mind). In recent writings, he defends a position known as digital philosophy. In the epistemology of mathematics, he claims that his findings in mathematical logic and algorithmic information theory show there are "mathematical facts that are true for no reason, that are true by accident". Chaitin proposes that mathematicians must abandon any hope of proving those mathematical facts and adopt a quasi-empirical methodology. == Honors == In 1995 he was given the degree of doctor of science honoris causa by the University of Maine. In 2002 he was given the title of honorary professor by the University of Buenos Aires in Argentina, where his parents were born and where Chaitin spent part of his youth. In 2007 he was given a Leibniz Medal by Wolfram Research. In 2009 he was given the degree of doctor of philosophy honoris causa by the National University of Córdoba. He was formerly a researcher at IBM's Thomas J. Watson Research Center and a professor at the Federal University of Rio de Janeiro. == Bibliography == Information, Randomness & Incompleteness (World Scientific 1987) (online) Algorithmic Information Theory (Cambridge University Press 1987) (online) Information-theoretic Incompleteness (World Scientific 1992) (online) The Limits of Mathematics (Springer-Verlag 1998) (online Archived 25 April 2023 at the Wayback Machine) The Unknowable (Springer-Verlag 1999) (online) Exploring Randomness (Springer-Verlag 2001) (online) Conversations with a Mathematician (Springer-Verlag 2002) (online) From Philosophy to Program Size (Tallinn Cybernetics Institute 2003) Meta Math!: The Quest for Omega (Pantheon Books 2005) (reprinted in UK as Meta Maths: The Quest for Omega, Atlantic Books 2006) (arXiv:math/0404335) Teoria algoritmica della complessità (G. Giappichelli Editore 2006) Thinking about Gödel & Turing (World Scientific 2007) (online Archived 29 April 2023 at the Wayback Machine) Mathematics, Complexity and Philosophy (Editorial Midas 2011) Gödel's Way (CRC Press 2012) Proving Darwin: Making Biology Mathematical (Pantheon Books 2012) (online) Philosophical Mathematics: Infinity, Incompleteness, Irreducibility (Academia.edu 2024) (online) == References == == Further reading == Pagallo, Ugo (2005), Introduzione alla filosofia digitale. Da Leibniz a Chaitin [Introduction to Digital Philosophy: From Leibniz to Chaitin] (in Italian), G. Giappichelli Editore, ISBN 978-88-348-5635-2, archived from the original on 22 July 2011, retrieved 16 April 2008 Calude, Cristian S., ed. (2007), Randomness and Complexity. From Leibniz to Chaitin, World Scientific, ISBN 978-981-277-082-0 Wuppuluri, Shyam; Doria, Francisco A., eds. (2020), Unravelling Complexity: The Life and Work of Gregory Chaitin, World Scientific, doi:10.1142/11270, ISBN 978-981-12-0006-9, S2CID 198790362 == External links == G J Chaitin Home Page from academia.edu G J Chaitin Home Page from UMaine.edu in the Internet Archive Archived 29 October 2013 at the Wayback Machine List of publications of G J Chaitin Video of lecture on metabiology: "Life as evolving software" on YouTube Video of lecture on "Leibniz, complexity and incompleteness" New Scientist article (March, 2001) on Chaitin, Omegas and Super-Omegas A short version of Chaitin's proof Gregory Chaitin extended film interview and transcripts for the 'Why Are We Here?' documentary series Chaitin Lisp on github
Wikipedia:Gregory Eskin#0	Gregory Eskin (Hebrew: גרגורי אסקין, Russian: Григорий Ильич Эскин, born 5 December 1936) is a Russian-Israeli-American mathematician, specializing in partial differential equations. Eskin received in 1963 his Ph.D. (Russian candidate's degree) from Moscow State University with thesis advisor Georgiy Shilov. In 1974 Eskin immigrated with his family to Israel and became a professor at the Hebrew University of Jerusalem. In 1983 he was an invited speaker at the International Congress of Mathematicians at Warsaw. In 1982 he with his family emigrated from Israel to the USA and he became a professor at UCLA. He was elected a Fellow of the American Mathematical Society in 2014. He is married to Marina Eskin, also a mathematician. Their son Alex is a professor of mathematics at the University of Chicago, their other son Eleazar is a professor of computer science and human genetics at UCLA, and their daughter Ascia is a researcher in the Department of Human Genetics, David Geffen UCLA School of Medicine. == Selected publications == === Articles === with Marko Iosifovich Vishik: Vishik, M. I.; Eskin, G. I. (1965). "Equations in convolutions in a bounded region". Russian Mathematical Surveys. 20 (3): 85–151. Bibcode:1965RuMaS..20...85V. doi:10.1070/RM1965v020n03ABEH001184. Eskin, Gregory (1977). "Parametrix and propagation of singularities for the interior mixed hyperbolic problem". Journal d'Analyse Mathématique. 32 (1): 17–62. doi:10.1007/BF02803574. S2CID 121083501. Eskin, G. (2006). "A new approach to hyperbolic inverse problems". Inverse Problems. 22 (3): 815–831. arXiv:math/0505452. Bibcode:2006InvPr..22..815E. doi:10.1088/0266-5611/22/3/005. S2CID 7160768. Eskin, Gregory (2015). "Aharonov-Bohm effect revisited". Rev. Math. Phys. 27 (2): 1530001–113. arXiv:1504.04784. Bibcode:2015RvMaP..2730001E. doi:10.1142/S0129055X15300010. S2CID 119589778. === Books === Краевые задачи для эллиптических псевдодифференциальных уравнений (Boundary problems for elliptic pseudodifferential equations) М.: Наука, (Moscow, Nauka) 1973. — 232 p. Boundary Value Problems for Elliptic Pseudodifferential Equations. American Mathematical Society, 2008. — 375 p. Lectures on Linear Partial Differential Equations. American Mathematical Society, 2011. — 410 p. == References ==
Wikipedia:Gregory Gutin#0	Gregory Z. Gutin (Hebrew: גרגורי גוטין; born 17 January 1957) is a scholar in theoretical computer science and discrete mathematics. He received his PhD in Mathematics in 1993 from Tel Aviv University under the supervision of Noga Alon. Since September 2000 Gutin has been Professor in Computer Science at Royal Holloway, University of London. Gutin's research interests are in algorithms and complexity, access control, graph theory and combinatorial optimization. == Publications == Gutin, G.; Punnen, A. P. (May 2006). The Traveling Salesman Problem and Its Variations. Springer. ISBN 978-0-306-48213-7. Bang-Jensen, Jørgen; Gutin, Gregory Z. (December 2008). Digraphs: Theory, Algorithms and Applications. Springer. ISBN 978-1-84800-998-1. Bang-Jensen, J.; Gutin, G. (2018). Classes of Directed Graphs. Springer. ISBN 978-3-319-71840-8. == Awards and honours == Gutin was the recipient of the Royal Society Wolfson Research Merit Award in 2014, and the best paper awards at SACMAT 2015, 2016 and 2021. In January 2017 there was a workshop celebrating Gutin's 60th birthday. In 2017, he became a member of Academia Europaea. == References == == External links == Official website
Wikipedia:Gregory coefficients#0	Gregory coefficients Gn, also known as reciprocal logarithmic numbers, Bernoulli numbers of the second kind, or Cauchy numbers of the first kind, are the rational numbers that occur in the Maclaurin series expansion of the reciprocal logarithm z ln ⁡ ( 1 + z ) = 1 + 1 2 z − 1 12 z 2 + 1 24 z 3 − 19 720 z 4 + 3 160 z 5 − 863 60480 z 6 + ⋯ = 1 + ∑ n = 1 ∞ G n z n , \| z \| < 1 . {\displaystyle {\begin{aligned}{\frac {z}{\ln(1+z)}}&=1+{\frac {1}{2}}z-{\frac {1}{12}}z^{2}+{\frac {1}{24}}z^{3}-{\frac {19}{720}}z^{4}+{\frac {3}{160}}z^{5}-{\frac {863}{60480}}z^{6}+\cdots \\&=1+\sum _{n=1}^{\infty }G_{n}z^{n}\,,\qquad \|z\|<1\,.\end{aligned}}} Gregory coefficients are alternating Gn = (−1)n−1\|Gn\| for n > 0 and decreasing in absolute value. These numbers are named after James Gregory who introduced them in 1670 in the numerical integration context. They were subsequently rediscovered by many mathematicians and often appear in works of modern authors, who do not always recognize them. == Numerical values == == Computation and representations == The simplest way to compute Gregory coefficients is to use the recurrence formula \| G n \| = − ∑ k = 1 n − 1 \| G k \| n + 1 − k + 1 n + 1 {\displaystyle \|G_{n}\|=-\sum _{k=1}^{n-1}{\frac {\|G_{k}\|}{n+1-k}}+{\frac {1}{n+1}}} with G1 = ⁠1/2⁠. Gregory coefficients may be also computed explicitly via the following differential n ! G n = [ d n d z n z ln ⁡ ( 1 + z ) ] z = 0 , {\displaystyle n!G_{n}=\left[{\frac {{\textrm {d}}^{n}}{{\textrm {d}}z^{n}}}{\frac {z}{\ln(1+z)}}\right]_{z=0},} or the integral G n = 1 n ! ∫ 0 1 x ( x − 1 ) ( x − 2 ) ⋯ ( x − n + 1 ) d x = ∫ 0 1 ( x n ) d x , {\displaystyle G_{n}={\frac {1}{n!}}\int _{0}^{1}x(x-1)(x-2)\cdots (x-n+1)\,dx=\int _{0}^{1}{\binom {x}{n}}\,dx,} which can be proved by integrating ( 1 + z ) x {\displaystyle (1+z)^{x}} between 0 and 1 with respect to x {\displaystyle x} , once directly and the second time using the binomial series expansion first. It implies the finite summation formula n ! G n = ∑ ℓ = 0 n s ( n , ℓ ) ℓ + 1 , {\displaystyle n!G_{n}=\sum _{\ell =0}^{n}{\frac {s(n,\ell )}{\ell +1}},} where s(n,ℓ) are the signed Stirling numbers of the first kind. and Schröder's integral formula G n = ( − 1 ) n − 1 ∫ 0 ∞ d x ( 1 + x ) n ( ln 2 ⁡ x + π 2 ) , {\displaystyle G_{n}=(-1)^{n-1}\int _{0}^{\infty }{\frac {dx}{(1+x)^{n}(\ln ^{2}x+\pi ^{2})}},} == Bounds and asymptotic behavior == The Gregory coefficients satisfy the bounds 1 6 n ( n − 1 ) < \| G n \| < 1 6 n , n > 2 , {\displaystyle {\frac {1}{6n(n-1)}}<{\big \|}G_{n}{\big \|}<{\frac {1}{6n}},\qquad n>2,} given by Johan Steffensen. These bounds were later improved by various authors. The best known bounds for them were given by Blagouchine. In particular, 1 n ln 2 n − 2 n ln 3 n ⩽ \| G n \| ⩽ 1 n ln 2 n − 2 γ n ln 3 n , n ⩾ 5 . {\displaystyle {\frac {\,1\,}{\,n\ln ^{2}\!n\,}}\,-\,{\frac {\,2\,}{\,n\ln ^{3}\!n\,}}\leqslant \,{\big \|}G_{n}{\big \|}\,\leqslant \,{\frac {\,1\,}{\,n\ln ^{2}\!n\,}}-{\frac {\,2\gamma \,}{\,n\ln ^{3}\!n\,}}\,,\qquad \quad n\geqslant 5\,.} Asymptotically, at large index n, these numbers behave as \| G n \| ∼ 1 n ln 2 ⁡ n , n → ∞ . {\displaystyle {\big \|}G_{n}{\big \|}\sim {\frac {1}{n\ln ^{2}n}},\qquad n\to \infty .} More accurate description of Gn at large n may be found in works of Van Veen, Davis, Coffey, Nemes and Blagouchine. == Series with Gregory coefficients == Series involving Gregory coefficients may be often calculated in a closed-form. Basic series with these numbers include ∑ n = 1 ∞ \| G n \| = 1 ∑ n = 1 ∞ G n = 1 ln ⁡ 2 − 1 ∑ n = 1 ∞ \| G n \| n = γ , {\displaystyle {\begin{aligned}&\sum _{n=1}^{\infty }{\big \|}G_{n}{\big \|}=1\\[2mm]&\sum _{n=1}^{\infty }G_{n}={\frac {1}{\ln 2}}-1\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n}}=\gamma ,\end{aligned}}} where γ = 0.5772156649... is Euler's constant. These results are very old, and their history may be traced back to the works of Gregorio Fontana and Lorenzo Mascheroni. More complicated series with the Gregory coefficients were calculated by various authors. Kowalenko, Alabdulmohsin and some other authors calculated ∑ n = 2 ∞ \| G n \| n − 1 = − 1 2 + ln ⁡ 2 π 2 − γ 2 ∑ n = 1 ∞ \| G n \| n + 1 = 1 − ln ⁡ 2. {\displaystyle {\begin{array}{l}\displaystyle \sum _{n=2}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n-1}}=-{\frac {1}{2}}+{\frac {\ln 2\pi }{2}}-{\frac {\gamma }{2}}\\[6mm]\displaystyle \displaystyle \sum _{n=1}^{\infty }\!{\frac {{\big \|}G_{n}{\big \|}}{n+1}}=1-\ln 2.\end{array}}} Alabdulmohsin also gives these identities with ∑ n = 0 ∞ ( − 1 ) n ( \| G 3 n + 1 \| + \| G 3 n + 2 \| ) = 3 π ∑ n = 0 ∞ ( − 1 ) n ( \| G 3 n + 2 \| + \| G 3 n + 3 \| ) = 2 3 π − 1 ∑ n = 0 ∞ ( − 1 ) n ( \| G 3 n + 3 \| + \| G 3 n + 4 \| ) = 1 2 − 3 π . {\displaystyle {\begin{aligned}&\sum _{n=0}^{\infty }(-1)^{n}({\big \|}G_{3n+1}{\big \|}+{\big \|}G_{3n+2}{\big \|})={\frac {\sqrt {3}}{\pi }}\\[2mm]&\sum _{n=0}^{\infty }(-1)^{n}({\big \|}G_{3n+2}{\big \|}+{\big \|}G_{3n+3}{\big \|})={\frac {2{\sqrt {3}}}{\pi }}-1\\[2mm]&\sum _{n=0}^{\infty }(-1)^{n}({\big \|}G_{3n+3}{\big \|}+{\big \|}G_{3n+4}{\big \|})={\frac {1}{2}}-{\frac {\sqrt {3}}{\pi }}.\end{aligned}}} Candelperger, Coppo and Young showed that ∑ n = 1 ∞ \| G n \| ⋅ H n n = π 2 6 − 1 , {\displaystyle \sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}\cdot H_{n}}{n}}={\frac {\pi ^{2}}{6}}-1,} where Hn are the harmonic numbers. Blagouchine provides the following identities ∑ n = 1 ∞ G n n = li ⁡ ( 2 ) − γ ∑ n = 3 ∞ \| G n \| n − 2 = − 1 8 + ln ⁡ 2 π 12 − ζ ′ ( 2 ) 2 π 2 ∑ n = 4 ∞ \| G n \| n − 3 = − 1 16 + ln ⁡ 2 π 24 − ζ ′ ( 2 ) 4 π 2 + ζ ( 3 ) 8 π 2 ∑ n = 1 ∞ \| G n \| n + 2 = 1 2 − 2 ln ⁡ 2 + ln ⁡ 3 ∑ n = 1 ∞ \| G n \| n + 3 = 1 3 − 5 ln ⁡ 2 + 3 ln ⁡ 3 ∑ n = 1 ∞ \| G n \| n + k = 1 k + ∑ m = 1 k ( − 1 ) m ( k m ) ln ⁡ ( m + 1 ) , k = 1 , 2 , 3 , … ∑ n = 1 ∞ \| G n \| n 2 = ∫ 0 1 − li ⁡ ( 1 − x ) + γ + ln ⁡ x x d x ∑ n = 1 ∞ G n n 2 = ∫ 0 1 li ⁡ ( 1 + x ) − γ − ln ⁡ x x d x , {\displaystyle {\begin{aligned}&\sum _{n=1}^{\infty }{\frac {G_{n}}{n}}=\operatorname {li} (2)-\gamma \\[2mm]&\sum _{n=3}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n-2}}=-{\frac {1}{8}}+{\frac {\ln 2\pi }{12}}-{\frac {\zeta '(2)}{\,2\pi ^{2}}}\\[2mm]&\sum _{n=4}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n-3}}=-{\frac {1}{16}}+{\frac {\ln 2\pi }{24}}-{\frac {\zeta '(2)}{4\pi ^{2}}}+{\frac {\zeta (3)}{8\pi ^{2}}}\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n+2}}={\frac {1}{2}}-2\ln 2+\ln 3\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n+3}}={\frac {1}{3}}-5\ln 2+3\ln 3\\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n+k}}={\frac {1}{k}}+\sum _{m=1}^{k}(-1)^{m}{\binom {k}{m}}\ln(m+1)\,,\qquad k=1,2,3,\ldots \\[2mm]&\sum _{n=1}^{\infty }{\frac {{\big \|}G_{n}{\big \|}}{n^{2}}}=\int _{0}^{1}{\frac {-\operatorname {li} (1-x)+\gamma +\ln x}{x}}\,dx\\[2mm]&\sum _{n=1}^{\infty }{\frac {G_{n}}{n^{2}}}=\int _{0}^{1}{\frac {\operatorname {li} (1+x)-\gamma -\ln x}{x}}\,dx,\end{aligned}}} where li(z) is the integral logarithm and ( k m ) {\displaystyle {\tbinom {k}{m}}} is the binomial coefficient. It is also known that the zeta function, the gamma function, the polygamma functions, the Stieltjes constants and many other special functions and constants may be expressed in terms of infinite series containing these numbers. == Generalizations == Various generalizations are possible for the Gregory coefficients. Many of them may be obtained by modifying the parent generating equation. For example, Van Veen consider ( ln ⁡ ( 1 + z ) z ) s = s ∑ n = 0 ∞ z n n ! K n ( s ) , \| z \| < 1 , {\displaystyle \left({\frac {\ln(1+z)}{z}}\right)^{s}=s\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}K_{n}^{(s)}\,,\qquad \|z\|<1\,,} and hence n ! G n = − K n ( − 1 ) {\displaystyle n!G_{n}=-K_{n}^{(-1)}} Equivalent generalizations were later proposed by Kowalenko and Rubinstein. In a similar manner, Gregory coefficients are related to the generalized Bernoulli numbers ( t e t − 1 ) s = ∑ k = 0 ∞ t k k ! B k ( s ) , \| t \| < 2 π , {\displaystyle \left({\frac {t}{e^{t}-1}}\right)^{s}=\sum _{k=0}^{\infty }{\frac {t^{k}}{k!}}B_{k}^{(s)},\qquad \|t\|<2\pi \,,} see, so that n ! G n = − B n ( n − 1 ) n − 1 {\displaystyle n!G_{n}=-{\frac {B_{n}^{(n-1)}}{n-1}}} Jordan defines polynomials ψn(s) such that z ( 1 + z ) s ln ⁡ ( 1 + z ) = ∑ n = 0 ∞ z n ψ n ( s ) , \| z \| < 1 , {\displaystyle {\frac {z(1+z)^{s}}{\ln(1+z)}}=\sum _{n=0}^{\infty }z^{n}\psi _{n}(s)\,,\qquad \|z\|<1\,,} and call them Bernoulli polynomials of the second kind. From the above, it is clear that Gn = ψn(0). Carlitz generalized Jordan's polynomials ψn(s) by introducing polynomials β ( z ln ⁡ ( 1 + z ) ) s ⋅ ( 1 + z ) x = ∑ n = 0 ∞ z n n ! β n ( s ) ( x ) , \| z \| < 1 , {\displaystyle \left({\frac {z}{\ln(1+z)}}\right)^{s}\!\!\cdot (1+z)^{x}=\sum _{n=0}^{\infty }{\frac {z^{n}}{n!}}\,\beta _{n}^{(s)}(x)\,,\qquad \|z\|<1\,,} and therefore n ! G n = β n ( 1 ) ( 0 ) {\displaystyle n!G_{n}=\beta _{n}^{(1)}(0)} Blagouchine introduced numbers Gn(k) such that n ! G n ( k ) = ∑ ℓ = 1 n s ( n , ℓ ) ℓ + k , {\displaystyle n!G_{n}(k)=\sum _{\ell =1}^{n}{\frac {s(n,\ell )}{\ell +k}},} obtained their generating function and studied their asymptotics at large n. Clearly, Gn = Gn(1). These numbers are strictly alternating Gn(k) = (-1)n-1\|Gn(k)\| and involved in various expansions for the zeta-functions, Euler's constant and polygamma functions. A different generalization of the same kind was also proposed by Komatsu c n ( k ) = ∑ ℓ = 0 n s ( n , ℓ ) ( ℓ + 1 ) k , {\displaystyle c_{n}^{(k)}=\sum _{\ell =0}^{n}{\frac {s(n,\ell )}{(\ell +1)^{k}}},} so that Gn = cn(1)/n! Numbers cn(k) are called by the author poly-Cauchy numbers. Coffey defines polynomials P n + 1 ( y ) = 1 n ! ∫ 0 y x ( 1 − x ) ( 2 − x ) ⋯ ( n − 1 − x ) d x {\displaystyle P_{n+1}(y)={\frac {1}{n!}}\int _{0}^{y}x(1-x)(2-x)\cdots (n-1-x)\,dx} and therefore \|Gn\| = Pn+1(1). == See also == Stirling polynomials Bernoulli polynomials of the second kind == References ==
Wikipedia:Gresham Professor of Geometry#0	The Professor of Geometry at Gresham College, London, gives free educational lectures to the general public. The college was founded for this purpose in 1597, when it created seven professorships; this was later increased to ten. Geometry is one of the original professorships as set out by the will of Thomas Gresham in 1575. The Professor of Geometry is appointed in partnership with the City of London Corporation. == List of Gresham Professors of Geometry == Note, the notation used in dates given as, for example, "1596⁄7" refer to the practice of dual dating dates between 1 January and 25 March. == References == Gresham College old website, Internet Archive List of professors Gresham College website Profile of current professor and list of past professors == Notes == == External links == '400 Years of Geometry at Gresham College', lecture by Robin Wilson at Gresham College, 14 May 2008 (available for download as PDF, audio and video files) == Further reading == Ward, John (1740). The Lives of the Professors of Gresham College. New York & London: Johnson Reprint Corporation.
Wikipedia:Griess algebra#0	In mathematics, the Griess algebra is a commutative non-associative algebra on a real vector space of dimension 196884 that has the Monster group M as its automorphism group. It is named after mathematician R. L. Griess, who constructed it in 1980 and subsequently used it in 1982 to construct M. The Monster fixes (vectorwise) a 1-space in this algebra and acts absolutely irreducibly on the 196883-dimensional orthogonal complement of this 1-space. (The Monster preserves the standard inner product on the 196884-space.) Griess's construction was later simplified by Jacques Tits and John H. Conway. The Griess algebra is the same as the degree 2 piece of the monster vertex algebra, and the Griess product is one of the vertex algebra products. == References == Conway, John Horton (1985), "A simple construction for the Fischer-Griess monster group", Inventiones Mathematicae, 79 (3): 513–540, Bibcode:1985InMat..79..513C, doi:10.1007/BF01388521, ISSN 0020-9910, MR 0782233 R. L. Griess Jr, The Friendly Giant, Inventiones Mathematicae 69 (1982), 1-102
Wikipedia:Grigori Milstein#0	Grigori N. Milstein (Russian: Григорий Нойхович Мильштейн; 6 June 1937 – 22 November 2023) was a Russian mathematician who made many important contributions to Stochastic Numerics, Estimation, Control, Stability theory, Financial Mathematics. == Biography == G.N. Milstein received his undergraduate degree in mathematics from the Ural State University (UrGU; Sverdlovsk, USSR), which is now Ural Federal University (Ekaterinburg, Russia). He completed his PhD studies at the same university. Milstein has been an assistant professor, associate professor and, after defending his DSc thesis, professor at the Faculty of Mathematics and Mechanics UrGU (then URFU). He also worked as senior researcher at the Weierstrass Institute for Applied Analysis and Stochastics (Berlin, Germany) and was a visiting professor at University of Leicester (Leicester, UK) and University of Manchester (Manchester, UK). == Research == Milstein was a world-leading expert in Stochastic Numerics, Estimation, Control, Stability, Financial Mathematics. He published four research monographs: The first of the listed books was the first monograph in the world published on the topic of numerical methods for stochastic differential equations. He also contributed to the second edition of R. Khasminskii "Stochastic Stability of Differential Equations", Springer, 2012. He has published more than 100 journal papers. In Milstein's early pioneering paper on Stochastic Numerics (1974,1975), he constructed a first-order mean-square method for SDEs that is known as Milstein method. In 1978, Milstein introduced weak-sense approximations of SDEs for the first time and proposed a number of weak schemes. These papers became classics and now are the basis of the modern theory of numerical integration of stochastic differential equations. In 1985-1987 Professor Milstein proved fundamental convergence theorems in the mean-square and weak sense, respectively, which became the foundation for constructing and analysing numerical methods for SDEs. == References == == External links == Grigori N. Milstein, List of publications Workshop Milstein's method: 50 years on, 30 June - 3 July 2025
Wikipedia:Grigorii Fikhtengol'ts#0	Grigorii Mikhailovich Fikhtengol'ts (Russian: Григо́рий Миха́йлович Фихтенго́льц, Ukrainian: Григорій Михайлович Фіхтенгольц, romanized: Hryhorii Mykhailovych Fikhtenholts; 8 June 1888 – 26 June 1959) was a Soviet mathematician working on real analysis and functional analysis. Fikhtengol'ts was one of the founders of the Leningrad school of real analysis. He was born in Odesa, Russian Empire in 1888, and graduated Odesa University in 1911. He authored a three-volume textbook titled "A Course of Differential and Integral Calculus". The textbook covers mathematical analysis of functions of one real variable, functions of many real variables, and complex functions. Due to the depth and precision of the material's presentation, the book holds a classical position in the mathematical literature. It has been translated into several languages, including German, Ukrainian, Polish, Chinese, Vietnamese, and Persian. However, no English translation has been completed yet. Fikhtengol'ts's books on analysis are widely used in Middle and Eastern European, as well as Chinese universities, due to their exceptionally detailed and well-organized presentation of material on mathematical analysis. For unknown reasons, these books have not gained the same level of fame in universities in other parts of the world. He was an Invited Speaker of the ICM in 1924 in Toronto. Leonid Kantorovich and Isidor Natanson were among his students. == Books == Grigorii Mikhailovich Fikhtengol'ts (1965). The Fundamentals of Mathematical Analysis. Vol. 1. Pergamon Press. ISBN 9781483139074. Grigorii Mikhailovich Fikhtengol'ts (1965). The Fundamentals of Mathematical Analysis. Vol. 2. Pergamon Press. ISBN 9781483154138. == References == Kantorovič, L. V.; Natanson, I. P. (1958). "Grigoriĭ Mihaĭlovič Fihtengolʹc (on his seventieth birthday)". Vestnik Leningrad. Univ. (in Russian). 13 (7): 5–13. MR 0099901. Kantorovič, L. V.; Natanson, I. P. (1959). "Grigoriĭ Mihaĭlovič Fihtengolʹc. Necrologue". Uspekhi Mat. Nauk (in Russian). 14 (5 (89)): 123–128. MR 0110626. Bogachev, V. I. (2005). "On the works of G. M. Fikhtengolʹts on the theory of the integral". Istor.-Mat. Issled. (2) (in Russian). 9 (44). MR 2365093. == External links == Grigorii Mikhailovich Fichtenholz Grigorii Fikhtengol'ts at the Mathematics Genealogy Project
Wikipedia:Griselda Pascual#0	Griselda, also spelled Grizelda, is a feminine given name from Germanic sources that is now used in English, Italian, and Spanish as well. According to the 1990 United States Census, the name was 1,066th in popularity among females in the United States. The name likely specifically stems from the Proto-Germanic language elements grīsaz, "grey", and hildiz, meaning "battle" (compare modern German grau and Held), thus literally "gray battle-maid". As a figure in European folklore, Griselda is noted for her patience and obedience and has been depicted in works of art, literature and opera. The name can also be spelled "Griselde", "Grisselda", "Grieselda", "Grizelda", "Gricelda", and "Criselda". Common nicknames include "Zelda", "Selda", "Grissy", "Gris", "Grisel", "Grizel" or "Crisel" People named Griselda or Grizelda include: Griselda Álvarez (1913–2009), first female governor in Mexico Griselda Báthory (1569–1590), Hungarian and Polish noblewoman Griselda Blanco (1943–2012), a former Medellín Cartel drug lord Grizelda Cjiekella (died 2015), South African politician Griselda Delgado del Carpio (born 1955), Bolivian bishop Griselda El Tayib (1925–2022), British-born visual artist and cultural anthropologist Griselda Gambaro (born 1928), Argentine writer Griselda González (born 1965), Argentine former long-distance runner Griselda Hinojosa (1875–1959), first female pharmacist in Chile Grizelda Kristiņa (1910–2013), last native speaker of the Livonian language Griselda Pascual (1926–2001), Spanish Catalan mathematician Griselda Pollock (born 1949), British art historian, cultural analyst and scholar Griselda Steevens (1653–1746), Irish philanthropist and benefactor of Dr Steevens' Hospital in Dublin Griselda Tessio (born 1947), vice-governor of the Argentine province of Santa Fe Grizelda Elizabeth Cottnam Tonge (1803–1825), Nova Scotian poet == References ==
Wikipedia:Grosshans subgroup#0	In mathematics, in the representation theory of algebraic groups, a Grosshans subgroup, named after Frank Grosshans, is an algebraic subgroup of an algebraic group that is an observable subgroup for which the ring of functions on the quotient variety is finitely generated. == References == == External links == Invariants of Unipotent subgroups
Wikipedia:Groupoid algebra#0	In abstract algebra, a magma, binar, or, rarely, groupoid is a basic kind of algebraic structure. Specifically, a magma consists of a set equipped with a single binary operation that must be closed by definition. No other properties are imposed. == History and terminology == The term groupoid was introduced in 1927 by Heinrich Brandt describing his Brandt groupoid. The term was then appropriated by B. A. Hausmann and Øystein Ore (1937) in the sense (of a set with a binary operation) used in this article. In a couple of reviews of subsequent papers in Zentralblatt, Brandt strongly disagreed with this overloading of terminology. The Brandt groupoid is a groupoid in the sense used in category theory, but not in the sense used by Hausmann and Ore. Nevertheless, influential books in semigroup theory, including Clifford and Preston (1961) and Howie (1995) use groupoid in the sense of Hausmann and Ore. Hollings (2014) writes that the term groupoid is "perhaps most often used in modern mathematics" in the sense given to it in category theory. According to Bergman and Hausknecht (1996): "There is no generally accepted word for a set with a not necessarily associative binary operation. The word groupoid is used by many universal algebraists, but workers in category theory and related areas object strongly to this usage because they use the same word to mean 'category in which all morphisms are invertible'. The term magma was used by Serre [Lie Algebras and Lie Groups, 1965]." It also appears in Bourbaki's Éléments de mathématique, Algèbre, chapitres 1 à 3, 1970. == Definition == A magma is a set M matched with an operation • that sends any two elements a, b ∈ M to another element, a • b ∈ M. The symbol • is a general placeholder for a properly defined operation. To qualify as a magma, the set and operation (M, •) must satisfy the following requirement (known as the magma or closure property): For all a, b in M, the result of the operation a • b is also in M. And in mathematical notation: a , b ∈ M ⟹ a ⋅ b ∈ M . {\displaystyle a,b\in M\implies a\cdot b\in M.} If • is instead a partial operation, then (M, •) is called a partial magma or, more often, a partial groupoid. == Morphism of magmas == A morphism of magmas is a function f : M → N that maps magma (M, •) to magma (N, ∗) that preserves the binary operation: f (x • y) = f(x) ∗ f(y). For example, with M equal to the positive real numbers and • as the geometric mean, N equal to the real number line, and ∗ as the arithmetic mean, a logarithm f is a morphism of the magma (M, •) to (N, ∗). proof: log ⁡ x y = log ⁡ x + log ⁡ y 2 {\displaystyle \log {\sqrt {xy}}\ =\ {\frac {\log x+\log y}{2}}} Note that these commutative magmas are not associative; nor do they have an identity element. This morphism of magmas has been used in economics since 1863 when W. Stanley Jevons calculated the rate of inflation in 39 commodities in England in his A Serious Fall in the Value of Gold Ascertained, page 7. == Notation and combinatorics == The magma operation may be applied repeatedly, and in the general, non-associative case, the order matters, which is notated with parentheses. Also, the operation • is often omitted and notated by juxtaposition: (a • (b • c)) • d ≡ (a(bc))d. A shorthand is often used to reduce the number of parentheses, in which the innermost operations and pairs of parentheses are omitted, being replaced just with juxtaposition: xy • z ≡ (x • y) • z. For example, the above is abbreviated to the following expression, still containing parentheses: (a • bc)d. A way to avoid completely the use of parentheses is prefix notation, in which the same expression would be written ••a•bcd. Another way, familiar to programmers, is postfix notation (reverse Polish notation), in which the same expression would be written abc••d•, in which the order of execution is simply left-to-right (no currying). The set of all possible strings consisting of symbols denoting elements of the magma, and sets of balanced parentheses is called the Dyck language. The total number of different ways of writing n applications of the magma operator is given by the Catalan number Cn. Thus, for example, C2 = 2, which is just the statement that (ab)c and a(bc) are the only two ways of pairing three elements of a magma with two operations. Less trivially, C3 = 5: ((ab)c)d, (a(bc))d, (ab)(cd), a((bc)d), and a(b(cd)). There are nn2 magmas with n elements, so there are 1, 1, 16, 19683, 4294967296, ... (sequence A002489 in the OEIS) magmas with 0, 1, 2, 3, 4, ... elements. The corresponding numbers of non-isomorphic magmas are 1, 1, 10, 3330, 178981952, ... (sequence A001329 in the OEIS) and the numbers of simultaneously non-isomorphic and non-antiisomorphic magmas are 1, 1, 7, 1734, 89521056, ... (sequence A001424 in the OEIS). == Free magma == A free magma MX on a set X is the "most general possible" magma generated by X (i.e., there are no relations or axioms imposed on the generators; see free object). The binary operation on MX is formed by wrapping each of the two operands in parentheses and juxtaposing them in the same order. For example: a • b = (a)(b), a • (a • b) = (a)((a)(b)), (a • a) • b = ((a)(a))(b). MX can be described as the set of non-associative words on X with parentheses retained. It can also be viewed, in terms familiar in computer science, as the magma of full binary trees with leaves labelled by elements of X. The operation is that of joining trees at the root. A free magma has the universal property such that if f : X → N is a function from X to any magma N, then there is a unique extension of f to a morphism of magmas f′ f′ : MX → N. == Types of magma == Magmas are not often studied as such; instead there are several different kinds of magma, depending on what axioms the operation is required to satisfy. Commonly studied types of magma include: Quasigroup: A magma where division is always possible. Loop: A quasigroup with an identity element. Semigroup: A magma where the operation is associative. Monoid: A semigroup with an identity element. Group: A magma with inverse, associativity, and an identity element. Note that each of divisibility and invertibility imply the cancellation property. Magmas with commutativity Commutative magma: A magma with commutativity. Commutative monoid: A monoid with commutativity. Abelian group: A group with commutativity. == Classification by properties == A magma (S, •), with x, y, u, z ∈ S, is called Medial If it satisfies the identity xy • uz ≡ xu • yz Left semimedial If it satisfies the identity xx • yz ≡ xy • xz Right semimedial If it satisfies the identity yz • xx ≡ yx • zx Semimedial If it is both left and right semimedial Left distributive If it satisfies the identity x • yz ≡ xy • xz Right distributive If it satisfies the identity yz • x ≡ yx • zx Autodistributive If it is both left and right distributive Commutative If it satisfies the identity xy ≡ yx Idempotent If it satisfies the identity xx ≡ x Unipotent If it satisfies the identity xx ≡ yy Zeropotent If it satisfies the identities xx • y ≡ xx ≡ y • xx Alternative If it satisfies the identities xx • y ≡ x • xy and x • yy ≡ xy • y Power-associative If the submagma generated by any element is associative Flexible if xy • x ≡ x • yx Associative If it satisfies the identity x • yz ≡ xy • z, called a semigroup A left unar If it satisfies the identity xy ≡ xz A right unar If it satisfies the identity yx ≡ zx Semigroup with zero multiplication, or null semigroup If it satisfies the identity xy ≡ uv Unital If it has an identity element Left-cancellative If, for all x, y, z, relation xy = xz implies y = z Right-cancellative If, for all x, y, z, relation yx = zx implies y = z Cancellative If it is both right-cancellative and left-cancellative A semigroup with left zeros If it is a semigroup and it satisfies the identity xy ≡ x A semigroup with right zeros If it is a semigroup and it satisfies the identity yx ≡ x Trimedial If any triple of (not necessarily distinct) elements generates a medial submagma Entropic If it is a homomorphic image of a medial cancellation magma. Central If it satisfies the identity xy • yz ≡ y == Number of magmas satisfying given properties == == Category of magmas == The category of magmas, denoted Mag, is the category whose objects are magmas and whose morphisms are magma homomorphisms. The category Mag has direct products, and there is an inclusion functor: Set ↪ Mag as trivial magmas, with operations given by projection x T y = y . More generally, because Mag is algebraic, it is a complete category. An important property is that an injective endomorphism can be extended to an automorphism of a magma extension, just the colimit of the (constant sequence of the) endomorphism. == See also == Magma category Universal algebra Magma computer algebra system, named after the object of this article. Commutative magma Algebraic structures whose axioms are all identities Groupoid algebra Hall set == References == == Further reading == Bruck, Richard Hubert (1971), A survey of binary systems (3rd ed.), Springer, ISBN 978-0-387-03497-3
Wikipedia:Groups, Geometry, and Dynamics#0	Groups, Geometry, and Dynamics is a quarterly peer-reviewed mathematics journal published quarterly by the European Mathematical Society. It was established in 2007 and covers all aspects of groups, group actions, geometry and dynamical systems. The journal is indexed by Mathematical Reviews and Zentralblatt MATH. Its 2009 MCQ was 0.65, and its 2012 impact factor is 0.867. == External links == Official website
Wikipedia:Grzegorz Rempala#0	Grzegorz (“Greg”) A. Rempala (Polish: Rempała; born March 19, 1968) is a Polish-American applied mathematician who works on the theory and applications of complex stochastic systems. == Biography == Rempala studied mathematics at the University of Warsaw from 1987 to 1991, and worked at the Computer Science Institute of the Polish Academy of Sciences from 1991 to 1992. In 1992 he moved to the US where in 1996 he completed his PhD thesis at Bowling Green State University. His advisor was Prof. Arjun K Gupta. In 1998 he nostrified his degree at the University of Warsaw in the Department of Mathematics. The chair of his nostrification committee was Prof. Stanisław Kwapień. In 2007 he received his habilitation from Warsaw University of Technology. From 1996 until 2008 Rempala was a professor (full professor from 2007) in the Department of Mathematics at the University of Louisville. In 2008 he joint the Department of Biostatistics at the Medical College of Georgia and in 2013 moved to The Ohio State University (OSU) where he is currently a professor in the Division of Biostatistics and in the Department of Mathematics. In 2016 he was named the interim director of the Mathematical Biosciences Institute at OSU, and served in this position until 2018. His wife, Helena Rempala, is a professor of clinical psychology in the Department of Psychiatry and Behavioral Health at OSU. They have two sons, Jaś and Antoś. He is a collaborator of the IBS Biomedical Mathematics Group. == Scientific interests == Rempala is known for his work on random matrices, in particular on a random permanent function. He has also established some results in nonparametric statistics related to central limit theorems for products of random variables. More recently, he has worked on mathematical models of chemical reactions, diversity of molecular populations and disease spread across contact networks. == See also == Computing the permanent Central limit theorem == References == == External links == Greg Rempala's Web Page
Wikipedia:Grzegorz Rozenberg#0	Grzegorz Rozenberg (born 14 March 1942, Warsaw) is a Polish and Dutch computer scientist. His primary research areas are natural computing, formal language and automata theory, graph transformations, and concurrent systems. He is referred to as the guru of natural computing, as he was promoting the vision of natural computing as a coherent scientific discipline already in the 1970s, gave this discipline its current name, and defined its scope. His research career spans over forty five years. He is a professor at the Leiden Institute of Advanced Computer Science of Leiden University, The Netherlands and adjoint professor at the department of computer science, University of Colorado at Boulder, USA. Rozenberg is also a performing magician, with the artist name Bolgani and specializing in close-up illusions. He is the father of well-known Dutch artist Dadara. == Education and career == Rozenberg received his Master and Engineer degrees in computer science from the Warsaw University of Technology in Warsaw, Poland. He obtained a Ph.D. in mathematics from the Polish Academy of Sciences also in Warsaw in 1968. Since then he has held full-time positions at the Polish Academy of Sciences, Warsaw, Poland (assistant professor), Utrecht University, The Netherlands (assistant professor), State University of New York at Buffalo, USA (associate professor), and University of Antwerp (UIA), Belgium (professor). Since 1979 he has been a professor of computer science at Leiden University, The Netherlands and adjoint professor at the Department of Computer Science of University of Colorado at Boulder, US. == Publications and editorial functions == Rozenberg has authored over 500 papers, 6 books, and (co-)edited over 100 books and special issues of scientific journals. He was also a (co-)editor of four handbooks: "Handbook of Formal Languages" (3 volumes, Springer-Verlag), "Handbook of Graph Grammars and Computing by Graph Transformation", "Handbook of Membrane Computing" (Oxford University Press), and the "Handbook of Natural Computing" (4 volumes, Springer-Verlag). He is on the editorial/advisory board of about 20 journals, and is the editor-in-chief and either the founder or a co-founder of the following journals and book series: International Journal on Natural Computing (Springer-Verlag), Theoretical Computer Science C: Theory of Natural Computing (Elsevier), Monographs in Theoretical Computer Science (Springer-Verlag), Texts in Theoretical Computer Science (Springer-Verlag), and Natural Computing book Series (Springer-Verlag). == Functions in the academic community == G. Rozenberg either founded or co-founded and/or was the chair of the following conferences: International Conference on Developments in Language Theory, International Conference on Graph Transformation, International Conference on Unconventional Computation, International Conference on Theory and Applications of Petri Nets, and the International Meeting on DNA Computing. Rozenberg was president of the European Association for Theoretical Computer Science from 1985 to 1994 (the longest term in that position) and the editor of the Bulletin of the European Association for Theoretical Computer Science from 1980 until 2003. He also was the president of the International Society for Nanoscale Science, Computation and Engineering, the director of European Molecular Computing Consortium, and the chair of European Educational Forum. == Awards and recognition == G. Rozenberg is a Foreign Member of the Finnish Academy of Sciences and Letters, a member of Academia Europaea, and the holder of Honorary Doctorates of the University of Turku, Finland, Technische Universität Berlin, Germany, the University of Bologna, Italy, the Swedish University Åbo Akademi in Turku, Finland, and the Warsaw University of Technology, Poland. He has received the Distinguished Achievements Award of the European Association for Theoretical Computer Science "in recognition of his outstanding scientific contributions to theoretical computer science". He is an ISI highly cited researcher. Several books and special issues of scientific journals have been dedicated to G. Rozenberg. Also an annual award granted by International Society for Nanoscale Science, Computation, and Engineering was named after G. Rozenberg. It is called Rozenberg Tulip Award and it is awarded for outstanding achievements in the field of Biomolecular Computing and Molecular Programming. In 2017, Grzegorz Rozenberg has been appointed Knight in the Order of the Netherlands Lion. == References == == External links == Home page of Grzegorz Rozenberg .
Wikipedia:Grzegorz Świątek#0	Grzegorz Świątek (born 1964) is a Polish mathematician, currently a professor at the Warsaw University of Technology. He is known for his contributions to dynamical systems. Świątek earned his PhD from the University of Warsaw under supervision of Michał Misiurewicz in 1987. Then he has held academic positions in Poland and the US (including the Pennsylvania State University). He is currently a professor at the Warsaw University of Technology. He published his scientific work in such journals as Annals of Mathematics, Inventiones Mathematicae and Duke Mathematical Journal. With Jacek Graczyk he provided a rigorous proof of the real Fatou conjecture. Świątek was an invited speaker at International Congress of Mathematicians in Berlin in 1998, and at the conference Dynamics, Equations and Applications in Kraków in 2019. In 2007, he received the Stefan Banach Prize of the Polish Mathematical Society. == References ==
Wikipedia:Gröbner basis#0	In mathematics, and more specifically in computer algebra, computational algebraic geometry, and computational commutative algebra, a Gröbner basis is a particular kind of generating set of an ideal in a polynomial ring K [ x 1 , … , x n ] {\displaystyle K[x_{1},\ldots ,x_{n}]} over a field K {\displaystyle K} . A Gröbner basis allows many important properties of the ideal and the associated algebraic variety to be deduced easily, such as the dimension and the number of zeros when it is finite. Gröbner basis computation is one of the main practical tools for solving systems of polynomial equations and computing the images of algebraic varieties under projections or rational maps. Gröbner basis computation can be seen as a multivariate, non-linear generalization of both Euclid's algorithm for computing polynomial greatest common divisors, and Gaussian elimination for linear systems. Gröbner bases were introduced by Bruno Buchberger in his 1965 Ph.D. thesis, which also included an algorithm to compute them (Buchberger's algorithm). He named them after his advisor Wolfgang Gröbner. In 2007, Buchberger received the Association for Computing Machinery's Paris Kanellakis Theory and Practice Award for this work. However, the Russian mathematician Nikolai Günther had introduced a similar notion in 1913, published in various Russian mathematical journals. These papers were largely ignored by the mathematical community until their rediscovery in 1987 by Bodo Renschuch et al. An analogous concept for multivariate power series was developed independently by Heisuke Hironaka in 1964, who named them standard bases. This term has been used by some authors to also denote Gröbner bases. The theory of Gröbner bases has been extended by many authors in various directions. It has been generalized to other structures such as polynomials over principal ideal rings or polynomial rings, and also some classes of non-commutative rings and algebras, like Ore algebras. == Tools == === Polynomial ring === Gröbner bases are primarily defined for ideals in a polynomial ring R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} over a field K. Although the theory works for any field, most Gröbner basis computations are done either when K is the field of rationals or the integers modulo a prime number. In the context of Gröbner bases, a nonzero polynomial in R = K [ x 1 , … , x n ] {\displaystyle R=K[x_{1},\ldots ,x_{n}]} is commonly represented as a sum c 1 M 1 + ⋯ + c m M m , {\displaystyle c_{1}M_{1}+\cdots +c_{m}M_{m},} where the c i {\displaystyle c_{i}} are nonzero elements of K, called coefficients, and the M i {\displaystyle M_{i}} are monomials (called power products by Buchberger and some of his followers) of the form x 1 a 1 ⋯ x n a n , {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}},} where the a i {\displaystyle a_{i}} are nonnegative integers. The vector A = [ a 1 , … , a n ] {\displaystyle A=[a_{1},\ldots ,a_{n}]} is called the exponent vector of the monomial. When the list X = [ x 1 , … , x n ] {\displaystyle X=[x_{1},\ldots ,x_{n}]} of the variables is fixed, the notation of monomials is often abbreviated as x 1 a 1 ⋯ x n a n = X A . {\displaystyle x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}=X^{A}.} Monomials are uniquely defined by their exponent vectors, and, when a monomial ordering (see below) is fixed, a polynomial is uniquely represented by the ordered list of the ordered pairs formed by an exponent vector and the corresponding coefficient. This representation of polynomials is especially efficient for Gröbner basis computation in computers, although it is less convenient for other computations such as polynomial factorization and polynomial greatest common divisor. If F = { f 1 , … , f k } {\displaystyle F=\{f_{1},\ldots ,f_{k}\}} is a finite set of polynomials in the polynomial ring R, the ideal generated by F is the set of linear combinations of elements of F with coefficients in R; that is the set of polynomials that can be written ∑ i = 1 k g i f i {\textstyle \sum _{i=1}^{k}g_{i}f_{i}} with g 1 , … , g k ∈ R . {\displaystyle g_{1},\ldots ,g_{k}\in R.} === Monomial ordering === All operations related to Gröbner bases require the choice of a total order on the monomials, with the following properties of compatibility with multiplication. For all monomials M, N, P, M ≤ N ⟺ M P ≤ N P {\displaystyle M\leq N\Longleftrightarrow MP\leq NP} M ≤ M P {\displaystyle M\leq MP} . A total order satisfying these condition is sometimes called an admissible ordering. These conditions imply that the order is a well-order, that is, every strictly decreasing sequence of monomials is finite. Although Gröbner basis theory does not depend on a particular choice of an admissible monomial ordering, three monomial orderings are especially important for the applications: Lexicographical ordering, commonly called lex or plex (for pure lexical ordering). Total degree reverse lexicographical ordering, commonly called degrevlex. Elimination ordering, lexdeg. Gröbner basis theory was initially introduced for the lexicographical ordering. It was soon realised that the Gröbner basis for degrevlex is almost always much easier to compute, and that it is almost always easier to compute a lex Gröbner basis by first computing the degrevlex basis and then using a "change of ordering algorithm". When elimination is needed, degrevlex is not convenient; both lex and lexdeg may be used but, again, many computations are relatively easy with lexdeg and almost impossible with lex. === Basic operations === ==== Leading term, coefficient and monomial ==== Once a monomial ordering is fixed, the terms of a polynomial (product of a monomial with its nonzero coefficient) are naturally ordered by decreasing monomials (for this order). This makes the representation of a polynomial as a sorted list of pairs coefficient–exponent vector a canonical representation of the polynomials (that is, two polynomials are equal if and only if they have the same representation). The first (greatest) term of a polynomial p for this ordering and the corresponding monomial and coefficient are respectively called the leading term, leading monomial and leading coefficient and denoted, in this article, lt(p), lm(p) and lc(p). Most polynomial operations related to Gröbner bases involve the leading terms. So, the representation of polynomials as sorted lists make these operations particularly efficient (reading the first element of a list takes a constant time, independently of the length of the list). ==== Polynomial operations ==== The other polynomial operations involved in Gröbner basis computations are also compatible with the monomial ordering; that is, they can be performed without reordering the result: The addition of two polynomials consists in a merge of the two corresponding lists of terms, with a special treatment in the case of a conflict (that is, when the same monomial appears in the two polynomials). The multiplication of a polynomial by a scalar consists of multiplying each coefficient by this scalar, without any other change in the representation. The multiplication of a polynomial by a monomial m consists of multiplying each monomial of the polynomial by m. This does not change the term ordering by definition of a monomial ordering. ==== Divisibility of monomials ==== Let M = x 1 a 1 ⋯ x n a n {\displaystyle M=x_{1}^{a_{1}}\cdots x_{n}^{a_{n}}} and N = x 1 b 1 ⋯ x n b n {\displaystyle N=x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}} be two monomials, with exponent vectors A = [ a 1 , … , a n ] {\displaystyle A=[a_{1},\ldots ,a_{n}]} and B = [ b 1 , … , b n ] . {\displaystyle B=[b_{1},\ldots ,b_{n}].} One says that M divides N, or that N is a multiple of M, if a i ≤ b i {\displaystyle a_{i}\leq b_{i}} for every i; that is, if A is componentwise not greater than B. In this case, the quotient N M {\textstyle {\frac {N}{M}}} is defined as N M = x 1 b 1 − a 1 ⋯ x n b n − a n . {\textstyle {\frac {N}{M}}=x_{1}^{b_{1}-a_{1}}\cdots x_{n}^{b_{n}-a_{n}}.} In other words, the exponent vector of N M {\textstyle {\frac {N}{M}}} is the componentwise subtraction of the exponent vectors of N and M. The greatest common divisor gcd(M, N) of M and N is the monomial x 1 min ( a 1 , b 1 ) ⋯ x n min ( a n , b n ) {\textstyle x_{1}^{\min(a_{1},b_{1})}\cdots x_{n}^{\min(a_{n},b_{n})}} whose exponent vector is the componentwise minimum of A and B. The least common multiple lcm(M, N) is defined similarly with max instead of min. One has lcm ⁡ ( M , N ) = M N gcd ( M , N ) . {\displaystyle \operatorname {lcm} (M,N)={\frac {MN}{\gcd(M,N)}}.} === Reduction === The reduction of a polynomial by other polynomials with respect to a monomial ordering is central to Gröbner basis theory. It is a generalization of both row reduction occurring in Gaussian elimination and division steps of the Euclidean division of univariate polynomials. When completed as much as possible, it is sometimes called multivariate division although its result is not uniquely defined. Lead-reduction is a special case of reduction that is easier to compute. It is fundamental for Gröbner basis computation, since general reduction is needed only at the end of a Gröbner basis computation, for getting a reduced Gröbner basis from a non-reduced one. Let an admissible monomial ordering be fixed, to which refers every monomial comparison that will occur in this section. A polynomial f is lead-reducible by another polynomial g if the leading monomial lm(f) is a multiple of lm(g). The polynomial f is reducible by g if some monomial of f is a multiple lm(g). (So, if f is lead-reducible by g, it is also reducible, but f may be reducible without being lead-reducible.) Suppose that f is reducible by g, and let cm be a term of f such that the monomial m is a multiple of lm(g). A one-step reduction of f by g consists of replacing f by red 1 ⁡ ( f , g ) = f − c lc ⁡ ( g ) m lm ⁡ ( g ) g . {\displaystyle \operatorname {red} _{1}(f,g)=f-{\frac {c}{\operatorname {lc} (g)}}\,{\frac {m}{\operatorname {lm} (g)}}\,g.} This operation removes the monomial m from f without changing the terms with a monomial greater than m (for the monomial ordering). In particular, a one step lead-reduction of f produces a polynomial all of whose monomials are smaller than lm(f). Given a finite set G of polynomials, one says that f is reducible or lead-reducible by G if it is reducible or lead-reducible, respectively, by at least one element g of G. In this case, a one-step reduction (resp. one-step lead-reduction) of f by G is any one-step reduction (resp. one-step lead-reduction) of f by an element of G. The (complete) reduction (resp. lead-reduction) of f by G consists of iterating one-step reductions (respect. one-step lead reductions) until getting a polynomial that is irreducible (resp. lead-irreducible) by G. It is sometimes called a normal form of f by G. In general this form is not uniquely defined because there are, in general, several elements of G that can be used for reducing f; this non-uniqueness is the starting point of Gröbner basis theory. The definition of the reduction shows immediately that, if h is a normal form of f by G, one has f = h + ∑ g ∈ G q g g , {\displaystyle f=h+\sum _{g\in G}q_{g}\,g,} where h is irreducible by G and the q g {\displaystyle q_{g}} are polynomials such that lm ⁡ ( q g g ) ≤ lm ⁡ ( f ) . {\displaystyle \operatorname {lm} (q_{g}\,g)\leq \operatorname {lm} (f).} In the case of univariate polynomials, if G consists of a single element g, then h is the remainder of the Euclidean division of f by g, and qg is the quotient. Moreover, the division algorithm is exactly the process of lead-reduction. For this reason, some authors use the term multivariate division instead of reduction. ==== Non uniqueness of reduction ==== In the example that follows, there are exactly two complete lead-reductions that produce two very different results. The fact that the results are irreducible (not only lead-irreducible) is specific to the example, although this is rather common with such small examples. In this two variable example, the monomial ordering that is used is the lexicographic order with x > y , {\displaystyle x>y,} and we consider the reduction of f = 2 x 3 − x 2 y + y 3 + 3 y {\displaystyle f=2x^{3}-x^{2}y+y^{3}+3y} , by G = { g 1 , g 2 } , {\displaystyle G=\{g_{1},g_{2}\},} with g 1 = x 2 + y 2 − 1 , g 2 = x y − 2. {\displaystyle {\begin{aligned}g_{1}&=x^{2}+y^{2}-1,\\g_{2}&=xy-2.\end{aligned}}} For the first reduction step, either the first or the second term of f may be reduced. However, the reduction of a term amounts to removing this term at the cost of adding new lower terms; if it is not the first reducible term that is reduced, it may occur that a further reduction adds a similar term, which must be reduced again. It is therefore always better to reduce first the largest (for the monomial order) reducible term; that is, in particular, to lead-reduce first until getting a lead-irreducible polynomial. The leading term 2 x 3 {\displaystyle 2x^{3}} of f is reducible by g 1 {\displaystyle g_{1}} and not by g 2 . {\displaystyle g_{2}.} So the first reduction step consists of multiplying g 1 {\displaystyle g_{1}} by −2x and adding the result to f: f → − 2 x g 1 f 1 = f − 2 x g 1 = − x 2 y − 2 x y 2 + 2 x + y 3 + 3 y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}=f-2xg_{1}=-x^{2}y-2xy^{2}+2x+y^{3}+3y.} The leading term − x 2 y {\displaystyle -x^{2}y} of f 1 {\displaystyle f_{1}} is a multiple of the leading monomials of both g 1 {\displaystyle g_{1}} and g 2 , {\displaystyle g_{2},} So, one has two choices for the second reduction step. If one chooses g 2 , {\displaystyle g_{2},} one gets a polynomial that can be reduced again by g 2 : {\displaystyle g_{2}\colon } f → − 2 x g 1 f 1 → x g 2 − 2 x y 2 + y 3 + 3 y → 2 y g 2 f 2 = y 3 − y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}\;\xrightarrow {xg_{2}} \;-2xy^{2}+y^{3}+3y\;\xrightarrow {2yg_{2}} \;f_{2}=y^{3}-y.} No further reduction is possible, so f 2 {\displaystyle f_{2}} is a complete reduction of f. One gets a different result with the other choice for the second step: f → − 2 x g 1 f 1 → y g 1 − 2 x y 2 + 2 x + 2 y 3 + 2 y → 2 y g 2 f 3 = 2 x + 2 y 3 − 2 y . {\displaystyle f\;\xrightarrow {\overset {}{-2xg_{1}}} \;f_{1}\;\xrightarrow {yg_{1}} \;-2xy^{2}+2x+2y^{3}+2y\;\xrightarrow {2yg_{2}} \;f_{3}=2x+2y^{3}-2y.} Again, the result f 3 {\displaystyle f_{3}} is irreducible, although only lead reductions were done. In summary, the complete reduction of f can result in either f 2 = y 3 − y {\displaystyle f_{2}=y^{3}-y} or f 3 = 2 x + 2 y 3 − 2 y . {\displaystyle f_{3}=2x+2y^{3}-2y.} It is for dealing with the problems set by this non-uniqueness that Buchberger introduced Gröbner bases and S-polynomials. Intuitively, 0 = f − f {\displaystyle 0=f-f} may be reduced to f 2 − f 3 . {\displaystyle f_{2}-f_{3}.} This implies that f 2 − f 3 {\displaystyle f_{2}-f_{3}} belongs to the ideal generated by G. So, this ideal is not changed by adding f 3 − f 2 {\displaystyle f_{3}-f_{2}} to G, and this allows more reductions. In particular, f 3 {\displaystyle f_{3}} can be reduced to f 2 {\displaystyle f_{2}} by f 3 − f 2 {\displaystyle f_{3}-f_{2}} and this restores the uniqueness of the reduced form. Here Buchberger's algorithm for Gröbner bases would begin by adding to G the polynomial g 3 = y g 1 − x g 2 = 2 x + y 3 − y . {\displaystyle g_{3}=yg_{1}-xg_{2}=2x+y^{3}-y.} This polynomial, called S-polynomial by Buchberger, is the difference of the one-step reductions of the least common multiple x 2 y {\displaystyle x^{2}y} of the leading monomials of g 1 {\displaystyle g_{1}} and g 2 {\displaystyle g_{2}} , by g 2 {\displaystyle g_{2}} and g 1 {\displaystyle g_{1}} respectively: g 3 = ( x 2 y − x 2 y l t ( g 2 ) g 2 ) − ( x 2 y − x 2 y l t ( g 1 ) g 1 ) = x 2 y l t ( g 1 ) g 1 − x 2 y l t ( g 2 ) g 2 {\displaystyle g_{3}=\left(x^{2}y-{\frac {x^{2}y}{lt(g_{2})}}g_{2}\right)-\left(x^{2}y-{\frac {x^{2}y}{lt(g_{1})}}g_{1}\right)={\frac {x^{2}y}{lt(g_{1})}}g_{1}-{\frac {x^{2}y}{lt(g_{2})}}g_{2}} . In this example, one has g 3 = f 3 − f 2 . {\displaystyle g_{3}=f_{3}-f_{2}.} This does not complete Buchberger's algorithm, as xy gives different results, when reduced by g 2 {\displaystyle g_{2}} or g 3 . {\displaystyle g_{3}.} === S-polynomial === Given monomial ordering, the S-polynomial or critical pair of two polynomials f and g is the polynomial S ( f , g ) = red 1 ⁡ ( l c m , g ) − red 1 ⁡ ( l c m , f ) {\displaystyle S(f,g)=\operatorname {red} _{1}(\mathrm {lcm} ,g)-\operatorname {red} _{1}(\mathrm {lcm} ,f)} ; where lcm denotes the least common multiple of the leading monomials of f and g. Using the definition of red 1 {\displaystyle \operatorname {red} _{1}} , this translates to: S ( f , g ) = ( l c m − 1 lc ⁡ ( g ) l c m lm ⁡ ( g ) g ) − ( l c m − 1 lc ⁡ ( f ) l c m lm ⁡ ( f ) f ) = 1 lc ⁡ ( f ) l c m lm ⁡ ( f ) f − 1 lc ⁡ ( g ) l c m lm ⁡ ( g ) g . {\displaystyle {\begin{aligned}S(f,g)&=\left(\mathrm {lcm} -{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (g)}}\,g\right)-\left(\mathrm {lcm} -{\frac {1}{\operatorname {lc} (f)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (f)}}\,f\right)\\&={\frac {1}{\operatorname {lc} (f)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (f)}}\,f-{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\mathrm {lcm} }{\operatorname {lm} (g)}}\,g\\\end{aligned}}.} Using the property that relates the lcm and the gcd, the S-polynomial can also be written as: S ( f , g ) = 1 lc ⁡ ( f ) lm ⁡ ( g ) g c d f − 1 lc ⁡ ( g ) lm ⁡ ( f ) g c d g ; {\displaystyle S(f,g)={\frac {1}{\operatorname {lc} (f)}}\,{\frac {\operatorname {lm} (g)}{\mathrm {gcd} }}\,f-{\frac {1}{\operatorname {lc} (g)}}\,{\frac {\operatorname {lm} (f)}{\mathrm {gcd} }}\,g;} where gcd denotes the greatest common divisor of the leading monomials of f and g. As the monomials that are reducible by both f and g are exactly the multiples of lcm, one can deal with all cases of non-uniqueness of the reduction by considering only the S-polynomials. This is a fundamental fact for Gröbner basis theory and all algorithms for computing them. For avoiding fractions when dealing with polynomials with integer coefficients, the S polynomial is often defined as S ( f , g ) = lc ⁡ ( g ) lm ⁡ ( g ) g c d f − lc ⁡ ( f ) lm ⁡ ( f ) g c d g ; {\displaystyle S(f,g)=\operatorname {lc} (g)\,{\frac {\operatorname {lm} (g)}{\mathrm {gcd} }}\,f-\operatorname {lc} (f)\,{\frac {\operatorname {lm} (f)}{\mathrm {gcd} }}\,g;} This does not change anything to the theory since the two polynomials are associates. == Definition == Let R = F [ x 1 , … , x n ] {\displaystyle R=F[x_{1},\ldots ,x_{n}]} be a polynomial ring over a field F. In this section, we suppose that an admissible monomial ordering has been fixed. Let G be a finite set of polynomials in R that generates an ideal I. The set G is a Gröbner basis (with respect to the monomial ordering), or, more precisely, a Gröbner basis of I if the ideal generated by the leading monomials of the polynomials in I equals the ideal generated by the leading monomials of G, or, equivalently, There are many characterizing properties, which can each be taken as an equivalent definition of Gröbner bases. For conciseness, in the following list, the notation "one-word/another word" means that one can take either "one-word" or "another word" for having two different characterizations of Gröbner bases. All the following assertions are characterizations of Gröbner bases: Counting the above definition, this provides 12 characterizations of Gröbner bases. The fact that so many characterizations are possible makes Gröbner bases very useful. For example, condition 3 provides an algorithm for testing ideal membership; condition 4 provides an algorithm for testing whether a set of polynomials is a Gröbner basis and forms the basis of Buchberger's algorithm for computing Gröbner bases; conditions 5 and 6 allow computing in R / I {\displaystyle R/I} in a way that is very similar to modular arithmetic. === Existence === For every admissible monomial ordering and every finite set G of polynomials, there is a Gröbner basis that contains G and generates the same ideal. Moreover, such a Gröbner basis may be computed with Buchberger's algorithm. This algorithm uses condition 4, and proceeds roughly as follows: for any two elements of G, compute the complete reduction by G of their S-polynomial, and add the result to G if it is not zero; repeat this operation with the new elements of G included until, eventually, all reductions produce zero. The algorithm terminates always because of Dickson's lemma or because polynomial rings are Noetherian (Hilbert's basis theorem). Condition 4 ensures that the result is a Gröbner basis, and the definitions of S-polynomials and reduction ensure that the generated ideal is not changed. The above method is an algorithm for computing Gröbner bases; however, it is very inefficient. Many improvements of the original Buchberger's algorithm, and several other algorithms have been proposed and implemented, which dramatically improve the efficiency. See § Algorithms and implementations, below. === Reduced Gröbner bases === A Gröbner basis is minimal if all leading monomials of its elements are irreducible by the other elements of the basis. Given a Gröbner basis of an ideal I, one gets a minimal Gröbner basis of I by removing the polynomials whose leading monomials are multiple of the leading monomial of another element of the Gröbner basis. However, if two polynomials of the basis have the same leading monomial, only one must be removed. So, every Gröbner basis contains a minimal Gröbner basis as a subset. All minimal Gröbner bases of a given ideal (for a fixed monomial ordering) have the same number of elements, and the same leading monomials, and the non-minimal Gröbner bases have more elements than the minimal ones. A Gröbner basis is reduced if every polynomial in it is irreducible by the other elements of the basis, and has 1 as leading coefficient. So, every reduced Gröbner basis is minimal, but a minimal Gröbner basis need not be reduced. Given a Gröbner basis of an ideal I, one gets a reduced Gröbner basis of I by first removing the polynomials that are lead-reducible by other elements of the basis (for getting a minimal basis); then replacing each element of the basis by the result of the complete reduction by the other elements of the basis; and, finally, by dividing each element of the basis by its leading coefficient. All reduced Gröbner bases of an ideal (for a fixed monomial ordering) are equal. It follows that two ideals are equal if and only if they have the same reduced Gröbner basis. Sometimes, reduced Gröbner bases are defined without the condition on the leading coefficients. In this case, the uniqueness of reduced Gröbner bases is true only up to the multiplication of polynomials by a nonzero constant. When working with polynomials over the field Q {\displaystyle \mathbb {Q} } of the rational numbers, it is useful to work only with polynomials with integer coefficients. In this case, the condition on the leading coefficients in the definition of a reduced basis may be replaced by the condition that all elements of the basis are primitive polynomials with integer coefficients, with positive leading coefficients. This restores the uniqueness of reduced bases. === Special cases === For every monomial ordering, the empty set of polynomials is the unique Gröbner basis of the zero ideal. For every monomial ordering, a set of polynomials that contains a nonzero constant is a Gröbner basis of the unit ideal (the whole polynomial ring). Conversely, every Gröbner basis of the unit ideal contains a nonzero constant. The reduced Gröbner basis of the unit is formed by the single polynomial 1. In the case of polynomials in a single variable, there is a unique admissible monomial ordering, the ordering by the degree. The minimal Gröbner bases are the singletons consisting of a single polynomial. The reduced Gröbner bases are the monic polynomials. == Example and counterexample == Let R = Q [ x , y ] {\displaystyle R=\mathbb {Q} [x,y]} be the ring of bivariate polynomials with rational coefficients and consider the ideal I = ⟨ f , g ⟩ {\displaystyle I=\langle f,g\rangle } generated by the polynomials f = x 2 − y {\displaystyle f=x^{2}-y} , g = x 3 − x {\displaystyle g=x^{3}-x} . By reducing g by f, one obtains a new polynomial k such that I = ⟨ f , k ⟩ : {\displaystyle I=\langle f,k\rangle :} k = g − x f = x y − x . {\displaystyle k=g-xf=xy-x.} None of f and k is reducible by the other, but xk is reducible by f, which gives another polynomial in I: h = x k − ( y − 1 ) f = y 2 − y . {\displaystyle h=xk-(y-1)f=y^{2}-y.} Under lexicographic ordering with x > y {\displaystyle x>y} we have lt(f) = x2 lt(k) = xy lt(h) = y2 As f, k and h belong to I, and none of them is reducible by the others, none of { f , k } , {\displaystyle \{f,k\},} { f , h } , {\displaystyle \{f,h\},} and { h , k } {\displaystyle \{h,k\}} is a Gröbner basis of I. On the other hand, {f, k, h} is a Gröbner basis of I, since the S-polynomials y f − x k = y ( x 2 − y ) − x ( x y − x ) = f − h y k − x h = y ( x y − x ) − x ( y 2 − y ) = 0 y 2 f − x 2 h = y ( y f − x k ) + x ( y k − x h ) {\displaystyle {\begin{aligned}yf-xk&=y(x^{2}-y)-x(xy-x)=f-h\\yk-xh&=y(xy-x)-x(y^{2}-y)=0\\y^{2}f-x^{2}h&=y(yf-xk)+x(yk-xh)\end{aligned}}} can be reduced to zero by f, k and h. The method that has been used here for finding h and k, and proving that {f, k, h} is a Gröbner basis is a direct application of Buchberger's algorithm. So, it can be applied mechanically to any similar example, although, in general, there are many polynomials and S-polynomials to consider, and the computation is generally too large for being done without a computer. == Properties and applications of Gröbner bases == Unless explicitly stated, all the results that follow are true for any monomial ordering (see that article for the definitions of the different orders that are mentioned below). It is a common misconception that the lexicographical order is needed for some of these results. On the contrary, the lexicographical order is, almost always, the most difficult to compute, and using it makes impractical many computations that are relatively easy with graded reverse lexicographic order (grevlex), or, when elimination is needed, the elimination order (lexdeg) which restricts to grevlex on each block of variables. === Equality of ideals === Reduced Gröbner bases are unique for any given ideal and any monomial ordering. Thus two ideals are equal if and only if they have the same (reduced) Gröbner basis (usually a Gröbner basis software always produces reduced Gröbner bases). === Membership and inclusion of ideals === The reduction of a polynomial f by the Gröbner basis G of an ideal I yields 0 if and only if f is in I. This allows to test the membership of an element in an ideal. Another method consists in verifying that the Gröbner basis of G∪{f} is equal to G. To test if the ideal I generated by f1, ..., fk is contained in the ideal J, it suffices to test that every fI is in J. One may also test the equality of the reduced Gröbner bases of J and J ∪ {f1, ...,fk}. === Solutions of a system of algebraic equations === Any set of polynomials may be viewed as a system of polynomial equations by equating the polynomials to zero. The set of the solutions of such a system depends only on the generated ideal, and, therefore does not change when the given generating set is replaced by the Gröbner basis, for any ordering, of the generated ideal. Such a solution, with coordinates in an algebraically closed field containing the coefficients of the polynomials, is called a zero of the ideal. In the usual case of rational coefficients, this algebraically closed field is chosen as the complex field. An ideal does not have any zero (the system of equations is inconsistent) if and only if 1 belongs to the ideal (this is Hilbert's Nullstellensatz), or, equivalently, if its Gröbner basis (for any monomial ordering) contains 1, or, also, if the corresponding reduced Gröbner basis is [1]. Given the Gröbner basis G of an ideal I, it has only a finite number of zeros, if and only if, for each variable x, G contains a polynomial with a leading monomial that is a power of x (without any other variable appearing in the leading term). If this is the case, then the number of zeros, counted with multiplicity, is equal to the number of monomials that are not multiples of any leading monomial of G. This number is called the degree of the ideal. When the number of zeros is finite, the Gröbner basis for a lexicographical monomial ordering provides, theoretically, a solution: the first coordinate of a solution is a root of the greatest common divisor of polynomials of the basis that depend only on the first variable. After substituting this root in the basis, the second coordinate of this solution is a root of the greatest common divisor of the resulting polynomials that depend only on the second variable, and so on. This solving process is only theoretical, because it implies GCD computation and root-finding of polynomials with approximate coefficients, which are not practicable because of numeric instability. Therefore, other methods have been developed to solve polynomial systems through Gröbner bases (see System of polynomial equations for more details). === Dimension, degree and Hilbert series === The dimension of an ideal I in a polynomial ring R is the Krull dimension of the ring R/I and is equal to the dimension of the algebraic set of the zeros of I. It is also equal to number of hyperplanes in general position which are needed to have an intersection with the algebraic set, which is a finite number of points. The degree of the ideal and of its associated algebraic set is the number of points of this finite intersection, counted with multiplicity. In particular, the degree of a hypersurface is equal to the degree of its definition polynomial. The dimension depend only on the set of the leading monomials of the Gröbner basis of the ideal for any monomial ordering. The same is true for the degree and degree-compatible monomial orderings; a monomial ordering is degree compatible is smaller for the degree implies smaller for the monomial ordering. The dimension is the maximal size of a subset S of the variables such that there is no leading monomial depending only on the variables in S. Thus, if the ideal has dimension 0, then for each variable x there is a leading monomial in the Gröbner basis that is a power of x. Both dimension and degree may be deduced from the Hilbert series of the ideal, which is the series ∑ i = 0 ∞ d i t i {\textstyle \sum _{i=0}^{\infty }d_{i}t^{i}} , where d i {\displaystyle d_{i}} is the number of monomials of degree i that are not multiple of any leading monomial in the Gröbner basis. The Hilbert series may be summed into a rational fraction ∑ i = 0 ∞ d i t i = P ( t ) ( 1 − t ) d , {\displaystyle \sum _{i=0}^{\infty }d_{i}t^{i}={\frac {P(t)}{(1-t)^{d}}},} where d is the dimension of the ideal and P ( t ) {\displaystyle P(t)} is a polynomial. The number P ( 1 ) {\displaystyle P(1)} is the degree of the algebraic set defined by the ideal, in the case of a homogeneous ideal or a monomial ordering compatible with the degree; that is, to compare two monomials, one compares their total degrees first. The dimension does not depend on the choice of a monomial ordering, although the Hilbert series and the polynomial P ( t ) {\displaystyle P(t)} may change with changes of the monomial ordering. However, for homogeneous ideals or monomial orderings compatible with the degree, the Hilbert series and the polynomial P ( t ) {\displaystyle P(t)} do not depend on the choice of monomial ordering. Most computer algebra systems that provide functions to compute Gröbner bases provide also functions for computing the Hilbert series, and thus also the dimension and the degree. === Elimination === The computation of Gröbner bases for an elimination monomial ordering allows computational elimination theory. This is based on the following theorem. Consider a polynomial ring K [ x 1 , … , x n , y 1 , … , y m ] = K [ X , Y ] , {\displaystyle K[x_{1},\ldots ,x_{n},y_{1},\ldots ,y_{m}]=K[X,Y],} in which the variables are split into two subsets X and Y. Let us also choose an elimination monomial ordering "eliminating" X, that is a monomial ordering for which two monomials are compared by comparing first the X-parts, and, in case of equality only, considering the Y-parts. This implies that a monomial containing an X-variable is greater than every monomial independent of X. If G is a Gröbner basis of an ideal I for this monomial ordering, then G ∩ K [ Y ] {\displaystyle G\cap K[Y]} is a Gröbner basis of I ∩ K [ Y ] {\displaystyle I\cap K[Y]} (this ideal is often called the elimination ideal). Moreover, G ∩ K [ Y ] {\displaystyle G\cap K[Y]} consists exactly of the polynomials of G whose leading terms belong to K[Y] (this makes the computation of G ∩ K [ Y ] {\displaystyle G\cap K[Y]} very easy, as only the leading monomials need to be checked). This elimination property has many applications, some described in the next sections. Another application, in algebraic geometry, is that elimination realizes the geometric operation of projection of an affine algebraic set into a subspace of the ambient space: with above notation, the (Zariski closure of) the projection of the algebraic set defined by the ideal I into the Y-subspace is defined by the ideal I ∩ K [ Y ] . {\displaystyle I\cap K[Y].} The lexicographical ordering such that x 1 > ⋯ > x n {\displaystyle x_{1}>\cdots >x_{n}} is an elimination ordering for every partition { x 1 , … , x k } , { x k + 1 , … , x n } . {\displaystyle \{x_{1},\ldots ,x_{k}\},\{x_{k+1},\ldots ,x_{n}\}.} Thus a Gröbner basis for this ordering carries much more information than usually necessary. This may explain why Gröbner bases for the lexicographical ordering are usually the most difficult to compute. === Intersecting ideals === If I and J are two ideals generated respectively by {f1, ..., fm} and {g1, ..., gk}, then a single Gröbner basis computation produces a Gröbner basis of their intersection I ∩ J. For this, one introduces a new indeterminate t, and one uses an elimination ordering such that the first block contains only t and the other block contains all the other variables (this means that a monomial containing t is greater than every monomial that does not contain t). With this monomial ordering, a Gröbner basis of I ∩ J consists in the polynomials that do not contain t, in the Gröbner basis of the ideal K = ⟨ t f 1 , … , t f m , ( 1 − t ) g 1 , … , ( 1 − t ) g k ⟩ . {\displaystyle K=\langle tf_{1},\ldots ,tf_{m},(1-t)g_{1},\ldots ,(1-t)g_{k}\rangle .} In other words, I ∩ J is obtained by eliminating t in K. This may be proven by observing that the ideal K consists of the polynomials ( a − b ) t + b {\displaystyle (a-b)t+b} such that a ∈ I {\displaystyle a\in I} and b ∈ J {\displaystyle b\in J} . Such a polynomial is independent of t if and only if a = b, which means that b ∈ I ∩ J . {\displaystyle b\in I\cap J.} === Implicitization of a rational curve === A rational curve is an algebraic curve that has a set of parametric equations of the form x 1 = f 1 ( t ) g 1 ( t ) ⋮ x n = f n ( t ) g n ( t ) , {\displaystyle {\begin{aligned}x_{1}&={\frac {f_{1}(t)}{g_{1}(t)}}\\&\;\;\vdots \\x_{n}&={\frac {f_{n}(t)}{g_{n}(t)}},\end{aligned}}} where f i ( t ) {\displaystyle f_{i}(t)} and g i ( t ) {\displaystyle g_{i}(t)} are univariate polynomials for 1 ≤ i ≤ n. One may (and will) suppose that f i ( t ) {\displaystyle f_{i}(t)} and g i ( t ) {\displaystyle g_{i}(t)} are coprime (they have no non-constant common factors). Implicitization consists in computing the implicit equations of such a curve. In case of n = 2, that is for plane curves, this may be computed with the resultant. The implicit equation is the following resultant: Res t ( g 1 x 1 − f 1 , g 2 x 2 − f 2 ) . {\displaystyle {\text{Res}}_{t}(g_{1}x_{1}-f_{1},g_{2}x_{2}-f_{2}).} Elimination with Gröbner bases allows to implicitize for any value of n, simply by eliminating t in the ideal ⟨ g 1 x 1 − f 1 , … , g n x n − f n ⟩ . {\displaystyle \langle g_{1}x_{1}-f_{1},\ldots ,g_{n}x_{n}-f_{n}\rangle .} If n = 2, the result is the same as with the resultant, if the map t ↦ ( x 1 , x 2 ) {\displaystyle t\mapsto (x_{1},x_{2})} is injective for almost every t. In the other case, the resultant is a power of the result of the elimination. === Saturation === When modeling a problem by polynomial equations, it is often assumed that some quantities are non-zero, so as to avoid degenerate cases. For example, when dealing with triangles, many properties become false if the triangle degenerates to a line segment, i.e. the length of one side is equal to the sum of the lengths of the other sides. In such situations, one cannot deduce relevant information from the polynomial system unless the degenerate solutions are ignored. More precisely, the system of equations defines an algebraic set which may have several irreducible components, and one must remove the components on which the degeneracy conditions are everywhere zero. This is done by saturating the equations by the degeneracy conditions, which may be done via the elimination property of Gröbner bases. ==== Definition of the saturation ==== The localization of a ring consists in adjoining to it the formal inverses of some elements. This section concerns only the case of a single element, or equivalently a finite number of elements (adjoining the inverses of several elements is equivalent to adjoining the inverse of their product). The localization of a ring R by an element f is the ring R f = R [ t ] / ( 1 − f t ) , {\displaystyle R_{f}=R[t]/(1-ft),} where t is a new indeterminate representing the inverse of f. The localization of an ideal I of R is the ideal I f = R f I {\displaystyle I_{f}=R_{f}I} of R f . {\displaystyle R_{f}.} When R is a polynomial ring, computing in R f {\displaystyle R_{f}} is not efficient because of the need to manage the denominators. Therefore, localization is usually replaced by the operation of saturation. The saturation with respect to f of an ideal I in R is the inverse image of R f I {\displaystyle R_{f}I} under the canonical map from R to R f . {\displaystyle R_{f}.} It is the ideal I : f ∞ = { g ∈ R ∣ ( ∃ k ∈ N ) f k g ∈ I } {\displaystyle I:f^{\infty }=\{g\in R\mid (\exists k\in \mathbb {N} )f^{k}g\in I\}} consisting in all elements of R whose product with some power of f belongs to I. If J is the ideal generated by I and 1−ft in R[t], then I : f ∞ = J ∩ R . {\displaystyle I:f^{\infty }=J\cap R.} It follows that, if R is a polynomial ring, a Gröbner basis computation eliminating t produces a Gröbner basis of the saturation of an ideal by a polynomial. The important property of the saturation, which ensures that it removes from the algebraic set defined by the ideal I the irreducible components on which the polynomial f is zero, is the following: The primary decomposition of I : f ∞ {\displaystyle I:f^{\infty }} consists of the components of the primary decomposition of I that do not contain any power of f. ==== Computation of the saturation ==== A Gröbner basis of the saturation by f of a polynomial ideal generated by a finite set of polynomials F, may be obtained by eliminating t in F ∪ { 1 − t f } , {\displaystyle F\cup \{1-tf\},} that is by keeping the polynomials independent of t in the Gröbner basis of F ∪ { 1 − t f } {\displaystyle F\cup \{1-tf\}} for an elimination ordering eliminating t. Instead of using F, one may also start from a Gröbner basis of F. Which method is most efficient depends on the problem. However, if the saturation does not remove any component, that is if the ideal is equal to its saturated ideal, computing first the Gröbner basis of F is usually faster. On the other hand, if the saturation removes some components, the direct computation may be dramatically faster. If one wants to saturate with respect to several polynomials f 1 , … , f k {\displaystyle f_{1},\ldots ,f_{k}} or with respect to a single polynomial which is a product f = f 1 ⋯ f k , {\displaystyle f=f_{1}\cdots f_{k},} there are three ways to proceed which give the same result but may have very different computation times (it depends on the problem which is the most efficient). Saturating by f = f 1 ⋯ f k {\displaystyle f=f_{1}\cdots f_{k}} in a single Gröbner basis computation. Saturating by f 1 , {\displaystyle f_{1},} then saturating the result by f 2 , {\displaystyle f_{2},} and so on. Adding to F or to its Gröbner basis the polynomials 1 − t 1 f 1 , … , 1 − t k f k , {\displaystyle 1-t_{1}f_{1},\ldots ,1-t_{k}f_{k},} and eliminating the t i {\displaystyle t_{i}} in a single Gröbner basis computation. === Effective Nullstellensatz === Hilbert's Nullstellensatz has two versions. The first one asserts that a set of polynomials has no common zeros over an algebraic closure of the field of the coefficients, if and only if 1 belongs to the generated ideal. This is easily tested with a Gröbner basis computation, because 1 belongs to an ideal if and only if 1 belongs to the Gröbner basis of the ideal, for any monomial ordering. The second version asserts that the set of common zeros (in an algebraic closure of the field of the coefficients) of an ideal is contained in the hypersurface of the zeros of a polynomial f, if and only if a power of f belongs to the ideal. This may be tested by saturating the ideal by f; in fact, a power of f belongs to the ideal if and only if the saturation by f provides a Gröbner basis containing 1. === Implicitization in higher dimension === By definition, an affine rational variety of dimension k may be described by parametric equations of the form x 1 = p 1 p 0 ⋮ x n = p n p 0 , {\displaystyle {\begin{aligned}x_{1}&={\frac {p_{1}}{p_{0}}}\\&\;\;\vdots \\x_{n}&={\frac {p_{n}}{p_{0}}},\end{aligned}}} where p 0 , … , p n {\displaystyle p_{0},\ldots ,p_{n}} are n+1 polynomials in the k variables (parameters of the parameterization) t 1 , … , t k . {\displaystyle t_{1},\ldots ,t_{k}.} Thus the parameters t 1 , … , t k {\displaystyle t_{1},\ldots ,t_{k}} and the coordinates x 1 , … , x n {\displaystyle x_{1},\ldots ,x_{n}} of the points of the variety are zeros of the ideal I = ⟨ p 0 x 1 − p 1 , … , p 0 x n − p n ⟩ . {\displaystyle I=\left\langle p_{0}x_{1}-p_{1},\ldots ,p_{0}x_{n}-p_{n}\right\rangle .} One could guess that it suffices to eliminate the parameters to obtain the implicit equations of the variety, as it has been done in the case of curves. Unfortunately this is not always the case. If the p i {\displaystyle p_{i}} have a common zero (sometimes called base point), every irreducible component of the non-empty algebraic set defined by the p i {\displaystyle p_{i}} is an irreducible component of the algebraic set defined by I. It follows that, in this case, the direct elimination of the t i {\displaystyle t_{i}} provides an empty set of polynomials. Therefore, if k>1, two Gröbner basis computations are needed to implicitize: Saturate I {\displaystyle I} by p 0 {\displaystyle p_{0}} to get a Gröbner basis G {\displaystyle G} Eliminate the t i {\displaystyle t_{i}} from G {\displaystyle G} to get a Gröbner basis of the ideal (of the implicit equations) of the variety. == Algorithms and implementations == Buchberger's algorithm is the oldest algorithm for computing Gröbner bases. It has been devised by Bruno Buchberger together with the Gröbner basis theory. It is straightforward to implement, but it appeared soon that raw implementations can solve only trivial problems. The main issues are the following ones: Even when the resulting Gröbner basis is small, the intermediate polynomials can be huge. It results that most of the computing time may be spent in memory management. So, specialized memory management algorithms may be a fundamental part of an efficient implementation. The integers occurring during a computation may be sufficiently large for making fast multiplication algorithms and multimodular arithmetic useful. For this reason, most optimized implementations use the GMPlibrary. Also, modular arithmetic, Chinese remainder theorem and Hensel lifting are used in optimized implementations The choice of the S-polynomials to reduce and of the polynomials used for reducing them is devoted to heuristics. As in many computational problems, heuristics cannot detect most hidden simplifications, and if heuristic choices are avoided, one may get a dramatic improvement of the algorithm efficiency. In most cases most S-polynomials that are computed are reduced to zero; that is, most computing time is spent to compute zero. The monomial ordering that is most often needed for the applications (pure lexicographic) is not the ordering that leads to the easiest computation, generally the ordering degrevlex. For solving 3. many improvements, variants and heuristics have been proposed before the introduction of F4 and F5 algorithms by Jean-Charles Faugère. As these algorithms are designed for integer coefficients or with coefficients in the integers modulo a prime number, Buchberger's algorithm remains useful for more general coefficients. Roughly speaking, F4 algorithm solves 3. by replacing many S-polynomial reductions by the row reduction of a single large matrix for which advanced methods of linear algebra can be used. This solves partially issue 4., as reductions to zero in Buchberger's algorithm correspond to relations between rows of the matrix to be reduced, and the zero rows of the reduced matrix correspond to a basis of the vector space of these relations. F5 algorithm improves F4 by introducing a criterion that allows reducing the size of the matrices to be reduced. This criterion is almost optimal, since the matrices to be reduced have full rank in sufficiently regular cases (in particular, when the input polynomials form a regular sequence). Tuning F5 for a general use is difficult, since its performances depend on an order on the input polynomials and a balance between the incrementation of the working polynomial degree and of the number of the input polynomials that are considered. To date (2022), there is no distributed implementation that is significantly more efficient than F4, but, over modular integers F5 has been used successfully for several cryptographic challenges; for example, for breaking HFE challenge. Issue 5. has been solved by the discovery of basis conversion algorithms that start from the Gröbner basis for one monomial ordering for computing a Gröbner basis for another monomial ordering. FGLM algorithm is such a basis conversion algorithm that works only in the zero-dimensional case (where the polynomials have a finite number of complex common zeros) and has a polynomial complexity in the number of common zeros. A basis conversion algorithm that works is the general case is the Gröbner walk algorithm. In its original form, FGLM may be the critical step for solving systems of polynomial equations because FGML does not take into account the sparsity of involved matrices. This has been fixed by the introduction of sparse FGLM algorithms. Most general-purpose computer algebra systems have implementations of one or several algorithms for Gröbner bases, often also embedded in other functions, such as for solving systems of polynomial equations or for simplifying trigonometric functions; this is the case, for example, of CoCoA, GAP, Macaulay 2, Magma, Maple, Mathematica, SINGULAR, SageMath and SymPy. When F4 is available, it is generally much more efficient than Buchberger's algorithm. The implementation techniques and algorithmic variants are not always documented, although they may have a dramatic effect on efficiency. Implementations of F4 and (sparse)-FGLM are included in the library Msolve. Beside Gröbner algorithms, Msolve contains fast algorithms for real-root isolation, and combines all these functions in an algorithm for the real solutions of systems of polynomial equations that outperforms dramatically the other software for this problem (Maple and Magma). Msolve is available on GitHub, and is interfaced with Julia, Maple and SageMath; this means that Msolve can be used directly from within these software environments. == Complexity == The complexity of the Gröbner basis computations is commonly evaluated in term of the number n of variables and the maximal degree d of the input polynomials. In the worst case, the main parameter of the complexity is the maximal degree of the elements of the resulting reduced Gröbner basis. More precisely, if the Gröbner basis contains an element of a large degree D, this element may contain Ω ( D n ) {\displaystyle \Omega (D^{n})} nonzero terms whose computation requires a time of Ω ( D n ) > D Ω ( n ) . {\displaystyle \Omega (D^{n})>D^{\Omega (n)}.} On the other hand, if all polynomials in the reduced Gröbner basis a homogeneous ideal have a degree of at most D, the Gröbner basis can be computed by linear algebra on the vector space of polynomials of degree less than 2D, which has a dimension O ( D n ) . {\displaystyle O(D^{n}).} So, the complexity of this computation is O ( D n ) O ( 1 ) = D O ( n ) . {\displaystyle O(D^{n})^{O(1)}=D^{O(n)}.} The worst-case complexity of a Gröbner basis computation is doubly exponential in n. More precisely, the complexity is upper bounded by a polynomial in d 2 n . {\textstyle d^{2^{n}}.} Using little o notation, it is therefore bounded by d 2 n + o ( n ) . {\textstyle d^{2^{n+o(n)}}.} On the other hand, examples have been given of reduced Gröbner bases containing polynomials of degree d 2 Ω ( n ) , {\textstyle d^{2^{\Omega (n)}},} or containing d 2 Ω ( n ) {\textstyle d^{2^{\Omega (n)}}} elements. As every algorithm for computing a Gröbner basis must write its result, this provides a lower bound of the complexity. Gröbner basis is EXPSPACE-complete. == Generalizations == The concept and algorithms of Gröbner bases have been generalized to submodules of free modules over a polynomial ring. In fact, if L is a free module over a ring R, then one may consider the direct sum R ⊕ L {\displaystyle R\oplus L} as a ring by defining the product of two elements of L to be 0. This ring may be identified with R [ e 1 , … , e l ] / ⟨ { e i e j \| 1 ≤ i ≤ j ≤ l } ⟩ {\displaystyle R[e_{1},\ldots ,e_{l}]/\left\langle \{e_{i}e_{j}\|1\leq i\leq j\leq l\}\right\rangle } , where e 1 , … , e l {\displaystyle e_{1},\ldots ,e_{l}} is a basis of L. This allows identifying a submodule of L generated by g 1 , … , g k {\displaystyle g_{1},\ldots ,g_{k}} with the ideal of R [ e 1 , … , e l ] {\displaystyle R[e_{1},\ldots ,e_{l}]} generated by g 1 , … , g k {\displaystyle g_{1},\ldots ,g_{k}} and the products e i e j {\displaystyle e_{i}e_{j}} , 1 ≤ i ≤ j ≤ l {\displaystyle 1\leq i\leq j\leq l} . If R is a polynomial ring, this reduces the theory and the algorithms of Gröbner bases of modules to the theory and the algorithms of Gröbner bases of ideals. The concept and algorithms of Gröbner bases have also been generalized to ideals over various rings, commutative or not, like polynomial rings over a principal ideal ring or Weyl algebras. == Areas of applications == === Error-Correcting Codes === Gröbner bases have been applied in the theory of error-correcting codes for algebraic decoding. By using Gröbner basis computation on various forms of error-correcting equations, decoding methods were developed for correcting errors of cyclic codes, affine variety codes, algebraic-geometric codes and even general linear block codes. Applying Gröbner basis in algebraic decoding is still a research area of channel coding theory. == See also == Bergman's diamond lemma, an extension of Gröbner bases to non-commutative rings Graver basis Janet basis Regular chains, an alternative way to represent algebraic sets == References == == Further reading == Adams, William W.; Loustaunau, Philippe (1994). An Introduction to Gröbner Bases. Graduate Studies in Mathematics. Vol. 3. American Mathematical Society. ISBN 0-8218-3804-0. Li, Huishi (2011). Gröbner Bases in Ring Theory. World Scientific. ISBN 978-981-4365-13-0. Becker, Thomas; Weispfenning, Volker (1998). Gröbner Bases: A Computational Approach to Commutative Algebra. Graduate Texts in Mathematics. Vol. 141. Springer. ISBN 0-387-97971-9. Buchberger, Bruno (1965). An Algorithm for Finding the Basis Elements of the Residue Class Ring of a Zero Dimensional Polynomial Ideal (PDF) (PhD). University of Innsbruck. — (2006). "Bruno Buchberger's PhD thesis 1965: An algorithm for finding the basis elements of the residue class ring of a zero dimensional polynomial ideal". Journal of Symbolic Computation. 41 (3–4). Translated by Abramson, M.: 471–511. doi:10.1016/j.jsc.2005.09.007. [This is Buchberger's thesis inventing Gröbner bases.] Buchberger, Bruno (1970). "An Algorithmic Criterion for the Solvability of a System of Algebraic Equations" (PDF). Aequationes Mathematicae. 4: 374–383. doi:10.1007/BF01844169. S2CID 189834323. (This is the journal publication of Buchberger's thesis.)Burchberger, B.; Winkler, F., eds. (26 February 1998). "An Algorithmic Criterion for the Solvability of a System of Algebraic Equations". Gröbner Bases and Applications. London Mathematical Society Lecture Note Series. Vol. 251. Cambridge University Press. pp. 535–545. ISBN 978-0-521-63298-0. Buchberger, Bruno; Kauers, Manuel (2010). "Gröbner Bases". Scholarpedia. 5 (10): 7763. Bibcode:2010SchpJ...5.7763B. doi:10.4249/scholarpedia.7763. Fröberg, Ralf (1997). An Introduction to Gröbner Bases. Wiley. ISBN 0-471-97442-0. Sturmfels, Bernd (November 2005). "What is ... a Gröbner Basis?" (PDF). Notices of the American Mathematical Society. 52 (10): 1199–1200, a brief introduction{{cite journal}}: CS1 maint: postscript (link) Shirshov, Anatoliĭ I. (1999). "Certain algorithmic problems for Lie algebras" (PDF). ACM SIGSAM Bulletin. 33 (2): 3–6. doi:10.1145/334714.334715. S2CID 37070503. (translated from Sibirsk. Mat. Zh. Siberian Mathematics Journal 3 (1962), 292–296). Aschenbrenner, Matthias; Hillar, Christopher (2007). "Finite generation of symmetric ideals". Transactions of the American Mathematical Society. 359 (11): 5171–92. arXiv:math/0411514. doi:10.1090/S0002-9947-07-04116-5. S2CID 5656701. (on infinite dimensional Gröbner bases for polynomial rings in infinitely many indeterminates). == External links == Faugère's own implementation of his F4 algorithm "Gröbner basis", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Buchberger, B. (2003). "Gröbner Bases: A Short Introduction for Systems Theorists" (PDF). In Moreno-Diaz, R.; Buchberger, B.; Freire, J. (eds.). Computer Aided Systems Theory — EUROCAST 2001: A Selection of Papers from the 8th International Workshop on Computer Aided Systems Theory. Springer. pp. 1–19. ISBN 978-3-540-45654-4. Buchberger, B.; Zapletal, A. "Gröbner Bases Bibliography". Comparative Timings Page for Gröbner Bases Software Prof. Bruno Buchberger Bruno Buchberger Weisstein, Eric W. "Gröbner Basis". MathWorld. Gröbner basis introduction on Scholarpedia
Wikipedia:Gröbner fan#0	In computer algebra, the Gröbner fan of an ideal in the ring of polynomials is a concept in the theory of Gröbner bases. It is defined to be a fan consisting of cones that correspond to different monomial orders on that ideal. The concept was introduced by Mora and Robbiano in 1988. The result is a weaker version of the result presented in the same issue of the journal by Bayer and Morrison. Gröbner fan is a base for the nowadays active field of tropical geometry. One implementation of the Gröbner fan is called Gfan, based on an article of Fukuda, et al. which is included in some computer algebra systems such as Singular, Macaulay2, and CoCoA. == See also == Gröbner basis Tropical geometry == References ==
Wikipedia:Guido Ascoli#0	Guido Ascoli (12 December 1887, in Livorno – 10 May 1957, in Torino) was an Italian mathematician, known for his contributions to the theory of partial differential equations, and for his works on the teaching of mathematics in secondary high schools. == Selected publications == Ascoli, G.; Burgatti, P.; Giraud, G. (1936), Equazioni alle derivate parziali dei tipi ellittico e parabolico (in Italian), Firenze: Sansoni Editore, pp. IV + 186, JFM 62.0547.04 (available from the "Edizione Nazionale Mathematica Italiana"). A book collecting the winning papers of the 1935 prize of the Annali della Reale Scuola Normale Superiore di Pisa. An English translation of the title reads as:-"Partial differential equations of elliptic and parabolic type". == Biographical references == O'Connor, John J.; Robertson, Edmund F., "Guido Ascoli", MacTutor History of Mathematics Archive, University of St Andrews Guido Ascoli Tricomi, Francesco (1957), "Guido Ascoli", Bollettino della Unione Matematica Italiana, Serie 3 (in Italian), 12 (3): 346–350, MR 0088442, Zbl 0077.00806, available from the Biblioteca Digitale Italiana di Matematica. Tricomi, F. G. (1962). "Guido Ascoli". Matematici italiani del primo secolo dello stato unitario. Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, series IV (in Italian). Vol. I. Torino. p. 120. Zbl 0132.24405. Archived from the original on 2012-08-25. Retrieved 2013-08-13.{{cite book}}: CS1 maint: bot: original URL status unknown (link) CS1 maint: location missing publisher (link). "Italian mathematicians of the first century of the unitary state" is an important historical memoir giving brief biographies of the Italian mathematicians who worked and lived between 1861 and 1961. Its content is available from the website of the.
Wikipedia:Guido Mislin#0	Guido Mislin (born April 13, 1941 in Basel) is a Swiss mathematician, academic and researcher. He is a Professor Emeritus of Mathematics at ETH Zurich. He is also associated with Ohio State University as a guest at Mathematics Department. Mislin's main area of research is algebraic topology, focusing especially on questions regarding general localization theory, as they occur in the context of homotopy theory. He has also conducted research in the field of cohomology of groups and algebraic K-Theory of group rings. He has published over 90 research articles and several books including Localization of Nilpotent Groups and Spaces and Proper Group Actions and the Baum-Connes Conjecture. == Education == Mislin completed his undergraduate studies and diploma in Mathematics in 1964 and received his Ph.D. in 1967 from ETH Zūrich. He then moved to U.S. and completed his post-doctoral studies from Cornell University and University of California, Berkeley. == Career == Following his post-doctoral studies, Mislin was appointed as an assistant professor at the Ohio State University. In 1972, he moved back to Switzerland and joined ETH Zurich as an Associate Professor of Mathematics. He was promoted to Professor of Mathematics in 1979. Mislin headed the Department of Mathematics from 1998 till 2002. He retired in 2006 and was gifted with Guido's Book of Conjectures, which is a collection of short notes written by 91 different authors. Mislin is associated with ETH Zurich as a Professor Emeritus of Mathematics. == Research == Mislin specializes in algebraic topology, and has conducted research focusing especially on questions regarding general localization theory. He has also worked on cohomology of groups and algebraic K-Theory of group rings. Mislin studied the cohomology of classifying spaces of complex Lie groups and related discrete groups. His work proved that the Generalized Isomorphism Conjecture is equivalent to a Finite Subgroup Conjecture, generalizing earlier results due to Mark Feshbach and John Milnor, without using Becker-Gottlieb transfer. He presented a theorem regarding constructing torsion classes in systemic manner, by using Chern classes of canonical representation. He discussed the results and also proved certain properties regarding the Chern classes of representations of cyclic groups. Mislin authored a paper in 1990s regarding group homomorphisms inducing mod-p cohomology isomorphisms and highlighted the conditions on p in group theoretic terms for p to induce an H'Z/p-isomorphism. He applied the concept of satellites in order to define Tate cohomology groups for an arbitrary group G and G-module M. Mislin focused on Bass conjecture and conducted a study to prove that the Bost Conjecture on the L1-assembly map for discrete groups implies the Bass Conjecture. He reformulated the weak Bass Conjecture as a comparison of ordinary and L2-Lefschetz numbers. Mislin studied and extended the work conducted by several authors on theory of topological localization. He presented new results to the theory which were then applied to other studies. Mislin also presented a periodicity theorem and proved the various properties of homotopy groups of K-theory localization. == Awards and honors == 1968 - Silver Medal, ETH Zurich == Bibliography == === Selected books === Localization of Nilpotent Groups and Spaces (1975) ISBN 978-1483258744 Proper Group Actions and the Baum-Connes Conjecture (Advanced Courses in Mathematics - CRM Barcelona) (2003) ISBN 978-3764304089 === Selected articles === Mislin, G. (1994). Tate cohomology for arbitrary groups via satellites. Topology and its Applications, 56(3), 293–300. Kropholler, P. H., & Mislin, G. (1998). Groups acting on finite dimensional spaces with finite stabilizers. Commentarii Mathematici Helvetici, 73(1), 122–136. Friedlander, E. M., & Mislin, G. (1984). Cohomology of classifying spaces of complex Lie groups and related discrete groups. Commentarii Mathematici Helvetici, 59(1), 347–361. Mislin, G. (1990). On group homomorphisms inducing mod-p cohomology isomorphisms. Commentarii Mathematici Helvetici, 65(1), 454–461. Mislin, G. (1974). Nilpotent groups with finite commutator subgroups. In Localization in Group Theory and Homotopy Theory (pp. 103–120). Springer, Berlin, Heidelberg. == References ==
Wikipedia:Guillermo Martínez (writer)#0	Guillermo Martínez (born 29 July 1962) is an Argentine novelist and short story writer. Martínez was born in Bahía Blanca, Argentina. He gained a PhD in mathematical logic at the University of Buenos Aires. After his degree in Argentina, he worked for two years in a postdoctoral position at the Mathematical Institute, Oxford. His most successful novel has been Crímenes Imperceptibles (Imperceptible Crimes), known in English as The Oxford Murders, written in 2003. In the same year, he was awarded the Planeta Prize for this novel, which has been translated into a number of languages. The book has appeared as a film in 2008, directed by Alex de la Iglesia, and starring John Hurt, Elijah Wood, Leonor Watling and Julie Cox. == Books == Vast Hell (Infierno grande, 1989) — short stories Regarding Roderer (Acerca de Roderer, 1993) — novel The Woman of the Master (La mujer del maestro, 1998) — novel Borges and Mathematics (Borges y las matemáticas, 2003) — essays The Oxford Murders (Crímenes imperceptibles, 2003) — novel The Immortality Formula (La fórmula de la inmortalidad, 2005) — essays The Book of Murder (La Muerte Lenta de Luciana B, 2007) — novel Gödel (para todos), 2009 — essay Lalu la luco, 2016 — novel The Oxford Brotherhood (Little, Brown, 2021) — novel ISBN 978-1-64313-877-0 == References == == External links == Guillermo Martínez website (in Spanish) Fantastic Fiction entry
Wikipedia:Gunduz Caginalp#0	Gunduz Caginalp (died December 7th, 2021) was a Turkish-born American mathematician whose research has also contributed over 100 papers to physics, materials science and economics/finance journals, including two with Michael Fisher and nine with Nobel Laureate Vernon Smith. He began his studies at Cornell University in 1970 and received an AB in 1973 "Cum Laude with Honors in All Subjects" and Phi Beta Kappa. In 1976 he received a master's degree, and in 1978 a PhD, both also at Cornell. He held positions at The Rockefeller University, Carnegie-Mellon University and the University of Pittsburgh (since 1984), where he was a professor of Mathematics until his death on December 7, 2021. He was born in Turkey, and spent his first seven years and ages 13–16 there, and the middle years in New York City. Caginalp and his wife Eva were married in 1992 and had three sons, Carey, Reggie and Ryan. He served as the Editor of the Journal of Behavioral Finance (1999–2003), and was an Associate Editor for numerous journals. He received awards from the National Science Foundation as well as private foundations. == Thesis and related research == Caginalp's PhD in Applied Mathematics at Cornell University (with thesis advisor Professor Michael Fisher) focused on surface free energy. Previous results by David Ruelle, Fisher, and Elliot Lieb in the 1960s had established that the free energy of a large system can be written as a product of the volume times a term f ∞ {\displaystyle f_{\infty }} (free energy per unit volume) that is independent of the size of the system, plus smaller terms. A remaining problem was to prove that there was a similar term associated with the surface. This was more difficult since the f ∞ {\displaystyle f_{\infty }} proofs relied on discarding terms that were proportional to the surface. A key result of Caginalp's thesis [1,2,3] is the proof that the free energy, F, of a lattice system occupying a region Ω {\displaystyle \Omega } with volume \| Ω \| {\displaystyle \|\Omega \|} and surface area \| ∂ Ω \| {\displaystyle \|\partial \Omega \|} can be written as F ( Ω ) = \| Ω \| f ∞ + \| ∂ Ω \| f x + . . . {\displaystyle F(\Omega )=\|\Omega \|f_{\infty }+\|\partial \Omega \|f_{x}+...} with f x {\displaystyle f_{x}} is the surface free energy (independent of \| Ω \| {\displaystyle \|\Omega \|} and \| ∂ Ω \| {\displaystyle \|\partial \Omega \|} ). Shortly after his PhD, Caginalp joined the Mathematical Physics group of James Glimm (2002 National Medal of Science recipient) at The Rockefeller University. In addition to working on mathematical statistical mechanics, he also proved existence theorems on nonlinear hyperbolic differential equations describing fluid flow. These papers were published in the Annals of Physics and the Journal of Differential Equations. == Developing phase field models == In 1980, Caginalp was the first recipient of the Zeev Nehari position established at Carnegie-Mellon University's Mathematical Sciences Department. At that time he began working on free boundary problems, e.g., problems in which there is an interface between two phases that must be determined as part of the solution to the problem. His original paper on this topic is the second most cited paper in a leading journal, Archive for Rational Mechanics and Analysis, during the subsequent quarter century. He had published over fifty papers on the phase field equations in mathematics, physics and materials journals. The focus of research in the mathematics and physics communities changed considerably during this period, and this perspective is widely used to derive macroscopic equations from a microscopic setting, as well as performing computations on dendritic growth and other phenomena. In the mathematics community during the previous century, the interface between two phases was generally studied via the Stefan model, in which temperature played a dual role, as the sign of the temperature determined the phase, so the interface is defined as the set of points at which the temperature is zero. Physically, however, the temperature at the interface was known to be proportional to the curvature, thereby preventing the temperature from fulfilling its dual role of the Stefan model. This suggested that an additional variable would be needed for a complete description of the interface. In the physics literature, the idea of an "order parameter" and mean field theory had been used by Landau in the 1940s to describe the region near the critical point (i.e., the region in which the liquid and solid phases become indistinguishable). However, the calculation of exact exponents in statistical mechanics showed that mean field theory was not reliable. There was speculation in the physics community that such a theory could be used to describe an ordinary phase transition. However, the fact that the order parameter could not produce the correct exponents in critical phenomena for which it was invented led to skepticism that it could produce results for normal phase transitions. The justification for an order parameter or mean field approach had been that the correlation length between atoms approaches infinity near the critical point. For an ordinary phase transition, the correlation length is typically just a few atomic lengths. Furthermore, in critical phenomena one is often trying to calculate the critical exponents, which should be independent of the details of the system (often called "universality"). In a typical interface problem, one is trying to calculate the interface position essentially exactly, so that one cannot "hide behind universality". In 1980 there seemed to be ample reason to be skeptical of the idea that an order parameter could be used to describe a moving interface between two phases of a material. Beyond the physical justifications, there remained issues related to the dynamics of an interface and the mathematics of the equations. For example, if one uses an order parameter, ϕ {\displaystyle \phi } , together with the temperature variable, T, in a system of parabolic equations, will an initial transition layer in ϕ {\displaystyle \phi } , describing the interface remain as such? One expects that will vary from -1 to +1 as one moves from the solid to the liquid and that the transition will be made on a spatial scale of ε {\displaystyle \varepsilon } , the physical thickness of the interface. The interface in the phase field system is then described by the level set of points on which ϕ {\displaystyle \phi } vanishes. The simplest model [4] can be written as a pair ( ϕ , T ) {\displaystyle (\phi ,T)} that satisfies the equations C P T t + l 2 ϕ = K Δ T α ε 2 ϕ t = ε 2 Δ ϕ + 1 2 ( ϕ − ϕ 3 ) + ε [ s ] E 3 σ ( T − T E ) {\displaystyle {\begin{array}{lcl}C_{P}T_{t}+{\frac {l}{2}}\phi =K\Delta T\\\alpha \varepsilon ^{2}\phi _{t}=\varepsilon ^{2}\Delta \phi +{\frac {1}{2}}(\phi -\phi ^{3})+{\frac {\varepsilon [s]_{E}}{3\sigma }}(T-T_{E})\end{array}}} where C P , l , α , σ , [ s ] E {\displaystyle C_{P},l,\alpha ,\sigma ,[s]_{E}} are physically measurable constants, and ε {\displaystyle \varepsilon } is the interface thickness. With the interface described as the level set of points where the phase variable vanishes, the model allows the interface to be identified without tracking, and is valid even if there are self-intersections. === Modeling === Using the phase field idea to model solidification so that the physical parameters could be identified was originally undertaken in [4]. === Alloys === A number of papers in collaboration with Weiqing Xie* and James Jones [5,6] have extended the modeling to alloy solid-liquid interfaces. === Basic theorems and analytical results === Initiated during the 1980s, these include the following. Given a set of physical parameters describing the material, namely latent heat, surface tension, etc., there is a phase field system of equations whose solutions formally approach those of the corresponding sharp interface system [4,7]. In fact it has been proven that a broad spectrum of interface problems are distinguished limits of the phase field equations. These include the classical Stefan model, the Cahn-Hilliard model, and motion by mean curvature. There exists a unique solution to this system of equations and the interface width is stable in time [4]. === Computational results === The earliest qualitative computations were done in collaboration with J.T. Lin in 1987. Since the true interface thickness, ε {\displaystyle \varepsilon } , is atomic length, realistic computations did not appear feasible without a new ansatz. One can write the phase field equations in a form where ε is the interface thickness and d 0 {\displaystyle d_{0}} the capillarity length (related to the surface tension), so that it is possible to vary ε {\displaystyle \varepsilon } as a free parameter without varying d 0 {\displaystyle d_{0}} if the scaling is done appropriately [4]. One can increase the size of epsilon and not change the motion of the interface significantly provided that d 0 {\displaystyle d_{0}} is fixed [8]. This means that computations with real parameters are feasible. Computations in collaboration with Dr. Bilgin Altundas* compared the numerical results with dendritic growth in microgravity conditions on the space shuttle [9]. === Phase field models of second order === As phase field models became a useful tool in materials science, the need for even better convergence (from the phase field to the sharp interface problems) became apparent. This led to the development of phase field models of second order, meaning that as the interface thickness, ε {\displaystyle \varepsilon } , becomes small, the difference in the interface of the phase field model and the interface of the related sharp interface model become second order in interface thickness, i.e., ε 2 {\displaystyle \varepsilon ^{2}} . In collaboration with Dr. Christof Eck, Dr. Emre Esenturk* and Profs. Xinfu Chen and Caginalp developed a new phase field model and proved that it was indeed second order [10, 11,12]. Numerical computations confirmed these results. == Application of renormalization group methods to differential equations == The philosophical perspective of the renormalization group (RG) initiated by Ken Wilson in the 1970s is that in a system with large degrees of freedom, one should be able to repeatedly average and adjust, or renormalize, at each step without changing the essential feature that one is trying to compute. In the 1990s Nigel Goldenfeld and collaborators began to investigate the possibility of using this idea for the Barenblatt equation. Caginalp further developed these ideas so that one can calculate the decay (in space and time) of solutions to a heat equation with nonlinearity [13] that satisfies a dimensional condition. The methods were also applied to interface problems and systems of parabolic differential equations with Huseyin Merdan. == Research in behavioral finance and experimental economics == Caginalp has been a leader in the newly developing field of Quantitative Behavioral Finance. The work has three main facets: (1) statistical time series modeling, (2) mathematical modeling using differential equations, and (3) laboratory experiments; comparison with models and world markets. His research is influenced by decades of experience as an individual investor and trader. === Statistical time series modeling === The efficient-market hypothesis (EMH) has been the dominant theory for financial markets for the past half century. It stipulates that asset prices are essentially random fluctuations about their fundamental value. As empirical evidence, its proponents cite market data that appears to be "white noise". Behavioral finance has challenged this perspective, citing large market upheavals such as the high-tech bubble and bust of 1998–2003, etc. The difficulty in establishing the key ideas of behavioral finance and economics has been the presence of "noise" in the market. Caginalp and others have made substantial progress toward surmounting this key difficulty. An early study by Caginalp and Constantine in 1995 showed that using the ratio of two clone closed-end funds, one can remove the noise associated with valuation. They showed that today's price is not likely to be yesterday's price (as indicated by EMH), or a pure continuation of the change during the previous time interval, but is halfway between those prices. Subsequent work with Ahmet Duran [14] examined the data involving large deviations between the price and net asset value of closed end funds, finding strong evidence that there is a subsequent movement in the opposite direction (suggesting overreaction). More surprisingly, there is a precursor to the deviation, which is usually a result of large changes in price in the absence of significant changes in value. Dr. Vladimira Ilieva and Mark DeSantis* focused on large scale data studies that effectively subtracted out the changes due to the net asset value of closed end funds [15]. Thus one could establish significant coefficients for price trend. The work with DeSantis was particularly noteworthy in two respects: (a) by standardizing the data, it became possible to compare the impact of price trend versus changes in money supply, for example; (b) the impact of the price trend was shown to be nonlinear, so that a small uptrend has a positive impact on prices (demonstrating underreaction), while a large uptrend has a negative influence. The measure of large or small is based upon the frequency of occurrence (measure in standard deviations). Using exchange traded funds (ETFs), they also showed (together with Akin Sayrak) that the concept of resistance—whereby a stock has retreats as it nears a yearly high—has strong statistical support [16]. The research shows the importance of two key ideas: (i) By compensating for much of the change in valuation, one can reduce the noise that obscures many behavioral and other influence on price dynamics; (ii) By examining nonlinearity (e.g., in the price trend effect) one can uncover influences that would be statistically insignificant upon examining only linear terms. === Mathematical modeling using differential equations === The asset flow differential approach involves understanding asset market dynamics. (I) Unlike the EMH, the model developed by Caginalp and collaborators since 1990 involves ingredients that were marginalized by the classical efficient market hypothesis: while price change depends on supply and demand for the asset (e.g., stock) the latter can depend on a variety of motivations and strategies, such as the recent price trend. Unlike the classical theories, there is no assumption of infinite arbitrage, which says that any small deviation from the true value (that is universally accepted since all participants have the same information) is quickly exploited by an (essentially) infinite capital managed by "informed" investors. Among the consequences of this theory is that equilibrium is not a unique price, but depends on the price history and strategies of the traders. Classical models of price dynamics are all built on the idea that there is infinite arbitrage capital. The Caginalp asset flow model introduced an important new concept of liquidity, L, or excess cash that is defined to be the total cash in the system divided by the total number of shares. (II) In subsequent years, these asset flow equations were generalized to include distinct groups with differing assessments of value, and distinct strategies and resources. For example, one group may be focused on trend (momentum) while another emphasizes value, and attempts to buy the stock when it is undervalued. (III) In collaboration with Duran these equations were studied in terms of optimization of parameters, rendering them a useful tool for practical implementation. (IV) More recently, David Swigon, DeSantis and Caginalp studied the stability of the asset flow equations and showed that instabilities, for example, flash crashes could occur as a result of traders utilizing momentum strategies together with shorter time scales [17, 18]. In recent years, there has been related work that is sometimes called "evolutionary finance". === Laboratory experiments; comparison with models and world markets === In the 1980s asset market experiments pioneered by Vernon Smith (2002 Economics Nobel Laureate) and collaborators provided a new tool to study microeconomics and finance. In particular these posed a challenge to classical economics by showing that participants when participants traded (with real money) an asset with a well defined value the price would soar well above the fundamental value that is defined by the experimenters. Repetition of this experiment under various conditions showed the robustness of the phenomenon. By designing new experiments, Profs. Caginalp, Smith and David Porter largely resolved this paradox through the framework of the asset flow equations. In particular, the bubble size (and more generally, the asset price) was highly correlated by the excess cash in the system, and momentum was also shown to be a factor [19]. In classical economics there would be just one quantity, namely the share price that has units of dollars per share. The experiments showed that this is distinct from the fundamental value per share. The liquidity, L, introduced by Caginalp and collaborators, is a third quantity that also has these units [20]. The temporal evolution of prices involves a complex relationship among these three variables, together with quantities reflecting the motivations of the traders that may involve price trend and other factors. Other studies have shown quantitatively that motivations in the experimental traders are similar to those in world markets. ∗ {\displaystyle \ast } - PhD student of Caginalp == References == Fisher, Michael E.; Caginalp, Gunduz (1977). "Wall and boundary free energies: I. Ferromagnetic scalar spin systems". Communications in Mathematical Physics. 56 (1). Springer Science and Business Media LLC: 11–56. Bibcode:1977CMaPh..56...11F. doi:10.1007/bf01611116. ISSN 0010-3616. S2CID 121460163. Caginalp, Gunduz; Fisher, Michael E. (1979). "Wall and boundary free energies: II. General domains and complete boundaries". Communications in Mathematical Physics. 65 (3). Springer Science and Business Media LLC: 247–280. Bibcode:1979CMaPh..65..247C. doi:10.1007/bf01197882. ISSN 0010-3616. S2CID 122609456. Caginalp, Gunduz (1980). "Wall and boundary free energies: III. Correlation decay and vector spin systems". Communications in Mathematical Physics. 76 (2). Springer Science and Business Media LLC: 149–163. Bibcode:1980CMaPh..76..149C. doi:10.1007/bf01212823. ISSN 0010-3616. S2CID 125456415. Caginalp, Gunduz (1986). "An analysis of a phase field model of a free boundary". Archive for Rational Mechanics and Analysis. 92 (3). Springer Science and Business Media LLC: 205–245. Bibcode:1986ArRMA..92..205C. doi:10.1007/bf00254827. ISSN 0003-9527. S2CID 121539936. (Earlier version: CMU Preprint 1982) Caginalp, G.; Xie, W. (1993-09-01). "Phase-field and sharp-interface alloy models". Physical Review E. 48 (3). American Physical Society (APS): 1897–1909. Bibcode:1993PhRvE..48.1897C. doi:10.1103/physreve.48.1897. ISSN 1063-651X. PMID 9960800. Caginalp, G.; Jones, J. (1995). "A Derivation and Analysis of Phase Field Models of Thermal Alloys". Annals of Physics. 237 (1). Elsevier BV: 66–107. Bibcode:1995AnPhy.237...66C. doi:10.1006/aphy.1995.1004. ISSN 0003-4916. Caginalp, Gunduz; Chen, Xinfu (1992). "Phase Field Equations in the Singular Limit of Sharp Interface Problems". On the Evolution of Phase Boundaries. The IMA Volumes in Mathematics and its Applications. Vol. 43. New York, NY: Springer New York. pp. 1–27. doi:10.1007/978-1-4613-9211-8_1. ISBN 978-1-4613-9213-2. ISSN 0940-6573. Caginalp, G.; Socolovsky, E.A. (1989). "Efficient computation of a sharp interface by spreading via phase field methods". Applied Mathematics Letters. 2 (2). Elsevier BV: 117–120. doi:10.1016/0893-9659(89)90002-5. ISSN 0893-9659. Altundas, Y. B.; Caginalp, G. (2003). "Computations of Dendrites in 3-D and Comparison with Microgravity Experiments". Journal of Statistical Physics. 110 (3/6). Springer Science and Business Media LLC: 1055–1067. Bibcode:2003JSP...110.1055A. doi:10.1023/a:1022140725763. ISSN 0022-4715. S2CID 8645350. "Rapidly converging phase field models via second order asymptotic". Discrete and Continuous Dynamical Systems, Series B: 142–152. 2005. Chen, Xinfu; Caginalp, G.; Eck, Christof (2006). "A rapidly converging phase field model". Discrete & Continuous Dynamical Systems - Series A. 15 (4). American Institute of Mathematical Sciences (AIMS): 1017–1034. doi:10.3934/dcds.2006.15.1017. ISSN 1553-5231. Chen, Xinfu; Caginalp, Gunduz; Esenturk, Emre (2011-10-01). "Interface Conditions for a Phase Field Model with Anisotropic and Non-Local Interactions". Archive for Rational Mechanics and Analysis. 202 (2). Springer Science and Business Media LLC: 349–372. Bibcode:2011ArRMA.202..349C. doi:10.1007/s00205-011-0429-8. ISSN 0003-9527. S2CID 29421680. Caginalp, G. (1996-01-01). "Renormalization group calculation of anomalous exponents for nonlinear diffusion". Physical Review E. 53 (1). American Physical Society (APS): 66–73. Bibcode:1996PhRvE..53...66C. doi:10.1103/physreve.53.66. ISSN 1063-651X. PMID 9964235. Duran, Ahmet; Caginalp, Gunduz (2007). "Overreaction diamonds: precursors and aftershocks for significant price changes". Quantitative Finance. 7 (3). Informa UK Limited: 321–342. doi:10.1080/14697680601009903. ISSN 1469-7688. S2CID 12127798. Caginalp, Gunduz; DeSantis, Mark (2011). "Nonlinearity in the dynamics of financial markets". Nonlinear Analysis: Real World Applications. 12 (2). Elsevier BV: 1140–1151. doi:10.1016/j.nonrwa.2010.09.008. ISSN 1468-1218. S2CID 5807976. "The Nonlinear Price Dynamics of U.S. Equity ETFs". Journal of Econometrics. 183 (2). 2014. SSRN 2584084. DeSantis, M.; Swigon, D.; Caginalp, G. (2012). "Nonlinear Dynamics and Stability in a Multigroup Asset Flow Model". SIAM Journal on Applied Dynamical Systems. 11 (3). Society for Industrial & Applied Mathematics (SIAM): 1114–1148. doi:10.1137/120862211. ISSN 1536-0040. S2CID 13919799. DeSantis, Mark; Swigon, David; Caginalp, Gunduz (July 1, 2011). "Are flash crashes caused by instabilities arising from rapid trading?". Wilmott Magazine. pp. 46–47. Caginalp, G.; Porter, D.; Smith, V. (1998-01-20). "Initial cash/asset ratio and asset prices: An experimental study". Proceedings of the National Academy of Sciences. 95 (2): 756–761. Bibcode:1998PNAS...95..756C. doi:10.1073/pnas.95.2.756. ISSN 0027-8424. PMC 18494. PMID 11038619. Caginalp, G.; Balenovich, D. (1999). Dewynne, J. N.; Howison, S. D.; Wilmott, P. (eds.). "Asset flow and momentum: deterministic and stochastic equations". Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences. 357 (1758). The Royal Society: 2119–2133. Bibcode:1999RSPTA.357.2119C. doi:10.1098/rsta.1999.0421. ISSN 1364-503X. S2CID 29969244. == External links == Download papers on economics/finance: http://ssrn.com/author=328612 List of papers on phase field equations: http://www.pitt.edu/~caginalp/pfpub8_10.pdf Archived 2012-10-14 at the Wayback Machine Download all papers from Personal home page Archived 2007-08-16 at the Wayback Machine List of publications, 2016 Archived 2017-11-22 at the Wayback Machine Publication List Archived 2011-06-06 at the Wayback Machine NY Times Article Newsletter: Quantitative Behavioral Finance Archived 2019-02-14 at the Wayback Machine
Wikipedia:Gunnar Kangro#0	Gunnar Kangro (November 21, 1913 – December 25, 1975, Tartu) was an Estonian mathematician. He worked mainly on summation theory. He taught various courses on mathematical analysis, functional analysis and algebra in University of Tartu and he has written several university textbooks. == Biography == Gunnar Kangro was born on November 21, 1913, in Tartu as the third and youngest child of a construction engineer and a building contractor. After graduating from Tartu Secondary Science School (Tartu Reaalgümnaasium) in 1931 he was admitted to the Faculty of Mathematics and Natural Sciences of Tartu University to study mathematics. After graduating the university in 1935 and finishing the compulsory service in Estonian Defence Forces he was appointed as junior assistant in the Laboratory of Mathematics and Mechanics in Tallinn Technical Institute (currently Tallinn University of Technology). He defended his Master's thesis in 1938 under the supervision of professor Hermann Jaakson in Tartu University. In 1940 he got a scholarship for doctoral studies from Tartu University. In July 1941, Kangro was drafted to the Red Army. He served until February 1942, when he was sent to Chelyabinsk Agricultural Mechanisation Institute. At the end of 1943, he was transferred to Moscow University, where he continued his research towards a Doctor's degree. Kangro returned to Estonia in autumn 1944, and from November onwards he started teaching at Tartu University. In 1947–48 he was the dean of the Faculty of Physics and Mathematics. He defended his Doctoral thesis in July 1947 and became a professor in 1951. In 1952–1959 he was the head of the Chair of Geometry, from 1959 until his death in 1975 he was the head of the Chair of Mathematical Analysis. == Research == In his Doctoral thesis he created a new theory of convergence, which generalized Borel theory of summation, and applied this to study problems connected to convergence of complex power series. He defined new summation methods and used them to characterize properties of the sum of power series and its analytic continuation. The star-shaped summability domains of power series obtained by these new summation methods enabled him to get results extending the applications of Borel summation methods in function theory. In the 1950s, he laid the basis on systematic treatment of summability factors together with German mathematicians Alexander Peyerimhoff and Wolfgang Jurkat. He combined ideas of modern functional analysis with classical analysis. Together with his student Simson Baron he started to describe the summability factors for double series. Considering applications to orthogonal series and Tauberian theorems, Kangro created a theory of summability with speed based on functional analysis, which helped him to solve several problems in function and summability theory. In addition to laying the basis for the new theory he also pointed out main directions for applications. This work was interrupted by his unexpected death in 1975. == Teaching == After the war Kangro had a great influence on modernizing the teaching of mathematics in Tartu State University. His courses on algebra and mathematical analysis reflected the changes taking place in these areas in the first half of the 20th century: function theory of polynomials was replaced by abstract algebra, mathematical analysis was based on axiomatic methods and set theory. His course on functional analysis became a starting point for a new research direction in numerical methods in Tartu. Kangro's main contribution was raising a new generation of mathematicians. He has supervised 23 Candidate's theses in mathematics. In addition to mathematical analysis he has also contributed to development of algebra, numerical methods and geometry in Estonia. Notable is his initiative in reorganization of mathematical higher education in University of Tartu in the 1960s in connection with increased need for computer experts. == Honors == 1961 member of Academy of Sciences of the Estonian SSR (now Estonian Academy of Sciences) 1965 Merited Scientist of the Estonian SSR == Books == 1948 Kõrgem algebra I, Teaduslik kirjastus, Tartu 1950 Kõrgem algebra II, Eesti Riiklik Kirjastus, Tallinn-Tartu 1962 Kõrgem algebra, Eesti Riiklik Kirjastus, Tallinn 1965 Matemaatiline analüüs I, Eesti Raamat, Tallinn 1968 Matemaatiline analüüs II, Valgus, Tallinn == Publications == Verallgemeinerte Theorie der absoluten Summierbarkeit. Tartu Ülikooli Toimetised, seeria A, 37 (1942). Theory of summability of sequences and series. Journal of Soviet Mathematics, 5, 1–45 (1976). == References ==
Wikipedia:Guofang Wei#0	Guofang Wei is a mathematician in the field of differential geometry. She is a professor at the University of California, Santa Barbara. == Education == Wei earned a doctorate in mathematics from the State University of New York at Stony Brook in 1989, under the supervision of Detlef Gromoll. Her dissertation produced fundamental new examples of manifolds with positive Ricci curvature and was published in the Bulletin of the American Mathematical Society. These examples were later expanded upon by Burkard Wilking. == Research == In addition to her work on the topology of manifolds with nonnegative Ricci curvature, she has completed work on the isometry groups of manifolds with negative Ricci curvature with coauthors Xianzhe Dai and Zhongmin Shen. She also has major work with Peter Petersen on manifolds with integral Ricci curvature bounds. Starting in 2000 Wei began working with Christina Sormani on limits of manifolds with lower Ricci curvature bounds using techniques of Jeff Cheeger and Tobias Colding, particularly Kenji Fukaya's metric measure convergence. The limit spaces in this setting are metric measure spaces. Wei was invited to present this work in a series of talks at the Seminaire Borel in Switzerland. Sormani and Wei also developed a notion called the covering spectrum of a Riemannian manifold. Dr. Wei has completed research with her student, Will Wylie, on smooth metric measure spaces and the Bakry–Emery Ricci tensor. Guofang Wei was twice invited to present her work at the prestigious Geometry Festival both in 1996 and 2009. == Outreach == In addition to conducting research, Guofang Wei has mentored the Dos Pueblos High School Math Team Archived 2011-07-14 at the Wayback Machine, which won second place in the International Shing-Tung Yau High School Math Awards competition in Beijing in 2008. == Awards and honors == In 2013 she became a fellow of the American Mathematical Society, for "contributions to global Riemannian geometry and its relation with Ricci curvature". == Selected publications == "Examples of complete manifolds of positive Ricci curvature with nilpotent isometry groups", Bull. Amer. Math. Soc. Vol. 19, no. 1 (1988), 311–313. doi:10.1090/S0273-0979-1988-15653-4 with X. Dai and Z. Shen, "Negative Ricci curvature and isometry group", Duke Math J. 76 (1994), 59–73. doi:10.1215/S0012-7094-94-07603-5 with X. Dai and R. Ye, "Smoothing Riemannian manifolds with Ricci curvature bounds", MANUSCR MATH, vol. 90, no. 1 (1996), 49–61. doi:10.1007/BF02568293 with P. Petersen, "Relative volume comparison with integral curvature bounds", GAFA 7 (1997), 1031–1045. doi:10.1007/s000390050036 with C. Sormani, "The covering spectrum of a compact length space", Journal of Diff. Geom. 67 (2004), 35–77. doi:10.4310/jdg/1099587729 with X. Dai and X. Wang, "On stability of Riemannian manifold with parallel spinors", Invent Math, vol. 161, no. 1, (2005) 151–176. doi:10.1007/s00222-004-0424-x with W. Wylie, "Comparison geometry for the Bakry–Emery Ricci tensor", Journal of Diff. Geom. 83, no. 2 (2009), 377–405. doi:10.4310/jdg/1261495336 == References == == External links == Guofang Wei at UCSB Seminaire Borel Geometry Festival 2009 Dos Pueblos High School Math Team wins Second Place in S T Yau HS Math Awards Guofang Wei at the Mathematics Genealogy Project
Wikipedia:Gurcharan Singh Gill#0	Gurcharan Singh Gill (born c. 1935) is a genealogist and mathematician who is claimed to be the first Sikh convert to Mormonism. He served as the first mission president of the Bangalore India Mission. Gill is a retired math professor of the Emiritus Faculty of Brigham Young University. == Family == Gill was born into an agricultural Sikh family in present-day Moga, Moga district in Punjab, India. His father was Ram Singh and mother was Basant Kaur Sidhu of Dhudike village. Early in his life, a brother (Ajaib Singh) and sister (Nasib) of his died due to illness, which had a lasting impact on the young Gill. Gill married Vilo Pratt, with the couple having seven children. After his marriage, his Mormon family-in-law assisted Gill with bringing his seven surviving brothers in India to the United States. Gill's grandson is James Goldberg. == Mormonism == After the death of Gill's two siblings, he became preoccupied with questions regarding what happens after death, feeling that the religions he was familiar with, namely Sikhism, Hinduism, and Islam, did not give satisfactory answers on the nature of the after-life. After arriving in the Fresno, California, the United States for his studies at Fresno State College, he began researching what Christian groups had to say about the matter and eventually came to discover the Church of Jesus Christ of Latter-day Saints. Gill did not agree with the Catholic and Protestant beliefs that people who had not been baptized in their lifetime would not be in heaven. However, Gill was convinced by the Mormon beliefs of the plan of salvation, premortal life, life after death, temples, and temple work, and the eternal link within a family, and became baptized as a Mormon. He also felt that the Mormon teachings complimented the Sikh beliefs he was raised with. He was baptized on January 7, 1956. In 1993, Gill returned to India, where he served as the first mission president of the Bangalore India Mission in Bangalore, India. In 2018, Gill reached out to Church leaders in Bengaluru, New Delhi, and Hyderabad to propose commemorating the mission's silver jubilee with discussions relating to family history and temple service. At the time, the closest Mormon temple was in Hong Kong. President Russell M. Nelson would later announce the plan for a proposed temple in Bengaluru. Approximately 10,000 Latter-day Saints live in India. == Genealogical research == After becoming a Mormon, Gill returned to his native India to conduct genealogical research into his family by scouring and locating native-sources. At the time, genealogical research was not very emphasized or promoted by Indian Latter-day saints members. The local, traditional genealogists of Gill's village had left for Pakistan during the 1947 partition, thus Gill had to at-first rely on oral-history narrated by his relatives to document his genealogy. Gill discovered in 1986 that tax-records in the Moga district were attached to a genealogical pedigree going back four generations, with records dating back to the 1850s. These records have been described as being one of the few surviving records of Punjabi genealogy, as census records in India (specifically enumeration data) were rarely preserved. The more recent records were written in Punjabi using Gurmukhi script and the older records were written in Urdu in Nastaliq script. The names of more than 250,000 individuals have been extracted from the records by Gill. The genealogical importance of such records for the purpose of family history research was raised by Gurcharan Singh Gill. The Shajjra Nasb (also known as Kursee Nama) records of villages of Moga district and partly of Firozpur district from 1887 to 1958 have been digitized by the Church of Jesus Christ of Latter-day Saints via FamilySearch and are available for online viewing. These records detail land ownership pedigrees for families of the village. On July 17, 2010, Gill held a workshop at the University of the Fraser Valley, Abbotsford, which touched-upon the objective of gaining permission from Deputy Commissioners and the Punjab government to preserve thee records electronically through the means of digitization and indexing at no cost to the districts or the Punjab government. Furthermore, he stressed upon the importance of translating the records into English. == Education and career == Gill left India to pursue studies in USA in 1954. Gill earned a bachelor's degree (B.Sc.) in Mathematics and Physics from BYU in 1958 and a master's degree (M.Sc., 1960) and doctorate (PhD, 1965) from the University of Utah, both in mathematics. Gill worked as a professor of mathematics at Brigham Young University from 1960 to 1963 and again from 1965 to 1999. After retiring in 1999, he dedicated himself to genealogical research, with an emphasis on the land records of Moga district. == Bibliography == Deeper Roots of the Gill, Bhatti, Sidhu, Brar, Tur, and Related Jat and Rajput Clans (2008) Extraction of Pedigrees from Revenue Records of Punjab, India (2016) == Notes == == References ==
Wikipedia:Gury Marchuk#0	Gury Ivanovich Marchuk (Russian: Гурий Иванович Марчук; 8 June 1925 – 24 March 2013) was a Soviet and Russian scientist in the fields of computational mathematics, and physics of atmosphere. Academician (since 1968); the President of the USSR Academy of Sciences in 1986–1991. Among his notable prizes are the USSR State Prize (1979), Demidov Prize (2004), Lomonosov Gold Medal (2004). Marchuk was born in Orenburg Oblast, Russia. A member of the Communist Party of the Soviet Union since 1947, Academician Marchuk was elected to the Central Committee of the Party as a candidate member in 1976 and as a full member in 1981. He was elected as deputy of the Supreme Soviet of the Union of Soviet Socialist Republics in 1979. He was appointed to succeed Vladimir Kirillin as chairman of the State Committee for Science and Technology (GKNT) in 1980. Marchuk was a proponent of the Integrated Long-Term Programme (ILTP) of Cooperation in Science & Technology that was established in 1987 as a scientific cooperative venture between India and the Soviet Union. The programme allowed the scientists of the countries to collaboratively undertake research in areas as diverse as healthcare and lasers. Marchuk co-chaired the programme's Joint Council with Prof. C.N.R. Rao for 25 years and was made an honorary member of India's National Academy of Sciences. In 2002, the Government of India conferred the Padma Bhushan on him. == Honours and awards == Hero of Socialist Labour (1975) Honorary Citizen of Obninsk (1985) Four Orders of Lenin (1967, 1971, 1975, 1985) Keldysh Gold Medal — for his work "The development and creation of new methods of mathematical modeling" (1981) Karpinski International Prize (1988) Chebyshev Gold Medal — for outstanding performance in mathematics (1996) Lomonosov Gold Medal (Moscow State University, 2004) - for his outstanding contribution to the creation of new models and methods for solving problems in the physics of nuclear reactors, the physics of the atmosphere and ocean, and immunology Cavalier silver sign "Property of Siberia" Lenin Prize in Science (1961) Friedman Prize (1975) USSR State Prize (1979) State Prize of the Russian Federation (2000) Demidov Prize (2004) Honorary Doctorates of the University of Toulouse (1973), Charles University (Prague, 1978), Dresden University of Technology (1978), Technical University of Budapest (1978) Foreign Member of the Bulgarian Academy of Sciences (1977), German Academy of Sciences at Berlin (1977), Czechoslovak Academy of Sciences (1977), Polish Academy of Sciences (1988) Order of Merit for the Fatherland, 2nd and 4th classes Jubilee Medal "300 Years of the Russian Navy" Medal "For the Victory over Germany in the Great Patriotic War 1941–1945" Medal "For Valiant Labour in the Great Patriotic War 1941-1945" Commander of the Legion of Honour Order of Georgi Dimitrov Padma Bhushan (2002) Vilhelm Bjerknes Medal (2008) == References == == External links == Gury Marchuk — scientific works on the website Math-Net.Ru Scientific biography (in Russian) [1].
Wikipedia:Gustaf Eneström#0	Gustaf Hjalmar Eneström (5 September 1852 – 10 June 1923) was a Swedish mathematician, statistician and historian of mathematics known for introducing the Eneström index, which is used to identify Euler's writings. Most historical scholars refer to the works of Euler by their Eneström index. Eneström received a Bachelor of Science (filosofie kandidat) degree from Uppsala university in 1871, received a position at Uppsala University Library in 1875, and at the National Library of Sweden in 1879. From 1884 to 1914, he was the publisher of the mathematical-historical journal Bibliotheca Mathematica, which he had founded and partially funded with his own means. Concerning the history of mathematics, he was known as critical to Moritz Cantor. With Soichi Kakeya, he is known for the Eneström-Kakeya theorem which determines an annulus containing the roots of a real polynomial. In 1923 George Sarton wrote, "No one has done more for the sound development of our studies". Sarton went on: "the very presence of Eneström obliged every scholar devoting himself to the history of mathematics to increase his circumspection and improve his work." Eneström has also developed an election method similar to Phragmen's voting rules. == Notes == == References == Lory, W. (1926). "Gustav Eneström". Isis. 8 (2): 313–320. doi:10.1086/358394.
Wikipedia:Gustav Herglotz#0	Gustav Herglotz (2 February 1881 – 22 March 1953) was a German Bohemian physicist best known for his works on the theory of relativity and seismology. == Biography == Gustav Ferdinand Joseph Wenzel Herglotz was born in Volary num. 28 to a public notary Gustav Herglotz (also a Doctor of Law) and his wife Maria née Wachtel. The family were Sudeten Germans. He studied mathematics and astronomy at the University of Vienna in 1899, and attended lectures by Ludwig Boltzmann. In this time of study, he had a friendship with his colleagues Paul Ehrenfest, Hans Hahn and Heinrich Tietze. In 1900 he went to the LMU Munich and achieved his Doctorate in 1902 under Hugo von Seeliger. Afterwards, he went to the University of Göttingen, where he habilitated under Felix Klein. In 1904 he became Privatdozent for Astronomy and Mathematics there, and in 1907 Professor extraordinarius. In 1908 he became Professor extraordinarius in Vienna, and in 1909 at the University of Leipzig. From 1925 (until becoming Emeritus in 1947) he again was in Göttingen as the successor of Carl Runge on the chair of applied mathematics. One of his students was Emil Artin. == Work == Herglotz worked in the fields of seismology, number theory, celestial mechanics, theory of electrons, special relativity, general relativity, hydrodynamics, refraction theory. In 1904, Herglotz defined relations for the electrodynamic potential which are also valid in special relativity even before that theory was fully developed. Hermann Minkowski (during a conversation reported by Arnold Sommerfeld) pointed out that the four-dimensional symmetry of electrodynamics is latently contained and mathematically applied in Herglotz' paper. In 1907, he became interested in the theory of earthquakes, and together with Emil Wiechert, he developed the Wiechert–Herglotz method for the determination of the velocity distribution of Earth's interior from the known propagation times of seismic waves (an inverse problem). There, Herglotz solved a special integral equation of Abelian type. The Herglotz–Noether theorem stated by Herglotz (1909) and independently by Fritz Noether (1909), was used by Herglotz to classify all possible forms of rotational motions satisfying Born rigidity. In the course of this work, Herglotz showed that the Lorentz transformations correspond to hyperbolic motions in R 3 {\displaystyle R_{3}} , by which he classified the one-parameter Lorentz transformations into loxodromic, parabolic, elliptic, and hyperbolic groups (see Möbius transformation#Lorentz transformation). In 1911, he formulated the Herglotz representation theorem which concerns holomorphic functions f on the unit disk D, with Re f ≥ 0 and f(0) = 1, represented as an integral over the boundary of D with respect to a probability measure μ. The theorem asserts that such a function exists if and only if there is a μ such that ∀ z ∈ D f ( z ) = ∫ ∂ D λ + z λ − z d μ ( λ ) . {\displaystyle \forall z\in D\ \ f(z)\ =\ \int _{\partial D}{\frac {\lambda +z}{\lambda -z}}\ d\mu (\lambda ).} The theorem also asserts that the probability measure is unique to f. In 1911, he formulated a relativistic theory of elasticity. In the course of that work, he obtained the vector Lorentz transformation for arbitrary velocities (see History of Lorentz transformations#Herglotz (1911)). In 1916, he also contributed to general relativity. Independently of previous work by Hendrik Lorentz (1916), he showed as to how the contracted Riemann tensor and the curvature invariant can be geometrically interpreted. == Selected works == Gesammelte Schriften / Gustav Herglotz, edited for d. Akad. d. Wiss. in Göttingen by Hans Schwerdtfeger. XL, 652 p., Vandenhoeck & Ruprecht, Göttingen 1979, ISBN 3-525-40720-3. Vorlesungen über die Mechanik der Kontinua / G. Herglotz, prepared by R. B. Guenther and H. Schwerdtfeger, Teubner-Archiv zur Mathematik; vol. 3, 251 p.: 1 Ill., graph. Darst.; 22 cm, Teubner, Leipzig 1985. Über die analytische Fortsetzung des Potentials ins Innere der anziehenden Massen, Preisschriften der Fürstlichen Jablonowskischen Gesellschaft zu Leipzig, VII, 52 pages, with 18 Fig.; Teubner, Leipzig (1914). Über das quadratische Reziprozitätsgesetz in imaginären quadratischen Zahlkörpern, Ber. über d. Verh. d. königl. sächs. Gesellsch. d. Wissensch. zu Leipzig, pp. 303–310 (1921). == See also == Acceleration (special relativity) Möbius transformation Spherical wave transformation Squeeze mapping Rindler coordinates Herglotz's variational principle == References == == External links == Media related to Gustav Herglotz at Wikimedia Commons Works by or about Gustav Herglotz at Wikisource Gustav Herglotz at the Mathematics Genealogy Project O'Connor, John J.; Robertson, Edmund F., "Gustav Herglotz", MacTutor History of Mathematics Archive, University of St Andrews Herglotz, Gustav (1881–1953) at the MathWorld Gustav Herglotz by Joachim Ritter and Sebastian Rost
Wikipedia:Gustav Lehrer#0	Gustav Lehrer (born 1947) is an Australian mathematician and researcher. He is known for his work in algebraic geometry, group theory, representation theory, and topology. Along with his doctoral student John Graham, Lehrer is credited with the discovery of cellular algebras. Lehrer is also noted for his parametrization of the characters of the finite special linear groups, the development of a theory for decomposition of characters induced from parabolic subgroups (with Robert Howlett), and the determination of the action of a complex reflection group on the cohomology of the complement of its reflecting hyperplanes. == Early life and education == Gustav Lehrer was born in 1947 in Munich, Germany. His parents were holocaust survivors originally from Poland, and the family emigrated to Australia when Lehrer was 3 years old. He completed his BSc degree in mathematics from the University of Sydney in 1967. He then completed his PhD in mathematics in 1971 at the University of Warwick under the supervision of James Alexander "Sandy" Green. His doctoral thesis was titled On the discrete series characters of linear groups. == Career == Lehrer served as a lecturer in the United Kingdom at the University of Warwick and the University of Birmingham from 1971 until returning to Australia in 1974 to take a position as a lecturer at the University of Sydney. In 1991, Lehrer was appointed head of the School of Mathematics and Statistics at the University of Sydney. From 1996 to 1998, Lehrer was Head of the Centre for Mathematics and its Applications (CMA) at the Australian National University. In 2007, he became of a member of the Mathematical Sciences Sectional Committee of the Australian Academy of Science. Through the course of his career, he was a guest lecturer worldwide, including the Institute for Mathematical Research in Zurich, the University of Aarhaus, the University of Essen, the Ecole Normale Superieure, and the Science University of Tokyo. Lehrer is a member of the Board of Governors of Tel Aviv University, and from 2011-2022 was the President of the Sydney Jewish Museum. He was a member of the Board of Trustees of Sydney Grammar School from 1995 to 2022. == Research == Much of Lehrer's research has centered around representation theory. His doctoral thesis dealt with the issue of representing finite Lie groups. Throughout his time in the UK, he researched linear groups. During this period, his work was influenced by David Mumford, who lectured at the University of Warwick during Lehrer's time there. Upon returning to Australia, Lehrer collaborated with Robert Howlett to solve Springer's decomposition problem, resulting in the development of the Howlett-Lehrer theory. This theory subsequently had applications in a number of other fields of mathematics, including contributing to the development of the Deligne-Lusztig theory. Lehrer contributed significantly to the problem of determining the geometry of the space of configurations of n distinct points in a complex plane, which is a problem in topology with relations to knot theory, developing novel algebraic geometric, topological, and analytical approaches. Lehrer is perhaps best known for the invention of cellular algebras with John Graham, which provide a means of "deforming" structures which split into more complicated structures which do not. Cellular algebras have a range of applications in physics, including the theory of quantum groups. In 2014, Lehrer solved the second fundamental problem of invariant theory of the orthogonal group, which had remained unsolved for 75 years. The problem involved the relationship between different quantities that remain the same after undergoing geometric transformation. == Awards and honors == He was made a fellow of the Australian Academy of Science in 1998, and received an Australian Research Council senior fellowship from 1999-2004. Lehrer was granted the Humboldt research award in 1999. In 2001, he won the Centenary medal "for service to Australian society and science in pure mathematics." In 2005, he was made a professorial fellow of the Australian Research Council. Lehrer also received a visiting fellowship at All Souls College, Oxford. He received the Hannan medal in 2015. In 2016, Lehrer became a Member of the Order of Australia (AM) for "significant service to tertiary mathematics education (as an academic and researcher), and to professional and community groups." Volume 311 of the Journal of Algebra was dedicated to Lehrer on his 60th birthday in 2009. == Personal life == Lehrer is married to Nanna Lehrer with three children; Lisa, Alex, and Eddie. == Selected works == Gustav Lehrer, Ruibin Zhang, The Brauer category and invariant theory, Journal of the European Mathematical Society 2015 Gustav Lehrer, Donald E. Taylor, Unitary Reflection Groups, Australian Mathematical Society Lecture Series 20, Cambridge University Press 2009 John J. Graham, Gustav Lehrer, The representation theory of affine Temperley-Lieb algebras, Enseignement Mathematique 1998 John J. Graham, Gustav Lehrer, Cellular Algebras, Inventiones Mathematicae 1996 Gustav Leher, Louis Solomon, On the action of the symmetric group on the cohomology of the complement of its reflecting hyperplanes, Journal of Algebra 1986 Robert Howlett, Gustav Lehrer, Induced cuspidal representations and generalised Hecke rings, Inventiones Mathematicae 1980 == References ==
Wikipedia:Gustave Dumas#0	Gustave Dumas (5 March 1872, L'Etivaz, Vaud, Switzerland – 11 July 1955) was a Swiss mathematician, specializing in algebraic geometry. Dumas received a baccalaureate degree from the University of Lausanne, then another baccalaureate degree from the Sorbonne, and in 1904 a doctoral degree from the Sorbonne with dissertation Sur les fonctions à caractère algébrique dans le voisinage d'un point donné. In 1906 he obtained his habilitation qualification from Zürich's Federal Polytechnic School with habilitation dissertation Sur quelques cas d'irréductibilité des polynômes à coefficients rationnels. From 1906 to 1913 Dumas taught higher mathematics at the Federal Polytechnic School. At the University of Lausanne's Engineering School, he became in 1913 a professor extraordinarius and in 1916 a professor ordinarius, retiring in 1942. At Lausanne he had an important influence on his student Georges de Rham, who became Dumas's assistant before graduating in 1925. Dumas served a two-year term as president of the Swiss Mathematical Society in 1922–1923. He was an Invited Speaker of the International Congress of Mathematicians in 1928 at Bologna. == Selected publications == "Sur quelques cas d'irréductibilité des polynômes à coefficients rationnels." Journal de Mathématiques Pures et Appliquées 2 (1906): 191–258. "Sur la résolution des singularités des surfaces." CR Acad. Sci. Paris 152 (1911): 682–684. "Sur le polygone de Newton et les courbes algébriques planes." Commentarii Mathematici Helvetici 1, no. 1 (1929): 120–141. doi:10.1007/BF01208360 == References ==
Wikipedia:Gustavo Ponce#0	Gustavo A. Ponce (born 20 April 1952 in Venezuela) is a Venezuelan mathematician. == Education and career == Ponce graduated from the Central University of Venezuela with a bachelor's degree in 1976. At the Courant Institute of Mathematical Sciences of New York University he graduated with a master's degree in 1980 and a Ph.D. in 1982 with thesis Long time stability of solutions of nonlinear evolution equations under the supervision of Sergiu Klainerman (and Louis Nirenberg). Ponce was a visiting lecturer at the University of California, Berkeley from 1982 to 1984, an assistant professor at the Central University of Venezuela from 1984 to 1986, and an assistant professor at the University of Chicago from 1986 to 1989. He was from 1989 to 1991 an associate professor at Pennsylvania State University and is since 1991 a full professor at the University of California, Santa Barbara. He was a visiting professor for brief periods at many academic institutions, including the University of Bonn in 1989, the University of Paris-Sud in 1997 (and again in 2003 and 2012), MSRI in 2001, the Instituto Nacional de Matemática Pura e Aplicada (IMPA) in 2002 (and again in 2010), the Institute for Advanced Study in 2004, the Institute Henri Poincaré in 2009, the Autonomous University of Madrid in 2011, the University of the Basque Country in 2015, and IHES in 2016. Ponce does research on nonlinear partial differential equations (PDEs) using PDE solutions to equations in mathematical physics, such as the Euler and Navier-Stokes equations of hydrodynamics. He was on the editorial boards of Transactions of the AMS from 2006 to 2014 and the Memoirs of the AMS from 2006 to 2014. In 1998 he was an Invited Speaker with talk On nonlinear dispersive equations at the International Congress of Mathematicians in Berlin. In 2012 he was elected a Fellow of the American Mathematical Society. == Selected publications == with Felipe Linares: Introduction to nonlinear dispersive equations, Universitext (2nd ed.), Springer, 2015, doi:10.1007/978-1-4939-2181-2, ISBN 978-1-4939-2180-5 with Sergiu Klainerman: Klainerman, S.; Ponce, Gustavo (1983), "Global, small amplitude solutions to nonlinear evolution equations", Communications on Pure and Applied Mathematics, 63: 133–141, doi:10.1002/cpa.3160360106 with Tosio Kato: Kato, Tosio; Ponce, Gustavo (1988), "Commutators estimates and the Euler and Navier-Stokes equations", Communications on Pure and Applied Mathematics, 41 (7): 891–907, doi:10.1002/cpa.3160410704 with Felipe Linares: Linares, Felipe; Ponce, Gustavo (1993), "On the Davey-Stewartson systems", Annales de l'Institut Henri Poincaré C, 10 (5): 523–548, doi:10.1016/S0294-1449(16)30203-7 with Carlos Kenig and Luis Vega: Kenig, Carlos; Ponce, Gustavo; Vega, Luis (1996), "A bilinear estimate with applications to KdV equation", Journal of the American Mathematical Society, 9 (2): 573–603, doi:10.1090/S0894-0347-96-00200-7, JSTOR 2152869 with Carlos Kenig and Luis Vega: "Smoothing effects and local theory theory for generalized nonlinear Schrödinger equations", Inventiones Mathematicae, 134: 489–545, 1998, doi:10.1007/s002220050272 with Carlos Kenig and Luis Vega: Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis (2004), "The Cauchy problem for quasi-linear Schrödinger equations", Inventiones Mathematicae, 158 (2): 343–388, doi:10.1007/s00222-004-0373-4 with Carlos Kenig, Christian Rolvung, and Luis Vega: Kenig, C.E.; Ponce, G.; Rolvung, C.; Vega, L. (2005), "Variable Coefficient Schrödinger flows for ultrahyperbolic operators", Advances in Mathematics, 196 (2): 373–486, arXiv:math/0503205, doi:10.1016/j.aim.2004.02.002 with A. Alexandrou Himonas, Gerard Misiolek, and Yong Zhou: Himonas, A. Alexandrou; Misiołek, Gerard; Ponce, Gustavo; Zhou, Yong (2007), "Persistence Properties and Unique Continuation of Solutions of the Camassa-Holm equation", Communications in Mathematical Physics, 271 (2): 511–522, arXiv:math/0604192, doi:10.1007/s00220-006-0172-4 with Luis Escauriaza, Carlos Kenig, and Luis Vega: Escauriaza, L.; Kenig, C. E.; Ponce, G.; Vega, L. (2006), "On uniqueness properties of solutions of Schrödinger equations", Communications in Partial Differential Equations, 31 (12): 1811–1823, arXiv:1110.4873, doi:10.1080/03605300500530446, S2CID 121929231 == References ==
Wikipedia:Gustavo Sannia#0	Gustavo Sannia (13 May 1875 – 21 December 1930) was an Italian mathematician working in differential geometry, projective geometry, and summation of series. He was the son of Achille Sannia, mathematician and senator of the Kingdom of Italy. == Biography == Gustavo Sannia was born in Naples. Sannia lived in Turin from 1902 to 1915 and from 1919 to 1922, first as an assistant to D'Ovidio and Fubini and later as a professor. From 1915 to 1919, he taught at the University of Cagliari. Sannia returned to Naples in 1924, where he would remain until his premature death. == Selected publications == "Deformazioni infinitesime delle curve inestendibili e corrispondenza per ortogonalità di elementi." Rendiconti del Circolo Matematico di Palermo (1884–1940) 21, no. 1 (1906): 229–256. doi:10.1007/BF03013473 "Nuova esposizione della geometria infinitesimale délle congruenze rettilinee." Annali di Matematica Pura ed Applicata (1898–1922) 15, no. 1 (1908): 143–185. doi:10.1007/BF02419759 "Nuovo metodo per lo studio delle congruenze e dei complessi di raggi." Rendiconti del Circolo Matematico di Palermo (1884–1940) 33, no. 1 (1912): 328–340. doi:10.1007/BF03015310 "Osservazioni sulla «Réclamation de priorité» del sig. Zindler." Annali di Matematica Pura ed Applicata (1898–1922) 19, no. 1 (1912): 57–59. doi:10.1007/BF02419391 "Su due forme differenziali che individuano una congruenza o un complesso di rette." Rendiconti del Circolo Matematico di Palermo (1884–1940) 33, no. 1 (1912): 67–74. doi:10.1007/BF03015288 "Sui differenziali totali di ordine superiore." Rendiconti del Circolo Matematico di Palermo (1884–1940) 36, no. 1 (1913): 305–316. doi:10.1007/BF03016036 "Nuovo metodo di sommazione delle serie: Estensione del metodo di Borel." Rendiconti del Circolo Matematico di Palermo (1884–1940) 42, no. 1 (1916): 303–322. doi:10.1007/BF03014904 "Riavvicinamento di geometrie differenziali delle superficie: metriche, affine, proiettiva." Annali di Matematica Pura ed Applicata (1898–1922) 31, no. 1 (1922): 165–189. doi:10.1007/BF02419749 "Nuova trattazione della geometria proiettivo-differenziale delle curve sghembe." Annali di Matematica Pura ed Applicata 3, no. 1 (1926): 1–25. doi:10.1007/BF02418644 == References == == Bibliography == G. F. Tricomi, Matematici italiani del primo secolo dello stato unitario, Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, 4th series, vol. 1, 1962. == External links == Gustavo Sannia at mathematica.sns.it
Wikipedia:Guy Hirsch#0	Guy Hirsch (20 September 1915 – 4 August 1993) was a Belgian mathematician and philosopher of mathematics, who worked on algebraic topology and epistemology of mathematics. He became a member of the Royal Flemish Academy of Belgium for Science and the Arts in 1973. He is known for the Leray–Hirsch theorem, a basic result on the algebraic topology of fiber bundles that he proved independently of Jean Leray in the late 1940s. == References == == External links == Guy Hirsch at the Mathematics Genealogy Project
Wikipedia:Guy Terjanian#0	Guy Terjanian is a French mathematician who has worked on algebraic number theory. He achieved his Ph.D. under Claude Chevalley in 1970, and at that time published a counterexample to the original form of a conjecture of Emil Artin, which suitably modified had just been proved as the Ax-Kochen theorem. In 1977, he proved that if p is an odd prime number, and the natural numbers x, y and z satisfy x 2 p + y 2 p = z 2 p {\displaystyle x^{2p}+y^{2p}=z^{2p}} , then 2p must divide x or y. == See also == Ax–Kochen theorem == References == == Further reading == math.unicaen.fr article Topic: Arithmetic & geometry
Wikipedia:Gwyneth Stallard#0	Gwyneth Mary Stallard is a British mathematician whose research concerns complex dynamics and the iteration of meromorphic functions. She is a professor of pure mathematics at the Open University. == Education and career == Stallard read mathematics at King's College, Cambridge, finishing in 1985, and earned her Ph.D. from Imperial College London in 1991. Her dissertation, Some problems in the iteration of meromorphic functions, was supervised by Irvine Noel Baker. She has spoken about the difficulty of finding postdoctoral research positions at a time when there were few such positions in England and the ties of her husband's job prevented her from moving abroad; she maintained her mathematical career at this stage by taking a temporary lectureship teaching engineering students at the University of Southampton. When she became a professor of mathematics at the Open University, she became the first woman to be a professor in the department. == Activism and recognition == Stallard won the Whitehead Prize in 2000, and describes this point as the moment when she became confident in her mathematical research abilities. Stallard was the chair of the Women in Mathematics Committee of the London Mathematical Society from 2006 to 2015, and in 2015 she was named an Officer of the Order of the British Empire for her work in support of women in mathematics. In 2016 she was given a special award by the Suffrage Science Scheme on Ada Lovelace Day in recognition of her work in this area. In 2016, the London Mathematical Society selected her as their Mary Cartwright Lecturer. == References == == External links == Home page
Wikipedia:György Hajós#0	In graph theory, a branch of mathematics, the Hajós construction is an operation on graphs named after György Hajós (1961) that may be used to construct any critical graph or any graph whose chromatic number is at least some given threshold. == The construction == Let G and H be two undirected graphs, vw be an edge of G, and xy be an edge of H. Then the Hajós construction forms a new graph that combines the two graphs by identifying vertices v and x into a single vertex, removing the two edges vw and xy, and adding a new edge wy. For example, let G and H each be a complete graph K4 on four vertices; because of the symmetry of these graphs, the choice of which edge to select from each of them is unimportant. In this case, the result of applying the Hajós construction is the Moser spindle, a seven-vertex unit distance graph that requires four colors. As another example, if G and H are cycle graphs of length p and q respectively, then the result of applying the Hajós construction is itself a cycle graph, of length p + q − 1. == Constructible graphs == A graph G is said to be k-constructible (or Hajós-k-constructible) when it formed in one of the following three ways: The complete graph Kk is k-constructible. Let G and H be any two k-constructible graphs. Then the graph formed by applying the Hajós construction to G and H is k-constructible. Let G be any k-constructible graph, and let u and v be any two non-adjacent vertices in G. Then the graph formed by combining u and v into a single vertex is also k-constructible. Equivalently, this graph may be formed by adding edge uv to the graph and then contracting it. == Connection to coloring == It is straightforward to verify that every k-constructible graph requires at least k colors in any proper graph coloring. Indeed, this is clear for the complete graph Kk, and the effect of identifying two nonadjacent vertices is to force them to have the same color as each other in any coloring, something that does not reduce the number of colors. In the Hajós construction itself, the new edge wy forces at least one of the two vertices w and y to have a different color than the combined vertex for v and x, so any proper coloring of the combined graph leads to a proper coloring of one of the two smaller graphs from which it was formed, which again causes it to require k colors. Hajós proved more strongly that a graph requires at least k colors, in any proper coloring, if and only if it contains a k-constructible graph as a subgraph. Equivalently, every k-critical graph (a graph that requires k colors but for which every proper subgraph requires fewer colors) is k-constructible. Alternatively, every graph that requires k colors may be formed by combining the Hajós construction, the operation of identifying any two nonadjacent vertices, and the operations of adding a vertex or edge to the given graph, starting from the complete graph Kk. A similar construction may be used for list coloring in place of coloring. == Constructibility of critical graphs == For k = 3, every k-critical graph (that is, every odd cycle) can be generated as a k-constructible graph such that all of the graphs formed in its construction are also k-critical. For k = 8, this is not true: a graph found by Catlin (1979) as a counterexample to Hajós's conjecture that k-chromatic graphs contain a subdivision of Kk, also serves as a counterexample to this problem. Subsequently, k-critical but not k-constructible graphs solely through k-critical graphs were found for all k ≥ 4. For k = 4, one such example is the graph obtained from the dodecahedron graph by adding a new edge between each pair of antipodal vertices == The Hajós number == Because merging two non-adjacent vertices reduces the number of vertices in the resulting graph, the number of operations needed to represent a given graph G using the operations defined by Hajós may exceed the number of vertices in G. More specifically, Mansfield & Welsh (1982) define the Hajós number h(G) of a k-chromatic graph G to be the minimum number of steps needed to construct G from Kk, where each step forms a new graph by combining two previously formed graphs, merging two nonadjacent vertices of a previously formed graph, or adding a vertex or edge to a previously formed graph. They showed that, for an n-vertex graph G with m edges, h(G) ≤ 2n2/3 − m + 1 − 1. If every graph has a polynomial Hajós number, this would imply that it is possible to prove non-colorability in nondeterministic polynomial time, and therefore imply that NP = co-NP, a conclusion considered unlikely by complexity theorists. However, it is not known how to prove non-polynomial lower bounds on the Hajós number without making some complexity-theoretic assumption, and if such a bound could be proven it would also imply the existence of non-polynomial bounds on certain types of Frege system in mathematical logic. The minimum size of an expression tree describing a Hajós construction for a given graph G may be significantly larger than the Hajós number of G, because a shortest expression for G may re-use the same graphs multiple times, an economy not permitted in an expression tree. There exist 3-chromatic graphs for which the smallest such expression tree has exponential size. == Other applications == Koester (1991) used the Hajós construction to generate an infinite set of 4-critical polyhedral graphs, each having more than twice as many edges as vertices. Similarly, Liu & Zhang (2006) used the construction, starting with the Grötzsch graph, to generate many 4-critical triangle-free graphs, which they showed to be difficult to color using traditional backtracking algorithms. In polyhedral combinatorics, Euler (2003) used the Hajós construction to generate facets of the stable set polytope. == Notes == == References == Catlin, P. A. (1979), "Hajós's graph-colouring conjecture: variations and counterexamples", Journal of Combinatorial Theory, Series B, 26 (2): 268–274, doi:10.1016/0095-8956(79)90062-5. Diestel, Reinhard (2006), Graph Theory, Graduate Texts in Mathematics, vol. 173 (3rd ed.), Springer-Verlag, pp. 117–118, ISBN 978-3-540-26183-4. Euler, Reinhardt (2003), "Hajós' construction and polytopes", Combinatorial optimization—Eureka, you shrink!, Lecture Notes in Computer Science, vol. 2570, Berlin: Springer-Verlag, pp. 39–47, doi:10.1007/3-540-36478-1_6, ISBN 978-3-540-00580-3, MR 2163949. Gravier, Sylvain (1996), "A Hajós-like theorem for list coloring", Discrete Mathematics, 152 (1–3): 299–302, doi:10.1016/0012-365X(95)00350-6, MR 1388650. Hajós, G. (1961), "Über eine Konstruktion nicht n-färbbarer Graphen", Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe, 10: 116–117. As cited by Jensen & Toft (1994). Iwama, Kazuo; Pitassi, Toniann (1995), "Exponential lower bounds for the tree-like Hajós calculus", Information Processing Letters, 54 (5): 289–294, doi:10.1016/0020-0190(95)00035-B, MR 1336013. Jensen, Tommy R.; Royle, Gordon F. (1999), "Hajós constructions of critical graphs", Journal of Graph Theory, 30 (1): 37–50, doi:10.1002/(SICI)1097-0118(199901)30:1<37::AID-JGT5>3.0.CO;2-V, MR 1658542. Jensen, Tommy R.; Toft, Bjarne (1994), Graph Coloring Problems (2nd ed.), John Wiley and Sons, ISBN 978-0-471-02865-9. Koester, Gerhard (1991), "On 4-critical planar graphs with high edge density", Discrete Mathematics, 98 (2): 147–151, doi:10.1016/0012-365X(91)90039-5, MR 1144633. Kubale, Marek (2004), Graph Colorings, Contemporary Mathematics, vol. 352, American Mathematical Society, p. 156, ISBN 978-0-8218-3458-9. Liu, Sheng; Zhang, Jian (2006), "Using Hajós' construction to generate hard graph 3-colorability instances", Artificial Intelligence and Symbolic Computation, Lecture Notes in Computer Science, vol. 4120, Springer-Verlag, pp. 211–225, doi:10.1007/11856290_19, ISBN 978-3-540-39728-1. Mansfield, A. J.; Welsh, D. J. A. (1982), "Some colouring problems and their complexity", Graph theory (Cambridge, 1981), North-Holland Math. Stud., vol. 62, Amsterdam: North-Holland, pp. 159–170, MR 0671913. Pitassi, Toniann; Urquhart, Alasdair (1995), "The complexity of the Hajós calculus", SIAM Journal on Discrete Mathematics, 8 (3): 464–483, CiteSeerX 10.1.1.30.5879, doi:10.1137/S089548019224024X, MR 1341550.
Wikipedia:Gábor J. Székely#0	Gábor J. Székely (Hungarian pronunciation: [ˈseːkɛj]; born February 4, 1947, in Budapest) is a Hungarian-American statistician/mathematician best known for introducing energy statistics (E-statistics). Examples include: the distance correlation, which is a bona fide dependence measure, equals zero exactly when the variables are independent; the distance skewness, which equals zero exactly when the probability distribution is diagonally symmetric; the E-statistic for normality test; and the E-statistic for clustering. Other important discoveries include the Hungarian semigroups, the location testing for Gaussian scale mixture distributions, the uncertainty principle of game theory, the half-coin which involves negative probability, and the solution of an old open problem of lottery mathematics: in a 5-from-90 lotto the minimum number of tickets one needs to buy to guarantee that at least one of these tickets has (at least) 2 matches is exactly 100. == Life and career == Székely attended the Eötvös Loránd University, Hungary graduating in 1970. His first advisor was Alfréd Rényi. Székely received his Ph.D. in 1971 from Eötvös Loránd University, the Candidate Degree in 1976 under the direction of Paul Erdős and Andrey Kolmogorov, and the Doctor of Science degree from the Hungarian Academy of Sciences in 1986. During the years 1970-1995 he has worked as a Professor in Eötvös Loránd University at the Department of Probability Theory and Statistics. Between 1985 and 1995 Székely was the first program manager of the Budapest Semesters in Mathematics. Between 1990 and 1997 he was the founding chair of the Department of Stochastics of the Budapest Institute of Technology (Technical University of Budapest) and editor-in-chief of Matematikai Lapok, the official journal of the János Bolyai Mathematical Society. In 1989 Székely was visiting professor at Yale University, and in 1990-91 he was the first Lukacs Distinguished Professor in Ohio. Since 1995 he has been teaching at the Bowling Green State University at the Department of Mathematics and Statistics. Székely was academic advisor of Morgan Stanley, NY, and Bunge, Chicago, helped to establish the Morgan Stanley Mathematical Modeling Centre in Budapest (2005) and the Bunge Mathematical Institute (BMI) in Warsaw (2006) to provide quantitative analysis to support the firms' global business. Since 2006 he is a Program Director of Statistics of the National Science Foundation, now retired. Székely is also Research Fellow of the Rényi Institute of Mathematics of the Hungarian Academy of Sciences. For an informal biographical sketch see Conversations with Gábor J. Székely == Awards == Rollo Davidson Prize of Cambridge University (1988) Elected Fellow of the International Statistical Institute (1996) Elected Fellow of the American Statistical Association (2000) Elected Fellow of the Institute of Mathematical Statistics (2010) == Books == Székely, G. J. (1986) Paradoxes in Probability Theory and Mathematical Statistics, Reidel. Ruzsa, I. Z. and Székely, G. J. (1988) Algebraic Probability Theory, Wiley. Székely, G. J. (editor) (1995) Contests in Higher Mathematics, Springer. Rao, C.R. and Székely, G.J. (editors) (2000) Statistics For The 21st Century: Methodologies For Applications Of The Future (Statistics, Textbooks And Monographs), New York, Marcel Dekker. Guoyan Zheng, Shuo Li, Székely, G. J.(2017)Statistical Shape and Deformation Analysis, 1st Edition, Academic Press. Székely, G.J. and Rizzo, M.L. (2023) The Energy of Data and Distance Correlation, Chapman and Hall/CRC Press, Monographs on Statistics and Applied Probability Volume 171 [1]. == Selected works == Székely, G. J. (1981–82) Why is 7 a mystical number? (in Hungarian) in: MIOK Évkönyv, 482-487, ed. Sándor Scheiber. Székely, G.J. and Ruzsa, I.Z. (1982) Intersections of traces of random walks with fixed sets, Annals of Probability 10, 132-136. Székely, G. J. and Ruzsa, I.Z. (1985) No distribution is prime, Z. Wahrscheinlichkeitstheorie verw. Geb. 70, 263-269. Székely, G. J. and Buczolich, Z. (1989) When is a weighted average of ordered sample elements a maximum likelihood estimator of the location parameter? Advances in Applied Mathematics 10, 439-456. [2] Székely, G. J, Bennett, C.D., and Glass, A. M. W. (2004) Fermat's last theorem for rational exponents, The American Mathematical Monthly 11/4, 322-329. Székely, G. J. (2006) Student's t-test for scale mixtures. Lecture Notes Monograph Series 49, Institute of Mathematical Statistics, 10-18. Székely, G. J., Rizzo, M. L. and Bakirov, N. K. (2007) Measuring and testing independence by correlation of distances, The Annals of Statistics, 35, 2769-2794. arXiv:0803.4101 Székely, G. J. and Rizzo, M.L. (2009) Brownian distance covariance, The Annals of Applied Statistics, 3/4, 1233-1308. arXiv:1010.0297 Rizzo, M. L. and Székely, G. J. (2010) DISCO analysis: A nonparametric extension of analysis of variance, The Annals of Applied Statistics, 4/2, 1034-1055. arXiv:1011.2288 Székely, G.J. and Rizzo, M.L. (2013) Energy statistics: statistics based on distances, Invited paper, Journal of Statistical Planning and Inference, 143/8, 1249-1272. Székely, G.J. and Rizzo, M.L. (2014) Partial distance correlation with methods for dissimilarities, The Annals of Statistics, 42/6, 2382-2412. == References == == External links == Székely's website Archived 2010-11-12 at the Wayback Machine. Gábor J. Székely at the Mathematics Genealogy Project
Wikipedia:Gábor Szegő#0	Gábor Szegő (Hungarian: [ˈɡaːbor ˈsɛɡøː]) (January 20, 1895 – August 7, 1985) was a Hungarian-American mathematician. He was one of the foremost mathematical analysts of his generation and made fundamental contributions to the theory of orthogonal polynomials and Toeplitz matrices building on the work of his contemporary Otto Toeplitz. == Life == Szegő was born in Kunhegyes, Austria-Hungary (today Hungary), into a Jewish family as the son of Adolf Szegő and Hermina Neuman. He married the chemist Anna Elisabeth Neményi in 1919, with whom he had two children. In 1912 he started studies in mathematical physics at the University of Budapest, with summer visits to the University of Berlin and the University of Göttingen, where he attended lectures by Frobenius and Hilbert, amongst others. In Budapest he was taught mainly by Fejér, Beke, Kürschák and Bauer and made the acquaintance of his future collaborators George Pólya and Michael Fekete. His studies were interrupted in 1915 by World War I, in which he served in the infantry, artillery and air corps. In 1918 while stationed in Vienna, he was awarded a doctorate by the University of Vienna for his work on Toeplitz determinants. He received his Privat-Dozent from the University of Berlin in 1921, where he stayed until being appointed as successor to Knopp at the University of Königsberg in 1926. Intolerable working conditions during the Nazi regime resulted in a temporary position at the Washington University in St. Louis, Missouri in 1936, before his appointment as chairman of the mathematics department at Stanford University in 1938, where he helped build up the department until his retirement in 1966. He died in Palo Alto, California. His doctoral students include Paul Rosenbloom and Joseph Ullman. The Gábor Szegö Prize, Szegő Gábor Primary School, and Szegő Gábor Matematikaverseny (a mathematics competition in his former school) are all named in his honor. == Works == Szegő's most important work was in analysis. He was one of the foremost analysts of his generation and made fundamental contributions to the theory of Toeplitz matrices and orthogonal polynomials. He wrote over 130 papers in several languages. Each of his four books, several written in collaboration with others, has become a classic in its field. The monograph Orthogonal polynomials, published in 1939, contains much of his research and has had a profound influence in many areas of applied mathematics, including theoretical physics, stochastic processes and numerical analysis. == Tutoring von Neumann == At the age of 15, the young John von Neumann, recognised as a mathematical prodigy, was sent to study advanced calculus under Szegő. On their first meeting, Szegő was so astounded by von Neumann's mathematical talent and speed that, as recalled by his wife, he came back home with tears in his eyes. Szegő subsequently visited the von Neumann house twice a week to tutor the child prodigy. Some of von Neumann's instant solutions to the problems in calculus posed by Szegő, sketched out on his father's stationery, are now on display at the von Neumann archive at Budapest. == Honours == Amongst the many honours received during his lifetime were: Julius König Prize of the Hungarian Mathematical Society (1928) Member of the Königsberger Gelehrten Gesellschaft (1928) Corresponding member of the Austrian Academy of Sciences in Vienna (1960) Honorary member of the Hungarian Academy of Sciences (1965) == Bibliography == The collected Papers of Gábor Szegő, 3 Vols (ed. Richard Askey), Birkhäuser, 1982, ISBN 3-7643-3063-5 Pólya, George; Szegő, Gábor (1972) [1925], Problems and Theorems in Analysis, 2 Vols, Springer-Verlag Szegő, Gábor (1933), Asymptotische Entwicklungen der Jacobischen Polynome, Niemeyer Szegő, Gábor (1939), Orthogonal Polynomials, American Mathematical Society; 2nd edn. 1955 Pólya, George; Szegő, Gábor (1951), Isoperimetric problems in mathematical physics, Annals of Mathematics Studies, vol. 27, Princeton University Press, ISBN 0691079889 {{citation}}: ISBN / Date incompatibility (help) Szegő, Gábor; Grenander, Ulf (1958), Toeplitz forms and their applications, Chelsea == Selected articles == Szegő, G. (1920). "Beiträge zur Theorie der Toeplitzschen Formen". Math. Z. 6 (3–4): 167–202. doi:10.1007/bf01199955. S2CID 118147030. Szegő, G. (1921). "Beiträge zur Theorie der Toeplitzschen Formen, II". Math. Z. 9 (3–4): 167–190. doi:10.1007/bf01279027. S2CID 125157848. Szegő, G. (1935). "A problem concerning orthogonal polynomials". Trans. Amer. Math. Soc. 37: 196–206. doi:10.1090/s0002-9947-1935-1501782-2. MR 1501782. Szego, Gabriel (1936). "Correction". Trans. Amer. Math. Soc. 39 (3): 500. doi:10.2307/1989765. JSTOR 1989765. MR 1501861. Szegő, Gabriel (1936). "Inequalities for the zeros of Legendre polynomials and related functions". Trans. Amer. Math. Soc. 39: 1–17. doi:10.1090/s0002-9947-1936-1501831-2. MR 1501831. Szegő, Gabriel (1936). "On some Hermitian forms associated with two given curves of the complex plane". Trans. Amer. Math. Soc. 40 (3): 450–461. doi:10.1090/s0002-9947-1936-1501884-1. MR 1501884. Szegő, G. (1940). "On the gradient of solid harmonic polynomials". Trans. Amer. Math. Soc. 47: 51–65. doi:10.1090/s0002-9947-1940-0000847-6. MR 0000847. with A. C. Schaeffer: Schaeffer, A. C.; Szegő, G. (1941). "Inequalities for harmonic polynomials in two and three dimensions". Trans. Amer. Math. Soc. 50 (2): 187–225. doi:10.1090/s0002-9947-1941-0005164-7. MR 0005164. Szegő, G. (1942). "On the oscillations of differential transforms. I". Trans. Amer. Math. Soc. 52 (3): 450–462. doi:10.1090/s0002-9947-1942-0007170-6. MR 0007170. Szegő, G. (1943). "On the oscillations of differential transforms. IV. Jacobi polynomials". Trans. Amer. Math. Soc. 53 (3): 463–468. doi:10.1090/s0002-9947-1943-0008100-4. MR 0008100. with Max Schiffer: Schiffer, M.; Szegő, G. (1949). "Virtual mass and polarization". Trans. Amer. Math. Soc. 67: 130–205. doi:10.1090/s0002-9947-1949-0033922-9. MR 0033922. Szegő, G. (1950). "On certain special sets of orthogonal polynomials". Proc. Amer. Math. Soc. 1 (6): 731–737. doi:10.1090/s0002-9939-1950-0042546-2. MR 0042546. with Albert Edrei: Edrei, A.; Szegő, G. (1953). "A note on the reciprocal of a Fourier series". Proc. Amer. Math. Soc. 4 (2): 323–329. doi:10.1090/s0002-9939-1953-0053267-7. MR 0053267. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Gábor Szegő", MacTutor History of Mathematics Archive, University of St Andrews Askey, Richard (17 May 1995). "Gábor Szegő - One hundred years (Topic #25)". Op-Sf Net. Gábor Szegő: 1895-1985, by Richard Askey and Paul Nevai
Wikipedia:Gérard Vergnaud#0	Gérard Vergnaud (8 February 1933 – 6 June 2021) was a French mathematician, philosopher, educator, and psychologist. He earned his doctorate from the International Center for Genetic Epistemology in Geneva under the supervision of Jean Piaget. Vergnaud was a professor emeritus of the Centre national de la recherche scientifique in Paris, where he was a researcher in mathematics. Among his most significant work has been the development of the Theory of Conceptual Fields, which describes how children develop an understanding of mathematics. Gérard Vergnaud graduated from HEC Paris in 1956 and from the University of Geneva in 1968. == References == == External links == Official website Gérard Vergnaud – Association pour la Recherche en Didactique des Mathématiques Ciencia al Día – "Horror a las matemáticas" (in Spanish) (accessed 29 January 2011).
Wikipedia:Günter Harder#0	Günter Harder (born 14 March 1938 in Ratzeburg) is a German mathematician, specializing in arithmetic geometry and number theory. == Education == Harder studied mathematics and physics in Hamburg and Göttingen. Simultaneously with the Staatsexamen in 1964 in Hamburg, he received his doctoral degree (Dr. rer. nat.) under Ernst Witt with a thesis Über die Galoiskohomologie der Tori. Two years later he completed his habilitation. == Career == After a one-year postdoc position at Princeton University and a position as an assistant professor at the University of Heidelberg, he became a professor ordinarius at the University of Bonn. With the exception of a six-year stay at the former Universität-Gesamthochschule Wuppertal, Harder remained at the University of Bonn until his retirement in 2003. From 1995 to 2006 he was one of the directors of the Max-Planck-Institut für Mathematik in Bonn. He was a visiting professor at Harvard University, Yale University, at Princeton's Institute for Advanced Study (IAS) (for the academic years 1966–1967, 1972–1973, 1986–1987, autumn of 1983, autumn of 2006), at the Institut des Hautes Études Scientifiques (I.H.É.S.) in Paris, at the Tata Institute of Fundamental Research in Mumbai, and at the Mathematical Sciences Research Institute (MSRI) at the University of California, Berkeley. For decades, Harder was known to German mathematicians as the Spiritus Rector for a mathematical workshop held for one week in spring and one week in autumn; the workshop, sponsored by the Mathematical Research Institute of Oberwolfach, introduced young mathematicians and scientists to important new developments in pure mathematics and mathematical sciences. Harder's doctoral students include Kai Behrend, Jörg Bewersdorff, Joachim Schwermer, and Maria Heep-Altiner. == Research == His research deals with arithmetic geometry, automorphic forms, Shimura varieties, motives, and algebraic number theory. He made foundational contributions to the Waldspurger formula. With Ina Kersten, he is a co-editor of the collected works of Ernst Witt. == Awards and honors == Harder was an invited speaker at the International Congress of Mathematicians in 1970 and gave a talk titled Semisimple group schemes over curves and automorphic functions and in 1990 with a talk titled Eisenstein cohomology of arithmetic groups and its applications to number theory. In 1988 he was awarded the Leibniz Prize by the Deutsche Forschungsgemeinschaft. In 2004 Harder received, with Friedhelm Waldhausen, the von Staudt Prize. == Selected publications == Harder, G. (1971). "A Gauss-Bonnet formula for discrete arithmetically defined groups". Annales scientifiques de l'École normale supérieure. 4 (3). Societe Mathematique de France: 409–455. doi:10.24033/asens.1217. ISSN 0012-9593. (online). Harder, G. (1974). "Chevalley Groups Over Function Fields and Automorphic Forms". The Annals of Mathematics. 100 (2). JSTOR: 249–306. doi:10.2307/1971073. ISSN 0003-486X. JSTOR 1971073. Harder, G.; Narasimhan, M. S. (1975). "On the cohomology groups of moduli spaces of vector bundles on curves". Mathematische Annalen. 212 (3). Springer Science and Business Media LLC: 215–248. doi:10.1007/bf01357141. ISSN 0025-5831. S2CID 117851906. (online) "Algebraische Zyklen auf Hilbert-Blumenthal-Flächen". Journal für die reine und angewandte Mathematik (Crelle's Journal). 1986 (366). Walter de Gruyter GmbH: 53–120. 1 March 1986. doi:10.1515/crll.1986.366.53. ISSN 0075-4102. S2CID 119657447. (online). Harder, G. (1987). "Eisenstein cohomology of arithmetic groups. The case GL2". Inventiones Mathematicae. 89 (1). Springer Science and Business Media LLC: 37–118. Bibcode:1987InMat..89...37H. doi:10.1007/bf01404673. ISSN 0020-9910. S2CID 121239087. Goresky, M.; Harder, G.; MacPherson, R. (1994). "Weighted cohomology". Inventiones Mathematicae. 116 (1). Springer Science and Business Media LLC: 139–213. Bibcode:1994InMat.116..139G. doi:10.1007/bf01231560. ISSN 0020-9910. S2CID 189831832. Harder, Günter (1993). "Eisensteinkohomologie und die Konstruktion gemischter Motive". Lecture Notes in Mathematics. Vol. 1562. Berlin, Heidelberg: Springer Berlin Heidelberg. doi:10.1007/bfb0090305. ISBN 978-3-540-57408-8. ISSN 0075-8434. Harder, Günter (2011). "Lectures on Algebraic Geometry I". Aspects of Mathematics. Vol. 35. Wiesbaden: Springer Fachmedien Wiesbaden. doi:10.1007/978-3-8348-8330-8. ISBN 978-3-8348-1844-7. ISSN 0179-2156. Harder, Günter (2011). Lectures on Algebraic Geometry II. Wiesbaden: Vieweg+Teubner. doi:10.1007/978-3-8348-8159-5. ISBN 978-3-8348-0432-7. Bruinier, Jan (2008). The 1-2-3 of modular forms : lectures at a summer school in Nordfjordeid, Norway. Berlin: Springer. ISBN 978-3-540-74117-6. OCLC 233973403. (contains Harder's contribution: Harder, Günter (2008). "Congruence Between a Siegel and an Elliptic Modular Form". The 1-2-3 of Modular Forms. Berlin, Heidelberg: Springer Berlin Heidelberg. pp. 247–262. doi:10.1007/978-3-540-74119-0_4. ISBN 978-3-540-74117-6.) == References == == External links == Homepage of Günter Harder at the Hausdorff Center of the University of Bonn Homepage of Günter Harder at the University of Bonn
Wikipedia:Günter Heimbeck#0	Günter Heimbeck (born 23 June 1946 in Gunzenhausen, Germany) is a German–Namibian retired professor of mathematics. His particular research interest is geometry; the Heimbeck Planes are named for him. Heimbeck probably is the first and only Namibian scholar to have a scientific sub-discipline carry his name. Heimbeck studied mathematics at University of Würzburg from 1965. He completed his PhD in 1974, and his habilitation in 1981. He then became lecturer at his alma mater. In 1985 Heimbeck emigrated to South Africa, where he taught at the University of the Witwatersrand in Johannesburg. In 1987 he took up a professorship for mathematics at the University of Namibia. Heimbeck is advisor to the Namibian Ministry of Education. == References ==
Wikipedia:Günter Pilz#0	Günter Pilz (born 1945 in Bad Hall, Upper Austria) is professor of mathematics at the Johannes Kepler University (JKU) Linz. Until his retirement in 2013 he was the head of the Institute of Algebra. == Vita == After studying mathematics and physics at the University of Vienna (1963–1967) and his PhD (1967), Günter Pilz was assistant professor at several institutions: at the department of mathematics of the University of Vienna (1966–1968), at the department of statistics at the Vienna University of Technology (1968–1969), as research associate at the department of mathematics, University of Arizona, United States (1969–1970) and at the department of mathematics at the University of Linz (1970–1974). In 1971, he received his Habilitation. In 1974, he was promoted to a professor of mathematics at the JKU. He was head of the department of mathematics in Linz (1980–1983 und 1987–1993) and head of the Institute of Algebra (since 1996). From 1996 to 2000, Günter Pilz was dean of studies at the Faculty of Science and Technology and from 2000 to 2007 vice rector for research. Also, he was the chairman in the “Forum Research” of the Austrian Conference of Rectors from 2001 to 2005 and was chosen in 2003 to be the Austrian representative for Research Integrity in the corresponding OECD group. His main area of research is the theory and applications of algebraic structures. He is honorary professor at the Shandong University of Technology in China and Honorary Doctor from the Ural State University in Ekaterinburg, Russia. He was also a member in the Council for Research and Technology in Upper Austria (2003–2008). == Main areas of research == Theory and applications of near-rings: these are "collections of objects with which one can calculate almost as well as with numbers". These near-rings have numerous applications, for instance in the construction of optimal designs for statistical experiments. == Prizes == Honorary medal in silver, by the County of Upper Austria (2007/08) == External links == The Institute of Algebra Publications by Günter Pilz
Wikipedia:H square#0	In mathematics and control theory, H2, or H-square is a Hardy space with square norm. It is a subspace of L2 space, and is thus a Hilbert space. In particular, it is a reproducing kernel Hilbert space. == On the unit circle == In general, elements of L2 on the unit circle are given by ∑ n = − ∞ ∞ a n e i n φ {\displaystyle \sum _{n=-\infty }^{\infty }a_{n}e^{in\varphi }} whereas elements of H2 are given by ∑ n = 0 ∞ a n e i n φ . {\displaystyle \sum _{n=0}^{\infty }a_{n}e^{in\varphi }.} The projection from L2 to H2 (by setting an = 0 when n < 0) is orthogonal. == On the half-plane == The Laplace transform L {\displaystyle {\mathcal {L}}} given by [ L f ] ( s ) = ∫ 0 ∞ e − s t f ( t ) d t {\displaystyle [{\mathcal {L}}f](s)=\int _{0}^{\infty }e^{-st}f(t)dt} can be understood as a linear operator L : L 2 ( 0 , ∞ ) → H 2 ( C + ) {\displaystyle {\mathcal {L}}:L^{2}(0,\infty )\to H^{2}\left(\mathbb {C} ^{+}\right)} where L 2 ( 0 , ∞ ) {\displaystyle L^{2}(0,\infty )} is the set of square-integrable functions on the positive real number line, and C + {\displaystyle \mathbb {C} ^{+}} is the right half of the complex plane. It is more; it is an isomorphism, in that it is invertible, and it isometric, in that it satisfies ‖ L f ‖ H 2 = 2 π ‖ f ‖ L 2 . {\displaystyle \\|{\mathcal {L}}f\\|_{H^{2}}={\sqrt {2\pi }}\\|f\\|_{L^{2}}.} The Laplace transform is "half" of a Fourier transform; from the decomposition L 2 ( R ) = L 2 ( − ∞ , 0 ) ⊕ L 2 ( 0 , ∞ ) {\displaystyle L^{2}(\mathbb {R} )=L^{2}(-\infty ,0)\oplus L^{2}(0,\infty )} one then obtains an orthogonal decomposition of L 2 ( R ) {\displaystyle L^{2}(\mathbb {R} )} into two Hardy spaces L 2 ( R ) = H 2 ( C − ) ⊕ H 2 ( C + ) . {\displaystyle L^{2}(\mathbb {R} )=H^{2}\left(\mathbb {C} ^{-}\right)\oplus H^{2}\left(\mathbb {C} ^{+}\right).} This is essentially the Paley-Wiener theorem. == See also == H∞ == References == Jonathan R. Partington, "Linear Operators and Linear Systems, An Analytical Approach to Control Theory", London Mathematical Society Student Texts 60, (2004) Cambridge University Press, ISBN 0-521-54619-2.
Wikipedia:H tree#0	In fractal geometry, the H tree is a fractal tree structure constructed from perpendicular line segments, each smaller by a factor of the square root of 2 from the next larger adjacent segment. It is so called because its repeating pattern resembles the letter "H". It has Hausdorff dimension 2, and comes arbitrarily close to every point in a rectangle. Its applications include VLSI design and microwave engineering. == Construction == An H tree can be constructed by starting with a line segment of arbitrary length, drawing two shorter segments at right angles to the first through its endpoints, and continuing in the same vein, reducing (dividing) the length of the line segments drawn at each stage by 2 {\displaystyle {\sqrt {2}}} . A variant of this construction could also be defined in which the length at each iteration is multiplied by a ratio less than 1 / 2 {\displaystyle 1/{\sqrt {2}}} , but for this variant the resulting shape covers only part of its bounding rectangle, with a fractal boundary. An alternative process that generates the same fractal set is to begin with a rectangle with sides in the ratio 1 : 2 {\displaystyle 1:{\sqrt {2}}} , and repeatedly bisect it into two smaller silver rectangles, at each stage connecting the two centroids of the two smaller rectangles by a line segment. A similar process can be performed with rectangles of any other shape, but the 1 : 2 {\displaystyle 1:{\sqrt {2}}} rectangle leads to the line segment size decreasing uniformly by a 2 {\displaystyle {\sqrt {2}}} factor at each step while for other rectangles the length will decrease by different factors at odd and even levels of the recursive construction. == Properties == The H tree is a self-similar fractal; its Hausdorff dimension is equal to 2. The points of the H tree come arbitrarily close to every point in a rectangle (the same as the starting rectangle in the constructing by centroids of subdivided rectangles). However, it does not include all points of the rectangle; for instance, the points on the perpendicular bisector of the initial line segment (other than the midpoint of this segment) are not included. == Applications == In VLSI design, the H tree may be used as the layout for a complete binary tree using a total area that is proportional to the number of nodes of the tree. Additionally, the H tree forms a space efficient layout for trees in graph drawing, and as part of a construction of a point set for which the sum of squared edge lengths of the traveling salesman tour is large. It is commonly used as a clock distribution network for routing timing signals to all parts of a chip with equal propagation delays to each part, and has also been used as an interconnection network for VLSI multiprocessors. The planar H tree can be generalized to the three-dimensional structure via adding line segments on the direction perpendicular to the H tree plane. The resultant three-dimensional H tree has Hausdorff dimension equal to 3. The planar H tree and its three-dimensional version have been found to constitute artificial electromagnetic atoms in photonic crystals and metamaterials and might have potential applications in microwave engineering. == Related sets == The H tree is an example of a fractal canopy, in which the angle between neighboring line segments is always 180 degrees. In its property of coming arbitrarily close to every point of its bounding rectangle, it also resembles a space-filling curve, although it is not itself a curve. Topologically, an H tree has properties similar to those of a dendroid. However, they are not dendroids: dendroids must be closed sets, and H trees are not closed (their closure is the whole rectangle). Variations of the same tree structure with thickened polygonal branches in place of the line segments of the H tree have been defined by Benoit Mandelbrot, and are sometimes called the Mandelbrot tree. In these variations, to avoid overlaps between the leaves of the tree and their thickened branches, the scale factor by which the size is reduced at each level must be slightly greater than 2 {\displaystyle {\sqrt {2}}} . == Notes == == References ==
Wikipedia:Hadamard product (entire functions)#0	In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic counterparts sinh and cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function f ( z ) {\displaystyle f(z)} has a root at w {\displaystyle w} , then f ( z ) / ( z − w ) {\displaystyle f(z)/(z-w)} , taking the limit value at w {\displaystyle w} , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be continued analytically to an entire function. A transcendental entire function is an entire function that is not a polynomial. Just as meromorphic functions can be viewed as a generalization of rational fractions, entire functions can be viewed as a generalization of polynomials. In particular, if for meromorphic functions one can generalize the factorization into simple fractions (the Mittag-Leffler theorem on the decomposition of a meromorphic function), then for entire functions there is a generalization of the factorization — the Weierstrass theorem on entire functions. == Properties == Every entire function f ( z ) {\displaystyle f(z)} can be represented as a single power series: f ( z ) = ∑ n = 0 ∞ a n z n {\displaystyle \ f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}\ } that converges everywhere in the complex plane, hence uniformly on compact sets. The radius of convergence is infinite, which implies that lim n → ∞ \| a n \| 1 n = 0 {\displaystyle \ \lim _{n\to \infty }\|a_{n}\|^{\frac {1}{n}}=0\ } or, equivalently, lim n → ∞ ln ⁡ \| a n \| n = − ∞ . {\displaystyle \ \lim _{n\to \infty }{\frac {\ln \|a_{n}\|}{n}}=-\infty ~.} Any power series satisfying this criterion will represent an entire function. If (and only if) the coefficients of the power series are all real then the function evidently takes real values for real arguments, and the value of the function at the complex conjugate of z {\displaystyle z} will be the complex conjugate of the value at z . {\displaystyle z~.} Such functions are sometimes called self-conjugate (the conjugate function, F ∗ ( z ) , {\displaystyle F^{*}(z),} being given by F ¯ ( z ¯ ) {\displaystyle {\bar {F}}({\bar {z}})} ). If the real part of an entire function is known in a neighborhood of a point then both the real and imaginary parts are known for the whole complex plane, up to an imaginary constant. For instance, if the real part is known in a neighborhood of zero, then we can find the coefficients for n > 0 {\displaystyle n>0} from the following derivatives with respect to a real variable r {\displaystyle r} : R e ⁡ { a n } = 1 n ! d n d r n R e ⁡ { f ( r ) } a t r = 0 I m ⁡ { a n } = 1 n ! d n d r n R e ⁡ { f ( r e − i π 2 n ) } a t r = 0 {\displaystyle {\begin{aligned}\operatorname {\mathcal {Re}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f(r)\ \right\}&&\quad \mathrm {at} \quad r=0\\\operatorname {\mathcal {Im}} \left\{\ a_{n}\ \right\}&={\frac {1}{n!}}{\frac {d^{n}}{dr^{n}}}\ \operatorname {\mathcal {Re}} \left\{\ f\left(r\ e^{-{\frac {i\pi }{2n}}}\right)\ \right\}&&\quad \mathrm {at} \quad r=0\end{aligned}}} (Likewise, if the imaginary part is known in a neighborhood then the function is determined up to a real constant.) In fact, if the real part is known just on an arc of a circle, then the function is determined up to an imaginary constant.} Note however that an entire function is not determined by its real part on all curves. In particular, if the real part is given on any curve in the complex plane where the real part of some other entire function is zero, then any multiple of that function can be added to the function we are trying to determine. For example, if the curve where the real part is known is the real line, then we can add i {\displaystyle i} times any self-conjugate function. If the curve forms a loop, then it is determined by the real part of the function on the loop since the only functions whose real part is zero on the curve are those that are everywhere equal to some imaginary number. The Weierstrass factorization theorem asserts that any entire function can be represented by a product involving its zeroes (or "roots"). The entire functions on the complex plane form an integral domain (in fact a Prüfer domain). They also form a commutative unital associative algebra over the complex numbers. Liouville's theorem states that any bounded entire function must be constant. As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function. Specifically, by the Casorati–Weierstrass theorem, for any transcendental entire function f {\displaystyle f} and any complex w {\displaystyle w} there is a sequence ( z m ) m ∈ N {\displaystyle (z_{m})_{m\in \mathbb {N} }} such that lim m → ∞ \| z m \| = ∞ , and lim m → ∞ f ( z m ) = w . {\displaystyle \ \lim _{m\to \infty }\|z_{m}\|=\infty ,\qquad {\text{and}}\qquad \lim _{m\to \infty }f(z_{m})=w~.} Picard's little theorem is a much stronger result: Any non-constant entire function takes on every complex number as value, possibly with a single exception. When an exception exists, it is called a lacunary value of the function. The possibility of a lacunary value is illustrated by the exponential function, which never takes on the value 0 {\displaystyle 0} . One can take a suitable branch of the logarithm of an entire function that never hits 0 {\displaystyle 0} , so that this will also be an entire function (according to the Weierstrass factorization theorem). The logarithm hits every complex number except possibly one number, which implies that the first function will hit any value other than 0 {\displaystyle 0} an infinite number of times. Similarly, a non-constant, entire function that does not hit a particular value will hit every other value an infinite number of times. Liouville's theorem is a special case of the following statement: == Growth == Entire functions may grow as fast as any increasing function: for any increasing function g : [ 0 , ∞ ) → [ 0 , ∞ ) {\displaystyle g:[0,\infty )\to [0,\infty )} there exists an entire function f {\displaystyle f} such that f ( x ) > g ( \| x \| ) {\displaystyle f(x)>g(\|x\|)} for all real x {\displaystyle x} . Such a function f {\displaystyle f} may be easily found of the form: f ( z ) = c + ∑ k = 1 ∞ ( z k ) n k {\displaystyle f(z)=c+\sum _{k=1}^{\infty }\left({\frac {z}{k}}\right)^{n_{k}}} for a constant c {\displaystyle c} and a strictly increasing sequence of positive integers n k {\displaystyle n_{k}} . Any such sequence defines an entire function f ( z ) {\displaystyle f(z)} , and if the powers are chosen appropriately we may satisfy the inequality f ( x ) > g ( \| x \| ) {\displaystyle f(x)>g(\|x\|)} for all real x {\displaystyle x} . (For instance, it certainly holds if one chooses c := g ( 2 ) {\displaystyle c:=g(2)} and, for any integer k ≥ 1 {\displaystyle k\geq 1} one chooses an even exponent n k {\displaystyle n_{k}} such that ( k + 1 k ) n k ≥ g ( k + 2 ) {\displaystyle \left({\frac {k+1}{k}}\right)^{n_{k}}\geq g(k+2)} ). == Order and type == The order (at infinity) of an entire function f ( z ) {\displaystyle f(z)} is defined using the limit superior as: ρ = lim sup r → ∞ ln ⁡ ( ln ⁡ ‖ f ‖ ∞ , B r ) ln ⁡ r , {\displaystyle \rho =\limsup _{r\to \infty }{\frac {\ln \left(\ln \\|f\\|_{\infty ,B_{r}}\right)}{\ln r}},} where B r {\displaystyle B_{r}} is the disk of radius r {\displaystyle r} and ‖ f ‖ ∞ , B r {\displaystyle \\|f\\|_{\infty ,B_{r}}} denotes the supremum norm of f ( z ) {\displaystyle f(z)} on B r {\displaystyle B_{r}} . The order is a non-negative real number or infinity (except when f ( z ) = 0 {\displaystyle f(z)=0} for all z {\displaystyle z} ). In other words, the order of f ( z ) {\displaystyle f(z)} is the infimum of all m {\displaystyle m} such that: f ( z ) = O ( exp ⁡ ( \| z \| m ) ) , as z → ∞ . {\displaystyle f(z)=O\left(\exp \left(\|z\|^{m}\right)\right),\quad {\text{as }}z\to \infty .} The example of f ( z ) = exp ⁡ ( 2 z 2 ) {\displaystyle f(z)=\exp(2z^{2})} shows that this does not mean f ( z ) = O ( exp ⁡ ( \| z \| m ) ) {\displaystyle f(z)=O(\exp(\|z\|^{m}))} if f ( z ) {\displaystyle f(z)} is of order m {\displaystyle m} . If 0 < ρ < ∞ , {\displaystyle 0<\rho <\infty ,} one can also define the type: σ = lim sup r → ∞ ln ⁡ ‖ f ‖ ∞ , B r r ρ . {\displaystyle \sigma =\limsup _{r\to \infty }{\frac {\ln \\|f\\|_{\infty ,B_{r}}}{r^{\rho }}}.} If the order is 1 and the type is σ {\displaystyle \sigma } , the function is said to be "of exponential type σ {\displaystyle \sigma } ". If it is of order less than 1 it is said to be of exponential type 0. If f ( z ) = ∑ n = 0 ∞ a n z n , {\displaystyle f(z)=\sum _{n=0}^{\infty }a_{n}z^{n},} then the order and type can be found by the formulas ρ = lim sup n → ∞ n ln ⁡ n − ln ⁡ \| a n \| ( e ρ σ ) 1 ρ = lim sup n → ∞ n 1 ρ \| a n \| 1 n {\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{-\ln \|a_{n}\|}}\\[6pt](e\rho \sigma )^{\frac {1}{\rho }}&=\limsup _{n\to \infty }n^{\frac {1}{\rho }}\|a_{n}\|^{\frac {1}{n}}\end{aligned}}} Let f ( n ) {\displaystyle f^{(n)}} denote the n {\displaystyle n} -th derivative of f {\displaystyle f} . Then we may restate these formulas in terms of the derivatives at any arbitrary point z 0 {\displaystyle z_{0}} : ρ = lim sup n → ∞ n ln ⁡ n n ln ⁡ n − ln ⁡ \| f ( n ) ( z 0 ) \| = ( 1 − lim sup n → ∞ ln ⁡ \| f ( n ) ( z 0 ) \| n ln ⁡ n ) − 1 ( ρ σ ) 1 ρ = e 1 − 1 ρ lim sup n → ∞ \| f ( n ) ( z 0 ) \| 1 n n 1 − 1 ρ {\displaystyle {\begin{aligned}\rho &=\limsup _{n\to \infty }{\frac {n\ln n}{n\ln n-\ln \|f^{(n)}(z_{0})\|}}=\left(1-\limsup _{n\to \infty }{\frac {\ln \|f^{(n)}(z_{0})\|}{n\ln n}}\right)^{-1}\\[6pt](\rho \sigma )^{\frac {1}{\rho }}&=e^{1-{\frac {1}{\rho }}}\limsup _{n\to \infty }{\frac {\|f^{(n)}(z_{0})\|^{\frac {1}{n}}}{n^{1-{\frac {1}{\rho }}}}}\end{aligned}}} The type may be infinite, as in the case of the reciprocal gamma function, or zero (see example below under § Order 1). Another way to find out the order and type is Matsaev's theorem. === Examples === Here are some examples of functions of various orders: ==== Order ρ ==== For arbitrary positive numbers ρ {\displaystyle \rho } and σ {\displaystyle \sigma } one can construct an example of an entire function of order ρ {\displaystyle \rho } and type σ {\displaystyle \sigma } using: f ( z ) = ∑ n = 1 ∞ ( e ρ σ n ) n ρ z n {\displaystyle f(z)=\sum _{n=1}^{\infty }\left({\frac {e\rho \sigma }{n}}\right)^{\frac {n}{\rho }}z^{n}} ==== Order 0 ==== Non-zero polynomials ∑ n = 0 ∞ 2 − n 2 z n {\displaystyle \sum _{n=0}^{\infty }2^{-n^{2}}z^{n}} ==== Order 1/4 ==== f ( z 4 ) {\displaystyle f({\sqrt[{4}]{z}})} where f ( u ) = cos ⁡ ( u ) + cosh ⁡ ( u ) {\displaystyle f(u)=\cos(u)+\cosh(u)} ==== Order 1/3 ==== f ( z 3 ) {\displaystyle f({\sqrt[{3}]{z}})} where f ( u ) = e u + e ω u + e ω 2 u = e u + 2 e − u 2 cos ⁡ ( 3 u 2 ) , with ω a complex cube root of 1 . {\displaystyle f(u)=e^{u}+e^{\omega u}+e^{\omega ^{2}u}=e^{u}+2e^{-{\frac {u}{2}}}\cos \left({\frac {{\sqrt {3}}u}{2}}\right),\quad {\text{with }}\omega {\text{ a complex cube root of 1}}.} ==== Order 1/2 ==== cos ⁡ ( a z ) {\displaystyle \cos \left(a{\sqrt {z}}\right)} with a ≠ 0 {\displaystyle a\neq 0} (for which the type is given by σ = \| a \| {\displaystyle \sigma =\|a\|} ) ==== Order 1 ==== exp ⁡ ( a z ) {\displaystyle \exp(az)} with a ≠ 0 {\displaystyle a\neq 0} ( σ = \| a \| {\displaystyle \sigma =\|a\|} ) sin ⁡ ( z ) {\displaystyle \sin(z)} cosh ⁡ ( z ) {\displaystyle \cosh(z)} the Bessel functions J n ( z ) {\displaystyle J_{n}(z)} and spherical Bessel functions j n ( z ) {\displaystyle j_{n}(z)} for integer values of n {\displaystyle n} the reciprocal gamma function 1 / Γ ( z ) {\displaystyle 1/\Gamma (z)} ( σ {\displaystyle \sigma } is infinite) ∑ n = 2 ∞ z n ( n ln ⁡ n ) n . ( σ = 0 ) {\displaystyle \sum _{n=2}^{\infty }{\frac {z^{n}}{(n\ln n)^{n}}}.\quad (\sigma =0)} ==== Order 3/2 ==== Airy function A i ( z ) {\displaystyle Ai(z)} ==== Order 2 ==== exp ⁡ ( a z 2 ) {\displaystyle \exp(az^{2})} with a ≠ 0 {\displaystyle a\neq 0} ( σ = \| a \| {\displaystyle \sigma =\|a\|} ) The Barnes G-function ( σ {\displaystyle \sigma } is infinite). ==== Order infinity ==== exp ⁡ ( exp ⁡ ( z ) ) {\displaystyle \exp(\exp(z))} == Genus == Entire functions of finite order have Hadamard's canonical representation (Hadamard factorization theorem): f ( z ) = z m e P ( z ) ∏ n = 1 ∞ ( 1 − z z n ) exp ⁡ ( z z n + ⋯ + 1 p ( z z n ) p ) , {\displaystyle f(z)=z^{m}e^{P(z)}\prod _{n=1}^{\infty }\left(1-{\frac {z}{z_{n}}}\right)\exp \left({\frac {z}{z_{n}}}+\cdots +{\frac {1}{p}}\left({\frac {z}{z_{n}}}\right)^{p}\right),} where z k {\displaystyle z_{k}} are those roots of f {\displaystyle f} that are not zero ( z k ≠ 0 {\displaystyle z_{k}\neq 0} ), m {\displaystyle m} is the order of the zero of f {\displaystyle f} at z = 0 {\displaystyle z=0} (the case m = 0 {\displaystyle m=0} being taken to mean f ( 0 ) ≠ 0 {\displaystyle f(0)\neq 0} ), P {\displaystyle P} a polynomial (whose degree we shall call q {\displaystyle q} ), and p {\displaystyle p} is the smallest non-negative integer such that the series ∑ n = 1 ∞ 1 \| z n \| p + 1 {\displaystyle \sum _{n=1}^{\infty }{\frac {1}{\|z_{n}\|^{p+1}}}} converges. The non-negative integer g = max { p , q } {\displaystyle g=\max\{p,q\}} is called the genus of the entire function f {\displaystyle f} . If the order ρ {\displaystyle \rho } is not an integer, then g = [ ρ ] {\displaystyle g=[\rho ]} is the integer part of ρ {\displaystyle \rho } . If the order is a positive integer, then there are two possibilities: g = ρ − 1 {\displaystyle g=\rho -1} or g = ρ {\displaystyle g=\rho } . For example, sin {\displaystyle \sin } , cos {\displaystyle \cos } and exp {\displaystyle \exp } are entire functions of genus g = ρ = 1 {\displaystyle g=\rho =1} . == Other examples == According to J. E. Littlewood, the Weierstrass sigma function is a 'typical' entire function. This statement can be made precise in the theory of random entire functions: the asymptotic behavior of almost all entire functions is similar to that of the sigma function. Other examples include the Fresnel integrals, the Jacobi theta function, and the reciprocal Gamma function. The exponential function and the error function are special cases of the Mittag-Leffler function. According to the fundamental theorem of Paley and Wiener, Fourier transforms of functions (or distributions) with bounded support are entire functions of order 1 {\displaystyle 1} and finite type. Other examples are solutions of linear differential equations with polynomial coefficients. If the coefficient at the highest derivative is constant, then all solutions of such equations are entire functions. For example, the exponential function, sine, cosine, Airy functions and Parabolic cylinder functions arise in this way. The class of entire functions is closed with respect to compositions. This makes it possible to study dynamics of entire functions. An entire function of the square root of a complex number is entire if the original function is even, for example cos ⁡ ( z ) {\displaystyle \cos({\sqrt {z}})} . If a sequence of polynomials all of whose roots are real converges in a neighborhood of the origin to a limit which is not identically equal to zero, then this limit is an entire function. Such entire functions form the Laguerre–Pólya class, which can also be characterized in terms of the Hadamard product, namely, f {\displaystyle f} belongs to this class if and only if in the Hadamard representation all z n {\displaystyle z_{n}} are real, ρ ≤ 1 {\displaystyle \rho \leq 1} , and P ( z ) = a + b z + c z 2 {\displaystyle P(z)=a+bz+cz^{2}} , where b {\displaystyle b} and c {\displaystyle c} are real, and c ≤ 0 {\displaystyle c\leq 0} . For example, the sequence of polynomials ( 1 − ( z − d ) 2 n ) n {\displaystyle \left(1-{\frac {(z-d)^{2}}{n}}\right)^{n}} converges, as n {\displaystyle n} increases, to exp ⁡ ( − ( z − d ) 2 ) {\displaystyle \exp(-(z-d)^{2})} . The polynomials 1 2 ( ( 1 + i z n ) n + ( 1 − i z n ) n ) {\displaystyle {\frac {1}{2}}\left(\left(1+{\frac {iz}{n}}\right)^{n}+\left(1-{\frac {iz}{n}}\right)^{n}\right)} have all real roots, and converge to cos ⁡ ( z ) {\displaystyle \cos(z)} . The polynomials ∏ m = 1 n ( 1 − z 2 ( ( m − 1 2 ) π ) 2 ) {\displaystyle \prod _{m=1}^{n}\left(1-{\frac {z^{2}}{\left(\left(m-{\frac {1}{2}}\right)\pi \right)^{2}}}\right)} also converge to cos ⁡ ( z ) {\displaystyle \cos(z)} , showing the buildup of the Hadamard product for cosine. == See also == Jensen's formula Carlson's theorem Exponential type Paley–Wiener theorem Wiman-Valiron theory == Notes == == References == == Sources ==
Wikipedia:Hadamard's lemma#0	In mathematics, Hadamard's lemma, named after Jacques Hadamard, is essentially a first-order form of Taylor's theorem, in which we can express a smooth, real-valued function exactly in a convenient manner. == Statement == === Proof === == Consequences and applications == == See also == Bump function – Smooth and compactly supported function Continuously differentiable – Mathematical function whose derivative existsPages displaying short descriptions of redirect targets Smoothness – Number of derivatives of a function (mathematics) Taylor's theorem – Approximation of a function by a truncated power series == Citations == == References == Nestruev, Jet (2002). Smooth manifolds and observables. Berlin: Springer. ISBN 0-387-95543-7. Nestruev, Jet (10 September 2020). Smooth Manifolds and Observables. Graduate Texts in Mathematics. Vol. 220. Cham, Switzerland: Springer Nature. ISBN 978-3-030-45649-8. OCLC 1195920718.
Wikipedia:Hafez Bashar al-Assad#0	Hafez Bashar al-Assad (Arabic: حافظ بشار الأسد; born 5 December 2001) is the eldest son of former Syrian president Bashar al-Assad and his wife Asma al-Assad. He was regarded as a potential successor to his father before the fall of the Assad regime on 8 December 2024. == Early and personal life == Assad was born in Damascus on 5 December 2001. He was named after his grandfather, former President of Syria Hafez al-Assad. In his youth, he attended a Montessori school alongside his sister and brother. They also attended a language school in the Baramkeh district of Damascus. Assad is fluent in Arabic and English. He also speaks Russian and studies Chinese. In 2020, Assad was placed under sanctions by the U.S. Department of State in connection with previous sanctions targeting his father's regime. As a result, he is not allowed to travel to or own assets in the United States. In December 2024, his family fled Syria following the fall of the Assad regime and joined him in Moscow, Russia. After the collapse of the Assad regime, he posted on Twitter about a series of events detailing his family's final moments before escaping to Moscow. He said, "There was no plan, not even a backup, to leave Damascus, let alone Syria." His Twitter account was suspended after he described the family's last days in the country before the rebel takeover. After the suspension, he appeared in a short video on 12 February 2025, to address concerns about whether the detailed post on social media was from a fake account or actually his. He clarified that both the Twitter account and Telegram were his, and that he didn't have any other accounts or use any other social media platforms. == Mathematics competitions == Assad has described mathematics to the media as his "childhood passion." He participated in the International Mathematical Olympiad, representing Syria, from 2016 to 2018. 2016 International Mathematical Olympiad (Hong Kong) – He placed 355th out of 602 overall and 4th out of 6 on the Syrian team. 2017 International Mathematical Olympiad (Rio de Janeiro) – He placed 528th out of 615 overall and 6th out of 6 on the Syrian team. 2018 International Mathematical Olympiad (Cluj-Napoca) – He placed 486th out of 594 overall and 6th out of 6 on the Syrian team. Assad's participation in the International Mathematical Olympiad has been criticized by some outside observers, viewing his inclusion on the Syrian team as the result of nepotism. However, this was denied by a representative of the Syrian government, claiming that Assad qualified in fair competition for a space on the team. == Public life == Assad received media attention in 2013 when a Facebook account attributed to him posted a criticism of the U.S. military in the wake of the Ghouta chemical attack. However, both Syrian opposition figures and outside media sources questioned whether the since-deleted account was authentic. During the 2017 IMO, Assad was interviewed by Brazilian newspaper O Globo. When asked if his father is a dictator, he defended him saying: "I know what kind of man my father is. People say a lot of things, many are blind. But that's not reality". He also took his father's side in conflict, saying: "It's not a civil war, it's people taking over our home. It's a war against the people. The population and the government are united against the invaders who are taking over the country." == Education == In 2015, Assad received his primary education certificate from Naeem Maasarani School in Damascus with a final grade of 2,983 out of 3,100. In 2018, Assad received his secondary education certificate from Sawa and Montessori School in Damascus with a final grade of 2,672 out of 2,900. In 2016, it was announced that Assad would complete his higher education in Russia. Before he matriculated to Moscow State University, he studied for a period of time at the Higher Institute for Applied Sciences and Technology in Damascus. In 2023, Assad graduated with a master's degree in mathematics from the Faculty of Mechanics and Mathematics of Moscow State University with honors. He graduated on an accelerated schedule, completing his final two years in a single year. He wrote his thesis on the topic of number theory. In November 2024, he successfully defended his doctoral dissertation on number theory, titled "Arithmetic Questions of Polynomials in Algebraic Number Fields", at Moscow State University. Assad dedicated his thesis "to the martyrs of the Syrian Arab Army, without whose selfless sacrifices none of us would exist". == References ==
Wikipedia:Hafner–Sarnak–McCurley constant#0	The Hafner–Sarnak–McCurley constant is a mathematical constant representing the probability that the determinants of two randomly chosen square integer matrices will be relatively prime. The probability depends on the matrix size, n, in accordance with the formula D ( n ) = ∏ k = 1 ∞ { 1 − [ 1 − ∏ j = 1 n ( 1 − p k − j ) ] 2 } , {\displaystyle D(n)=\prod _{k=1}^{\infty }\left\{1-\left[1-\prod _{j=1}^{n}(1-p_{k}^{-j})\right]^{2}\right\},} where pk is the kth prime number. The constant is the limit of this expression as n approaches infinity. Its value is roughly 0.3532363719... (sequence A085849 in the OEIS). == References == Finch, S. R. (2003), "§2.5 Hafner–Sarnak–McCurley Constant", Mathematical Constants, Cambridge, England: Cambridge University Press, pp. 110–112, ISBN 0-521-81805-2. Flajolet, P. & Vardi, I. (1996), "Zeta Function Expansions of Classical Constants", Unpublished manuscript. Hafner, J. L.; Sarnak, P. & McCurley, K. (1993), "Relatively Prime Values of Polynomials", in Knopp, M. & Seingorn, M. (eds.), A Tribute to Emil Grosswald: Number Theory and Related Analysis, Providence, RI: Amer. Math. Soc., ISBN 0-8218-5155-1. Vardi, I. (1991), Computational Recreations in Mathematica, Redwood City, CA: Addison–Wesley, ISBN 0-201-52989-0. == External links == Weisstein, Eric W., "Hafner-Sarnak-McCurley Constant", MathWorld
Wikipedia:Hafnian#0	In mathematics, the hafnian is a scalar function of a symmetric matrix that generalizes the permanent. The hafnian was named by Eduardo R. Caianiello "to mark the fruitful period of stay in Copenhagen (Hafnia in Latin)." == Definition == The hafnian of a 2 n × 2 n {\displaystyle 2n\times 2n} symmetric matrix A {\displaystyle A} is defined as haf ⁡ ( A ) = ∑ ρ ∈ P 2 n 2 ∏ { i , j } ∈ ρ A i , j , {\displaystyle \operatorname {haf} (A)=\sum _{\rho \in P_{2n}^{2}}\prod _{\{i,j\}\in \rho }A_{i,j},} where P 2 n 2 {\displaystyle P_{2n}^{2}} is the set of all partitions of the set { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2 {\displaystyle 2} . This definition is similar to that of the Pfaffian, but differs in that the signatures of the permutations are not taken into account. Thus the relationship of the hafnian to the Pfaffian is the same as relationship of the permanent to the determinant. == Basic properties == The hafnian may also be defined as haf ⁡ ( A ) = 1 n ! 2 n ∑ σ ∈ S 2 n ∏ i = 1 n A σ ( 2 i − 1 ) , σ ( 2 i ) , {\displaystyle \operatorname {haf} (A)={\frac {1}{n!2^{n}}}\sum _{\sigma \in S_{2n}}\prod _{i=1}^{n}A_{\sigma (2i-1),\sigma (2i)},} where S 2 n {\displaystyle S_{2n}} is the symmetric group on { 1 , 2 , . . . , 2 n } {\displaystyle \{1,2,...,2n\}} . The two definitions are equivalent because if σ ∈ S 2 n {\displaystyle \sigma \in S_{2n}} , then { { σ ( 2 i − 1 ) , σ ( 2 i ) } : i ∈ { 1 , . . . , n } } {\displaystyle \{\{\sigma (2i-1),\sigma (2i)\}:i\in \{1,...,n\}\}} is a partition of { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2, and as σ {\displaystyle \sigma } ranges over S 2 n {\displaystyle S_{2n}} , each such partition is counted exactly n ! 2 n {\displaystyle n!2^{n}} times. Note that this argument relies on the symmetry of A {\displaystyle A} , without which the original definition is not well-defined. The hafnian of an adjacency matrix of a graph is the number of perfect matchings (also known as 1-factors) in the graph. This is because a partition of { 1 , 2 , … , 2 n } {\displaystyle \{1,2,\dots ,2n\}} into subsets of size 2 can also be thought of as a perfect matching in the complete graph K 2 n {\displaystyle K_{2n}} . The hafnian can also be thought of as a generalization of the permanent, since the permanent can be expressed as per ⁡ ( C ) = haf ⁡ ( 0 C C T 0 ) {\displaystyle \operatorname {per} (C)=\operatorname {haf} {\begin{pmatrix}0&C\\C^{\mathsf {T}}&0\end{pmatrix}}} . Just as the hafnian counts the number of perfect matchings in a graph given its adjacency matrix, the permanent counts the number of matchings in a bipartite graph given its biadjacency matrix. The hafnian is also related to moments of multivariate Gaussian distributions. By Wick's probability theorem, the hafnian of a real 2 n × 2 n {\displaystyle 2n\times 2n} symmetric matrix may expressed as haf ⁡ ( A ) = E ( X 1 , … , X 2 n ) ∼ N ( 0 , A + λ I ) [ X 1 … X 2 n ] , {\displaystyle \operatorname {haf} (A)=\mathbb {E} _{\left(X_{1},\dots ,X_{2n}\right)\sim {\mathcal {N}}\left(0,A+\lambda I\right)}\left[X_{1}\dots X_{2n}\right],} where λ {\displaystyle \lambda } is any number large enough to make A + λ I {\displaystyle A+\lambda I} positive semi-definite. Note that the hafnian does not depend on the diagonal entries of the matrix, and the expectation on the right-hand side does not depend on λ {\displaystyle \lambda } . == Generating function == Let S = ( A C C T B ) = S T {\displaystyle S={\begin{pmatrix}A&C\\C^{\mathsf {T}}&B\end{pmatrix}}=S^{\mathsf {T}}} be an arbitrary complex symmetric 2 m × 2 m {\displaystyle 2m\times 2m} matrix composed of four m × m {\displaystyle m\times m} blocks A = A T {\displaystyle A=A^{\mathsf {T}}} , B = B T {\displaystyle B=B^{\mathsf {T}}} , C {\displaystyle C} and C T {\displaystyle C^{\mathsf {T}}} . Let z 1 , … , z m {\displaystyle z_{1},\ldots ,z_{m}} be a set of m {\displaystyle m} independent variables, and let Z = ( 0 diag ( z 1 , z 2 , … , z m ) diag ( z 1 , z 2 , … , z m ) 0 ) {\displaystyle Z={\begin{pmatrix}0&{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})\\{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})&0\end{pmatrix}}} be an antidiagonal block matrix composed of entries z j {\displaystyle z_{j}} (each one is presented twice, one time per nonzero block). Let I {\displaystyle I} denote the identity matrix. Then the following identity holds: 1 det ( I − Z S ) = ∑ { n k } haf ⁡ S ~ ( { n k } ) ∏ k = 1 m z k n k n k ! {\displaystyle {\frac {1}{\sqrt {\det {\big (}I-ZS{\big )}}}}=\sum _{\{n_{k}\}}\operatorname {haf} {\tilde {S}}(\{n_{k}\})\prod _{k=1}^{m}{\frac {z_{k}^{n_{k}}}{n_{k}!}}} where the right-hand side involves hafnians of ( 2 ∑ k n k ) × ( 2 ∑ k n k ) {\displaystyle {\Big (}2\sum _{k}n_{k}{\Big )}\times {\Big (}2\sum _{k}n_{k}{\Big )}} matrices S ~ ( { n k } ) = ( A ~ ( { n k } ) C ~ ( { n k } ) C ~ T ( { n k } ) B ~ ( { n k } ) ) {\displaystyle {\tilde {S}}(\{n_{k}\})={\begin{pmatrix}{\tilde {A}}(\{n_{k}\})&{\tilde {C}}(\{n_{k}\})\\{\tilde {C}}^{\mathsf {T}}(\{n_{k}\})&{\tilde {B}}(\{n_{k}\})\\\end{pmatrix}}} , whose blocks A ~ ( { n k } ) {\displaystyle {\tilde {A}}(\{n_{k}\})} , B ~ ( { n k } ) {\displaystyle {\tilde {B}}(\{n_{k}\})} , C ~ ( { n k } ) {\displaystyle {\tilde {C}}(\{n_{k}\})} and C ~ T ( { n k } ) {\displaystyle {\tilde {C}}^{\mathsf {T}}(\{n_{k}\})} are built from the blocks A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} and C T {\displaystyle C^{\mathsf {T}}} respectively in the way introduced in MacMahon's Master theorem. In particular, A ~ ( { n k } ) {\displaystyle {\tilde {A}}(\{n_{k}\})} is a matrix built by replacing each entry A k , t {\displaystyle A_{k,t}} in the matrix A {\displaystyle A} with a n k × n t {\displaystyle n_{k}\times n_{t}} block filled with A k , t {\displaystyle A_{k,t}} ; the same scheme is applied to B {\displaystyle B} , C {\displaystyle C} and C T {\displaystyle C^{\mathsf {T}}} . The sum ∑ { n k } {\displaystyle \sum _{\{n_{k}\}}} runs over all k {\displaystyle k} -tuples of non-negative integers, and it is assumed that haf ⁡ S ~ ( { n k = 0 \| k = 1 … m } ) = 1 {\displaystyle \operatorname {haf} {\tilde {S}}(\{n_{k}=0\|k=1\ldots m\})=1} . The identity can be proved by means of multivariate Gaussian integrals and Wick's probability theorem. The expression in the left-hand side, 1 / det ( I − Z S ) {\displaystyle 1{\Big /}{\sqrt {\det {\big (}I-ZS{\big )}}}{\Big .}} , is in fact a multivariate generating function for a series of hafnians, and the right-hand side constitutes its multivariable Taylor expansion in the vicinity of the point z 1 = … = z m = 0. {\displaystyle z_{1}=\ldots =z_{m}=0.} As a consequence of the given relation, the hafnian of a symmetric 2 m × 2 m {\displaystyle 2m\times 2m} matrix S {\displaystyle S} can be represented as the following mixed derivative of the order m {\displaystyle m} : haf ⁡ S = ( ∏ k = 1 m ∂ ∂ z k ) 1 det ( I − Z S ) \| z j = 0 . {\displaystyle \operatorname {haf} S={\bigg (}\prod _{k=1}^{m}{\frac {\partial }{\partial z_{k}}}{\bigg )}{\Bigg .}{\frac {1}{\sqrt {\det {\big (}I-ZS{\big )}}}}{\Bigg \vert }_{z_{j}=0}.} The hafnian generating function identity written above can be considered as a hafnian generalization of MacMahon's Master theorem, which introduces the generating function for matrix permanents and has the following form in terms of the introduced notation: 1 det ( I − diag ( z 1 , z 2 , … , z m ) C ) = ∑ { n k } per ⁡ C ~ ( { n k } ) ∏ k = 1 m z k n k n k ! {\displaystyle {\frac {1}{\det {\big (}I-{\textrm {diag}}(z_{1},z_{2},\ldots ,z_{m})C{\big )}}}=\sum _{\{n_{k}\}}\operatorname {per} {\tilde {C}}(\{n_{k}\})\prod _{k=1}^{m}{\frac {z_{k}^{n_{k}}}{n_{k}!}}} Note that MacMahon's Master theorem comes as a simple corollary from the hafnian generating function identity due to the relation per ⁡ ( C ) = haf ⁡ ( 0 C C T 0 ) {\displaystyle \operatorname {per} (C)=\operatorname {haf} {\begin{pmatrix}0&C\\C^{\mathsf {T}}&0\end{pmatrix}}} . == Non-negativity == If C {\displaystyle C} is a Hermitian positive semi-definite n × n {\displaystyle n\times n} matrix and B {\displaystyle B} is a complex symmetric n × n {\displaystyle n\times n} matrix, then haf ⁡ ( B C C ¯ B ¯ ) ≥ 0 , {\displaystyle \operatorname {haf} {\begin{pmatrix}B&C\\{\overline {C}}&{\overline {B}}\end{pmatrix}}\geq 0,} where C ¯ {\displaystyle {\overline {C}}} denotes the complex conjugate of C {\displaystyle C} . A simple way to see this when ( C B B ¯ C ¯ ) {\displaystyle {\begin{pmatrix}C&B\\{\overline {B}}&{\overline {C}}\\\end{pmatrix}}} is positive semi-definite is to observe that, by Wick's probability theorem, haf ⁡ ( B C C ¯ B ¯ ) = E [ \| X 1 … X n \| 2 ] {\displaystyle \operatorname {haf} {\begin{pmatrix}B&C\\{\overline {C}}&{\overline {B}}\\\end{pmatrix}}=\mathbb {E} \left[\left\|X_{1}\dots X_{n}\right\|^{2}\right]} when ( X 1 , … , X n ) {\displaystyle \left(X_{1},\dots ,X_{n}\right)} is a complex normal random vector with mean 0 {\displaystyle 0} , covariance matrix C {\displaystyle C} and relation matrix B {\displaystyle B} . This result is a generalization of the fact that the permanent of a Hermitian positive semi-definite matrix is non-negative. This corresponds to the special case B = 0 {\displaystyle B=0} using the relation per ⁡ ( C ) = haf ⁡ ( 0 C C T 0 ) {\displaystyle \operatorname {per} (C)=\operatorname {haf} {\begin{pmatrix}0&C\\C^{\mathsf {T}}&0\end{pmatrix}}} . == Loop hafnian == The loop hafnian of an m × m {\displaystyle m\times m} symmetric matrix is defined as lhaf ⁡ ( A ) = ∑ ρ ∈ M ∏ ( i , j ) ∈ ρ A i , j {\displaystyle \operatorname {lhaf} (A)=\sum _{\rho \in {\mathcal {M}}}\prod _{(i,j)\in \rho }A_{i,j}} where M {\displaystyle {\mathcal {M}}} is the set of all perfect matchings of the complete graph on m {\displaystyle m} vertices with loops, i.e., the set of all ways to partition the set { 1 , 2 , … , m } {\displaystyle \{1,2,\dots ,m\}} into pairs or singletons (treating a singleton ( i ) {\displaystyle (i)} as the pair ( i , i ) {\displaystyle (i,i)} ). Thus the loop hafnian depends on the diagonal entries of the matrix, unlike the hafnian. Furthermore, the loop hafnian can be non-zero when m {\displaystyle m} is odd. The loop hafnian can be used to count the total number of matchings in a graph (perfect or non-perfect), also known as its Hosoya index. Specifically, if one takes the adjacency matrix of a graph and sets the diagonal elements to 1, then the loop hafnian of the resulting matrix is equal to the total number of matchings in the graph. The loop hafnian can also be thought of as incorporating a mean into the interpretation of the hafnian as a multivariate Gaussian moment. Specifically, by Wick's probability theorem again, the loop hafnian of a real m × m {\displaystyle m\times m} symmetric matrix can be expressed as lhaf ⁡ ( A ) = E ( X 1 , … , X m ) ∼ N ( diag ⁡ ( A ) , A + λ I ) [ X 1 … X m ] , {\displaystyle \operatorname {lhaf} (A)=\mathbb {E} _{\left(X_{1},\dots ,X_{m}\right)\sim {\mathcal {N}}\left(\operatorname {diag} \left(A\right),A+\lambda I\right)}\left[X_{1}\dots X_{m}\right],} where λ {\displaystyle \lambda } is any number large enough to make A + λ I {\displaystyle A+\lambda I} positive semi-definite. == Computation == Computing the hafnian of a (0,1)-matrix is #P-complete, because computing the permanent of a (0,1)-matrix is #P-complete. The hafnian of a 2 n × 2 n {\displaystyle 2n\times 2n} matrix can be computed in O ( n 3 2 n ) {\displaystyle O(n^{3}2^{n})} time. If the entries of a matrix are non-negative, then its hafnian can be approximated to within an exponential factor in polynomial time. == See also == Permanent Pfaffian Boson sampling == References ==
Wikipedia:Hagop Panossian#0	Hagop Panossian (Armenian: Յակոբ Փանոսեան; born 8 June 1946) is an Armenian aerospace engineer, academic and philanthropist with over 30 years of experience in rocket engine control and modeling, large space structures, actuation systems, failure detection, stochastic systems, vibration damping and optimal and adaptive control. Since founding the ARPA Institute in 1992, he has served as its president and remains actively involved in its wide range of initiatives. == Early life and education == Panossian was born in Anjar, Lebanon, and received his primary and secondary education from the Armenian Evangelical Secondary School of Anjar. In 1969, he graduated with a Bachelor of Science in mathematics from the American University of Beirut. For five years, Panossian taught mathematics and sciences at the Calousd Gulbengian Secondary School, before moving to the United States at the age of 28. He earned a master's degree in applied mathematics from the University of South Carolina in 1974, and a doctorate in engineering from the University of California, Los Angeles in 1981. == Career == === Work in aerospace engineering === From 1981 to 1987, Panossian worked at Textron, and since 1987, he has worked at Rockwell International, Rocketdyne, Boeing, and Pratt & Whitney, specialising in rocket systems and emerging high-frequency oscillations for space shuttle engines. Notably, he programmed the control law for the X33 Aerospike engine, and proposed an innovative method for managing high-frequency oscillations in engine parts and ensuring stability at low temperatures. In 1987, he was selected by the Fulbright Association as an exchange scientist in Armenia for four months and taught on automatic control systems at Yerevan State University and Polytechnic universities. In 2008, he was elected a foreign member of the National Academy of Sciences of Armenia. Panossian has served as an adjunct professor at California State University, Northridge in the Mechanical Engineering Department. He is also an Associate Technical Fellow of the American Institute of Aeronautics and Astronatics, a Senior Member of the Institute of Electrical and Electronics Engineers, and a Fellow of the Institute for the Advancement of Engineering (IAE). === Non-profit organizations === Panossian has founded two non-profit philanthropic organizations in Los Angeles. In 1983, he founded the Armenian Engineers and Scientists of America (AESA) and served as its president in 1987 and 1988. In 1992, he founded the ARPA Institute (Analysis, Research & Planning for Armenia) and continues to serve as its president, coordinating lectures, invention competitions, awards and other programs. The Institute promotes international cooperation between the Republic of Armenia and the Armenian diaspora through consulting, analysis and research across various fields including education, economics, medicine, law, history and technologies. Panossian has been actively involved in Armenia, leading initiatives through the ARPA Institute. He has helped in the modernization of Armenia's blood services system and education of youth about health risks of smoking and substance abuse. Through his initiatives, the first class 1000 cleanroom was established in the Alikhanyan National Science Laboratory, the first DNA sequencer and various other instruments were donated to the Institute of Molecular Biology, and valuable scientific devices and instruments to institutes of the National Academy of Sciences of Armenia. Additionally, he organizes, through ARPA, an annual invention competition for young Armenian scientists and monthly lectures and/or panel discussion in Los Angeles, featuring specialists in various fields. == Personal life == Panossian is married to Ani (who has passed away in 2015) and has two sons, Armen and Baruir, and a daughter, Lorig. == Honours == Panossian has received numerous awards, including: Rocketdyne President's Award for "Outstanding Achievements in Problem Resolving Through Applying NOPD to Graphic Division Printing Press Cylinders", Rockwell International, (July, 1992) "Engineer of the Year" award for "distinguished contributions in developing non obstructive particle damping techniques for reducing severe structural vibrations in a wide range of product applications", Rockwell International (February, 1993) Rocketdyne "Engineer of the Year" award "In recognition of outstanding professional contributions to Rocketdyne, the community and to engineering progress" (January, 1993) "Distinguished Engineering Achievements" award for "Outstanding contributions to industry, education and government and the entire engineering community", Institute for the Advancement of Engineering (March, 1993) Resolution for "remarkable achievements and contributions to the community", Los Angeles City Council Proclamation (April, 2012) == Publications == Some of his publications include: Uncertainty Management In Modeling and Control of Large Flexible Structures (1984) Optimal Stochastic Modeling and Control of Flexible Structures (1988) Real-time failure control (SAFD) (1990) X-33 attitude control using the XRS-2200 linear aerospike engine (1999) Optimized Non-Obstructive Particle Damping (NOPD) Treatment for Composite Honeycomb Structures (2006) Non-Obstructive Particle Damping: New Experiences and Capabilities (2008) Modeling Techniques for Evaluating the Effectiveness of Particle Damping in Turbomachinery (2009) == References == == External links == National Academy of Sciences of Armenia Hagop Panossian on YouTube ARPA Institute Mousa Ler Online ResearchGate
Wikipedia:Hahn–Banach theorem#0	In functional analysis, the Hahn–Banach theorem is a central result that allows the extension of bounded linear functionals defined on a vector subspace of some vector space to the whole space. The theorem also shows that there are sufficient continuous linear functionals defined on every normed vector space in order to study the dual space. Another version of the Hahn–Banach theorem is known as the Hahn–Banach separation theorem or the hyperplane separation theorem, and has numerous uses in convex geometry. == History == The theorem is named for the mathematicians Hans Hahn and Stefan Banach, who proved it independently in the late 1920s. The special case of the theorem for the space C [ a , b ] {\displaystyle C[a,b]} of continuous functions on an interval was proved earlier (in 1912) by Eduard Helly, and a more general extension theorem, the M. Riesz extension theorem, from which the Hahn–Banach theorem can be derived, was proved in 1923 by Marcel Riesz. The first Hahn–Banach theorem was proved by Eduard Helly in 1912 who showed that certain linear functionals defined on a subspace of a certain type of normed space ( C N {\displaystyle \mathbb {C} ^{\mathbb {N} }} ) had an extension of the same norm. Helly did this through the technique of first proving that a one-dimensional extension exists (where the linear functional has its domain extended by one dimension) and then using induction. In 1927, Hahn defined general Banach spaces and used Helly's technique to prove a norm-preserving version of Hahn–Banach theorem for Banach spaces (where a bounded linear functional on a subspace has a bounded linear extension of the same norm to the whole space). In 1929, Banach, who was unaware of Hahn's result, generalized it by replacing the norm-preserving version with the dominated extension version that uses sublinear functions. Whereas Helly's proof used mathematical induction, Hahn and Banach both used transfinite induction. The Hahn–Banach theorem arose from attempts to solve infinite systems of linear equations. This is needed to solve problems such as the moment problem, whereby given all the potential moments of a function one must determine if a function having these moments exists, and, if so, find it in terms of those moments. Another such problem is the Fourier cosine series problem, whereby given all the potential Fourier cosine coefficients one must determine if a function having those coefficients exists, and, again, find it if so. Riesz and Helly solved the problem for certain classes of spaces (such as L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} and C ( [ a , b ] ) {\displaystyle C([a,b])} ) where they discovered that the existence of a solution was equivalent to the existence and continuity of certain linear functionals. In effect, they needed to solve the following problem: (The vector problem) Given a collection ( f i ) i ∈ I {\displaystyle \left(f_{i}\right)_{i\in I}} of bounded linear functionals on a normed space X {\displaystyle X} and a collection of scalars ( c i ) i ∈ I , {\displaystyle \left(c_{i}\right)_{i\in I},} determine if there is an x ∈ X {\displaystyle x\in X} such that f i ( x ) = c i {\displaystyle f_{i}(x)=c_{i}} for all i ∈ I . {\displaystyle i\in I.} If X {\displaystyle X} happens to be a reflexive space then to solve the vector problem, it suffices to solve the following dual problem: (The functional problem) Given a collection ( x i ) i ∈ I {\displaystyle \left(x_{i}\right)_{i\in I}} of vectors in a normed space X {\displaystyle X} and a collection of scalars ( c i ) i ∈ I , {\displaystyle \left(c_{i}\right)_{i\in I},} determine if there is a bounded linear functional f {\displaystyle f} on X {\displaystyle X} such that f ( x i ) = c i {\displaystyle f\left(x_{i}\right)=c_{i}} for all i ∈ I . {\displaystyle i\in I.} Riesz went on to define L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} space ( 1 < p < ∞ {\displaystyle 1<p<\infty } ) in 1910 and the ℓ p {\displaystyle \ell ^{p}} spaces in 1913. While investigating these spaces he proved a special case of the Hahn–Banach theorem. Helly also proved a special case of the Hahn–Banach theorem in 1912. In 1910, Riesz solved the functional problem for some specific spaces and in 1912, Helly solved it for a more general class of spaces. It wasn't until 1932 that Banach, in one of the first important applications of the Hahn–Banach theorem, solved the general functional problem. The following theorem states the general functional problem and characterizes its solution. The Hahn–Banach theorem can be deduced from the above theorem. If X {\displaystyle X} is reflexive then this theorem solves the vector problem. == Hahn–Banach theorem == A real-valued function f : M → R {\displaystyle f:M\to \mathbb {R} } defined on a subset M {\displaystyle M} of X {\displaystyle X} is said to be dominated (above) by a function p : X → R {\displaystyle p:X\to \mathbb {R} } if f ( m ) ≤ p ( m ) {\displaystyle f(m)\leq p(m)} for every m ∈ M . {\displaystyle m\in M.} For this reason, the following version of the Hahn–Banach theorem is called the dominated extension theorem. The theorem remains true if the requirements on p {\displaystyle p} are relaxed to require only that p {\displaystyle p} be a convex function: p ( t x + ( 1 − t ) y ) ≤ t p ( x ) + ( 1 − t ) p ( y ) for all 0 < t < 1 and x , y ∈ X . {\displaystyle p(tx+(1-t)y)\leq tp(x)+(1-t)p(y)\qquad {\text{ for all }}0<t<1{\text{ and }}x,y\in X.} A function p : X → R {\displaystyle p:X\to \mathbb {R} } is convex and satisfies p ( 0 ) ≤ 0 {\displaystyle p(0)\leq 0} if and only if p ( a x + b y ) ≤ a p ( x ) + b p ( y ) {\displaystyle p(ax+by)\leq ap(x)+bp(y)} for all vectors x , y ∈ X {\displaystyle x,y\in X} and all non-negative real a , b ≥ 0 {\displaystyle a,b\geq 0} such that a + b ≤ 1. {\displaystyle a+b\leq 1.} Every sublinear function is a convex function. On the other hand, if p : X → R {\displaystyle p:X\to \mathbb {R} } is convex with p ( 0 ) ≥ 0 , {\displaystyle p(0)\geq 0,} then the function defined by p 0 ( x ) = def inf t > 0 p ( t x ) t {\displaystyle p_{0}(x)\;{\stackrel {\scriptscriptstyle {\text{def}}}{=}}\;\inf _{t>0}{\frac {p(tx)}{t}}} is positively homogeneous (because for all x {\displaystyle x} and r > 0 {\displaystyle r>0} one has p 0 ( r x ) = inf t > 0 p ( t r x ) t = r inf t > 0 p ( t r x ) t r = r inf τ > 0 p ( τ x ) τ = r p 0 ( x ) {\displaystyle p_{0}(rx)=\inf _{t>0}{\frac {p(trx)}{t}}=r\inf _{t>0}{\frac {p(trx)}{tr}}=r\inf _{\tau >0}{\frac {p(\tau x)}{\tau }}=rp_{0}(x)} ), hence, being convex, it is sublinear. It is also bounded above by p 0 ≤ p , {\displaystyle p_{0}\leq p,} and satisfies F ≤ p 0 {\displaystyle F\leq p_{0}} for every linear functional F ≤ p . {\displaystyle F\leq p.} So the extension of the Hahn–Banach theorem to convex functionals does not have a much larger content than the classical one stated for sublinear functionals. If F : X → R {\displaystyle F:X\to \mathbb {R} } is linear then F ≤ p {\displaystyle F\leq p} if and only if − p ( − x ) ≤ F ( x ) ≤ p ( x ) for all x ∈ X , {\displaystyle -p(-x)\leq F(x)\leq p(x)\quad {\text{ for all }}x\in X,} which is the (equivalent) conclusion that some authors write instead of F ≤ p . {\displaystyle F\leq p.} It follows that if p : X → R {\displaystyle p:X\to \mathbb {R} } is also symmetric, meaning that p ( − x ) = p ( x ) {\displaystyle p(-x)=p(x)} holds for all x ∈ X , {\displaystyle x\in X,} then F ≤ p {\displaystyle F\leq p} if and only \| F \| ≤ p . {\displaystyle \|F\|\leq p.} Every norm is a seminorm and both are symmetric balanced sublinear functions. A sublinear function is a seminorm if and only if it is a balanced function. On a real vector space (although not on a complex vector space), a sublinear function is a seminorm if and only if it is symmetric. The identity function R → R {\displaystyle \mathbb {R} \to \mathbb {R} } on X := R {\displaystyle X:=\mathbb {R} } is an example of a sublinear function that is not a seminorm. === For complex or real vector spaces === The dominated extension theorem for real linear functionals implies the following alternative statement of the Hahn–Banach theorem that can be applied to linear functionals on real or complex vector spaces. The theorem remains true if the requirements on p {\displaystyle p} are relaxed to require only that for all x , y ∈ X {\displaystyle x,y\in X} and all scalars a {\displaystyle a} and b {\displaystyle b} satisfying \| a \| + \| b \| ≤ 1 , {\displaystyle \|a\|+\|b\|\leq 1,} p ( a x + b y ) ≤ \| a \| p ( x ) + \| b \| p ( y ) . {\displaystyle p(ax+by)\leq \|a\|p(x)+\|b\|p(y).} This condition holds if and only if p {\displaystyle p} is a convex and balanced function satisfying p ( 0 ) ≤ 0 , {\displaystyle p(0)\leq 0,} or equivalently, if and only if it is convex, satisfies p ( 0 ) ≤ 0 , {\displaystyle p(0)\leq 0,} and p ( u x ) ≤ p ( x ) {\displaystyle p(ux)\leq p(x)} for all x ∈ X {\displaystyle x\in X} and all unit length scalars u . {\displaystyle u.} A complex-valued functional F {\displaystyle F} is said to be dominated by p {\displaystyle p} if \| F ( x ) \| ≤ p ( x ) {\displaystyle \|F(x)\|\leq p(x)} for all x {\displaystyle x} in the domain of F . {\displaystyle F.} With this terminology, the above statements of the Hahn–Banach theorem can be restated more succinctly: Hahn–Banach dominated extension theorem: If p : X → R {\displaystyle p:X\to \mathbb {R} } is a seminorm defined on a real or complex vector space X , {\displaystyle X,} then every dominated linear functional defined on a vector subspace of X {\displaystyle X} has a dominated linear extension to all of X . {\displaystyle X.} In the case where X {\displaystyle X} is a real vector space and p : X → R {\displaystyle p:X\to \mathbb {R} } is merely a convex or sublinear function, this conclusion will remain true if both instances of "dominated" (meaning \| F \| ≤ p {\displaystyle \|F\|\leq p} ) are weakened to instead mean "dominated above" (meaning F ≤ p {\displaystyle F\leq p} ). Proof The following observations allow the Hahn–Banach theorem for real vector spaces to be applied to (complex-valued) linear functionals on complex vector spaces. Every linear functional F : X → C {\displaystyle F:X\to \mathbb {C} } on a complex vector space is completely determined by its real part Re ⁡ F : X → R {\displaystyle \;\operatorname {Re} F:X\to \mathbb {R} \;} through the formula F ( x ) = Re ⁡ F ( x ) − i Re ⁡ F ( i x ) for all x ∈ X {\displaystyle F(x)\;=\;\operatorname {Re} F(x)-i\operatorname {Re} F(ix)\qquad {\text{ for all }}x\in X} and moreover, if ‖ ⋅ ‖ {\displaystyle \\|\cdot \\|} is a norm on X {\displaystyle X} then their dual norms are equal: ‖ F ‖ = ‖ Re ⁡ F ‖ . {\displaystyle \\|F\\|=\\|\operatorname {Re} F\\|.} In particular, a linear functional on X {\displaystyle X} extends another one defined on M ⊆ X {\displaystyle M\subseteq X} if and only if their real parts are equal on M {\displaystyle M} (in other words, a linear functional F {\displaystyle F} extends f {\displaystyle f} if and only if Re ⁡ F {\displaystyle \operatorname {Re} F} extends Re ⁡ f {\displaystyle \operatorname {Re} f} ). The real part of a linear functional on X {\displaystyle X} is always a real-linear functional (meaning that it is linear when X {\displaystyle X} is considered as a real vector space) and if R : X → R {\displaystyle R:X\to \mathbb {R} } is a real-linear functional on a complex vector space then x ↦ R ( x ) − i R ( i x ) {\displaystyle x\mapsto R(x)-iR(ix)} defines the unique linear functional on X {\displaystyle X} whose real part is R . {\displaystyle R.} If F {\displaystyle F} is a linear functional on a (complex or real) vector space X {\displaystyle X} and if p : X → R {\displaystyle p:X\to \mathbb {R} } is a seminorm then \| F \| ≤ p if and only if Re ⁡ F ≤ p . {\displaystyle \|F\|\,\leq \,p\quad {\text{ if and only if }}\quad \operatorname {Re} F\,\leq \,p.} Stated in simpler language, a linear functional is dominated by a seminorm p {\displaystyle p} if and only if its real part is dominated above by p . {\displaystyle p.} The proof above shows that when p {\displaystyle p} is a seminorm then there is a one-to-one correspondence between dominated linear extensions of f : M → C {\displaystyle f:M\to \mathbb {C} } and dominated real-linear extensions of Re ⁡ f : M → R ; {\displaystyle \operatorname {Re} f:M\to \mathbb {R} ;} the proof even gives a formula for explicitly constructing a linear extension of f {\displaystyle f} from any given real-linear extension of its real part. Continuity A linear functional F {\displaystyle F} on a topological vector space is continuous if and only if this is true of its real part Re ⁡ F ; {\displaystyle \operatorname {Re} F;} if the domain is a normed space then ‖ F ‖ = ‖ Re ⁡ F ‖ {\displaystyle \\|F\\|=\\|\operatorname {Re} F\\|} (where one side is infinite if and only if the other side is infinite). Assume X {\displaystyle X} is a topological vector space and p : X → R {\displaystyle p:X\to \mathbb {R} } is sublinear function. If p {\displaystyle p} is a continuous sublinear function that dominates a linear functional F {\displaystyle F} then F {\displaystyle F} is necessarily continuous. Moreover, a linear functional F {\displaystyle F} is continuous if and only if its absolute value \| F \| {\displaystyle \|F\|} (which is a seminorm that dominates F {\displaystyle F} ) is continuous. In particular, a linear functional is continuous if and only if it is dominated by some continuous sublinear function. === Proof === The Hahn–Banach theorem for real vector spaces ultimately follows from Helly's initial result for the special case where the linear functional is extended from M {\displaystyle M} to a larger vector space in which M {\displaystyle M} has codimension 1. {\displaystyle 1.} This lemma remains true if p : X → R {\displaystyle p:X\to \mathbb {R} } is merely a convex function instead of a sublinear function. The lemma above is the key step in deducing the dominated extension theorem from Zorn's lemma. When M {\displaystyle M} has countable codimension, then using induction and the lemma completes the proof of the Hahn–Banach theorem. The standard proof of the general case uses Zorn's lemma although the strictly weaker ultrafilter lemma (which is equivalent to the compactness theorem and to the Boolean prime ideal theorem) may be used instead. Hahn–Banach can also be proved using Tychonoff's theorem for compact Hausdorff spaces (which is also equivalent to the ultrafilter lemma) The Mizar project has completely formalized and automatically checked the proof of the Hahn–Banach theorem in the HAHNBAN file. == Continuous extension theorem == The Hahn–Banach theorem can be used to guarantee the existence of continuous linear extensions of continuous linear functionals. In category-theoretic terms, the underlying field of the vector space is an injective object in the category of locally convex vector spaces. On a normed (or seminormed) space, a linear extension F {\displaystyle F} of a bounded linear functional f {\displaystyle f} is said to be norm-preserving if it has the same dual norm as the original functional: ‖ F ‖ = ‖ f ‖ . {\displaystyle \\|F\\|=\\|f\\|.} Because of this terminology, the second part of the above theorem is sometimes referred to as the "norm-preserving" version of the Hahn–Banach theorem. Explicitly: === Proof of the continuous extension theorem === The following observations allow the continuous extension theorem to be deduced from the Hahn–Banach theorem. The absolute value of a linear functional is always a seminorm. A linear functional F {\displaystyle F} on a topological vector space X {\displaystyle X} is continuous if and only if its absolute value \| F \| {\displaystyle \|F\|} is continuous, which happens if and only if there exists a continuous seminorm p {\displaystyle p} on X {\displaystyle X} such that \| F \| ≤ p {\displaystyle \|F\|\leq p} on the domain of F . {\displaystyle F.} If X {\displaystyle X} is a locally convex space then this statement remains true when the linear functional F {\displaystyle F} is defined on a proper vector subspace of X . {\displaystyle X.} Proof for normed spaces A linear functional f {\displaystyle f} on a normed space is continuous if and only if it is bounded, which means that its dual norm ‖ f ‖ = sup { \| f ( m ) \| : ‖ m ‖ ≤ 1 , m ∈ domain ⁡ f } {\displaystyle \\|f\\|=\sup\{\|f(m)\|:\\|m\\|\leq 1,m\in \operatorname {domain} f\}} is finite, in which case \| f ( m ) \| ≤ ‖ f ‖ ‖ m ‖ {\displaystyle \|f(m)\|\leq \\|f\\|\\|m\\|} holds for every point m {\displaystyle m} in its domain. Moreover, if c ≥ 0 {\displaystyle c\geq 0} is such that \| f ( m ) \| ≤ c ‖ m ‖ {\displaystyle \|f(m)\|\leq c\\|m\\|} for all m {\displaystyle m} in the functional's domain, then necessarily ‖ f ‖ ≤ c . {\displaystyle \\|f\\|\leq c.} If F {\displaystyle F} is a linear extension of a linear functional f {\displaystyle f} then their dual norms always satisfy ‖ f ‖ ≤ ‖ F ‖ {\displaystyle \\|f\\|\leq \\|F\\|} so that equality ‖ f ‖ = ‖ F ‖ {\displaystyle \\|f\\|=\\|F\\|} is equivalent to ‖ F ‖ ≤ ‖ f ‖ , {\displaystyle \\|F\\|\leq \\|f\\|,} which holds if and only if \| F ( x ) \| ≤ ‖ f ‖ ‖ x ‖ {\displaystyle \|F(x)\|\leq \\|f\\|\\|x\\|} for every point x {\displaystyle x} in the extension's domain. This can be restated in terms of the function ‖ f ‖ ‖ ⋅ ‖ : X → R {\displaystyle \\|f\\|\,\\|\cdot \\|:X\to \mathbb {R} } defined by x ↦ ‖ f ‖ ‖ x ‖ , {\displaystyle x\mapsto \\|f\\|\,\\|x\\|,} which is always a seminorm: A linear extension of a bounded linear functional f {\displaystyle f} is norm-preserving if and only if the extension is dominated by the seminorm ‖ f ‖ ‖ ⋅ ‖ . {\displaystyle \\|f\\|\,\\|\cdot \\|.} Applying the Hahn–Banach theorem to f {\displaystyle f} with this seminorm ‖ f ‖ ‖ ⋅ ‖ {\displaystyle \\|f\\|\,\\|\cdot \\|} thus produces a dominated linear extension whose norm is (necessarily) equal to that of f , {\displaystyle f,} which proves the theorem: === Non-locally convex spaces === The continuous extension theorem might fail if the topological vector space (TVS) X {\displaystyle X} is not locally convex. For example, for 0 < p < 1 , {\displaystyle 0<p<1,} the Lebesgue space L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is a complete metrizable TVS (an F-space) that is not locally convex (in fact, its only convex open subsets are itself L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} and the empty set) and the only continuous linear functional on L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is the constant 0 {\displaystyle 0} function (Rudin 1991, §1.47). Since L p ( [ 0 , 1 ] ) {\displaystyle L^{p}([0,1])} is Hausdorff, every finite-dimensional vector subspace M ⊆ L p ( [ 0 , 1 ] ) {\displaystyle M\subseteq L^{p}([0,1])} is linearly homeomorphic to Euclidean space R dim ⁡ M {\displaystyle \mathbb {R} ^{\dim M}} or C dim ⁡ M {\displaystyle \mathbb {C} ^{\dim M}} (by F. Riesz's theorem) and so every non-zero linear functional f {\displaystyle f} on M {\displaystyle M} is continuous but none has a continuous linear extension to all of L p ( [ 0 , 1 ] ) . {\displaystyle L^{p}([0,1]).} However, it is possible for a TVS X {\displaystyle X} to not be locally convex but nevertheless have enough continuous linear functionals that its continuous dual space X ∗ {\displaystyle X^{}} separates points; for such a TVS, a continuous linear functional defined on a vector subspace might have a continuous linear extension to the whole space. If the TVS X {\displaystyle X} is not locally convex then there might not exist any continuous seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } defined on X {\displaystyle X} (not just on M {\displaystyle M} ) that dominates f , {\displaystyle f,} in which case the Hahn–Banach theorem can not be applied as it was in the above proof of the continuous extension theorem. However, the proof's argument can be generalized to give a characterization of when a continuous linear functional has a continuous linear extension: If X {\displaystyle X} is any TVS (not necessarily locally convex), then a continuous linear functional f {\displaystyle f} defined on a vector subspace M {\displaystyle M} has a continuous linear extension F {\displaystyle F} to all of X {\displaystyle X} if and only if there exists some continuous seminorm p {\displaystyle p} on X {\displaystyle X} that dominates f . {\displaystyle f.} Specifically, if given a continuous linear extension F {\displaystyle F} then p := \| F \| {\displaystyle p:=\|F\|} is a continuous seminorm on X {\displaystyle X} that dominates f ; {\displaystyle f;} and conversely, if given a continuous seminorm p : X → R {\displaystyle p:X\to \mathbb {R} } on X {\displaystyle X} that dominates f {\displaystyle f} then any dominated linear extension of f {\displaystyle f} to X {\displaystyle X} (the existence of which is guaranteed by the Hahn–Banach theorem) will be a continuous linear extension. == Geometric Hahn–Banach (the Hahn–Banach separation theorems) == The key element of the Hahn–Banach theorem is fundamentally a result about the separation of two convex sets: { − p ( − x − n ) − f ( n ) : n ∈ M } , {\displaystyle \{-p(-x-n)-f(n):n\in M\},} and { p ( m + x ) − f ( m ) : m ∈ M } . {\displaystyle \{p(m+x)-f(m):m\in M\}.} This sort of argument appears widely in convex geometry, optimization theory, and economics. Lemmas to this end derived from the original Hahn–Banach theorem are known as the Hahn–Banach separation theorems. They are generalizations of the hyperplane separation theorem, which states that two disjoint nonempty convex subsets of a finite-dimensional space R n {\displaystyle \mathbb {R} ^{n}} can be separated by some affine hyperplane, which is a fiber (level set) of the form f − 1 ( s ) = { x : f ( x ) = s } {\displaystyle f^{-1}(s)=\{x:f(x)=s\}} where f ≠ 0 {\displaystyle f\neq 0} is a non-zero linear functional and s {\displaystyle s} is a scalar. When the convex sets have additional properties, such as being open or compact for example, then the conclusion can be substantially strengthened: Then following important corollary is known as the Geometric Hahn–Banach theorem or Mazur's theorem (also known as Ascoli–Mazur theorem). It follows from the first bullet above and the convexity of M . {\displaystyle M.} Mazur's theorem clarifies that vector subspaces (even those that are not closed) can be characterized by linear functionals. === Supporting hyperplanes === Since points are trivially convex, geometric Hahn–Banach implies that functionals can detect the boundary of a set. In particular, let X {\displaystyle X} be a real topological vector space and A ⊆ X {\displaystyle A\subseteq X} be convex with Int ⁡ A ≠ ∅ . {\displaystyle \operatorname {Int} A\neq \varnothing .} If a 0 ∈ A ∖ Int ⁡ A {\displaystyle a_{0}\in A\setminus \operatorname {Int} A} then there is a functional that is vanishing at a 0 , {\displaystyle a_{0},} but supported on the interior of A . {\displaystyle A.} Call a normed space X {\displaystyle X} smooth if at each point x {\displaystyle x} in its unit ball there exists a unique closed hyperplane to the unit ball at x . {\displaystyle x.} Köthe showed in 1983 that a normed space is smooth at a point x {\displaystyle x} if and only if the norm is Gateaux differentiable at that point. === Balanced or disked neighborhoods === Let U {\displaystyle U} be a convex balanced neighborhood of the origin in a locally convex topological vector space X {\displaystyle X} and suppose x ∈ X {\displaystyle x\in X} is not an element of U . {\displaystyle U.} Then there exists a continuous linear functional f {\displaystyle f} on X {\displaystyle X} such that sup \| f ( U ) \| ≤ \| f ( x ) \| . {\displaystyle \sup \|f(U)\|\leq \|f(x)\|.} == Applications == The Hahn–Banach theorem is the first sign of an important philosophy in functional analysis: to understand a space, one should understand its continuous functionals. For example, linear subspaces are characterized by functionals: if X is a normed vector space with linear subspace M (not necessarily closed) and if z {\displaystyle z} is an element of X not in the closure of M, then there exists a continuous linear map f : X → K {\displaystyle f:X\to \mathbf {K} } with f ( m ) = 0 {\displaystyle f(m)=0} for all m ∈ M , {\displaystyle m\in M,} f ( z ) = 1 , {\displaystyle f(z)=1,} and ‖ f ‖ = dist ⁡ ( z , M ) − 1 . {\displaystyle \\|f\\|=\operatorname {dist} (z,M)^{-1}.} (To see this, note that dist ⁡ ( ⋅ , M ) {\displaystyle \operatorname {dist} (\cdot ,M)} is a sublinear function.) Moreover, if z {\displaystyle z} is an element of X, then there exists a continuous linear map f : X → K {\displaystyle f:X\to \mathbf {K} } such that f ( z ) = ‖ z ‖ {\displaystyle f(z)=\\|z\\|} and ‖ f ‖ ≤ 1. {\displaystyle \\|f\\|\leq 1.} This implies that the natural injection J {\displaystyle J} from a normed space X into its double dual V ∗ ∗ {\displaystyle V^{}} is isometric. That last result also suggests that the Hahn–Banach theorem can often be used to locate a "nicer" topology in which to work. For example, many results in functional analysis assume that a space is Hausdorff or locally convex. However, suppose X is a topological vector space, not necessarily Hausdorff or locally convex, but with a nonempty, proper, convex, open set M. Then geometric Hahn–Banach implies that there is a hyperplane separating M from any other point. In particular, there must exist a nonzero functional on X — that is, the continuous dual space X ∗ {\displaystyle X^{}} is non-trivial. Considering X with the weak topology induced by X ∗ , {\displaystyle X^{},} then X becomes locally convex; by the second bullet of geometric Hahn–Banach, the weak topology on this new space separates points. Thus X with this weak topology becomes Hausdorff. This sometimes allows some results from locally convex topological vector spaces to be applied to non-Hausdorff and non-locally convex spaces. === Partial differential equations === The Hahn–Banach theorem is often useful when one wishes to apply the method of a priori estimates. Suppose that we wish to solve the linear differential equation P u = f {\displaystyle Pu=f} for u , {\displaystyle u,} with f {\displaystyle f} given in some Banach space X. If we have control on the size of u {\displaystyle u} in terms of ‖ f ‖ X {\displaystyle \\|f\\|_{X}} and we can think of u {\displaystyle u} as a bounded linear functional on some suitable space of test functions g , {\displaystyle g,} then we can view f {\displaystyle f} as a linear functional by adjunction: ( f , g ) = ( u , P ∗ g ) . {\displaystyle (f,g)=(u,P^{}g).} At first, this functional is only defined on the image of P , {\displaystyle P,} but using the Hahn–Banach theorem, we can try to extend it to the entire codomain X. The resulting functional is often defined to be a weak solution to the equation. === Characterizing reflexive Banach spaces === === Example from Fredholm theory === To illustrate an actual application of the Hahn–Banach theorem, we will now prove a result that follows almost entirely from the Hahn–Banach theorem. The above result may be used to show that every closed vector subspace of R N {\displaystyle \mathbb {R} ^{\mathbb {N} }} is complemented because any such space is either finite dimensional or else TVS–isomorphic to R N . {\displaystyle \mathbb {R} ^{\mathbb {N} }.} == Generalizations == General template There are now many other versions of the Hahn–Banach theorem. The general template for the various versions of the Hahn–Banach theorem presented in this article is as follows: p : X → R {\displaystyle p:X\to \mathbb {R} } is a sublinear function (possibly a seminorm) on a vector space X , {\displaystyle X,} M {\displaystyle M} is a vector subspace of X {\displaystyle X} (possibly closed), and f {\displaystyle f} is a linear functional on M {\displaystyle M} satisfying \| f \| ≤ p {\displaystyle \|f\|\leq p} on M {\displaystyle M} (and possibly some other conditions). One then concludes that there exists a linear extension F {\displaystyle F} of f {\displaystyle f} to X {\displaystyle X} such that \| F \| ≤ p {\displaystyle \|F\|\leq p} on X {\displaystyle X} (possibly with additional properties). === For seminorms === So for example, suppose that f {\displaystyle f} is a bounded linear functional defined on a vector subspace M {\displaystyle M} of a normed space X , {\displaystyle X,} so its the operator norm ‖ f ‖ {\displaystyle \\|f\\|} is a non-negative real number. Then the linear functional's absolute value p := \| f \| {\displaystyle p:=\|f\|} is a seminorm on M {\displaystyle M} and the map q : X → R {\displaystyle q:X\to \mathbb {R} } defined by q ( x ) = ‖ f ‖ ‖ x ‖ {\displaystyle q(x)=\\|f\\|\,\\|x\\|} is a seminorm on X {\displaystyle X} that satisfies p ≤ q \| M {\displaystyle p\leq q{\big \vert }_{M}} on M . {\displaystyle M.} The Hahn–Banach theorem for seminorms guarantees the existence of a seminorm P : X → R {\displaystyle P:X\to \mathbb {R} } that is equal to \| f \| {\displaystyle \|f\|} on M {\displaystyle M} (since P \| M = p = \| f \| {\displaystyle P{\big \vert }_{M}=p=\|f\|} ) and is bounded above by P ( x ) ≤ ‖ f ‖ ‖ x ‖ {\displaystyle P(x)\leq \\|f\\|\,\\|x\\|} everywhere on X {\displaystyle X} (since P ≤ q {\displaystyle P\leq q} ). === Geometric separation === === Maximal dominated linear extension === If S = { s } {\displaystyle S=\{s\}} is a singleton set (where s ∈ X {\displaystyle s\in X} is some vector) and if F : X → R {\displaystyle F:X\to \mathbb {R} } is such a maximal dominated linear extension of f : M → R , {\displaystyle f:M\to \mathbb {R} ,} then F ( s ) = inf m ∈ M [ f ( s ) + p ( s − m ) ] . {\displaystyle F(s)=\inf _{m\in M}[f(s)+p(s-m)].} === Vector valued Hahn–Banach === === Invariant Hahn–Banach === A set Γ {\displaystyle \Gamma } of maps X → X {\displaystyle X\to X} is commutative (with respect to function composition ∘ {\displaystyle \,\circ \,} ) if F ∘ G = G ∘ F {\displaystyle F\circ G=G\circ F} for all F , G ∈ Γ . {\displaystyle F,G\in \Gamma .} Say that a function f {\displaystyle f} defined on a subset M {\displaystyle M} of X {\displaystyle X} is Γ {\displaystyle \Gamma } -invariant if L ( M ) ⊆ M {\displaystyle L(M)\subseteq M} and f ∘ L = f {\displaystyle f\circ L=f} on M {\displaystyle M} for every L ∈ Γ . {\displaystyle L\in \Gamma .} This theorem may be summarized: Every Γ {\displaystyle \Gamma } -invariant continuous linear functional defined on a vector subspace of a normed space X {\displaystyle X} has a Γ {\displaystyle \Gamma } -invariant Hahn–Banach extension to all of X . {\displaystyle X.} === For nonlinear functions === The following theorem of Mazur–Orlicz (1953) is equivalent to the Hahn–Banach theorem. The following theorem characterizes when any scalar function on X {\displaystyle X} (not necessarily linear) has a continuous linear extension to all of X . {\displaystyle X.} == Converse == Let X be a topological vector space. A vector subspace M of X has the extension property if any continuous linear functional on M can be extended to a continuous linear functional on X, and we say that X has the Hahn–Banach extension property (HBEP) if every vector subspace of X has the extension property. The Hahn–Banach theorem guarantees that every Hausdorff locally convex space has the HBEP. For complete metrizable topological vector spaces there is a converse, due to Kalton: every complete metrizable TVS with the Hahn–Banach extension property is locally convex. On the other hand, a vector space X of uncountable dimension, endowed with the finest vector topology, then this is a topological vector spaces with the Hahn–Banach extension property that is neither locally convex nor metrizable. A vector subspace M of a TVS X has the separation property if for every element of X such that x ∉ M , {\displaystyle x\not \in M,} there exists a continuous linear functional f {\displaystyle f} on X such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} and f ( m ) = 0 {\displaystyle f(m)=0} for all m ∈ M . {\displaystyle m\in M.} Clearly, the continuous dual space of a TVS X separates points on X if and only if { 0 } , {\displaystyle \{0\},} has the separation property. In 1992, Kakol proved that any infinite dimensional vector space X, there exist TVS-topologies on X that do not have the HBEP despite having enough continuous linear functionals for the continuous dual space to separate points on X. However, if X is a TVS then every vector subspace of X has the extension property if and only if every vector subspace of X has the separation property. == Relation to axiom of choice and other theorems == The proof of the Hahn–Banach theorem for real vector spaces (HB) commonly uses Zorn's lemma, which in the axiomatic framework of Zermelo–Fraenkel set theory (ZF) is equivalent to the axiom of choice (AC). It was discovered by Łoś and Ryll-Nardzewski and independently by Luxemburg that HB can be proved using the ultrafilter lemma (UL), which is equivalent (under ZF) to the Boolean prime ideal theorem (BPI). BPI is strictly weaker than the axiom of choice and it was later shown that HB is strictly weaker than BPI. The ultrafilter lemma is equivalent (under ZF) to the Banach–Alaoglu theorem, which is another foundational theorem in functional analysis. Although the Banach–Alaoglu theorem implies HB, it is not equivalent to it (said differently, the Banach–Alaoglu theorem is strictly stronger than HB). However, HB is equivalent to a certain weakened version of the Banach–Alaoglu theorem for normed spaces. The Hahn–Banach theorem is also equivalent to the following statement: (∗): On every Boolean algebra B there exists a "probability charge", that is: a non-constant finitely additive map from B {\displaystyle B} into [ 0 , 1 ] . {\displaystyle [0,1].} (BPI is equivalent to the statement that there are always non-constant probability charges which take only the values 0 and 1.) In ZF, the Hahn–Banach theorem suffices to derive the existence of a non-Lebesgue measurable set. Moreover, the Hahn–Banach theorem implies the Banach–Tarski paradox. For separable Banach spaces, D. K. Brown and S. G. Simpson proved that the Hahn–Banach theorem follows from WKL0, a weak subsystem of second-order arithmetic that takes a form of Kőnig's lemma restricted to binary trees as an axiom. In fact, they prove that under a weak set of assumptions, the two are equivalent, an example of reverse mathematics. == See also == Farkas' lemma – Solvability theorem for finite systems of linear inequalities Fichera's existence principle – Theorem in functional analysis M. Riesz extension theorem – theorem in mathematics, proved by Marcel RieszPages displaying wikidata descriptions as a fallback Separating axis theorem – On the existence of hyperplanes separating disjoint convex setsPages displaying short descriptions of redirect targets Vector-valued Hahn–Banach theorems == Notes == Proofs == References == == Bibliography == Adasch, Norbert; Ernst, Bruno; Keim, Dieter (1978). Topological Vector Spaces: The Theory Without Convexity Conditions. Lecture Notes in Mathematics. Vol. 639. Berlin New York: Springer-Verlag. ISBN 978-3-540-08662-8. OCLC 297140003. Banach, Stefan (1932). Théorie des Opérations Linéaires [Theory of Linear Operations] (PDF). Monografie Matematyczne (in French). Vol. 1. Warszawa: Subwencji Funduszu Kultury Narodowej. Zbl 0005.20901. Archived from the original (PDF) on 2014-01-11. Retrieved 2020-07-11. Berberian, Sterling K. (1974). Lectures in Functional Analysis and Operator Theory. Graduate Texts in Mathematics. Vol. 15. New York: Springer. ISBN 978-0-387-90081-0. OCLC 878109401. Bourbaki, Nicolas (1987) [1981]. Topological Vector Spaces: Chapters 1–5. Éléments de mathématique. Translated by Eggleston, H.G.; Madan, S. Berlin New York: Springer-Verlag. ISBN 3-540-13627-4. OCLC 17499190. Conway, John (1990). A course in functional analysis. Graduate Texts in Mathematics. Vol. 96 (2nd ed.). New York: Springer-Verlag. ISBN 978-0-387-97245-9. OCLC 21195908. Edwards, Robert E. (1995). Functional Analysis: Theory and Applications. New York: Dover Publications. ISBN 978-0-486-68143-6. OCLC 30593138. Grothendieck, Alexander (1973). Topological Vector Spaces. Translated by Chaljub, Orlando. New York: Gordon and Breach Science Publishers. ISBN 978-0-677-30020-7. OCLC 886098. Jarchow, Hans (1981). Locally convex spaces. Stuttgart: B.G. Teubner. ISBN 978-3-519-02224-4. OCLC 8210342. Köthe, Gottfried (1983) [1969]. Topological Vector Spaces I. Grundlehren der mathematischen Wissenschaften. Vol. 159. Translated by Garling, D.J.H. New York: Springer Science & Business Media. ISBN 978-3-642-64988-2. MR 0248498. OCLC 840293704. Łoś, Jerzy; Ryll-Nardzewski, Czesław (1951). "On the application of Tychonoff's theorem in mathematical proofs". Fundamenta Mathematicae. 38 (1): 233–237. doi:10.4064/fm-38-1-233-237. ISSN 0016-2736. Retrieved 7 July 2022. Luxemburg, W. A. J. (1962). "Two Applications of the Method of Construction by Ultrapowers to Analysis". Bulletin of the American Mathematical Society. 68 (4). American Mathematical Society: 416–419. doi:10.1090/s0002-9904-1962-10824-6. ISSN 0273-0979. Narici, Lawrence (2007). "On the Hahn-Banach Theorem". Advanced Courses of Mathematical Analysis II (PDF). World Scientific. pp. 87–122. doi:10.1142/9789812708441_0006. ISBN 978-981-256-652-2. Retrieved 7 July 2022. Narici, Lawrence; Beckenstein, Edward (1997). "The Hahn–Banach Theorem: The Life and Times". Topology and Its Applications. 77 (2): 193–211. doi:10.1016/s0166-8641(96)00142-3. Narici, Lawrence; Beckenstein, Edward (2011). Topological Vector Spaces. Pure and applied mathematics (Second ed.). Boca Raton, FL: CRC Press. ISBN 978-1584888666. OCLC 144216834. Pincus, David (1972). "Independence of the prime ideal theorem from the Hahn Banach theorem" (PDF). Bulletin of the American Mathematical Society. 78 (5). American Mathematical Society: 766–770. doi:10.1090/s0002-9904-1972-13025-8. ISSN 0273-0979. Retrieved 7 July 2022. Pincus, David (1974). "The strength of the Hahn-Banach theorem". In Hurd, A.; Loeb, P. (eds.). Victoria Symposium on Nonstandard Analysis. Lecture Notes in Mathematics. Vol. 369. Berlin, Heidelberg: Springer. pp. 203–248. doi:10.1007/bfb0066014. ISBN 978-3-540-06656-9. ISSN 0075-8434. Reed, Michael and Simon, Barry, Methods of Modern Mathematical Physics, Vol. 1, Functional Analysis, Section III.3. Academic Press, San Diego, 1980. ISBN 0-12-585050-6. Reed, Michael; Simon, Barry (1980). Functional Analysis (revised and enlarged ed.). Boston, MA: Academic Press. ISBN 978-0-12-585050-6. Robertson, Alex P.; Robertson, Wendy J. (1980). Topological Vector Spaces. Cambridge Tracts in Mathematics. Vol. 53. Cambridge England: Cambridge University Press. ISBN 978-0-521-29882-7. OCLC 589250. Rudin, Walter (1991). Functional Analysis. International Series in Pure and Applied Mathematics. Vol. 8 (Second ed.). New York, NY: McGraw-Hill Science/Engineering/Math. ISBN 978-0-07-054236-5. OCLC 21163277. Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 8 (Second ed.). New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135. Schmitt, Lothar M (1992). "An Equivariant Version of the Hahn–Banach Theorem". Houston J. Of Math. 18: 429–447. Schechter, Eric (1996). Handbook of Analysis and Its Foundations. San Diego, CA: Academic Press. ISBN 978-0-12-622760-4. OCLC 175294365. Swartz, Charles (1992). An introduction to Functional Analysis. New York: M. Dekker. ISBN 978-0-8247-8643-4. OCLC 24909067. Tao, Terence, The Hahn–Banach theorem, Menger's theorem, and Helly's theorem Trèves, François (2006) [1967]. Topological Vector Spaces, Distributions and Kernels. Mineola, N.Y.: Dover Publications. ISBN 978-0-486-45352-1. OCLC 853623322. Wilansky, Albert (2013). Modern Methods in Topological Vector Spaces. Mineola, New York: Dover Publications, Inc. ISBN 978-0-486-49353-4. OCLC 849801114. Wittstock, Gerd, Ein operatorwertiger Hahn-Banach Satz, J. of Functional Analysis 40 (1981), 127–150 Zeidler, Eberhard, Applied Functional Analysis: main principles and their applications, Springer, 1995. "Hahn–Banach theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
Wikipedia:Haidao Suanjing#0	Haidao Suanjing (海島算經; The Island Mathematical Manual) was written by the Chinese mathematician Liu Hui of the Three Kingdoms era (220–280) as an extension of chapter 9 of The Nine Chapters on the Mathematical Art. During the Tang dynasty, this appendix was taken out from The Nine Chapters on the Mathematical Art as a separate book, titled Haidao suanjing (Sea Island Mathematical Manual), named after problem No 1 "Looking at a sea island." In the time of the early Tang dynasty, Haidao Suanjing was selected into one of The Ten Computational Canons as the official mathematical texts for imperial examinations in mathematics. == Content == This book contained many practical problems of surveying using geometry. This work provided detailed instructions on how to measure distances and heights with tall surveyor's poles and horizontal bars fixed at right angles to them. The units of measurement were 1 li = 180 zhang = 1800 chi, 1 zhang = 10 chi = 100 cun, 1 bu (step) = 6 chi, 1 chi = 10 cun. Calculation was carried out with place value decimal Rod calculus. Liu Hui used his rectangle in right angle triangle theorem as the mathematical basis for survey. The setup is pictured on the right. By invoking his "in-out-complement" principle, he proved that the area of two inscribed rectangles in the two complementary right angle triangles have equal area, thus C E ⋅ A F = F B ⋅ B C {\displaystyle CE\cdot AF=FB\cdot BC} === Survey of sea island === Now we are surveying a sea island. Set up two 3-zhang poles at 1000 steps apart; let the two poles and the island be in a straight line. Step back from the front post 123 steps. With eye on ground level, the tip of the pole is on a straight line with the peak of island. Step back 127 steps from the rear pole. Eye on ground level also aligns with the tip of pole and tip of island. What is the height of the island, and what is the distance to the pole? Answer: The height of the island is 4 li and 55 steps, and it is 102 li and 150 steps from the pole. Method: Let the numerator equal to the height of pole multiplied by the separation of poles, let denominator be the difference of offsets, add the quotient to the height of pole to obtain the height of island. As the distance of front pole to the island could not be measured directly, Liu Hui set up two poles of same height at a known distance apart and made two measurements. The pole was perpendicular to the ground, eye view from ground level when the tip of pole was on a straight line sight with the peak of island, the distance of eye to the pole was called front offset = D G {\displaystyle DG} , similarly, the back offset = F H {\displaystyle FH} , difference of offsets = F H − D G {\displaystyle FH-DG} . Pole height C D = 30 {\displaystyle CD=30} chi Front pole offset D G = 123 {\displaystyle DG=123} steps Back pole offset F H = 127 {\displaystyle FH=127} steps Difference of offset = F H − D G {\displaystyle FH-DG} Distance between the poles = D F {\displaystyle DF} Height of island = A B {\displaystyle AB} Distance of front pole to island = B D {\displaystyle BD} Using his principle of inscribe rectangle in right angle triangle for △ A B G {\displaystyle \triangle ABG} and △ A B H {\displaystyle \triangle ABH} , he obtained: Height of island A B = C D × D F F H − D G + C D {\displaystyle AB={\tfrac {CD\times DF}{FH-DG}}+CD} Distance of front pole to island B D = D G × D F F H − D G {\displaystyle BD={\tfrac {DG\times DF}{FH-DG}}} . === Height of a hill top pine tree === A pine tree of unknown height grows on the hill. Set up two poles of 2 zhang each, one at front and one at the rear 50 steps in between. Let the rear pole align with the front pole. Step back 7 steps and 4 chi, view the tip of pine tree from the ground till it aligns in a straight line with the tip of the pole. Then view the tree trunk, the line of sight intersects the poles at 2 chi and 8 cun from its tip . Step back 8 steps and 5 chi from the rear pole, the view from ground also aligns with tree top and pole top. What is the height of the pine tree, and what is its distance from the pole ? Answer: the height of the pine is 12 zhang 2 chi 8 cun, the distance of mountain from the pole is 1 li and (28 + 4/7) steps. Method: let the numerator be the product of separation of the poles and intersection from tip of pole, let the denominator be the difference of offsets. Add the height of pole to the quotient to obtain the height of pine tree. === The size of a square city wall viewed afar === We are viewing from the south a square city of unknown size. Set up an east gnome and a west pole, 6 zhang apart, linked with a rope at eye level. Let the east pole aligned with the NE and SE corners. Move back 5 steps from the north gnome, watch the NW corner of the city, the line of sight intersects the rope at 2 zhang 2 chi and 6.5 cun from the east end. Step back northward 13 steps and 2 chi, watch the NW corner of the city, the line of sight just aligns with the west pole. What is the length of the square city, and what is its distance to the pole? Answer: The length of the square city is 3 li, 43 and 3/4 steps; the distance of the city to the pole is 4 li and 45 steps. === The depth of a ravine (using hence-forward cross-bars) === === The height of a building on a plain seen from a hill === === The breadth of a river-mouth seen from a distance on land === === The depth of a transparent pool === === The width of a river as seen from a hill === === The size of a city seen from a mountain === == Studies and translations == The 19th century British Protestant Christian missionary Alexander Wylie in his article "Jottings on the Sciences of Chinese Mathematics" published in North China Herald 1852, was the first person to introduce Sea Island Mathematical Manual to the West. In 1912, Japanese mathematic historian Yoshio Mikami published The Development of Mathematics in China and Japan, chapter 5 was dedicated to this book. A French mathematician translated the book into French in 1932. In 1986 Ang Tian Se and Frank Swetz translated Haidao into English. After comparing the development of surveying in China and the West, Frank Swetz concluded that "in the endeavours of mathematical surveying, China's accomplishments exceeded those realized in the West by about one thousand years." == References ==
Wikipedia:Haim Shapira#0	Haim-Moshe Shapira (Hebrew: חיים משה שפירא; 26 March 1902 – 16 July 1970) was a key Israeli politician in the early days of the state's existence. A signatory of Israel's declaration of independence, he served continuously as a minister from the country's foundation in 1948 until his death in 1970 apart from a brief spell in the late 1950s. == Biography == Haim-Moshe Shapira was born to Zalman Shapira and Rosa Krupnik in the Russian Empire in Grodno in what is today Belarus. He was educated in heder and a yeshiva, where he organised a youth group called Bnei Zion (Sons of Zion). He worked in the Education and Culture department of the National Jewish Council in Kaunas (now in Lithuania), and in 1919 set up the Young Mizrachi, which became a leading player in the religious Zionist youth movement in Lithuania. In 1922 he started work as a teacher at an ultra-orthodox school in Vilnius, and also served on the board of the Mizrachi group in the city. Between 1923 and 1924 he was active in the Young Mizrachi group in Warsaw, before attending a Rabbinical Seminary in Berlin between 1924 and 1925. In 1925 he was a delegate at the Zionist Congress, where he was elected onto the executive committee. In the same year he immigrated to Mandatory Palestine. In 1928 he was elected onto the Central Committee of the Hapoel HaMizrachi movement, and also served as a member of the World Mizrachi committee. In 1936 he was elected as a member of the Zionist Directorate and became a director of the Aliyah department of the Jewish Agency, a role he filled until 1948. In 1938 he was sent on a special mission to try to save Jews in Austria following the takeover by Nazi Germany. == Political career == Shapira was one of the people to sign Israel's declaration of independence, and was immediately appointed Minister of Health and Minister of Immigration in David Ben-Gurion's provisional government. In Israel's first elections in 1949, Shapira won a seat as a member of the United Religious Front bloc, an alliance of Agudat Yisrael, Poalei Agudat Yisrael, Mizrachi and his Hapoel HaMizrachi party. He was reappointed to his previous ministerial posts, and also became Minister of Internal Affairs. After the 1951 elections in which Hapoel HaMizrachi ran as an independent party, Shapira was appointed Minister of Internal Affairs and Minister of Religions. Following a cabinet reshuffle in 1952, he lost the Internal Affairs portfolio, but was appointed Minister of Welfare instead. Another reshuffle in 1955 saw him regain the Internal Affairs portfolio. The 1955 elections saw Mizrachi and Hapoel HaMizrachi run as a combined bloc, the National Religious Front, which later became the National Religious Party (NRP). Shapira was reappointed Minister of Religions and Minister of Welfare. In 1957 he was seriously injured by a hand grenade thrown into the Knesset by Moshe Dwek, but survived. He and all other NRP ministers resigned from the cabinet in July 1958, marking the only spell he spent out of office during his time in Israel. Following the 1959 elections, Shapira returned to the cabinet as Minister of Internal Affairs. After the early elections in 1961, he re-added the health portfolio to his roles. After elections in 1965 Shapira became just Internal Affairs Minister, a role he retained again after the 1969 elections. He died in office on 16 July 1970. === Positions on the Arab–Israeli conflict === Shapira belonged to the dovish camp of religious Zionism. This camp held considerable power before the Six-Day War, but was weakened significantly after the war in favor of hawkish Gush Emunim, whose spiritual leader was Rabbi Zvi Yehuda Kook. Before the state of Israel was founded, Shapira opposed the dissident military organizations, Irgun and the Stern Gang, although he resigned in response to the attack on the Irgun arms ship Altalena, ordered by David Ben-Gurion. Before the United Nations voted in favor of the United Nations Partition Plan for Palestine, Shapira took a minority position in his movement, supporting the plan. When military actions were debated during the 1948 Arab–Israeli War, Shapira voiced moderate and careful positions. After the state was founded, he supported giving a hundred thousand Palestinian refugees the option to return to Israel in exchange for a peace accord. Regarding the Deir Yassin and Qibya affairs, he said, "It's wrong from a Jewish perspective. Jews should not act like that". His opinion differed from others in his party, including Zalman Shragai. Shapira supported retreat from the Sinai Peninsula after the 1956 Suez Crisis. He said: "A bit more modesty, a bit less vanity and pride won't be unhelpful to us". In this context, he cited the decision of Rabbi Yochanan ben Zakai to negotiate with the Romans. Shapira was the most vocal of the ministers opposing a pre-emptive Israeli attack before the Six Day War. "How dare you go to war when all the circumstances are against us", he said to the IDF's chief of staff, Yitzhak Rabin. The other National Religious Party ministers joined Shapira in this stance. During the war, he opposed opening a new front in the Golan Heights. Despite his moderate worldview, he acted to include the right wing parties in the government on the eve of the war. This effort resulted in the establishment of a national unity government. After the war, Shapira voiced support for the settlement movement but warned that future peace agreements would be based on territorial concessions. However, he believed that discussions were meaningless as long as the Arabs refused to consider peace with Israel. He was more determined about Jerusalem – "The eternal capital should not be taken from the eternal nation". When the pupils of Rabbi Zvi Yehuda Kook expressed indignation at his moderate worldview, he replied, "we should not distance ourselves from our few friends in the world". He cited the opinion of Rabbi Joseph B. Soloveitchik, who said that questions of territorial concessions should be decided by those who are experts in the fields of defense and national security. When Moshe Dayan demanded the annexation of the West Bank to Israel, Shapira opposed it. Dayan wondered, "How can a religious Jew be so yielding". He remarked that Shapira's opinion differed from that of other party members. == Injury == On 29 October 1957 Moshe Dwek, a 26 year old disgruntled citizen, entered the main hall of the Knesset and threw a hand grenade towards the seats of the government ministers, wounding Shapira, Prime Minister David Ben-Gurion, Foreign Minister Golda Meir and Transportation Minister Moshe Carmel. Dwek claimed the Jewish Agency had not helped him sufficiently. Ben-Gurion was wounded in his hands and foot by shrapnel, Carmel suffered a broken arm, Meir was treated for minor injuries. However, Shapiro was more seriously injured and had to undergo several operations to remove shrapnel from his stomach and head. In September 1958, on the day before Yom Kippur, Dwek sent Shapira a letter of apology. Shapiro accepted the apology. == References == == External links == Haim-Moshe Shapira on the Knesset website
Wikipedia:Hairy ball theorem#0	The hairy ball theorem of algebraic topology (sometimes called the hedgehog theorem in Europe) states that there is no nonvanishing continuous tangent vector field on even-dimensional n-spheres. For the ordinary sphere, or 2‑sphere, if f is a continuous function that assigns a vector in ℝ3 to every point p on a sphere such that f(p) is always tangent to the sphere at p, then there is at least one pole, a point where the field vanishes (a p such that f(p) = 0). The theorem was first proved by Henri Poincaré for the 2-sphere in 1885, and extended to higher even dimensions in 1912 by Luitzen Egbertus Jan Brouwer. The theorem has been expressed colloquially as "you can't comb a hairy ball flat without creating a cowlick" or "you can't comb the hair on a coconut". == Counting zeros == Every zero of a vector field has a (non-zero) "index", and it can be shown that the sum of all of the indices at all of the zeros must be two, because the Euler characteristic of the 2-sphere is two. Therefore, there must be at least one zero. This is a consequence of the Poincaré–Hopf theorem. In the case of the torus, the Euler characteristic is 0; and it is possible to "comb a hairy doughnut flat". In this regard, it follows that for any compact regular 2-dimensional manifold with non-zero Euler characteristic, any continuous tangent vector field has at least one zero. == Application to computer graphics == A common problem in computer graphics is to generate a non-zero vector in ℝ3 that is orthogonal to a given non-zero vector. There is no single continuous function that can do this for all non-zero vector inputs. This is a corollary of the hairy ball theorem. To see this, consider the given vector as the radius of a sphere and note that finding a non-zero vector orthogonal to the given one is equivalent to finding a non-zero vector that is tangent to the surface of that sphere where it touches the radius. However, the hairy ball theorem says there exists no continuous function that can do this for every point on the sphere (equivalently, for every given vector). == Lefschetz connection == There is a closely related argument from algebraic topology, using the Lefschetz fixed-point theorem. Since the Betti numbers of a 2-sphere are 1, 0, 1, 0, 0, ... the Lefschetz number (total trace on homology) of the identity mapping is 2. By integrating a vector field we get (at least a small part of) a one-parameter group of diffeomorphisms on the sphere; and all of the mappings in it are homotopic to the identity. Therefore, they all have Lefschetz number 2, also. Hence they have fixed points (since the Lefschetz number is nonzero). Some more work would be needed to show that this implies there must actually be a zero of the vector field. It does suggest the correct statement of the more general Poincaré-Hopf index theorem. == Corollary == A consequence of the hairy ball theorem is that any continuous function that maps an even-dimensional sphere into itself has either a fixed point or a point that maps onto its own antipodal point. This can be seen by transforming the function into a tangential vector field as follows. Let s be the function mapping the sphere to itself, and let v be the tangential vector function to be constructed. For each point p, construct the stereographic projection of s(p) with p as the point of tangency. Then v(p) is the displacement vector of this projected point relative to p. According to the hairy ball theorem, there is a p such that v(p) = 0, so that s(p) = p. This argument breaks down only if there exists a point p for which s(p) is the antipodal point of p, since such a point is the only one that cannot be stereographically projected onto the tangent plane of p. A further corollary is that any even-dimensional projective space has the fixed-point property. This follows from the previous result by lifting continuous functions of R P 2 n {\displaystyle \mathbb {RP} ^{2n}} into itself to functions of S 2 n {\displaystyle S^{2n}} into itself. == Higher dimensions == The connection with the Euler characteristic χ suggests the correct generalisation: the 2n-sphere has no non-vanishing vector field for n ≥ 1. The difference between even and odd dimensions is that, because the only nonzero Betti numbers of the m-sphere are b0 and bm, their alternating sum χ is 2 for m even, and 0 for m odd. Indeed it is easy to see that an odd-dimensional sphere admits a non-vanishing tangent vector field through a simple process of considering coordinates of the ambient even-dimensional Euclidean space R 2 n {\displaystyle \mathbb {R} ^{2n}} in pairs. Namely, one may define a tangent vector field to S 2 n − 1 {\displaystyle S^{2n-1}} by specifying a vector field v : R 2 n → R 2 n {\displaystyle v:\mathbb {R} ^{2n}\to \mathbb {R} ^{2n}} given by v ( x 1 , … , x 2 n ) = ( x 2 , − x 1 , … , x 2 n , − x 2 n − 1 ) . {\displaystyle v(x_{1},\dots ,x_{2n})=(x_{2},-x_{1},\dots ,x_{2n},-x_{2n-1}).} In order for this vector field to restrict to a tangent vector field to the unit sphere S 2 n − 1 ⊂ R 2 n {\displaystyle S^{2n-1}\subset \mathbb {R} ^{2n}} it is enough to verify that the dot product with a unit vector of the form x = ( x 1 , … , x 2 n ) {\displaystyle x=(x_{1},\dots ,x_{2n})} satisfying ‖ x ‖ = 1 {\displaystyle \\|x\\|=1} vanishes. Due to the pairing of coordinates, one sees v ( x 1 , … , x 2 n ) ∙ ( x 1 , … , x 2 n ) = ( x 2 x 1 − x 1 x 2 ) + ⋯ + ( x 2 n x 2 n − 1 − x 2 n − 1 x 2 n ) = 0. {\displaystyle v(x_{1},\dots ,x_{2n})\bullet (x_{1},\dots ,x_{2n})=(x_{2}x_{1}-x_{1}x_{2})+\cdots +(x_{2n}x_{2n-1}-x_{2n-1}x_{2n})=0.} For a 2n-sphere, the ambient Euclidean space is R 2 n + 1 {\displaystyle \mathbb {R} ^{2n+1}} which is odd-dimensional, and so this simple process of pairing coordinates is not possible. Whilst this does not preclude the possibility that there may still exist a tangent vector field to the even-dimensional sphere which does not vanish, the hairy ball theorem demonstrates that in fact there is no way of constructing such a vector field. == Physical exemplifications == The hairy ball theorem has numerous physical exemplifications. For example, rotation of a rigid ball around its fixed axis gives rise to a continuous tangential vector field of velocities of the points located on its surface. This field has two zero-velocity points, which disappear after drilling the ball completely through its center, thereby converting the ball into the topological equivalent of a torus, a body to which the “hairy ball” theorem does not apply. The hairy ball theorem may be successfully applied for the analysis of the propagation of electromagnetic waves, in the case when the wave-front forms a surface, topologically equivalent to a sphere (the surface possessing the Euler characteristic χ = 2). At least one point on the surface at which vectors of electric and magnetic fields equal zero will necessarily appear. On certain 2-spheres of parameter space for electromagnetic waves in plasmas (or other complex media), these type of "cowlicks" or "bald points" also appear, which indicates that there exists topological excitation, i.e., robust waves that are immune to scattering and reflections, in the systems. If one idealizes the wind in the Earth's atmosphere as a tangent-vector field, then the hairy ball theorem implies that given any wind at all on the surface of the Earth, there must at all times be a cyclone somewhere. Note, however, that wind can move vertically in the atmosphere, so the idealized case is not meteorologically sound. (What is true is that for every "shell" of atmosphere around the Earth, there must be a point on the shell where the wind is not moving horizontally.) The theorem also has implications in computer modeling (including video game design), in which a common problem is to compute a non-zero 3-D vector that is orthogonal (i.e., perpendicular) to a given one; the hairy ball theorem implies that there is no single continuous function that accomplishes this task. == See also == Fixed-point theorem Intermediate value theorem Vector fields on spheres == References == == Further reading == Eisenberg, Murray; Guy, Robert (1979), "A Proof of the Hairy Ball Theorem", The American Mathematical Monthly, 86 (7): 571–574, doi:10.2307/2320587, JSTOR 2320587 Jarvis, Tyler; Tanton, James (2004), "The Hairy Ball Theorem via Sperner's Lemma", American Mathematical Monthly, 111 (7): 599–603, doi:10.1080/00029890.2004.11920120, JSTOR 4145162, S2CID 29784803 Richeson, David S. (2008), "Combing the Hair on a Coconut", Euler's Gem: The Polyhedron Formula and the Birth of Topology, Princeton University Press, pp. 202–218, ISBN 978-0-691-12677-7 == External links == Weisstein, Eric W. "Hairy Ball Theorem". MathWorld.
Wikipedia:Hajek projection#0	In statistics, Hájek projection of a random variable T {\displaystyle T} on a set of independent random vectors X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} is a particular measurable function of X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} that, loosely speaking, captures the variation of T {\displaystyle T} in an optimal way. It is named after the Czech statistician Jaroslav Hájek . == Definition == Given a random variable T {\displaystyle T} and a set of independent random vectors X 1 , … , X n {\displaystyle X_{1},\dots ,X_{n}} , the Hájek projection T ^ {\displaystyle {\hat {T}}} of T {\displaystyle T} onto { X 1 , … , X n } {\displaystyle \{X_{1},\dots ,X_{n}\}} is given by T ^ = E ⁡ ( T ) + ∑ i = 1 n [ E ⁡ ( T ∣ X i ) − E ⁡ ( T ) ] = ∑ i = 1 n E ⁡ ( T ∣ X i ) − ( n − 1 ) E ⁡ ( T ) {\displaystyle {\hat {T}}=\operatorname {E} (T)+\sum _{i=1}^{n}\left[\operatorname {E} (T\mid X_{i})-\operatorname {E} (T)\right]=\sum _{i=1}^{n}\operatorname {E} (T\mid X_{i})-(n-1)\operatorname {E} (T)} == Properties == Hájek projection T ^ {\displaystyle {\hat {T}}} is an L 2 {\displaystyle L^{2}} projection of T {\displaystyle T} onto a linear subspace of all random variables of the form ∑ i = 1 n g i ( X i ) {\displaystyle \sum _{i=1}^{n}g_{i}(X_{i})} , where g i : R d → R {\displaystyle g_{i}:\mathbb {R} ^{d}\to \mathbb {R} } are arbitrary measurable functions such that E ⁡ ( g i 2 ( X i ) ) < ∞ {\displaystyle \operatorname {E} (g_{i}^{2}(X_{i}))<\infty } for all i = 1 , … , n {\displaystyle i=1,\dots ,n} E ⁡ ( T ^ ∣ X i ) = E ⁡ ( T ∣ X i ) {\displaystyle \operatorname {E} ({\hat {T}}\mid X_{i})=\operatorname {E} (T\mid X_{i})} and hence E ⁡ ( T ^ ) = E ⁡ ( T ) {\displaystyle \operatorname {E} ({\hat {T}})=\operatorname {E} (T)} Under some conditions, asymptotic distributions of the sequence of statistics T n = T n ( X 1 , … , X n ) {\displaystyle T_{n}=T_{n}(X_{1},\dots ,X_{n})} and the sequence of its Hájek projections T ^ n = T ^ n ( X 1 , … , X n ) {\displaystyle {\hat {T}}_{n}={\hat {T}}_{n}(X_{1},\dots ,X_{n})} coincide, namely, if Var ⁡ ( T n ) / Var ⁡ ( T ^ n ) → 1 {\displaystyle \operatorname {Var} (T_{n})/\operatorname {Var} ({\hat {T}}_{n})\to 1} , then T n − E ⁡ ( T n ) Var ⁡ ( T n ) − T ^ n − E ⁡ ( T ^ n ) Var ⁡ ( T ^ n ) {\displaystyle {\frac {T_{n}-\operatorname {E} (T_{n})}{\sqrt {\operatorname {Var} (T_{n})}}}-{\frac {{\hat {T}}_{n}-\operatorname {E} ({\hat {T}}_{n})}{\sqrt {\operatorname {Var} ({\hat {T}}_{n})}}}} converges to zero in probability. == References ==
Wikipedia:Hajer Bahouri#0	Hajer Bahouri (born 30 March 1958, in Tunis) is a Franco-Tunisian mathematician who is interested in partial differential equations. She is Director of Research at the National Center for Scientific Research and the Laboratory of Analysis and Applied Mathematics at the University Paris-Est-Créteil-Val-de-Marne. == Career == From 1977, Bahouri studied mathematics at the University of Tunis, graduating in 1979; she then received the President's Award. She studied in Paris and obtained a Master of Advanced Studies in 1980 at the University Paris-Sud and a doctorate in 1982, under the direction of Serge Alinhac, with a thesis entitled Uniqueness and non-uniqueness of the Cauchy problem for real symbol operators. Then she devoted herself to research at École Polytechnique; from 1984 to 1988, she was a lecturer at the University of Paris-Sud and Rennes-I. In 1987, she obtained her doctoral degree (thesis) at the University of Paris-Sud (Uniqueness, non-uniqueness and Hölder continuity of the Cauchy problem for partial differential equations. Propagation of the wavefront Cρ for nonlinear equations). Starting in 1988 she was a professor at the Tunis University, where she directed, from 2003, the laboratory of partial differential equations. From 2002 to 2004, she was also a lecturer at École Polytechnique. Since 2010, she has been Research Director of the National Center for Scientific Research at the University Paris-Est-Créteil-Val-de-Marne (Laboratory of Analysis and Applied Mathematics). == Awards == In 2002, she was a guest speaker at the International Congress of Mathematicians in Beijing, with Jean-Yves Chemin (Quasilinear wave equations and microlocal analysis). In 2001, she received the Tunisian Medal of Merit and, in 2016, she won the Paul Doistau-Émile Blutet Prize. == Selected publications == " High Frequency Approximation of Solutions to Critical Nonlinear Wave Equations", 1997 Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël (2011). Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Vol. 343. Berlin, Heidelberg: Springer. ISBN 978-3-642-16830-7. OCLC 704397128. "Phase-space Analysis and Pseudodifferential Calculus on the Heisenberg Group", 2012 == References == This article incorporates text available under the CC BY-SA 3.0 license. This article incorporates text available under the CC BY-SA 3.0 license.
Wikipedia:Half-transitive graph#0	In the mathematical field of graph theory, a half-transitive graph is a graph that is both vertex-transitive and edge-transitive, but not symmetric. In other words, a graph is half-transitive if its automorphism group acts transitively upon both its vertices and its edges, but not on ordered pairs of linked vertices. Every connected symmetric graph must be vertex-transitive and edge-transitive, and the converse is true for graphs of odd degree, so that half-transitive graphs of odd degree do not exist. However, there do exist half-transitive graphs of even degree. The smallest half-transitive graph is the Holt graph, with degree 4 and 27 vertices. == References ==
Wikipedia:Halil Mete Soner#0	Halil Mete Soner is a Turkish American mathematician born in Ankara and is the Normal John Sollenberger Professor at Princeton University. Soner's research interests are nonlinear partial differential equations; asymptotic analysis of Ginzburg-Landau type systems, viscosity solutions, and mathematical finance. Currently he is working on mean field games and control and related nonlinear partial differential equations on Wasserstein spaces. == Education == After graduating from the Ankara Science High School, he started his university education at the Middle East Technical University in Ankara, later transferred to Boğaziçi University, Istanbul in 1977. He received a B.Sc. in mathematics and another in electrical engineering simultaneously in 1981, both in first-rank. Soner then attended Brown University in Providence, RI, U.S. on a research fellowship, where he obtained his M.Sc. (1983) and Ph.D. (1986) in applied mathematics. == Career == In 1985, Soner was research associate at the Institute for Mathematics and Applied Sciences in Minneapolis, MN and, assistant professor and then professor between 1986-1998 in the Department of Mathematical Sciences at the Carnegie Mellon University in Pittsburgh, PA. During 1997-1998 he was Research Associate at the Feza Gursey Institute for Basic Sciences in Istanbul and visiting professor of Mathematics at the Boğaziçi University, Istanbul and the University of Paris, Paris, France. From 1998 for two years, Soner was “Paul M. Whythes `55” Professor of Finance and Engineering in the Department of Operations Research and Financial Engineering at Princeton University. Then, he moved to Koç University where he served as the Dean of the College of Administrative Sciences and Economics until September 2007. From 2007 until 2009 he was the Isik Inselbag Professor of Finance in Sabancı University. From 2009 to 2019 he was a Professor of Financial Mathematics at ETH Zürich. Currently he is the chair of the department of Operations Research and Financial Engineering at Princeton. The meeting METE held in ETH Zurich in 2018 provides an account of his contributions. Soner co-authored a book, with Wendell Fleming, on viscosity solutions and stochastic control; Controlled Markov Processes and Viscosity Solutions (Springer-Verlag) in 1993, (second edition 2006). He authored or co-authored papers on nonlinear partial differential equations, viscosity solutions, stochastic optimal control, mathematical finance, martingale optimal transport and mean field games and control. == Awards == He received the TÜBITAK-TWAS Science award in 2002, was the recipient of an ERC Advanced Investigators Grant in 2009 and the Alexander von Humboldt Foundation Research Award in 2014 and was elected as a SIAM Fellow in 2015. == Personal life == He lives in Princeton with his wife Serpil and they have a son, Mehmet Ali. == References == == External links == His homepage
Wikipedia:Hall algebra#0	In mathematics, the Hall algebra is an associative algebra with a basis corresponding to isomorphism classes of finite abelian p-groups. It was first discussed by Steinitz (1901) but forgotten until it was rediscovered by Philip Hall (1959), both of whom published no more than brief summaries of their work. The Hall polynomials are the structure constants of the Hall algebra. The Hall algebra plays an important role in the theory of Masaki Kashiwara and George Lusztig regarding canonical bases in quantum groups. Ringel (1990) generalized Hall algebras to more general categories, such as the category of representations of a quiver. == Construction == A finite abelian p-group M is a direct sum of cyclic p-power components C p λ i , {\displaystyle C_{p^{\lambda _{i}}},} where λ = ( λ 1 , λ 2 , … ) {\displaystyle \lambda =(\lambda _{1},\lambda _{2},\ldots )} is a partition of n {\displaystyle n} called the type of M. Let g μ , ν λ ( p ) {\displaystyle g_{\mu ,\nu }^{\lambda }(p)} be the number of subgroups N of M such that N has type ν {\displaystyle \nu } and the quotient M/N has type μ {\displaystyle \mu } . Hall proved that the functions g are polynomial functions of p with integer coefficients. Thus we may replace p with an indeterminate q, which results in the Hall polynomials g μ , ν λ ( q ) ∈ Z [ q ] . {\displaystyle g_{\mu ,\nu }^{\lambda }(q)\in \mathbb {Z} [q].\,} Hall next constructs an associative ring H {\displaystyle H} over Z [ q ] {\displaystyle \mathbb {Z} [q]} , now called the Hall algebra. This ring has a basis consisting of the symbols u λ {\displaystyle u_{\lambda }} and the structure constants of the multiplication in this basis are given by the Hall polynomials: u μ u ν = ∑ λ g μ , ν λ ( q ) u λ . {\displaystyle u_{\mu }u_{\nu }=\sum _{\lambda }g_{\mu ,\nu }^{\lambda }(q)u_{\lambda }.\,} It turns out that H is a commutative ring, freely generated by the elements u 1 n {\displaystyle u_{\mathbf {1} ^{n}}} corresponding to the elementary p-groups. The linear map from H to the algebra of symmetric functions defined on the generators by the formula u 1 n ↦ q − n ( n − 1 ) / 2 e n {\displaystyle u_{\mathbf {1} ^{n}}\mapsto q^{-n(n-1)/2}e_{n}\,} (where en is the nth elementary symmetric function) uniquely extends to a ring homomorphism and the images of the basis elements u λ {\displaystyle u_{\lambda }} may be interpreted via the Hall–Littlewood symmetric functions. Specializing q to 0, these symmetric functions become Schur functions, which are thus closely connected with the theory of Hall polynomials. == References == Hall, Philip (1959), "The algebra of partitions", Proceedings of the 4th Canadian mathematical congress, Banff, pp. 147–159 George Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, Journal of the American Mathematical Society 4 (1991), no. 2, 365–421. Macdonald, Ian G. (1995), Symmetric functions and Hall polynomials, Oxford Mathematical Monographs (2nd ed.), The Clarendon Press Oxford University Press, ISBN 978-0-19-853489-1, MR 1354144 Ringel, Claus Michael (1990), "Hall algebras and quantum groups", Inventiones Mathematicae, 101 (3): 583–591, Bibcode:1990InMat.101..583R, doi:10.1007/BF01231516, MR 1062796, S2CID 120480847 Schiffmann, Olivier (2012), "Lectures on Hall algebras", Geometric methods in representation theory. II, Sémin. Congr., vol. 24-II, Paris: Soc. Math. France, pp. 1–141, arXiv:math/0611617, Bibcode:2006math.....11617S, MR 3202707 Steinitz, Ernst (1901), "Zur Theorie der Abel'schen Gruppen", Jahresbericht der Deutschen Mathematiker-Vereinigung, 9: 80–85
Wikipedia:Hall–Littlewood polynomials#0	In mathematics, the Hall–Littlewood polynomials are symmetric functions depending on a parameter t and a partition λ. They are Schur functions when t is 0 and monomial symmetric functions when t is 1 and are special cases of Macdonald polynomials. They were first defined indirectly by Philip Hall using the Hall algebra, and later defined directly by Dudley E. Littlewood (1961). == Definition == The Hall–Littlewood polynomial P is defined by P λ ( x 1 , … , x n ; t ) = ( ∏ i ≥ 0 ∏ j = 1 m ( i ) 1 − t 1 − t j ) ∑ w ∈ S n w ( x 1 λ 1 ⋯ x n λ n ∏ i < j x i − t x j x i − x j ) , {\displaystyle P_{\lambda }(x_{1},\ldots ,x_{n};t)=\left(\prod _{i\geq 0}\prod _{j=1}^{m(i)}{\frac {1-t}{1-t^{j}}}\right){\sum _{w\in S_{n}}w\left(x_{1}^{\lambda _{1}}\cdots x_{n}^{\lambda _{n}}\prod _{i<j}{\frac {x_{i}-tx_{j}}{x_{i}-x_{j}}}\right)},} where λ is a partition of at most n with elements λi, and m(i) elements equal to i, and Sn is the symmetric group of order n!. As an example, P 42 ( x 1 , x 2 ; t ) = x 1 4 x 2 2 + x 1 2 x 2 4 + ( 1 − t ) x 1 3 x 2 3 {\displaystyle P_{42}(x_{1},x_{2};t)=x_{1}^{4}x_{2}^{2}+x_{1}^{2}x_{2}^{4}+(1-t)x_{1}^{3}x_{2}^{3}} === Specializations === We have that P λ ( x ; 1 ) = m λ ( x ) {\displaystyle P_{\lambda }(x;1)=m_{\lambda }(x)} , P λ ( x ; 0 ) = s λ ( x ) {\displaystyle P_{\lambda }(x;0)=s_{\lambda }(x)} and P λ ( x ; − 1 ) = P λ ( x ) {\displaystyle P_{\lambda }(x;-1)=P_{\lambda }(x)} where the latter is the Schur P polynomials. == Properties == Expanding the Schur polynomials in terms of the Hall–Littlewood polynomials, one has s λ ( x ) = ∑ μ K λ μ ( t ) P μ ( x , t ) {\displaystyle s_{\lambda }(x)=\sum _{\mu }K_{\lambda \mu }(t)P_{\mu }(x,t)} where K λ μ ( t ) {\displaystyle K_{\lambda \mu }(t)} are the Kostka–Foulkes polynomials. Note that as t = 1 {\displaystyle t=1} , these reduce to the ordinary Kostka coefficients. A combinatorial description for the Kostka–Foulkes polynomials was given by Lascoux and Schützenberger, K λ μ ( t ) = ∑ T ∈ S S Y T ( λ , μ ) t c h a r g e ( T ) {\displaystyle K_{\lambda \mu }(t)=\sum _{T\in SSYT(\lambda ,\mu )}t^{\mathrm {charge} (T)}} where "charge" is a certain combinatorial statistic on semistandard Young tableaux, and the sum is taken over the set S S Y T ( λ , μ ) {\displaystyle SSYT(\lambda ,\mu )} of all semi-standard Young tableaux T with shape λ and type μ. == See also == Hall polynomial == References == I.G. Macdonald (1979). Symmetric Functions and Hall Polynomials. Oxford University Press. pp. 101–104. ISBN 0-19-853530-9. D.E. Littlewood (1961). "On certain symmetric functions". Proceedings of the London Mathematical Society. 43: 485–498. doi:10.1112/plms/s3-11.1.485. == External links == Weisstein, Eric W. "Hall–Littlewood Polynomial". MathWorld.
Wikipedia:Hamlet Isakhanli#0	Hamlet Abdulla Oglu Isayev (Azerbaijani: Hamlet Abdulla oğlu İsayev, IPA: [hɑmˈlet ɑbdulˈlɑ oɣˈlu iˈsɑjef]; born March 1, 1948) is an Azerbaijani polymath, mathematician, professor, poet, translator, entrepreneur, author, and specialist in science, culture, and the history of education. He founded Khazar University and served as president from 1991 to 2010, and is currently chairman of the university's board of directors and trustees. He founded Dunya School, founded a publishing house in Baku, Azerbaijan, and translates poetry, lectures, and edits. He is a founding member of the Eurasian Academy. "Hamlet Isakhanli" (Azerbaijani: Hamlet İsaxanlı, IPA: [hɑmˈlet isɑxɑnˈlɯ]) is his penname used in publications in the humanities, social sciences, and poetry. His works in mathematics are published under the surname Isayev (Г.А. Исаев, Hamlet Isaev or Hamlet A.Isayev). He is also known as Hamlet Isakhanli (Hamlet Isaxanli, Hamlet İsaxanlı)). As a polymath, Isakhanli's academic and literary works cover a broad range of fields. == Biography == Isakhanli finished high school with a gold medal and was admitted to the Faculty of Mechanics and Mathematics at Azerbaijan State University (BSU) in 1965. In 1970, he graduated with honors, and was admitted to postgraduate studies at BSU, before attending Lomonosov Moscow State University for graduate studies and research. In 1973, he received a Ph.D. in physical-mathematical sciences. From 1973 to 1983, he was a professor at Azerbaijan State Institute of Oil and Chemistry (currently Azerbaijan State Oil Academy), and chaired the department of mathematics at the Baku campus of Leningrad Institute of Economics and Finance. Isakhanli is a member of the Hajibayramli clan. He was married to Naila Isayeva. == Research == Isakhanli is a co-author and co-editor of the Khazar English-Azerbaijani Comprehensive Dictionary in six volumes, of which the first three volumes have been published. Isakhanli discusses in his writings the problems of poetry and philosophy, political science and international relations, history, and journalism. == Poetry == Isakhanli has translated poetry written in English, Russian and French, into Azerbaijani, particularly the poems of George Gordon Byron, W. Blake, Robert Herrick, and Gérard de Nerval, Vasily Zhukovsky, Evgeny Baratynsky, Fyodor Tyutchev, Afanasy Fet, Sergei Yesenin, Nikolay Gumilev, Anna Akhmatova, and Alla Akhundova. The lyrics of his poems have been used for composing songs and musical performances. == Reforms in education == Isakhanli studied educational organizations in the West, and in the late 1980s, wrote about an educational crisis in the Soviet Union and Azerbaijan, offering solutions in articles published in newspapers—and broadcast on radio and TV—in Azerbaijan and Russia. == Khazar University and Dunya School == Khazar University (previously Azerbaijan University), established on March 18, 1991, was the first private university, and one of the first officially established universities, stablished in Azerbaijan in the post-Soviet era. It features a Western-type curricula, a flexible credit accumulation system, a student-centered model of education, application of modern methods of management, broad and effective international relations, and local and foreign specialists. Dunya School, established by Isakhanli in 1998, is affiliated with Khazar University, and provides pre-school, primary, secondary, and high school level students with bilingual education in Azerbaijani and English languages, offering an International Baccalaureate and IB Diploma Program. == Awards == In 2010, Isakhanli received the National Khazar Award, in recognition of his contribution to Azerbaijani education, and for founding a university that meets international standards. == Publications == "The Life of the Scholar And Founder" by Fuad Tanriverdiyev, Baku, 1997.(in Azerbaijani) "What I Brought To This World" by Knyaz Aslan and Vahid Omarli, Baku, 2005 (in Azerbaijani and Russian). "In Search of Khazar" by Hamlet Isakhanli, Baku, 2006 (in English, Azerbaijani and Russian). "On the Education System in Transition Economy. A View From Azerbaijan" by Hamlet Isakhanli, Baku, 2006. Emily Van Buskirk, "Current Trends in Education in Azerbaijan - A discussion with Professor Hamlet Isakhanli", Caspian Studies Program, Harvard University, April 25, 2001, Accessed on April 14, 2006. Hamlet Isakhanli - Assembly of Science and Art. 1-10. Khazar University Press, Baku, 2008 (in Azerbaijani) Lyudmila Lavrova. "An invitation to traveling to the poetry world - "Contrasts"". Preface to the book by Hamlet Isakhanli, published in Moscow in Russian: Гамлет Исаханлы «Контрасты». Kнигa cтихотворений" (В переводах Аллы Ахундовой). "ИзoгpaфЪ", Москва, 2006). The Social and Political Context of Sovietization and Collectivization Period in the Central Transcaucasia and Isakhan Revolt (in Azeri) / Proceedings of the Conference "Socio-political Thought of the XX century". Baku, May 12, 1996. – Baku: Khazar University Press, 1996. – p. 1-16. Also // Georgia newspaper. – Tbilisi. – 1996. – No 34-42. Also // Khalg newspaper. – Baku. – 1996. – No 148-150. Fragments From the History of Thought and Education (in Azeri) // Khazar View, 1998–1999. No 57-64. Negotiations on Nagorno-Karabagh: where do we go from here? / Summary and transcript from a Panel Discussion held on April 23, 2001. Caspian Studies’ Program, John F. Kennedy School of Government, Harvard University. Minority Education Policy in Azerbaijan and Iran / Hamlet Isakhanli, Val D. Rust, Afgan Abdullayev, Marufa Madatova, Inna Grudskaya, Younes Vahdati // Journal of Azerbaijani Studies. – 2002. – Vol. 5. – No 3–4. – p. 3–78. Karakmazli, D. Taste of a Fig: Poems and Translations. Baku: Uyurd, 2005./Poems of Hamlet Isakhanli translated from Azeri into Russian – pp. 180–190. Perspectives on the United States (Hamlet Isakhanli, Anar Ahmadov) in "Global Perspectives on the United States: a Nation by Nation Survey". Vol. 1. – Great Barrington: Berkshire Publishing Group, 2007. – p. 31-33. Hamlet Isakhanli. What is Happening in the Modern World in the Field of Higher Education and How "The State Program to Reform the Higher Education System of the Azerbaijan Republic for the period of 2008-2012" May Best be Carried Out? (in Azeri). Khazar University Press, Baku, Azerbaijan, 2008. An Interesting Person (in Azeri) / Interview was led by X. Macidoglu // 525 newspaper. – 2008. –January 29 (No 15). – p. 4. Hamlet Isakhanli: "A creative person desires neither his house be high, nor his name..." /Interview with Hamlet Isakhanli (in Azeri) // 525 newspaper. – 2008. – October 11 (No.187). – p. 20. Nagiyev R. Formula of Vengeance (in Azeri) // Kommunist. – 1987. – July 2. – p. 6. Odinets V. At the Universities of Canada (in Russian) // Sovetskiy Ekonomist. – 1989. – January 27. – p. 4. Hamlet Isayev (Issakhanly) // Who's Who in the Former Soviet Union. – Surrey, England: Debrett's Peerage, Ltd., 1994. – p. 22-23. Seyidova I. "Khazar" - is a Lake which Can be Called a Sea (in Russian) // Consulting and Business. – Baku. – 1996. – p. 82-83. Who Dedicates Himself to Work and Regards it as a Field of Creativity will Never be Exhausted and Become Discouraged ... (in Azeri) / Interview was led by V. Omarov // Khazar View – 1997. – No 32. – p. 3-5. Mustafayev C. Poetry Gets Mingled with Intellect (A foreword to Hamlet Isakhanli's book "Contrasts") (in Azeri). – Baku: Khazar University Press, 2001. – 15 p. Suleymanli M. Inner Expression of the Great Word (in Azeri) // 525 newspaper – 2002. – September 3. – p. 7. and Hemistichs Taking Reader Together: While Reading the Book "Contrasts" by Hamlet Isakhanli (in Azeri) // Khazar View. – 2002. – September 15 (No 129). – p. 6-9. Nazirli K. Poetic World of the Well-known Scholar. On Hamlet Isakhanli's Book "Contrasts" (in Azeri) // Khazar View. – 2003.– No 152. – p. 14-15. Metin Turan. New Azerbaijan Poetry and Hamlet Isakhanli (in Turkish) // LİTTERA. Edebiyat Yazıları. – Ankara. – 2004. – v.14. – p. 163-168. Also (in Azeri) // 525 newspaper. – 2005. – June 28. – p. 7. A Spouse of "Contrasts" Author and the Target of the "Emotions Turned out into Poetry" (in Azeri) / Inter¬view was led by S. Gulten // "Qadin Dunyasi" newspaper. – 2004. – November 5–18 (No 21). – p. 9 Nazirli K. Hamlet Isakhanli's Sensitive Poetry (in Aze¬ri) // Khazar View. – 2005. – November 1 (No 196). – p. 21-23. Also // 525 newspaper. – 2005. – October 12. – p. 7. "When you are Straightforward, Sincere, and GoodWilled, you Also Feel an Admiration of Surrounding you People" (in Azeri) / X. Ahmadov, Z. Aliyeva // "Geopolitika" newspaper. – 2007. – No 38. – p. 12. Isayev (Isakhanli) Hamlet Abdulla (in English and Russian) / M.B. Babayev // Azerbaijan Mathematicians XX Century. – Baku: Oka Ofset, 2007. – p. 77. Mecidoglu Kh. The Beloved One of the God (in Azeri) // Khazar View. – 2008. – January 1 (No 242). – p. 8-11. Mecidoglu Kh. About my friend (in Azeri) // 525-ci qazet– December 20, 2008. – (No. 235). – p. 26. Mecidoglu Kh. If We don't Pass From Aspirations to Deeds/ Our Hopes will Vanish and Become in Vain (in Azeri) // Khazar View. – 2008. – March 1 (No 246). – p. 7-9. Nazarli T. The Way Taking its Start From the Com¬mand of Soul (in Azeri) // Azerbaijan. – 2008. – March 19 (No 61). Khazar View. – 2008. – 01 aprel (No 248). – p. 14-16. == References == == External links == Homepage of Hamlet Isakhanli (in English, in Azeri, in Russian, and in German) Khazar University – a research-oriented university (in English and in Azeri) Khazar University Institutional Repository (KUIR)
Wikipedia:Hamming space#0	In statistics and coding theory, a Hamming space is usually the set of all 2 N {\displaystyle 2^{N}} binary strings of length N, where different binary strings are considered to be adjacent when they differ only in one position. The total distance between any two binary strings is then the total number of positions at which the corresponding bits are different, called the Hamming distance. Hamming spaces are named after American mathematician Richard Hamming, who introduced the concept in 1950. They are used in the theory of coding signals and transmission. More generally, a Hamming space can be defined over any alphabet (set) Q as the set of words of a fixed length N with letters from Q. If Q is a finite field, then a Hamming space over Q is an N-dimensional vector space over Q. In the typical, binary case, the field is thus GF(2) (also denoted by Z2). In coding theory, if Q has q elements, then any subset C (usually assumed of cardinality at least two) of the N-dimensional Hamming space over Q is called a q-ary code of length N; the elements of C are called codewords. In the case where C is a linear subspace of its Hamming space, it is called a linear code. A typical example of linear code is the Hamming code. Codes defined via a Hamming space necessarily have the same length for every codeword, so they are called block codes when it is necessary to distinguish them from variable-length codes that are defined by unique factorization on a monoid. The Hamming distance endows a Hamming space with a metric, which is essential in defining basic notions of coding theory such as error detecting and error correcting codes. Hamming spaces over non-field alphabets have also been considered, especially over finite rings (most notably over Z4) giving rise to modules instead of vector spaces and ring-linear codes (identified with submodules) instead of linear codes. The typical metric used in this case the Lee distance. There exist a Gray isometry between Z 2 2 m {\displaystyle \mathbb {Z} _{2}^{2m}} (i.e. GF(22m)) with the Hamming distance and Z 4 m {\displaystyle \mathbb {Z} _{4}^{m}} (also denoted as GR(4,m)) with the Lee distance. == References ==
Wikipedia:Hannes Keller#0	Hannes Keller (20 September 1934 – 1 December 2022(2022-12-01) (aged 88)) was a Swiss physicist, mathematician, deep diving pioneer, and entrepreneur. In 1962, he reached a depth of 1,000 feet (300 m) in open ocean. In the 1970s through the 1980s, Keller made himself a name as an entrepreneur in the IT industry. Keller was also an amateur classical pianist who produced two CDs and occasionally performed for audiences of up to 2000 people. == Deep diving == Keller was born in Winterthur, Switzerland. He studied philosophy, mathematics, and theoretical physics at the University of Zurich. He became interested in deep diving and developed tables for mixed-gas decompression, supported by Albert A. Bühlmann who suggested suitable gases. Keller successfully tested his idea in Lake Zurich, where he reached a depth of 400 feet (120 m), and Lake Maggiore, where he reached a depth of 728 feet (222 m). On 3 December 1962, he set a new world record when he reached a depth of 1,000 feet (300 m) off the coast of Santa Catalina Island, California, together with Peter Small. This major achievement was overshadowed by the tragic end of the mission: Keller was lucky to survive while Peter Small and Chris Whittaker, a young UCLA student and supporting diver, lost their lives. In the following years, navies and hospitals bought decompression chambers constructed by Keller. == Career after diving == In the 1970s, Keller sold his own line of computers and in the 1980s became a leading vendor of IBM PCs in Switzerland. He developed a series of software products (Witchpen, Ways for Windows, and Wizardmaker) which provided automatic spell checking, literal machine translation, and macro recording. Keller used to run Visipix the largest fine art and photo museum online with 1.3 million exhibits, all with free copyrights for any use. After 2005, Keller was a full-time artist. In 2009, Keller joined the advisory board for the United States Historical Diving Society. Keller died on 1 December 2022 in Niederglatt, Switzerland, at the age of 88. == References == == Further reading == Keller, H (1975). "Diving 2001. presented at Oceans 2000 in 1973". South Pacific Underwater Medicine Society Journal. 5 (3). ISSN 0813-1988. OCLC 16986801. Archived from the original on 15 April 2013. Retrieved 12 September 2008.
Wikipedia:Hans Bruun Nielsen#0	Hans Bruun Nielsen (1943–2015) was a mathematician and Associate Professor of Technical University of Denmark specializing in Numerical analysis and the application of numerical methods. == Book == Eldén, Wittmeyer-Koch, Nielsen Introduction to Numerical Computation - analysis and MATLAB illustrations, 2004 Studentlitteratur ([1]), ISBN 91-44-03727-9 The book's Contents page Formulae from the book (E.g. order of Cholesky decomposition computations, maximum error in Chebyshev interpolation, and maximum stable step length using Runge-Kutta.) == External links == The Hans Bruun Nielsen Informatics and Mathematical Modelling page Nielsen's DACE software download page Design and Analysis of Computer Experiments is a freely distributable MATLAB toolbox for Kriging approximations to computer models. immoptibox: matlab toolbox for optimization and data fitting More software by HBN CV Obituary by Kaj Madsen
Wikipedia:Hans Heilbronn#0	Hans Arnold Heilbronn (8 October 1908 – 28 April 1975) was a mathematician. == Education == He was born into a German-Jewish family. He was a student at the universities of Berlin, Freiburg and Göttingen, where he met Edmund Landau, who supervised his doctorate. In his thesis, he improved a result of Hoheisel on the size of prime gaps. == Life == Heilbronn fled Germany for Britain in 1933 due to the rise of Nazism. He arrived in Cambridge, then found accommodation in Manchester and eventually was offered a position at Bristol University, where he stayed for about one and a half years. There he proved that the class number of the number field Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} tends to plus infinity as d {\displaystyle d} does, as well as, in collaboration with Edward Linfoot, that there are at most ten quadratic number fields of the form Q ( − d ) {\displaystyle \mathbb {Q} ({\sqrt {-d}})} , d {\displaystyle d} a natural number, with class number 1. On invitation of Louis Mordell he moved back to Manchester in 1934, but left again only one year later, accepting the Bevan Fellowship at Trinity College, Cambridge. In Cambridge Heilbronn published several joint papers with Harold Davenport, in one of which they devised a new variant of the Hardy-Littlewood circle method, now sometimes referred to as the Davenport-Heilbronn method, proving that for any indefinite diagonal form f {\displaystyle f} of degree k {\displaystyle k} in more than n = 2 k {\displaystyle n=2^{k}} variables whose coefficients are not all in rational ratio there exists x {\displaystyle x} in Z n {\displaystyle \mathbb {Z} ^{n}} such that \| f ( x ) \| {\displaystyle \|f(x)\|} is arbitrarily small. During the Second World War he was briefly interned as an enemy alien but released to serve in the British Army. In 1946 he returned to Bristol, becoming Henry Overton Wills Professor of Mathematics. He was elected a Fellow of the Royal Society in 1951 and was president of the London Mathematical Society from 1959 to 1961. Heilbronn and his wife moved to North America in 1964. He stayed at the California Institute of Technology for a while, then moved on to Toronto, where he was Professor of Mathematics at the University of Toronto from 1964 to 1975. He became a Canadian citizen in 1970. His PhD students include Inder Chowla, Tom Callahan and Albrecht Fröhlich. The Heilbronn Institute for Mathematical Research is named in his honour. == See also == Class number problem Deuring–Heilbronn phenomenon Heilbronn triangle problem Heilbronn set Heilbronn Institute for Mathematical Research List of German Canadians == References ==
Wikipedia:Hans Munthe-Kaas#0	Hans Zanna Munthe-Kaas (born 28 March 1961) is a Norwegian mathematician working at UiT The Arctic University of Norway and the University of Bergen. The main focus of is work lies in the area of computational mathematics in the borderland between pure and applied mathematics and computer science. He took his PhD at the Norwegian Institute of Technology in 1989, called to full Professor 1997 and has since 2005 been Professor of Mathematics. Since 2021 he is mainly working at UiT The Arctic University of Norway in Tromsø where he is co-director of the newly established Lie-Størmer Center for fundamental structures in computational and pure mathematics. == Research == The work of Munthe-Kaas is centred on applications of differential geometry and Lie group techniques in geometric integration and structure preserving algorithms for numerical integration of differential equations. A central aspect is the analysis of structure preservation by algebraic and combinatorial techniques (B-series and Lie–Butcher series). In the mid-1990s Munthe-Kaas developed what are now known as Runge–Kutta–Munthe-Kaas methods, a generalisation of Runge–Kutta methods to integration of differential equations evolving on Lie groups. The analysis of numerical Lie group integrators leads to the study of new types of formal power series for flows on manifolds (Lie–Butcher series). Lie–Butcher theory combines classical B-series with Lie-series. == Honors and awards == Munthe-Kaas received Exxon Mobil Award for best PhD at NTNU, 1989, and the Carl-Erik Frōberg Prize in Numerical Mathematics 1996 for the paper "Lie–Butcher theory for Runge–Kutta Methods". Munthe-Kaas is elected member of Academia Europaea, the Norwegian Academy of Science and Letters, the Royal Norwegian Society of Sciences and Letters and the Norwegian Academy of Technological Sciences. Munthe-Kaas is the chair of the international Abel prize committee (2018–2022), he is President of the Scientific Council of Centre International de Mathématiques Pures et Appliquées (CIMPA) (2017–present) and he is Editor-in-Chief of Journal Foundations of Computational Mathematics (2017–present). Munthe-Kaas was secretary of Foundations of Computational Mathematics (2005–2011) and member of the Board of the Abel Prize in Mathematics (2010–2018). == Personal life == Munthe-Kaas married Antonella Zanna, an Italian numerical analyst, in 1997; they have four children and a dog. == References ==
Wikipedia:Hans Rademacher#0	Hans Adolph Rademacher (German: [ˈʁaːdəmaxɐ]; 3 April 1892 – 7 February 1969) was a German-born American mathematician, known for work in mathematical analysis and number theory. == Biography == Rademacher received his Ph.D. in 1916 from Georg-August-Universität Göttingen; Constantin Carathéodory supervised his dissertation. In 1919, he became privatdozent under Constantin Carathéodory at University of Berlin. In 1922, he became an assistant professor at the University of Hamburg, where he supervised budding mathematicians like Theodor Estermann. He was dismissed from his position at the University of Breslau by the Nazis in 1933 due to his public support of the Weimar Republic, and emigrated from Europe in 1934. After leaving Germany, he moved to Philadelphia and worked at the University of Pennsylvania until his retirement in 1962; he held the Thomas A. Scott Professorship of Mathematics at Pennsylvania from 1956 to 1962. Rademacher had a number of well-known students, including George Andrews, Paul T. Bateman, Theodor Estermann and Emil Grosswald. == Research == Rademacher performed research in analytic number theory, mathematical genetics, the theory of functions of a real variable, and quantum theory. Most notably, he developed the theory of Dedekind sums. In 1937 Rademacher discovered an exact convergent series for the partition function P(n), the number of integer partitions of a number, improving upon Ramanujan's asymptotic non-convergent series and validating Ramanujan's supposition that an exact series representation existed. == Awards and honors == With his retirement from the University of Pennsylvania, a group of mathematicians provided the seed funding for The Hans A. Rademacher Instructorships, and honored him with an honorary degree as Doctor of Science. Rademacher is the co-author (with Otto Toeplitz) of the popular mathematics book The Enjoyment of Mathematics, published in German in 1930 and still in print. == Works == with Otto Toeplitz: Von Zahlen und Figuren. 1930. 2nd edn. 1933. Springer 2001, ISBN 3-540-63303-0. The Enjoyment of Mathematics. Von Zahlen und Figuren translated into English by Herbert Zuckerman, Princeton University Press, 1957 with Ernst Steinitz Vorlesungen über die Theorie der Polyeder- unter Einschluss der Elemente der Topologie. Springer 1932, 1976. Generalization of the Reciprocity Formula for Dedekind Sums. In: Duke Math. Journal. Vol. 21, 1954, pp. 391–397. Lectures on analytic number theory. 1955. Lectures on elementary number theory. Blaisdell, New York 1964, Krieger 1977. with Grosswald: Dedekind sums. Carus Mathematical Monographs 1972. Topics in analytic number theory. ed. Grosswald. Springer Verlag, 1973 (Grundlehren der mathematischen Wissenschaften). Collected papers. 2 vols. ed. Grosswald. MIT press, 1974. Higher mathematics from an elementary point of view. Birkhäuser 1983. == Further reading == George E. Andrews, David M. Bressoud, L. Alayne Parson (eds.) The Rademacher legacy to mathematics. American Mathematical Society, 1994. Lexikon bedeutender Mathematiker. Deutsch, Thun, Frankfurt am Main, ISBN 3-8171-1164-9. Tom Apostol: Introduction to Analytical number theory. Springer Tom Apostol: Modular functions and Dirichlet Series in Number Theory. Springer Berndt, Bruce C. (1992). "Hans Rademacher (1892–1969)" (PDF). Acta Arithmetica. 61 (3): 209–231. doi:10.4064/aa-61-3-209-225. Retrieved 2009-02-07. Obituary and list of publications. == See also == Hadamard transform Rademacher's contour Rademacher complexity Rademacher function Rademacher–Menchov theorem Rademacher's series Rademacher system Rademacher distribution Rademacher's theorem == References == == External links == O'Connor, John J.; Robertson, Edmund F. "Hans Rademacher". MacTutor History of Mathematics Archive. University of St Andrews. Hans Rademacher at the Mathematics Genealogy Project
Wikipedia:Hans Riesel#0	Hans Ivar Riesel (28 May 1929 in Stockholm – 21 December 2014) was a Swedish mathematician who discovered the 18th Mersenne prime in 1957 using the computer BESK: 23217-1, comprising 969 digits. He held the record for the largest known prime from 1957 to 1961, when Alexander Hurwitz discovered a larger one. Riesel also discovered the Riesel numbers as well as developing the Lucas–Lehmer–Riesel test. After having worked at the Swedish Board for Computing Machinery, he was awarded his Ph.D. from Stockholm University in 1969 for his thesis Contributions to numerical number theory, and in the same year joined the Royal Institute of Technology as a senior lecturer and associate professor. == Career and research == After completing an engineering degree at Kungliga Tekniska högskolan (KTH) in 1953, Riesel joined the state-run BESK computer project. Using nights and weekends on the machine he coded a self-checking Lucas–Lehmer routine in machine language, feeding exponents from punched paper tape; on 24 September 1957 the program halted with a zero residue for p = 3,217, identifying 23217 − 1 (969 digits) as the largest known prime of the day. Intrigued by numbers that resist such searches, he proved in 1956 that there exist odd integers k for which k·2ⁿ − 1 is composite for every n ≥ 1, inaugurating the study of what are now called Riesel numbers. He later generalised the Lucas–Lehmer test to these sequences, publishing the Lucas–Lehmer–Riesel test in 1981; this test remains the work-horse algorithm for the PrimeGrid distributed-computing project. Appointed senior lecturer at KTH in 1969, Riesel launched Sweden's first graduate course on computational number theory and supervised nine PhD theses on fast modular arithmetic and discrete logarithms. His monograph Prime Numbers and Computer Methods for Factorization (Birkhäuser, 1985; 2nd ed. 1994) synthesised that course and became a standard reference for early RSA cryptosystem implementers. Outside academia he co-founded the non-profit Stockholm Computer Association, promoting open access to idle mainframe time for scientific projects. He retired in 1994 but continued to maintain the Riesel Sieve webpages, coordinating a volunteer effort that has eliminated all but ten candidate Riesel numbers below 10,000. == Selected publications == Riesel, Hans (1994). Prime Numbers and Computer Methods for factorization. Progress in Mathematics. Vol. 126 (2nd ed.). Boston, MA: Birkhäuser. ISBN 0-8176-3743-5. Zbl 0821.11001. == References == == External links == Riesel's web page Obituary
Wikipedia:Hans Rådström#0	Hans Vilhem Rådström (1919–1970) was a Swedish mathematician who worked on complex analysis, continuous groups, convex sets, set-valued analysis, and game theory. From 1952, he was lektor (assistant professor) at Stockholm University, and from 1969, he was Professor of Applied Mathematics at Linköping University. == Early life == Hans Rådström was the son of the writer and editor Karl Johan Rådström, and the elder brother of the writer and journalist Pär Rådström. Rådström studied mathematics and obtained his Ph.D. under the joint supervision of Torsten Carleman and Fritz Carlson. His early work pertained to the theory of functions of a complex variable, particularly, complex dynamics. He was appointed lektor (assistant professor) at Stockholm University in 1952. Later, he was associated with the Royal Institute of Technology in Stockholm. In 1952 he became co-editor of the Scandinavian popular-mathematics journal Nordisk Matematisk Tidskrift. He also edited the Swedish edition of The Scientific American Book of Mathematical Puzzles and Diversions, a recreational mathematics book by Martin Gardner. == Set-valued analysis == Rådström was interested in Hilbert's fifth problem on the analyticity of the continuous operation of topological groups. The solution of this problem by Andrew Gleason used constructions of subsets of topological vector spaces, (rather than simply points), and inspired Rådström's research on set-valued analysis. He visited the Institute for Advanced Study (IAS) in Princeton from 1948 to 1950, where he co-organized a seminar on convexity. Together with Olof Hanner, who, like Rådström, would earn his Ph.D. from Stockholm University in 1952, he improved Werner Fenchel's version of Carathéodory's lemma. In the 1950s, he obtained important results on convex sets. He proved the Rådström embedding theorem, which implies that the collection of all nonempty compact convex subsets of a normed real vector-space (endowed with the Hausdorff distance) can be isometrically embedded as a convex cone in a normed real vector-space. Under the embedding, the nonempty compact convex sets are mapped to points in the range space. In Rådström's construction, this embedding is additive and positively homogeneous. Rådström's approach used ideas from the theory of topological semi-groups. Later, Lars Hörmander proved a variant of this theorem for locally convex topological vector spaces using the support function (of convex analysis); in Hörmander's approach, the range of the embedding was the Banach lattice L1, and the embedding was isotone. Rådström characterized the generators of continuous semigroups of sets as compact convex sets. == Students == Rådström's Ph.D. students included Per Enflo and Martin Ribe, both of whom wrote Ph.D. theses in functional analysis. In the uniform and Lipschitz categories of topological vector spaces, Enflo's results concerned spaces with local convexity, especially Banach spaces. In 1970, Hans Rådström died of a heart attack. Enflo supervised one of Rådström's Linköping students, Lars-Erik Andersson, from 1970–1971, helping him with his 1972 thesis, On connected subgroups of Banach spaces, on Hilbert's fifth problem for complete, normed spaces. The Swedish functional analyst Edgar Asplund, then Professor of Mathematics at Aarhus University in Denmark, assisted Ribe as supervisor of his 1972 thesis, before dying of cancer in 1974. Ribe's results concerned topological vector spaces without assuming local convexity; Ribe constructed a counter-example to naive extensions of the Hahn–Banach theorem to topological vector spaces that lack local convexity. == References == == External links == Hans Rådström at the Mathematics Genealogy Project Hans Rådström at Mathematical Reviews
Wikipedia:Hans Zantema#0	Hans Zantema (1956 - 28 January 2025) was a Dutch mathematician and computer scientist, and professor at Radboud University in Nijmegen, known for his work on termination analysis. == Biography == Born in Goingarijp, the Netherlands, Zantema received his PhD in algebraic number theory in 1983 at the University of Amsterdam under supervision of Hendrik Lenstra Jr. for the thesis, entitled "Integer Valued Polynomials in Algebraic Number Theory." After graduation, Zantema spent a few years of employment in the industry before he switched to computer science: from 1987 to 2000 at Utrecht University and since 2000 at Eindhoven University of Technology. Since 2007 he was also a part-time full professor at Radboud University in Nijmegen. His main achievements are in term rewriting systems, in particular in automatically proving termination of term rewriting. His name is attached to Zantema's problem, namely whether the string rewrite system 0011 -> 111000 terminates. He also contributed to the theory and especially the visualisation of streams. This led to the book "Playing with Infinity". == Selected publications == Zantema, Hans. 1983. Integer Valued Polynomials in Algebraic Number Theory. PhD thesis Zantema, Hans. 2007 De achterkant van Sudoku. Oplossen, programmeren en ontwerpen. Aramith Hersengymnastiek. Articles, a selection: Zantema, Hans. "Termination of term rewriting: interpretation and type elimination." Journal of Symbolic Computation 17.1 (1994): 23–50. Zantema, Hans. "Termination of term rewriting by semantic labelling." Fundamenta Informaticae 24.1 (1995): 89-105. Endrullis, Jörg, Johannes Waldmann, and Hans Zantema. "Matrix interpretations for proving termination of term rewriting Archived 2014-10-22 at the Wayback Machine." Journal of Automated Reasoning 40.2-3 (2008): 195–220. == References == == External links == Hans Zantema's homepage at tue.nl
Wikipedia:Hans-Bjørn Foxby#0	Hans-Bjørn Foxby (1947 – 2014) was a Danish mathematician, and a professor of mathematics at University of Copenhagen. Foxby classes are named after him. Foxby’s research was in commutative algebra. He died from Alzheimer's disease on 8 April 2014. == References ==
Wikipedia:Harald Helfgott#0	Harald Andrés Helfgott (born 25 November 1977) is a Peruvian mathematician working in number theory. Helfgott is a researcher (directeur de recherche) at the CNRS at the Institut Mathématique de Jussieu, Paris. He is best known for submitting a proof, now widely accepted but not yet fully published, of Goldbach's weak conjecture. == Early life and education == Helfgott was born on 25 November 1977 in Lima, Peru. He graduated from Brandeis University in 1998 (BA, summa cum laude). He received his Ph.D. from Princeton University in 2003 under the direction of Henryk Iwaniec and Peter Sarnak, with the thesis Root numbers and the parity problem. == Career == Helfgott was a post-doctoral Gibbs Assistant Professor at Yale University from 2003 to 2004. He was then a post-doctoral fellow at CRM–ISM–Université de Montréal from 2004 to 2006. Helfgott was a Lecturer, Senior Lecturer, and then Reader at the University of Bristol from 2006 to 2011. He has been a researcher at the CNRS since 2010, initially as a chargé de recherche première classe at the École normale supérieure before becoming a directeur de recherche deuxième classe at the Institut Mathématique de Jussieu in 2014. He was also an Alexander von Humboldt Professor at the University of Göttingen from 2015 to 2022. == Research == In 2013, he released two papers claiming to be a proof of Goldbach's weak conjecture; the claim is now broadly accepted. In 2017 Helfgott spotted a subtle error in the proof of the quasipolynomial time algorithm for the graph isomorphism problem that was announced by László Babai in 2015. Babai subsequently fixed his proof. == Awards == In 2008, Helfgott was awarded the Leverhulme Mathematics Prize for his work on number theory, diophantine geometry and group theory. In June 2010, Helfgott received the Whitehead Prize by the London Mathematical Society for his contributions to number theory, including work on Möbius sums in two variables, integral points on elliptic curves, and for his work on growth and expansion of multiplication of sets in SL2(Fp). In February 2011, Helfgott was awarded the Adams Prize jointly with Tom Sanders. In August 2013, Helfgott received an Honorary Professorship from National University of San Marcos in Lima, Peru. In 2014, he was an invited speaker at the International Congress of Mathematicians in Seoul and in 2015 he won a Humboldt Professorship. He was included in the 2019 class of fellows of the American Mathematical Society "for contributions to analytic number theory, additive combinatorics and combinatorial group theory". == Publications == "Zentralblatt MATH". == References == == External links == Photographs Archived 2013-10-06 at the Wayback Machine, July 2013. Harald Helfgott at the Mathematics Genealogy Project Videos of Harald Helfgott in the AV-Portal of the German National Library of Science and Technology
Wikipedia:Harald Niederreiter#0	Harald G. Niederreiter (born June 7, 1944) is an Austrian mathematician known for his work in discrepancy theory, algebraic geometry, quasi-Monte Carlo methods, and cryptography. == Education and career == Niederreiter was born on June 7, 1944, in Vienna, and grew up in Salzburg. He began studying mathematics at the University of Vienna in 1963, and finished his doctorate there in 1969, with a thesis on discrepancy in compact abelian groups supervised by Edmund Hlawka. He began his academic career as an assistant professor at the University of Vienna, but soon moved to Southern Illinois University. During this period he also visited the University of Illinois at Urbana-Champaign, Institute for Advanced Study, and University of California, Los Angeles. In 1978 he moved again, becoming the head of a new mathematics department at the University of the West Indies in Jamaica. In 1981 he returned to Austria for a post at the Austrian Academy of Sciences, where from 1989 to 2000 he served as director of the Institutes of Information Processing and Discrete Mathematics. In 2001 he became a professor at the National University of Singapore. In 2009 he returned to Austria again, to the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences. He also worked from 2010 to 2011 as a professor at the King Fahd University of Petroleum and Minerals in Saudi Arabia. == Research == Niederreiter's initial research interests were in the abstract algebra of abelian groups and finite fields, subjects also represented by his later book Finite Fields (with Rudolf Lidl, 1983). From his doctoral thesis onwards, he also incorporated discrepancy theory and the theory of uniformly distributed sets in metric spaces into his study of these subjects. In 1970, Niederreiter began to work on numerical analysis and random number generation, and in 1974 he published the book Uniform Distribution of Sequences. Combining his work on pseudorandom numbers with the Monte Carlo method, he did pioneering research in the quasi-Monte Carlo method in the late 1970s, and again later published a book on the topic, Random Number Generation and Quasi-Monte Carlo Methods (1995). Niederreiter's interests in pseudorandom numbers also led him to study stream ciphers in the 1980s, and this interest branched out into other areas of cryptography such as public key cryptography. The Niederreiter cryptosystem, an encryption system based on error-correcting codes that can also be used for digital signatures, was developed by him in 1986. His work in cryptography is represented by his book Algebraic Geometry in Coding Theory and Cryptography (with C. P. Xing, 2009). Returning to pure mathematics, Niederreiter has also made contributions to algebraic geometry with the discovery of many dense curves over finite fields, and published the book Rational Points on Curves over Finite Fields: Theory and Applications (with C. P. Xing, 2001). == Awards and honors == Niederreiter is a member of the Austrian Academy of Sciences and the German Academy of Sciences Leopoldina. In 1998 he was an invited speaker at the International Congress of Mathematicians, and won the Kardinal Innitzer Prize. He became a fellow of the American Mathematical Society in 2013. Niederreiter's book Random Number Generation and Quasi-Monte Carlo Methods won the Outstanding Simulation Publication Award. In 2014, a workshop in honor of Niederreiter's 70th birthday was held at the Johann Radon Institute for Computational and Applied Mathematics of the Austrian Academy of Sciences, and a Festschrift was published in his honor. == References ==
Wikipedia:Haran's diamond theorem#0	In mathematics, the Haran diamond theorem gives a general sufficient condition for a separable extension of a Hilbertian field to be Hilbertian. == Statement of the diamond theorem == Let K be a Hilbertian field and L a separable extension of K. Assume there exist two Galois extensions N and M of K such that L is contained in the compositum NM, but is contained in neither N nor M. Then L is Hilbertian. The name of the theorem comes from the pictured diagram of fields, and was coined by Jarden. == Some corollaries == === Weissauer's theorem === This theorem was firstly proved using non-standard methods by Weissauer. It was reproved by Fried using standard methods. The latter proof led Haran to his diamond theorem. Weissauer's theorem Let K be a Hilbertian field, N a Galois extension of K, and L a finite proper extension of N. Then L is Hilbertian. Proof using the diamond theorem If L is finite over K, it is Hilbertian; hence we assume that L/K is infinite. Let x be a primitive element for L/N, i.e., L = N(x). Let M be the Galois closure of K(x). Then all the assumptions of the diamond theorem are satisfied, hence L is Hilbertian. === Haran–Jarden condition === Another, preceding to the diamond theorem, sufficient permanence condition was given by Haran–Jarden: Theorem. Let K be a Hilbertian field and N, M two Galois extensions of K. Assume that neither contains the other. Then their compositum NM is Hilbertian. This theorem has a very nice consequence: Since the field of rational numbers, Q is Hilbertian (Hilbert's irreducibility theorem), we get that the algebraic closure of Q is not the compositum of two proper Galois extensions. == References == Haran, Dan (1999), "Hilbertian fields under separable algebraic extensions", Inventiones Mathematicae, 137 (1): 113–126, Bibcode:1999InMat.137..113H, doi:10.1007/s002220050325, MR 1702139, S2CID 120002473, Zbl 0933.12003. Fried, Michael D.; Jarden, Moshe (2008), Field Arithmetic, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3 Folge, vol. 11 (3rd revised ed.), Berlin: Springer-Verlag, ISBN 978-3-540-77269-9, MR 2445111, Zbl 1145.12001.
Wikipedia:Hardy field#0	In mathematics, a Hardy field is a field consisting of germs of real-valued functions at infinity that are closed under differentiation. They are named after the English mathematician G. H. Hardy. == Definition == Initially at least, Hardy fields were defined in terms of germs of real functions at infinity. Specifically we consider a collection H of functions that are defined for all large real numbers, that is functions f that map (u,∞) to the real numbers R, for some real number u depending on f. Here and in the rest of the article we say a function has a property "eventually" if it has the property for all sufficiently large x, so for example we say a function f in H is eventually zero if there is some real number U such that f(x) = 0 for all x ≥ U. We can form an equivalence relation on H by saying f is equivalent to g if and only if f − g is eventually zero. The equivalence classes of this relation are called germs at infinity. If H forms a field under the usual addition and multiplication of functions then so will H modulo this equivalence relation under the induced addition and multiplication operations. Moreover, if every function in H is eventually differentiable and the derivative of any function in H is also in H then H modulo the above equivalence relation is called a Hardy field. Elements of a Hardy field are thus equivalence classes and should be denoted, say, [f]∞ to denote the class of functions that are eventually equal to the representative function f. However, in practice the elements are normally just denoted by the representatives themselves, so instead of [f]∞ one would just write f. == Examples == If F is a subfield of R then we can consider it as a Hardy field by considering the elements of F as constant functions, that is by considering the number α in F as the constant function fα that maps every x in R to α. This is a field since F is, and since the derivative of every function in this field is 0 which must be in F it is a Hardy field. A less trivial example of a Hardy field is the field of rational functions on R, denoted R(x). This is the set of functions of the form P(x)/Q(x) where P and Q are polynomials with real coefficients. Since the polynomial Q can have only finitely many zeros by the fundamental theorem of algebra, such a rational function will be defined for all sufficiently large x, specifically for all x larger than the largest real root of Q. Adding and multiplying rational functions gives more rational functions, and the quotient rule shows that the derivative of rational function is again a rational function, so R(x) forms a Hardy field. Another example is the field of functions that can be expressed using the standard arithmetic operations, exponents, and logarithms, and are well-defined on some interval of the form ( x , ∞ ) {\displaystyle (x,\infty )} . Such functions are sometimes called Hardy L-functions. Much bigger Hardy fields (that contain Hardy L-functions as a subfield) can be defined using transseries. == Properties == Every element of a Hardy field is eventually either strictly positive, strictly negative, or zero. This follows fairly immediately from the facts that the elements in a Hardy field are eventually differentiable and hence continuous and eventually either have a multiplicative inverse or are zero. This means periodic functions such as the sine and cosine functions cannot exist in Hardy fields. This avoidance of periodic functions also means that every element in a Hardy field has a (possibly infinite) limit at infinity, so if f is an element of H, then lim x → ∞ f ( x ) {\displaystyle \lim _{x\rightarrow \infty }f(x)} exists in R ∪ {−∞,+∞}. It also means we can place an ordering on H by saying f < g if g − f is eventually strictly positive. Note that this is not the same as stating that f < g if the limit of f is less than the limit of g. For example, if we consider the germs of the identity function f(x) = x and the exponential function g(x) = ex then since g(x) − f(x) > 0 for all x we have that g > f. But they both tend to infinity. In this sense the ordering tells us how quickly all the unbounded functions diverge to infinity. Even finite limits being equal is not enough: consider f(x) = 1/x and g(x) = 0. == In model theory == The modern theory of Hardy fields doesn't restrict to real functions but to those defined in certain structures expanding real closed fields. Indeed, if R is an o-minimal expansion of a field, then the set of unary definable functions in R that are defined for all sufficiently large elements forms a Hardy field denoted H(R). The properties of Hardy fields in the real setting still hold in this more general setting. == References ==
Wikipedia:Hardy notation#0	Big O notation is a mathematical notation that describes the limiting behavior of a function when the argument tends towards a particular value or infinity. Big O is a member of a family of notations invented by German mathematicians Paul Bachmann, Edmund Landau, and others, collectively called Bachmann–Landau notation or asymptotic notation. The letter O was chosen by Bachmann to stand for Ordnung, meaning the order of approximation. In computer science, big O notation is used to classify algorithms according to how their run time or space requirements grow as the input size grows. In analytic number theory, big O notation is often used to express a bound on the difference between an arithmetical function and a better understood approximation; a famous example of such a difference is the remainder term in the prime number theorem. Big O notation is also used in many other fields to provide similar estimates. Big O notation characterizes functions according to their growth rates: different functions with the same asymptotic growth rate may be represented using the same O notation. The letter O is used because the growth rate of a function is also referred to as the order of the function. A description of a function in terms of big O notation usually only provides an upper bound on the growth rate of the function. Associated with big O notation are several related notations, using the symbols o, Ω, ω, and Θ, to describe other kinds of bounds on asymptotic growth rates. == Formal definition == Let f , {\displaystyle f,} the function to be estimated, be a real or complex valued function, and let g , {\displaystyle g,} the comparison function, be a real valued function. Let both functions be defined on some unbounded subset of the positive real numbers, and g ( x ) {\displaystyle g(x)} be non-zero (often, but not necessarily, strictly positive) for all large enough values of x . {\displaystyle x.} One writes f ( x ) = O ( g ( x ) ) as x → ∞ {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}x\to \infty } and it is read " f ( x ) {\displaystyle f(x)} is big O of g ( x ) {\displaystyle g(x)} " or more often " f ( x ) {\displaystyle f(x)} is of the order of g ( x ) {\displaystyle g(x)} " if the absolute value of f ( x ) {\displaystyle f(x)} is at most a positive constant multiple of the absolute value of g ( x ) {\displaystyle g(x)} for all sufficiently large values of x . {\displaystyle x.} That is, f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}} if there exists a positive real number M {\displaystyle M} and a real number x 0 {\displaystyle x_{0}} such that \| f ( x ) \| ≤ M \| g ( x ) \| for all x ≥ x 0 . {\displaystyle \|f(x)\|\leq M\ \|g(x)\|\quad {\text{ for all }}x\geq x_{0}~.} In many contexts, the assumption that we are interested in the growth rate as the variable x {\displaystyle \ x\ } goes to infinity or to zero is left unstated, and one writes more simply that f ( x ) = O ( g ( x ) ) . {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}.} The notation can also be used to describe the behavior of f {\displaystyle f} near some real number a {\displaystyle a} (often, a = 0 {\displaystyle a=0} ): we say f ( x ) = O ( g ( x ) ) as x → a {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a} if there exist positive numbers δ {\displaystyle \delta } and M {\displaystyle M} such that for all defined x {\displaystyle x} with 0 < \| x − a \| < δ , {\displaystyle 0<\|x-a\|<\delta ,} \| f ( x ) \| ≤ M \| g ( x ) \| . {\displaystyle \|f(x)\|\leq M\|g(x)\|.} As g ( x ) {\displaystyle g(x)} is non-zero for adequately large (or small) values of x , {\displaystyle x,} both of these definitions can be unified using the limit superior: f ( x ) = O ( g ( x ) ) as x → a {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}\quad {\text{ as }}\ x\to a} if lim sup x → a \| f ( x ) \| \| g ( x ) \| < ∞ . {\displaystyle \limsup _{x\to a}{\frac {\left\|f(x)\right\|}{\left\|g(x)\right\|}}<\infty .} And in both of these definitions the limit point a {\displaystyle a} (whether ∞ {\displaystyle \infty } or not) is a cluster point of the domains of f {\displaystyle f} and g , {\displaystyle g,} i. e., in every neighbourhood of a {\displaystyle a} there have to be infinitely many points in common. Moreover, as pointed out in the article about the limit inferior and limit superior, the lim sup x → a {\displaystyle \textstyle \limsup _{x\to a}} (at least on the extended real number line) always exists. In computer science, a slightly more restrictive definition is common: f {\displaystyle f} and g {\displaystyle g} are both required to be functions from some unbounded subset of the positive integers to the nonnegative real numbers; then f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O{\bigl (}g(x){\bigr )}} if there exist positive integer numbers M {\displaystyle M} and n 0 {\displaystyle n_{0}} such that \| f ( n ) \| ≤ M \| g ( n ) \| {\displaystyle \|f(n)\|\leq M\|g(n)\|} for all n ≥ n 0 . {\displaystyle n\geq n_{0}.} == Example == In typical usage the O notation is asymptotical, that is, it refers to very large x. In this setting, the contribution of the terms that grow "most quickly" will eventually make the other ones irrelevant. As a result, the following simplification rules can be applied: If f(x) is a sum of several terms, if there is one with largest growth rate, it can be kept, and all others omitted. If f(x) is a product of several factors, any constants (factors in the product that do not depend on x) can be omitted. For example, let f(x) = 6x4 − 2x3 + 5, and suppose we wish to simplify this function, using O notation, to describe its growth rate as x approaches infinity. This function is the sum of three terms: 6x4, −2x3, and 5. Of these three terms, the one with the highest growth rate is the one with the largest exponent as a function of x, namely 6x4. Now one may apply the second rule: 6x4 is a product of 6 and x4 in which the first factor does not depend on x. Omitting this factor results in the simplified form x4. Thus, we say that f(x) is a "big O" of x4. Mathematically, we can write f(x) = O(x4). One may confirm this calculation using the formal definition: let f(x) = 6x4 − 2x3 + 5 and g(x) = x4. Applying the formal definition from above, the statement that f(x) = O(x4) is equivalent to its expansion, \| f ( x ) \| ≤ M x 4 {\displaystyle \|f(x)\|\leq Mx^{4}} for some suitable choice of a real number x0 and a positive real number M and for all x > x0. To prove this, let x0 = 1 and M = 13. Then, for all x > x0: \| 6 x 4 − 2 x 3 + 5 \| ≤ 6 x 4 + \| 2 x 3 \| + 5 ≤ 6 x 4 + 2 x 4 + 5 x 4 = 13 x 4 {\displaystyle {\begin{aligned}\|6x^{4}-2x^{3}+5\|&\leq 6x^{4}+\|2x^{3}\|+5\\&\leq 6x^{4}+2x^{4}+5x^{4}\\&=13x^{4}\end{aligned}}} so \| 6 x 4 − 2 x 3 + 5 \| ≤ 13 x 4 . {\displaystyle \|6x^{4}-2x^{3}+5\|\leq 13x^{4}.} == Use == Big O notation has two main areas of application: In mathematics, it is commonly used to describe how closely a finite series approximates a given function, especially in the case of a truncated Taylor series or asymptotic expansion. In computer science, it is useful in the analysis of algorithms. In both applications, the function g(x) appearing within the O(·) is typically chosen to be as simple as possible, omitting constant factors and lower order terms. There are two formally close, but noticeably different, usages of this notation: infinite asymptotics infinitesimal asymptotics. This distinction is only in application and not in principle, however—the formal definition for the "big O" is the same for both cases, only with different limits for the function argument. === Infinite asymptotics === Big O notation is useful when analyzing algorithms for efficiency. For example, the time (or the number of steps) it takes to complete a problem of size n might be found to be T(n) = 4n2 − 2n + 2. As n grows large, the n2 term will come to dominate, so that all other terms can be neglected —for instance when n = 500, the term 4n2 is 1000 times as large as the 2n term. Ignoring the latter would have negligible effect on the expression's value for most purposes. Further, the coefficients become irrelevant if we compare to any other order of expression, such as an expression containing a term n3 or n4. Even if T(n) = 1,000,000n2, if U(n) = n3, the latter will always exceed the former once n grows larger than 1,000,000, viz. T(1,000,000) = 1,000,0003 = U(1,000,000). Additionally, the number of steps depends on the details of the machine model on which the algorithm runs, but different types of machines typically vary by only a constant factor in the number of steps needed to execute an algorithm. So the big O notation captures what remains: we write either T ( n ) = O ( n 2 ) {\displaystyle T(n)=O(n^{2})} or T ( n ) ∈ O ( n 2 ) {\displaystyle T(n)\in O(n^{2})} and say that the algorithm has order of n2 time complexity. The sign "=" is not meant to express "is equal to" in its normal mathematical sense, but rather a more colloquial "is", so the second expression is sometimes considered more accurate (see the "Equals sign" discussion below) while the first is considered by some as an abuse of notation. === Infinitesimal asymptotics === Big O can also be used to describe the error term in an approximation to a mathematical function. The most significant terms are written explicitly, and then the least-significant terms are summarized in a single big O term. Consider, for example, the exponential series and two expressions of it that are valid when x is small: e x = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + ⋯ for all finite x = 1 + x + x 2 2 + O ( x 3 ) as x → 0 = 1 + x + O ( x 2 ) as x → 0 {\displaystyle {\begin{aligned}e^{x}&=1+x+{\frac {x^{2}}{2!}}+{\frac {x^{3}}{3!}}+{\frac {x^{4}}{4!}}+\dotsb &&{\text{for all finite }}x\\[4pt]&=1+x+{\frac {x^{2}}{2}}+O(x^{3})&&{\text{as }}x\to 0\\[4pt]&=1+x+O(x^{2})&&{\text{as }}x\to 0\end{aligned}}} The middle expression (the one with O(x3)) means the absolute-value of the error ex − (1 + x + x2/2) is at most some constant times \|x3!\| when x is close enough to 0. == Properties == If the function f can be written as a finite sum of other functions, then the fastest growing one determines the order of f(n). For example, f ( n ) = 9 log ⁡ n + 5 ( log ⁡ n ) 4 + 3 n 2 + 2 n 3 = O ( n 3 ) as n → ∞ . {\displaystyle f(n)=9\log n+5(\log n)^{4}+3n^{2}+2n^{3}=O(n^{3})\qquad {\text{as }}n\to \infty .} In particular, if a function may be bounded by a polynomial in n, then as n tends to infinity, one may disregard lower-order terms of the polynomial. The sets O(nc) and O(cn) are very different. If c is greater than one, then the latter grows much faster. A function that grows faster than nc for any c is called superpolynomial. One that grows more slowly than any exponential function of the form cn is called subexponential. An algorithm can require time that is both superpolynomial and subexponential; examples of this include the fastest known algorithms for integer factorization and the function nlog n. We may ignore any powers of n inside of the logarithms. The set O(log n) is exactly the same as O(log(nc)). The logarithms differ only by a constant factor (since log(nc) = c log n) and thus the big O notation ignores that. Similarly, logs with different constant bases are equivalent. On the other hand, exponentials with different bases are not of the same order. For example, 2n and 3n are not of the same order. Changing units may or may not affect the order of the resulting algorithm. Changing units is equivalent to multiplying the appropriate variable by a constant wherever it appears. For example, if an algorithm runs in the order of n2, replacing n by cn means the algorithm runs in the order of c2n2, and the big O notation ignores the constant c2. This can be written as c2n2 = O(n2). If, however, an algorithm runs in the order of 2n, replacing n with cn gives 2cn = (2c)n. This is not equivalent to 2n in general. Changing variables may also affect the order of the resulting algorithm. For example, if an algorithm's run time is O(n) when measured in terms of the number n of digits of an input number x, then its run time is O(log x) when measured as a function of the input number x itself, because n = O(log x). === Product === f 1 = O ( g 1 ) and f 2 = O ( g 2 ) ⇒ f 1 f 2 = O ( g 1 g 2 ) {\displaystyle f_{1}=O(g_{1}){\text{ and }}f_{2}=O(g_{2})\Rightarrow f_{1}f_{2}=O(g_{1}g_{2})} f ⋅ O ( g ) = O ( f g ) {\displaystyle f\cdot O(g)=O(fg)} === Sum === If f 1 = O ( g 1 ) {\displaystyle f_{1}=O(g_{1})} and f 2 = O ( g 2 ) {\displaystyle f_{2}=O(g_{2})} then f 1 + f 2 = O ( max ( \| g 1 \| , \| g 2 \| ) ) {\displaystyle f_{1}+f_{2}=O(\max(\|g_{1}\|,\|g_{2}\|))} . It follows that if f 1 = O ( g ) {\displaystyle f_{1}=O(g)} and f 2 = O ( g ) {\displaystyle f_{2}=O(g)} then f 1 + f 2 ∈ O ( g ) {\displaystyle f_{1}+f_{2}\in O(g)} . === Multiplication by a constant === Let k be a nonzero constant. Then O ( \| k \| ⋅ g ) = O ( g ) {\displaystyle O(\|k\|\cdot g)=O(g)} . In other words, if f = O ( g ) {\displaystyle f=O(g)} , then k ⋅ f = O ( g ) . {\displaystyle k\cdot f=O(g).} == Multiple variables == Big O (and little o, Ω, etc.) can also be used with multiple variables. To define big O formally for multiple variables, suppose f {\displaystyle f} and g {\displaystyle g} are two functions defined on some subset of R n {\displaystyle \mathbb {R} ^{n}} . We say f ( x ) is O ( g ( x ) ) as x → ∞ {\displaystyle f(\mathbf {x} ){\text{ is }}O(g(\mathbf {x} ))\quad {\text{ as }}\mathbf {x} \to \infty } if and only if there exist constants M {\displaystyle M} and C > 0 {\displaystyle C>0} such that \| f ( x ) \| ≤ C \| g ( x ) \| {\displaystyle \|f(\mathbf {x} )\|\leq C\|g(\mathbf {x} )\|} for all x {\displaystyle \mathbf {x} } with x i ≥ M {\displaystyle x_{i}\geq M} for some i . {\displaystyle i.} Equivalently, the condition that x i ≥ M {\displaystyle x_{i}\geq M} for some i {\displaystyle i} can be written ‖ x ‖ ∞ ≥ M {\displaystyle \\|\mathbf {x} \\|_{\infty }\geq M} , where ‖ x ‖ ∞ {\displaystyle \\|\mathbf {x} \\|_{\infty }} denotes the Chebyshev norm. For example, the statement f ( n , m ) = n 2 + m 3 + O ( n + m ) as n , m → ∞ {\displaystyle f(n,m)=n^{2}+m^{3}+O(n+m)\quad {\text{ as }}n,m\to \infty } asserts that there exist constants C and M such that \| f ( n , m ) − ( n 2 + m 3 ) \| ≤ C \| n + m \| {\displaystyle \|f(n,m)-(n^{2}+m^{3})\|\leq C\|n+m\|} whenever either m ≥ M {\displaystyle m\geq M} or n ≥ M {\displaystyle n\geq M} holds. This definition allows all of the coordinates of x {\displaystyle \mathbf {x} } to increase to infinity. In particular, the statement f ( n , m ) = O ( n m ) as n , m → ∞ {\displaystyle f(n,m)=O(n^{m})\quad {\text{ as }}n,m\to \infty } (i.e., ∃ C ∃ M ∀ n ∀ m ⋯ {\displaystyle \exists C\,\exists M\,\forall n\,\forall m\,\cdots } ) is quite different from ∀ m : f ( n , m ) = O ( n m ) as n → ∞ {\displaystyle \forall m\colon ~f(n,m)=O(n^{m})\quad {\text{ as }}n\to \infty } (i.e., ∀ m ∃ C ∃ M ∀ n ⋯ {\displaystyle \forall m\,\exists C\,\exists M\,\forall n\,\cdots } ). Under this definition, the subset on which a function is defined is significant when generalizing statements from the univariate setting to the multivariate setting. For example, if f ( n , m ) = 1 {\displaystyle f(n,m)=1} and g ( n , m ) = n {\displaystyle g(n,m)=n} , then f ( n , m ) = O ( g ( n , m ) ) {\displaystyle f(n,m)=O(g(n,m))} if we restrict f {\displaystyle f} and g {\displaystyle g} to [ 1 , ∞ ) 2 {\displaystyle [1,\infty )^{2}} , but not if they are defined on [ 0 , ∞ ) 2 {\displaystyle [0,\infty )^{2}} . This is not the only generalization of big O to multivariate functions, and in practice, there is some inconsistency in the choice of definition. == Matters of notation == === Equals sign === The statement "f(x) is O[g(x)]" as defined above is usually written as f(x) = O[g(x)]. Some consider this to be an abuse of notation, since the use of the equals sign could be misleading as it suggests a symmetry that this statement does not have. As de Bruijn says, O[x] = O[x2] is true but O[x2] = O[x] is not. Knuth describes such statements as "one-way equalities", since if the sides could be reversed, "we could deduce ridiculous things like n = n2 from the identities n = O[n2] and n2 = O[n2]". In another letter, Knuth also pointed out that the equality sign is not symmetric with respect to such notations [as, in this notation,] mathematicians customarily use the '=' sign as they use the word 'is' in English: Aristotle is a man, but a man isn't necessarily Aristotle. For these reasons, it would be more precise to use set notation and write f(x) ∈ O[g(x)] – read as: "f(x) is an element of O[g(x)]", or "f(x) is in the set O[g(x)]" – thinking of O[g(x)] as the class of all functions h(x) such that \|h(x)\| ≤ C \|g(x)\| for some positive real number C. However, the use of the equals sign is customary. === Other arithmetic operators === Big O notation can also be used in conjunction with other arithmetic operators in more complicated equations. For example, h(x) + O(f(x)) denotes the collection of functions having the growth of h(x) plus a part whose growth is limited to that of f(x). Thus, g ( x ) = h ( x ) + O ( f ( x ) ) {\displaystyle g(x)=h(x)+O(f(x))} expresses the same as g ( x ) − h ( x ) = O ( f ( x ) ) . {\displaystyle g(x)-h(x)=O(f(x)).} ==== Example ==== Suppose an algorithm is being developed to operate on a set of n elements. Its developers are interested in finding a function T(n) that will express how long the algorithm will take to run (in some arbitrary measurement of time) in terms of the number of elements in the input set. The algorithm works by first calling a subroutine to sort the elements in the set and then perform its own operations. The sort has a known time complexity of O(n2), and after the subroutine runs the algorithm must take an additional 55n3 + 2n + 10 steps before it terminates. Thus the overall time complexity of the algorithm can be expressed as T(n) = 55n3 + O(n2). Here the terms 2n + 10 are subsumed within the faster-growing O(n2). Again, this usage disregards some of the formal meaning of the "=" symbol, but it does allow one to use the big O notation as a kind of convenient placeholder. === Multiple uses === In more complicated usage, O(·) can appear in different places in an equation, even several times on each side. For example, the following are true for n → ∞ {\displaystyle n\to \infty } : ( n + 1 ) 2 = n 2 + O ( n ) , ( n + O ( n 1 / 2 ) ) ⋅ ( n + O ( log ⁡ n ) ) 2 = n 3 + O ( n 5 / 2 ) , n O ( 1 ) = O ( e n ) . {\displaystyle {\begin{aligned}(n+1)^{2}&=n^{2}+O(n),\\(n+O(n^{1/2}))\cdot (n+O(\log n))^{2}&=n^{3}+O(n^{5/2}),\\n^{O(1)}&=O(e^{n}).\end{aligned}}} The meaning of such statements is as follows: for any functions which satisfy each O(·) on the left side, there are some functions satisfying each O(·) on the right side, such that substituting all these functions into the equation makes the two sides equal. For example, the third equation above means: "For any function f(n) = O(1), there is some function g(n) = O(en) such that nf(n) = g(n)". In terms of the "set notation" above, the meaning is that the class of functions represented by the left side is a subset of the class of functions represented by the right side. In this use the "=" is a formal symbol that unlike the usual use of "=" is not a symmetric relation. Thus for example nO(1) = O(en) does not imply the false statement O(en) = nO(1). === Typesetting === Big O is typeset as an italicized uppercase "O", as in the following example: O ( n 2 ) {\displaystyle O(n^{2})} . In TeX, it is produced by simply typing 'O' inside math mode. Unlike Greek-named Bachmann–Landau notations, it needs no special symbol. However, some authors use the calligraphic variant O {\displaystyle {\mathcal {O}}} instead. == Orders of common functions == Here is a list of classes of functions that are commonly encountered when analyzing the running time of an algorithm. In each case, c is a positive constant and n increases without bound. The slower-growing functions are generally listed first. The statement f ( n ) = O ( n ! ) {\displaystyle f(n)=O(n!)} is sometimes weakened to f ( n ) = O ( n n ) {\displaystyle f(n)=O\left(n^{n}\right)} to derive simpler formulas for asymptotic complexity. For any k > 0 {\displaystyle k>0} and c > 0 {\displaystyle c>0} , O ( n c ( log ⁡ n ) k ) {\displaystyle O(n^{c}(\log n)^{k})} is a subset of O ( n c + ε ) {\displaystyle O(n^{c+\varepsilon })} for any ε > 0 {\displaystyle \varepsilon >0} , so may be considered as a polynomial with some bigger order. == Related asymptotic notations == Big O is widely used in computer science. Together with some other related notations, it forms the family of Bachmann–Landau notations. === Little-o notation === Intuitively, the assertion "f(x) is o(g(x))" (read "f(x) is little-o of g(x)" or "f(x) is of inferior order to g(x)") means that g(x) grows much faster than f(x), or equivalently f(x) grows much slower than g(x). As before, let f be a real or complex valued function and g a real valued function, both defined on some unbounded subset of the positive real numbers, such that g(x) is strictly positive for all large enough values of x. One writes f ( x ) = o ( g ( x ) ) as x → ∞ {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to \infty } if for every positive constant ε there exists a constant x 0 {\displaystyle x_{0}} such that \| f ( x ) \| ≤ ε g ( x ) for all x ≥ x 0 . {\displaystyle \|f(x)\|\leq \varepsilon g(x)\quad {\text{ for all }}x\geq x_{0}.} For example, one has 2 x = o ( x 2 ) {\displaystyle 2x=o(x^{2})} and 1 / x = o ( 1 ) , {\displaystyle 1/x=o(1),} both as x → ∞ . {\displaystyle x\to \infty .} The difference between the definition of the big-O notation and the definition of little-o is that while the former has to be true for at least one constant M, the latter must hold for every positive constant ε, however small. In this way, little-o notation makes a stronger statement than the corresponding big-O notation: every function that is little-o of g is also big-O of g, but not every function that is big-O of g is little-o of g. For example, 2 x 2 = O ( x 2 ) {\displaystyle 2x^{2}=O(x^{2})} but 2 x 2 ≠ o ( x 2 ) {\displaystyle 2x^{2}\neq o(x^{2})} . If g(x) is nonzero, or at least becomes nonzero beyond a certain point, the relation f ( x ) = o ( g ( x ) ) {\displaystyle f(x)=o(g(x))} is equivalent to lim x → ∞ f ( x ) g ( x ) = 0 {\displaystyle \lim _{x\to \infty }{\frac {f(x)}{g(x)}}=0} (and this is in fact how Landau originally defined the little-o notation). Little-o respects a number of arithmetic operations. For example, if c is a nonzero constant and f = o ( g ) {\displaystyle f=o(g)} then c ⋅ f = o ( g ) {\displaystyle c\cdot f=o(g)} , and if f = o ( F ) {\displaystyle f=o(F)} and g = o ( G ) {\displaystyle g=o(G)} then f ⋅ g = o ( F ⋅ G ) . {\displaystyle f\cdot g=o(F\cdot G).} if f = o ( F ) {\displaystyle f=o(F)} and g = o ( G ) {\displaystyle g=o(G)} then f + g = o ( F + G ) {\displaystyle f+g=o(F+G)} It also satisfies a transitivity relation: if f = o ( g ) {\displaystyle f=o(g)} and g = o ( h ) {\displaystyle g=o(h)} then f = o ( h ) . {\displaystyle f=o(h).} Little-o can also be generalized to the finite case: f ( x ) = o ( g ( x ) ) as x → x 0 {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}} if f ( x ) = α ( x ) g ( x ) {\displaystyle f(x)=\alpha (x)g(x)} for some α ( x ) {\displaystyle \alpha (x)} with lim x → x 0 α ( x ) = 0 {\displaystyle \lim _{x\to x_{0}}\alpha (x)=0} . Or, if g(x) is nonzero in a neighbourhood around x 0 {\displaystyle x_{0}} : f ( x ) = o ( g ( x ) ) as x → x 0 {\displaystyle f(x)=o(g(x))\quad {\text{ as }}x\to x_{0}} if lim x → x 0 f ( x ) g ( x ) = 0 {\displaystyle \lim _{x\to x_{0}}{\frac {f(x)}{g(x)}}=0} . This definition especially useful in the computation of limits using Taylor series. For example: sin ⁡ x = x − x 3 3 ! + … = x + o ( x 2 ) as x → 0 {\displaystyle \sin x=x-{\frac {x^{3}}{3!}}+\ldots =x+o(x^{2}){\text{ as }}x\to 0} , so lim x → 0 sin ⁡ x x = lim x → 0 x + o ( x 2 ) x = lim x → 0 1 + o ( x ) = 1 {\displaystyle \lim _{x\to 0}{\frac {\sin x}{x}}=\lim _{x\to 0}{\frac {x+o(x^{2})}{x}}=\lim _{x\to 0}1+o(x)=1} === Big Omega notation === Another asymptotic notation is Ω {\displaystyle \Omega } , read "big omega". There are two widespread and incompatible definitions of the statement f ( x ) = Ω ( g ( x ) ) {\displaystyle f(x)=\Omega (g(x))} as x → a {\displaystyle x\to a} , where a is some real number, ∞ {\displaystyle \infty } , or − ∞ {\displaystyle -\infty } , where f and g are real functions defined in a neighbourhood of a, and where g is positive in this neighbourhood. The Hardy–Littlewood definition is used mainly in analytic number theory, and the Knuth definition mainly in computational complexity theory; the definitions are not equivalent. ==== The Hardy–Littlewood definition ==== In 1914 G.H. Hardy and J.E. Littlewood introduced the new symbol Ω , {\displaystyle \ \Omega \ ,} which is defined as follows: f ( x ) = Ω ( g ( x ) ) {\displaystyle f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim sup x → ∞ \| f ( x ) g ( x ) \| > 0 . {\displaystyle \quad \limsup _{x\to \infty }\ \left\|{\frac {\ f(x)\ }{g(x)}}\right\|>0~.} Thus f ( x ) = Ω ( g ( x ) ) {\displaystyle ~f(x)=\Omega {\bigl (}\ g(x)\ {\bigr )}~} is the negation of f ( x ) = o ( g ( x ) ) . {\displaystyle ~f(x)=o{\bigl (}\ g(x)\ {\bigr )}~.} In 1916 the same authors introduced the two new symbols Ω R {\displaystyle \ \Omega _{R}\ } and Ω L , {\displaystyle \ \Omega _{L}\ ,} defined as: f ( x ) = Ω R ( g ( x ) ) {\displaystyle f(x)=\Omega _{R}{\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim sup x → ∞ f ( x ) g ( x ) > 0 ; {\displaystyle \quad \limsup _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}>0\ ;} f ( x ) = Ω L ( g ( x ) ) {\displaystyle f(x)=\Omega _{L}{\bigl (}\ g(x)\ {\bigr )}\quad } as x → ∞ {\displaystyle \quad x\to \infty \quad } if lim inf x → ∞ f ( x ) g ( x ) < 0 . {\displaystyle \quad ~\liminf _{x\to \infty }\ {\frac {\ f(x)\ }{g(x)}}<0~.} These symbols were used by E. Landau, with the same meanings, in 1924. Authors that followed Landau, however, use a different notation for the same definitions: The symbol Ω R {\displaystyle \ \Omega _{R}\ } has been replaced by the current notation Ω + {\displaystyle \ \Omega _{+}\ } with the same definition, and Ω L {\displaystyle \ \Omega _{L}\ } became Ω − . {\displaystyle \ \Omega _{-}~.} These three symbols Ω , Ω + , Ω − , {\displaystyle \ \Omega \ ,\Omega _{+}\ ,\Omega _{-}\ ,} as well as f ( x ) = Ω ± ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{\pm }{\bigl (}\ g(x)\ {\bigr )}\ } (meaning that f ( x ) = Ω + ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{+}{\bigl (}\ g(x)\ {\bigr )}\ } and f ( x ) = Ω − ( g ( x ) ) {\displaystyle \ f(x)=\Omega _{-}{\bigl (}\ g(x)\ {\bigr )}\ } are both satisfied), are now currently used in analytic number theory. ===== Simple examples ===== We have sin ⁡ x = Ω ( 1 ) {\displaystyle \sin x=\Omega (1)\quad } as x → ∞ , {\displaystyle \quad x\to \infty \ ,} and more precisely sin ⁡ x = Ω ± ( 1 ) {\displaystyle \sin x=\Omega _{\pm }(1)\quad } as x → ∞ . {\displaystyle \quad x\to \infty ~.} We have 1 + sin ⁡ x = Ω ( 1 ) {\displaystyle 1+\sin x=\Omega (1)\quad } as x → ∞ , {\displaystyle \quad x\to \infty \ ,} and more precisely 1 + sin ⁡ x = Ω + ( 1 ) {\displaystyle 1+\sin x=\Omega _{+}(1)\quad } as x → ∞ ; {\displaystyle \quad x\to \infty \ ;} however 1 + sin ⁡ x ≠ Ω − ( 1 ) {\displaystyle 1+\sin x\neq \Omega _{-}(1)\quad } as x → ∞ . {\displaystyle \quad x\to \infty ~.} ==== The Knuth definition ==== In 1976 Donald Knuth published a paper to justify his use of the Ω {\displaystyle \Omega } -symbol to describe a stronger property. Knuth wrote: "For all the applications I have seen so far in computer science, a stronger requirement ... is much more appropriate". He defined f ( x ) = Ω ( g ( x ) ) ⇔ g ( x ) = O ( f ( x ) ) {\displaystyle f(x)=\Omega (g(x))\Leftrightarrow g(x)=O(f(x))} with the comment: "Although I have changed Hardy and Littlewood's definition of Ω {\displaystyle \Omega } , I feel justified in doing so because their definition is by no means in wide use, and because there are other ways to say what they want to say in the comparatively rare cases when their definition applies." === Family of Bachmann–Landau notations === The limit definitions assume g ( n ) > 0 {\displaystyle g(n)>0} for sufficiently large n {\displaystyle n} . The table is (partly) sorted from smallest to largest, in the sense that o , O , Θ , ∼ , {\displaystyle o,O,\Theta ,\sim ,} (Knuth's version of) Ω , ω {\displaystyle \Omega ,\omega } on functions correspond to < , ≤ , ≈ , = , {\displaystyle <,\leq ,\approx ,=,} ≥ , > {\displaystyle \geq ,>} on the real line (the Hardy–Littlewood version of Ω {\displaystyle \Omega } , however, doesn't correspond to any such description). Computer science uses the big O {\displaystyle O} , big Theta Θ {\displaystyle \Theta } , little o {\displaystyle o} , little omega ω {\displaystyle \omega } and Knuth's big Omega Ω {\displaystyle \Omega } notations. Analytic number theory often uses the big O {\displaystyle O} , small o {\displaystyle o} , Hardy's ≍ {\displaystyle \asymp } , Hardy–Littlewood's big Omega Ω {\displaystyle \Omega } (with or without the +, − or ± subscripts) and ∼ {\displaystyle \sim } notations. The small omega ω {\displaystyle \omega } notation is not used as often in analysis. === Use in computer science === Informally, especially in computer science, the big O notation often can be used somewhat differently to describe an asymptotic tight bound where using big Theta Θ notation might be more factually appropriate in a given context. For example, when considering a function T(n) = 73n3 + 22n2 + 58, all of the following are generally acceptable, but tighter bounds (such as numbers 2 and 3 below) are usually strongly preferred over looser bounds (such as number 1 below). T(n) = O(n100) T(n) = O(n3) T(n) = Θ(n3) The equivalent English statements are respectively: T(n) grows asymptotically no faster than n100 T(n) grows asymptotically no faster than n3 T(n) grows asymptotically as fast as n3. So while all three statements are true, progressively more information is contained in each. In some fields, however, the big O notation (number 2 in the lists above) would be used more commonly than the big Theta notation (items numbered 3 in the lists above). For example, if T(n) represents the running time of a newly developed algorithm for input size n, the inventors and users of the algorithm might be more inclined to put an upper asymptotic bound on how long it will take to run without making an explicit statement about the lower asymptotic bound. === Other notation === In their book Introduction to Algorithms, Cormen, Leiserson, Rivest and Stein consider the set of functions f which satisfy f ( n ) = O ( g ( n ) ) ( n → ∞ ) . {\displaystyle f(n)=O(g(n))\quad (n\to \infty )~.} In a correct notation this set can, for instance, be called O(g), where O ( g ) = { f : there exist positive constants c and n 0 such that 0 ≤ f ( n ) ≤ c g ( n ) for all n ≥ n 0 } . {\displaystyle O(g)=\{f:{\text{there exist positive constants}}~c~{\text{and}}~n_{0}~{\text{such that}}~0\leq f(n)\leq cg(n){\text{ for all }}n\geq n_{0}\}.} The authors state that the use of equality operator (=) to denote set membership rather than the set membership operator (∈) is an abuse of notation, but that doing so has advantages. Inside an equation or inequality, the use of asymptotic notation stands for an anonymous function in the set O(g), which eliminates lower-order terms, and helps to reduce inessential clutter in equations, for example: 2 n 2 + 3 n + 1 = 2 n 2 + O ( n ) . {\displaystyle 2n^{2}+3n+1=2n^{2}+O(n).} === Extensions to the Bachmann–Landau notations === Another notation sometimes used in computer science is Õ (read soft-O), which hides polylogarithmic factors. There are two definitions in use: some authors use f(n) = Õ(g(n)) as shorthand for f(n) = O(g(n) logk n) for some k, while others use it as shorthand for f(n) = O(g(n) logk g(n)). When g(n) is polynomial in n, there is no difference; however, the latter definition allows one to say, e.g. that n 2 n = O ~ ( 2 n ) {\displaystyle n2^{n}={\tilde {O}}(2^{n})} while the former definition allows for log k ⁡ n = O ~ ( 1 ) {\displaystyle \log ^{k}n={\tilde {O}}(1)} for any constant k. Some authors write O* for the same purpose as the latter definition. Essentially, it is big O notation, ignoring logarithmic factors because the growth-rate effects of some other super-logarithmic function indicate a growth-rate explosion for large-sized input parameters that is more important to predicting bad run-time performance than the finer-point effects contributed by the logarithmic-growth factor(s). This notation is often used to obviate the "nitpicking" within growth-rates that are stated as too tightly bounded for the matters at hand (since logk n is always o(nε) for any constant k and any ε > 0). Also, the L notation, defined as L n [ α , c ] = e ( c + o ( 1 ) ) ( ln ⁡ n ) α ( ln ⁡ ln ⁡ n ) 1 − α , {\displaystyle L_{n}[\alpha ,c]=e^{(c+o(1))(\ln n)^{\alpha }(\ln \ln n)^{1-\alpha }},} is convenient for functions that are between polynomial and exponential in terms of ln ⁡ n {\displaystyle \ln n} . == Generalizations and related usages == The generalization to functions taking values in any normed vector space is straightforward (replacing absolute values by norms), where f and g need not take their values in the same space. A generalization to functions g taking values in any topological group is also possible. The "limiting process" x → xo can also be generalized by introducing an arbitrary filter base, i.e. to directed nets f and g. The o notation can be used to define derivatives and differentiability in quite general spaces, and also (asymptotical) equivalence of functions, f ∼ g ⟺ ( f − g ) ∈ o ( g ) {\displaystyle f\sim g\iff (f-g)\in o(g)} which is an equivalence relation and a more restrictive notion than the relationship "f is Θ(g)" from above. (It reduces to lim f / g = 1 if f and g are positive real valued functions.) For example, 2x is Θ(x), but 2x − x is not o(x). == History (Bachmann–Landau, Hardy, and Vinogradov notations) == The symbol O was first introduced by number theorist Paul Bachmann in 1894, in the second volume of his book Analytische Zahlentheorie ("analytic number theory"). The number theorist Edmund Landau adopted it, and was thus inspired to introduce in 1909 the notation o; hence both are now called Landau symbols. These notations were used in applied mathematics during the 1950s for asymptotic analysis. The symbol Ω {\displaystyle \Omega } (in the sense "is not an o of") was introduced in 1914 by Hardy and Littlewood. Hardy and Littlewood also introduced in 1916 the symbols Ω R {\displaystyle \Omega _{R}} ("right") and Ω L {\displaystyle \Omega _{L}} ("left"), precursors of the modern symbols Ω + {\displaystyle \Omega _{+}} ("is not smaller than a small o of") and Ω − {\displaystyle \Omega _{-}} ("is not larger than a small o of"). Thus the Omega symbols (with their original meanings) are sometimes also referred to as "Landau symbols". This notation Ω {\displaystyle \Omega } became commonly used in number theory at least since the 1950s. The symbol ∼ {\displaystyle \sim } , although it had been used before with different meanings, was given its modern definition by Landau in 1909 and by Hardy in 1910. Just above on the same page of his tract Hardy defined the symbol ≍ {\displaystyle \asymp } , where f ( x ) ≍ g ( x ) {\displaystyle f(x)\asymp g(x)} means that both f ( x ) = O ( g ( x ) ) {\displaystyle f(x)=O(g(x))} and g ( x ) = O ( f ( x ) ) {\displaystyle g(x)=O(f(x))} are satisfied. The notation is still currently used in analytic number theory. In his tract Hardy also proposed the symbol ≍ − {\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} } , where f ≍ − g {\displaystyle f\mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} g} means that f ∼ K g {\displaystyle f\sim Kg} for some constant K ≠ 0 {\displaystyle K\not =0} . In the 1970s the big O was popularized in computer science by Donald Knuth, who proposed the different notation f ( x ) = Θ ( g ( x ) ) {\displaystyle f(x)=\Theta (g(x))} for Hardy's f ( x ) ≍ g ( x ) {\displaystyle f(x)\asymp g(x)} , and proposed a different definition for the Hardy and Littlewood Omega notation. Two other symbols coined by Hardy were (in terms of the modern O notation) f ≼ g ⟺ f = O ( g ) {\displaystyle f\preccurlyeq g\iff f=O(g)} and f ≺ g ⟺ f = o ( g ) ; {\displaystyle f\prec g\iff f=o(g);} (Hardy however never defined or used the notation ≺ ≺ {\displaystyle \prec \!\!\prec } , nor ≪ {\displaystyle \ll } , as it has been sometimes reported). Hardy introduced the symbols ≼ {\displaystyle \preccurlyeq } and ≺ {\displaystyle \prec } (as well as the already mentioned other symbols) in his 1910 tract "Orders of Infinity", and made use of them only in three papers (1910–1913). In his nearly 400 remaining papers and books he consistently used the Landau symbols O and o. Hardy's symbols ≼ {\displaystyle \preccurlyeq } and ≺ {\displaystyle \prec } (as well as ≍ − {\displaystyle \mathbin {\,\asymp \;\;\;\;\!\!\!\!\!\!\!\!\!\!\!\!\!-} } ) are not used anymore. On the other hand, in the 1930s, the Russian number theorist Ivan Matveyevich Vinogradov introduced his notation ≪ {\displaystyle \ll } , which has been increasingly used in number theory instead of the O {\displaystyle O} notation. We have f ≪ g ⟺ f = O ( g ) , {\displaystyle f\ll g\iff f=O(g),} and frequently both notations are used in the same paper. The big-O originally stands for "order of" ("Ordnung", Bachmann 1894), and is thus a Latin letter. Neither Bachmann nor Landau ever call it "Omicron". The symbol was much later on (1976) viewed by Knuth as a capital omicron, probably in reference to his definition of the symbol Omega. The digit zero should not be used. == See also == Asymptotic computational complexity Asymptotic expansion: Approximation of functions generalizing Taylor's formula Asymptotically optimal algorithm: A phrase frequently used to describe an algorithm that has an upper bound asymptotically within a constant of a lower bound for the problem Big O in probability notation: Op, op Limit inferior and limit superior: An explanation of some of the limit notation used in this article Master theorem (analysis of algorithms): For analyzing divide-and-conquer recursive algorithms using big O notation Nachbin's theorem: A precise method of bounding complex analytic functions so that the domain of convergence of integral transforms can be stated Order of approximation Order of accuracy Computational complexity of mathematical operations == References and notes == === Notes === == Further reading == Hardy, G. H. (1910). Orders of Infinity: The 'Infinitärcalcül' of Paul du Bois-Reymond. Cambridge University Press. Knuth, Donald (1997). "1.2.11: Asymptotic Representations". Fundamental Algorithms. The Art of Computer Programming. Vol. 1 (3rd ed.). Addison-Wesley. ISBN 978-0-201-89683-1. Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "3.1: Asymptotic notation". Introduction to Algorithms (2nd ed.). MIT Press and McGraw-Hill. ISBN 978-0-262-03293-3. Sipser, Michael (1997). Introduction to the Theory of Computation. PWS Publishing. pp. 226–228. ISBN 978-0-534-94728-6. Avigad, Jeremy; Donnelly, Kevin (2004). Formalizing O notation in Isabelle/HOL (PDF). International Joint Conference on Automated Reasoning. doi:10.1007/978-3-540-25984-8_27. Black, Paul E. (11 March 2005). Black, Paul E. (ed.). "big-O notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "little-o notation". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "ω". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. Black, Paul E. (17 December 2004). Black, Paul E. (ed.). "Θ". Dictionary of Algorithms and Data Structures. U.S. National Institute of Standards and Technology. Retrieved December 16, 2006. == External links == Growth of sequences — OEIS (Online Encyclopedia of Integer Sequences) Wiki Introduction to Asymptotic Notations Big-O Notation – What is it good for An example of Big O in accuracy of central divided difference scheme for first derivative A Gentle Introduction to Algorithm Complexity Analysis
Wikipedia:Haridatta#0	Haridatta (c. 683 CE) was an astronomer-mathematician of Kerala, India, who is believed to be the promulgator of the Parahita system of astronomical computations. This system of computations is widely popular in Kerala and Tamil Nadu. According to legends, Haridatta promulgated the Parahita system on the occasion of the Mamankam held in the year 683 CE. Mamankam was a 12-yearly festival held in Thirunnavaya on the banks of the Bharathapuzha river. The distinctive contribution of Haridatta, apart from his resolving the Aryabhatiya calculations and using the Katapayadi system of numerals is the corrections he introduced to the values of the mean and true positions, the velocity, etc., of the moon and other planets as obtained from Aryabhata's constants. This correction is called the Sakabda-samskara since it applied from the date of Aryabhata in the Saka era 444, at which date his constants gave accurate results. == Parahita system == The Parahita system of computations introduced by Haridatta was a simplification of the system propounded in Aryabhatiya by Aryabhata. Haridatta introduced the following simplifications. The system was called Parahita meaning suitable for the common man because it simplified astronomical computations and made it accessible for practice even for ordinary persons. Haridatta dispensed with the numerical symbolism used by Aryabhata and replaced it with the more flexible Katapayadi system. In this system, letters are used to represent digits and these letters are then used to invent meaningful words and sentences to denote specific numbers. These words and sentences could be remembered with much less effort. Computations in Indian astronomy involved long numbers representing various parameters associated with the several celestial objects which are applicable for a Mahayuga, a period of 4,320,000 years. To avoid computations with these large numbers, Haridatta introduced a smaller Yuga, called a Dhijagannupura-yuga, of 576 years or 210,389 days (which 1/7500 th part of a Mahyuga) and accurately determined the zero corrections for this sub-Yuga for the mean motion of the several planets. These corrections were then used to compute the mean planets for any given date. == Works of Haridatta == Scholars have been able to identify only two works as authored by Haridatta. One of them, titled Grahacaranibandhana, is the basic manual of computations of the Parahita system of astronomy. This was unearthed by K.V. Sarma and was published in 1954. The other work titled Mahamarganibandhana is no longer extant. == See also == Indian astronomy Indian mathematics Indian mathematicians History of mathematics List of astronomers and mathematicians of the Kerala school == References ==
Wikipedia:Harish-Chandra class#0	In mathematics, Harish-Chandra's class is a class of Lie groups used in representation theory. Harish-Chandra's class contains all semisimple connected linear Lie groups and is closed under natural operations, most importantly, the passage to Levi subgroups. This closure property is crucial for many inductive arguments in representation theory of Lie groups, whereas the classes of semisimple or connected semisimple Lie groups are not closed in this sense. == Definition == A Lie group G with the Lie algebra g is said to be in Harish-Chandra's class if it satisfies the following conditions: g is a reductive Lie algebra (the product of a semisimple and abelian Lie algebra). The Lie group G has only a finite number of connected components. The adjoint action of any element of G on g is given by an action of an element of the connected component of the Lie group of Lie algebra automorphisms of the complexification g⊗C. The subgroup Gss of G generated by the image of the semisimple part gss=[g,g] of the Lie algebra g under the exponential map has finite center. == References == A. W. Knapp, Structure theory of semisimple Lie groups, in ISBN 0-8218-0609-2
Wikipedia:Harish-Chandra isomorphism#0	In mathematics, the Harish-Chandra isomorphism, introduced by Harish-Chandra (1951), is an isomorphism of commutative rings constructed in the theory of Lie algebras. The isomorphism maps the center Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} of the universal enveloping algebra U ( g ) {\displaystyle U({\mathfrak {g}})} of a reductive Lie algebra g {\displaystyle {\mathfrak {g}}} to the elements S ( h ) W {\displaystyle S({\mathfrak {h}})^{W}} of the symmetric algebra S ( h ) {\displaystyle S({\mathfrak {h}})} of a Cartan subalgebra h {\displaystyle {\mathfrak {h}}} that are invariant under the Weyl group W {\displaystyle W} . == Introduction and setting == Let g {\displaystyle {\mathfrak {g}}} be a semisimple Lie algebra, h {\displaystyle {\mathfrak {h}}} its Cartan subalgebra and λ , μ ∈ h ∗ {\displaystyle \lambda ,\mu \in {\mathfrak {h}}^{}} be two elements of the weight space (where h ∗ {\displaystyle {\mathfrak {h}}^{}} is the dual of h {\displaystyle {\mathfrak {h}}} ) and assume that a set of positive roots Φ + {\displaystyle \Phi _{+}} have been fixed. Let V λ {\displaystyle V_{\lambda }} and V μ {\displaystyle V_{\mu }} be highest weight modules with highest weights λ {\displaystyle \lambda } and μ {\displaystyle \mu } respectively. === Central characters === The g {\displaystyle {\mathfrak {g}}} -modules V λ {\displaystyle V_{\lambda }} and V μ {\displaystyle V_{\mu }} are representations of the universal enveloping algebra U ( g ) {\displaystyle U({\mathfrak {g}})} and its center acts on the modules by scalar multiplication (this follows from the fact that the modules are generated by a highest weight vector). So, for v ∈ V λ {\displaystyle v\in V_{\lambda }} and x ∈ Z ( U ( g ) ) {\displaystyle x\in {\mathcal {Z}}(U({\mathfrak {g}}))} , x ⋅ v := χ λ ( x ) v {\displaystyle x\cdot v:=\chi _{\lambda }(x)v} and similarly for V μ {\displaystyle V_{\mu }} , where the functions χ λ , χ μ {\displaystyle \chi _{\lambda },\,\chi _{\mu }} are homomorphisms from Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} to scalars called central characters. == Statement of Harish-Chandra theorem == For any λ , μ ∈ h ∗ {\displaystyle \lambda ,\mu \in {\mathfrak {h}}^{}} , the characters χ λ = χ μ {\displaystyle \chi _{\lambda }=\chi _{\mu }} if and only if λ + δ {\displaystyle \lambda +\delta } and μ + δ {\displaystyle \mu +\delta } are on the same orbit of the Weyl group of h ∗ {\displaystyle {\mathfrak {h}}^{}} , where δ {\displaystyle \delta } is the half-sum of the positive roots, sometimes known as the Weyl vector. Another closely related formulation is that the Harish-Chandra homomorphism from the center of the universal enveloping algebra Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} to S ( h ) W {\displaystyle S({\mathfrak {h}})^{W}} (the elements of the symmetric algebra of the Cartan subalgebra fixed by the Weyl group) is an isomorphism. === Explicit isomorphism === More explicitly, the isomorphism can be constructed as the composition of two maps, one from Z = Z ( U ( g ) ) {\displaystyle {\mathfrak {Z}}={\mathcal {Z}}(U({\mathfrak {g}}))} to U ( h ) = S ( h ) , {\displaystyle U({\mathfrak {h}})=S({\mathfrak {h}}),} and another from S ( h ) {\displaystyle S({\mathfrak {h}})} to itself. The first is a projection γ : Z → S ( h ) {\displaystyle \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})} . For a choice of positive roots Φ + {\displaystyle \Phi _{+}} , defining n + = ⨁ α ∈ Φ + g α , n − = ⨁ α ∈ Φ − g α {\displaystyle n^{+}=\bigoplus _{\alpha \in \Phi _{+}}{\mathfrak {g}}_{\alpha },n^{-}=\bigoplus _{\alpha \in \Phi _{-}}{\mathfrak {g}}_{\alpha }} as the corresponding positive nilpotent subalgebra and negative nilpotent subalgebra respectively, due to the Poincaré–Birkhoff–Witt theorem there is a decomposition U ( g ) = U ( h ) ⊕ ( U ( g ) n + + n − U ( g ) ) . {\displaystyle U({\mathfrak {g}})=U({\mathfrak {h}})\oplus (U({\mathfrak {g}}){\mathfrak {n}}^{+}+{\mathfrak {n}}^{-}U({\mathfrak {g}})).} If z ∈ Z {\displaystyle z\in {\mathfrak {Z}}} is central, then in fact z ∈ U ( h ) ⊕ ( U ( g ) n + ∩ n − U ( g ) ) . {\displaystyle z\in U({\mathfrak {h}})\oplus (U({\mathfrak {g}}){\mathfrak {n}}^{+}\cap {\mathfrak {n}}^{-}U({\mathfrak {g}})).} The restriction of the projection U ( g ) → U ( h ) {\displaystyle U({\mathfrak {g}})\rightarrow U({\mathfrak {h}})} to the centre is γ : Z → S ( h ) {\displaystyle \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})} , and is a homomorphism of algebras. This is related to the central characters by χ λ ( x ) = γ ( x ) ( λ ) {\displaystyle \chi _{\lambda }(x)=\gamma (x)(\lambda )} The second map is the twist map τ : S ( h ) → S ( h ) {\displaystyle \tau :S({\mathfrak {h}})\rightarrow S({\mathfrak {h}})} . On h {\displaystyle {\mathfrak {h}}} viewed as a subspace of U ( h ) {\displaystyle U({\mathfrak {h}})} it is defined τ ( h ) = h − δ ( h ) 1 {\displaystyle \tau (h)=h-\delta (h)1} with δ {\displaystyle \delta } the Weyl vector. Then γ ~ = τ ∘ γ : Z → S ( h ) {\displaystyle {\tilde {\gamma }}=\tau \circ \gamma :{\mathfrak {Z}}\rightarrow S({\mathfrak {h}})} is the isomorphism. The reason this twist is introduced is that χ λ {\displaystyle \chi _{\lambda }} is not actually Weyl-invariant, but it can be proven that the twisted character χ ~ λ = χ λ − δ {\displaystyle {\tilde {\chi }}_{\lambda }=\chi _{\lambda -\delta }} is. == Applications == The theorem has been used to obtain a simple Lie algebraic proof of Weyl's character formula for finite-dimensional irreducible representations. The proof has been further simplified by Victor Kac, so that only the quadratic Casimir operator is required; there is a corresponding streamlined treatment proof of the character formula in the second edition of Humphreys (1978, pp. 143–144). Further, it is a necessary condition for the existence of a non-zero homomorphism of some highest weight modules (a homomorphism of such modules preserves central character). A simple consequence is that for Verma modules or generalized Verma modules V λ {\displaystyle V_{\lambda }} with highest weight λ {\displaystyle \lambda } , there exist only finitely many weights μ {\displaystyle \mu } for which a non-zero homomorphism V λ → V μ {\displaystyle V_{\lambda }\rightarrow V_{\mu }} exists. == Fundamental invariants == For g {\displaystyle {\mathfrak {g}}} a simple Lie algebra, let r {\displaystyle r} be its rank, that is, the dimension of any Cartan subalgebra h {\displaystyle {\mathfrak {h}}} of g {\displaystyle {\mathfrak {g}}} . H. S. M. Coxeter observed that S ( h ) W {\displaystyle S({\mathfrak {h}})^{W}} is isomorphic to a polynomial algebra in r {\displaystyle r} variables (see Chevalley–Shephard–Todd theorem for a more general statement). Therefore, the center of the universal enveloping algebra of a simple Lie algebra is isomorphic to a polynomial algebra. The degrees of the generators of the algebra are the degrees of the fundamental invariants given in the following table. The number of the fundamental invariants of a Lie group is equal to its rank. Fundamental invariants are also related to the cohomology ring of a Lie group. In particular, if the fundamental invariants have degrees d 1 , ⋯ , d r {\displaystyle d_{1},\cdots ,d_{r}} , then the generators of the cohomology ring have degrees 2 d 1 − 1 , ⋯ , 2 d r − 1 {\displaystyle 2d_{1}-1,\cdots ,2d_{r}-1} . Due to this, the degrees of the fundamental invariants can be calculated from the Betti numbers of the Lie group and vice versa. In another direction, fundamental invariants are related to cohomology of the classifying space. The cohomology ring H ∗ ( B G , R ) {\displaystyle H^{*}(BG,\mathbb {R} )} is isomorphic to a polynomial algebra on generators with degrees 2 d 1 , ⋯ , 2 d r {\displaystyle 2d_{1},\cdots ,2d_{r}} . == Examples == If g {\displaystyle {\mathfrak {g}}} is the Lie algebra s l ( 2 , R ) {\displaystyle {\mathfrak {sl}}(2,\mathbb {R} )} , then the center of the universal enveloping algebra is generated by the Casimir invariant of degree 2, and the Weyl group acts on the Cartan subalgebra, which is isomorphic to R {\displaystyle \mathbb {R} } , by negation, so the invariant of the Weyl group is the square of the generator of the Cartan subalgebra, which is also of degree 2. For g = A 2 = s l ( 3 , C ) {\displaystyle {\mathfrak {g}}=A_{2}={\mathfrak {sl}}(3,\mathbb {C} )} , the Harish-Chandra isomorphism says Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} is isomorphic to a polynomial algebra of Weyl-invariant polynomials in two variables h 1 , h 2 {\displaystyle h_{1},h_{2}} (since the Cartan subalgebra is two-dimensional). For A 2 {\displaystyle A_{2}} , the Weyl group is S 3 ≅ D 6 {\displaystyle S_{3}\cong D_{6}} which acts on the CSA in the standard representation. Since the Weyl group acts by reflections, they are isometries and so the degree 2 polynomial f 2 ( h 1 , h 2 ) = h 1 2 + h 2 2 {\displaystyle f_{2}(h_{1},h_{2})=h_{1}^{2}+h_{2}^{2}} is Weyl-invariant. The contours of the degree 3 Weyl-invariant polynomial (for a particular choice of standard representation where one of the reflections is across the x-axis) are shown below. These two polynomials generate the polynomial algebra, and are the fundamental invariants for A 2 {\displaystyle A_{2}} . For all the Lie algebras in the classification, there is a fundamental invariant of degree 2, the quadratic Casimir. In the isomorphism, these correspond to a degree 2 polynomial on the CSA. Since the Weyl group acts by reflections on the CSA, they are isometries, so the degree 2 invariant polynomial is f 2 ( h ) = h 1 2 + ⋯ + h r 2 {\displaystyle f_{2}(\mathbf {h} )=h_{1}^{2}+\cdots +h_{r}^{2}} where r {\displaystyle r} is the dimension of the CSA h {\displaystyle {\mathfrak {h}}} , also known as the rank of the Lie algebra. For g = A 1 = s l ( 2 , C ) {\displaystyle {\mathfrak {g}}=A_{1}={\mathfrak {sl}}(2,\mathbb {C} )} , the Cartan subalgebra is one-dimensional, and the Harish-Chandra isomorphism says Z ( U ( g ) ) {\displaystyle {\mathcal {Z}}(U({\mathfrak {g}}))} is isomorphic to the algebra of Weyl-invariant polynomials in a single variable h {\displaystyle h} . The Weyl group is S 2 {\displaystyle S_{2}} acting as reflection, with non-trivial element acting on polynomials by h ↦ − h {\displaystyle h\mapsto -h} . The subalgebra of Weyl-invariant polynomials in the full polynomial algebra K [ h ] {\displaystyle K[h]} is therefore only the even polynomials, generated by f 2 ( h ) = h 2 {\displaystyle f_{2}(h)=h^{2}} . For g = B 2 = s o ( 5 ) = s p ( 4 ) {\displaystyle {\mathfrak {g}}=B_{2}={\mathfrak {so}}(5)={\mathfrak {sp}}(4)} , the Weyl group is D 8 {\displaystyle D_{8}} , acting on two coordinates h 1 , h 2 {\displaystyle h_{1},h_{2}} , and is generated (non-minimally) by four reflections, which act on coordinates as ( h 1 ↦ − h 1 , h 2 ↦ h 2 ) , ( h 1 ↦ h 1 , h 2 ↦ − h 2 ) , ( h 1 ↦ h 2 , h 2 ↦ h 1 ) , ( h 1 ↦ − h 2 , h 2 ↦ h 1 ) {\displaystyle (h_{1}\mapsto -h_{1},h_{2}\mapsto h_{2}),(h_{1}\mapsto h_{1},h_{2}\mapsto -h_{2}),(h_{1}\mapsto h_{2},h_{2}\mapsto h_{1}),(h_{1}\mapsto -h_{2},h_{2}\mapsto h_{1})} . Any invariant quartic must be even in both h 1 {\displaystyle h_{1}} and h 2 {\displaystyle h_{2}} , and invariance under exchange of coordinates means any invariant quartic can be written f 4 ( h 1 , h 2 ) = a h 1 4 + b h 1 2 h 2 2 + a h 2 4 . {\displaystyle f_{4}(h_{1},h_{2})=ah_{1}^{4}+bh_{1}^{2}h_{2}^{2}+ah_{2}^{4}.} Despite this being a two-dimensional vector space, this contributes only one new fundamental invariant as f 2 ( h 1 , h 2 ) 2 {\displaystyle f_{2}(h_{1},h_{2})^{2}} lies in the space. In this case, there is no unique choice of quartic invariant as any polynomial with b ≠ 2 a {\displaystyle b\neq 2a} (and a , b {\displaystyle a,b} not both zero) suffices. == Generalization to affine Lie algebras == The above result holds for reductive, and in particular semisimple Lie algebras. There is a generalization to affine Lie algebras shown by Feigin and Frenkel showing that an algebra known as the Feigin–Frenkel center is isomorphic to a W-algebra associated to the Langlands dual Lie algebra L g {\displaystyle ^{L}{\mathfrak {g}}} . The Feigin–Frenkel center of an affine Lie algebra g ^ {\displaystyle {\hat {\mathfrak {g}}}} is not exactly the center of the universal enveloping algebra Z ( U ( g ^ ) ) {\displaystyle {\mathcal {Z}}(U({\hat {\mathfrak {g}}}))} . They are elements S {\displaystyle S} of the vacuum affine vertex algebra at critical level k = − h ∨ {\displaystyle k=-h^{\vee }} , where h ∨ {\displaystyle h^{\vee }} is the dual Coxeter number for g {\displaystyle {\mathfrak {g}}} which are annihilated by the positive loop algebra g [ t ] {\displaystyle {\mathfrak {g}}[t]} part of g ^ {\displaystyle {\hat {\mathfrak {g}}}} , that is, Z ( g ^ ) := { S ∈ V cri ( g ) \| g [ t ] S = 0 } {\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}}):=\{S\in V_{\text{cri}}({\mathfrak {g}})\|{\mathfrak {g}}[t]S=0\}} where V cri ( g ) {\displaystyle V_{\text{cri}}({\mathfrak {g}})} is the affine vertex algebra at the critical level. Elements of this center are also known as singular vectors or Segal–Sugawara vectors. The isomorphism in this case is an isomorphism between the Feigin–Frenkel center and the W-algebra constructed associated to the Langlands dual Lie algebra by Drinfeld–Sokolov reduction: Z ( g ^ ) ≅ W ( L g ) . {\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}})\cong {\mathcal {W}}(^{L}{\mathfrak {g}}).} There is also a description of Z ( g ^ ) {\displaystyle {\mathfrak {Z}}({\hat {\mathfrak {g}}})} as a polynomial algebra in a finite number of countably infinite families of generators, ∂ n S i , i = 1 , ⋯ , l , n ≥ 0 {\displaystyle \partial ^{n}S_{i},i=1,\cdots ,l,n\geq 0} , where S i , i = 1 , ⋯ , l {\displaystyle S_{i},i=1,\cdots ,l} have degrees d i + 1 , i = 1 , ⋯ , l {\displaystyle d_{i}+1,i=1,\cdots ,l} and ∂ {\displaystyle \partial } is the (negative of) the natural derivative operator on the loop algebra. == See also == Translation functor Infinitesimal character == Notes == == External resources == Notes on the Harish-Chandra isomorphism == References ==
Wikipedia:Harm Bart#0	Harm Bart (born 5 August 1942) is a Dutch mathematician, economist, and Professor of Mathematics at the Erasmus University Rotterdam, particularly known for his work on "factorization problems for matrix and operator functions." == Biography == Born in Enkhuizen, Bart started his study at the Vrije Universiteit in Amsterdam in 1960. Here he received his BA in Mathematics and Astronomy in 1964, his MA in Mathematics with a minor in Dogmatics in 1969, and his PhD in 1973 with the thesis "Meromorphic operator valued functions" under supervision of Rien Kaashoek. After graduation Bart started his academic career at the Faculty of Mathematics at the Vrije Universiteit as Assistant Professor in 1973, and Associate Professor at the Faculty of Mathematics and Computer Science in 1977. In 1982 he was appointed Professor at the Faculty of Mathematics and Computer Science of the Eindhoven University of Technology. In 1984 he became Extraordinary Professor of Mathematics and since 1985 Professor of Mathematics at the Erasmus School of Economics of the Erasmus University Rotterdam. From 1987 to 1992 he was Co-Director of the Econometric Institute, first with Teun Kloek and later with Ton Vorst. From 1996 to 2000 he was also Dean of the Erasmus School of Economics. In 2004 Bart was decorated as Officer in the Order of Orange-Nassau. He has been elected Fellow of the Stieltjes Institute for Mathematics, and Fellow of the Tinbergen Institute. == Publications == Bart authored and co-authored three books and over fifty articles. 1973. Meromorphic Operator Valued Functions. Vrije Universiteit, Amsterdam 1979. Minimal Factorization of Matrix and Operator Functions, Operator Theory: Advances and Applications, Vol. 1 With I. Gohberg and M.A. Kaashoek. Birkhauser Verlag 2008. Factorization of matrix and operator functions: The State Space Method (Operator Theory: Advances and Applications / Linear Operators and Linear Systems). With Israel Gohberg, Marinus A. Kaashoek and André C.M. Ran. Vol. 178. Springer. Articles, a selection: Bart, Harm, and Seymour Goldberg. "Characterizations of almost periodic strongly continuous groups and semigroups." Mathematische Annalen 236.2 (1978): 105–116. Bart, H., Gohberg, I., Kaashoek, M. A., & Van Dooren, P. (1980). Factorizations of transfer functions. SIAM Journal on Control and Optimization, 18(6), 675–696. Bart, Harm, and H. Hoogland. "Complementary triangular forms of pairs of matrices, realizations with prescribed main matrices, and complete factorization of rational matrix functions." Linear Algebra and its Applications 103 (1988): 193–228. == References == == External links == Harm Bart Curriculum Vitae at few.eur.nl Harm Bart at the Mathematics Genealogy Project
Wikipedia:Harmonic differential#0	In mathematics, a real differential one-form ω on a surface is called a harmonic differential if ω and its conjugate one-form, written as ω∗, are both closed. == Explanation == Consider the case of real one-forms defined on a two dimensional real manifold. Moreover, consider real one-forms that are the real parts of complex differentials. Let ω = A dx + B dy, and formally define the conjugate one-form to be ω∗ = A dy − B dx. == Motivation == There is a clear connection with complex analysis. Let us write a complex number z in terms of its real and imaginary parts, say x and y respectively, i.e. z = x + iy. Since ω + iω∗ = (A − iB)(dx + i dy), from the point of view of complex analysis, the quotient (ω + iω∗)/dz tends to a limit as dz tends to 0. In other words, the definition of ω∗ was chosen for its connection with the concept of a derivative (analyticity). Another connection with the complex unit is that (ω∗)∗ = −ω (just as i2 = −1). For a given function f, let us write ω = df, i.e. ω = ⁠∂f/∂x⁠ dx + ⁠∂f/∂y⁠ dy, where ∂ denotes the partial derivative. Then (df)∗ = ⁠∂f/∂x⁠ dy − ⁠∂f/∂y⁠ dx. Now d((df)∗) is not always zero, indeed d((df)∗) = Δf dx dy, where Δf = ⁠∂2f/∂x2⁠ + ⁠∂2f/∂y2⁠. == Cauchy–Riemann equations == As we have seen above: we call the one-form ω harmonic if both ω and ω∗ are closed. This means that ⁠∂A/∂y⁠ = ⁠∂B/∂x⁠ (ω is closed) and ⁠∂B/∂y⁠ = −⁠∂A/∂x⁠ (ω∗ is closed). These are called the Cauchy–Riemann equations on A − iB. Usually they are expressed in terms of u(x, y) + iv(x, y) as ⁠∂u/∂x⁠ = ⁠∂v/∂y⁠ and ⁠∂v/∂x⁠ = −⁠∂u/∂y⁠. == Notable results == A harmonic differential (one-form) is precisely the real part of an (analytic) complex differential.: 172 To prove this one shows that u + iv satisfies the Cauchy–Riemann equations exactly when u + iv is locally an analytic function of x + iy. Of course an analytic function w(z) = u + iv is the local derivative of something (namely ∫w(z) dz). The harmonic differentials ω are (locally) precisely the differentials df of solutions f to Laplace's equation Δf = 0.: 172 If ω is a harmonic differential, so is ω∗.: 172 == See also == De Rham cohomology == References ==
Wikipedia:Harmonic polynomial#0	In mathematics, a polynomial p {\displaystyle p} whose Laplacian is zero is termed a harmonic polynomial. The harmonic polynomials form a subspace of the vector space of polynomials over the given field. In fact, they form a graded subspace. For the real field ( R {\displaystyle \mathbb {R} } ), the harmonic polynomials are important in mathematical physics. The Laplacian is the sum of second-order partial derivatives with respect to each of the variables, and is an invariant differential operator under the action of the orthogonal group via the group of rotations. The standard separation of variables theorem states that every multivariate polynomial over a field can be decomposed as a finite sum of products of a radial polynomial and a harmonic polynomial. This is equivalent to the statement that the polynomial ring is a free module over the ring of radial polynomials. == Examples == Consider a degree- d {\displaystyle d} univariate polynomial p ( x ) := ∑ k = 0 d a k x k {\displaystyle p(x):=\textstyle \sum _{k=0}^{d}a_{k}x^{k}} . In order to be harmonic, this polynomial must satisfy 0 = ∂ 2 ∂ x 2 p ( x ) = ∑ k = 2 d k ( k − 1 ) a k x k − 2 {\displaystyle 0={\tfrac {\partial ^{2}}{\partial x^{2}}}p(x)=\sum _{k=2}^{d}k(k-1)a_{k}x^{k-2}} at all points x ∈ R {\displaystyle x\in \mathbb {R} } . In particular, when d = 2 {\displaystyle d=2} , we have a polynomial p ( x ) = a 0 + a 1 x + a 2 x 2 {\displaystyle p(x)=a_{0}+a_{1}x+a_{2}x^{2}} , which must satisfy the condition a 2 = 0 {\displaystyle a_{2}=0} . Hence, the only harmonic polynomials of one (real) variable are affine functions x ↦ a 0 + a 1 x {\displaystyle x\mapsto a_{0}+a_{1}x} . In the multivariable case, one finds nontrivial spaces of harmonic polynomials. Consider for instance the bivariate quadratic polynomial p ( x , y ) := a 0 , 0 + a 1 , 0 x + a 0 , 1 y + a 1 , 1 x y + a 2 , 0 x 2 + a 0 , 2 y 2 , {\displaystyle p(x,y):=a_{0,0}+a_{1,0}x+a_{0,1}y+a_{1,1}xy+a_{2,0}x^{2}+a_{0,2}y^{2},} where a 0 , 0 , a 1 , 0 , a 0 , 1 , a 1 , 1 , a 2 , 0 , a 0 , 2 {\displaystyle a_{0,0},a_{1,0},a_{0,1},a_{1,1},a_{2,0},a_{0,2}} are real coefficients. The Laplacian of this polynomial is given by Δ p ( x , y ) = ∂ 2 ∂ x 2 p ( x , y ) + ∂ 2 ∂ y 2 p ( x , y ) = 2 ( a 2 , 0 + a 0 , 2 ) . {\displaystyle \Delta p(x,y)={\tfrac {\partial ^{2}}{\partial x^{2}}}p(x,y)+{\tfrac {\partial ^{2}}{\partial y^{2}}}p(x,y)=2(a_{2,0}+a_{0,2}).} Hence, in order for p ( x , y ) {\displaystyle p(x,y)} to be harmonic, its coefficients need only satisfy the relationship a 2 , 0 = − a 0 , 2 {\displaystyle a_{2,0}=-a_{0,2}} . Equivalently, all (real) quadratic bivariate harmonic polynomials are linear combinations of the polynomials 1 , x , y , x y , x 2 − y 2 . {\displaystyle 1,\quad x,\quad y,\quad xy,\quad x^{2}-y^{2}.} Note that, as in any vector space, there are other choices of basis for this same space of polynomials. A basis for real bivariate harmonic polynomials up to degree 6 is given as follows: ϕ 0 ( x , y ) = 1 ϕ 1 , 1 ( x , y ) = x ϕ 1 , 2 ( x , y ) = y ϕ 2 , 1 ( x , y ) = x y ϕ 2 , 2 ( x , y ) = x 2 − y 2 ϕ 3 , 1 ( x , y ) = y 3 − 3 x 2 y ϕ 3 , 2 ( x , y ) = x 3 − 3 x y 2 ϕ 4 , 1 ( x , y ) = x 3 y − x y 3 ϕ 4 , 2 ( x , y ) = − x 4 + 6 x 2 y 2 − y 4 ϕ 5 , 1 ( x , y ) = 5 x 4 y − 10 x 2 y 3 + y 5 ϕ 5 , 2 ( x , y ) = x 5 − 10 x 3 y 2 + 5 x y 4 ϕ 6 , 1 ( x , y ) = 3 x 5 y − 10 x 3 y 3 + 3 x y 5 ϕ 6 , 2 ( x , y ) = − x 6 + 15 x 4 y 2 − 15 x 2 y 4 + y 6 {\displaystyle {\begin{aligned}\phi _{0}(x,y)&=1\\\phi _{1,1}(x,y)&=x&\phi _{1,2}(x,y)&=y\\\phi _{2,1}(x,y)&=xy&\phi _{2,2}(x,y)&=x^{2}-y^{2}\\\phi _{3,1}(x,y)&=y^{3}-3x^{2}y&\phi _{3,2}(x,y)&=x^{3}-3xy^{2}\\\phi _{4,1}(x,y)&=x^{3}y-xy^{3}&\phi _{4,2}(x,y)&=-x^{4}+6x^{2}y^{2}-y^{4}\\\phi _{5,1}(x,y)&=5x^{4}y-10x^{2}y^{3}+y^{5}&\phi _{5,2}(x,y)&=x^{5}-10x^{3}y^{2}+5xy^{4}\\\phi _{6,1}(x,y)&=3x^{5}y-10x^{3}y^{3}+3xy^{5}&\phi _{6,2}(x,y)&=-x^{6}+15x^{4}y^{2}-15x^{2}y^{4}+y^{6}\end{aligned}}} == See also == Harmonic function Spherical harmonics Zonal spherical harmonics Multilinear polynomial == References == Lie Group Representations of Polynomial Rings by Bertram Kostant published in the American Journal of Mathematics Vol 85 No 3 (July 1963) doi:10.2307/2373130
Wikipedia:Harold Scott MacDonald Coxeter#0	Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated at the University of Cambridge, with student visits to Princeton University. He worked for 60 years at the University of Toronto in Canada, from 1936 until his retirement in 1996, becoming a full professor there in 1948. His many honours included membership in the Royal Society of Canada, the Royal Society, and the Order of Canada. He was an author of 12 books, including The Fifty-Nine Icosahedra (1938) and Regular Polytopes (1947). Many concepts in geometry and group theory are named after him, including the Coxeter graph, Coxeter groups, Coxeter's loxodromic sequence of tangent circles, Coxeter–Dynkin diagrams, and the Todd–Coxeter algorithm. == Biography == Coxeter was born in Kensington, England, to Harold Samuel Coxeter and Lucy (née Gee). His father had taken over the family business of Coxeter & Son, manufacturers of surgical instruments and compressed gases (including a mechanism for anaesthetising surgical patients with nitrous oxide), but was able to retire early and focus on sculpting and baritone singing; Lucy Coxeter was a portrait and landscape painter who had attended the Royal Academy of Arts. A maternal cousin was the architect Sir Giles Gilbert Scott. In his youth, Coxeter composed music and was an accomplished pianist at the age of 10. He felt that mathematics and music were intimately related, outlining his ideas in a 1962 article on "Music and Mathematics" in the Canadian Music Journal. He was educated at King Alfred School, London, and St George's School, Harpenden, where his best friend was John Flinders Petrie, later a mathematician for whom Petrie polygons were named. He was accepted at King's College, Cambridge, in 1925, but decided to spend a year studying in hopes of gaining admittance to Trinity College, where the standard of mathematics was higher. Coxeter won an entrance scholarship and went to Trinity in 1926 to read mathematics. There he earned his BA (as Senior Wrangler) in 1928, and his doctorate in 1931. In 1932 he went to Princeton University for a year as a Rockefeller Fellow, where he worked with Hermann Weyl, Oswald Veblen, and Solomon Lefschetz. Returning to Trinity for a year, he attended Ludwig Wittgenstein's seminars on the philosophy of mathematics. In 1934 he spent a further year at Princeton as a Procter Fellow. In 1936 Coxeter moved to the University of Toronto. In 1938 he and P. Du Val, H. T. Flather, and John Flinders Petrie published The Fifty-Nine Icosahedra with University of Toronto Press. In 1940 Coxeter edited the eleventh edition of Mathematical Recreations and Essays, originally published by W. W. Rouse Ball in 1892. He was elevated to professor in 1948. He was elected a Fellow of the Royal Society of Canada in 1948 and a Fellow of the Royal Society in 1950. He met M. C. Escher in 1954 and the two became lifelong friends; his work on geometric figures helped inspire some of Escher's works, particularly the Circle Limit series based on hyperbolic tessellations. He also inspired some of the innovations of Buckminster Fuller. Coxeter, M. S. Longuet-Higgins and J. C. P. Miller were the first to publish the full list of uniform polyhedra (1954). He worked for 60 years at the University of Toronto and published twelve books. == Personal life == Coxeter was a vegetarian. He attributed his longevity to his vegetarian diet, daily exercise such as fifty press-ups and standing on his head for fifteen minutes each morning, and consuming a nightly cocktail made from Kahlúa (a coffee liqueur), peach schnapps, and soy milk. == Awards == Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor. He was made a Fellow of the Royal Society in 1950 and in 1997 he was awarded their Sylvester Medal. In 1990, he became a Foreign Member of the American Academy of Arts and Sciences and in 1997 was made a Companion of the Order of Canada. In 1973 he received the Jeffery–Williams Prize. A festschrift in his honour, The Geometric Vein, was published in 1982. It contained 41 essays on geometry, based on a symposium for Coxeter held at Toronto in 1979. A second such volume, The Coxeter Legacy, was published in 2006 based on a Toronto Coxeter symposium held in 2004. == Works == 1940: "Regular and Semi-Regular Polytopes I", Mathematische Zeitschrift 46: 380–407, MR 2,10 doi:10.1007/BF01181449 1942: Non-Euclidean Geometry (1st edition), (2nd ed, 1947), (3rd ed, 1957), (4th ed, 1961), (5th ed, 1965), University of Toronto Press (6th ed, 1998), MAA, ISBN 978-0-88385-522-5. 1954: (with Michael S. Longuet-Higgins and J. C. P. Miller) "Uniform Polyhedra", Philosophical Transactions of the Royal Society A 246: 401–50 doi:10.1098/rsta.1954.0003 1949: The Real Projective Plane 1957: (with W. O. J. Moser) Generators and Relations for Discrete Groups 1980: Second edition, Springer-Verlag ISBN 0-387-09212-9 1961: Introduction to Geometry, (2nd paperback edition 1989, ISBN 978-0-471-50458-0.) 1963: Regular Polytopes (2nd edition), Macmillan Company 1967: (with S. L. Greitzer) Geometry Revisited 1970: Twisted honeycombs (American Mathematical Society, 1970, Regional conference series in mathematics Number 4, ISBN 0-8218-1653-5) 1973: Regular Polytopes, (3rd edition), Dover edition, ISBN 0-486-61480-8 1974: Projective Geometry (2nd edition) 1974: Regular Complex Polytopes, Cambridge University Press, ISBN 978-0-521-20125-4. 1981: (with R. Frucht and D. L. Powers), Zero-Symmetric Graphs, Academic Press, ISBN 978-0-12-194580-0. 1985: "Regular and Semi-Regular Polytopes II", Mathematische Zeitschrift 188: 559–591 1987 Projective Geometry (1987) ISBN 978-0-387-40623-7 1988: "Regular and Semi-Regular Polytopes III", Mathematische Zeitschrift 200: 3–45 1995: F. Arthur Sherk, Peter McMullen, Anthony C. Thompson and Asia Ivić Weiss, editors: Kaleidoscopes — Selected Writings of H. S. M. Coxeter. John Wiley and Sons, ISBN 0-471-01003-0. 1999: The Beauty of Geometry: Twelve Essays, Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 2011: The Fifty-Nine Icosahedra, Tarquin Group, ISBN 978-1-907550-08-9 == See also == Coxeter–James Prize Spiral similarity All pages with titles containing Coxeter == References == == Further reading == Davis, Chandler; Ellers, Erich W, eds. (2006). The Coxeter Legacy: Reflections and Projections. Providence, R.I.: American Mathematical Society. ISBN 978-0-8218-3722-1. OCLC 62282754. Roberts, Siobhan (2006). King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry. New York: Walker & Company. ISBN 978-0-8027-1499-2. OCLC 71436884. == External links == Harold Scott MacDonald Coxeter archival papers held at the University of Toronto Archives and Records Management Services Harold Scott MacDonald Coxeter at the Mathematics Genealogy Project H. S. M. Coxeter (1907–2003), Erich W. Ellers, Branko Grünbaum, Peter McMullen, Asia Ivic Weiss Notices of the AMS: Volume 50, Number 10. www.donaldcoxeter.com www.math.yorku.ca/dcoxeter webpages dedicated to him (in development) Jaron's World: Shapes in Other Dimensions, Discover mag., Apr 2007 The Mathematics in the Art of M.C. Escher video of a lecture by H.S.M. Coxeter, April 28, 2000.
Wikipedia:Harry Dym#0	Harry Dym (Hebrew: הארי דים; January 26, 1938 – July 18, 2024) was an Israeli-born American mathematician at the Weizmann Institute of Science, Israel. Dym's research interests included operator theory, interpolation theory, and inverse problems. Dym earned his Ph.D. in 1965 from the Massachusetts Institute of Technology, under the supervision of Henry McKean. He introduced the Dym equation, which bears his name. Dym died on July 18, 2024, at the age of 86. == Works == as editor with Bernd Fritzsche, Victor Katsnelson, and Bernd Kirstein: Topics in Interpolation Theory, Birkhäuser 1997 Linear Algebra in Action, American Mathematical Society 2007 with H. P. McKean: Fourier Series and Integrals, Academic Press 1974 with H. P. McKean: Gaussian processes, function theory, and the inverse spectral problem, Academic Press 1976, Dover 2008 J {\displaystyle J} contractive matrix functions, reproducing kernel Hilbert spaces and interpolation, AMS 1989 as editor: Topics in Analysis and Operator Theory, Birkhäuser 1989 with Vladimir Bolotnikov: On boundary interpolation for matrix valued Schur functions, AMS 2006 with Damir Z. Arov: J {\displaystyle J} -contractive matrix valued functions and related topics, Cambridge University Press 2008 == Sources == Daniel Alpay, Israel Gohberg, Victor Vinnikov (Herausgeber) Interpolation theory, systems theory, and related topics: the Harry Dym anniversary volume, Birkhäuser 2002. == References == == External links == Homepage [1] Harry Dym's 85th birthday Zoom party
Wikipedia:Harry Trentelman#0	Harry Trentelman is a full professor in Systems and Control at the Johann Bernoulli Institute for Mathematics and Computer Science of the University of Groningen. From 1985 to 1991 he served as an assistant professor and as an associate professor at the Mathematics Department of the Eindhoven University of Technology, the Netherlands. He obtained his PhD degree in Mathematics from the University of Groningen in 1985. His Ph.D. thesis was titled "Almost Invariant Subspaces and High Gain Feedback Mathematics Subject Classification: 93—Systems theory; control" which he defended following studying for it under mentorship from Jan Camiel Willems. Trentelman serves as a senior editor of the IEEE Transactions on Automatic Control and as an associate editor of Automatica. He is past associate editor of the SIAM Journal on Control and Optimization and Systems and Control Letters. Trentelman was named Fellow of the Institute of Electrical and Electronics Engineers (IEEE) in 2015 for "contributions to geometric theory of linear systems and behavioral models". == References == == External links == Harry Trentelman publications indexed by Google Scholar
Wikipedia:Hartley kernel#0	In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse. The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983. The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use. == Definition == The Hartley transform of a function f ( t ) {\displaystyle f(t)} is defined by: H ( ω ) = { H f } ( ω ) = 1 2 π ∫ − ∞ ∞ f ( t ) cas ⁡ ( ω t ) d t , {\displaystyle H(\omega )=\left\{{\mathcal {H}}f\right\}(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{\infty }f(t)\operatorname {cas} (\omega t)\,\mathrm {d} t\,,} where ω {\displaystyle \omega } can in applications be an angular frequency and cas ⁡ ( t ) = cos ⁡ ( t ) + sin ⁡ ( t ) = 2 sin ⁡ ( t + π / 4 ) = 2 cos ⁡ ( t − π / 4 ) , {\displaystyle \operatorname {cas} (t)=\cos(t)+\sin(t)={\sqrt {2}}\sin(t+\pi /4)={\sqrt {2}}\cos(t-\pi /4)\,,} is the cosine-and-sine (cas) or Hartley kernel. In engineering terms, this transform takes a signal (function) from the time-domain to the Hartley spectral domain (frequency domain). === Inverse transform === The Hartley transform has the convenient property of being its own inverse (an involution): f = { H { H f } } . {\displaystyle f=\{{\mathcal {H}}\{{\mathcal {H}}f\}\}\,.} === Conventions === The above is in accord with Hartley's original definition, but (as with the Fourier transform) various minor details are matters of convention and can be changed without altering the essential properties: Instead of using the same transform for forward and inverse, one can remove the 1 / 2 π {\displaystyle {1}/{\sqrt {2\pi }}} from the forward transform and use 1 / 2 π {\displaystyle {1}/{2\pi }} for the inverse—or, indeed, any pair of normalizations whose product is 1 / 2 π {\displaystyle {1}/{2\pi }} . (Such asymmetrical normalizations are sometimes found in both purely mathematical and engineering contexts.) One can also use 2 π ν t {\displaystyle 2\pi \nu t} instead of ω t {\displaystyle \omega t} (i.e., frequency instead of angular frequency), in which case the 1 / 2 π {\displaystyle {1}/{\sqrt {2\pi }}} coefficient is omitted entirely. One can use cos − sin {\displaystyle \cos -\sin } instead of cos + sin {\displaystyle \cos +\sin } as the kernel. == Relation to Fourier transform == This transform differs from the classic Fourier transform F ( ω ) = F { f ( t ) } ( ω ) {\displaystyle F(\omega )={\mathcal {F}}\{f(t)\}(\omega )} in the choice of the kernel. In the Fourier transform, we have the exponential kernel, exp ⁡ ( − i ω t ) = cos ⁡ ( ω t ) − i sin ⁡ ( ω t ) {\displaystyle \exp \left({-\mathrm {i} \omega t}\right)=\cos(\omega t)-\mathrm {i} \sin(\omega t)} , where i {\displaystyle \mathrm {i} } is the imaginary unit. The two transforms are closely related, however, and the Fourier transform (assuming it uses the same 1 / 2 π {\displaystyle 1/{\sqrt {2\pi }}} normalization convention) can be computed from the Hartley transform via: F ( ω ) = H ( ω ) + H ( − ω ) 2 − i H ( ω ) − H ( − ω ) 2 . {\displaystyle F(\omega )={\frac {H(\omega )+H(-\omega )}{2}}-\mathrm {i} {\frac {H(\omega )-H(-\omega )}{2}}\,.} That is, the real and imaginary parts of the Fourier transform are simply given by the even and odd parts of the Hartley transform, respectively. Conversely, for real-valued functions f ( t ) {\displaystyle f(t)} , the Hartley transform is given from the Fourier transform's real and imaginary parts: { H f } = ℜ { F f } − ℑ { F f } = ℜ { F f ⋅ ( 1 + i ) } , {\displaystyle \{{\mathcal {H}}f\}=\Re \{{\mathcal {F}}f\}-\Im \{{\mathcal {F}}f\}=\Re \{{\mathcal {F}}f\cdot (1+\mathrm {i} )\}\,,} where ℜ {\displaystyle \Re } and ℑ {\displaystyle \Im } denote the real and imaginary parts. == Properties == The Hartley transform is a real linear operator, and is symmetric (and Hermitian). From the symmetric and self-inverse properties, it follows that the transform is a unitary operator (indeed, orthogonal). Convolution using Hartley transforms is f ( x ) ∗ g ( x ) = F ( ω ) G ( ω ) + F ( − ω ) G ( ω ) + F ( ω ) G ( − ω ) − F ( − ω ) G ( − ω ) 2 {\displaystyle f(x)*g(x)={\frac {F(\omega )G(\omega )+F(-\omega )G(\omega )+F(\omega )G(-\omega )-F(-\omega )G(-\omega )}{2}}} where F ( ω ) = { H f } ( ω ) {\displaystyle F(\omega )=\{{\mathcal {H}}f\}(\omega )} and G ( ω ) = { H g } ( ω ) {\displaystyle G(\omega )=\{{\mathcal {H}}g\}(\omega )} Similar to the Fourier transform, the Hartley transform of an even/odd function is even/odd, respectively. === cas === The properties of the Hartley kernel, for which Hartley introduced the name cas for the function (from cosine and sine) in 1942, follow directly from trigonometry, and its definition as a phase-shifted trigonometric function cas ⁡ ( t ) = 2 sin ⁡ ( t + π / 4 ) = sin ⁡ ( t ) + cos ⁡ ( t ) {\displaystyle \operatorname {cas} (t)={\sqrt {2}}\sin(t+\pi /4)=\sin(t)+\cos(t)} . For example, it has an angle-addition identity of: 2 cas ⁡ ( a + b ) = cas ⁡ ( a ) cas ⁡ ( b ) + cas ⁡ ( − a ) cas ⁡ ( b ) + cas ⁡ ( a ) cas ⁡ ( − b ) − cas ⁡ ( − a ) cas ⁡ ( − b ) . {\displaystyle 2\operatorname {cas} (a+b)=\operatorname {cas} (a)\operatorname {cas} (b)+\operatorname {cas} (-a)\operatorname {cas} (b)+\operatorname {cas} (a)\operatorname {cas} (-b)-\operatorname {cas} (-a)\operatorname {cas} (-b)\,.} Additionally: cas ⁡ ( a + b ) = cos ⁡ ( a ) cas ⁡ ( b ) + sin ⁡ ( a ) cas ⁡ ( − b ) = cos ⁡ ( b ) cas ⁡ ( a ) + sin ⁡ ( b ) cas ⁡ ( − a ) , {\displaystyle \operatorname {cas} (a+b)={\cos(a)\operatorname {cas} (b)}+{\sin(a)\operatorname {cas} (-b)}=\cos(b)\operatorname {cas} (a)+\sin(b)\operatorname {cas} (-a)\,,} and its derivative is given by: cas ′ ⁡ ( a ) = d d a cas ⁡ ( a ) = cos ⁡ ( a ) − sin ⁡ ( a ) = cas ⁡ ( − a ) . {\displaystyle \operatorname {cas} '(a)={\frac {d}{da}}\operatorname {cas} (a)=\cos(a)-\sin(a)=\operatorname {cas} (-a)\,.} == See also == cis (mathematics) Fractional Fourier transform == References == Bracewell, Ronald N. (1986). Written at Stanford, California, USA. The Hartley Transform. Oxford Engineering Science Series. Vol. 19 (1 ed.). New York, NY, USA: Oxford University Press, Inc. ISBN 0-19-503969-6. (NB. Also translated into German and Russian.) Bracewell, Ronald N. (1994). "Aspects of the Hartley transform". Proceedings of the IEEE. 82 (3): 381–387. doi:10.1109/5.272142. Millane, Rick P. (1994). "Analytic properties of the Hartley transform". Proceedings of the IEEE. 82 (3): 413–428. doi:10.1109/5.272146. == Further reading == Olnejniczak, Kraig J.; Heydt, Gerald T., eds. (March 1994). "Scanning the Special Section on the Hartley transform". Special Issue on Hartley transform. Vol. 82. Proceedings of the IEEE. pp. 372–380. Retrieved 2017-10-31. (NB. Contains extensive bibliography.)
Wikipedia:Hartman–Grobman theorem#0	In mathematics, in the study of dynamical systems, the Hartman–Grobman theorem or linearisation theorem is a theorem about the local behaviour of dynamical systems in the neighbourhood of a hyperbolic equilibrium point. It asserts that linearisation—a natural simplification of the system—is effective in predicting qualitative patterns of behaviour. The theorem owes its name to Philip Hartman and David M. Grobman. The theorem states that the behaviour of a dynamical system in a domain near a hyperbolic equilibrium point is qualitatively the same as the behaviour of its linearization near this equilibrium point, where hyperbolicity means that no eigenvalue of the linearization has real part equal to zero. Therefore, when dealing with such dynamical systems one can use the simpler linearization of the system to analyse its behaviour around equilibria. == Main theorem == Consider a system evolving in time with state u ( t ) ∈ R n {\displaystyle u(t)\in \mathbb {R} ^{n}} that satisfies the differential equation d u / d t = f ( u ) {\displaystyle du/dt=f(u)} for some smooth map f : R n → R n {\displaystyle f\colon \mathbb {R} ^{n}\to \mathbb {R} ^{n}} . Now suppose the map has a hyperbolic equilibrium state u ∗ ∈ R n {\displaystyle u^{}\in \mathbb {R} ^{n}} : that is, f ( u ∗ ) = 0 {\displaystyle f(u^{})=0} and the Jacobian matrix A = [ ∂ f i / ∂ x j ] {\displaystyle A=[\partial f_{i}/\partial x_{j}]} of f {\displaystyle f} at state u ∗ {\displaystyle u^{}} has no eigenvalue with real part equal to zero. Then there exists a neighbourhood N {\displaystyle N} of the equilibrium u ∗ {\displaystyle u^{}} and a homeomorphism h : N → R n {\displaystyle h\colon N\to \mathbb {R} ^{n}} , such that h ( u ∗ ) = 0 {\displaystyle h(u^{})=0} and such that in the neighbourhood N {\displaystyle N} the flow of d u / d t = f ( u ) {\displaystyle du/dt=f(u)} is topologically conjugate by the continuous map U = h ( u ) {\displaystyle U=h(u)} to the flow of its linearisation d U / d t = A U {\displaystyle dU/dt=AU} . A like result holds for iterated maps, and for fixed points of flows or maps on manifolds. A mere topological conjugacy does not provide geometric information about the behavior near the equilibrium. Indeed, neighborhoods of any two equilibria are topologically conjugate so long as the dimensions of the contracting directions (negative eigenvalues) match and the dimensions of the expanding directions (positive eigenvalues) match. But the topological conjugacy in this context does provide the full geometric picture. In effect, the nonlinear phase portrait near the equilibrium is a thumbnail of the phase portrait of the linearized system. This is the meaning of the following regularity results, and it is illustrated by the saddle equilibrium in the example below. Even for infinitely differentiable maps f {\displaystyle f} , the homeomorphism h {\displaystyle h} need not to be smooth, nor even locally Lipschitz. However, it turns out to be Hölder continuous, with exponent arbitrarily close to 1. Moreover, on a surface, i.e., in dimension 2, the linearizing homeomorphism and its inverse are continuously differentiable (with, as in the example below, the differential at the equilibrium being the identity) but need not be C 2 {\displaystyle C^{2}} . And in any dimension, if f {\displaystyle f} has Hölder continuous derivative, then the linearizing homeomorphism is differentiable at the equilibrium and its differential at the equilibrium is the identity. The Hartman–Grobman theorem has been extended to infinite-dimensional Banach spaces, non-autonomous systems d u / d t = f ( u , t ) {\displaystyle du/dt=f(u,t)} (potentially stochastic), and to cater for the topological differences that occur when there are eigenvalues with zero or near-zero real-part. == Example == The algebra necessary for this example is easily carried out by a web service that computes normal form coordinate transforms of systems of differential equations, autonomous or non-autonomous, deterministic or stochastic. Consider the 2D system in variables u = ( y , z ) {\displaystyle u=(y,z)} evolving according to the pair of coupled differential equations d y d t = − 3 y + y z and d z d t = z + y 2 . {\displaystyle {\frac {dy}{dt}}=-3y+yz\quad {\text{and}}\quad {\frac {dz}{dt}}=z+y^{2}.} By direct computation it can be seen that the only equilibrium of this system lies at the origin, that is u ∗ = 0 {\displaystyle u^{}=0} . The coordinate transform, u = h − 1 ( U ) {\displaystyle u=h^{-1}(U)} where U = ( Y , Z ) {\displaystyle U=(Y,Z)} , given by y ≈ Y + Y Z + 1 42 Y 3 + 1 2 Y Z 2 z ≈ Z − 1 7 Y 2 − 1 3 Y 2 Z {\displaystyle {\begin{aligned}y&\approx Y+YZ+{\tfrac {1}{42}}Y^{3}+{\tfrac {1}{2}}YZ^{2}\\[5pt]z&\approx Z-{\tfrac {1}{7}}Y^{2}-{\tfrac {1}{3}}Y^{2}Z\end{aligned}}} is a smooth map between the original u = ( y , z ) {\displaystyle u=(y,z)} and new U = ( Y , Z ) {\displaystyle U=(Y,Z)} coordinates, at least near the equilibrium at the origin. In the new coordinates the dynamical system transforms to its linearisation d Y d t = − 3 Y and d Z d t = Z . {\displaystyle {\frac {dY}{dt}}=-3Y\quad {\text{and}}\quad {\frac {dZ}{dt}}=Z.} That is, a distorted version of the linearisation gives the original dynamics in some finite neighbourhood. == See also == Linear approximation Stable manifold theorem == References == == Further reading == Irwin, Michael C. (2001). "Linearization". Smooth Dynamical Systems. World Scientific. pp. 109–142. ISBN 981-02-4599-8. Perko, Lawrence (2001). Differential Equations and Dynamical Systems (Third ed.). New York: Springer. pp. 119–127. ISBN 0-387-95116-4. Robinson, Clark (1995). Dynamical Systems : Stability, Symbolic Dynamics, and Chaos. Boca Raton: CRC Press. pp. 156–165. ISBN 0-8493-8493-1. == External links == Coayla-Teran, E.; Mohammed, S.; Ruffino, P. (February 2007). "Hartman–Grobman Theorems along Hyperbolic Stationary Trajectories". Discrete and Continuous Dynamical Systems. 17 (2): 281–292. doi:10.3934/dcds.2007.17.281. Teschl, Gerald (2012). Ordinary Differential Equations and Dynamical Systems. Providence: American Mathematical Society. ISBN 978-0-8218-8328-0. "The Most Addictive Theorem in Applied Mathematics". Scientific American.
Wikipedia:Hasan Abu-Libdeh#0	Hasan Abu-Libdeh (Arabic: حسن أبو لبدة; born 1954) is a Palestinian statistician and politician, who founded the Palestinian Central Bureau of Statistics in 1993. He served in the Palestinian National Authority as Minister of Labour, Social Affairs, and National Economy. == Biography == Hasan Abu-Libdeh was born in Arrabeh, West Bank in 1954. He completed a Bachelor's degree in mathematics at Birzeit University in 1979, and an M.Sc. in mathematical statistics at Stanford University in 1981. Abu-Libdeh later received an M.Sc. in applied statistics in 1986 and a Ph.D. in biostatistics in 1988 from Cornell University. He worked as assistant professor at Birzeit University from 1988 to 1991. Abu-Libdeh founded the Palestinian Central Bureau of Statistics in 1993, becoming its first director and administering its controversial first census in 1997, which he called "as important as the intifada". Alongside his ministerial positions in the Palestinian Authority, Abu-Libdeh served as Deputy Director of the Palestinian Economic Council for Development and Reconstruction and twice as Cabinet Secretary. He also worked as chief executive of the 2008 Palestine Investment Conference. On 29 November 2011, the Palestinian prosecutor-general charged Abu Libdeh with corruption, with charges including breach of trust, fraud, insider trading, and embezzlement of public funds. == References ==
Wikipedia:Hasan Tahsini#0	Hoxhë Hasan Tahsini or simply Hoxha Tahsim (7 April 1811 – 3 July 1881) was an Albanian alim, astronomer, mathematician and philosopher. He was the first rector of Istanbul University and one of the founders of the Central Committee for Defending Albanian Rights. Tahsini is regarded as one of the most prominent scholars of the Ottoman Empire of the 19th century. == Early life == Hasan Tahsini was born in 1811 in the village of Ninat, Konispol, then part of the Ottoman Empire. His father Osman Efendi Rushiti was a member of the ulama. When he was young he worked as tutor to the sons of Hayrullah Efendi, Minister of Education of the Ottoman Empire. Hayrullah Efendi later appointed Tahsini to the staff of the Ottoman school of Paris, where Tahsini taught Turkish and religious sciences, while also being the imam of the Ottoman embassy and a student of mathematics and natural sciences at the University of Paris. He studied in Paris for twelve years after being sent there by Resid Pasha, who was trying to create a Westernized ulama elite. In 1869 Tahsini returned in the Ottoman Empire to accompany the body of Fuad Pasha who had died in Nice. His brother is Ferik Hoxha, his nephew is Xhaferr Hoxha, the father of Bilal Xhaferi. == Istanbul University == In 1870 he became the first rector of the newly established Istanbul University, where he gave lectures on physics, astronomy and psychology. The government appointed Tahsini as rector as it was believed that he could establish a balance between western European and Islamic methods and ideologies. However, at that time Tahsini's scientific research and unreserved liberalism led to him being frequently attacked by conservative ulama circles The attacks against Tahsini began when he conducted experiments in order to illustrate the notion of vacuum to his students. Tahsini placed a pigeon underneath a glass bell, emptied the receptacle and the pigeon eventually suffocated proving Tahsini's theory. The conservative circles considered Tahsini's experiment as evidence of witchcraft and performance of magic. After the experiment he was declared a heretic through a fatwa, dismissed from the university, and disallowed to give lectures. The university was also closed for a period because Jamal-al-Din Afghani, another professor influenced by Tahsini, supported his theories. == Works == Tahsini wrote the first Turkish language treatise on psychology titled Psychology or the Science of Soul, a work influenced by modernism and the first book whose title contained the word psychology. He also wrote the first Turkish-language book on modern astronomy being also the first popular science book in Turkish. Other works of Tahsini in the Turkish language include a translation of Constantin François de Chassebœuf's Loi Naturelle. Tahsini was a prominent member of the Central Committee for Defending Albanian Rights established in Istanbul, 1877. The Committee for Defending Albanian Rights appointed Tahsini along with Sami Frashëri, Pashko Vasa and Jani Vreto to create an Albanian alphabet which by 19 March 1879 the group approved Frashëri's 36 letter alphabet consisting mostly of Latin characters. Tahsini during the process had worked with Sami Frasheri, one of the most important figures of the Albanian National Awakening to develop a unique alphabet of the Albanian language. According to Tahsini the alphabet was developed in such way that each letter required the least hand movements to be written. == Sources ==
Wikipedia:Hasibun Naher#0	Hasibun Naher is a Bangladeshi applied mathematics researcher and educator. In February 2018, she was one of five young women from developing countries to receive the OWSD-Elsevier Foundation Award. Her research has included the application of mathematics to tsunamis in order to improve predictions of how they develop. She is currently associate professor of mathematics at BRAC University, Dhaka. In 2019, she became a laureate of the Asian Scientist 100 by the Asian Scientist. == Biography == Dr. Hasibun Naher completed her Bachelor's of Science as well as her Master of Sciences in Mathematics from Jahangirnagar University, Savar, Bangladesh and secured first class in both. She went on to receive her PhD from the school of Mathematical Sciences, at the Universiti Sains Malaysia (USM), Penang, Malaysia. She taught undergraduate courses at the School of Mathematical Sciences, USM from 2011 to 2013. Naher joined BRAC University in 2007 and is an Associate Professor in mathematics at their Department of Mathematics and Natural Sciences. == Publications == === Books === Naher, H. (2005). Secondary algebra (1st ed.). Dhaka: Mullick, A. A. Uraka Book House. Naher, H. (2014). New travelling wave solutions of some nonlinear partial differential equations via extension of (G’/G) - expansion method (1st ed.). Penang: Universiti Sains Malaysia. Naher, H. (2006). Juniour algebra (1st ed.). Dhaka: Mullick, A. A. Uraka Book House. === Journal articles === Naher, H., & Tanim, T. (2018). Active learning strategies in mathematics and science. 21st Century Education Forum @ Harvard 2018, 16(1), 18–31. Retrieved from https://www.21caf.org/21cefharvard-cp.html Khan, A. T., & Naher, H. (2018). Some new non-travelling wave solutions of the fisher equation with nonlinear auxiliary equation. Oriental Journal of Physical Science, 7(9). https://doi.org/10.13005/OJPS03.02.04 Naher, H., & Abdullah, F. A. (2017). New Generalized (G’/G)-expansion Method to the Zhiber-Shabat Equation and Liouville Equations. In M. N. (Ed.), 1st International Conference on Applied and Industrial Mathematics and Statistics 2017, ICoAIMS 2017 (Vol. 890). https://doi.org/10.1088/1742-6596/890/1/012018 Hassan, Q. M. U., Naher, H., Abdullah, F., & Mohyud-Din, S. T. (2016). Solutions of the nonlinear evolution equation via the generalized riccati equation mapping together with the (Gʹ=g)-expansion method. Journal of Computational Analysis and Applications, 21(1), 62–82. Naher, H., & Abdullah, F. A. (2016). Further extension of the generalized and improved (G’/G)-expansion method for nonlinear evolution equation. Journal of the Association of Arab Universities for Basic and Applied Sciences, 19, 52–58. https://doi.org/10.1016/j.jaubas.2014.05.005 Naher, H. (2015). New approach of (G′/G)-expansion method and new approach of generalized (G′/G)-expansion method for ZKBBM equation. Journal of the Egyptian Mathematical Society, 23(1), 42–48. https://doi.org/10.1016/j.joems.2014.03.005 Naher, H., Abdullah, F. A., & Bekir, A. (2015). Some new traveling wave solutions of the modified Benjamin-Bona-Mahony equation via the improved (G’/G)-expansion method. New Trends in Mathematical Sciences, 3(1), 78–89. Naher, H., Abdullah, F. A., & Rashid, A. (2014). Some new solutions of the (3+1)-Dimensional Jimbo-Miwa equation via the improved (G’/G)-Expansion method. Journal of Computational Analysis and Applications, 17(2), 287–296. Naher, H., & Abdullah, F. A. (2014). New approach of (G’/G)-expansion method for RLW equation. Research Journal of Applied Sciences, Engineering and Technology, 7(23), 4864–4871. https://doi.org/10.19026/rjaset.7.876 Naher, H., & Abdullah, F. A. (2014). Some new solutions of the (1+1)-dimensional PDE via the improved (G ′/ G)-expansion method. 21st National Symposium on Mathematical Sciences: Germination of Mathematical Sciences Education and Research Towards Global Sustainability, SKSM 21, 1605, 446–451. https://doi.org/10.1063/1.4887630 Naher, H., Abdullah, F. A., & Rashid, A. (2014). The generalized riccati equation together with the (G’ / G) -Expansion method for the (3+1)-dimensional modified KDV-zakharov-kuznetsov equation. UPB Scientific Bulletin, Series A: Applied Mathematics and Physics, 76(3), 77–90. Naher, H., & Abdullah, F. A. (2014). The improved (G’/ G)-expansion method to the (2+1)-dimensional breaking soliton equation. Journal of Computational Analysis and Applications, 16(2), 220–235. Naher, H., Abdullah, F. A., & Mohyud-Din, S. T. (2013). Extended generalized Riccati equation mapping method for the fifth-order Sawada-Kotera equation. AIP Advances, 3(5). https://doi.org/10.1063/1.4804433 Naher, H., Abdullah, F. A., & Akbar, M. A. (2013). Generalized and Improved (G′/G)-Expansion Method for (3+1)-Dimensional Modified KdV-Zakharov-Kuznetsev Equation. PLoS ONE, 8(5). https://doi.org/10.1371/journal.pone.0064618 Naher, H., Abdullah, F. A., Akbar, M. A., & Yildirim, A. (2013). The extended generalized Riccati equation mapping method for the (1+1)-dimensional modified KdV equation. World Applied Sciences Journal, 25(4), 543–553. https://doi.org/10.5829/idosi.wasj.2013.25.04.2980 Naher, H., & Abdullah, F. A. (2013). New approach of (G′G)-expansion method and new approach of generalized (G′G)-expansion method for nonlinear evolution equation. AIP Advances, 3(3). https://doi.org/10.1063/1.4794947 Naher, H., Abdullah, F. A., & Akbar, M. A. (2012). New traveling wave solutions of the higher dimensional nonlinear partial differential equation by the exp-function method. Journal of Applied Mathematics, 2012. https://doi.org/10.1155/2012/575387 Naher, H., & Abdullah, F. A. (2012). The improved (G’/G) -expansion method for the (2+1)-dimensional modified Zakharov-Kuznetsov equation. Journal of Applied Mathematics, 2012. https://doi.org/10.1155/2012/438928 Naher, H., & Abdullah, F. A. (2012). Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G′/G)-expansion method. Mathematical Problems in Engineering, 2012. https://doi.org/10.1155/2012/871724 Naher, H., & Abdullah, F. A. (2012). Some new traveling wave solutions of the nonlinear reaction diffusion equation by using the improved (G’/G) -expansion method. Mathematical Problems in Engineering, 2012, 17. https://doi.org/10.1155/2012/871724 Naher, H., & Abdullah, F. A. (2012). New traveling wave solutions by the extended generalized Riccati equation mapping method of the (2 + 1) -dimensional evolution equation. Journal of Applied Mathematics, 2012. https://doi.org/10.1155/2012/486458 Naher, H., Abdullah, F. A., & Bekir, A. (2012). Abundant traveling wave solutions of the compound KdV-Burgers equation via the improved (G′/G)-expansion method. AIP Advances, 2(4). https://doi.org/10.1063/1.4769751 Naher, H., Abdullah, F. A., & Ali Akbar, M. (2012). New traveling wave solutions of the higher dimensional nonlinear evolution equation by the improved (G′/G) expansion method. World Applied Sciences Journal, 16(1), 11–21. Naher, H., & Abdullah, F. A. (2012). The modified benjamin-bona-mahony equation via the extended generalized riccati equation mapping method. Applied Mathematical Sciences, 6(109–112), 5495–5512. Naher, H., Abdullah, F. A., & Akbar, M. A. (2011). The (G’/G)-expansion method for abundant traveling wave solutions of Caudrey-Dodd-Gibbon equation. Mathematical Problems in Engineering, 2011. https://doi.org/10.1155/2011/218216 Naher, H., Abdullah, F. A., & Ali Akbar, M. (2011). The exp-function method for new exact solutions of the nonlinear partial differential equations. International Journal of Physical Sciences, 6(29), 6706–6716. https://doi.org/10.5897/IJPS11.1026 === Conferences === Naher, H., & Tanim, T. (2018). Active learning strategies in mathematics and science. 21st Century Education Forum @ Harvard 2018, 16(1), 18–31. Retrieved from https://www.21caf.org/21cefharvard-cp.html Naher, H., Tanim, T., & Sultana, N. (2018). Active learning to promote student engagement in undergraduate level. 10th ICRTEL 2018 – International Conference on Research in Teaching, Education & Learning. Bali, Indonesia: International Journal of Social Sciences. Naher, H., & Abdullah, F. A. (2017). New Generalized (G’/G)-expansion Method to the Zhiber-Shabat Equation and Liouville Equations. In M. N. (Ed.), 1st International Conference on Applied and Industrial Mathematics and Statistics 2017, ICoAIMS 2017 (Vol. 890). https://doi.org/10.1088/1742-6596/890/1/012018 Naher, H. (2016). Analytical solutions of coupled nonlinear evolution equations in mathematical physics. The 3rd Computational Mathematics and Applications Conference (CMA 2016). Bangkok, Thailand. Naher, H. (2015). New generalized (G’/G)-expansion method to the higher dimensional nonlinear evolution equation. The 5th World Congress on Engineering and Technology (CET 2015). Retrieved from http://www.engii.org/cet2015/ShowKeyNoteSpeakerDetails.aspx?personID=4178 Naher, H., & Abdullah, F. A. (2014). Some new solutions of the (1+1)-dimensional PDE via the improved (G ′/ G)-expansion method. 21st National Symposium on Mathematical Sciences: Germination of Mathematical Sciences Education and Research Towards Global Sustainability, SKSM 21, 1605, 446–451. https://doi.org/10.1063/1.4887630 Naher, H. (2013). Some new solutions of the (1+1)-dimensional PDE via the improved (G′/G)-expansion method. Proceedings of the 21st National Symposium on Mathematical Sciences (SKSM21. Penag, Malaysia: Germination of Mathematical Sciences Education and Research, towards Global Sustainability. Naher, H., & Abdullah, F. A. (2012). New exact traveling wave solutions for the higher dimensional nonlinear evolution equation by using the improved (G’/G)-expansion method. 2nd Regional Conference on Applied and Engineering Mathematics (RCAEM-II), 681–684. === Online === Academic Impact. (2019). Women and girls in science podcast series: Mathematician Hasibun Naher. Retrieved from Academic Impact website: https://academicimpact.un.org/content/women-and-girls-science-podcast-se... 26 fantastic female scientists. (2019). Retrieved from Asian Scientist website: https://www.asianscientist.com/2019/03/features/26-fantastic-female-scie... Chan, J. (2018). Asia’s rising scientists: Hasibun Naher. Retrieved from Asian Scientist website: https://www.asianscientist.com/2018/09/features/asias-rising-scientists-... The Daily Star. (2018). Bangladeshi scholar wins int’l award. Retrieved from The Daily Star website: https://www.thedailystar.net/city/bangladeshi-scholar-wins-intl-award-15... Dhaka Tribune. (2018). Bangladeshi academic wins international award. Retrieved from Dhaka Tribune website: https://www.dhakatribune.com/bangladesh/2018/02/19/hasibun-naher-2018-ow... Bert, A., & Francescon, D. (2018). Elsevier at #AAASmtg: live updates with Women in Science winners: 5 researchers from developing countries are preparing to accept OWSD-Elsevier Foundation Awards for their work in the physical sciences. Retrieved from Elsevier website: https://www.elsevier.com/connect/elsevier-at-aaasmtg-live-updates-with-w... Rueda, A. (2018). Prize awarded to women scientists from developing world. Retrieved from SciDev.Net website: https://www.scidev.net/global/gender/news/prize-awarded-to-women-scienti... The Elsevier Foundation. (2018). Five women scientists in developing countries win 2018 OWSD-Elsevier Foundation Awards. Retrieved from The Elsevier Foundation website: https://elsevierfoundation.org/five-women-scientists-in-developing-count... Bert, A. (2018). For this scientist, her name was her destiny: Dr. Hasibun Naher of Bangladesh builds mathematical models to predict tsunamis and earthquakes. Retrieved from Elsevier website: https://www.elsevier.com/connect/for-this-scientist-her-name-was-her-des... Five women scientists win 2018 OWSD-Elsevier Foundation Awards. (2018). Retrieved from The World Academy of Sciences website: https://twas.org/article/five-women-scientists-win-2018-owsd-elsevier-fo... Prize awarded to women scientists from developing world. (2018). Retrieved from The Institute of Pharmaceutical Sciences (TIPS) website: http://tips.tums.ac.ir/cpages/mainpage.asp?I=S1M7P1C217 Bangladeshi academic wins international award. (2018). Retrieved from Daily Ajker Ograbani website: http://ajkerograbani.com/en/bangladeshi-academic-wins-international-award/ == References ==
Wikipedia:Hassan Ugail#0	Hassan Ugail (born 24 September 1970) is a Maldivian mathematician and computer scientist. He is a professor of visual computing at the Faculty of Engineering and Informatics at the University of Bradford. == Early life and education == Hassan Ugail was born in Hithadhoo, Addu City, in Seenu Atoll, Maldives. In 1987, he moved to Malé to continue his education at the English Preparatory And Secondary School and at the Centre for Higher Secondary Education. In 1992, he received a British Council scholarship to continue his studies in the UK. Ugail received a B.Sc. degree in Mathematics in 1995 and a postgraduate certificate in 1996, both from King's College London. He earned his PhD in Visual Computing at the University of Leeds in 1999. His doctoral research focused on the application of partial differential equations in interactive surface design. == Career == After completing his PhD, Ugail worked as a post-doctoral research fellow at the Department of Applied Mathematics at University of Leeds until September 2002. He then became a lecturer at the School of Informatics at the University of Bradford. He was appointed as a senior lecturer in April 2005, and became a professor in 2009. In 2010, Ugail received the Vice-Chancellor's Excellence in Knowledge Transfer Award from the University of Bradford. In 2011, Ugail received the Maldives National Award for Innovation for his work in the field of visual computing. Ugail is the director of the Centre for Visual Computing at the University of Bradford. Ugail is known for his work on computer-based human face analysis including facial recognition, face ageing, emotion analysis and lie detection. In 2018, Ugail worked with Bellingcat journalists to verify the identities of two suspected Russian spies involved in the Salisbury Novichok poisoning case. In 2019, he helped the Commission on Deaths and Disappearances to investigate cold cases such as Ahmed Rilwan, Yameen Rasheed and Afrasheem Ali via his lie detection services as well as his face-recognition system. In 2020, BBC News investigators consulted Ugail as an expert in facial mapping to identify an alleged Nazi war criminal. As of 2023, Ugail's team is working on image analysis as part of a project to assess the quality of human organs for transplant. The project is supported by NHS Blood and Transplant, Quality in Organ Donation biobank, and the National Institute for Health and Care Research. == Bibliography == Ugail, Hassan (2022). Deep Learning in Visual Computing, Explanations and Examples. Ugail, Hassan (2020). Multidisciplinary Data Visualization. Ugail, Hassan (2019). Computational Techniques for Human Smile Analysis. Ugail, Hassan (2011). Partial Differential Equations for Geometric Design. == References == == External links == University of Bradford profile ResearchGate profile
Wikipedia:Hasse derivative#0	In mathematics, the Hasse derivative is a generalisation of the derivative which allows the formulation of Taylor's theorem in coordinate rings of algebraic varieties. == Definition == Let k[X] be a polynomial ring over a field k. The r-th Hasse derivative of Xn is D ( r ) X n = ( n r ) X n − r , {\displaystyle D^{(r)}X^{n}={\binom {n}{r}}X^{n-r},} if n ≥ r and zero otherwise. In characteristic zero we have D ( r ) = 1 r ! ( d d X ) r . {\displaystyle D^{(r)}={\frac {1}{r!}}\left({\frac {\mathrm {d} }{\mathrm {d} X}}\right)^{r}\ .} == Properties == The Hasse derivative is a generalized derivation on k[X] and extends to a generalized derivation on the function field k(X), satisfying an analogue of the product rule D ( r ) ( f g ) = ∑ i = 0 r D ( i ) ( f ) D ( r − i ) ( g ) {\displaystyle D^{(r)}(fg)=\sum _{i=0}^{r}D^{(i)}(f)D^{(r-i)}(g)} and an analogue of the chain rule. Note that the D ( r ) {\displaystyle D^{(r)}} are not themselves derivations in general, but are closely related. A form of Taylor's theorem holds for a function f defined in terms of a local parameter t on an algebraic variety: f = ∑ r D ( r ) ( f ) ⋅ t r . {\displaystyle f=\sum _{r}D^{(r)}(f)\cdot t^{r}\ .} == Notes == == References == Goldschmidt, David M. (2003). Algebraic functions and projective curves. Graduate Texts in Mathematics. Vol. 215. New York, NY: Springer-Verlag. ISBN 0-387-95432-5. Zbl 1034.14011.
Wikipedia:Hasse–Schmidt derivation#0	In mathematics, a Hasse–Schmidt derivation is an extension of the notion of a derivation. The concept was introduced by Schmidt & Hasse (1937). == Definition == For a (not necessarily commutative nor associative) ring B and a B-algebra A, a Hasse–Schmidt derivation is a map of B-algebras D : A → A [ [ t ] ] {\displaystyle D:A\to A[\![t]\!]} taking values in the ring of formal power series with coefficients in A. This definition is found in several places, such as Gatto & Salehyan (2016, §3.4), which also contains the following example: for A being the ring of infinitely differentiable functions (defined on, say, Rn) and B=R, the map f ↦ exp ⁡ ( t d d x ) f ( x ) = f + t d f d x + t 2 2 d 2 f d x 2 + ⋯ {\displaystyle f\mapsto \exp \left(t{\frac {d}{dx}}\right)f(x)=f+t{\frac {df}{dx}}+{\frac {t^{2}}{2}}{\frac {d^{2}f}{dx^{2}}}+\cdots } is a Hasse–Schmidt derivation, as follows from applying the Leibniz rule iteratedly. == Equivalent characterizations == Hazewinkel (2012) shows that a Hasse–Schmidt derivation is equivalent to an action of the bialgebra NSymm = Z ⟨ Z 1 , Z 2 , … ⟩ {\displaystyle \operatorname {NSymm} =\mathbf {Z} \langle Z_{1},Z_{2},\ldots \rangle } of noncommutative symmetric functions in countably many variables Z1, Z2, ...: the part D i : A → A {\displaystyle D_{i}:A\to A} of D which picks the coefficient of t i {\displaystyle t^{i}} , is the action of the indeterminate Zi. == Applications == Hasse–Schmidt derivations on the exterior algebra A = ⋀ M {\textstyle A=\bigwedge M} of some B-module M have been studied by Gatto & Salehyan (2016, §4). Basic properties of derivations in this context lead to a conceptual proof of the Cayley–Hamilton theorem. See also Gatto & Scherbak (2015). == References == Gatto, Letterio; Salehyan, Parham (2016), Hasse–Schmidt derivations on Grassmann algebras, Springer, doi:10.1007/978-3-319-31842-4, ISBN 978-3-319-31842-4, MR 3524604 Gatto, Letterio; Scherbak, Inna (2015), Remarks on the Cayley-Hamilton Theorem, arXiv:1510.03022 Hazewinkel, Michiel (2012), "Hasse–Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions", Axioms, 1 (2): 149–154, arXiv:1110.6108, doi:10.3390/axioms1020149, S2CID 15969581 Schmidt, F.K.; Hasse, H. (1937), "Noch eine Begründung der Theorie der höheren Differentialquotienten in einem algebraischen Funktionenkörper einer Unbestimmten. (Nach einer brieflichen Mitteilung von F.K. Schmidt in Jena)", J. Reine Angew. Math., 1937 (177): 215–237, doi:10.1515/crll.1937.177.215, ISSN 0075-4102, MR 1581557, S2CID 120317012
Wikipedia:Hat notation#0	A "hat" (circumflex (ˆ)), placed over a symbol is a mathematical notation with various uses. == Estimated value == In statistics, a circumflex (ˆ), called a "hat", is used to denote an estimator or an estimated value. For example, in the context of errors and residuals, the "hat" over the letter ε ^ {\displaystyle {\hat {\varepsilon }}} indicates an observable estimate (the residuals) of an unobservable quantity called ε {\displaystyle \varepsilon } (the statistical errors). Another example of the hat operator denoting an estimator occurs in simple linear regression. Assuming a model of y i = β 0 + β 1 x i + ε i {\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i}} , with observations of independent variable data x i {\displaystyle x_{i}} and dependent variable data y i {\displaystyle y_{i}} , the estimated model is of the form y ^ i = β ^ 0 + β ^ 1 x i {\displaystyle {\hat {y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}x_{i}} where ∑ i ( y i − y ^ i ) 2 {\displaystyle \sum _{i}(y_{i}-{\hat {y}}_{i})^{2}} is commonly minimized via least squares by finding optimal values of β ^ 0 {\displaystyle {\hat {\beta }}_{0}} and β ^ 1 {\displaystyle {\hat {\beta }}_{1}} for the observed data. == Hat matrix == In statistics, the hat matrix H projects the observed values y of response variable to the predicted values ŷ: y ^ = H y . {\displaystyle {\hat {\mathbf {y} }}=H\mathbf {y} .} == Cross product == In screw theory, one use of the hat operator is to represent the cross product operation. Since the cross product is a linear transformation, it can be represented as a matrix. The hat operator takes a vector and transforms it into its equivalent matrix. a × b = a ^ b {\displaystyle \mathbf {a} \times \mathbf {b} =\mathbf {\hat {a}} \mathbf {b} } For example, in three dimensions, a × b = [ a x a y a z ] × [ b x b y b z ] = [ 0 − a z a y a z 0 − a x − a y a x 0 ] [ b x b y b z ] = a ^ b {\displaystyle \mathbf {a} \times \mathbf {b} ={\begin{bmatrix}a_{x}\\a_{y}\\a_{z}\end{bmatrix}}\times {\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}={\begin{bmatrix}0&-a_{z}&a_{y}\\a_{z}&0&-a_{x}\\-a_{y}&a_{x}&0\end{bmatrix}}{\begin{bmatrix}b_{x}\\b_{y}\\b_{z}\end{bmatrix}}=\mathbf {\hat {a}} \mathbf {b} } == Unit vector == In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in v ^ {\displaystyle {\hat {\mathbf {v} }}} (pronounced "v-hat"). This is especially common in physics context. == Fourier transform == The Fourier transform of a function f {\displaystyle f} is traditionally denoted by f ^ {\displaystyle {\hat {f}}} . == Operator == In quantum mechanics, operators are denoted with hat notation. For instance, see the time-independent Schrödinger equation, where the Hamiltonian operator is denoted H ^ {\displaystyle {\hat {H}}} . H ^ ψ = E ψ {\displaystyle {\hat {H}}\psi =E\psi } == See also == Exterior algebra – Algebra associated to any vector space Glossary of mathematical symbols Top-hat filter – signal filtering techniquePages displaying wikidata descriptions as a fallback Circumflex – Diacritic (◌̂) in European scripts == References ==

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