source
stringlengths 16
98
| text
stringlengths 40
168k
|
|---|---|
Wikipedia:Hausdorff dimension#0
|
In mathematics, Hausdorff dimension is a measure of roughness, or more specifically, fractal dimension, that was introduced in 1918 by mathematician Felix Hausdorff. For instance, the Hausdorff dimension of a single point is zero, of a line segment is 1, of a square is 2, and of a cube is 3. That is, for sets of points that define a smooth shape or a shape that has a small number of corners—the shapes of traditional geometry and science—the Hausdorff dimension is an integer agreeing with the usual sense of dimension, also known as the topological dimension. However, formulas have also been developed that allow calculation of the dimension of other less simple objects, where, solely on the basis of their properties of scaling and self-similarity, one is led to the conclusion that particular objects—including fractals—have non-integer Hausdorff dimensions. Because of the significant technical advances made by Abram Samoilovitch Besicovitch allowing computation of dimensions for highly irregular or "rough" sets, this dimension is also commonly referred to as the Hausdorff–Besicovitch dimension. More specifically, the Hausdorff dimension is a dimensional number associated with a metric space, i.e. a set where the distances between all members are defined. The dimension is drawn from the extended real numbers, R ¯ {\displaystyle {\overline {\mathbb {R} }}} , as opposed to the more intuitive notion of dimension, which is not associated to general metric spaces, and only takes values in the non-negative integers. In mathematical terms, the Hausdorff dimension generalizes the notion of the dimension of a real vector space. That is, the Hausdorff dimension of an n-dimensional inner product space equals n. This underlies the earlier statement that the Hausdorff dimension of a point is zero, of a line is one, etc., and that irregular sets can have noninteger Hausdorff dimensions. For instance, the Koch snowflake shown at right is constructed from an equilateral triangle; in each iteration, its component line segments are divided into 3 segments of unit length, the newly created middle segment is used as the base of a new equilateral triangle that points outward, and this base segment is then deleted to leave a final object from the iteration of unit length of 4. That is, after the first iteration, each original line segment has been replaced with N=4, where each self-similar copy is 1/S = 1/3 as long as the original. Stated another way, we have taken an object with Euclidean dimension, D, and reduced its linear scale by 1/3 in each direction, so that its length increases to N=SD. This equation is easily solved for D, yielding the ratio of logarithms (or natural logarithms) appearing in the figures, and giving—in the Koch and other fractal cases—non-integer dimensions for these objects. The Hausdorff dimension is a successor to the simpler, but usually equivalent, box-counting or Minkowski–Bouligand dimension. == Intuition == The intuitive concept of dimension of a geometric object X is the number of independent parameters one needs to pick out a unique point inside. However, any point specified by two parameters can be instead specified by one, because the cardinality of the real plane is equal to the cardinality of the real line (this can be seen by an argument involving interweaving the digits of two numbers to yield a single number encoding the same information). The example of a space-filling curve shows that one can even map the real line to the real plane surjectively (taking one real number into a pair of real numbers in a way so that all pairs of numbers are covered) and continuously, so that a one-dimensional object completely fills up a higher-dimensional object. Every space-filling curve hits some points multiple times and does not have a continuous inverse. It is impossible to map two dimensions onto one in a way that is continuous and continuously invertible. The topological dimension, also called Lebesgue covering dimension, explains why. This dimension is the greatest integer n such that in every covering of X by small open balls there is at least one point where n + 1 balls overlap. For example, when one covers a line with short open intervals, some points must be covered twice, giving dimension n = 1. But topological dimension is a very crude measure of the local size of a space (size near a point). A curve that is almost space-filling can still have topological dimension one, even if it fills up most of the area of a region. A fractal has an integer topological dimension, but in terms of the amount of space it takes up, it behaves like a higher-dimensional space. The Hausdorff dimension measures the local size of a space taking into account the distance between points, the metric. Consider the number N(r) of balls of radius at most r required to cover X completely. When r is very small, N(r) grows polynomially with 1/r. For a sufficiently well-behaved X, the Hausdorff dimension is the unique number d such that N(r) grows as 1/rd as r approaches zero. More precisely, this defines the box-counting dimension, which equals the Hausdorff dimension when the value d is a critical boundary between growth rates that are insufficient to cover the space, and growth rates that are overabundant. For shapes that are smooth, or shapes with a small number of corners, the shapes of traditional geometry and science, the Hausdorff dimension is an integer agreeing with the topological dimension. But Benoit Mandelbrot observed that fractals, sets with noninteger Hausdorff dimensions, are found everywhere in nature. He observed that the proper idealization of most rough shapes one sees is not in terms of smooth idealized shapes, but in terms of fractal idealized shapes: Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. For fractals that occur in nature, the Hausdorff and box-counting dimension coincide. The packing dimension is yet another similar notion which gives the same value for many shapes, but there are well-documented exceptions where all these dimensions differ. == Formal definition == The formal definition of the Hausdorff dimension is arrived at by defining first the d-dimensional Hausdorff measure, a fractional-dimension analogue of the Lebesgue measure. First, an outer measure is constructed: Let X {\displaystyle X} be a metric space. If S ⊂ X {\displaystyle S\subset X} and d ∈ [ 0 , ∞ ) {\displaystyle d\in [0,\infty )} , H δ d ( S ) = inf { ∑ i = 1 ∞ ( diam U i ) d : ⋃ i = 1 ∞ U i ⊇ S , diam U i < δ } , {\displaystyle H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},} where the infimum is taken over all countable covers U {\displaystyle U} of S {\displaystyle S} . The Hausdorff d-dimensional outer measure is then defined as H d ( S ) = lim δ → 0 H δ d ( S ) {\displaystyle {\mathcal {H}}^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S)} , and the restriction of the mapping to measurable sets justifies it as a measure, called the d {\displaystyle d} -dimensional Hausdorff Measure. === Hausdorff dimension === The Hausdorff dimension dim H ( X ) {\displaystyle \dim _{\operatorname {H} }{(X)}} of X {\displaystyle X} is defined by dim H ( X ) := inf { d ≥ 0 : H d ( X ) = 0 } . {\displaystyle \dim _{\operatorname {H} }{(X)}:=\inf\{d\geq 0:{\mathcal {H}}^{d}(X)=0\}.} This is the same as the supremum of the set of d ∈ [ 0 , ∞ ) {\displaystyle d\in [0,\infty )} such that the d {\displaystyle d} -dimensional Hausdorff measure of X {\displaystyle X} is infinite (except that when this latter set of numbers d {\displaystyle d} is empty the Hausdorff dimension is zero). === Hausdorff content === The d {\displaystyle d} -dimensional unlimited Hausdorff content of S {\displaystyle S} is defined by C H d ( S ) := H ∞ d ( S ) = inf { ∑ k = 1 ∞ ( diam U k ) d : ⋃ k = 1 ∞ U k ⊇ S } {\displaystyle C_{H}^{d}(S):=H_{\infty }^{d}(S)=\inf \left\{\sum _{k=1}^{\infty }(\operatorname {diam} U_{k})^{d}:\bigcup _{k=1}^{\infty }U_{k}\supseteq S\right\}} In other words, C H d ( S ) {\displaystyle C_{H}^{d}(S)} has the construction of the Hausdorff measure where the covering sets are allowed to have arbitrarily large sizes. (Here we use the standard convention that inf ∅ = ∞ {\displaystyle \inf \varnothing =\infty } .) The Hausdorff measure and the Hausdorff content can both be used to determine the dimension of a set, but if the measure of the set is non-zero, their actual values may disagree. == Examples == Countable sets have Hausdorff dimension 0. The Euclidean space R n {\displaystyle \mathbb {R} ^{n}} has Hausdorff dimension n {\displaystyle n} , and the circle S 1 {\displaystyle S^{1}} has Hausdorff dimension 1. Fractals often are spaces whose Hausdorff dimension strictly exceeds the topological dimension. For example, the Cantor set, a zero-dimensional topological space, is a union of two copies of itself, each copy shrunk by a factor 1/3; hence, it can be shown that its Hausdorff dimension is ln(2)/ln(3) ≈ 0.63. The Sierpinski triangle is a union of three copies of itself, each copy shrunk by a factor of 1/2; this yields a Hausdorff dimension of ln(3)/ln(2) ≈ 1.58. These Hausdorff dimensions are related to the "critical exponent" of the Master theorem for solving recurrence relations in the analysis of algorithms. Space-filling curves like the Peano curve have the same Hausdorff dimension as the space they fill. The trajectory of Brownian motion in dimension 2 and above is conjectured to be Hausdorff dimension 2. Lewis Fry Richardson performed detailed experiments to measure the approximate Hausdorff dimension for various coastlines. His results have varied from 1.02 for the coastline of South Africa to 1.25 for the west coast of Great Britain. == Properties of Hausdorff dimension == === Hausdorff dimension and inductive dimension === Let X be an arbitrary separable metric space. There is a topological notion of inductive dimension for X which is defined recursively. It is always an integer (or +∞) and is denoted dimind(X). Theorem. Suppose X is non-empty. Then dim H a u s ( X ) ≥ dim ind ( X ) . {\displaystyle \dim _{\mathrm {Haus} }(X)\geq \dim _{\operatorname {ind} }(X).} Moreover, inf Y dim Haus ( Y ) = dim ind ( X ) , {\displaystyle \inf _{Y}\dim _{\operatorname {Haus} }(Y)=\dim _{\operatorname {ind} }(X),} where Y ranges over metric spaces homeomorphic to X. In other words, X and Y have the same underlying set of points and the metric dY of Y is topologically equivalent to dX. These results were originally established by Edward Szpilrajn (1907–1976), e.g., see Hurewicz and Wallman, Chapter VII. === Hausdorff dimension and Minkowski dimension === The Minkowski dimension is similar to, and at least as large as, the Hausdorff dimension, and they are equal in many situations. However, the set of rational points in [0, 1] has Hausdorff dimension zero and Minkowski dimension one. There are also compact sets for which the Minkowski dimension is strictly larger than the Hausdorff dimension. === Hausdorff dimensions and Frostman measures === If there is a measure μ defined on Borel subsets of a metric space X such that μ(X) > 0 and μ(B(x, r)) ≤ rs holds for some constant s > 0 and for every ball B(x, r) in X, then dimHaus(X) ≥ s. A partial converse is provided by Frostman's lemma. === Behaviour under unions and products === If X = ⋃ i ∈ I X i {\displaystyle X=\bigcup _{i\in I}X_{i}} is a finite or countable union, then dim Haus ( X ) = sup i ∈ I dim Haus ( X i ) . {\displaystyle \dim _{\operatorname {Haus} }(X)=\sup _{i\in I}\dim _{\operatorname {Haus} }(X_{i}).} This can be verified directly from the definition. If X and Y are non-empty metric spaces, then the Hausdorff dimension of their product satisfies dim Haus ( X × Y ) ≥ dim Haus ( X ) + dim Haus ( Y ) . {\displaystyle \dim _{\operatorname {Haus} }(X\times Y)\geq \dim _{\operatorname {Haus} }(X)+\dim _{\operatorname {Haus} }(Y).} This inequality can be strict. It is possible to find two sets of dimension 0 whose product has dimension 1. In the opposite direction, it is known that when X and Y are Borel subsets of Rn, the Hausdorff dimension of X × Y is bounded from above by the Hausdorff dimension of X plus the upper packing dimension of Y. These facts are discussed in Mattila (1995). == Self-similar sets == Many sets defined by a self-similarity condition have dimensions which can be determined explicitly. Roughly, a set E is self-similar if it is the fixed point of a set-valued transformation ψ, that is ψ(E) = E, although the exact definition is given below. Theorem. Suppose ψ i : R n → R n , i = 1 , … , m {\displaystyle \psi _{i}:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n},\quad i=1,\ldots ,m} are each a contraction mapping on Rn with contraction constant ri < 1. Then there is a unique non-empty compact set A such that A = ⋃ i = 1 m ψ i ( A ) . {\displaystyle A=\bigcup _{i=1}^{m}\psi _{i}(A).} The theorem follows from Stefan Banach's contractive mapping fixed point theorem applied to the complete metric space of non-empty compact subsets of Rn with the Hausdorff distance. === The open set condition === To determine the dimension of the self-similar set A (in certain cases), we need a technical condition called the open set condition (OSC) on the sequence of contractions ψi. There is an open set V with compact closure, such that ⋃ i = 1 m ψ i ( V ) ⊆ V , {\displaystyle \bigcup _{i=1}^{m}\psi _{i}(V)\subseteq V,} where the sets in union on the left are pairwise disjoint. The open set condition is a separation condition that ensures the images ψi(V) do not overlap "too much". Theorem. Suppose the open set condition holds and each ψi is a similitude, that is a composition of an isometry and a dilation around some point. Then the unique fixed point of ψ is a set whose Hausdorff dimension is s where s is the unique solution of ∑ i = 1 m r i s = 1. {\displaystyle \sum _{i=1}^{m}r_{i}^{s}=1.} The contraction coefficient of a similitude is the magnitude of the dilation. In general, a set E which is carried onto itself by a mapping A ↦ ψ ( A ) = ⋃ i = 1 m ψ i ( A ) {\displaystyle A\mapsto \psi (A)=\bigcup _{i=1}^{m}\psi _{i}(A)} is self-similar if and only if the intersections satisfy the following condition: H s ( ψ i ( E ) ∩ ψ j ( E ) ) = 0 , {\displaystyle H^{s}\left(\psi _{i}(E)\cap \psi _{j}(E)\right)=0,} where s is the Hausdorff dimension of E and Hs denotes s-dimensional Hausdorff measure. This is clear in the case of the Sierpinski gasket (the intersections are just points), but is also true more generally: Theorem. Under the same conditions as the previous theorem, the unique fixed point of ψ is self-similar. == See also == List of fractals by Hausdorff dimension Examples of deterministic fractals, random and natural fractals. Assouad dimension, another variation of fractal dimension that, like Hausdorff dimension, is defined using coverings by balls Intrinsic dimension Packing dimension Fractal dimension == References == == Further reading == == External links == Hausdorff dimension at Encyclopedia of Mathematics Hausdorff measure at Encyclopedia of Mathematics
|
Wikipedia:Hausdorff measure#0
|
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in R n {\displaystyle \mathbb {R} ^{n}} or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite. Likewise, the one-dimensional Hausdorff measure of a simple curve in R n {\displaystyle \mathbb {R} ^{n}} is equal to the length of the curve, and the two-dimensional Hausdorff measure of a Lebesgue-measurable subset of R 2 {\displaystyle \mathbb {R} ^{2}} is proportional to the area of the set. Thus, the concept of the Hausdorff measure generalizes the Lebesgue measure and its notions of counting, length, and area. It also generalizes volume. In fact, there are d-dimensional Hausdorff measures for any d ≥ 0, which is not necessarily an integer. These measures are fundamental in geometric measure theory. They appear naturally in harmonic analysis or potential theory. == Definition == Let ( X , ρ ) {\displaystyle (X,\rho )} be a metric space. For any subset U ⊂ X {\displaystyle U\subset X} , let diam U {\displaystyle \operatorname {diam} U} denote its diameter, that is diam U := sup { ρ ( x , y ) : x , y ∈ U } , diam ∅ := 0. {\displaystyle \operatorname {diam} U:=\sup\{\rho (x,y):x,y\in U\},\quad \operatorname {diam} \emptyset :=0.} Let S {\displaystyle S} be any subset of X , {\displaystyle X,} and δ > 0 {\displaystyle \delta >0} a real number. Define H δ d ( S ) = inf { ∑ i = 1 ∞ ( diam U i ) d : ⋃ i = 1 ∞ U i ⊇ S , diam U i < δ } , {\displaystyle H_{\delta }^{d}(S)=\inf \left\{\sum _{i=1}^{\infty }(\operatorname {diam} U_{i})^{d}:\bigcup _{i=1}^{\infty }U_{i}\supseteq S,\operatorname {diam} U_{i}<\delta \right\},} where the infimum is over all countable covers of S {\displaystyle S} by sets U i ⊂ X {\displaystyle U_{i}\subset X} satisfying diam U i < δ {\displaystyle \operatorname {diam} U_{i}<\delta } . Note that H δ d ( S ) {\displaystyle H_{\delta }^{d}(S)} is monotone nonincreasing in δ {\displaystyle \delta } since the larger δ {\displaystyle \delta } is, the more collections of sets are permitted, making the infimum not larger. Thus, lim δ → 0 H δ d ( S ) {\displaystyle \lim _{\delta \to 0}H_{\delta }^{d}(S)} exists but may be infinite. Let H d ( S ) := sup δ > 0 H δ d ( S ) = lim δ → 0 H δ d ( S ) . {\displaystyle H^{d}(S):=\sup _{\delta >0}H_{\delta }^{d}(S)=\lim _{\delta \to 0}H_{\delta }^{d}(S).} It can be seen that H d ( S ) {\displaystyle H^{d}(S)} is an outer measure (more precisely, it is a metric outer measure). By Carathéodory's extension theorem, its restriction to the σ-field of Carathéodory-measurable sets is a measure. It is called the d {\displaystyle d} -dimensional Hausdorff measure of S {\displaystyle S} . Due to the metric outer measure property, all Borel subsets of X {\displaystyle X} are H d {\displaystyle H^{d}} measurable. In the above definition the sets in the covering are arbitrary. However, we can require the covering sets to be open or closed, or in normed spaces even convex, that will yield the same H δ d ( S ) {\displaystyle H_{\delta }^{d}(S)} numbers, hence the same measure. In R n {\displaystyle \mathbb {R} ^{n}} restricting the covering sets to be balls may change the measures but does not change the dimension of the measured sets. == Properties of Hausdorff measures == Note that if d is a positive integer, the d-dimensional Hausdorff measure of R d {\displaystyle \mathbb {R} ^{d}} is a rescaling of the usual d-dimensional Lebesgue measure λ d {\displaystyle \lambda _{d}} , which is normalized so that the Lebesgue measure of the unit cube [0,1]d is 1. In fact, for any Borel set E, λ d ( E ) = 2 − d α d H d ( E ) , {\displaystyle \lambda _{d}(E)=2^{-d}\alpha _{d}H^{d}(E),} where 2 − d {\displaystyle 2^{-d}} scales diameter to radius; while α d {\displaystyle \alpha _{d}} is the volume of the unit d-ball with radius one, which can be expressed using Euler's gamma function α d = Γ ( 1 2 ) d Γ ( d 2 + 1 ) = π d / 2 Γ ( d 2 + 1 ) . {\displaystyle \alpha _{d}={\frac {\Gamma \left({\frac {1}{2}}\right)^{d}}{\Gamma \left({\frac {d}{2}}+1\right)}}={\frac {\pi ^{d/2}}{\Gamma \left({\frac {d}{2}}+1\right)}}.} This is λ d ( E ) = β d H d ( E ) {\displaystyle \lambda _{d}(E)=\beta _{d}H^{d}(E)} , where β d = 2 − d α d {\displaystyle \beta _{d}=2^{-d}\alpha _{d}} is the volume of the d-ball with diameter one. Remark. Some authors adopt a definition of Hausdorff measure slightly different from the one chosen here, the difference being that the value H d ( E ) {\displaystyle H^{d}(E)} defined above is multiplied by the factor β d {\displaystyle \beta _{d}} , so that Hausdorff d-dimensional measure coincides exactly with Lebesgue measure in the case of Euclidean space. == Relation with Hausdorff dimension == It turns out that H d ( S ) {\displaystyle H^{d}(S)} may have a finite, nonzero value for at most one d {\displaystyle d} . That is, the Hausdorff Measure is zero for any value above a certain dimension and infinity below a certain dimension, analogous to the idea that the area of a line is zero and the length of a 2D shape is in some sense infinity. This leads to one of several possible equivalent definitions of the Hausdorff dimension: dim H a u s ( S ) = inf { d ≥ 0 : H d ( S ) = 0 } = sup { d ≥ 0 : H d ( S ) = ∞ } , {\displaystyle \dim _{\mathrm {Haus} }(S)=\inf\{d\geq 0:H^{d}(S)=0\}=\sup\{d\geq 0:H^{d}(S)=\infty \},} where we take inf ∅ = + ∞ {\displaystyle \inf \emptyset =+\infty } and sup ∅ = 0 {\displaystyle \sup \emptyset =0} . Note that it is not guaranteed that the Hausdorff measure must be finite and nonzero for some d, and indeed the measure at the Hausdorff dimension may still be zero; in this case, the Hausdorff dimension still acts as a change point between measures of zero and infinity. == Generalizations == In geometric measure theory and related fields, the Minkowski content is often used to measure the size of a subset of a metric measure space. For suitable domains in Euclidean space, the two notions of size coincide, up to overall normalizations depending on conventions. More precisely, a subset of R n {\displaystyle \mathbb {R} ^{n}} is said to be m {\displaystyle m} -rectifiable if it is the image of a bounded set in R m {\displaystyle \mathbb {R} ^{m}} under a Lipschitz function. If m < n {\displaystyle m<n} , then the m {\displaystyle m} -dimensional Minkowski content of a closed m {\displaystyle m} -rectifiable subset of R n {\displaystyle \mathbb {R} ^{n}} is equal to 2 − m α m {\displaystyle 2^{-m}\alpha _{m}} times the m {\displaystyle m} -dimensional Hausdorff measure (Federer 1969, Theorem 3.2.39, pp 275). In fractal geometry, some fractals with Hausdorff dimension d {\displaystyle d} have zero or infinite d {\displaystyle d} -dimensional Hausdorff measure. For example, almost surely the image of planar Brownian motion has Hausdorff dimension 2 and its two-dimensional Hausdorff measure is zero. In order to "measure" the "size" of such sets, the following variation on the notion of the Hausdorff measure can be considered: In the definition of the measure ( diam U i ) d {\displaystyle (\operatorname {diam} U_{i})^{d}} is replaced with ϕ ( diam U i ) , {\displaystyle \phi (\operatorname {diam} U_{i}),} where ϕ {\displaystyle \phi } is any monotone increasing function satisfying ϕ ( 0 ) = 0. {\displaystyle \phi (0)=0.} This is the Hausdorff measure of S {\displaystyle S} with gauge function ϕ , {\displaystyle \phi ,} or ϕ {\displaystyle \phi } -Hausdorff measure. A d {\displaystyle d} -dimensional set S {\displaystyle S} may satisfy H d ( S ) = 0 , {\displaystyle H^{d}(S)=0,} but H ϕ ( S ) ∈ ( 0 , ∞ ) {\displaystyle H^{\phi }(S)\in (0,\infty )} with an appropriate ϕ . {\displaystyle \phi .} Examples of gauge functions include ϕ ( t ) = t 2 log log 1 t or ϕ ( t ) = t 2 log 1 t log log log 1 t . {\displaystyle \phi (t)=t^{2}\log \log {\frac {1}{t}}\quad {\text{or}}\quad \phi (t)=t^{2}\log {\frac {1}{t}}\log \log \log {\frac {1}{t}}.} The former gives almost surely positive and σ {\displaystyle \sigma } -finite measure to the Brownian path in R n {\displaystyle \mathbb {R} ^{n}} when n > 2 {\displaystyle n>2} , and the latter when n = 2 {\displaystyle n=2} . == See also == Hausdorff dimension Geometric measure theory Measure theory Outer measure == References == Evans, Lawrence C.; Gariepy, Ronald F. (1992), Measure Theory and Fine Properties of Functions, CRC Press. Federer, Herbert (1969), Geometric Measure Theory, Springer-Verlag, ISBN 3-540-60656-4. Hausdorff, Felix (1918), "Dimension und äusseres Mass" (PDF), Mathematische Annalen, 79 (1–2): 157–179, doi:10.1007/BF01457179, S2CID 122001234. Morgan, Frank (1988), Geometric Measure Theory, Academic Press. Rogers, C. A. (1998), Hausdorff measures, Cambridge Mathematical Library (3rd ed.), Cambridge University Press, ISBN 0-521-62491-6 Szpilrajn, E (1937), "La dimension et la mesure" (PDF), Fundamenta Mathematicae, 28: 81–89, doi:10.4064/fm-28-1-81-89. == External links == Hausdorff dimension at Encyclopedia of Mathematics Hausdorff measure at Encyclopedia of Mathematics
|
Wikipedia:Hausdorff paradox#0
|
The Hausdorff paradox is a paradox in mathematics named after Felix Hausdorff. It involves the sphere S 2 {\displaystyle {S^{2}}} (the surface of a 3-dimensional ball in R 3 {\displaystyle {\mathbb {R} ^{3}}} ). It states that if a certain countable subset is removed from S 2 {\displaystyle {S^{2}}} , then the remainder can be divided into three disjoint subsets A , B {\displaystyle {A,B}} and C {\displaystyle {C}} such that A , B , C {\displaystyle {A,B,C}} and B ∪ C {\displaystyle {B\cup C}} are all congruent. In particular, it follows that on S 2 {\displaystyle S^{2}} there is no finitely additive measure defined on all subsets such that the measure of congruent sets is equal (because this would imply that the measure of B ∪ C {\displaystyle {B\cup C}} is simultaneously 1 / 3 {\displaystyle 1/3} , 1 / 2 {\displaystyle 1/2} , and 2 / 3 {\displaystyle 2/3} of the non-zero measure of the whole sphere). The paradox was published in Mathematische Annalen in 1914 and also in Hausdorff's book, Grundzüge der Mengenlehre, the same year. The proof of the much more famous Banach–Tarski paradox uses Hausdorff's ideas. The proof of this paradox relies on the axiom of choice. This paradox shows that there is no finitely additive measure on a sphere defined on all subsets which is equal on congruent pieces. (Hausdorff first showed in the same paper the easier result that there is no countably additive measure defined on all subsets.) The structure of the group of rotations on the sphere plays a crucial role here – the statement is not true on the plane or the line. In fact, as was later shown by Banach, it is possible to define an "area" for all bounded subsets in the Euclidean plane (as well as "length" on the real line) in such a way that congruent sets will have equal "area". (This Banach measure, however, is only finitely additive, so it is not a measure in the full sense, but it equals the Lebesgue measure on sets for which the latter exists.) This implies that if two open subsets of the plane (or the real line) are equi-decomposable then they have equal area. == See also == Banach–Tarski paradox – Geometric theorem Paradoxes of set theory == References == == Further reading == Hausdorff, Felix (1914). "Bemerkung über den Inhalt von Punktmengen". Mathematische Annalen. 75 (3): 428–434. doi:10.1007/bf01563735. S2CID 123243365. (Original article; in German) Hausdorff, Felix (1914). Grundzüge der Mengenlehre (in German). == External links == Hausdorff Paradox on ProofWiki
|
Wikipedia:Haya Freedman#0
|
Haya Freedman (Hebrew: חיה פרידמן; 1923–2005) was a Polish-born Israeli mathematician known for her research on the Tamari lattice and on ring theory, and as a teacher of mathematics at the London School of Economics. == Early life and education == Haya Freedman was born in Lviv, which at that time was part of Poland, and at the age of ten moved to Mandatory Palestine. She earned a master's degree from the Hebrew University of Jerusalem, studying abstract algebra there under the supervision of Jacob Levitzki. She began doctoral studies under Dov Tamari in the early 1950s, doing research on the Tamari lattice that she would much later publish with Tamari. However, at that time her husband wanted to shift his own research from mathematics to computer science, and as part of that shift decided to move to England. Freedman moved with him in 1956, breaking off her studies. Instead, she completed a PhD at Queen Mary College in 1960, under the supervision of Kurt Hirsch. == Academic career == In 1965, Freedman became a faculty member in mathematics in Birkbeck College, University of London. In 1966, Cyril Offord founded the sub-department of mathematics at the London School of Economics, and she became one of the founding faculty members there. She retired in 1989. == Legacy == In her honour, the London School of Economics offers an annual prize, the Haya Freedman Prize, for the best dissertation in the applied mathematics MSc. == References ==
|
Wikipedia:Haya Kaspi#0
|
Haya Kaspi (Hebrew: חיה כספי; born 6 October 1948) is an Israeli operations researcher, statistician, and probability theorist. She is a professor emeritus of industrial engineering and management at the Technion – Israel Institute of Technology. == Education and career == Kaspi was born in HaOgen. She earned a bachelor's degree in mathematics at the Hebrew University of Jerusalem in 1971, and a master's degree in applied mathematics at the Technion – Israel Institute of Technology in 1974. Next, she went to the US for her doctoral studies, completing a Ph.D. in operations research at Cornell University in 1979. Her dissertation, Ladder Sets of Markov Additive Processes, was supervised by N. U. Prabhu. After postdoctoral study at Princeton University, she returned to the Technion in 1980 as a lecturer. She was promoted to full professor in 1997. == Recognition == In 2008, Kaspi was selected as a Fellow of the Institute of Mathematical Statistics "for contributions to the general theory of Markov processes and its applications, to the theory of Markov local time; and for excellence in teaching and editorial work". In 2011, Kaspi and Nathalie Eisenbaum shared the Itô Prize of the Bernoulli Society for Mathematical Statistics and Probability for their joint work on permanental point processes (processes whose joint intensity can be represented as a permanent). == References == == External links == Home page Haya Kaspi publications indexed by Google Scholar
|
Wikipedia:Haynsworth inertia additivity formula#0
|
In mathematics, the Haynsworth inertia additivity formula, discovered by Emilie Virginia Haynsworth (1916–1985), concerns the number of positive, negative, and zero eigenvalues of a Hermitian matrix and of block matrices into which it is partitioned. The inertia of a Hermitian matrix H is defined as the ordered triple I n ( H ) = ( π ( H ) , ν ( H ) , δ ( H ) ) {\displaystyle \mathrm {In} (H)=\left(\pi (H),\nu (H),\delta (H)\right)} whose components are respectively the numbers of positive, negative, and zero eigenvalues of H. Haynsworth considered a partitioned Hermitian matrix H = [ H 11 H 12 H 12 ∗ H 22 ] {\displaystyle H={\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}} where H11 is nonsingular and H12* is the conjugate transpose of H12. The formula states: I n [ H 11 H 12 H 12 ∗ H 22 ] = I n ( H 11 ) + I n ( H / H 11 ) {\displaystyle \mathrm {In} {\begin{bmatrix}H_{11}&H_{12}\\H_{12}^{\ast }&H_{22}\end{bmatrix}}=\mathrm {In} (H_{11})+\mathrm {In} (H/H_{11})} where H/H11 is the Schur complement of H11 in H: H / H 11 = H 22 − H 12 ∗ H 11 − 1 H 12 . {\displaystyle H/H_{11}=H_{22}-H_{12}^{\ast }H_{11}^{-1}H_{12}.} == Generalization == If H11 is singular, we can still define the generalized Schur complement, using the Moore–Penrose inverse H 11 + {\displaystyle H_{11}^{+}} instead of H 11 − 1 {\displaystyle H_{11}^{-1}} . The formula does not hold if H11 is singular. However, a generalization has been proven in 1974 by Carlson, Haynsworth and Markham, to the effect that π ( H ) ≥ π ( H 11 ) + π ( H / H 11 ) {\displaystyle \pi (H)\geq \pi (H_{11})+\pi (H/H_{11})} and ν ( H ) ≥ ν ( H 11 ) + ν ( H / H 11 ) {\displaystyle \nu (H)\geq \nu (H_{11})+\nu (H/H_{11})} . Carlson, Haynsworth and Markham also gave sufficient and necessary conditions for equality to hold. == See also == Block matrix pseudoinverse Sylvester's law of inertia == Notes and references ==
|
Wikipedia:Hayyim Selig Slonimski#0
|
Ḥayyim Selig ben Ya'akov Slonimski (Yiddish: חַיִּים זֶעלִיג בֶּן יַעֲקֹב סלאָנימסקי; March 31, 1810 – May 15, 1904), also known by his acronym ḤaZaS (חז״ס), was a Hebrew publisher, mathematician, astronomer, inventor, science writer, and rabbi. He was among the first to write books on science for a broad Jewish audience, and was the founder of Ha-Tsfira, the first Hebrew-language newspaper with an emphasis on the sciences. == Biography == Ḥayyim Selig Slonimski was born in Bialystok, in the Grodno Governorate of the Russian Empire (present-day Poland), the oldest son of Rabbi Avraham Ya'akov Bishka and Leah (Neches) Bishka. His father belonged to a family of rabbis, writers, publishers and printers, and his mother was the daughter of Rabbi Yeḥiel Neches, an owner of a well-known beit midrash in Bialystok. Slonimski had a traditional Jewish upbringing and Talmudic education; without a formal secular education, Slonimski taught himself mathematics, astronomy, and foreign languages. An advocate for the education of Eastern European Jews in the sciences, Slonimski introduced a vocabulary of technical terms created partly by himself into the Hebrew language. At age 24 he finished writing a textbook on mathematics, but due to lack of funds, only the first part of which was published in 1834 under the title Mosedei Ḥokhmah.: 180 The following year, Slonimski released Sefer Kokhva de-Shavit (1835), a collection of essays on Halley's comet and other astronomy-related topics such as the laws of Kepler and Newton's laws of motion.: 180 In 1838 Slonimski settled in Warsaw, where he became acquainted with mathematician and inventor Abraham Stern (1768–1842), whose youngest daughter Sarah Gitel he would later marry in 1842. There he published another astronomical work, the highly popular Toldot ha-Shamayim (1838). He also tried his hand at the applied sciences, and a number of his technological inventions received recognition and awards. The most notable of his inventions was his calculating machine, created in 1842 based on his tables, which he exhibited to the St. Petersburg Academy of Sciences, and for which he was awarded the 1844 Demidov Prize of 2,500 rubles by the Russian Academy of Sciences. He also received a title of honorary citizen, which granted him the right to live outside of the Pale of Settlement to which Jews were normally restricted. In 1844 he published a new formula in Crelle's Journal for calculating the Jewish calendar. In 1853 he invented a chemical process for plating iron vessels with lead to prevent corrosion, and in 1856 a device for simultaneously sending multiple telegrams using just one telegraphic wire. The system of multiple telegraphy perfected by Lord Kelvin in 1858 was based on Slonimski's discovery. Slonimski lived between 1846 and 1858 in Tomaszów Mazowiecki, an industrial town in central Poland. He corresponded with several scientists, notably Alexander von Humboldt, and wrote a sketch of Humboldt's life. In February 1862 in Warsaw, Slonimski launched Ha-Tsfira, the first Hebrew newspaper in Poland, and was the publisher, editor, and chief contributor. It ceased publication after six months due to his departure on the eve of the January Uprising from Warsaw to Zhitomir, the capital of the Ukrainian province Volhynia.: 6 There Slonimski was appointed as principal of the rabbinical seminary in Zhitomir and as government censor of Hebrew books. After the seminary was closed by the Russian government in 1874, Slonimski resumed the publication of Ha-Tsfira, first in Berlin and then again in Warsaw, after he obtained the necessary permission from the tsarist government. The newspaper would quickly become a central cultural institution of Polish Jewry. He died in Warsaw on May 15, 1904. == The Stalin controversy == In 1952 Josef Stalin made a speech in which, among other things, he claimed that it was a Russian who had beat out America in the 19th century in the development of the telegraph. While Stalin's claim was mocked in the United States, Slonimsky's grandson, the musicologist Nicolas Slonimsky, was able to confirm the accuracy of some of Stalin's claims. == Major works == Mosede Ḥokmah (1834), on the fundamental principles of higher algebra Sefer Kukba di-Shebit (1835), essays on the Halley comet and on astronomy in general Toledot ha-Shamayim (1838), on astronomy and optics Yesode ha-'Ibbur (1852), on the Jewish calendar system and its history Meẓi'ut ha-Nefesh ve-Ḳiyyumah (1852), on the immortality of the soul Ot Zikkaron (1858), a biographical sketch of Alexander von Humboldt == See also == Slonimski's Theorem == References == This article incorporates text from a publication now in the public domain: Isidore Singer and Judah David Eisenstein (1901–1906). "Slonimski, Ḥayyim Selig". In Singer, Isidore; et al. (eds.). The Jewish Encyclopedia. New York: Funk & Wagnalls. === Footnotes === == External links == Works by or about Hayyim Selig Slonimski at the Internet Archive
|
Wikipedia:Hazel Perfect#0
|
Hazel Perfect (circa 1927 – 8 July 2015) was a British mathematician specialising in combinatorics. == Contributions == Perfect was known for inventing gammoids,[AMG] for her work with Leon Mirsky on doubly stochastic matrices,[SP2] for her three books Topics in Geometry,[TIG] Topics in Algebra,[TIA] and Independence Theory in Combinatorics,[ITC] and for her work as a translator (from an earlier German translation) of Pavel Alexandrov's book An Introduction to the Theory of Groups (Hafner, 1959).[ITG] The Perfect–Mirsky conjecture, named after Perfect and Leon Mirsky, concerns the region of the complex plane formed by the eigenvalues of doubly stochastic matrices. Perfect and Mirsky conjectured that for n × n {\displaystyle n\times n} matrices this region is the union of regular polygons of up to n {\displaystyle n} sides, having the roots of unity of each degree up to n {\displaystyle n} as vertices. Perfect and Mirsky proved their conjecture for n ≤ 3 {\displaystyle n\leq 3} ; it was subsequently shown to be true for n = 4 {\displaystyle n=4} and false for n = 5 {\displaystyle n=5} , but remains open for larger values of n {\displaystyle n} .[SP2] == Education and career == Perfect earned a master's degree through Westfield College (a constituent college for women in the University of London) in 1949, with a thesis on The Reduction of Matrices to Canonical Form. In the 1950s, Perfect was a lecturer at University College of Swansea; she collaborated with Gordon Petersen, a visitor to Swansea at that time, on their translation of Alexandrov's book. She completed her Ph.D. at the University of London in 1969; her dissertation was Studies in Transversal Theory with Particular Reference to Independence Structures and Graphs. She became a reader in mathematics at the University of Sheffield. == Selected publications == === Books === === Research papers === === Translation === == References ==
|
Wikipedia:Heath-Brown–Moroz constant#0
|
The Heath-Brown–Moroz constant C, named for Roger Heath-Brown and Boris Moroz, is defined as C = ∏ p ( 1 − 1 p ) 7 ( 1 + 7 p + 1 p 2 ) = 0.001317641... {\displaystyle C=\prod _{p}\left(1-{\frac {1}{p}}\right)^{7}\left(1+{\frac {7p+1}{p^{2}}}\right)=0.001317641...} where p runs over the primes. == Application == This constant is part of an asymptotic estimate for the distribution of rational points of bounded height on the cubic surface X03=X1X2X3. Let H be a positive real number and N(H) the number of solutions to the equation X03=X1X2X3 with all the Xi non-negative integers less than or equal to H and their greatest common divisor equal to 1. Then N ( H ) = C ⋅ H ( log H ) 6 4 × 6 ! + O ( H ( log H ) 5 ) {\displaystyle N(H)=C\cdot {\frac {H(\log H)^{6}}{4\times 6!}}+O(H(\log H)^{5})} . == References == == External links == Wolfram Mathworld's article
|
Wikipedia:Hecke algebra#0
|
In mathematics, the Hecke algebra is the algebra generated by Hecke operators, which are named after Erich Hecke. == Properties == The algebra is a commutative ring. In the classical elliptic modular form theory, the Hecke operators Tn with n coprime to the level acting on the space of cusp forms of a given weight are self-adjoint with respect to the Petersson inner product. Therefore, the spectral theorem implies that there is a basis of modular forms that are eigenfunctions for these Hecke operators. Each of these basic forms possesses an Euler product. More precisely, its Mellin transform is the Dirichlet series that has Euler products with the local factor for each prime p is the reciprocal of the Hecke polynomial, a quadratic polynomial in p−s. In the case treated by Mordell, the space of cusp forms of weight 12 with respect to the full modular group is one-dimensional. It follows that the Ramanujan form has an Euler product and establishes the multiplicativity of τ(n). == Generalizations == The classical Hecke algebra has been generalized to other settings, such as the Hecke algebra of a locally compact group and spherical Hecke algebra that arise when modular forms and other automorphic forms are viewed using adelic groups. These play a central role in the Langlands correspondence. The derived Hecke algebra is a further generalization of Hecke algebras to derived functors. It was introduced by Peter Schneider in 2015 who, together with Rachel Ollivier, used them to study the p-adic Langlands correspondence. It is the subject of several conjectures on the cohomology of arithmetic groups by Akshay Venkatesh and his collaborators. == See also == Abstract algebra Wiles's proof of Fermat's Last Theorem == Notes == == References == Bump, Daniel (1997). Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics. Vol. 55. Cambridge University Press. Serre, Jean-Pierre (1973). A Course in Arithmetic. Graduate Texts in Mathematics. New York: Springer Science+Business Media.
|
Wikipedia:HegartyMaths#0
|
HegartyMaths was an educational subscription tool used by schools in the United Kingdom. It was sometimes used as a replacement for general mathematics homework tasks. Its creator, Colin Hegarty, was the UK Teacher of the Year in 2015 and shortlisted for the Varkey Foundation's Global Teacher Prize in 2016. == Usage == HegartyMaths covered a variety of topics and had 943 tasks to complete. A task included an educational video with an explanation and examples on the topic. Afterwards, there was a quiz to complete, containing topic specific questions. The site was regularly updated and more topics were added to keep up with the General Certificate of Secondary Education (GCSE) mathematics curriculum. Students could complete tasks by themselves, or teachers could assign these tasks to students to complete as homework or for revision purposes and then track the student's progress. == History == HegartyMaths was created by co-founders and teachers Colin Hegarty and Brian Arnold. In 2011 they started to make maths videos on YouTube to support their own classes with maths homework and revision. Since the videos were freely available on YouTube, students from all over the country and the world started using the videos too. In 2012 Colin won £15,000 of funding from education charity SHINE, through its Let Teachers SHINE competition, to make a website to host the videos and make more content. The original website, launched on 12 July 2013, was called mathswebsite.com. It was built to contain free maths videos to assist students in revision and is still accessible today. In February 2016, a new site was launched, HegartyMaths.com. In 2019, Colin Hegarty sold HegartyMaths to Sparx Learning (a company selling GCSE revision packages), then Sparx Maths, for an undisclosed sum. Colin became part of the leadership team for Sparx and continued to lead development on HegartyMaths. HegartyMaths has since been shut down and now redirects to Sparx Maths. The Sparx Learning website claims this happened in 2022, after "incorporating its best features and learnings into Sparx Maths". At the beginning of 2023 Colin was announced as the new CEO of Sparx Learning. == References == == External links == hegartymaths.com mathswebsite.com
|
Wikipedia:Height function#0
|
A height function is a function that quantifies the complexity of mathematical objects. In Diophantine geometry, height functions quantify the size of solutions to Diophantine equations and are typically functions from a set of points on algebraic varieties (or a set of algebraic varieties) to the real numbers. For instance, the classical or naive height over the rational numbers is typically defined to be the maximum of the numerators and denominators of the coordinates (e.g. 7 for the coordinates (3/7, 1/2)), but in a logarithmic scale. == Significance == Height functions allow mathematicians to count objects, such as rational points, that are otherwise infinite in quantity. For instance, the set of rational numbers of naive height (the maximum of the numerator and denominator when expressed in lowest terms) below any given constant is finite despite the set of rational numbers being infinite. In this sense, height functions can be used to prove asymptotic results such as Baker's theorem in transcendental number theory which was proved by Alan Baker (1966, 1967a, 1967b). In other cases, height functions can distinguish some objects based on their complexity. For instance, the subspace theorem proved by Wolfgang M. Schmidt (1972) demonstrates that points of small height (i.e. small complexity) in projective space lie in a finite number of hyperplanes and generalizes Siegel's theorem on integral points and solution of the S-unit equation. Height functions were crucial to the proofs of the Mordell–Weil theorem and Faltings's theorem by Weil (1929) and Faltings (1983) respectively. Several outstanding unsolved problems about the heights of rational points on algebraic varieties, such as the Manin conjecture and Vojta's conjecture, have far-reaching implications for problems in Diophantine approximation, Diophantine equations, arithmetic geometry, and mathematical logic. == History == An early form of height function was proposed by Giambattista Benedetti (c. 1563), who argued that the consonance of a musical interval could be measured by the product of its numerator and denominator (in reduced form); see Giambattista Benedetti § Music. Heights in Diophantine geometry were initially developed by André Weil and Douglas Northcott beginning in the 1920s. Innovations in 1960s were the Néron–Tate height and the realization that heights were linked to projective representations in much the same way that ample line bundles are in other parts of algebraic geometry. In the 1970s, Suren Arakelov developed Arakelov heights in Arakelov theory. In 1983, Faltings developed his theory of Faltings heights in his proof of Faltings's theorem. == Height functions in Diophantine geometry == === Naive height === Classical or naive height is defined in terms of ordinary absolute value on homogeneous coordinates. It is typically a logarithmic scale and therefore can be viewed as being proportional to the "algebraic complexity" or number of bits needed to store a point. It is typically defined to be the logarithm of the maximum absolute value of the vector of coprime integers obtained by multiplying through by a lowest common denominator. This may be used to define height on a point in projective space over Q, or of a polynomial, regarded as a vector of coefficients, or of an algebraic number, from the height of its minimal polynomial. The naive height of a rational number x = p/q (in lowest terms) is multiplicative height H ( p / q ) = max { | p | , | q | } {\displaystyle H(p/q)=\max\{|p|,|q|\}} logarithmic height: h ( p / q ) = log H ( p / q ) {\displaystyle h(p/q)=\log H(p/q)} Therefore, the naive multiplicative and logarithmic heights of 4/10 are 5 and log(5), for example. The naive height H of an elliptic curve E given by y2 = x3 + Ax + B is defined to be H(E) = log max(4|A|3, 27|B|2). === Néron–Tate height === The Néron–Tate height, or canonical height, is a quadratic form on the Mordell–Weil group of rational points of an abelian variety defined over a global field. It is named after André Néron, who first defined it as a sum of local heights, and John Tate, who defined it globally in an unpublished work. === Weil height === Let X be a projective variety over a number field K. Let L be a line bundle on X. One defines the Weil height on X with respect to L as follows. First, suppose that L is very ample. A choice of basis of the space Γ ( X , L ) {\displaystyle \Gamma (X,L)} of global sections defines a morphism ϕ from X to projective space, and for all points p on X, one defines h L ( p ) := h ( ϕ ( p ) ) {\displaystyle h_{L}(p):=h(\phi (p))} , where h is the naive height on projective space. For fixed X and L, choosing a different basis of global sections changes h L {\displaystyle h_{L}} , but only by a bounded function of p. Thus h L {\displaystyle h_{L}} is well-defined up to addition of a function that is O(1). In general, one can write L as the difference of two very ample line bundles L1 and L2 on X and define h L := h L 1 − h L 2 , {\displaystyle h_{L}:=h_{L_{1}}-h_{L_{2}},} which again is well-defined up to O(1). ==== Arakelov height ==== The Arakelov height on a projective space over the field of algebraic numbers is a global height function with local contributions coming from Fubini–Study metrics on the Archimedean fields and the usual metric on the non-Archimedean fields. It is the usual Weil height equipped with a different metric. === Faltings height === The Faltings height of an abelian variety defined over a number field is a measure of its arithmetic complexity. It is defined in terms of the height of a metrized line bundle. It was introduced by Faltings (1983) in his proof of the Mordell conjecture. == Height functions in algebra == === Height of a polynomial === For a polynomial P of degree n given by P = a 0 + a 1 x + a 2 x 2 + ⋯ + a n x n , {\displaystyle P=a_{0}+a_{1}x+a_{2}x^{2}+\cdots +a_{n}x^{n},} the height H(P) is defined to be the maximum of the magnitudes of its coefficients: H ( P ) = max i | a i | . {\displaystyle H(P)={\underset {i}{\max }}\,|a_{i}|.} One could similarly define the length L(P) as the sum of the magnitudes of the coefficients: L ( P ) = ∑ i = 0 n | a i | . {\displaystyle L(P)=\sum _{i=0}^{n}|a_{i}|.} ==== Relation to Mahler measure ==== The Mahler measure M(P) of P is also a measure of the complexity of P. The three functions H(P), L(P) and M(P) are related by the inequalities ( n ⌊ n / 2 ⌋ ) − 1 H ( P ) ≤ M ( P ) ≤ H ( P ) n + 1 ; {\displaystyle {\binom {n}{\lfloor n/2\rfloor }}^{-1}H(P)\leq M(P)\leq H(P){\sqrt {n+1}};} L ( p ) ≤ 2 n M ( p ) ≤ 2 n L ( p ) ; {\displaystyle L(p)\leq 2^{n}M(p)\leq 2^{n}L(p);} H ( p ) ≤ L ( p ) ≤ ( n + 1 ) H ( p ) {\displaystyle H(p)\leq L(p)\leq (n+1)H(p)} where ( n ⌊ n / 2 ⌋ ) {\displaystyle \scriptstyle {\binom {n}{\lfloor n/2\rfloor }}} is the binomial coefficient. == Height functions in automorphic forms == One of the conditions in the definition of an automorphic form on the general linear group of an adelic algebraic group is moderate growth, which is an asymptotic condition on the growth of a height function on the general linear group viewed as an affine variety. == Other height functions == The height of an irreducible rational number x = p/q, q > 0 is | p | + q {\displaystyle |p|+q} (this function is used for constructing a bijection between N {\displaystyle \mathbb {N} } and Q {\displaystyle \mathbb {Q} } ). == See also == abc conjecture Birch and Swinnerton-Dyer conjecture Elliptic Lehmer conjecture Heath-Brown–Moroz constant Height of a formal group law Height zeta function Raynaud's isogeny theorem == References == == Sources == Baker, Alan (1966). "Linear forms in the logarithms of algebraic numbers. I". Mathematika. 13 (2): 204–216. doi:10.1112/S0025579300003971. ISSN 0025-5793. MR 0220680. Baker, Alan (1967a). "Linear forms in the logarithms of algebraic numbers. II". Mathematika. 14: 102–107. doi:10.1112/S0025579300008068. ISSN 0025-5793. MR 0220680. Baker, Alan (1967b). "Linear forms in the logarithms of algebraic numbers. III". Mathematika. 14 (2): 220–228. doi:10.1112/S0025579300003843. ISSN 0025-5793. MR 0220680. Baker, Alan; Wüstholz, Gisbert (2007). Logarithmic Forms and Diophantine Geometry. New Mathematical Monographs. Vol. 9. Cambridge University Press. p. 3. ISBN 978-0-521-88268-2. Zbl 1145.11004. Bombieri, Enrico; Gubler, Walter (2006). Heights in Diophantine Geometry. New Mathematical Monographs. Vol. 4. Cambridge University Press. ISBN 978-0-521-71229-3. Zbl 1130.11034. Borwein, Peter (2002). Computational Excursions in Analysis and Number Theory. CMS Books in Mathematics. Springer-Verlag. pp. 2, 3, 14148. ISBN 0-387-95444-9. Zbl 1020.12001. Bump, Daniel (1998). Automorphic Forms and Representations. Cambridge Studies in Advanced Mathematics. Vol. 55. Cambridge University Press. p. 300. ISBN 9780521658188. Cornell, Gary; Silverman, Joseph H. (1986). Arithmetic geometry. New York: Springer. ISBN 0387963111. → Contains an English translation of Faltings (1983) Faltings, Gerd (1983). "Endlichkeitssätze für abelsche Varietäten über Zahlkörpern" [Finiteness theorems for abelian varieties over number fields]. Inventiones Mathematicae (in German). 73 (3): 349–366. Bibcode:1983InMat..73..349F. doi:10.1007/BF01388432. MR 0718935. S2CID 121049418. Faltings, Gerd (1991). "Diophantine approximation on abelian varieties". Annals of Mathematics. 123 (3): 549–576. doi:10.2307/2944319. JSTOR 2944319. MR 1109353. Fili, Paul; Petsche, Clayton; Pritsker, Igor (2017). "Energy integrals and small points for the Arakelov height". Archiv der Mathematik. 109 (5): 441–454. arXiv:1507.01900. doi:10.1007/s00013-017-1080-x. S2CID 119161942. Mahler, K. (1963). "On two extremum properties of polynomials". Illinois Journal of Mathematics. 7 (4): 681–701. doi:10.1215/ijm/1255645104. Zbl 0117.04003. Néron, André (1965). "Quasi-fonctions et hauteurs sur les variétés abéliennes". Annals of Mathematics (in French). 82 (2): 249–331. doi:10.2307/1970644. JSTOR 1970644. MR 0179173. Schinzel, Andrzej (2000). Polynomials with special regard to reducibility. Encyclopedia of Mathematics and Its Applications. Vol. 77. Cambridge: Cambridge University Press. p. 212. ISBN 0-521-66225-7. Zbl 0956.12001. Schmidt, Wolfgang M. (1972). "Norm form equations". Annals of Mathematics. Second Series. 96 (3): 526–551. doi:10.2307/1970824. JSTOR 1970824. MR 0314761. Lang, Serge (1988). Introduction to Arakelov theory. New York: Springer-Verlag. ISBN 0-387-96793-1. MR 0969124. Zbl 0667.14001. Lang, Serge (1997). Survey of Diophantine Geometry. Springer-Verlag. ISBN 3-540-61223-8. Zbl 0869.11051. Weil, André (1929). "L'arithmétique sur les courbes algébriques". Acta Mathematica. 52 (1): 281–315. doi:10.1007/BF02592688. MR 1555278. Silverman, Joseph H. (1994). Advanced Topics in the Arithmetic of Elliptic Curves. New York: Springer. ISBN 978-1-4612-0851-8. Vojta, Paul (1987). Diophantine Approximations and Value Distribution Theory. Lecture Notes in Mathematics. Vol. 1239. Berlin, New York: Springer-Verlag. doi:10.1007/BFb0072989. ISBN 978-3-540-17551-3. MR 0883451. Zbl 0609.14011. Kolmogorov, Andrey; Fomin, Sergei (1957). Elements of the Theory of Functions and Functional Analysis. New York: Graylock Press. == External links == Polynomial height at Mathworld
|
Wikipedia:Heine's identity#0
|
In mathematical analysis, Heine's identity, named after Heinrich Eduard Heine is a Fourier expansion of a reciprocal square root which Heine presented as 1 z − cos ψ = 2 π ∑ m = − ∞ ∞ Q m − 1 2 ( z ) e i m ψ {\displaystyle {\frac {1}{\sqrt {z-\cos \psi }}}={\frac {\sqrt {2}}{\pi }}\sum _{m=-\infty }^{\infty }Q_{m-{\frac {1}{2}}}(z)e^{im\psi }} where Q m − 1 2 {\displaystyle Q_{m-{\frac {1}{2}}}} is a Legendre function of the second kind, which has degree, m − 1⁄2, a half-integer, and argument, z, real and greater than one. This expression can be generalized for arbitrary half-integer powers as follows ( z − cos ψ ) n − 1 2 = 2 π ( z 2 − 1 ) n 2 Γ ( 1 2 − n ) ∑ m = − ∞ ∞ Γ ( m − n + 1 2 ) Γ ( m + n + 1 2 ) Q m − 1 2 n ( z ) e i m ψ , {\displaystyle (z-\cos \psi )^{n-{\frac {1}{2}}}={\sqrt {\frac {2}{\pi }}}{\frac {(z^{2}-1)^{\frac {n}{2}}}{\Gamma ({\frac {1}{2}}-n)}}\sum _{m=-\infty }^{\infty }{\frac {\Gamma (m-n+{\frac {1}{2}})}{\Gamma (m+n+{\frac {1}{2}})}}Q_{m-{\frac {1}{2}}}^{n}(z)e^{im\psi },} where Γ {\displaystyle \scriptstyle \,\Gamma } is the Gamma function. == References ==
|
Wikipedia:Heine–Cantor theorem#0
|
In mathematics, the Heine–Cantor theorem states that a continuous function between two metric spaces is uniformly continuous if its domain is compact. The theorem is named after Eduard Heine and Georg Cantor. An important special case of the Cantor theorem is that every continuous function from a closed bounded interval to the real numbers is uniformly continuous. For an alternative proof in the case of M = [ a , b ] {\displaystyle M=[a,b]} , a closed interval, see the article Non-standard calculus. == See also == Cauchy-continuous function == External links == Heine–Cantor theorem at PlanetMath. Proof of Heine–Cantor theorem at PlanetMath.
|
Wikipedia:Heinrich Kleisli#0
|
In category theory, a Kleisli category is a category naturally associated to any monad T. It is equivalent to the category of free T-algebras. The Kleisli category is one of two extremal solutions to the question: "Does every monad arise from an adjunction?" The other extremal solution is the Eilenberg–Moore category. Kleisli categories are named for the mathematician Heinrich Kleisli. == Formal definition == Let ⟨T, η, μ⟩ be a monad over a category C. The Kleisli category of C is the category CT whose objects and morphisms are given by O b j ( C T ) = O b j ( C ) , H o m C T ( X , Y ) = H o m C ( X , T Y ) . {\displaystyle {\begin{aligned}\mathrm {Obj} ({{\mathcal {C}}_{T}})&=\mathrm {Obj} ({\mathcal {C}}),\\\mathrm {Hom} _{{\mathcal {C}}_{T}}(X,Y)&=\mathrm {Hom} _{\mathcal {C}}(X,TY).\end{aligned}}} That is, every morphism f: X → T Y in C (with codomain TY) can also be regarded as a morphism in CT (but with codomain Y). Composition of morphisms in CT is given by g ∘ T f = μ Z ∘ T g ∘ f : X → T Y → T 2 Z → T Z {\displaystyle g\circ _{T}f=\mu _{Z}\circ Tg\circ f:X\to TY\to T^{2}Z\to TZ} where f: X → T Y and g: Y → T Z. The identity morphism is given by the monad unit η: i d X = η X {\displaystyle \mathrm {id} _{X}=\eta _{X}} . An alternative way of writing this, which clarifies the category in which each object lives, is used by Mac Lane. We use very slightly different notation for this presentation. Given the same monad and category C {\displaystyle C} as above, we associate with each object X {\displaystyle X} in C {\displaystyle C} a new object X T {\displaystyle X_{T}} , and for each morphism f : X → T Y {\displaystyle f\colon X\to TY} in C {\displaystyle C} a morphism f ∗ : X T → Y T {\displaystyle f^{*}\colon X_{T}\to Y_{T}} . Together, these objects and morphisms form our category C T {\displaystyle C_{T}} , where we define composition, also called Kleisli composition, by g ∗ ∘ T f ∗ = ( μ Z ∘ T g ∘ f ) ∗ . {\displaystyle g^{*}\circ _{T}f^{*}=(\mu _{Z}\circ Tg\circ f)^{*}.} Then the identity morphism in C T {\displaystyle C_{T}} , the Kleisli identity, is i d X T = ( η X ) ∗ . {\displaystyle \mathrm {id} _{X_{T}}=(\eta _{X})^{*}.} == Extension operators and Kleisli triples == Composition of Kleisli arrows can be expressed succinctly by means of the extension operator (–)# : Hom(X, TY) → Hom(TX, TY). Given a monad ⟨T, η, μ⟩ over a category C and a morphism f : X → TY let f ♯ = μ Y ∘ T f . {\displaystyle f^{\sharp }=\mu _{Y}\circ Tf.} Composition in the Kleisli category CT can then be written g ∘ T f = g ♯ ∘ f . {\displaystyle g\circ _{T}f=g^{\sharp }\circ f.} The extension operator satisfies the identities: η X ♯ = i d T X f ♯ ∘ η X = f ( g ♯ ∘ f ) ♯ = g ♯ ∘ f ♯ {\displaystyle {\begin{aligned}\eta _{X}^{\sharp }&=\mathrm {id} _{TX}\\f^{\sharp }\circ \eta _{X}&=f\\(g^{\sharp }\circ f)^{\sharp }&=g^{\sharp }\circ f^{\sharp }\end{aligned}}} where f : X → TY and g : Y → TZ. It follows trivially from these properties that Kleisli composition is associative and that ηX is the identity. In fact, to give a monad is to give a Kleisli triple ⟨T, η, (–)#⟩, i.e. A function T : o b ( C ) → o b ( C ) {\displaystyle T\colon \mathrm {ob} (C)\to \mathrm {ob} (C)} ; For each object A {\displaystyle A} in C {\displaystyle C} , a morphism η A : A → T ( A ) {\displaystyle \eta _{A}\colon A\to T(A)} ; For each morphism f : A → T ( B ) {\displaystyle f\colon A\to T(B)} in C {\displaystyle C} , a morphism f ♯ : T ( A ) → T ( B ) {\displaystyle f^{\sharp }\colon T(A)\to T(B)} such that the above three equations for extension operators are satisfied. == Kleisli adjunction == Kleisli categories were originally defined in order to show that every monad arises from an adjunction. That construction is as follows. Let ⟨T, η, μ⟩ be a monad over a category C and let CT be the associated Kleisli category. Using Mac Lane's notation mentioned in the “Formal definition” section above, define a functor F: C → CT by F X = X T {\displaystyle FX=X_{T}\;} F ( f : X → Y ) = ( η Y ∘ f ) ∗ {\displaystyle F(f\colon X\to Y)=(\eta _{Y}\circ f)^{*}} and a functor G : CT → C by G Y T = T Y {\displaystyle GY_{T}=TY\;} G ( f ∗ : X T → Y T ) = μ Y ∘ T f {\displaystyle G(f^{*}\colon X_{T}\to Y_{T})=\mu _{Y}\circ Tf\;} One can show that F and G are indeed functors and that F is left adjoint to G. The counit of the adjunction is given by ε Y T = ( i d T Y ) ∗ : ( T Y ) T → Y T . {\displaystyle \varepsilon _{Y_{T}}=(\mathrm {id} _{TY})^{*}:(TY)_{T}\to Y_{T}.} Finally, one can show that T = GF and μ = GεF so that ⟨T, η, μ⟩ is the monad associated to the adjunction ⟨F, G, η, ε⟩. === Showing that GF = T === For any object X in category C: ( G ∘ F ) ( X ) = G ( F ( X ) ) = G ( X T ) = T X . {\displaystyle {\begin{aligned}(G\circ F)(X)&=G(F(X))\\&=G(X_{T})\\&=TX.\end{aligned}}} For any f : X → Y {\displaystyle f:X\to Y} in category C: ( G ∘ F ) ( f ) = G ( F ( f ) ) = G ( ( η Y ∘ f ) ∗ ) = μ Y ∘ T ( η Y ∘ f ) = μ Y ∘ T η Y ∘ T f = id T Y ∘ T f = T f . {\displaystyle {\begin{aligned}(G\circ F)(f)&=G(F(f))\\&=G((\eta _{Y}\circ f)^{*})\\&=\mu _{Y}\circ T(\eta _{Y}\circ f)\\&=\mu _{Y}\circ T\eta _{Y}\circ Tf\\&={\text{id}}_{TY}\circ Tf\\&=Tf.\end{aligned}}} Since ( G ∘ F ) ( X ) = T X {\displaystyle (G\circ F)(X)=TX} is true for any object X in C and ( G ∘ F ) ( f ) = T f {\displaystyle (G\circ F)(f)=Tf} is true for any morphism f in C, then G ∘ F = T {\displaystyle G\circ F=T} . Q.E.D. == References == Mac Lane, Saunders (1998). Categories for the Working Mathematician. Graduate Texts in Mathematics. Vol. 5 (2nd ed.). Springer. ISBN 0-387-98403-8. Zbl 0906.18001. Pedicchio, Maria Cristina; Tholen, Walter, eds. (2004). Categorical foundations. Special topics in order, topology, algebra, and sheaf theory. Encyclopedia of Mathematics and Its Applications. Vol. 97. Cambridge University Press. ISBN 0-521-83414-7. Zbl 1034.18001. Riehl, Emily (2016). Category Theory in Context (PDF). Dover Publications. ISBN 978-0-486-80903-8. OCLC 1006743127. Riguet, Jacques; Guitart, Rene (1992). "Enveloppe Karoubienne et categorie de Kleisli". Cahiers de Topologie et Géométrie Différentielle Catégoriques. 33 (3): 261–6. MR 1186950. Zbl 0767.18008. == External links == Kleisli category at the nLab
|
Wikipedia:Heinrich Martin Weber#0
|
Heinrich Martin Weber (5 March 1842, Heidelberg, Germany – 17 May 1913, Straßburg, Alsace-Lorraine, German Empire, now Strasbourg, France) was a German mathematician. Weber's main work was in algebra, number theory, and analysis. He is best known for his text Lehrbuch der Algebra published in 1895 and much of it is his original research in algebra and number theory. His work Theorie der algebraischen Functionen einer Veränderlichen (with Dedekind) established an algebraic foundation for Riemann surfaces, allowing a purely algebraic formulation of the Riemann–Roch theorem. Weber's research papers were numerous, most of them appearing in Crelle's Journal or Mathematische Annalen. He was the editor of Riemann's collected works. Weber was born in Heidelberg, Baden, and entered the University of Heidelberg in 1860. In 1866 he became a privatdozent, and in 1869 he was appointed as extraordinary professor at that school. Weber also taught in Zürich at the Federal Polytechnic Institute (today the ETH Zurich), at the University of Königsberg, and at the Technische Hochschule in Charlottenburg (today Technische Universität Berlin). His final post was at the Kaiser-Wilhelm-Universität Straßburg, Alsace-Lorraine, where he died. In 1893 in Chicago, his paper Zur Theorie der ganzzahligen algebraischen Gleichungen was read (but not by him) at the International Mathematical Congress held in connection with the World's Columbian Exposition. In 1895 and in 1904 he was president of the Deutsche Mathematiker-Vereinigung. His doctoral students include Heinrich Brandt, E. V. Huntington, Louis Karpinski, and Friedrich Levi. == Publications == with Richard Dedekind: Theorie der algebraischen Functionen einer Veränderlichen. J. Reine Angew. Math. 92 (1882) 181–290 Elliptische Functionen und algebraische Zahlen. Braunschweig 1891 Encyklopädie der Elementar-Mathematik. Ein Handbuch für Lehrer und Studierende. Leipzig 1903/07, (Vol. 1, Vol. 2, Vol. 3) (in German) with Bernhard Riemann (i.e. partly based on Riemann's lectures): Die partiellen Differential-Gleichungen der mathematischen Physik. Braunschweig 1900-01 Lehrbuch der Algebra. Braunschweig 1924, ed. Robert Fricke Weber, Heinrich Martin (1981) [1895], Lehrbuch der Algebra (in German), vol. 1 (3rd ed.), New York: AMS Chelsea Publishing, ISBN 978-0-8218-3258-5 Weber, Heinrich Martin (1981) [1895], Lehrbuch der Algebra (in German), vol. 2 (3rd ed.), New York: AMS Chelsea Publishing, ISBN 978-0-8218-2647-8 Weber, Heinrich Martin (1981) [1898], Lehrbuch der Algebra (in German), vol. 3 (3rd ed.), New York: AMS Chelsea Publishing, ISBN 978-0-8218-2971-4 The third volume is an expanded version of his earlier book "Elliptische Functionen und algebraische Zahlen". == References == O'Connor, John J.; Robertson, Edmund F., "Heinrich Martin Weber", MacTutor History of Mathematics Archive, University of St Andrews Schappacher, Norbert (1998), "On the history of Hilbert's twelfth problem: a comedy of errors", Matériaux pour l'histoire des mathématiques au XXe siècle (Nice, 1996), Sémin. Congr., vol. 3, Paris: Société Mathématique de France, pp. 243–273, ISBN 978-2-85629-065-1, MR 1640262 Voss, A. (1914), "Heinrich Weber.", Jahresbericht der Deutschen Mathematiker-Vereinigung (in German), 23: 431–444, ISSN 0012-0456 Heinrich Martin Weber at the Mathematics Genealogy Project
|
Wikipedia:Heinrich Tietze#0
|
Heinrich Franz Friedrich Tietze (August 31, 1880 – February 17, 1964) was an Austrian mathematician, famous for the Tietze extension theorem on functions from topological spaces to the real numbers. He also developed the Tietze transformations for group presentations, and was the first to pose the group isomorphism problem. Tietze's graph is also named after him; it describes the boundaries of a subdivision of the Möbius strip into six mutually-adjacent regions, found by Tietze as part of an extension of the four color theorem to non-orientable surfaces. == Education and career == Tietze was the son of Emil Tietze and the grandson of Franz Ritter von Hauer, both of whom were Austrian geologists. He was born in Schleinz, Austria-Hungary, and studied mathematics at the Technische Hochschule in Vienna beginning in 1898. After additional studies in Munich, he returned to Vienna, completing his doctorate in 1904 and his habilitation in 1908. From 1910 until 1918 Tietze taught mathematics in Brno, and was promoted to ordinary professor in 1913. He served in the Austrian army during World War I, and then returned to Brno, but in 1919 he took a position at the University of Erlangen, and then in 1925 moved again to the University of Munich, where he remained for the rest of his career. One of his doctoral students was Georg Aumann. Tietze retired in 1950, and died in Munich, West Germany. == Awards and honors == Tietze was a fellow of the Bavarian Academy of Sciences and a fellow of the Austrian Academy of Sciences. == Publications == Tietze, Heinrich (1957), "Über Schachturnier-Tabellen", Mathematische Zeitschrift, 67: 188–202, doi:10.1007/bf01258856, S2CID 123135166, archived from the original on January 11, 2013 Tietze, Heinrich (1910), "Einige Bemerkungen zum Problem des Kartenfärbens auf einseitigen Flächen", DMV Annual Report, archived from the original on January 11, 2013 Tietze, Heinrich (1915), "Über Funktionen, die auf einer abgeschlossenen Menge stetig sind", Journal für die reine und angewandte Mathematik, 145, archived from the original on 2012-07-08 Über die mit Lineal und Zirkel und die mit dem rechten Zeichenwinkel lösbaren Konstruktionsaufgaben, Mathematische Zeitschrift vol.46, 1940 mit Leopold Vietoris Beziehungen zwischen den verschiedenen Zweigen der Topologie, Enzyklopädie der Mathematischen Wissenschaften 1929 Über die Anzahl der stabilen Ruhelagen eines Würfels, Elemente der Mathematik vol.3, 1948 Über die topologische Invarianten mehrdimensionaler Mannigfaltigkeiten, Monatshefte für Mathematik und Physik, vol. 19, 1908, p.1-118 Über Simony Knoten und Simony Ketten mit vorgeschriebenen singulären Primzahlen für die Figur und für ihr Spiegelbild, Mathematische Zeitschrift vol.49, 1943, p.351 (Knot theory) Tietze, Heinrich (1965) [1959], Famous problems of mathematics. Solved and unsolved mathematical problems from antiquity to modern times., New York: Graylock Press, MR 0181558 == References == == External links == Heinrich Tietze at the Mathematics Genealogy Project
|
Wikipedia:Heinz Engl#0
|
Heinz Werner Engl (born 28 March 1953) is an Austrian mathematician who served as the rector of the University of Vienna. Engl was born in Linz. He studied at the Johannes Kepler University of Linz, where he earned an engineering diploma in technical mathematics in 1975, a doctorate in 1977, and a habilitation in 1979. He worked at the University of Linz starting in 1976 as an assistant professor, was promoted and tenured in 1981, and became a full professor in 1988. His research in this period concerned inverse problems in applied mathematics, with students including Barbara Kaltenbacher. He became vice-rector of the University of Vienna in 2007, and rector in 2011. With Martin Hanke and Andreas Neubauer he is the author of the book Regularization of Inverse Problems (Mathematics and its Applications 375, Kluwer Academic Publishers, 1996). Engl won the Theodor Körner Prize in 1978, the Wilhelm Exner Medal in 1998, and the ICIAM Pioneer Prize (jointly with Ingrid Daubechies) in 2007. He became a corresponding member of the Austrian Academy of Sciences in 2000, and a full member in 2003. He became a fellow of the Society for Industrial and Applied Mathematics in 2009, an inaugural fellow of the American Mathematical Society in 2012, and a member of Academia Europaea in 2013. He is also a member of the European Academy of Sciences and Arts. In 2012, Saarland University awarded him an honorary doctorate. He was a member of the Rotary Club Linz Süd from 1994 to 2012 and has been a member of the Rotary Club Vienna ever since. Heinz Engl is married and, according to his own statement, grandfather of Eleon Engl-Misirlisoy, the New Year's baby 2020 of the US capital Washington D.C. == References ==
|
Wikipedia:Heisook Lee#0
|
Heisook Lee (Korean: 이혜숙, born 1948) is a South Korean mathematician and activist for gender equality in mathematics. She is retired as a professor of mathematics and dean at Ewha Womans University. Her mathematical research has concerned abstract algebra and algebraic coding theory, including work on self-dual codes and bent functions. Lee graduated from Ewha Womans University in 1971. After a master's degree in 1974 from the University of British Columbia in Canada, she completed a Ph.D. in 1978 from Queen's University at Kingston, also in Canada. Her dissertation, The Brauer Group of an Integral Scheme, was supervised by Morris Orzech. After postdoctoral research in Germany at the University of Regensburg, she returned to Ewha Womans University in 1980, as a professor of mathematics. She became dean of natural sciences and dean of research affairs from 1997 to 2001, and dean of graduate studies from 2006 to 2008. Lee became founding editor of Communications of the Korean Mathematical Society in 1986. From 1994 to 1996 she was editor in chief of the Journal of the Korean Mathematical Society. She was the second president of the Korea Federation of Women's Science & Technology Associations (KOFWST), serving from 2006 to 2007. She founded the Center for Women in Science, Engineering, and Technology (WISET), now the Korean Foundation for Women in Science, Engineering, and Technology, and became its first president, serving from 2013 to 2016. She is a professor emeritus at Ewha Womans University, and president of the Korea Center for Gendered Innovations (GISTeR). == References ==
|
Wikipedia:Helen Byrne#0
|
Helen M. Byrne is a mathematician based at the University of Oxford. She is Professor of Mathematical Biology in the university's Mathematical Institute and a Professorial Fellow in Mathematics at Keble College. Her work involves developing mathematical models to describe biomedical systems including tumours. She was awarded the 2019 Society for Mathematical Biology Leah Edelstein-Keshet Prize for exceptional scientific achievements and for mentoring other scientists and was appointed a Fellow of the Society in 2021. == Early life and education == Byrne attended Manchester High School for Girls. Eventually she studied mathematics at Newnham College, Cambridge, where she became interested in the applications of mathematics to real-world problems. She moved to Wadham College, Oxford for her graduate studies, where she earned a master's degree in Mathematical Modelling and Numerical Analysis. She remained at Oxford for her doctoral degree in applied mathematics. She was appointed as a postdoctoral fellow at the cyclotron unit at Hammersmith Hospital. There, she started working in mathematical and theoretical biology. The biomedical questions she worked on included fitting mathematical models to positron emission tomography scans to evaluate oxygen and glucose transport and consumption within solid tumours. After hearing Mark Chaplain talk about tumours at a conference she realised she could use her mathematical skills to study tumour growth. == Research and career == Byrne worked with Mark Chaplain at the University of Bath from 1993. She joined the University of Manchester Institute of Science and Technology as a lecturer in 1996. In 1998 Byrne joined the University of Nottingham, where she was promoted to Professor of Applied Mathematics in 2003. She was involved with the development of the Nottingham Centre for Mathematical Medicine and Biology, which she directed from 1999 to 2011. She joined the faculty at the University of Oxford in 2011 where she was made Professor of Mathematical Biology based in the Mathematical Institute. Her research has considered mathematical models to describe biological tissue. She has explored how oxygen levels impact biological function, developing complex models that can describe disease progression. She was part of a team who demonstrated that cell cannibalism is involved in the development of inflammatory diseases. Byrne was appointed Director of Equality and Diversity in the Mathematical, Physical and Life Sciences (MPLS) Division from 2016 to 2020. In 2018 she was awarded the Society for Mathematical Biology Leah Edelstein-Keshet Prize, being appointed a fellow of the society in 2021. Byrne is co-director of the University of Liverpool 3D BioNet (an interdisciplinary network looking at how cells grow in three dimensions) and was on the management group of the Engineering and Physical Sciences Research Council Cyclops Healthcare Network which ran from 2016 to 2019. She is a member of the IBS Biomedical Mathematics Group. === Selected publications === Vipond, Oliver; Bull, Joshua A.; Macklin, Philip S.; Tillmann, Ulrike; Pugh, Christopher W.; Byrne, Helen M.; Harrington, Heather A. (2021-10-12). "Multiparameter persistent homology landscapes identify immune cell spatial patterns in tumors". Proceedings of the National Academy of Sciences. 118 (41): e2102166118. Bibcode:2021PNAS..11802166V. doi:10.1073/pnas.2102166118. ISSN 0027-8424. PMC 8522280. PMID 34625491. Nardini, John T.; Stolz, Bernadette J.; Flores, Kevin B.; Harrington, Heather A.; Byrne, Helen M. (2021-06-28). Chaplain, Mark (ed.). "Topological data analysis distinguishes parameter regimes in the Anderson-Chaplain model of angiogenesis". PLOS Computational Biology. 17 (6): e1009094. arXiv:2101.00523. Bibcode:2021PLSCB..17E9094N. doi:10.1371/journal.pcbi.1009094. ISSN 1553-7358. PMC 8270459. PMID 34181657. Maini, P. K.; Byrne, H. M.; Alarcón, T. (2003). "A cellular automaton model for tumour growth in inhomogeneous environment". Journal of Theoretical Biology. 225 (2): 257–274. Bibcode:2003JThBi.225..257A. doi:10.1016/S0022-5193(03)00244-3. PMID 14575659. Preziosi, Luigi; Byrne, H. M. (2003). "Modelling solid tumour growth using the theory of mixtures". Mathematical Medicine and Biology. 20 (4): 341–366. doi:10.1093/imammb/20.4.341. PMID 14969384. Byrne, Helen M. (2010). "Dissecting cancer through mathematics: from the cell to the animal model". Nature Reviews Cancer. 10 (3): 221–230. doi:10.1038/nrc2808. PMID 20179714. S2CID 24616792. == Personal life == Whilst a graduate student at Oxford, she competed for OUWLRC in the Henley Boat Races in 1990 and 1991, earning a half blue each time. == References == == External links == Helen Byrne publications indexed by Google Scholar
|
Wikipedia:Helen Calkins#0
|
Helen Calkins (1893–1970) was an American mathematician and professor, and one of the few women to earn a PhD in mathematics in the United States before World War II. == Biography == Helen Calkins was born on October 20, 1893, to Anna Burns Schermerhorn and Addison Niles Calkins, in Quincy, Illinois. The eldest of two daughters, she was a student at Quincy High School from 1908–1912 and then she attended Knox College in Galesburg, Illinois, starting in 1912, graduating in 1916 with a special honor in mathematics. Her first job was teaching math at a junior high school in Quincy. In 1917 she taught at the senior high school in Jacksonville, Illinois for a year before returning to Knox College as a math instructor 1918–1920. In February 1932 at Columbia University (1920–1921) she earned her master's degree with the thesis: The unity in mathematics, as illustrated by a certain differential equation. In February 1932, Calkins was awarded her PhD from Cornell; her advisors were: Charles F. Roos and David Clinton Gillespie. Her dissertation was titled, Some Implicit Functional Theorems, which concerned "a problem of maximizing a functional not of the ordinary calculus of variations type." Roos referred to it writing "I think that Miss Calkins' thesis is of unusual interest." Calkins joined the math department at the Pennsylvania College for Women (now Chatham University) in Pittsburgh, where she was usually the only member of the mathematics department that she headed. She enjoyed service on faculty committees, especially those concerning course curriculum and the library. With the start of World War II in 1941, Calkins became a statistician in engineering defense training at the Pennsylvania State College. In 1943, she spent a year at the University of Minnesota, to teach math to pre-flight cadets. Calkins retired as professor emeritus in 1957, from the school then known as Chatham College. Helen Calkins died on June 17, 1970, at the Good Samaritan Home in Quincy, Illinois of heart disease. She was 76 years of age. == Memberships == According to Judy Green, Calkins belonged to the Delta Delta Delta social sorority, the Pi Lambda Theta education honor and professional association, the College Club of Pittsburgh, and the Daughters of the American Revolution as well as several professional societies. American Mathematical Society Mathematical Association of America Sigma Delta Epsilon American Association of University Professors Phi Mu Epsilon == References ==
|
Wikipedia:Helen Infeld#0
|
Helen Infeld (1907–1993) was a mathematics professor and one of the few women to earn a doctorate in mathematics in the United States before World War II. For her anti-fascist political views, which were viewed as pro-communist, she was forced to leave Canada with her family and move to Poland to escape the consequences of McCarthyism in North America. == Biography == Helen Mary Schlauch was born July 20, 1907, in The Bronx, in New York City, and was the third child of Margaret Brosnahan and William Storb Schlauch. Her older sister Margaret Schlauch studied the classics and, like Helen, eventually became a university professor. === Education === Helen Schlauch attended high school in Hasbrouck Heights, New Jersey, and graduated in 1924. In New York City, she enrolled at Washington Square College of New York University completing her studies in 1928 with a mathematics major and English and psychology minors. At the College, she was president of the Pan Hellenic Council her junior year. Schlauch received her master's degree in 1929 at Cornell University in Ithaca, New York, supervised by Walter Buckingham Carver. Her thesis was titled: Mixed systems of linear equations and inequalities. While she worked as a faculty member at Hunter College, she began her doctoral studies at Cornell University. Her 1933 PhD dissertation, written under Virgil Snyder's supervision, concerned algebraic geometry titled, On the Normal Rational N-Ic. In May 1932, Helen married Leonard Palmer Adams, a doctoral student at Cornell. She had briefly changed her name to Helen Schlauch Adams when she applied for the Cornell PhD program. Helen and Leonard divorced in 1936 and had no children. === Second marriage === She was on the faculty of Hunter College from 1931 to 1941. During a 1938 meeting of the American Mathematical Society (AMS), she met Professor Leopold Infeld, a Polish-born theoretical physicist, who had taught for eight years in Jewish secondary schools in Poland after receiving his PhD in 1921 from the University of Krakow. According to Green, "When [Leopold] met Helen Adams he had just published The Evolution of Physics with Albert Einstein and was about to go to Canada to teach at the University of Toronto on the applied mathematics faculty." Leopold moved alone to Toronto but from 1938 to 1939, he regularly visited New York to see Helen. The pair were married in New Jersey on April 12, 1939, and they went on to have two children, both born in Toronto, Eric (sometimes spelled Eryk) and Joan. Helen Infeld held several positions at the University of Toronto. After the end of World War II, the University established the Ajax campus by using a massive decommissioned munitions plant to accommodate Canada's veterans returning from the War. Helen taught Calculus there for 3+1⁄2 years but when the campus closed in 1949, she became unemployed. After World War II and the attack on Hiroshima, Helen's husband, Leopold, protested against the armament race (echoing the beliefs of his collaborator, Albert Eistein, who was a "convinced pacifist"). His rhetoric caused some people to suggest that Leopold and Helen were Soviet spies. Published attacks against the couple caused them to become concerned for their family's safety so they decided to leave Canada. === Emigration === In 1950, the family departed Toronto for Warsaw, Poland, which was Leopold's birthplace but for Helen, this was her first trip overseas. In 1951, Helen's sister, Margaret Schlauch, who was a faculty member at New York University, left the United States to join the Infeld family in Poland, "saying she wished to avoid persecution for pro-Communist views." Margaret became a linguistics professor at Warsaw University and headed the English department. In 1958, the Canadian government canceled the citizenship of the Infeld's children when they were 15 and 18. Years later, the hostilities between the Infelds and the University of Toronto cooled and, according to a 1971 obituary, they received "assurance from the Secretary of State that citizenship would be restored to Eric and Joan should they wish to take up residence in Canada." Until 1982, Helen Infeld served as editor of Poland (A Monthly), the English-language version of a periodical devoted to cultural events in Poland. She died in Warsaw on July 6, 1993, at 85 and was buried in Powazki municipal cemetery. == Selected honors == Gold Cross of Merit (1954) The Chevalier Cross, Order of Polonia Restituta (1970) == Memberships == According to Green, Helen Infeld was active in several organizations. American Mathematical Society, (AMS) Mathematical Association of America, (MMA) Phi Beta Kappa Pi Mu Epsilon Polish-Icelandic Society == References ==
|
Wikipedia:Helen Popova Alderson#0
|
Helen Popova Alderson (1924–1972) was a Soviet and British mathematician and mathematics translator known for her research on quasigroups and on higher reciprocity laws. == Life == Alderson was born on 14 May 1924 in Baku, then part of the Soviet Union, to a family of two academics from Moscow. Her father, a neurophysiologist, had been a student of Ivan Pavlov. She began studying mathematics at Moscow University in 1937, when she was only 13. She had to break off her studies because of World War II, moving to Paris as a refugee with her family. After the war, she returned to study at the University of Edinburgh. She completed a Ph.D. there in 1951; her dissertation was Logarithmetics of Non-Associative Algebras. After leaving mathematical research to raise two children in Cambridge, she was funded by the Calouste Gulbenkian Foundation with a Fellowship at Lucy Cavendish College, Cambridge, beginning in the late 1960s. At Cambridge, she worked with J. W. S. Cassels. She died on 5 November 1972, from complications of kidney disease. == Research == In the theory of higher reciprocity laws, Alderson published necessary and sufficient conditions for 2 and 3 to be seventh powers, in modular arithmetic modulo a given prime number p {\displaystyle p} .[7X] According to Smith (1976), "plain quasigroups were first studied by Helen Popova-Alderson, in a series of papers dating back to the early fifties". Smith cites in particular a posthumous paper (Alderson 1974)[FPQ] and its references. In this context, a quasigroup is a mathematical structure consisting of a set of elements and a binary operation that does not necessarily obey the associative law, but where (like a group) this operation can be inverted. Being plain involves having only a finite number of elements and no non-trivial subalgebras. == Translation == As well as Russian, English, and French, Alderson spoke Polish, Czech, and some German. She became the English translator of Elementary Number Theory, a textbook originally published in Russian in 1937 by B. A. Venkov. Her translation was published by Wolters-Noordhoff of Groningen in 1970. As well as the original text, it includes footnotes by Alderson updating the material with new developments in number theory.[ENT] == Selected publications == == References ==
|
Wikipedia:Helen Wilson (mathematician)#0
|
Helen Jane Wilson, (born 1973), is a British mathematician and the first female Head of Mathematics at University College London (UCL). Her research focuses on the theoretical and numerical modelling of the flow of non-Newtonian fluids such as polymeric materials and particle suspensions. == Early life and education == Wilson was born in Warrington. Her father, Leslie Knight Wilson was a chartered accountant; her mother, Brenda (née Naylor) a French teacher. She attended Broomfields Junior School and Bridgewater High School. Wilson studied at Clare College, Cambridge, completing a BA, Certificate of Advanced Study in Mathematics (later converted to an MMath) and PhD in mathematics. Her PhD thesis, titled "Shear Flow Instabilities in Viscoelastic Fluids", was supervised by John Rallison. On graduation she moved to the University of Colorado at Boulder, where she began research on suspension mechanics with Rob Davis in the Chemical Engineering department. == Mathematical work == In 2000 Wilson returned to the UK to take up a lectureship in Applied Mathematics at the University of Leeds. In 2004 she moved to UCL, where she is Professor of Applied Mathematics and as of September 2018, Head of Department. Wilson is the first female to hold the position of Head of Mathematics at UCL. === Research in fluid mechanics === Wilson's PhD thesis and early papers focused on instabilities in viscoelastic fluids. She predicted a new instability in channel flow of a shear-thinning fluid which was later discovered experimentally by another group and on which she still works. She has also worked on instabilities in shear-banding flows and in more complex geometries. Her other major research interest, besides viscoelasticity, is suspension mechanics, and in particular the effect of particle contacts on fluid rheology. Her most recent projects draw these two fields together, investigating the interaction of solid particles with their complex material environment in fields ranging from healthcare to engineering. Her academic publications are listed on the UCL site. One of her best-known publications is the paper "The fluid dynamics of the chocolate fountain", co-authored with Adam Townsend. Unusually for a mathematical paper, this was covered in the Washington Post. === Knowledge transfer === Wilson gave the 2019 Joint London Mathematical Society Annual Lecture on "Toothpaste, custard and chocolate: mathematics gets messy". Problem plastics & how mathematics can help, published in UCL Science and presented at Mathematics Works (Oct 2007). Public lecture: From gases to gloops: Instabilities in fluids in the UCL Lunch Hour Lecture series on 23 February 2016. === Non-technical articles === Case study for the Royal Society on how a supportive employer can support a mother on her return to work. Blog post and BBC World TV news interview commenting on the award of the Fields Medal to a female mathematician for the first time. The D'Hondt method Explained: brief explanation of an easier way to understand the allocation of seats at the European elections === Books === Practical Analytical Methods for PDEs in volume 1 of the LTCC Advanced Mathematics series, World Scientific, 2015. In 2016, Wilson co-authored with Dame Celia Hoyles a chapter of the book "Mathematics: How It Shaped Our World" == Recognition == Wilson was president (for the 2015–2017 term) of the British Society of Rheology the first woman to hold this position. In 2014 she was a member of the subject panel for Mathematics on ALCAB (the A Level Content Advisory Board), advising on the reforms to A Level Mathematics for first teaching in September 2016. She was a Council Member and is now the Vice-President (Learned Societies) of the Institute of Mathematics and its Applications. == References ==
|
Wikipedia:Helena Nussenzveig Lopes#0
|
Helena Judith Nussenzveig Lopes is a Brazilian mathematician, known for her work on the Euler equations for incompressible flow in fluid dynamics. Since February 2025, CIMPA president. She is a professor titular in the Department of Mathematical Methods at the Federal University of Rio de Janeiro. == Education and career == Nussenzveig Lopes was born in Brazil, the daughter of physicist Herch Moysés Nussenzveig. She earned her Ph.D. from the University of California, Berkeley in 1991. Her dissertation, An Estimate of the Hausdorff Dimension of a Concentration Set for the 2D Incompressible Euler Equations, was jointly supervised by Ronald DiPerna and Lawrence C. Evans. From 1992 to 2012, she belonged to the faculty of the University of Campinas. She moved to the Federal University of Rio de Janeiro as a full professor in 2012, and headed the department of mathematics there from 2014 to 2016. Shechaired the Society for Industrial and Applied Mathematics Activity Group on Analysis of Partial Differential Equations for 2015–2016. == Recognition == In 2010 she was awarded the National Order of Scientific Merit. In 2016 she became a SIAM Fellow "for advances in analysis of weak solutions of incompressible Euler equations and for advancing applied mathematics in Brazil and internationally".. She was one of the invited speakers in the Partial Differential Equations Section of the 2018 International Congress of Mathematicians. In 2019, she was elected to the Brazilian Academy of Sciences. She was elected to the 2020 Class of Fellows of the American Mathematical Society "for contributions to the analysis of weak solutions of incompressible Euler equations and for advancing applied mathematics in Brazil and internationally." She was also awarded the 2020 World Academy of Sciences award in Mathematics and was elected for membership in the Academy in 2022. == References == == External links == Home page Helena Nussenzveig Lopes publications indexed by Google Scholar
|
Wikipedia:Helene Stähelin#0
|
Helene Stähelin (18 July 1891 Wintersingen – 30 December 1970 Basel) was a Swiss mathematician, teacher, and peace activist. Between 1948 and 1967, she was president of the Swiss section of the Women's International League for Peace and Freedom and its representative in the Swiss Peace Council. == Early life and scientific work == She was one of twelve children of the parson Gustav Stähelin (1858–1934) and his wife Luise, née Lieb. In 1894, the family moved from Wintersingen to Allschwil. Helene Stähelin attended the Töchterschule Basel and the Universities Basel and Göttingen. In 1922, she became teacher of mathematics and natural sciences at the Töchterinstitut(de) in Ftan. In 1924, she obtained her Dr.phil. degree from Basel University for her dissertation Die charakteristischen Zahlen analytischer Kurven auf dem Kegel zweiter Ordnung und ihrer Studyschen Bildkurven, advised by Hans Mohrmann and Otto Spiess. In 1926, she became a member of the Swiss Mathematical Society. Between 1934 and 1956, Helene Stähelin worked as teacher at the Protestant secondary school in Zug. After her pensioning she returned to Basel, where she assisted for several years to Otto Spiess' editing the Bernoulli family letters. == Political activism == Being a pacifist, Helene Stähelin committed herself to the Women's International League for Peace and Freedom (Internationale Frauenliga für Frieden und Freiheit, IFFF) and its struggle against scientific warfare. She was president of the IFFF's Swiss section in 1947–1967, the main issues were the United Nations Organization, nuclear weapons, and the Vietnam War. Due to her peace activism, she was watched by Swiss authorities in the mid 1950s, her file at the Swiss Public Prosecutor General was kept secret until 1986. Helene Stähelin also was active towards Women's suffrage in Switzerland. Although Stähelin herself never experienced the female vote. == See also == List of peace activists == References ==
|
Wikipedia:Helge Holden#0
|
Helge Holden (born 28 September 1956) is a Norwegian mathematician working in the field of differential equations and mathematical physics. He was Praeses of the Royal Norwegian Society of Sciences and Letters from 2014 to 2016. He earned the dr.philos. degree at the University of Oslo in 1985. The title of his dissertation with Raphael Høegh-Krohn was Point Interactions and the Short-Range Expansion. A Solvable Model in Quantum Mechanics and Its Approximation. He was appointed professor at the Norwegian Institute of Technology (now: the Norwegian University of Science and Technology ) in 1991. His research interests are Differential equations, mathematical physics (in particular hyperbolic conservation laws and completely integrable systems), Stochastic analysis, and flow in porous media. In 2014 he became Chairman of the board of the Abel Prize fund. He was elected Secretary General of the International Mathematical Union (IMU) for the period 2019–2022. == Awards and honors == He is a member of the Royal Norwegian Society of Sciences and Letters, the Norwegian Academy of Science and Letters and of the Norwegian Academy of Technological Sciences. In 2013 he became a fellow of the American Mathematical Society, for "contributions to partial differential equations". He was elected as a fellow of the Society for Industrial and Applied Mathematics in 2017, "for contributions to nonlinear partial differential equations and related fields, to research administration, and to the dissemination of mathematics". In 2022 he was awarded the Gunnerus Medal for his academic achievement. The Gunnerus Medal is awarded by The Royal Norwegian Society of Sciences and Letters. == References == == External links == Official website Helge Holden at the Mathematics Genealogy Project List of publications by Helge Holden in CRIStin
|
Wikipedia:Helju Rebane#0
|
Helju Rebane (born 18 July 1948) is an Estonian writer. She writes mainly prose and science fiction in the Estonian and Russian languages. She was born in Tallinn. Her father was philosopher Jaan Rebane and her uncles were physicist and former president of the Academy of Sciences of the ESSR Karl Rebane, physicist Toomas Rebane, and mathematician Jüri Rebane. She graduated from Tartu State University Tartu with a degree in theoretical mathematics in 1971. From 1972 until 1973, she worked in the department of logic and psychology at the university. Later she studied logic at Moscow University. In Moscow, she was as a lecturer at the Institute of Management Problems of the Scientific and Technical Committee of the USSR from 1974 until 1980, and as a senior engineer at the Ministry of Health Computing Center from 1981 until 1983. Rebane made her writing debut in the journal Looming in 1981 with the story Väike kohvik. In 1983 she won a prize in the story competition run by the literary journal Noorus. == Works == 1986 story "Väike kohvik". Eesti Raamat, 110 pp 2011 "Город на Альтрусе: фантастическая повесть и рассказы". Воронеж, 2011. 207 pp 2017 "50 рассказов". Москва: Ridero, 288 pp 2017 "Кот в лабиринте: рассказы". Москва: Ridero, 207 pp 2021 story "Õige valik". Fantaasia, 181 pp == References ==
|
Wikipedia:Helly's selection theorem#0
|
In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician Eduard Helly. A more general version of the theorem asserts compactness of the space BVloc of functions locally of bounded total variation that are uniformly bounded at a point. The theorem has applications throughout mathematical analysis. In probability theory, the result implies compactness of a tight family of measures. == Statement of the theorem == Let (fn)n ∈ N be a sequence of increasing functions mapping a real interval I into the real line R, and suppose that it is uniformly bounded: there are a,b ∈ R such that a ≤ fn ≤ b for every n ∈ N. Then the sequence (fn)n ∈ N admits a pointwise convergent subsequence. == Proof == === Step 1. An increasing function f on an interval I has at most countably many points of discontinuity. === Let A = { x ∈ I : f ( y ) ↛ f ( x ) as y → x } {\displaystyle A=\{x\in I:f(y)\not \rightarrow f(x){\text{ as }}y\rightarrow x\}} , i.e. the set of discontinuities, then since f is increasing, any x in A satisfies f ( x − ) ≤ f ( x ) ≤ f ( x + ) {\displaystyle f(x^{-})\leq f(x)\leq f(x^{+})} , where f ( x − ) = lim y ↑ x f ( y ) {\displaystyle f(x^{-})=\lim \limits _{y\uparrow x}f(y)} , f ( x + ) = lim y ↓ x f ( y ) {\displaystyle f(x^{+})=\lim \limits _{y\downarrow x}f(y)} , hence by discontinuity, f ( x − ) < f ( x + ) {\displaystyle f(x^{-})<f(x^{+})} . Since the set of rational numbers is dense in R, ∏ x ∈ A [ ( f ( x − ) , f ( x + ) ) ∩ Q ] {\displaystyle \prod _{x\in A}[{\bigl (}f(x^{-}),f(x^{+}){\bigr )}\cap \mathrm {Q} ]} is non-empty. Thus the axiom of choice indicates that there is a mapping s from A to Q. It is sufficient to show that s is injective, which implies that A has a non-larger cardinity than Q, which is countable. Suppose x1,x2∈A, x1<x2, then f ( x 1 − ) < f ( x 1 + ) ≤ f ( x 2 − ) < f ( x 2 + ) {\displaystyle f(x_{1}^{-})<f(x_{1}^{+})\leq f(x_{2}^{-})<f(x_{2}^{+})} , by the construction of s, we have s(x1)<s(x2). Thus s is injective. === Step 2. Inductive Construction of a subsequence converging at discontinuities and rationals. === Let A n = { x ∈ I ; f n ( y ) ↛ f n ( x ) as y → x } {\displaystyle A_{n}=\{x\in I;f_{n}(y)\not \rightarrow f_{n}(x){\text{ as }}y\to x\}} , i.e. the discontinuities of fn, A = ( ∪ n ∈ N A n ) ∪ ( I ∩ Q ) {\displaystyle A=(\cup _{n\in \mathrm {N} }A_{n})\cup (\mathrm {I} \cap \mathrm {Q} )} , then A is countable, and it can be denoted as {an: n∈N}. By the uniform boundedness of (fn)n ∈ N and B-W theorem, there is a subsequence (f(1)n)n ∈ N such that (f(1)n(a1))n ∈ N converges. Suppose (f(k)n)n ∈ N has been chosen such that (f(k)n(ai))n ∈ N converges for i=1,...,k, then by uniform boundedness, there is a subsequence (f(k+1)n)n ∈ N of (f(k)n)n ∈ N, such that (f(k+1)n(ak+1))n ∈ N converges, thus (f(k+1)n)n ∈ N converges for i=1,...,k+1. Let g k = f k ( k ) {\displaystyle g_{k}=f_{k}^{(k)}} , then gk is a subsequence of fn that converges pointwise in A. === Step 3. gk converges in I except possibly in an at most countable set. === Let h k ( x ) = sup a ≤ x , a ∈ A g k ( a ) {\displaystyle h_{k}(x)=\sup _{a\leq x,a\in A}g_{k}(a)} , then , hk(a)=gk(a) for a∈A, hk is increasing, let h ( x ) = lim sup k → ∞ h k ( x ) {\displaystyle h(x)=\limsup \limits _{k\rightarrow \infty }h_{k}(x)} , then h is increasing, since supremes and limits of increasing functions are increasing, and h ( a ) = lim k → ∞ g k ( a ) {\displaystyle h(a)=\lim \limits _{k\rightarrow \infty }g_{k}(a)} for a∈ A by Step 2. By Step 1, h has at most countably many discontinuities. We will show that gk converges at all continuities of h. Let x be a continuity of h, q,r∈ A, q<x<r, then g k ( q ) − h ( r ) ≤ g k ( x ) − h ( x ) ≤ g k ( r ) − h ( q ) {\displaystyle g_{k}(q)-h(r)\leq g_{k}(x)-h(x)\leq g_{k}(r)-h(q)} ,hence lim sup k → ∞ ( g k ( x ) − h ( x ) ) ≤ lim sup k → ∞ ( g k ( r ) − h ( q ) ) = h ( r ) − h ( q ) {\displaystyle \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(r)-h(q){\bigr )}=h(r)-h(q)} h ( q ) − h ( r ) = lim inf k → ∞ ( g k ( q ) − h ( r ) ) ≤ lim inf k → ∞ ( g k ( x ) − h ( x ) ) {\displaystyle h(q)-h(r)=\liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(q)-h(r){\bigr )}\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}} Thus, h ( q ) − h ( r ) ≤ lim inf k → ∞ ( g k ( x ) − h ( x ) ) ≤ lim sup k → ∞ ( g k ( x ) − h ( x ) ) ≤ h ( r ) − h ( q ) {\displaystyle h(q)-h(r)\leq \liminf \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq \limsup \limits _{k\rightarrow \infty }{\bigl (}g_{k}(x)-h(x){\bigr )}\leq h(r)-h(q)} Since h is continuous at x, by taking the limits q ↑ x , r ↓ x {\displaystyle q\uparrow x,r\downarrow x} , we have h ( q ) , h ( r ) → h ( x ) {\displaystyle h(q),h(r)\rightarrow h(x)} , thus lim k → ∞ g k ( x ) = h ( x ) {\displaystyle \lim \limits _{k\rightarrow \infty }g_{k}(x)=h(x)} === Step 4. Choosing a subsequence of gk that converges pointwise in I === This can be done with a diagonal process similar to Step 2. With the above steps we have constructed a subsequence of (fn)n ∈ N that converges pointwise in I. == Generalisation to BVloc == Let U be an open subset of the real line and let fn : U → R, n ∈ N, be a sequence of functions. Suppose that (fn) has uniformly bounded total variation on any W that is compactly embedded in U. That is, for all sets W ⊆ U with compact closure W̄ ⊆ U, sup n ∈ N ( ‖ f n ‖ L 1 ( W ) + ‖ d f n d t ‖ L 1 ( W ) ) < + ∞ , {\displaystyle \sup _{n\in \mathbf {N} }\left(\|f_{n}\|_{L^{1}(W)}+\|{\frac {\mathrm {d} f_{n}}{\mathrm {d} t}}\|_{L^{1}(W)}\right)<+\infty ,} where the derivative is taken in the sense of tempered distributions. Then, there exists a subsequence fnk, k ∈ N, of fn and a function f : U → R, locally of bounded variation, such that fnk converges to f pointwise almost everywhere; and fnk converges to f locally in L1 (see locally integrable function), i.e., for all W compactly embedded in U, lim k → ∞ ∫ W | f n k ( x ) − f ( x ) | d x = 0 ; {\displaystyle \lim _{k\to \infty }\int _{W}{\big |}f_{n_{k}}(x)-f(x){\big |}\,\mathrm {d} x=0;} : 132 and, for W compactly embedded in U, ‖ d f d t ‖ L 1 ( W ) ≤ lim inf k → ∞ ‖ d f n k d t ‖ L 1 ( W ) . {\displaystyle \left\|{\frac {\mathrm {d} f}{\mathrm {d} t}}\right\|_{L^{1}(W)}\leq \liminf _{k\to \infty }\left\|{\frac {\mathrm {d} f_{n_{k}}}{\mathrm {d} t}}\right\|_{L^{1}(W)}.} : 122 == Further generalizations == There are many generalizations and refinements of Helly's theorem. The following theorem, for BV functions taking values in Banach spaces, is due to Barbu and Precupanu: Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that zn is a uniformly bounded sequence in BV([0, T]; X) with zn(t) ∈ E for all n ∈ N and t ∈ [0, T]. Then there exists a subsequence znk and functions δ, z ∈ BV([0, T]; X) such that for all t ∈ [0, T], ∫ [ 0 , t ) Δ ( d z n k ) → δ ( t ) ; {\displaystyle \int _{[0,t)}\Delta (\mathrm {d} z_{n_{k}})\to \delta (t);} and, for all t ∈ [0, T], z n k ( t ) ⇀ z ( t ) ∈ E ; {\displaystyle z_{n_{k}}(t)\rightharpoonup z(t)\in E;} and, for all 0 ≤ s < t ≤ T, ∫ [ s , t ) Δ ( d z ) ≤ δ ( t ) − δ ( s ) . {\displaystyle \int _{[s,t)}\Delta (\mathrm {d} z)\leq \delta (t)-\delta (s).} == See also == Bounded variation Fraňková-Helly selection theorem Total variation == References == Rudin, W. (1976). Principles of Mathematical Analysis. International Series in Pure and Applied Mathematics (Third ed.). New York: McGraw-Hill. 167. ISBN 978-0070542358. Barbu, V.; Precupanu, Th. (1986). Convexity and optimization in Banach spaces. Mathematics and its Applications (East European Series). Vol. 10 (Second Romanian ed.). Dordrecht: D. Reidel Publishing Co. xviii+397. ISBN 90-277-1761-3. MR860772
|
Wikipedia:Helmholtz decomposition#0
|
In physics and mathematics, the Helmholtz decomposition theorem or the fundamental theorem of vector calculus states that certain differentiable vector fields can be resolved into the sum of an irrotational (curl-free) vector field and a solenoidal (divergence-free) vector field. In physics, often only the decomposition of sufficiently smooth, rapidly decaying vector fields in three dimensions is discussed. It is named after Hermann von Helmholtz. == Definition == For a vector field F ∈ C 1 ( V , R n ) {\displaystyle \mathbf {F} \in C^{1}(V,\mathbb {R} ^{n})} defined on a domain V ⊆ R n {\displaystyle V\subseteq \mathbb {R} ^{n}} , a Helmholtz decomposition is a pair of vector fields G ∈ C 1 ( V , R n ) {\displaystyle \mathbf {G} \in C^{1}(V,\mathbb {R} ^{n})} and R ∈ C 1 ( V , R n ) {\displaystyle \mathbf {R} \in C^{1}(V,\mathbb {R} ^{n})} such that: F ( r ) = G ( r ) + R ( r ) , G ( r ) = − ∇ Φ ( r ) , ∇ ⋅ R ( r ) = 0. {\displaystyle {\begin{aligned}\mathbf {F} (\mathbf {r} )&=\mathbf {G} (\mathbf {r} )+\mathbf {R} (\mathbf {r} ),\\\mathbf {G} (\mathbf {r} )&=-\nabla \Phi (\mathbf {r} ),\\\nabla \cdot \mathbf {R} (\mathbf {r} )&=0.\end{aligned}}} Here, Φ ∈ C 2 ( V , R ) {\displaystyle \Phi \in C^{2}(V,\mathbb {R} )} is a scalar potential, ∇ Φ {\displaystyle \nabla \Phi } is its gradient, and ∇ ⋅ R {\displaystyle \nabla \cdot \mathbf {R} } is the divergence of the vector field R {\displaystyle \mathbf {R} } . The irrotational vector field G {\displaystyle \mathbf {G} } is called a gradient field and R {\displaystyle \mathbf {R} } is called a solenoidal field or rotation field. This decomposition does not exist for all vector fields and is not unique. == History == The Helmholtz decomposition in three dimensions was first described in 1849 by George Gabriel Stokes for a theory of diffraction. Hermann von Helmholtz published his paper on some hydrodynamic basic equations in 1858, which was part of his research on the Helmholtz's theorems describing the motion of fluid in the vicinity of vortex lines. Their derivation required the vector fields to decay sufficiently fast at infinity. Later, this condition could be relaxed, and the Helmholtz decomposition could be extended to higher dimensions. For Riemannian manifolds, the Helmholtz-Hodge decomposition using differential geometry and tensor calculus was derived. The decomposition has become an important tool for many problems in theoretical physics, but has also found applications in animation, computer vision as well as robotics. == Three-dimensional space == Many physics textbooks restrict the Helmholtz decomposition to the three-dimensional space and limit its application to vector fields that decay sufficiently fast at infinity or to bump functions that are defined on a bounded domain. Then, a vector potential A {\displaystyle A} can be defined, such that the rotation field is given by R = ∇ × A {\displaystyle \mathbf {R} =\nabla \times \mathbf {A} } , using the curl of a vector field. Let F {\displaystyle \mathbf {F} } be a vector field on a bounded domain V ⊆ R 3 {\displaystyle V\subseteq \mathbb {R} ^{3}} , which is twice continuously differentiable inside V {\displaystyle V} , and let S {\displaystyle S} be the surface that encloses the domain V {\displaystyle V} with outward surface normal n ^ ′ {\displaystyle \mathbf {\hat {n}} '} . Then F {\displaystyle \mathbf {F} } can be decomposed into a curl-free component and a divergence-free component as follows: F = − ∇ Φ + ∇ × A , {\displaystyle \mathbf {F} =-\nabla \Phi +\nabla \times \mathbf {A} ,} where Φ ( r ) = 1 4 π ∫ V ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ ⋅ F ( r ′ ) | r − r ′ | d S ′ A ( r ) = 1 4 π ∫ V ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ − 1 4 π ∮ S n ^ ′ × F ( r ′ ) | r − r ′ | d S ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\cdot {\frac {\mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} S'\\[8pt]\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int _{V}{\frac {\nabla '\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'-{\frac {1}{4\pi }}\oint _{S}\mathbf {\hat {n}} '\times {\frac {\mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} S'\end{aligned}}} and ∇ ′ {\displaystyle \nabla '} is the nabla operator with respect to r ′ {\displaystyle \mathbf {r'} } , not r {\displaystyle \mathbf {r} } . If V = R 3 {\displaystyle V=\mathbb {R} ^{3}} and is therefore unbounded, and F {\displaystyle \mathbf {F} } vanishes faster than 1 / r {\displaystyle 1/r} as r → ∞ {\displaystyle r\to \infty } , then one has Φ ( r ) = 1 4 π ∫ R 3 ∇ ′ ⋅ F ( r ′ ) | r − r ′ | d V ′ A ( r ) = 1 4 π ∫ R 3 ∇ ′ × F ( r ′ ) | r − r ′ | d V ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla '\cdot \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\\[8pt]\mathbf {A} (\mathbf {r} )&={\frac {1}{4\pi }}\int _{\mathbb {R} ^{3}}{\frac {\nabla '\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V'\end{aligned}}} This holds in particular if F {\displaystyle \mathbf {F} } is twice continuously differentiable in R 3 {\displaystyle \mathbb {R} ^{3}} and of bounded support. === Derivation === === Solution space === If ( Φ 1 , A 1 ) {\displaystyle (\Phi _{1},{\mathbf {A} _{1}})} is a Helmholtz decomposition of F {\displaystyle \mathbf {F} } , then ( Φ 2 , A 2 ) {\displaystyle (\Phi _{2},{\mathbf {A} _{2}})} is another decomposition if, and only if, Φ 1 − Φ 2 = λ {\displaystyle \Phi _{1}-\Phi _{2}=\lambda \quad } and A 1 − A 2 = A λ + ∇ φ , {\displaystyle \quad \mathbf {A} _{1}-\mathbf {A} _{2}={\mathbf {A} }_{\lambda }+\nabla \varphi ,} where λ {\displaystyle \lambda } is a harmonic scalar field, A λ {\displaystyle {\mathbf {A} }_{\lambda }} is a vector field which fulfills ∇ × A λ = ∇ λ , {\displaystyle \nabla \times {\mathbf {A} }_{\lambda }=\nabla \lambda ,} φ {\displaystyle \varphi } is a scalar field. Proof: Set λ = Φ 2 − Φ 1 {\displaystyle \lambda =\Phi _{2}-\Phi _{1}} and B = A 2 − A 1 {\displaystyle {\mathbf {B} =A_{2}-A_{1}}} . According to the definition of the Helmholtz decomposition, the condition is equivalent to − ∇ λ + ∇ × B = 0 {\displaystyle -\nabla \lambda +\nabla \times \mathbf {B} =0} . Taking the divergence of each member of this equation yields ∇ 2 λ = 0 {\displaystyle \nabla ^{2}\lambda =0} , hence λ {\displaystyle \lambda } is harmonic. Conversely, given any harmonic function λ {\displaystyle \lambda } , ∇ λ {\displaystyle \nabla \lambda } is solenoidal since ∇ ⋅ ( ∇ λ ) = ∇ 2 λ = 0. {\displaystyle \nabla \cdot (\nabla \lambda )=\nabla ^{2}\lambda =0.} Thus, according to the above section, there exists a vector field A λ {\displaystyle {\mathbf {A} }_{\lambda }} such that ∇ λ = ∇ × A λ {\displaystyle \nabla \lambda =\nabla \times {\mathbf {A} }_{\lambda }} . If A ′ λ {\displaystyle {\mathbf {A} '}_{\lambda }} is another such vector field, then C = A λ − A ′ λ {\displaystyle \mathbf {C} ={\mathbf {A} }_{\lambda }-{\mathbf {A} '}_{\lambda }} fulfills ∇ × C = 0 {\displaystyle \nabla \times {\mathbf {C} }=0} , hence C = ∇ φ {\displaystyle C=\nabla \varphi } for some scalar field φ {\displaystyle \varphi } . === Fields with prescribed divergence and curl === The term "Helmholtz theorem" can also refer to the following. Let C be a solenoidal vector field and d a scalar field on R3 which are sufficiently smooth and which vanish faster than 1/r2 at infinity. Then there exists a vector field F such that ∇ ⋅ F = d and ∇ × F = C ; {\displaystyle \nabla \cdot \mathbf {F} =d\quad {\text{ and }}\quad \nabla \times \mathbf {F} =\mathbf {C} ;} if additionally the vector field F vanishes as r → ∞, then F is unique. In other words, a vector field can be constructed with both a specified divergence and a specified curl, and if it also vanishes at infinity, it is uniquely specified by its divergence and curl. This theorem is of great importance in electrostatics, since Maxwell's equations for the electric and magnetic fields in the static case are of exactly this type. The proof is by a construction generalizing the one given above: we set F = ∇ ( G ( d ) ) − ∇ × ( G ( C ) ) , {\displaystyle \mathbf {F} =\nabla ({\mathcal {G}}(d))-\nabla \times ({\mathcal {G}}(\mathbf {C} )),} where G {\displaystyle {\mathcal {G}}} represents the Newtonian potential operator. (When acting on a vector field, such as ∇ × F, it is defined to act on each component.) === Weak formulation === The Helmholtz decomposition can be generalized by reducing the regularity assumptions (the need for the existence of strong derivatives). Suppose Ω is a bounded, simply-connected, Lipschitz domain. Every square-integrable vector field u ∈ (L2(Ω))3 has an orthogonal decomposition: u = ∇ φ + ∇ × A {\displaystyle \mathbf {u} =\nabla \varphi +\nabla \times \mathbf {A} } where φ is in the Sobolev space H1(Ω) of square-integrable functions on Ω whose partial derivatives defined in the distribution sense are square integrable, and A ∈ H(curl, Ω), the Sobolev space of vector fields consisting of square integrable vector fields with square integrable curl. For a slightly smoother vector field u ∈ H(curl, Ω), a similar decomposition holds: u = ∇ φ + v {\displaystyle \mathbf {u} =\nabla \varphi +\mathbf {v} } where φ ∈ H1(Ω), v ∈ (H1(Ω))d. === Derivation from the Fourier transform === Note that in the theorem stated here, we have imposed the condition that if F {\displaystyle \mathbf {F} } is not defined on a bounded domain, then F {\displaystyle \mathbf {F} } shall decay faster than 1 / r {\displaystyle 1/r} . Thus, the Fourier transform of F {\displaystyle \mathbf {F} } , denoted as G {\displaystyle \mathbf {G} } , is guaranteed to exist. We apply the convention F ( r ) = ∭ G ( k ) e i k ⋅ r d V k {\displaystyle \mathbf {F} (\mathbf {r} )=\iiint \mathbf {G} (\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}} The Fourier transform of a scalar field is a scalar field, and the Fourier transform of a vector field is a vector field of same dimension. Now consider the following scalar and vector fields: G Φ ( k ) = i k ⋅ G ( k ) ‖ k ‖ 2 G A ( k ) = i k × G ( k ) ‖ k ‖ 2 Φ ( r ) = ∭ G Φ ( k ) e i k ⋅ r d V k A ( r ) = ∭ G A ( k ) e i k ⋅ r d V k {\displaystyle {\begin{aligned}G_{\Phi }(\mathbf {k} )&=i{\frac {\mathbf {k} \cdot \mathbf {G} (\mathbf {k} )}{\|\mathbf {k} \|^{2}}}\\\mathbf {G} _{\mathbf {A} }(\mathbf {k} )&=i{\frac {\mathbf {k} \times \mathbf {G} (\mathbf {k} )}{\|\mathbf {k} \|^{2}}}\\[8pt]\Phi (\mathbf {r} )&=\iiint G_{\Phi }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\\\mathbf {A} (\mathbf {r} )&=\iiint \mathbf {G} _{\mathbf {A} }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\end{aligned}}} Hence G ( k ) = − i k G Φ ( k ) + i k × G A ( k ) F ( r ) = − ∭ i k G Φ ( k ) e i k ⋅ r d V k + ∭ i k × G A ( k ) e i k ⋅ r d V k = − ∇ Φ ( r ) + ∇ × A ( r ) {\displaystyle {\begin{aligned}\mathbf {G} (\mathbf {k} )&=-i\mathbf {k} G_{\Phi }(\mathbf {k} )+i\mathbf {k} \times \mathbf {G} _{\mathbf {A} }(\mathbf {k} )\\[6pt]\mathbf {F} (\mathbf {r} )&=-\iiint i\mathbf {k} G_{\Phi }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}+\iiint i\mathbf {k} \times \mathbf {G} _{\mathbf {A} }(\mathbf {k} )e^{i\mathbf {k} \cdot \mathbf {r} }dV_{k}\\&=-\nabla \Phi (\mathbf {r} )+\nabla \times \mathbf {A} (\mathbf {r} )\end{aligned}}} === Longitudinal and transverse fields === A terminology often used in physics refers to the curl-free component of a vector field as the longitudinal component and the divergence-free component as the transverse component. This terminology comes from the following construction: Compute the three-dimensional Fourier transform F ^ {\displaystyle {\hat {\mathbf {F} }}} of the vector field F {\displaystyle \mathbf {F} } . Then decompose this field, at each point k, into two components, one of which points longitudinally, i.e. parallel to k, the other of which points in the transverse direction, i.e. perpendicular to k. So far, we have F ^ ( k ) = F ^ t ( k ) + F ^ l ( k ) {\displaystyle {\hat {\mathbf {F} }}(\mathbf {k} )={\hat {\mathbf {F} }}_{t}(\mathbf {k} )+{\hat {\mathbf {F} }}_{l}(\mathbf {k} )} k ⋅ F ^ t ( k ) = 0. {\displaystyle \mathbf {k} \cdot {\hat {\mathbf {F} }}_{t}(\mathbf {k} )=0.} k × F ^ l ( k ) = 0 . {\displaystyle \mathbf {k} \times {\hat {\mathbf {F} }}_{l}(\mathbf {k} )=\mathbf {0} .} Now we apply an inverse Fourier transform to each of these components. Using properties of Fourier transforms, we derive: F ( r ) = F t ( r ) + F l ( r ) {\displaystyle \mathbf {F} (\mathbf {r} )=\mathbf {F} _{t}(\mathbf {r} )+\mathbf {F} _{l}(\mathbf {r} )} ∇ ⋅ F t ( r ) = 0 {\displaystyle \nabla \cdot \mathbf {F} _{t}(\mathbf {r} )=0} ∇ × F l ( r ) = 0 {\displaystyle \nabla \times \mathbf {F} _{l}(\mathbf {r} )=\mathbf {0} } Since ∇ × ( ∇ Φ ) = 0 {\displaystyle \nabla \times (\nabla \Phi )=0} and ∇ ⋅ ( ∇ × A ) = 0 {\displaystyle \nabla \cdot (\nabla \times \mathbf {A} )=0} , we can get F t = ∇ × A = 1 4 π ∇ × ∫ V ∇ ′ × F | r − r ′ | d V ′ {\displaystyle \mathbf {F} _{t}=\nabla \times \mathbf {A} ={\frac {1}{4\pi }}\nabla \times \int _{V}{\frac {\nabla '\times \mathbf {F} }{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'} F l = − ∇ Φ = − 1 4 π ∇ ∫ V ∇ ′ ⋅ F | r − r ′ | d V ′ {\displaystyle \mathbf {F} _{l}=-\nabla \Phi =-{\frac {1}{4\pi }}\nabla \int _{V}{\frac {\nabla '\cdot \mathbf {F} }{\left|\mathbf {r} -\mathbf {r} '\right|}}\mathrm {d} V'} so this is indeed the Helmholtz decomposition. == Generalization to higher dimensions == === Matrix approach === The generalization to d {\displaystyle d} dimensions cannot be done with a vector potential, since the rotation operator and the cross product are defined (as vectors) only in three dimensions. Let F {\displaystyle \mathbf {F} } be a vector field on a bounded domain V ⊆ R d {\displaystyle V\subseteq \mathbb {R} ^{d}} which decays faster than | r | − δ {\displaystyle |\mathbf {r} |^{-\delta }} for | r | → ∞ {\displaystyle |\mathbf {r} |\to \infty } and δ > 2 {\displaystyle \delta >2} . The scalar potential is defined similar to the three dimensional case as: Φ ( r ) = − ∫ R d div ( F ( r ′ ) ) K ( r , r ′ ) d V ′ = − ∫ R d ∑ i ∂ F i ∂ r i ( r ′ ) K ( r , r ′ ) d V ′ , {\displaystyle \Phi (\mathbf {r} )=-\int _{\mathbb {R} ^{d}}\operatorname {div} (\mathbf {F} (\mathbf {r} '))K(\mathbf {r} ,\mathbf {r} ')\mathrm {d} V'=-\int _{\mathbb {R} ^{d}}\sum _{i}{\frac {\partial F_{i}}{\partial r_{i}}}(\mathbf {r} ')K(\mathbf {r} ,\mathbf {r} ')\mathrm {d} V',} where as the integration kernel K ( r , r ′ ) {\displaystyle K(\mathbf {r} ,\mathbf {r} ')} is again the fundamental solution of Laplace's equation, but in d-dimensional space: K ( r , r ′ ) = { 1 2 π log | r − r ′ | d = 2 , 1 d ( 2 − d ) V d | r − r ′ | 2 − d otherwise , {\displaystyle K(\mathbf {r} ,\mathbf {r} ')={\begin{cases}{\frac {1}{2\pi }}\log {|\mathbf {r} -\mathbf {r} '|}&d=2,\\{\frac {1}{d(2-d)V_{d}}}|\mathbf {r} -\mathbf {r} '|^{2-d}&{\text{otherwise}},\end{cases}}} with V d = π d 2 / Γ ( d 2 + 1 ) {\displaystyle V_{d}=\pi ^{\frac {d}{2}}/\Gamma {\big (}{\tfrac {d}{2}}+1{\big )}} the volume of the d-dimensional unit balls and Γ ( r ) {\displaystyle \Gamma (\mathbf {r} )} the gamma function. For d = 3 {\displaystyle d=3} , V d {\displaystyle V_{d}} is just equal to 4 π 3 {\displaystyle {\frac {4\pi }{3}}} , yielding the same prefactor as above. The rotational potential is an antisymmetric matrix with the elements: A i j ( r ) = ∫ R d ( ∂ F i ∂ x j ( r ′ ) − ∂ F j ∂ x i ( r ′ ) ) K ( r , r ′ ) d V ′ . {\displaystyle A_{ij}(\mathbf {r} )=\int _{\mathbb {R} ^{d}}\left({\frac {\partial F_{i}}{\partial x_{j}}}(\mathbf {r} ')-{\frac {\partial F_{j}}{\partial x_{i}}}(\mathbf {r} ')\right)K(\mathbf {r} ,\mathbf {r} ')\mathrm {d} V'.} Above the diagonal are ( d 2 ) {\displaystyle \textstyle {\binom {d}{2}}} entries which occur again mirrored at the diagonal, but with a negative sign. In the three-dimensional case, the matrix elements just correspond to the components of the vector potential A = [ A 1 , A 2 , A 3 ] = [ A 23 , A 31 , A 12 ] {\displaystyle \mathbf {A} =[A_{1},A_{2},A_{3}]=[A_{23},A_{31},A_{12}]} . However, such a matrix potential can be written as a vector only in the three-dimensional case, because ( d 2 ) = d {\displaystyle \textstyle {\binom {d}{2}}=d} is valid only for d = 3 {\displaystyle d=3} . As in the three-dimensional case, the gradient field is defined as G ( r ) = − ∇ Φ ( r ) . {\displaystyle \mathbf {G} (\mathbf {r} )=-\nabla \Phi (\mathbf {r} ).} The rotational field, on the other hand, is defined in the general case as the row divergence of the matrix: R ( r ) = [ ∑ k ∂ r k A i k ( r ) ; 1 ≤ i ≤ d ] . {\displaystyle \mathbf {R} (\mathbf {r} )=\left[\sum \nolimits _{k}\partial _{r_{k}}A_{ik}(\mathbf {r} );{1\leq i\leq d}\right].} In three-dimensional space, this is equivalent to the rotation of the vector potential. === Tensor approach === In a d {\displaystyle d} -dimensional vector space with d ≠ 3 {\displaystyle d\neq 3} , − 1 4 π | r − r ′ | {\textstyle -{\frac {1}{4\pi \left|\mathbf {r} -\mathbf {r} '\right|}}} can be replaced by the appropriate Green's function for the Laplacian, defined by ∇ 2 G ( r , r ′ ) = ∂ ∂ r μ ∂ ∂ r μ G ( r , r ′ ) = δ d ( r − r ′ ) {\displaystyle \nabla ^{2}G(\mathbf {r} ,\mathbf {r} ')={\frac {\partial }{\partial r_{\mu }}}{\frac {\partial }{\partial r_{\mu }}}G(\mathbf {r} ,\mathbf {r} ')=\delta ^{d}(\mathbf {r} -\mathbf {r} ')} where Einstein summation convention is used for the index μ {\displaystyle \mu } . For example, G ( r , r ′ ) = 1 2 π ln | r − r ′ | {\textstyle G(\mathbf {r} ,\mathbf {r} ')={\frac {1}{2\pi }}\ln \left|\mathbf {r} -\mathbf {r} '\right|} in 2D. Following the same steps as above, we can write F μ ( r ) = ∫ V F μ ( r ′ ) ∂ ∂ r μ ∂ ∂ r μ G ( r , r ′ ) d d r ′ = δ μ ν δ ρ σ ∫ V F ν ( r ′ ) ∂ ∂ r ρ ∂ ∂ r σ G ( r , r ′ ) d d r ′ {\displaystyle F_{\mu }(\mathbf {r} )=\int _{V}F_{\mu }(\mathbf {r} '){\frac {\partial }{\partial r_{\mu }}}{\frac {\partial }{\partial r_{\mu }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '=\delta _{\mu \nu }\delta _{\rho \sigma }\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\rho }}}{\frac {\partial }{\partial r_{\sigma }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '} where δ μ ν {\displaystyle \delta _{\mu \nu }} is the Kronecker delta (and the summation convention is again used). In place of the definition of the vector Laplacian used above, we now make use of an identity for the Levi-Civita symbol ε {\displaystyle \varepsilon } , ε α μ ρ ε α ν σ = ( d − 2 ) ! ( δ μ ν δ ρ σ − δ μ σ δ ν ρ ) {\displaystyle \varepsilon _{\alpha \mu \rho }\varepsilon _{\alpha \nu \sigma }=(d-2)!(\delta _{\mu \nu }\delta _{\rho \sigma }-\delta _{\mu \sigma }\delta _{\nu \rho })} which is valid in d ≥ 2 {\displaystyle d\geq 2} dimensions, where α {\displaystyle \alpha } is a ( d − 2 ) {\displaystyle (d-2)} -component multi-index. This gives F μ ( r ) = δ μ σ δ ν ρ ∫ V F ν ( r ′ ) ∂ ∂ r ρ ∂ ∂ r σ G ( r , r ′ ) d d r ′ + 1 ( d − 2 ) ! ε α μ ρ ε α ν σ ∫ V F ν ( r ′ ) ∂ ∂ r ρ ∂ ∂ r σ G ( r , r ′ ) d d r ′ {\displaystyle F_{\mu }(\mathbf {r} )=\delta _{\mu \sigma }\delta _{\nu \rho }\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\rho }}}{\frac {\partial }{\partial r_{\sigma }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '+{\frac {1}{(d-2)!}}\varepsilon _{\alpha \mu \rho }\varepsilon _{\alpha \nu \sigma }\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\rho }}}{\frac {\partial }{\partial r_{\sigma }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '} We can therefore write F μ ( r ) = − ∂ ∂ r μ Φ ( r ) + ε μ ρ α ∂ ∂ r ρ A α ( r ) {\displaystyle F_{\mu }(\mathbf {r} )=-{\frac {\partial }{\partial r_{\mu }}}\Phi (\mathbf {r} )+\varepsilon _{\mu \rho \alpha }{\frac {\partial }{\partial r_{\rho }}}A_{\alpha }(\mathbf {r} )} where Φ ( r ) = − ∫ V F ν ( r ′ ) ∂ ∂ r ν G ( r , r ′ ) d d r ′ A α = 1 ( d − 2 ) ! ε α ν σ ∫ V F ν ( r ′ ) ∂ ∂ r σ G ( r , r ′ ) d d r ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&=-\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\nu }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '\\A_{\alpha }&={\frac {1}{(d-2)!}}\varepsilon _{\alpha \nu \sigma }\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\sigma }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '\end{aligned}}} Note that the vector potential is replaced by a rank- ( d − 2 ) {\displaystyle (d-2)} tensor in d {\displaystyle d} dimensions. Because G ( r , r ′ ) {\displaystyle G(\mathbf {r} ,\mathbf {r} ')} is a function of only r − r ′ {\displaystyle \mathbf {r} -\mathbf {r} '} , one can replace ∂ ∂ r μ → − ∂ ∂ r μ ′ {\displaystyle {\frac {\partial }{\partial r_{\mu }}}\rightarrow -{\frac {\partial }{\partial r'_{\mu }}}} , giving Φ ( r ) = ∫ V F ν ( r ′ ) ∂ ∂ r ν ′ G ( r , r ′ ) d d r ′ A α = − 1 ( d − 2 ) ! ε α ν σ ∫ V F ν ( r ′ ) ∂ ∂ r σ ′ G ( r , r ′ ) d d r ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&=\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r'_{\nu }}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '\\A_{\alpha }&=-{\frac {1}{(d-2)!}}\varepsilon _{\alpha \nu \sigma }\int _{V}F_{\nu }(\mathbf {r} '){\frac {\partial }{\partial r_{\sigma }'}}G(\mathbf {r} ,\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '\end{aligned}}} Integration by parts can then be used to give Φ ( r ) = − ∫ V G ( r , r ′ ) ∂ ∂ r ν ′ F ν ( r ′ ) d d r ′ + ∮ S G ( r , r ′ ) F ν ( r ′ ) n ^ ν ′ d d − 1 r ′ A α = 1 ( d − 2 ) ! ε α ν σ ∫ V G ( r , r ′ ) ∂ ∂ r σ ′ F ν ( r ′ ) d d r ′ − 1 ( d − 2 ) ! ε α ν σ ∮ S G ( r , r ′ ) F ν ( r ′ ) n ^ σ ′ d d − 1 r ′ {\displaystyle {\begin{aligned}\Phi (\mathbf {r} )&=-\int _{V}G(\mathbf {r} ,\mathbf {r} '){\frac {\partial }{\partial r'_{\nu }}}F_{\nu }(\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '+\oint _{S}G(\mathbf {r} ,\mathbf {r} ')F_{\nu }(\mathbf {r} '){\hat {n}}'_{\nu }\,\mathrm {d} ^{d-1}\mathbf {r} '\\A_{\alpha }&={\frac {1}{(d-2)!}}\varepsilon _{\alpha \nu \sigma }\int _{V}G(\mathbf {r} ,\mathbf {r} '){\frac {\partial }{\partial r_{\sigma }'}}F_{\nu }(\mathbf {r} ')\,\mathrm {d} ^{d}\mathbf {r} '-{\frac {1}{(d-2)!}}\varepsilon _{\alpha \nu \sigma }\oint _{S}G(\mathbf {r} ,\mathbf {r} ')F_{\nu }(\mathbf {r} '){\hat {n}}'_{\sigma }\,\mathrm {d} ^{d-1}\mathbf {r} '\end{aligned}}} where S = ∂ V {\displaystyle S=\partial V} is the boundary of V {\displaystyle V} . These expressions are analogous to those given above for three-dimensional space. For a further generalization to manifolds, see the discussion of Hodge decomposition below. == Differential forms == The Hodge decomposition is closely related to the Helmholtz decomposition, generalizing from vector fields on R3 to differential forms on a Riemannian manifold M. Most formulations of the Hodge decomposition require M to be compact. Since this is not true of R3, the Hodge decomposition theorem is not strictly a generalization of the Helmholtz theorem. However, the compactness restriction in the usual formulation of the Hodge decomposition can be replaced by suitable decay assumptions at infinity on the differential forms involved, giving a proper generalization of the Helmholtz theorem. == Extensions to fields not decaying at infinity == Most textbooks only deal with vector fields decaying faster than | r | − δ {\displaystyle |\mathbf {r} |^{-\delta }} with δ > 1 {\displaystyle \delta >1} at infinity. However, Otto Blumenthal showed in 1905 that an adapted integration kernel can be used to integrate fields decaying faster than | r | − δ {\displaystyle |\mathbf {r} |^{-\delta }} with δ > 0 {\displaystyle \delta >0} , which is substantially less strict. To achieve this, the kernel K ( r , r ′ ) {\displaystyle K(\mathbf {r} ,\mathbf {r} ')} in the convolution integrals has to be replaced by K ′ ( r , r ′ ) = K ( r , r ′ ) − K ( 0 , r ′ ) {\displaystyle K'(\mathbf {r} ,\mathbf {r} ')=K(\mathbf {r} ,\mathbf {r} ')-K(0,\mathbf {r} ')} . With even more complex integration kernels, solutions can be found even for divergent functions that need not grow faster than polynomial. For all analytic vector fields that need not go to zero even at infinity, methods based on partial integration and the Cauchy formula for repeated integration can be used to compute closed-form solutions of the rotation and scalar potentials, as in the case of multivariate polynomial, sine, cosine, and exponential functions. == Uniqueness of the solution == In general, the Helmholtz decomposition is not uniquely defined. A harmonic function H ( r ) {\displaystyle H(\mathbf {r} )} is a function that satisfies Δ H ( r ) = 0 {\displaystyle \Delta H(\mathbf {r} )=0} . By adding H ( r ) {\displaystyle H(\mathbf {r} )} to the scalar potential Φ ( r ) {\displaystyle \Phi (\mathbf {r} )} , a different Helmholtz decomposition can be obtained: G ′ ( r ) = ∇ ( Φ ( r ) + H ( r ) ) = G ( r ) + ∇ H ( r ) , R ′ ( r ) = R ( r ) − ∇ H ( r ) . {\displaystyle {\begin{aligned}\mathbf {G} '(\mathbf {r} )&=\nabla (\Phi (\mathbf {r} )+H(\mathbf {r} ))=\mathbf {G} (\mathbf {r} )+\nabla H(\mathbf {r} ),\\\mathbf {R} '(\mathbf {r} )&=\mathbf {R} (\mathbf {r} )-\nabla H(\mathbf {r} ).\end{aligned}}} For vector fields F {\displaystyle \mathbf {F} } , decaying at infinity, it is a plausible choice that scalar and rotation potentials also decay at infinity. Because H ( r ) = 0 {\displaystyle H(\mathbf {r} )=0} is the only harmonic function with this property, which follows from Liouville's theorem, this guarantees the uniqueness of the gradient and rotation fields. This uniqueness does not apply to the potentials: In the three-dimensional case, the scalar and vector potential jointly have four components, whereas the vector field has only three. The vector field is invariant to gauge transformations and the choice of appropriate potentials known as gauge fixing is the subject of gauge theory. Important examples from physics are the Lorenz gauge condition and the Coulomb gauge. An alternative is to use the poloidal–toroidal decomposition. == Applications == === Electrodynamics === The Helmholtz theorem is of particular interest in electrodynamics, since it can be used to write Maxwell's equations in the potential image and solve them more easily. The Helmholtz decomposition can be used to prove that, given electric current density and charge density, the electric field and the magnetic flux density can be determined. They are unique if the densities vanish at infinity and one assumes the same for the potentials. === Fluid dynamics === In fluid dynamics, the Helmholtz projection plays an important role, especially for the solvability theory of the Navier-Stokes equations. If the Helmholtz projection is applied to the linearized incompressible Navier-Stokes equations, the Stokes equation is obtained. This depends only on the velocity of the particles in the flow, but no longer on the static pressure, allowing the equation to be reduced to one unknown. However, both equations, the Stokes and linearized equations, are equivalent. The operator P Δ {\displaystyle P\Delta } is called the Stokes operator. === Dynamical systems theory === In the theory of dynamical systems, Helmholtz decomposition can be used to determine "quasipotentials" as well as to compute Lyapunov functions in some cases. For some dynamical systems such as the Lorenz system (Edward N. Lorenz, 1963), a simplified model for atmospheric convection, a closed-form expression of the Helmholtz decomposition can be obtained: r ˙ = F ( r ) = [ a ( r 2 − r 1 ) , r 1 ( b − r 3 ) − r 2 , r 1 r 2 − c r 3 ] . {\displaystyle {\dot {\mathbf {r} }}=\mathbf {F} (\mathbf {r} )={\big [}a(r_{2}-r_{1}),r_{1}(b-r_{3})-r_{2},r_{1}r_{2}-cr_{3}{\big ]}.} The Helmholtz decomposition of F ( r ) {\displaystyle \mathbf {F} (\mathbf {r} )} , with the scalar potential Φ ( r ) = a 2 r 1 2 + 1 2 r 2 2 + c 2 r 3 2 {\displaystyle \Phi (\mathbf {r} )={\tfrac {a}{2}}r_{1}^{2}+{\tfrac {1}{2}}r_{2}^{2}+{\tfrac {c}{2}}r_{3}^{2}} is given as: G ( r ) = [ − a r 1 , − r 2 , − c r 3 ] , {\displaystyle \mathbf {G} (\mathbf {r} )={\big [}-ar_{1},-r_{2},-cr_{3}{\big ]},} R ( r ) = [ + a r 2 , b r 1 − r 1 r 3 , r 1 r 2 ] . {\displaystyle \mathbf {R} (\mathbf {r} )={\big [}+ar_{2},br_{1}-r_{1}r_{3},r_{1}r_{2}{\big ]}.} The quadratic scalar potential provides motion in the direction of the coordinate origin, which is responsible for the stable fix point for some parameter range. For other parameters, the rotation field ensures that a strange attractor is created, causing the model to exhibit a butterfly effect. === Medical Imaging === In magnetic resonance elastography, a variant of MR imaging where mechanical waves are used to probe the viscoelasticity of organs, the Helmholtz decomposition is sometimes used to separate the measured displacement fields into its shear component (divergence-free) and its compression component (curl-free). In this way, the complex shear modulus can be calculated without contributions from compression waves. === Computer animation and robotics === The Helmholtz decomposition is also used in the field of computer engineering. This includes robotics, image reconstruction but also computer animation, where the decomposition is used for realistic visualization of fluids or vector fields. == See also == Clebsch representation for a related decomposition of vector fields Darwin Lagrangian for an application Poloidal–toroidal decomposition for a further decomposition of the divergence-free component ∇ × A {\displaystyle \nabla \times \mathbf {A} } . Scalar–vector–tensor decomposition Hodge theory generalizing Helmholtz decomposition Polar factorization theorem Helmholtz–Leray decomposition used for defining the Leray projection == Notes == == References ==
|
Wikipedia:Helmut H. Schaefer#0
|
Helmut Heinrich Schaefer (February 14, 1925 in Großenhain, Weimar Republic – December 16, 2005 in Tübingen, Germany) was a German mathematician, who worked primarily in functional analysis. His two best known scientific monographs are titled Topological Vector Spaces (1966) and Banach Lattices and Positive Operators (1974). The first of these was subsequently translated into Spanish and Russian. The second made him an internationally recognized and leading scholar in this particular field of mathematics. (Roquette & Wolff, 2006) == Education and career == As teenager, Helmut Schaefer attended the Sankt Afra boarding school for gifted children in Meissen, Germany on a merit-based scholarship. In 1943, then 18, he was recruited to serve as interpreter of Anglo-American intelligence. After the war he studied mathematics at TU Dresden and University of Leipzig, where he earned his doctorate in 1951 and his habilitation in 1954. Prof. Ernst Hölder served as his academic advisor in Leipzig. In 1956 he accepted an offer from the University of Halle as professor of mathematics. In 1957, Schaefer, his wife and two children escaped from East Germany to the Federal Republic. For one year, he worked under Prof. Gottfried Köthe at the University of Mainz. In 1958 he became Associate Professor at Washington State University at Pullman and a few years later he, his wife and now three children moved on to the University of Michigan at Ann Arbor. Then in 1963 he accepted an offer from the University of Tübingen in Germany where he remained until his retirement in 1990. In Tübingen he served two terms as department head. Interrupting this period on several occasions and following retirement in Tübingen he spent a number of one-year terms or semesters as visiting or full professor at various American universities, including the University of Illinois Urbana-Champaign, the University of Maryland at College Park, the California Institute of Technology in Pasadena, Texas A&M University at College Station, and Florida Atlantic University at Boca Raton. He remained active in mathematical research until the year 1999, at which point he completely dedicated himself to his lifelong hobby of astronomy, especially astrophotography. In 1978, Helmut Schaefer was accepted as full member of the Mathematics and Natural Sciences Class of the Heidelberg Academy of Sciences. Earlier, he was admitted to the Academy of Sciences in Zaragoza (Spain). Over the years, he was able to attract many students to functional analysis, combining an expectation of high achievement with a tolerant, humorous, and factual attitude. Ten of his doctoral students went on to become professors at various universities in Germany and the U.S. (Roquette & Wolff, 2006) His doctoral students include Wolfgang Arendt, Rainer Nagel, and Bertram John Walsh. == Textbooks == 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. == References == Peter Roquette and Manfred Wolff (2006), Helmut Schaefer 1925 – 2005, Obituary, Jahrbuch 2005 der Heidelberger Akademie der Wissenschaften Monographs by Helmut H. Schaefer as catalogued by the German National Library Helmut H. Schaefer at the Mathematics Genealogy Project Schaefer, Helmut H.; Wolff, Manfred P. (1999). Topological Vector Spaces. GTM. Vol. 3. New York, NY: Springer New York Imprint Springer. ISBN 978-1-4612-7155-0. OCLC 840278135.
|
Wikipedia:Hemant Mehta#0
|
Hemant Mehta (; born February 25, 1983) is an American author, blogger, YouTuber and atheist activist. Mehta is a regular speaker at atheist events, and he has been a board member of charitable organizations such as the Secular Student Alliance and the Foundation Beyond Belief. Mehta used to run the Friendly Atheist blog on Patheos, in which he and his associates published articles several times a day, and he also co-hosts a weekly podcast called the Friendly Atheist Podcast. The blog stopped its activities on Patheos from December 14, 2021, because Mehta and some of his other associates moved to a new platform called OnlySky (Onlysky.media). Mehta as of 2024 also publishes a newsletter. On April 1, 2020, Mehta won his first appearance on the television game show Jeopardy! == Biography == Mehta was born in Chicago, Illinois in 1983. He graduated from the University of Illinois at Chicago in 2004 with a double degree in math and biology and began teaching in 2007. He acquired a master's degree in math education from DePaul University in 2010 and a national board certification in teaching in 2012. He taught high school math at Neuqua Valley High School until 2014 when he announced on Facebook and his blog that he had submitted his resignation to the school, citing that "As much as I love being in the classroom, the opportunities online are just a lot greater right now, and I don’t want to have any regrets down the road about not taking this chance while I have it." After his resignation, he stayed on as the head coach of the school's speech team. Mehta was raised in the Jain faith. He became an atheist as a teenager. Seeking to learn more about what motivated many Americans to be religious, he decided to attend and take notes at a number of churches across the United States. He based his choice of churches to attend on the results of an eBay auction in which he offered his bidders, "I am an atheist. You can bid on where I go to church or a temple or a mosque, etc." Mehta's experiences at the churches became the basis for his book, I Sold My Soul on eBay. == Activism == Hemant Mehta established a secular student group, Students WithOut Religious Dogma (SWORD), at the University of Illinois at Chicago while earning dual degrees there. Later, still in college, he was board chair for the Secular Student Alliance. He interned at the Center for Inquiry where he became familiar with a lot of the national organizations and leaders in the activist world at that time. He is on the board of directors for the Foundation Beyond Belief, a non-profit charitable organization. Mehta is a regular speaker at freethought and skeptical events around the U.S. He attempts to build bridges of understanding between believers and non-believers through his blog, The Friendly Atheist. Due in part to his positive message, he is also invited to speak at atheist events such as the one he presented at The Reason Rally or at the American Atheists Annual Convention in March 2012. A vocal advocate of building an atheist community, Mehta's activism includes fundraising for charitable causes. He helped establish and serves on the board of the Foundation Beyond Belief, which has raised more than $2,400,000 since it launched in 2010. He also established a church cleanup fund in response to reports of church vandalism in Bend, Oregon, in 2012. The vandals tagged the church with allusions to the Church of the Flying Spaghetti Monster, and Mehta's readers contributed nearly $3,000 in one day to help clean up the damage. Mehta writes for the "On Faith" column in The Washington Post and has been featured in a New York Times debate on prayer. When he has been asked about his beliefs, he has stated: "Simply put, I have never seen any evidence of 'God's work' in action. I've seen what people think is God's work, but which actually has perfectly natural explanations. I believe that most people are good, even when nobody's looking. I believe our best path to discovering the truth lies in science, not religion." In June 2013, Mehta wrote for the "Room for Debate" series in the New York Times, where he argued that "There’s a very real downside to praying. It lulls believers into a false sense of accomplishment." In July 2013, he held an open discussion at the Oak Hills Church in San Antonio, Texas, where he explained his frustration at being confronted time and again with the same arguments for believing in God. He also commented that "Many Christians had negative stereotypes about atheists that prevented fruitful conversation." Mehta was a guest on CNN on August 20, 2013, to discuss the rise of atheism among the millennial generation. Also in 2013, he began publishing "The Atheist Voice" series of video discussions on YouTube, which had over 215 thousand subscribers in 2020. He started a second channel, Friendly Atheist, in 2019, and has since gained almost 100 thousand subscribers. Mehta is the co-host, with Jessica Bluemke, of a weekly podcast named the Friendly Atheist Podcast, which has produced over 270 episodes as of June 2019. == Published works == In January 2006, Mehta posted an auction on eBay where he explained his background in atheism and offered to go to the worship services of the winning bidder's choosing. The auction ended on February 3, 2006, with a final bid of $504 from Jim Henderson, a minister from Seattle, Washington. Mehta later donated that money to the Secular Student Alliance, a non-profit organization for which he served as chair of the board of directors. Nearly a month after the auction, an article about Mehta's experiences appeared on the front page of The Wall Street Journal,. He was featured in the Chicago Sun-Times, the Seattle Times, and the Village Voice, and on National Public Radio. Henderson asked Mehta to visit a variety of churches and write about the experiences on Henderson's website, offthemap.com. Mehta eventually wrote about his visits at nine different churches as well as two additional pieces dealing with atheist conventions and Christian media. I Sold My Soul on eBay contains Mehta's observations and critiques of the churches along with background on how he became an atheist. Other published works include The Young Atheist's Survival Guide published in 2012 and The Friendly Atheist: Thoughts on the Role of Religion in Politics and Media published in 2013. The former is aimed at students, teachers, and parents who may face ostracism due to their lack of religious belief. The latter is intended as a study guide for the many things written by Mehta. In August 2014, Mehta announced his latest project, God is an Abusive Boyfriend (and you should break up), based on his YouTube series, The Atheist Voice. However, Mehta cancelled the project after receiving negative feedback from his readers. == References == == External links == Friendly Atheist newsletter YouTube video of speech at Reason Rally, March 2012 Interview on The Young Atheist’s Survival Guide on the Token Skeptic podcast (Ep. 149)
|
Wikipedia:Hemicontinuity#0
|
In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of upper and lower semicontinuity of single-valued functions to set-valued functions. A set-valued function that is both upper and lower hemicontinuous is said to be continuous in an analogy to the property of the same name for single-valued functions. To explain both notions, consider a sequence a of points in a domain, and a sequence b of points in the range. We say that b corresponds to a if each point in b is contained in the image of the corresponding point in a. Upper hemicontinuity requires that, for any convergent sequence a in a domain, and for any convergent sequence b that corresponds to a, the image of the limit of a contains the limit of b. Lower hemicontinuity requires that, for any convergent sequence a in a domain, and for any point x in the image of the limit of a, there exists a sequence b that corresponds to a subsequence of a, that converges to x. == Examples == The image on the right shows a function that is not lower hemicontinuous at x. To see this, let a be a sequence that converges to x from the left. The image of x is a vertical line that contains some point (x,y). But every sequence b that corresponds to a is contained in the bottom horizontal line, so it cannot converge to y. In contrast, the function is upper hemicontinuous everywhere. For example, considering any sequence a that converges to x from the left or from the right, and any corresponding sequence b, the limit of b is contained in the vertical line that is the image of the limit of a. The image on the left shows a function that is not upper hemicontinuous at x. To see this, let a be a sequence that converges to x from the right. The image of a contains vertical lines, so there exists a corresponding sequence b in which all elements are bounded away from f(x). The image of the limit of a contains a single point f(x), so it does not contain the limit of b. In contrast, that function is lower hemicontinuous everywhere. For example, for any sequence a that converges to x, from the left or from the right, f(x) contains a single point, and there exists a corresponding sequence b that converges to f(x). == Definitions == === Upper hemicontinuity === A set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is said to be upper hemicontinuous at a point a ∈ A {\displaystyle a\in A} if, for every open V ⊂ B {\displaystyle V\subset B} with Γ ( a ) ⊂ V , {\displaystyle \Gamma (a)\subset V,} there exists a neighbourhood U {\displaystyle U} of a {\displaystyle a} such that for all x ∈ U , {\displaystyle x\in U,} Γ ( x ) {\displaystyle \Gamma (x)} is a subset of V . {\displaystyle V.} === Lower hemicontinuity === A set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is said to be lower hemicontinuous at the point a ∈ A {\displaystyle a\in A} if for every open set V {\displaystyle V} intersecting Γ ( a ) , {\displaystyle \Gamma (a),} there exists a neighbourhood U {\displaystyle U} of a {\displaystyle a} such that Γ ( x ) {\displaystyle \Gamma (x)} intersects V {\displaystyle V} for all x ∈ U . {\displaystyle x\in U.} (Here V {\displaystyle V} intersects S {\displaystyle S} means nonempty intersection V ∩ S ≠ ∅ {\displaystyle V\cap S\neq \varnothing } ). === Continuity === If a set-valued function is both upper hemicontinuous and lower hemicontinuous, it is said to be continuous. == Properties == === Upper hemicontinuity === ==== Sequential characterization ==== As an example, look at the image at the right, and consider sequence a in the domain that converges to x (either from the left or from the right). Then, any sequence b that satisfies the requirements converges to some point in f(x). ==== Closed graph theorem ==== The graph of a set-valued function Γ : A ⇉ B {\displaystyle \Gamma :A\rightrightarrows B} is the set defined by G r ( Γ ) = { ( a , b ) ∈ A × B : b ∈ Γ ( a ) } . {\displaystyle Gr(\Gamma )=\{(a,b)\in A\times B:b\in \Gamma (a)\}.} The graph of Γ {\displaystyle \Gamma } is the set of all a ∈ A {\displaystyle a\in A} such that Γ ( a ) {\displaystyle \Gamma (a)} is not empty. === Lower hemicontinuity === ==== Sequential characterization ==== ==== Open graph theorem ==== A set-valued function Γ : A → B {\displaystyle \Gamma :A\to B} is said to have open lower sections if the set Γ − 1 ( b ) = { a ∈ A : b ∈ Γ ( a ) } {\displaystyle \Gamma ^{-1}(b)=\{a\in A:b\in \Gamma (a)\}} is open in A {\displaystyle A} for every b ∈ B . {\displaystyle b\in B.} If Γ {\displaystyle \Gamma } values are all open sets in B , {\displaystyle B,} then Γ {\displaystyle \Gamma } is said to have open upper sections. If Γ {\displaystyle \Gamma } has an open graph Gr ( Γ ) , {\displaystyle \operatorname {Gr} (\Gamma ),} then Γ {\displaystyle \Gamma } has open upper and lower sections and if Γ {\displaystyle \Gamma } has open lower sections then it is lower hemicontinuous. === Operations Preserving Hemicontinuity === Set-theoretic, algebraic and topological operations on set-valued functions (like union, composition, sum, convex hull, closure) usually preserve the type of continuity. But this should be taken with appropriate care since, for example, there exists a pair of lower hemicontinuous set-valued functions whose intersection is not lower hemicontinuous. This can be fixed upon strengthening continuity properties: if one of those lower hemicontinuous multifunctions has open graph then their intersection is again lower hemicontinuous. === Function Selections === Crucial to set-valued analysis (in view of applications) are the investigation of single-valued selections and approximations to set-valued functions. Typically lower hemicontinuous set-valued functions admit single-valued selections (Michael selection theorem, Bressan–Colombo directionally continuous selection theorem, Fryszkowski decomposable map selection). Likewise, upper hemicontinuous maps admit approximations (e.g. Ancel–Granas–Górniewicz–Kryszewski theorem). == Other concepts of continuity == The upper and lower hemicontinuity might be viewed as usual continuity: (For the notion of hyperspace compare also power set and function space). Using lower and upper Hausdorff uniformity we can also define the so-called upper and lower semicontinuous maps in the sense of Hausdorff (also known as metrically lower / upper semicontinuous maps). == See also == Differential inclusion Hausdorff distance – Distance between two metric-space subsets Semicontinuity – Property of functions which is weaker than continuityPages displaying short descriptions of redirect targets Selection theorem - a theorem about constructing a single-valued function from a set-valued function. == Notes == == References == Aliprantis, Charalambos D.; Border, Kim C. (2006). Infinite Dimensional Analysis: A Hitchhiker's Guide (Third ed.). Berlin: Springer Science & Business Media. ISBN 978-3-540-29587-7. OCLC 262692874. Aubin, Jean-Pierre; Cellina, Arrigo (1984). Differential Inclusions: Set-Valued Maps and Viability Theory. Grundl. der Math. Wiss. Vol. 264. Berlin: Springer. ISBN 0-387-13105-1. Aubin, Jean-Pierre; Frankowska, Hélène (1990). Set-Valued Analysis. Basel: Birkhäuser. ISBN 3-7643-3478-9. Deimling, Klaus (1992). Multivalued Differential Equations. Walter de Gruyter. ISBN 3-11-013212-5. Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (1995). Microeconomic Analysis. New York: Oxford University Press. pp. 949–951. ISBN 0-19-507340-1. Ok, Efe A. (2007). Real Analysis with Economic Applications. Princeton University Press. pp. 216–226. ISBN 978-0-691-11768-3.
|
Wikipedia:Henda Swart#0
|
Hendrika Cornelia Scott (Henda) Swart FRSSAf (born 1939, died February 2016 [age 77-78]) was a South African mathematician, a professor emeritus of mathematics at the University of KwaZulu-Natal and a professor at the University of Cape Town == Personal life == Born Hendrika Cornelia Scott, she married John Henry Swart. They had three children Christine, Sandra and Gustav. == Career == Swart began teaching at the University of Natal in 1962. She was the first person to earn a doctorate in mathematics from Stellenbosch University, in 1971, with a dissertation on the geometry of projective planes supervised by Kurt-Rüdiger Kannenberg. In 1977, her research interests shifted from geometry to graph theory, which she continued to publish in for the rest of her career. She was the editor-in-chief of the journal Utilitas Mathematica, and was vice president of the Institute of Combinatorics and its Applications. In 1996 she became a fellow of the Royal Society of South Africa. Swart was a part-time lecturer at the University of Cape town from 2014 until her death. == Publications == She published under the name Henda C Swart. She published nearly 100 papers from 1980 to 2018. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Henda Swart", MacTutor History of Mathematics Archive, University of St Andrews
|
Wikipedia:Heneri Dzinotyiweyi#0
|
Heneri Amos Murima Dzinotyiweyi is a Zimbabwean mathematician and politician. A former University of Zimbabwe Dean of Science, he is the Movement for Democratic Change-Tsvangirai member of parliament for Budiriro in Harare. On 10 February 2009, Morgan Tsvangirai designated Dzinotyiweyi for the position of Minister of Science and Technology Development as part of the Zimbabwe Government of National Unity of 2009. == References ==
|
Wikipedia:Henk Broer#0
|
Hendrik Wolter "Henk" Broer (born 18 February 1950, Diever) is a Dutch mathematician known for contributions to the theory of nonlinear dynamical systems. He was professor at the University of Groningen between 1981 and 2015. == Biography == Broer was granted a doctorate in the faculty of mathematics and natural sciences in 1979 under the supervision of Floris Takens for a thesis entitled Bifurcations of singularities in volume preserving vector fields. He was a professor at the University of Groningen, the Netherlands, from 1981 until his retirement in 2015. In 1985, he spent a semester as a guest professor of Boston University, Massachusetts. He is a long term member of the Royal Dutch Mathematical Society, serving as chairman from 2007 to 2009. == Awards and honors == In 2008, Broer became a member of the Royal Netherlands Academy of Arts and Sciences. In 2015, he was made a Knight of the Order of the Netherlands Lion. == Books == Broer has written a number of books and published over 120 articles on his subject, including the following: 1991. With F. Dumortier, S.J. van Strien and F.Takens. Structures in dynamics: finite dimensional deterministic studies. Studies in Mathematical Physics 2 North Holland 1995. With J. van de Craats and F. Verhulst. Chaostheorie - Het einde van de voorspelbaarheid? Epsilon Uitgaven 35 [Reprinted as Het einde van de voorspelbarheid? Chaostheorie, ideeën en toepassingen. Aramith Uitgevers - Epsilon Uitgaven 35 1996. With G.B. Huitema and M.B. Servryuk. Quasi-periodic tori in families of dynamical systems: order amidst chaos. Springer LNM 1645 1999. Meetkunde en Fysica, met differentiaalvormen en integraalstellingen. Epsilon Uitgaven 44 2003. With I. Hoveijn, G.A. Lunter and G. Vegter. Bifurcations in Hamiltonian systems. Springer LNM 1806 2011. With F.Takens, Dynamical Systems and Chaos. Epsilon Uitgaven 64; Appl. Math. Sciences 172, Springer-Verlag 2013. Hemelverschijnselen nabij de horizon, naar Minnaert en Wegener, Bernoulli en Hamilton. Epsilon Uitgaven 77 2016. Near the horizon: an invitation to geometric optics. The Carus Mathematical Monographs 33 MAA ISBN 0883851423; summary: "Near the Horizon: An Invitation to Geometric Optics by Henk W. Broer". Bookstore, American Mathematical Society. == Edited books == 1990. With F. Verhulst. Dynamische systemen en chaos: een revolutie vanuit de wiskunde. Epsilon Uitgaven 14 2001. With B. Krauskopf and G. Vegter. Global Analysis of Dynamical Systems. Festschrift dedicated to Floris Takens for his 60th birthday. Institute of Physics Publishing ISBN 0750308036 2005. With F. Dumortier, J. Mawhin, A. Vanderbauwhede and S.M. Verduyn Lunel. Equadiff 2003. Proceedings International Conference on Differential Equations, Hasselt 2003. World Scientific ISBN 981-256-169-2 2010. With B. Hasselblatt and F. Takens. Handbook of Dynamical Systems. Volume 3. North Holland ISBN 978-0-444-53141-4 == References == == External links == Henk Broer at the Mathematics Genealogy Project Official website
|
Wikipedia:Henk Lombaers#0
|
Henk Joseph Maria (Henk) Lombaers (Doorn, 1920 – 29 August 2007) was a Dutch mathematician, Professor at Delft University of Technology and a pioneer in the field of operations research in the Netherlands. == Life and work == Lombaers undertook teacher training. He studied chemistry in Amsterdam but had to terminate his studies prematurely due to the German occupation. From 1945 to 1956 he was employed by the Royal Netherlands Army, where he was involved in the introduction of radar-aircraft artillery. In 1956 he joined Koninklijke Hoogovens as an operational researcher. At Hoogovens he conducted research on the transhipment capacity of the port installations using computer simulations. He also looked into the statistical aspects of simulation. In 1968 he was appointed Professor of Quantitative Aspects of Business Administration at the Technical University of Delft, where he held his inaugural address on "Voorzien en verzinnen” (Forecast and Invention). In 1985 he retired with the speech, entitled "Daar moet je maar niet te hard op rekenen" (You shouldn't count on it too much). In Delft Lombaers was succeeded by Joop Evers, who had been Professor at the University of Twente. The PhD students under Lombaers were Jack P.C. Kleijnen (Emeritus Professor of Simulation and Information Systems at Tilburg University (TiU), who graduated in 1971, Hendrik van der Meerendonk also in 1971, Johannes Botman in 1981, and Jaap van den Herik in 1983. == Selected publications == 1968. Voorzien en verzinnen. Rotterdam : Universitaire Pers Rotterdam 1969. Trio-logie: Variaties over een thema uit de bedrijfsleer. With Pierre Malotaux and Jan in 't Veld. 1969. Operationele research in Nederland. Het Spectrum. 1970. Project planning by network analysis; proceedings of the second international congress Amsterdam, the Netherlands, 6–10 October 1969. H.J.M. Lombaers (ed.). 1985. Daar moet je maar niet te hard op rekenen. Delftse Universitaire Pers 1985. Operations research : praktijk en theorie : symposium ter gelegenheid van het afscheid van prof. H.J.M. Lombaers gehouden op 6 juni 1985 te Delft. Wil Heins (ed.). Delftse Universitaire Pers. == References == == External links == Henk Lombaers at the Mathematics Genealogy Project
|
Wikipedia:Henk Tijms#0
|
Henk Tijms (Beverwijk, April 23, 1944) is a Dutch mathematician and Emeritus Professor of Operations Research at the VU University Amsterdam. He studied mathematics in Amsterdam where he graduated from the University of Amsterdam in 1972 under supervision of Gijsbert de Leve. Tijms is the author of several articles on applied mathematics and stochastics and books on probability. His best-known books are Stochastic Modeling and Analysis (Wiley, 1986) and Understanding Probability (Cambridge University Press, 2004). On October 12, 2008, Tijms became the first non-American to receive the INFORMS Expository Writing Award. The award honoured his achievements in the field of mathematics. == References ==
|
Wikipedia:Henk Zijm#0
|
Willem Hendrik Maria (Henk) Zijm (born 3 May 1952) is a Dutch mathematician, and Professor Production and Supply Chain Management and Emeritus Rector Magnificus (2005–2009) at the University of Twente. == Biography == Born in Driehuizen, Texel, Zijm received both his BSc in mathematics, physics and astronomy in 1977, and his MSc cum laude in applied mathematics at the University of Amsterdam. In 1982 he received his Phd in operations research at the Eindhoven University of Technology under supervision of Jaap Wessels and Gerhard Willem Veltkamp with a thesis entitled "Nonnegative Matrices in Dynamic Programming." Zijm had started his academic career in 1981 as assistant professor at the Department of Actuarial Sciences and Econometrics of the University of Amsterdam. From 1983 to 1990 he worked in industry at Philips in Eindhoven as consultant in the fields of Operations Research, Logistic Management and Manufacturing Planning and Control. In 1987 he was appointed part-time as professor in mathematical models for operations management at the Eindhoven University of Technology. In 1990 he moved to the University of Twente, where he became professor in production and operations management. Zijm supervised around 150 master students and more than 30 PhD students, among them were Ivo Adan (1991), Geert-Jan van Houtum (1995), Erwin Hans (2001) and Nelly Litvak (2002). From 2000 to 2002, Zijm also directed the Center for Telematics and Information Technology (CTIT). From 2002 to 2004, he chaired the Faculty of Electrical Engineering, Mathematics and Computer Science, and from 2005 to 2009 he was Rector Magnificus of the University of Twente as successor of Frans van Vught. Since 2009, he has been professor in production and supply chain management, and since 2011, director of the Dutch Institute for Advanced Logistics (DINALOG) in Breda. Zijm retired in 2018. == Publications == Zijm has authored and co-authored numerous publications. A selection: Meester, Geatse, and Henk Zijm. "Multi-resource scheduling of an FMC in discrete parts manufacturing." Operations Research Proceedings 1993. Springer Berlin Heidelberg, 1994. 137–137. Zijm, Henk, and Geert-Jan Van Houtum. "On multi-stage production/inventory systems under stochastic demand." International Journal of Production Economics 35.1 (1994): 391–400. Buitenhek, Ronald, Geert‐Jan van Houtum, and Henk Zijm. "AMVA‐based solution procedures for open queueing networks with population constraints." Annals of Operations Research 93.1–4 (2000): 15–40. Zijm, W. Henk, and Zeynep Müge Avşar. "Capacitated two-indenture models for repairable item systems." International Journal of Production Economics 81 (2003): 573–588. Avsar, Zeynep Muge, and W. Henk Zijm. "Capacitated two-echelon inventory models for repairable item systems." Analysis and modeling of manufacturing systems. Springer US, 2003. 1–36. == References == == External links == [1] at utwente.nl
|
Wikipedia:Henri Darmon#0
|
Henri Rene Darmon (born 22 October 1965) is a French-Canadian mathematician. He is a number theorist who works on Hilbert's 12th problem and its relation with the Birch–Swinnerton-Dyer conjecture. He is currently a professor of mathematics at McGill University. == Career == Darmon received his BSc from McGill University in 1987 and his PhD from Harvard University in 1991 under supervision of Benedict Gross. From 1991 to 1996, he held positions in Princeton University. Since 1994, he has been a professor at McGill University. == Awards == Darmon was elected to the Royal Society of Canada in 2003. In 2008, he was awarded the Royal Society of Canada's John L. Synge Award. He received the 2017 AMS Cole Prize in Number Theory "for his contributions to the arithmetic of elliptic curves and modular forms", and the 2017 CRM-Fields-PIMS Prize, which is awarded in recognition of exceptional research achievement in the mathematical sciences. He was elected as a Fellow of the American Mathematical Society, in the 2025 class of fellows. == Personal life == Darmon is married to Galia Dafni, also a mathematician at Concordia University. == References == == External links == Professor Darmon's webpage
|
Wikipedia:Henri Dulac#0
|
Henri Claudius Rosarius Dulac (3 October 1870, Fayence – 2 September 1955, Fayence) was a French mathematician. == Life == Born in Fayence, France, Dulac graduated from École Polytechnique (Paris, class of 1892) and obtained a Doctorate in Mathematics. He started to teach a class of mathematic analysis at University, in Grenoble (France), Algiers (today Algeria) and Poitiers (France). Holder of a pulpit in pure mathematics in the Sciences University of Lyon (France) in 1911, his teaching was suspended during the first world war (1914 – 1918) and he had to serve as officer in the French army. After the war, he became holder of a pulpit of differential and integral calculus and also taught in École Centrale Lyon. He became examiner at École Polytechnique (Paris) and President of the admission jury. Awarded Officer of Legion d'honneur, the French order established by Napoleon and associate member of the French Academy of Sciences, he published part of Euler's works and contributed to the research through many publications in France and abroad. Father of 3 children, Anie (1901–1935), bachelor in mathematics, Jean (1903–2005), graduate of École Polytechnique, 1921 and Robert (1904–1996), graduate of polytechnique, 1922; he died in Fayence, France, in 1955. == Work == Among his publications: Recherches sur les points singuliers des équations différentielles (Journal of École Polytechnique, 1904). Intégrales d'une équation différentielle (Annales University of Grenoble, 1905). Sur les Points dicritiques (Journal of mathématics, 1906). Sur les séries de Mac-Laurin à plusieurs variables (Acta Mathematica, 1906). Détermination et intégration d'une classe d'équations différentielles (Bulletin of mathematical sciences). Intégrales passant par un point singulier (Rendeconti del Circolo, 1911). Sur les points singuliers (Annales, Toulouse, 1912). Solutions d'un système d'équations différentielles (Bulletin of the mathematical society, 1913). Sur les cycles limites (Bulletin of the mathematical society, 1923). Points singuliers des équations différentielles (Mémorial des sciences mathématiques, 1932). Courbes définies par une équation différentielle du premier ordre (Mémorial des sciences mathématiques, 1934). His researches are still mentioned or challenged by international university PHD students and professors, even a hundred years after being published. As an example: The Center Variety of Polynomial Differential Systems – Abdu Salam Jarrah, Faculté des sciences mathématiques, Université du Nouveau Mexique, USA (2001). Complete Polynomial Vector Fields on C2 – Julio Rebelo, Institute for Mathematical Sciences, SUNY, New York, USA (Oct. 2002). Dimension Increase and Splitting for Poincaré-Dulac Normal forms – Giuseppe Gaeta, Faculté de Mathématique de l'Université de Milan and Sebastian Walcher, chaire de Mathématique, Aix La Chapelle, Journal of Non linear Mathematical Physices (2005). Sources: Technica, n° 190, Nov. 1955, École Centrale Lyon, French Academy of Sciences, updated by Louis Boisgibault, his great grandson. == See also == Bendixson–Dulac theorem Hilbert's sixteenth problem Transseries == External links == Works by or about Henri Dulac at the Internet Archive
|
Wikipedia:Henri Fehr#0
|
Henri Fehr (Zurich, 2 February 1870 – Geneva, 2 November 1954) was a Swiss mathematician who was key to the foundation and organisation of national and international mathematical societies and journals. He studied mathematics in Switzerland and France, earning his doctorate at the University of Geneva in 1899 with a thesis on the application of Grassmann's vector methods to differential geometry. In 1910 he co-founded the Swiss Mathematical Society and established its flagship journals, Commentarii Mathematici Helvetici and, with Charles-Ange Laisant, L'Enseignement mathématique, sitting on their editorial and financial committees. Fehr spent his career at the University of Geneva—rising to dean and rector—and from the foundation of the International Commission on Mathematical Instruction in 1908 until his death in 1954 he served as its general secretary; he also held the vice-presidency of the International Mathematical Union from 1924 to 1932. == Education and the Swiss Mathematical Society == Fehr studied mathematics in Switzerland and France, taking his doctorate at the University of Geneva in 1899 with a thesis entitled Application de la Méthode Vectorielle de Grassmann à la Géométrie Infinitésimale (Paris: Georges Carré & C. Naud, 1899; 94 pp.), in which he applied Grassmann's vectorial methods to problems of infinitesimal differential geometry. In May–June 1910 he joined Rudolf Fueter and Marcel Grossmann in issuing the founding appeal for the Swiss Mathematical Society (SMG), serving as its vice-president in 1911–12 and president in 1912–13. He drafted the 700-page report on Swiss mathematical instruction (1910–13) that underpinned the SMG's Foundation for the Advancement of Mathematical Sciences and sat on its editorial and financial committees, securing both federal grants and subscription income. Under SMG auspices he also established the research journal Commentarii Mathematici Helvetici and, with Charles-Ange Laisant, the pedagogical journal L'Enseignement mathématique. == Academic career and international service == Fehr spent his professional life at the University of Geneva, advancing from professor of geometry and algebra to dean, vice-rector and finally rector. A noted pedagogue, he championed the social dimensions of mathematical education and teacher training. From the foundation of ICMI in 1908 until his death in 1954 he served as general secretary of the International Commission on Mathematical Instruction (ICMI), editing its organ L’Enseignement mathématique for 55 years and being elected Honorary President of ICMI in 1952. He was invited to speak at the ICM in Heidelberg (1904), Rome (1908), Cambridge (1912), Toronto (1924), and Zürich (1932). In June 1928, as Vice-President of the International Mathematical Union (IMU; a post he held 1924–1932) and ICMI general secretary, he was consulted by Ettore Bortolotti—secretary of the Italian Mathematical Union—about objections by IMU Secretary Gabriel Koenigs to the Bologna Congress invitations. Bortolotti's letter makes clear that Fehr's steadfast internationalism ensured mathematicians were invited "sans distinction de nationalité" despite formal protests and the risk of German boycott. == Publications == Application de la méthode vectorielle de Grassmann à la géométrie infinitésmale. Paris: G. Carré & C. Naud. 1899. 2nd ed. 1907. with Théodore Flournoy and Édouard Claparède: Enquête sur la methode de travail des mathématiciens. Paris: Gauthier-Villars. 1908. Der Mathematische Unterricht in der Schweiz. Geneva: A. Kündig. 1910. == References == == External links == Works by or about Henri Fehr at the Internet Archive Fonds Henri Fehr, Bibliothèque de Genève, Catalogue des manuscrits
|
Wikipedia:Henri Hogbe Nlend#0
|
Henri Hogbe Nlend (born 23 December 1939) is a Cameroonian mathematician, university professor, former government minister and presidential candidate. == Biography == Henri Hogbe Nlend was a professor at the University of Yaoundé, and at the University of Bordeaux. In 1976, at a meeting of the International Mathematical Union it was decided to form an African Mathematical Union. Hogbe Nlend was elected as its first president, a post he held until 1986. The AMU was partially funded from another organization in Paris, which was also chaired by Hogbe Nlend. It is said that he was good at raising funds and that meetings were held twice a year. Hogbe Nlend was a candidate in the presidential election held on 12 October 1997, which was boycotted by the major opposition parties, and placed second, although he received only 2.9% of the vote. The winning candidate, incumbent President Paul Biya, appointed Nlend as Minister of Scientific and Technical Research after the election. His textbook on the theory of duality topology-bornology and its use in functional analysis has been described as a classic. Henri Hogbe Nlend is a member of the Historical Cameroon Party, the Union of the Peoples of Cameroon (Union des Populations du Cameroun) and leader of one faction of this party. Hogbe Nlend fell out with Augustin Frederic Kodock, the Secretary-General of another UPC faction, in 2002. At the time of the July 2007 parliamentary election, Charly Gabriel Mbock, member of Hogbe Nlend UPC faction and outgoing UPC parliamentary deputy, resigned from the UPC and joined a new party, National Movement Party, vowing to carry on the struggle for which UPC has stood for, but this was disbanded when differences were resolved a year later, in a reconciliation meeting with the Hogbe Nlend UPC faction. (Kodock claimed in a press conference that Mbock had insufficient support to move this new party forward as he lacked the 500 signatures required by law). In reality, the 1990 Law to constitute an entity for legalisation as a political party in Cameroon does not include this requirement. Henri is a foundry fellow of the African Science Academy. == Selected bibliography == Théorie des bornologies et applications, (in French) Lecture Notes in Mathematics, Vol. 213. Springer-Verlag, Berlin-New York, 1971. v+168 pp. Bornologies and functional analysis, Translated from the French by V. B. Moscatelli. North-Holland Mathematics Studies, Vol. 26. Notas de Matemática, No. 62. [Notes on Mathematics, No. 62] North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977. xii+144 pp. ISBN 0-7204-0712-5 editor Functional analysis and its applications. Papers from the International School held in Nice, August 25—September 20, 1986. . ICPAM Lecture Notes. World Scientific Publishing Co., Singapore, 1988. viii+380 pp. ISBN 978-9971-5-0545-5 (47-06) == References ==
|
Wikipedia:Henri Villat#0
|
Henri René Pierre Villat (French: [vila]; 24 December 1879 – 19 March 1972) was a French mathematician. He was professor of fluid mechanics at the University of Paris from 1927 until his death. Villat became a member of the French Academy of Sciences in 1932, and its president in 1948. == References == == External links == Henri Villat at the Mathematics Genealogy Project
|
Wikipedia:Henry Farquharson#0
|
Major Francis Edward Henry Farquharson VC (25 March 1837 – 12 September 1875) was a Scottish recipient of the Victoria Cross, the highest and most prestigious award for gallantry in the face of the enemy that can be awarded to British and Commonwealth forces. == Early life == He was born in Glasgow on 25 March 1837 the son of Robert Farquharson, a thread manufacturer living at 7 St James Place. == Details == He was 20 years old, and a lieutenant in the 42nd Regiment of Foot (later The Black Watch (Royal Highlanders)), British Army during the Indian Mutiny when the following deed took place on 9 March 1858 at Lucknow, India for which he was awarded the VC: For conspicuous bravery, when engaged before Lucknow, on the 9th March, 1858, in having led a portion of his Company, stormed a bastion mounting two guns, and spiked the guns, by which the advanced position, held during the night of the 9th of March, was rendered secure from the fire of Artillery. Lieutenant Farquharson was severely wounded, while holding an advanced position, on the morning of the 10th of March. == Later life == He later achieved the rank of major. He fell ill during the Ashanti campaign of 1874 and retired from active service. He died at Dundrige in Harberton in Devonshire on 12 September 1875. He is buried a few metres east of the entrance to St Andrew's Church in Harberton.A stained glass window to his memory lies within the church. His Victoria Cross is displayed at the Black Watch Museum in Perth, Scotland together with his four other campaign medals: the Crimea Medal (Sebastopol); Indian Mutiny Medal (Lucknow); Ashanti Medal (Coomassie); and the Turkish Crimea Medal. == Artistic recognition == He was painted with fellow officers by Orlando Norie. == References == Scotland's Forgotten Valour (Graham Ross, 1995) The Register of the Victoria Cross (This England, 1997) Monuments to Courage (David Harvey, 1999) == External links == Location of grave and VC medal (Devonshire) Francis Farquharson at Find a Grave
|
Wikipedia:Henry Marshall Tory#0
|
Henry Marshall Tory (January 11, 1864 – February 6, 1947) was the first president of the University of Alberta (1908–1928), the first president of the Khaki University, the first president of the National Research Council (1928–1935), and the first president of Carleton College (1942–1947). His brother was James Cranswick Tory, Lieutenant Governor of Nova Scotia (1925–1930). == Early life == Born on a farm near Guysborough, in Guysborough County, Nova Scotia, his mother was a major factor in his educational ambition. At 22, he registered for Honours Mathematics and Physics in 1886 at McGill University and received an Honours B.A. with gold medal in 1890, despite his mother's desire for him to attend Mount Allison University to study Arts and Theology. After graduating, he studied theology and received a B.D. from Wesleyan College, affiliated with McGill. He spent the next two years preaching at a church. In 1893, he married Annie Gertrude Frost of Knowlton, Quebec, who has never had any mentions outside of being a good hostess. == Career == Tory became a lecturer in mathematics at McGill University in 1893, and he received an M.A. in Mathematics in 1896. He received a D.Sc. degree in 1903 and was promoted to associate professor of mathematics. In 1906, he set up the McGill University College of British Columbia which became the University of British Columbia in 1915. In 1905 during a stop in Edmonton, he had a chance meeting with Alexander Cameron Rutherford, future Premier of Alberta. The two quickly became friends and found they shared ideas concerning the importance of establishing new publicly funded and non-denominational universities in Canada. When Rutherford founded the University of Alberta two years later, he asked Tory to serve as president. He accepted and served in the position from 1908 to 1929. During World War I, Tory, initially somewhat reluctantly, became a colonel in the Canadian Expeditionary Force in 1916. After a tour of the front lines in France, he returned to England and proceeded to set up and run what came to be known as the Khaki University, enrolling over 50,000 Canadian student soldiers by the end of the Great War. Tory returned to Alberta in 1919 and resumed his position as president of the University of Alberta. Nearing retirement, on June 1, 1928, he accepted an appointment as the first President of the Council and Chief Executive Officer of the National Research Laboratories (which was later called the National Research Council of Canada). From 1939 to 1940, he was president of the Royal Society of Canada, just after his wife's death in 1938. From 1942 until his death in 1947, he was the first president of Carleton College (which later became Carleton University). == Legacy == The Henry Marshall Tory Building and the Tory Theatre at the University of Alberta were named in his honour, as was the Tory Building at Carleton University. The Henry Marshall Tory Medal at the University of British Columbia was established in 1941. == University Histories == William Hardy Alexander, The University of Alberta: A Retrospect 1908-1929 Walter Johns, History of the University of Alberta John Macdonald, The History of the University of Alberta, 1908-1958 Scott Rollans Echoes in the Halls: An Unofficial History of the University of Alberta (Association of Professors Emeriti of the U of A, University Of Alberta, 1999) Ellen Schoeck, I Was There: A Century of Alumni Stories about the University of Alberta, 1906–2006 William C. Gibson Wesbrook & His University (Vancouver: University of British Columbia Press) George Woodcock & Tim Fitzharris. The University of British Columbia – A Souvenir. (Toronto: Oxford University Press, 1986). H. Blair Neatby Creating Carleton: The Shaping of a University (Montreal: McGill-Queen's University Press, October 1, 2002) Paul Axelrod Scholars and Dollars: Politics, Economics, and the Universities of Ontario 1945-1980 (Toronto: University of Toronto Press, September 1, 1982) == References == Phillipson, Donald J.C. (16 December 2013). "Henry Marshall Tory". The Canadian Encyclopedia. Historica Canada. == External links == Henry Marshall Tory Fonds
|
Wikipedia:Henry N. Tisdale#0
|
Henry Nehemiah Tisdale (born 1944) is an American retired academic administrator, educator, and mathematician. He served as the 8th president of Claflin University, a historically black university in Orangeburg, South Carolina from 1994 to 2019. During his tenure, Tisdale oversaw significant academic and infrastructural growth at the institution. == Biography == === Early life and education === Henry Nehemiah Tisdale was born on January 13, 1944, in Kingstree, South Carolina. He received his elementary education at St. Paul Elementary and Junior High School (1st grade) and Cane Branch School. He then attended Tomlinson High School before transferring to St. Mark Elementary and High School, where he graduated as Valedictorian in 1961. He earned a Bachelor of Science degree in mathematics from Claflin University in 1965, graduating magna cum laude. He continued his education at Temple University, where he received a Master of Education (Ed.M.) degree in 1967. Tisdale pursued further studies in mathematics at Dartmouth College, where he earned both a Master of Arts (M.A.) in 1975 and a Doctor of Philosophy (Ph.D.) in 1978. His doctoral dissertation, titled "On Methods for Solving Optimal Stopping Problems", was supervised by Professor James Laurie Snell. He was the first African American to receive a doctorate in mathematics from Dartmouth. === Personal life === He was married to Alice née Carson, who died in July 2020. In 1996, Alice Carson Tisdale was honored by the Claflin University Board of Trustees by being the namesake of the University’s honors college. They had two children — a son and a daughter. Their daughter Danica Tisdale was crowned Miss Georgia in 2004. == Career == Tisdale began his professional career as a mathematics educator in the Philadelphia School District, where he taught from 1965 to 1969. In 1969, he moved on to Delaware State University where he held various academic positions for 24 years: Professor of Mathematics, 1969-1985; Assistant Director of Institutional Research and Planning, 1978-1985; Assistant Academic Dean for Administration, 1986 to 1987; Senior Vice President and Chief Academic Officer, 1987 to 1994. In 1994, Tisdale was elected as the 8th president of Claflin University, where he worked from June 1994 to June 2019. During his time as president he constructed many campus buildings including the Living and Learning Center, the Legacy Plaza, the Music Center, a new student residential hall, and a new university chapel. He also strengthened the school endowment and faculty. He is a member of the Mathematical Association of America. === Honors === In 2008, Claflin University, under the leadership of Tisdale, was named the number one HBCU by Forbes, as well as ranking in the top 4% of U.S. colleges and universities. In 2014, Tisdale was honored with South Caroline's Order of the Palmetto, the state's highest civilian honor. In 2019, Tisdale received Columbia World Affairs Council's Global Vision Award, which is given "to a leader whose contributions have made a significant impact on projecting South Carolina globally." == References == == External links == Profile at Claflin University's Athletics Hall of Fame
|
Wikipedia:Henry O. Pollak#0
|
Henry Otto Pollak (born December 13, 1927) is an Austrian-American mathematician who has made significant contributions to operator theory, signal analysis, graph theory, and computational geometry == Research == In several papers with David Slepian and Henry Landau, Pollak developed the theory of what are now known as the Landau–Pollak–Slepian operators on simultaneously time-limited and band-limited functions in operator theory. This work marked an early form of wavelet-based signal analysis. With Ronald Graham he is the namesake of the Graham–Pollak theorem in graph theory, a result on partitioning the edges of complete graphs into complete bipartite graphs that they published in the early 1970s. With Edgar Gilbert he is the namesake of the Gilbert–Pollak conjecture relating Steiner trees to Euclidean minimum spanning trees in computational geometry. After they formulated this problem in 1968, it was believed to be proven by Du and Hwang in the early 1990s, but the proof was later determined to be flawed and the problem remains open. == Life and career == Born in Vienna, Austria, the only child of a lawyer, Pollak fled the Nazis with his family in 1939, first to England and then in 1940 to the US. He received his BS in Mathematics (1947) from Yale University. While at Yale, he participated in the William Lowell Putnam Mathematical Competition and was on the team representing Yale University (along with Murray Gell-Mann and Murray Gerstenhaber) that won the second prize in 1947. He earned an M.A. and Ph.D. (1951) degree in mathematics from Harvard University, the latter on the thesis Some Estimates for Extremal Distance advised by Lars Ahlfors. Pollak then joined Bell Labs (1951), where he later became director of the Mathematics and Statistics Research Center. He has held teaching positions in the mathematics department at Columbia University. == Awards == Fellow of the American Association for the Advancement of Science (1971) Earle Raymond Hedrick lecturer (1973) Mathematical Association of America chair of New Jersey section (1958–59), governor (1961–63) and president (1975–76). Honorary doctorate from Bowdoin College (1977) Honorary doctorate from Eindhoven University of Technology (1981) Mathematical Association of America (MAA) Meritorious Service Award (1990) MAA Gung and Hu Distinguished Service to Mathematics Award (1993) National Council of Teachers of Mathematics Lifetime Achievement Award (2010) Mathematical Association of America Mary P. Dolciani Award in 2020. == Selected publications == Slepian, D.; Pollak, H. O. (1961). "Prolate spheroidal wave functions, Fourier analysis and uncertainty. I". The Bell System Technical Journal. 40: 43–63. doi:10.1002/j.1538-7305.1961.tb03976.x. MR 0140732. Landau, H. J.; Pollak, H. O. (1961). "Prolate spheroidal wave functions, Fourier analysis and uncertainty. II". The Bell System Technical Journal. 40: 65–84. doi:10.1002/j.1538-7305.1961.tb03977.x. MR 0140733. Landau, H. J.; Pollak, H. O. (1962), "Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals", The Bell System Technical Journal, 41: 1295–1336, doi:10.1002/j.1538-7305.1962.tb03279.x, MR 0147686 Gilbert, E. N.; Pollak, H. O. (1968). "Steiner minimal trees". SIAM Journal on Applied Mathematics. 16: 1–29. doi:10.1137/0116001. MR 0223269. Graham, R. L.; Pollak, H. O. (1971). "On the addressing problem for loop switching". The Bell System Technical Journal. 50 (8): 2495–2519. doi:10.1002/j.1538-7305.1971.tb02618.x. MR 0289210.</ref> Graham, R. L.; Pollak, H. O. (1972). "On embedding graphs in squashed cubes". Graph theory and applications (Proc. Conf., Western Michigan Univ., Kalamazoo, Mich., 1972; dedicated to the memory of J. W. T. Youngs). Lecture Notes in Mathematics. Vol. 303. pp. 99–110. MR 0332576. == References ==
|
Wikipedia:Henry Stapp#0
|
Henry Pierce Stapp (born March 23, 1928) is an American mathematical physicist, known for his work in quantum mechanics, particularly the development of axiomatic S-matrix theory, the proofs of strong nonlocality properties, and the place of free will in the orthodox quantum mechanics of John von Neumann. == Biography == Stapp received his PhD in particle physics at the University of California, Berkeley, under the supervision of Nobel laureates Emilio Segrè and Owen Chamberlain. In 1958, Stapp was invited by Wolfgang Pauli to ETH Zurich to work with him personally on basic problems in quantum mechanics. When Pauli died in December 1958, Stapp studied von Neumann's book on quantum mechanics, and on the basis of that work composed an article entitled "Mind, Matter and Quantum Mechanics", which was not submitted for publication; but the title became the title of his 1993 book. In 1969 Stapp was invited by Werner Heisenberg to work with him at the Max Planck Institute in Munich. In 1976 Stapp was invited by J.A. Wheeler to work with him on problems in the foundations of quantum mechanics. Dr. Stapp has published many papers pertaining to the non-local aspects of quantum mechanics and Bell's theorem, including three books. Stapp has worked also in a number of conventional areas of high energy physics, including analysis of the scattering of polarized protons, parity violation, and S-matrix theory. == Research == Some of Stapp's work concerns the implications of quantum mechanics. He has argued for the relevance of quantum mechanics to consciousness and free will. Stapp favors consciousness causes collapse, the idea that quantum wave functions collapse only when they interact with consciousness as a consequence of "orthodox" quantum mechanics. He argues that quantum wave functions collapse when conscious minds select one among the alternative quantum possibilities. His hypothesis of how mind may interact with matter via quantum processes in the brain differs from that of Roger Penrose and Stuart Hameroff's orchestrated objective reduction. While they postulate quantum computing in the microtubules in brain neurons, Stapp postulates a more global collapse, a 'mind like' wave-function collapse that exploits certain aspects of the quantum Zeno effect within the synapses. Stapp's view of the neural correlate of attention is explained in his book, Mindful Universe: Quantum Mechanics and the Participating Observer (2007). Stapp has claimed that consciousness is fundamental to the universe. In this book he credits John von Neumann's Mathematical Foundations of Quantum Mechanics (1932) with providing an orthodox quantum mechanics demonstrating mathematically the essential role of quantum physics in the mind. Stapp has taken interest in the work of Alfred North Whitehead. He has proposed what he calls a "revised Whiteheadianism". He has also written a chapter "Whiteheadian Process and Quantum Theory" (pp. 92–102) in the book Physics and Whitehead: Quantum, Process, and Experience (2003). His philosophy has been described as being influenced by both Heisenberg's physical realism and Bohr's idealism. A form of panpsychism Philosopher Gordon Globus noted that "Stapp unhesitatingly descends into panexperientialism". Stapp has co-authored papers with Jeffrey M. Schwartz. Schwartz has connected the work of Stapp with the concept of "mental force" and spiritual practices of Buddhism. == Reception == Stapp's work has drawn criticism from scientists such as David Bourget and Danko Georgiev. Recent papers and a book by Georgiev criticize Stapp's model in two aspects: (1) The mind in Stapp's model does not have its own wavefunction or density matrix, but nevertheless can act upon the brain using projection operators. Such usage is not compatible with standard quantum mechanics because one can attach any number of ghostly minds to any point in space that act upon physical quantum systems with any projection operators. Therefore, Stapp's model does not build upon "the prevailing principles of physics", but negates them. (2) Stapp's claim that quantum Zeno effect is robust against environmental decoherence directly contradicts a basic theorem in quantum information theory according to which acting with projection operators upon the density matrix of a quantum system can never decrease the von Neumann entropy of the system, but can only increase it. Stapp has responded to Bourget and Georgiev stating that the allegations of errors are incorrect. == Selected publications == Stapp, H; Schwartz, J. M; Beauregard, M. (2005). Quantum theory in neuroscience and psychology: A neurophysical model of mind-brain interaction. Philosophical Transactions of the Royal Society of London, Series B. 360 (1458): 1309-1327. Full paper Stapp, H; Schwartz, J. M; Beauregard, M. (2004). The volitional influence of the mind on the brain, with special reference to emotional self-regulation. In Beauregard, M. (Ed.). Consciousness, emotional self-regulation, and the brain, Philadelphia, PA: John Benjamins Publishing Company, Chapter 7. ISBN 90-272-5187-8 Stapp, H. (2009). Mind, Matter and Quantum Mechanics (The Frontiers Collection). Springer. ISBN 978-3-540-89653-1 Stapp, H. (2011). Mindful Universe: Quantum Mechanics and the Participating Observer. Springer. ISBN 978-3-642-18075-0 Stapp, H. (2017). Quantum Theory and Free Will: How Mental Intentions Translate into Bodily Actions. Springer. ISBN 978-3-319-58301-3 == See also == Epistemological Letters Consciousness causes collapse Quantum mind Quantum Zeno effect == References == == Further reading == Donald, M. On the Work of Henry P. Stapp. Streater, R. F. Quantum Theory on the Brain. Ludwig, K. (1995). Why the Difference Between Quantum and Classical Physics is Irrelevant to the Mind/Body Problem. Psyche 2 (16). == External links == List of papers by Stapp on LBNL server Stapp at the Chopra Foundation Archived 2014-12-09 at the Wayback Machine
|
Wikipedia:Henry Vuibert#0
|
Désiré-Henry Vuibert (21 August 1857 – 27 November 1945) was a French mathematician and publisher of technical books and journals, and founder of the French publishing house Vuibert. He was a publisher of the same class as Louis Hachette and Pierre Larousse, and is said to have begun his company in 1876.: 780 His book Les Anaglyphes geometriques described "Vuibert's principle of anaglyphic vision" based on a "procedure invented by Louis Ducos du Houron, which consisted in printing, in superimposition, pairs of stereoscopic views, taken in complementary colors". This book and the concepts therein are said to have "inspired Marcel Duchamp's interest in anaglyphs". Les Anaglyphes geometriques "set the standard" for representation of 3D in two dimensions and offered a "grand tour of shape" that influenced both artists and mathematicians alike. Also, according to one account, it was Vuibert, not Eutaris, who first worked out what is now called a Taylor circle. == References == == External links == "Vuibert, H. (Henry), 1857". The Online Books Page. Retrieved 2024-03-22.
|
Wikipedia:Henry Wallman#0
|
Henry "Hank" Wallman (1915–1992) was an American mathematician, known for his work in lattice theory, dimension theory, topology, and electronic circuit design. A native of Brooklyn and a 1933 graduate of Brooklyn College, Wallman received his Ph.D. in mathematics from Princeton University in 1937, under the supervision of Solomon Lefschetz and became a faculty member at the Massachusetts Institute of Technology, where he was associated with the Radiation Laboratory. During World War II he did classified work at MIT, possibly involving radar. In 1948, he left MIT to become a professor of electrotechnics at the Chalmers University of Technology in Gothenburg, Sweden, which awarded him the Chalmers medal in 1980 and where he eventually retired. In 1950 he was elected as a foreign member to the Swedish Royal Academy. He was elected a member of the Royal Swedish Academy of Engineering Sciences in 1960 and of the Royal Swedish Academy of Engineering Sciences in 1970. The disjunction property of Wallman is named after Wallman, as is the Wallman compactification, and he co-authored an important monograph on dimension theory with Witold Hurewicz. Wallman was also a radio enthusiast, and in the postwar period co-authored a book comprehensively documenting what was known at the time about vacuum tube amplification technology, including new developments such as showing how the central limit theorem could be used to describe the rise time of cascaded circuits. At Chalmers, Wallman helped build the Electronic Differential Analyser, an early example of an analog computer, and performed pioneering research in biomedical engineering combining video displays with X-ray imaging. == References ==
|
Wikipedia:Henry Wilbraham#0
|
Henry Wilbraham (25 July 1825 – 13 February 1883) was an English mathematician. He is known for discovering and explaining the Gibbs phenomenon nearly fifty years before J. Willard Gibbs did. Gibbs and Maxime Bôcher, as well as nearly everyone else, were unaware of Wilbraham's paper on the Gibbs phenomenon. == Biography == Henry Wilbraham was born to George and Lady Anne Wilbraham at Delamere, Cheshire. His family was privileged, with his father a parliamentarian and his mother the daughter of the Earl Fortescue. He attended Harrow School before being admitted to Trinity College, Cambridge at the age of 16. He received a BA in 1846 and an MA in 1849 from Cambridge. At the age of 22 he published his paper on the Gibbs phenomenon. He remained at Trinity as a Fellow until 1856. In 1864 he married Mary Jane Marriott, and together they had seven children. In the last years of his life, he was the District Registrar of the Chancery Court at Manchester. == References == Paul J. Nahin, Dr. Euler's Fabulous Formula, Princeton University Press, 2006. Ch. 4, Sect. 4. Wilbraham, Henry (1848), "On a certain periodic function", The Cambridge and Dublin Mathematical Journal, 3: 198–201 Hewitt, Edwin; Hewitt, Robert E. (1979). "The Gibbs–Wilbraham phenomenon: An episode in Fourier analysis". Archive for History of Exact Sciences. 21 (2): 129–160. doi:10.1007/BF00330404. S2CID 119355426.
|
Wikipedia:Herbert Fleischner#0
|
Herbert Fleischner (born 29 January 1944 in London) is an Austrian mathematician. == Education and career == Fleischner moved to Vienna with his parents in 1946. He attended primary and secondary school in Vienna, graduating in 1962. After that he studied mathematics and physics at the University of Vienna; his main teachers were Nikolaus Hofreiter and Edmund Hlawka. He obtained his PhD degree in 1968; his official PhD supervisor was Edmund Hlawka, and his PhD thesis was entitled Sätze über Eulersche Graphen mit speziellen Eigenschaften, Sätze über die Existenz von Hamiltonschen Linien. However, Herbert Izbicki was the actual supervisor since he was a graph theorist. Fleischner started his academic career as an assistant at the Technical University of Vienna. The academic years 1970/71 and 1972/72 he spent at SUNY Binghamton as postdoctoral research associate and assistant professor; 1972/73 he spent at the Institute for Advanced Study as visiting member on the basis of an NSF grant. After that he returned to Vienna and started working at the Austrian Academy of Sciences (ÖAW), first at the Institute for Information Processing, then at the Institute of Discrete Mathematics. He worked at the ÖAW until the end of 2002, but took leaves to work at Memphis State University (now Memphis University, 1977), MIT (1978, Max Kade Grant), University of Zimbabwe (Academic Staff Development Project sponsored by Österreichischer Entwicklungskooperation and UNESCO, 1997–1999), West Virginia University (2002). He als worked at Texas A&M University (SS 2003 und SS 2006). Fleischner’s research focuses mainly on graph theoretical topics such as hamiltonian and eulerian graphs. One of his main achievements is the proof of the theorem according to which the square of every two-connected graph has a Hamiltonian cycle. This result (now known as Fleischner's theorem) had been submitted in 1971 and was published in 1974. Another milestone in his research was the solution of the „Cycle plus Triangles Problems“ posed by Paul Erdős; its solution came about in cooperation with Michael Stiebitz (TU Ilmenau). Fleischner published more than 90 papers in various mathematical journals; his Erdős-number is 2. His friendship with the Austrian painter de:Robert Lettner resulted in a cooperation in which certain graphs were transformed into paintings called mutations. During 2002-2007 he was Chairman of the Committee for Developing Countries of the European Mathematical Society (EMS-CDC). == Publications == Eulerian Graphs and Related Topics: Part 1, Volume 1 (= Annals of Discrete Mathematics Band 45). Elsevier, Juli 1990, ISBN 978-0-444-88395-7. Eulerian Graphs and Related Topics: Part 1, Volume 2 (= Annals of Discrete Mathematics Band 50). Elsevier, Juni 1991, ISBN 978-0-444-89110-5. Эйлеровы графы и смежные вопросы. Москва: Мир (2002), ISBN 5-03-003115-4. (Russian translation of Eulerian Graphs and Related Topics: Part 1, Volume 1) == External links == Entry at the University of Technology Vienna == References ==
|
Wikipedia:Herchel Smith Professor of Pure Mathematics#0
|
The Herchel Smith Professorship of Pure Mathematics is a professorship in pure mathematics at the University of Cambridge. It was established in 2004 by a benefaction from Herchel Smith "of £14.315m, to be divided into five equal parts, to support the full endowment of five Professorships in the fields of Pure Mathematics, Physics, Biochemistry, Molecular Biology, and Molecular Genetics." When the position was advertised in 2004, the first holder was expected to focus on mathematical analysis. == List of Herchel Smith Professors of Pure Mathematics == 2006–2013 Ben J. Green 2019–present Pierre Raphael == References ==
|
Wikipedia:Herman Madsen#0
|
Vilhelm Herman Oluf Madsen (11 April 1844 – 14 June 1917) was a Danish politician, minister, army officer, businessman and inventor who served as War Minister in the 1901–1905 Deuntzer Cabinet. == Career == Madsen began his military career in 1859 and served in the Second War of Schleswig of 1864 as a lieutenant. In 1896, at the rank of colonel, Madsen was responsible for the adoption of the Madsen machine gun by the Danish army in 1902 and widely exported. He also constructed the Madsen 20 mm anti-aircraft cannon. As Minister of War in the Cabinet of J. H. Deuntzer from 1901 to 1905, he supported the Fortification of Copenhagen, which contributed to the conflict that led to the split of the Venstre Reform Party as the left wing of the party left the party in protest to form the Radikale Venstre. Madsen became a general in 1903 and was elected to the Folketing in 1909. == Personal life == Madsen was the father of the physician Thorvald Madsen. He was interested in mathematics and was the president of Danish Mathematical Society from 1903 to 1910. == References == == Sources == Bjørn A. Nielsen, Den danske hærs rekylgeværer : system V.H.O. Madsen og J.A.N. Rasmussen, Statens Forsvarshistoriske Museum, 2008. (Forsvarshistoriske skrifter, nr. 6). Article "Madsen, Vilhelm Herman Oluf", pp. 23–26, in: Dansk Biografisk Lexikon, 1. ed, vol. XI, 1897
|
Wikipedia:Herman Valentiner#0
|
Herman Valentiner (8 May 1850 – 17 September 1913) was a Danish mathematician who introduced the Valentiner group in 1889. Valentiner earned his Ph.D. in 1881 from the University of Copenhagen with a thesis on space curves, and took a teaching position. However, soon afterwards he moved to a Danish life insurance company. == References == == Further reading == Juel, C. (1913), "Herman Valentiner, Født 8. Maj 1850 - død 17. Sept. 1913", NYT Tidsskr. For Mat. B, 24: 65–69 Zeuthen, H. G., Herman Valentiner, Dansk biografisk Lexikon == External links == Herman Valentiner at the Mathematics Genealogy Project
|
Wikipedia:Herman ring#0
|
In the mathematical discipline known as complex dynamics, the Herman ring is a Fatou component where the rational function is conformally conjugate to an irrational rotation of the standard annulus. == Formal definition == Namely if ƒ possesses a Herman ring U with period p, then there exists a conformal mapping ϕ : U → { ζ : 0 < r < | ζ | < 1 } {\displaystyle \phi :U\rightarrow \{\zeta :0<r<|\zeta |<1\}} and an irrational number θ {\displaystyle \theta } , such that ϕ ∘ f ∘ p ∘ ϕ − 1 ( ζ ) = e 2 π i θ ζ . {\displaystyle \phi \circ f^{\circ p}\circ \phi ^{-1}(\zeta )=e^{2\pi i\theta }\zeta .} So the dynamics on the Herman ring is simple. == Name == It was introduced by, and later named after, Michael Herman (1979) who first found and constructed this type of Fatou component. == Function == Polynomials do not have Herman rings. Rational functions can have Herman rings. According to the result of Shishikura, if a rational function ƒ possesses a Herman ring, then the degree of ƒ is at least 3. Transcendental entire maps do not have them meromorphic functions can possess Herman rings. Herman rings for transcendental meromorphic functions have been studied by T. Nayak. According to a result of Nayak, if there is an omitted value for such a function then Herman rings of period 1 or 2 do not exist. Also, it is proved that if there is only a single pole and at least an omitted value, the function has no Herman ring of any period. == Examples == === Herman and parabolic basin === Here is an example of a rational function which possesses a Herman ring. f ( z ) = e 2 π i τ z 2 ( z − 4 ) 1 − 4 z {\displaystyle f(z)={\frac {e^{2\pi i\tau }z^{2}(z-4)}{1-4z}}} where τ = 0.6151732 … {\displaystyle \tau =0.6151732\dots } such that the rotation number of ƒ on the unit circle is ( 5 − 1 ) / 2 {\displaystyle ({\sqrt {5}}-1)/2} . The picture shown on the right is the Julia set of ƒ: the curves in the white annulus are the orbits of some points under the iterations of ƒ while the dashed line denotes the unit circle. There is an example of rational function that possesses a Herman ring, and some periodic parabolic Fatou components at the same time. === Period 2 Herman ring === Further, there is a rational function which possesses a Herman ring with period 2. Here the expression of this rational function is g a , b , c ( z ) = z 2 ( z − a ) z − b + c , {\displaystyle g_{a,b,c}(z)={\frac {z^{2}(z-a)}{z-b}}+c,\,} where a = 0.17021425 + 0.12612303 i , b = 0.17115266 + 0.12592514 i , c = − 1.18521775 − 0.16885254 i . {\displaystyle {\begin{aligned}a&=0.17021425+0.12612303i,\\b&=0.17115266+0.12592514i,\\c&=-1.18521775-0.16885254i.\end{aligned}}} This example was constructed by quasiconformal surgery from the quadratic polynomial h ( z ) = z 2 − 1 − e 5 π i 4 {\displaystyle h(z)=z^{2}-1-{\frac {e^{{\sqrt {5}}\pi i}}{4}}} which possesses a Siegel disk with period 2. The parameters a, b, c are calculated by trial and error. Letting a = 0.14285933 + 0.06404502 i , b = 0.14362386 + 0.06461542 i , and c = − 0.18242894 − 0.81957139 i , {\displaystyle {\begin{aligned}a&=0.14285933+0.06404502i,\\b&=0.14362386+0.06461542i,{\text{ and}}\\c&=-0.18242894-0.81957139i,\end{aligned}}} then the period of one of the Herman ring of ga,b,c is 3. Shishikura also given an example: a rational function which possesses a Herman ring with period 2, but the parameters showed above are different from his. === Period 5 Herman ring === So there is a question: How to find the formulas of the rational functions which possess Herman rings with higher period? This question can be answered (for any period > 0) by using the Mandelbrot set for the rational functions ga,b,c. The classic Mandelbrot set (for quadratic polynomials) is approximated by iterating the critical point for each such polynomial, and identifying the polynomials for which the iterates of the critical point do not converge to infinity. Similarly a Mandelbrot set can be defined for the set of rational functions ga,b,c by distinguishing between the values of (a,b,c) in complex 3-space for which all the three critical points (i.e. points where the derivative vanishes) of the function converge to infinity, and the values whose critical points do not all converge to infinity. For each value of a and b, the Mandelbrot set for ga,b,c can be calculated in the plane of complex values c. When a and b are nearly equal, this set approximates the classic Mandelbrot set for quadratic polynomials, because ga,b,c is equal to x2 + c when a=b. In the classic Mandelbrot set, Siegel discs can be approximated by choosing points along the edge of the Mandelbrot set with irrational winding number having continued fraction expansion with bounded denominators. The irrational numbers are of course only approximated in their computer representation. These denominators can be identified by the sequence of nodes along the edge of the Mandelbrot set approaching the point. Similarly, Herman rings can be identified in a Mandelbrot set of rational functions by observing a series of nodes arranged on both sides of a curve, and choosing points along that curve, avoiding the attached nodes, thereby obtaining a desired sequence of denominators in the continued fraction expansion of the rotation number. The following illustrates a planar slice of the Mandelbrot set of ga,b,c with |a-b| = .0001, and with c centered at a value of c which identifies a 5-cycle of Siegel discs in the classic Mandelbrot set. The image above uses a =0.12601278 +.0458649i, b= .12582484 +.045796497i, and is centered at a value of c = 0.3688 -.3578, which is near 5-cycles of Siegel discs in the classic Mandelbrot set. In the above image, a 5-cycle of Herman rings can be approximated by choosing a point c along the above illustrated curve having nodes on both sides, for which ga,b,c has approximately the desired winding number, using values as follows: a = .12601278 + .0458649 i , b = .12582484 + .045796497 i , and c = 0.37144067 − .35829275 i , {\displaystyle {\begin{aligned}a&=.12601278+.0458649i,\\b&=.12582484+.045796497i,{\text{ and}}\\c&=0.37144067-.35829275i,\end{aligned}}} The resulting 5-cycle of Herman rings is illustrated below: == See also == Douady rabbit Siegel disc == References ==
|
Wikipedia:Hermann Flaschka#0
|
Hermann Flaschka (25 March 1945 – 18 March 2021) was an Austrian-American mathematical physicist and Professor of Mathematics at the University of Arizona, known for his important contributions in completely integrable systems (soliton equations). == Childhood == Flaschka had lived in the USA since his family immigrated when he was a teenager. They lived in Atlanta, GA. His father Hermenegild Arved Flaschka (1915 - 1999) taught Chemistry at Georgia Tech. Hermann graduated from Druid Hills High School with the class of 1962 and received his Bachelor's degree at Georgia Tech in 1967. Among other achievements there he also received the "William Gilmer Perry Awards for Freshman English" in 1963, despite the fact that he's not a native speaker. == Career == He received his Ph.D. from the Massachusetts Institute of Technology in 1970. His advisor was Gilbert Strang and the title of his thesis Asymptotic Expansions and Hyperbolic Equations with Multiple Characteristics. He then worked as post-doc at the Carnegie Mellon University until 1972. He was a professor at the University of Arizona until his retirement in 2017. He lectured as visiting professor at several institutions, among them the Clarkson University (1978/79), the Kyoto RIMS (1980/81) and the École Polytechnique Fédérale de Lausanne (2002). In 1995 he received the Norbert Wiener Prize in Applied Mathematics. In 2012 he became a fellow of the American Mathematical Society. == Work == He made important contributions to the theory of completely integrable systems in particular the Toda lattice and the Korteweg–de Vries equation. In 1980 he co-founded Physica D: Nonlinear Phenomena for which he also served as co-editor for many years. Publisher Elsevier now lists him as honorary editor. == References == == External links == 1995 Norbert Wiener Prize in Applied Mathematics, Notices AMS Homepage Hermann Flaschka at the Mathematics Genealogy Project
|
Wikipedia:Hermann Kinkelin#0
|
Hermann Kinkelin (11 November 1832 – 1 January 1913) was a Swiss mathematician and politician. == Life == His family came from Lindau on Lake Constance. He studied at the Universities of Zurich, Lausanne, and Munich. In 1865 he became professor of mathematics at the University of Basel, where until his retirement in 1908, the full burden of teaching of mathematics was his responsibility. In 1867 he was naturalized in Basel. He was also a statistician, he founded the Swiss Statistical Society and the Statistical-economic society in Basel and led the 1870 and 1880 Federal census in Basel. Kinkelin's works dealt with the gamma function, infinite series, and solid geometry of the axonometric. Kinkelin produced more than 60 publications in actuarial mathematics and statistics. He was a founder of the Basel "mortality and age checkout" (later "Patria, Swiss life insurance company Mutual") and the Swiss Statistical Society, of which he was a member during 1877–86. Hermann Kinkelin died in Basel on 1 January 1913. == Publications == Investigation into the formula n F ( n x ) = f ( x ) + f ( x + 1 n ) + f ( x + 2 n ) + … f ( x + n − 1 n ) . {\displaystyle \scriptstyle nF(nx)=f(x)+f(x+{\frac {1}{n}})+f(x+{\frac {2}{n}})+\ldots f(x+{\frac {n-1}{n}}).} Archiv der Mathematik und Physik 22, 1854, pp. 189–224 (Google Books, dito) The fundamental equations of the Γ(x) function, Mitteilungen der Naturforschenden Gesellschaft in Bern 385 und 386, 1857, pp. 1–11 (Internet-Archiv, dito) On some infinite series, Mitteilungen der Naturforschenden Gesellschaft in Bern 419 und 420, 1858, pp. 89–104 (Internet Archive) About a transcendent relatives of the gamma function and its application to the integral calculus, Journal für die reine und angewandte Mathematik 57, 1860, pp. 122–138 (GDZ) The oblique axonometric projection, Vierteljahrsschrift der Naturforschenden Gesellschaft in Zürich 6, 1861, pp. 358–367 (Google Books) On the Theory of Prismoides, Archiv der Mathematik und Physik 39, 1862, p. 181–186 (Google Books, dito, dito) Proof of three sybling expressions of the triangle, Archiv der Mathematik und Physik 39, 1862, pp. 186–188 (Google Books, dito, dito) New evidence of the presence complex roots in an algebraic equation, Mathematische Annalen 1, 1869, pp. 502–506 (Google Books, GDZ, Jahrbuch-Rezension) The calculation of the Christian Easter, Zeitschrift für Mathematik und Physik 15, 1870, pp. 217–228 (Internet-Archiv) Lecture in Die Basler Mathematiker Daniel Bernoulli und Leonhard Euler, Verhandlungen der Naturforschenden Gesellschaft in Basel 7 (Anhang), 1884, pp. 51–71 (Internet-Archiv) Constructions of the centers of curvature of conics, Zeitschrift für Mathematik und Physik 40, 1895, pp. 58–59 (Internet-Archiv, Jahrbuch-Rezension) About the gamma function, Verhandlungen der Naturforschenden Gesellschaft in Basel 16, 1903, pp. 309–328 (Internet-Archiv, dito) === Monographs === General theory of harmonic series with applications to number theory, Schweighauser, Basel 1862 (Google Books) Short notice of the metric weights and measures, 1876; Nachdruck: Andreas Mächler, Riehen 2006, ISBN 3-905837-02-1 == References == Johann Jakob Burckhardt: Kinkelin. Hermann. In: Neue Deutsche Biographie (NDB). Band 11, Duncker & Humblot, Berlin 1977, pp. 625 (Digitalisat). H. Fäh, in: Verhh. d. Schweizer. Naturforschenden Ges., 96. J.verslg., 1913 (P); G. Schärtlin, Erinnerungen an H. K., ebd.; R. Flatt, Verz. d. gedr. Veröff. v. H. K., ebd.; ders., in: Basler Jb. 1914(P); HBLS (P). Hermann Wichers: Kinkelin, Hermann in Historischen Lexikon der Schweiz Hermann Kinkelin in History of Social Security in Switzerland
|
Wikipedia:Hermine Agavni Kalustyan#0
|
Hermine Agavni Kalustyan (Armenian: Հերմինէ Աղաւնի Գալուստեան, 1914 – 3 September 1989) was a Turkish-Armenian mathematician, educator, and politician. == Early life and education == Kalustyan was born in 1914 in Istanbul, Turkey. She graduated from Paris High School Teacher Training School and from Istanbul University Mathematics Department. From 1932 to 1936, she was at Ecole Normale Superieure to study math. In 1941, she became the first woman in Turkey to obtain a Ph.D. degree in mathematics. She wrote her dissertation titled "Conformal depiction and the movement of an object" in the Istanbul University under Richard von Mises and William Prager. == Career == Between 1948 and 1973, Kalustyan was appointed principal at Esayan Armenian High School. She also taught mathematics at the Getronagan and Galatasaray lyceums in Istanbul, Turkey. In 1961, she became the republic's first non-Muslim minority woman to serve in parliament with her appointment to the transitional parliament (1960–1961) that constructed the 1961 Constitution. Afterwards, she became a member of the CHP on February 18, 1961. In 1975, she published an Armenian book titled, "Towards the Past and Now: Towards Fezaya", and also moved to France. == Select publications == Ali Çakırbaş (June 2017), Non-Muslim MPs and Their Activities in the Republican Era (1923-1964), Nevşehir: Nevşehir Hacı Bektaş Veli University Social Sciences Institute, p. 293, archived from the original on 12 May 2021, access date: 12 May 2021, PhD Thesis Nihal Esen (2016), Non-Muslim Members of Parliament in the Republican Era, Elazığ: Fırat University Social Sciences Institute, p. 47, archived from the original on 12 May 2021, access date: 12 May 2021, Master's Thesis Grand National Assembly of Turkey Album 1920-2010, Volume 4 (1960-1983), p. 1641, archived from the original on 21 January 2016, access date: 12 May 2021 == References ==
|
Wikipedia:Hermite interpolation#0
|
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than n such that the polynomial and its first few derivatives have the same values at m (fewer than n) given points as the given function and its first few derivatives at those points. The number of pieces of information, function values and derivative values, must add up to n {\displaystyle n} . Hermite's method of interpolation is closely related to the Newton's interpolation method, in that both can be derived from the calculation of divided differences. However, there are other methods for computing a Hermite interpolating polynomial. One can use linear algebra, by taking the coefficients of the interpolating polynomial as unknowns, and writing as linear equations the constraints that the interpolating polynomial must satisfy. For another method, see Chinese remainder theorem § Hermite interpolation. For yet another method, see, which uses contour integration. == Statement of the problem == In the restricted formulation studied in, Hermite interpolation consists of computing a polynomial of degree as low as possible that matches an unknown function both in observed value, and the observed value of its first m derivatives. This means that n(m + 1) values ( x 0 , y 0 ) , ( x 1 , y 1 ) , … , ( x n − 1 , y n − 1 ) , ( x 0 , y 0 ′ ) , ( x 1 , y 1 ′ ) , … , ( x n − 1 , y n − 1 ′ ) , ⋮ ⋮ ⋮ ( x 0 , y 0 ( m ) ) , ( x 1 , y 1 ( m ) ) , … , ( x n − 1 , y n − 1 ( m ) ) {\displaystyle {\begin{matrix}(x_{0},y_{0}),&(x_{1},y_{1}),&\ldots ,&(x_{n-1},y_{n-1}),\\[1ex](x_{0},y_{0}'),&(x_{1},y_{1}'),&\ldots ,&(x_{n-1},y_{n-1}'),\\[1ex]\vdots &\vdots &&\vdots \\[1.2ex](x_{0},y_{0}^{(m)}),&(x_{1},y_{1}^{(m)}),&\ldots ,&(x_{n-1},y_{n-1}^{(m)})\end{matrix}}} must be known. The resulting polynomial has a degree less than n(m + 1). (In a more general case, there is no need for m to be a fixed value; that is, some points may have more known derivatives than others. In this case the resulting polynomial has a degree less than the number of data points.) Let us consider a polynomial P(x) of degree less than n(m + 1) with indeterminate coefficients; that is, the coefficients of P(x) are n(m + 1) new variables. Then, by writing the constraints that the interpolating polynomial must satisfy, one gets a system of n(m + 1) linear equations in n(m + 1) unknowns. In general, such a system has exactly one solution. In, Charles Hermite used contour integration to prove that this is effectively the case here, and to find the unique solution, provided that the xi are pairwise different. The Hermite interpolation problem is a problem of linear algebra that has the coefficients of the interpolation polynomial as unknown variables and a confluent Vandermonde matrix as its matrix. The general methods of linear algebra, and specific methods for confluent Vandermonde matrices are often used for computing the interpolation polynomial. Another method is described below. == Using Chinese remainder theorem == Let k be a positive integer, m 1 , … , m k {\displaystyle m_{1},\ldots ,m_{k}} be nonnegative integers, and values x 1 , … , x k {\displaystyle x_{1},\ldots ,x_{k}} that are real numbers or belong to any other field of characteristic zero. Hermite interpolation problem consists of finding a polynomial f such that f ( x i ) = y i , 0 , f ′ ( x i ) = y i , 1 , … , f m i ( x i ) = y i , m i {\displaystyle f(x_{i})=y_{i,0},f'(x_{i})=y_{i,1},\ldots ,f^{m_{i}}(x_{i})=y_{i,m_{i}}} for i = 1 , … , k {\displaystyle i=1,\ldots ,k} , where the y i , j {\displaystyle y_{i,j}} are given values in the same field as the x i {\displaystyle x_{i}} . These conditions implies that the Taylor polynomial of f of degree m i {\displaystyle m_{i}} at x i {\displaystyle x_{i}} is ∑ j = 0 m y i , j i ! ( x − x i ) j . {\displaystyle \sum _{j=0}^{m}{\frac {y_{i,j}}{i!}}(x-x_{i})^{j}.} In other words, the desired polynomial f is congruent to this polynomial modulo ( x − x i ) m i + 1 {\displaystyle (x-x_{i})^{m_{i}+1}} . The Chinese remainder theorem for polynomials implies that there is exactly one solution of degree less than n = ∑ i = 0 k ( m i + 1 ) . {\textstyle n=\sum _{i=0}^{k}(m_{i}+1).} Moreover, this solution can be computed with O ( n 2 ) {\displaystyle O(n^{2})} arithmetic operations, or even faster with fast polynomial multiplication. This approach does not works in positive characteristic, because of the denominators of the coefficients of the Taylor polynomial. The approach through divided differences, below, works in every characteristic. == Using divided differences == === Simple case when all k=2 === When using divided differences to calculate the Hermite polynomial of a function f, the first step is to copy each point m times. (Here we will consider the simplest case m = 1 {\displaystyle m=1} for all points.) Therefore, given n + 1 {\displaystyle n+1} data points x 0 , x 1 , x 2 , … , x n {\displaystyle x_{0},x_{1},x_{2},\ldots ,x_{n}} , and values f ( x 0 ) , f ( x 1 ) , … , f ( x n ) {\displaystyle f(x_{0}),f(x_{1}),\ldots ,f(x_{n})} and f ′ ( x 0 ) , f ′ ( x 1 ) , … , f ′ ( x n ) {\displaystyle f'(x_{0}),f'(x_{1}),\ldots ,f'(x_{n})} for a function f {\displaystyle f} that we want to interpolate, we create a new dataset z 0 , z 1 , … , z 2 n + 1 {\displaystyle z_{0},z_{1},\ldots ,z_{2n+1}} such that z 2 i = z 2 i + 1 = x i . {\displaystyle z_{2i}=z_{2i+1}=x_{i}.} Now, we create a divided differences table for the points z 0 , z 1 , … , z 2 n + 1 {\displaystyle z_{0},z_{1},\ldots ,z_{2n+1}} . However, for some divided differences, z i = z i + 1 ⟹ f [ z i , z i + 1 ] = f ( z i + 1 ) − f ( z i ) z i + 1 − z i = 0 0 {\displaystyle z_{i}=z_{i+1}\implies f[z_{i},z_{i+1}]={\frac {f(z_{i+1})-f(z_{i})}{z_{i+1}-z_{i}}}={\frac {0}{0}}} which is undefined. In this case, the divided difference is replaced by f ′ ( z i ) {\displaystyle f'(z_{i})} . All others are calculated normally. === A more general case when k>2 === In the general case, suppose a given point x i {\displaystyle x_{i}} has k derivatives. Then the dataset z 0 , z 1 , … , z N {\displaystyle z_{0},z_{1},\ldots ,z_{N}} contains k identical copies of x i {\displaystyle x_{i}} . When creating the table, divided differences of j = 2 , 3 , … , k {\displaystyle j=2,3,\ldots ,k} identical values will be calculated as f ( j ) ( x i ) j ! . {\displaystyle {\frac {f^{(j)}(x_{i})}{j!}}.} For example, f [ x i , x i , x i ] = f ″ ( x i ) 2 {\displaystyle f[x_{i},x_{i},x_{i}]={\frac {f''(x_{i})}{2}}} f [ x i , x i , x i , x i ] = f ( 3 ) ( x i ) 6 {\displaystyle f[x_{i},x_{i},x_{i},x_{i}]={\frac {f^{(3)}(x_{i})}{6}}} etc. A fast algorithm for the fully general case is given in. A slower but more numerically stable algorithm is described in. === Example === Consider the function f ( x ) = x 8 + 1 {\displaystyle f(x)=x^{8}+1} . Evaluating the function and its first two derivatives at x ∈ { − 1 , 0 , 1 } {\displaystyle x\in \{-1,0,1\}} , we obtain the following data: Since we have two derivatives to work with, we construct the set { z i } = { − 1 , − 1 , − 1 , 0 , 0 , 0 , 1 , 1 , 1 } {\displaystyle \{z_{i}\}=\{-1,-1,-1,0,0,0,1,1,1\}} . Our divided difference table is then: z 0 = − 1 f [ z 0 ] = 2 f ′ ( z 0 ) 1 = − 8 z 1 = − 1 f [ z 1 ] = 2 f ″ ( z 1 ) 2 = 28 f ′ ( z 1 ) 1 = − 8 f [ z 3 , z 2 , z 1 , z 0 ] = − 21 z 2 = − 1 f [ z 2 ] = 2 f [ z 3 , z 2 , z 1 ] = 7 15 f [ z 3 , z 2 ] = − 1 f [ z 4 , z 3 , z 2 , z 1 ] = − 6 − 10 z 3 = 0 f [ z 3 ] = 1 f [ z 4 , z 3 , z 2 ] = 1 5 4 f ′ ( z 3 ) 1 = 0 f [ z 5 , z 4 , z 3 , z 2 ] = − 1 − 2 − 1 z 4 = 0 f [ z 4 ] = 1 f ″ ( z 4 ) 2 = 0 1 2 1 f ′ ( z 4 ) 1 = 0 f [ z 6 , z 5 , z 4 , z 3 ] = 1 2 1 z 5 = 0 f [ z 5 ] = 1 f [ z 6 , z 5 , z 4 ] = 1 5 4 f [ z 6 , z 5 ] = 1 f [ z 7 , z 6 , z 5 , z 4 ] = 6 10 z 6 = 1 f [ z 6 ] = 2 f [ z 7 , z 6 , z 5 ] = 7 15 f ′ ( z 6 ) 1 = 8 f [ z 8 , z 7 , z 6 , z 5 ] = 21 z 7 = 1 f [ z 7 ] = 2 f ″ ( z 7 ) 2 = 28 f ′ ( z 7 ) 1 = 8 z 8 = 1 f [ z 8 ] = 2 {\displaystyle {\begin{array}{llcclrrrrr}z_{0}=-1&f[z_{0}]=2&&&&&&&&\\&&{\frac {f'(z_{0})}{1}}=-8&&&&&&&\\z_{1}=-1&f[z_{1}]=2&&{\frac {f''(z_{1})}{2}}=28&&&&&&\\&&{\frac {f'(z_{1})}{1}}=-8&&f[z_{3},z_{2},z_{1},z_{0}]=-21&&&&&\\z_{2}=-1&f[z_{2}]=2&&f[z_{3},z_{2},z_{1}]=7&&15&&&&\\&&f[z_{3},z_{2}]=-1&&f[z_{4},z_{3},z_{2},z_{1}]=-6&&-10&&&\\z_{3}=0&f[z_{3}]=1&&f[z_{4},z_{3},z_{2}]=1&&5&&4&&\\&&{\frac {f'(z_{3})}{1}}=0&&f[z_{5},z_{4},z_{3},z_{2}]=-1&&-2&&-1&\\z_{4}=0&f[z_{4}]=1&&{\frac {f''(z_{4})}{2}}=0&&1&&2&&1\\&&{\frac {f'(z_{4})}{1}}=0&&f[z_{6},z_{5},z_{4},z_{3}]=1&&2&&1&\\z_{5}=0&f[z_{5}]=1&&f[z_{6},z_{5},z_{4}]=1&&5&&4&&\\&&f[z_{6},z_{5}]=1&&f[z_{7},z_{6},z_{5},z_{4}]=6&&10&&&\\z_{6}=1&f[z_{6}]=2&&f[z_{7},z_{6},z_{5}]=7&&15&&&&\\&&{\frac {f'(z_{6})}{1}}=8&&f[z_{8},z_{7},z_{6},z_{5}]=21&&&&&\\z_{7}=1&f[z_{7}]=2&&{\frac {f''(z_{7})}{2}}=28&&&&&&\\&&{\frac {f'(z_{7})}{1}}=8&&&&&&&\\z_{8}=1&f[z_{8}]=2&&&&&&&&\\\end{array}}} and the generated polynomial is P ( x ) = 2 − 8 ( x + 1 ) + 28 ( x + 1 ) 2 − 21 ( x + 1 ) 3 + 15 x ( x + 1 ) 3 − 10 x 2 ( x + 1 ) 3 + 4 x 3 ( x + 1 ) 3 − 1 x 3 ( x + 1 ) 3 ( x − 1 ) + x 3 ( x + 1 ) 3 ( x − 1 ) 2 = 2 − 8 + 28 − 21 − 8 x + 56 x − 63 x + 15 x + 28 x 2 − 63 x 2 + 45 x 2 − 10 x 2 − 21 x 3 + 45 x 3 − 30 x 3 + 4 x 3 + x 3 + x 3 + 15 x 4 − 30 x 4 + 12 x 4 + 2 x 4 + x 4 − 10 x 5 + 12 x 5 − 2 x 5 + 4 x 5 − 2 x 5 − 2 x 5 − x 6 + x 6 − x 7 + x 7 + x 8 = x 8 + 1. {\displaystyle {\begin{aligned}P(x)&=2-8(x+1)+28(x+1)^{2}-21(x+1)^{3}+15x(x+1)^{3}-10x^{2}(x+1)^{3}\\&\quad {}+4x^{3}(x+1)^{3}-1x^{3}(x+1)^{3}(x-1)+x^{3}(x+1)^{3}(x-1)^{2}\\&=2-8+28-21-8x+56x-63x+15x+28x^{2}-63x^{2}+45x^{2}-10x^{2}-21x^{3}\\&\quad {}+45x^{3}-30x^{3}+4x^{3}+x^{3}+x^{3}+15x^{4}-30x^{4}+12x^{4}+2x^{4}+x^{4}\\&\quad {}-10x^{5}+12x^{5}-2x^{5}+4x^{5}-2x^{5}-2x^{5}-x^{6}+x^{6}-x^{7}+x^{7}+x^{8}\\&=x^{8}+1.\end{aligned}}} by taking the coefficients from the diagonal of the divided difference table, and multiplying the kth coefficient by ∏ i = 0 k − 1 ( x − z i ) {\textstyle \prod _{i=0}^{k-1}(x-z_{i})} , as we would when generating a Newton polynomial. ==== Quintic Hermite interpolation ==== The quintic Hermite interpolation based on the function ( f {\displaystyle f} ), its first ( f ′ {\displaystyle f'} ) and second derivatives ( f ″ {\displaystyle f''} ) at two different points ( x 0 {\displaystyle x_{0}} and x 1 {\displaystyle x_{1}} ) can be used for example to interpolate the position of an object based on its position, velocity and acceleration. The general form is given by p ( x ) = f ( x 0 ) + f ′ ( x 0 ) ( x − x 0 ) + 1 2 f ″ ( x 0 ) ( x − x 0 ) 2 + f ( x 1 ) − f ( x 0 ) − f ′ ( x 0 ) ( x 1 − x 0 ) − 1 2 f ″ ( x 0 ) ( x 1 − x 0 ) 2 ( x 1 − x 0 ) 3 ( x − x 0 ) 3 + 3 f ( x 0 ) − 3 f ( x 1 ) + 2 ( f ′ ( x 0 ) + 1 2 f ′ ( x 1 ) ) ( x 1 − x 0 ) + 1 2 f ″ ( x 0 ) ( x 1 − x 0 ) 2 ( x 1 − x 0 ) 4 ( x − x 0 ) 3 ( x − x 1 ) + 6 f ( x 1 ) − 6 f ( x 0 ) − 3 ( f ′ ( x 0 ) + f ′ ( x 1 ) ) ( x 1 − x 0 ) + 1 2 ( f ″ ( x 1 ) − f ″ ( x 0 ) ) ( x 1 − x 0 ) 2 ( x 1 − x 0 ) 5 ( x − x 0 ) 3 ( x − x 1 ) 2 . {\displaystyle {\begin{aligned}p(x)&=f(x_{0})+f'(x_{0})(x-x_{0})+{\frac {1}{2}}f''(x_{0})(x-x_{0})^{2}+{\frac {f(x_{1})-f(x_{0})-f'(x_{0})(x_{1}-x_{0})-{\frac {1}{2}}f''(x_{0})(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{3}}}(x-x_{0})^{3}\\&+{\frac {3f(x_{0})-3f(x_{1})+2\left(f'(x_{0})+{\frac {1}{2}}f'(x_{1})\right)(x_{1}-x_{0})+{\frac {1}{2}}f''(x_{0})(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{4}}}(x-x_{0})^{3}(x-x_{1})\\&+{\frac {6f(x_{1})-6f(x_{0})-3\left(f'(x_{0})+f'(x_{1})\right)(x_{1}-x_{0})+{\frac {1}{2}}\left(f''(x_{1})-f''(x_{0})\right)(x_{1}-x_{0})^{2}}{(x_{1}-x_{0})^{5}}}(x-x_{0})^{3}(x-x_{1})^{2}.\end{aligned}}} == Error == Call the calculated polynomial H and original function f. Consider first the real-valued case. Evaluating a point x ∈ [ x 0 , x n ] {\displaystyle x\in [x_{0},x_{n}]} , the error function is f ( x ) − H ( x ) = f ( K ) ( c ) K ! ∏ i ( x − x i ) k i , {\displaystyle f(x)-H(x)={\frac {f^{(K)}(c)}{K!}}\prod _{i}(x-x_{i})^{k_{i}},} where c is an unknown within the range [ x 0 , x N ] {\displaystyle [x_{0},x_{N}]} , K is the total number of data-points, and k i {\displaystyle k_{i}} is the number of derivatives known at each x i {\displaystyle x_{i}} . The degree of the polynomial on the right is thus one higher than the degree bound for H ( x ) {\displaystyle H(x)} . Furthermore, the error and all its derivatives up to the k i − 1 {\displaystyle k_{i}-1} st order is zero at each node, as it should be. In the complex case, as described for example on p. 360 in, f ( z ) − H ( z ) = w ( z ) 2 π i ∮ C f ( ζ ) w ( ζ ) ( ζ − z ) d ζ {\displaystyle f(z)-H(z)={\frac {w(z)}{2\pi i}}\oint _{C}{\frac {f(\zeta )}{w(\zeta )(\zeta -z)}}d\zeta } where the contour C {\displaystyle C} encloses z {\displaystyle z} and all the nodes x i {\displaystyle x_{i}} , and the node polynomial is w ( z ) = ∏ i ( z − x i ) k i {\displaystyle w(z)=\prod _{i}(z-x_{i})^{k_{i}}} . == See also == Cubic Hermite spline Newton series, also known as finite differences Neville's schema Bernstein polynomials == References == == External links == Hermites Interpolating Polynomial at Mathworld
|
Wikipedia:Hermite ring#0
|
In algebra, the term Hermite ring (after Charles Hermite) has been applied to three different objects. According to Kaplansky (1949) (p. 465), a ring is right Hermite if, for every two elements a and b of the ring, there is an element d of the ring and an invertible 2×2 matrix M over the ring such that (a b)M = (d 0), and the term left Hermite is defined similarly. Matrices over such a ring can be put in Hermite normal form by right multiplication by a square invertible matrix (Kaplansky (1949), p. 468.) Lam (2006) (appendix to §I.4) calls this property K-Hermite, using Hermite instead in the sense given below. According to Lam (1978) (§I.4, p. 26), a ring is right Hermite if any finitely generated stably free right module over the ring is free. This is equivalent to requiring that any row vector (b1,...,bn) of elements of the ring which generate it as a right module (i.e., b1R + ... + bnR = R) can be completed to a (not necessarily square) invertible matrix by adding some number of rows. The criterion of being left Hermite can be defined similarly. Lissner (1965) (p. 528) earlier called a commutative ring with this property an H-ring. According to Cohn (2006) (§0.4), a ring is Hermite if, in addition to every stably free (left) module being free, it has invariant basis number. All commutative rings which are Hermite in the sense of Kaplansky are also Hermite in the sense of Lam, but the converse is not necessarily true. All Bézout domains are Hermite in the sense of Kaplansky, and a commutative ring which is Hermite in the sense of Kaplansky is also a Bézout ring (Lam (2006), pp. 39-40.) The Hermite ring conjecture, introduced by Lam (1978) (p. xi), states that if R is a commutative Hermite ring, then the polynomial ring R[x] is also a Hermite ring. == References == Cohn, P. M. (2000), "From Hermite rings to Sylvester domains", Proceedings of the American Mathematical Society, 128 (7): 1899–1904, doi:10.1090/S0002-9939-99-05189-8, ISSN 0002-9939, MR 1646314 Cohn, P. M. (2006), Free Ideal Rings and Localization in General Rings, Cambridge University Press, ISBN 9780521853378 Kaplansky, Irving (1949), "Elementary divisors and modules", Transactions of the American Mathematical Society, 66 (2): 464–491, doi:10.2307/1990591, ISSN 0002-9947, JSTOR 1990591, MR 0031470 Lam, T. Y. (1978), Serre's Conjecture, Lecture Notes in Mathematics, vol. 635, Berlin, New York: Springer-Verlag, doi:10.1007/BFb0068340, ISBN 978-3-540-08657-4, MR 0485842 Lam, T. Y. (2006), Serre's Problem on Projective Modules, Springer Monographs in Mathematics, Berlin, Heidelberg: Springer-Verlag, doi:10.1007/978-3-540-34575-6, ISBN 978-3-540-23317-6 Lissner, David (1965), "Outer product rings", Transactions of the American Mathematical Society, 116: 526–535, doi:10.2307/1994132, ISSN 0002-9947, JSTOR 1994132, MR 0186687
|
Wikipedia:Hermite's identity#0
|
In mathematics, Hermite's identity, named after Charles Hermite, gives the value of a summation involving the floor function. It states that for every real number x and for every positive integer n the following identity holds: ∑ k = 0 n − 1 ⌊ x + k n ⌋ = ⌊ n x ⌋ . {\displaystyle \sum _{k=0}^{n-1}\left\lfloor x+{\frac {k}{n}}\right\rfloor =\lfloor nx\rfloor .} == Proofs == === Proof by algebraic manipulation === Split x {\displaystyle x} into its integer part and fractional part, x = ⌊ x ⌋ + { x } {\displaystyle x=\lfloor x\rfloor +\{x\}} . There is exactly one k ′ ∈ { 1 , … , n } {\displaystyle k'\in \{1,\ldots ,n\}} with ⌊ x ⌋ = ⌊ x + k ′ − 1 n ⌋ ≤ x < ⌊ x + k ′ n ⌋ = ⌊ x ⌋ + 1. {\displaystyle \lfloor x\rfloor =\left\lfloor x+{\frac {k'-1}{n}}\right\rfloor \leq x<\left\lfloor x+{\frac {k'}{n}}\right\rfloor =\lfloor x\rfloor +1.} By subtracting the same integer ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } from inside the floor operations on the left and right sides of this inequality, it may be rewritten as 0 = ⌊ { x } + k ′ − 1 n ⌋ ≤ { x } < ⌊ { x } + k ′ n ⌋ = 1. {\displaystyle 0=\left\lfloor \{x\}+{\frac {k'-1}{n}}\right\rfloor \leq \{x\}<\left\lfloor \{x\}+{\frac {k'}{n}}\right\rfloor =1.} Therefore, 1 − k ′ n ≤ { x } < 1 − k ′ − 1 n , {\displaystyle 1-{\frac {k'}{n}}\leq \{x\}<1-{\frac {k'-1}{n}},} and multiplying both sides by n {\displaystyle n} gives n − k ′ ≤ n { x } < n − k ′ + 1. {\displaystyle n-k'\leq n\,\{x\}<n-k'+1.} Now if the summation from Hermite's identity is split into two parts at index k ′ {\displaystyle k'} , it becomes ∑ k = 0 n − 1 ⌊ x + k n ⌋ = ∑ k = 0 k ′ − 1 ⌊ x ⌋ + ∑ k = k ′ n − 1 ( ⌊ x ⌋ + 1 ) = n ⌊ x ⌋ + n − k ′ = n ⌊ x ⌋ + ⌊ n { x } ⌋ = ⌊ n ⌊ x ⌋ + n { x } ⌋ = ⌊ n x ⌋ . {\displaystyle {\begin{aligned}\sum _{k=0}^{n-1}\left\lfloor x+{\frac {k}{n}}\right\rfloor &=\sum _{k=0}^{k'-1}\lfloor x\rfloor +\sum _{k=k'}^{n-1}(\lfloor x\rfloor +1)=n\,\lfloor x\rfloor +n-k'\\[8pt]&=n\,\lfloor x\rfloor +\lfloor n\,\{x\}\rfloor =\left\lfloor n\,\lfloor x\rfloor +n\,\{x\}\right\rfloor =\lfloor nx\rfloor .\end{aligned}}} === Proof using functions === Consider the function f ( x ) = ⌊ x ⌋ + ⌊ x + 1 n ⌋ + … + ⌊ x + n − 1 n ⌋ − ⌊ n x ⌋ {\displaystyle f(x)=\lfloor x\rfloor +\left\lfloor x+{\frac {1}{n}}\right\rfloor +\ldots +\left\lfloor x+{\frac {n-1}{n}}\right\rfloor -\lfloor nx\rfloor } Then the identity is clearly equivalent to the statement f ( x ) = 0 {\displaystyle f(x)=0} for all real x {\displaystyle x} . But then we find, f ( x + 1 n ) = ⌊ x + 1 n ⌋ + ⌊ x + 2 n ⌋ + … + ⌊ x + 1 ⌋ − ⌊ n x + 1 ⌋ = f ( x ) {\displaystyle f\left(x+{\frac {1}{n}}\right)=\left\lfloor x+{\frac {1}{n}}\right\rfloor +\left\lfloor x+{\frac {2}{n}}\right\rfloor +\ldots +\left\lfloor x+1\right\rfloor -\lfloor nx+1\rfloor =f(x)} Where in the last equality we use the fact that ⌊ x + p ⌋ = ⌊ x ⌋ + p {\displaystyle \lfloor x+p\rfloor =\lfloor x\rfloor +p} for all integers p {\displaystyle p} . But then f {\displaystyle f} has period 1 / n {\displaystyle 1/n} . It then suffices to prove that f ( x ) = 0 {\displaystyle f(x)=0} for all x ∈ [ 0 , 1 / n ) {\displaystyle x\in [0,1/n)} . But in this case, the integral part of each summand in f {\displaystyle f} is equal to 0. We deduce that the function is indeed 0 for all real inputs x {\displaystyle x} . == References ==
|
Wikipedia:Hernando Burgos-Soto#0
|
Hernando Burgos Soto is a Canadian (Colombian born) writer and mathematician, professor of mathematics at George Brown College. He is the author of several math papers in which he introduced some mathematics concepts and extended to tangles some celebrated results of knot theory about the Khovanov homology and the Jones polynomial. During his career as a mathematician, his interests have included Mathematical Statistics, Knot Theory, Algebraic Topology and more recently Mathematical Finance. He is comfortable writing in English and Spanish. When writing in Spanish, he works in the area of prose fiction writing short stories. Some of his short stories were published at the website Cuentos y Cuentos. == Education == Professor Burgos Soto holds a BSc.Ed in mathematics at the University of Atlántico and a MSc in Mathematics from the University of Valle. He earned a PhD in mathematics from National University of Colombia in 2009, and completed his dissertation at University of Toronto under the guidance of professor Dror Bar-Natan. == Research == His works include regression diagnostic analysis for General Linear Models, extension to tangles of Morwen Thistlethwaite's result on the alternation of the Jones polynomial for alternating links, and a Lee's result on Khovanov homology for links, that states that the Khovanov homology for alternating links is supported in two lines. During his works Professor Burgos has introduced some Mathematical concepts such as: Gravity Information in a tangle diagram, Alternating planar algebras, and Rotation Number of Smoothings. == Selected Publication == Burgos-Soto, Hernando (2010) The Jones Polynomial of Alternating Tangles, Journal of Knot Theory and Its Ramifications, Volume 19, Issue 11, Nov. 2010, pages 1487–1505 == References ==
|
Wikipedia:Herta Freitag#0
|
Herta Freitag (née Taussig; December 6, 1908 – January 25, 2000) was an Austrian-American mathematician, a professor of mathematics at Hollins College, known for her work on the Fibonacci numbers. == Life == She was born as Herta Taussig in Vienna, earning a master's degree from the University of Vienna in 1934. She took a teaching position at the university. However, her father (the editor of Die Neue Freie Presse) had publicly opposed the Nazis. Herta and her parents decided to move to a summer cottage in the mountains outside Vienna, to give themselves some time to make plans for the future. Herta's brother, Walter Taussig, a musician, was touring the United States and decided to remain in the U.S. (Walter later became an assistant conductor for the Metropolitan Opera Company.) Herta and her parents immediately started to work on finding a sponsor to bring them to the United States. However, even when they identified a possible sponsor, they had to wait until their quota number was called up. In 1938, she and her parents emigrated to England. She took a job as a maid as British immigration laws prevented her from entering the country as a teacher. In 1944, she, her brother, and her mother moved to the United States. (Her father had died a year earlier in England). She began teaching mathematics again at the Greer School in Dutchess County, New York. She earned a second master's degree in 1948 from Columbia University, and a doctorate from Columbia in 1953. Meanwhile, in 1948, she had joined the faculty at Hollins, where she eventually became a full professor and department chair. In 1962 she served as a section president for the Mathematical Association of America, the first woman in her section to do so. She retired in 1971, but returned to teaching again in 1979 after the death of her husband, Arthur Freitag, whom she had married in 1950. == Recognition == Freitag was named a Fellow of the American Association for the Advancement of Science in 1959. After her retirement, she became a frequent contributor to the Fibonacci Quarterly, and the journal honored her in 1996 by dedicating an issue to her on the occasion of her 89th birthday (89 being a Fibonacci number). == References ==
|
Wikipedia:Hervé Moulin#0
|
Hervé Moulin (born 1950 in Paris) is a French mathematician who is the Donald J. Robertson Chair of Economics at the Adam Smith Business School at the University of Glasgow. He is known for his research contributions in mathematical economics, in particular in the fields of mechanism design, social choice, game theory and fair division. He has written five books and over 100 peer-reviewed articles. Moulin was the George A. Peterkin Professor of Economics at Rice University (from 1999 to 2013):, the James B. Duke Professor of Economics at Duke University (from 1989 to 1999), the University Distinguished Professor at Virginia Tech (from 1987 to 1989), and Academic Supervisor at Higher School of Economics in St. Petersburg, Russia (from 2015 to 2022). He is a fellow of the Econometric Society since 1983, and the president of the Game Theory Society for the term 2016 - 2018. He also served as president of the Society for Social Choice and Welfare for the period of 1998 to 1999. He became a Fellow of the Royal Society of Edinburgh in 2015. Moulin's research has been supported in part by seven grants from the US National Science Foundation. He collaborates as an adviser with the fair division website Spliddit, created by Ariel Procaccia. On the occasion of his 65th birthday, the Paris School of Economics and the Aix-Marseille University organised a conference in his honor, with Peyton Young, William Thomson, Salvador Barbera, and Moulin himself among the speakers. == Biography == Moulin obtained his undergraduate degree from the École Normale Supérieure in Paris in 1971 and his doctoral degree in Mathematics at the University of Paris-IX in 1975 with a thesis on zero-sum games, which was published in French at the Mémoires de la Société Mathématique de France and in English in the Journal of Mathematical Analysis and its Applications. On 1979, he published a seminal paper in Econometrica introducing the notion of dominance solvable games. Dominance solvability is a solution concept for games which is based on an iterated procedure of deletion of dominated strategies by all participants. Dominance solvability is a stronger concept than Nash equilibrium because it does not require ex-ante coordination. Its only requirement is iterated common knowledge of rationality. His work on this concept was mentioned in Eric Maskin's Nobel Prize Lecture. One year later he proved an interesting result concerning the famous Gibbard-Satterthwaite Theorem, which states that any voting procedure on the universal domain of preferences whose range contains more than two alternatives is either dictatorial or manipulable. Moulin proved that it is possible to define non-dictatorial and non-manipulable social choice functions in the restricted domain of single-peaked preferences, i.e. those in which there is a unique best option, and other options are better as they are closer to the favorite one. Moreover, he provided a characterization of such rules. This paper inspired a whole literature on achieving strategy-proofness and fairness (even in a weak form as non-dictatorial schemes) on restricted domains of preferences. Moulin is also known for his seminal work in cost sharing and assignment problems. In particular, jointly with Anna Bogomolnaia, he proposed the probabilistic-serial procedure as a solution to the fair random assignment problem, which consists of dividing several goods among a number of persons. Probabilistic serial allows each person to "eat" her favorite shares, hence defining a probabilistic outcome. It always produces an outcome which is unambiguously efficient ex-ante, and thus has a strong claim over the popular random priority. The paper was published in 2001 in the Journal of Economic Theory. By summer of 2016, the article had 395 citations. He has been credited as the first proposer of the famous beauty contest game, also known as the guessing game, which shows that players fail to anticipate strategic behavior from other players. Experiments testing the equilibrium prediction of this game started the field of experimental economics. In July 2018 Moulin was elected Fellow of the British Academy (FBA). == Coauthors == Moulin has published work jointly with Matthew O. Jackson, Scott Shenker, and Anna Bogomolnaia, among many other academics. == See also == List of economists == References == == External links == Hervé Moulin's Personal Website List of Hervé Moulin's Publications at IDEAS REPEC Hervé Moulin at DBLP Bibliography Server
|
Wikipedia:Hessian matrix#0
|
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or, ∇ ∇ {\displaystyle \nabla \nabla } or ∇ ⊗ ∇ {\displaystyle \nabla \otimes \nabla } or D 2 {\displaystyle D^{2}} . == Definitions and properties == Suppose f : R n → R {\displaystyle f:\mathbb {R} ^{n}\to \mathbb {R} } is a function taking as input a vector x ∈ R n {\displaystyle \mathbf {x} \in \mathbb {R} ^{n}} and outputting a scalar f ( x ) ∈ R . {\displaystyle f(\mathbf {x} )\in \mathbb {R} .} If all second-order partial derivatives of f {\displaystyle f} exist, then the Hessian matrix H {\displaystyle \mathbf {H} } of f {\displaystyle f} is a square n × n {\displaystyle n\times n} matrix, usually defined and arranged as H f = [ ∂ 2 f ∂ x 1 2 ∂ 2 f ∂ x 1 ∂ x 2 ⋯ ∂ 2 f ∂ x 1 ∂ x n ∂ 2 f ∂ x 2 ∂ x 1 ∂ 2 f ∂ x 2 2 ⋯ ∂ 2 f ∂ x 2 ∂ x n ⋮ ⋮ ⋱ ⋮ ∂ 2 f ∂ x n ∂ x 1 ∂ 2 f ∂ x n ∂ x 2 ⋯ ∂ 2 f ∂ x n 2 ] . {\displaystyle \mathbf {H} _{f}={\begin{bmatrix}{\dfrac {\partial ^{2}f}{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{1}\,\partial x_{n}}}\\[2.2ex]{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{2}\,\partial x_{n}}}\\[2.2ex]\vdots &\vdots &\ddots &\vdots \\[2.2ex]{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}f}{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}f}{\partial x_{n}^{2}}}\end{bmatrix}}.} That is, the entry of the ith row and the jth column is ( H f ) i , j = ∂ 2 f ∂ x i ∂ x j . {\displaystyle (\mathbf {H} _{f})_{i,j}={\frac {\partial ^{2}f}{\partial x_{i}\,\partial x_{j}}}.} If furthermore the second partial derivatives are all continuous, the Hessian matrix is a symmetric matrix by the symmetry of second derivatives. The determinant of the Hessian matrix is called the Hessian determinant. The Hessian matrix of a function f {\displaystyle f} is the Jacobian matrix of the gradient of the function f {\displaystyle f} ; that is: H ( f ( x ) ) = J ( ∇ f ( x ) ) . {\displaystyle \mathbf {H} (f(\mathbf {x} ))=\mathbf {J} (\nabla f(\mathbf {x} )).} == Applications == === Inflection points === If f {\displaystyle f} is a homogeneous polynomial in three variables, the equation f = 0 {\displaystyle f=0} is the implicit equation of a plane projective curve. The inflection points of the curve are exactly the non-singular points where the Hessian determinant is zero. It follows by Bézout's theorem that a cubic plane curve has at most 9 inflection points, since the Hessian determinant is a polynomial of degree 3. === Second-derivative test === The Hessian matrix of a convex function is positive semi-definite. Refining this property allows us to test whether a critical point x {\displaystyle x} is a local maximum, local minimum, or a saddle point, as follows: If the Hessian is positive-definite at x , {\displaystyle x,} then f {\displaystyle f} attains an isolated local minimum at x . {\displaystyle x.} If the Hessian is negative-definite at x , {\displaystyle x,} then f {\displaystyle f} attains an isolated local maximum at x . {\displaystyle x.} If the Hessian has both positive and negative eigenvalues, then x {\displaystyle x} is a saddle point for f . {\displaystyle f.} Otherwise the test is inconclusive. This implies that at a local minimum the Hessian is positive-semidefinite, and at a local maximum the Hessian is negative-semidefinite. For positive-semidefinite and negative-semidefinite Hessians the test is inconclusive (a critical point where the Hessian is semidefinite but not definite may be a local extremum or a saddle point). However, more can be said from the point of view of Morse theory. The second-derivative test for functions of one and two variables is simpler than the general case. In one variable, the Hessian contains exactly one second derivative; if it is positive, then x {\displaystyle x} is a local minimum, and if it is negative, then x {\displaystyle x} is a local maximum; if it is zero, then the test is inconclusive. In two variables, the determinant can be used, because the determinant is the product of the eigenvalues. If it is positive, then the eigenvalues are both positive, or both negative. If it is negative, then the two eigenvalues have different signs. If it is zero, then the second-derivative test is inconclusive. Equivalently, the second-order conditions that are sufficient for a local minimum or maximum can be expressed in terms of the sequence of principal (upper-leftmost) minors (determinants of sub-matrices) of the Hessian; these conditions are a special case of those given in the next section for bordered Hessians for constrained optimization—the case in which the number of constraints is zero. Specifically, the sufficient condition for a minimum is that all of these principal minors be positive, while the sufficient condition for a maximum is that the minors alternate in sign, with the 1 × 1 {\displaystyle 1\times 1} minor being negative. === Critical points === If the gradient (the vector of the partial derivatives) of a function f {\displaystyle f} is zero at some point x , {\displaystyle \mathbf {x} ,} then f {\displaystyle f} has a critical point (or stationary point) at x . {\displaystyle \mathbf {x} .} The determinant of the Hessian at x {\displaystyle \mathbf {x} } is called, in some contexts, a discriminant. If this determinant is zero then x {\displaystyle \mathbf {x} } is called a degenerate critical point of f , {\displaystyle f,} or a non-Morse critical point of f . {\displaystyle f.} Otherwise it is non-degenerate, and called a Morse critical point of f . {\displaystyle f.} The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The eigenvalues of the Hessian at that point are the principal curvatures of the function, and the eigenvectors are the principal directions of curvature. (See Gaussian curvature § Relation to principal curvatures.) === Use in optimization === Hessian matrices are used in large-scale optimization problems within Newton-type methods because they are the coefficient of the quadratic term of a local Taylor expansion of a function. That is, y = f ( x + Δ x ) ≈ f ( x ) + ∇ f ( x ) T Δ x + 1 2 Δ x T H ( x ) Δ x {\displaystyle y=f(\mathbf {x} +\Delta \mathbf {x} )\approx f(\mathbf {x} )+\nabla f(\mathbf {x} )^{\mathsf {T}}\Delta \mathbf {x} +{\frac {1}{2}}\,\Delta \mathbf {x} ^{\mathsf {T}}\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} } where ∇ f {\displaystyle \nabla f} is the gradient ( ∂ f ∂ x 1 , … , ∂ f ∂ x n ) . {\displaystyle \left({\frac {\partial f}{\partial x_{1}}},\ldots ,{\frac {\partial f}{\partial x_{n}}}\right).} Computing and storing the full Hessian matrix takes Θ ( n 2 ) {\displaystyle \Theta \left(n^{2}\right)} memory, which is infeasible for high-dimensional functions such as the loss functions of neural nets, conditional random fields, and other statistical models with large numbers of parameters. For such situations, truncated-Newton and quasi-Newton algorithms have been developed. The latter family of algorithms use approximations to the Hessian; one of the most popular quasi-Newton algorithms is BFGS. Such approximations may use the fact that an optimization algorithm uses the Hessian only as a linear operator H ( v ) , {\displaystyle \mathbf {H} (\mathbf {v} ),} and proceed by first noticing that the Hessian also appears in the local expansion of the gradient: ∇ f ( x + Δ x ) = ∇ f ( x ) + H ( x ) Δ x + O ( ‖ Δ x ‖ 2 ) {\displaystyle \nabla f(\mathbf {x} +\Delta \mathbf {x} )=\nabla f(\mathbf {x} )+\mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} +{\mathcal {O}}(\|\Delta \mathbf {x} \|^{2})} Letting Δ x = r v {\displaystyle \Delta \mathbf {x} =r\mathbf {v} } for some scalar r , {\displaystyle r,} this gives H ( x ) Δ x = H ( x ) r v = r H ( x ) v = ∇ f ( x + r v ) − ∇ f ( x ) + O ( r 2 ) , {\displaystyle \mathbf {H} (\mathbf {x} )\,\Delta \mathbf {x} =\mathbf {H} (\mathbf {x} )r\mathbf {v} =r\mathbf {H} (\mathbf {x} )\mathbf {v} =\nabla f(\mathbf {x} +r\mathbf {v} )-\nabla f(\mathbf {x} )+{\mathcal {O}}(r^{2}),} that is, H ( x ) v = 1 r [ ∇ f ( x + r v ) − ∇ f ( x ) ] + O ( r ) {\displaystyle \mathbf {H} (\mathbf {x} )\mathbf {v} ={\frac {1}{r}}\left[\nabla f(\mathbf {x} +r\mathbf {v} )-\nabla f(\mathbf {x} )\right]+{\mathcal {O}}(r)} so if the gradient is already computed, the approximate Hessian can be computed by a linear (in the size of the gradient) number of scalar operations. (While simple to program, this approximation scheme is not numerically stable since r {\displaystyle r} has to be made small to prevent error due to the O ( r ) {\displaystyle {\mathcal {O}}(r)} term, but decreasing it loses precision in the first term.) Notably regarding Randomized Search Heuristics, the evolution strategy's covariance matrix adapts to the inverse of the Hessian matrix, up to a scalar factor and small random fluctuations. This result has been formally proven for a single-parent strategy and a static model, as the population size increases, relying on the quadratic approximation. === Other applications === The Hessian matrix is commonly used for expressing image processing operators in image processing and computer vision (see the Laplacian of Gaussian (LoG) blob detector, the determinant of Hessian (DoH) blob detector and scale space). It can be used in normal mode analysis to calculate the different molecular frequencies in infrared spectroscopy. It can also be used in local sensitivity and statistical diagnostics. == Generalizations == === Bordered Hessian === A bordered Hessian is used for the second-derivative test in certain constrained optimization problems. Given the function f {\displaystyle f} considered previously, but adding a constraint function g {\displaystyle g} such that g ( x ) = c , {\displaystyle g(\mathbf {x} )=c,} the bordered Hessian is the Hessian of the Lagrange function Λ ( x , λ ) = f ( x ) + λ [ g ( x ) − c ] {\displaystyle \Lambda (\mathbf {x} ,\lambda )=f(\mathbf {x} )+\lambda [g(\mathbf {x} )-c]} : H ( Λ ) = [ ∂ 2 Λ ∂ λ 2 ∂ 2 Λ ∂ λ ∂ x ( ∂ 2 Λ ∂ λ ∂ x ) T ∂ 2 Λ ∂ x 2 ] = [ 0 ∂ g ∂ x 1 ∂ g ∂ x 2 ⋯ ∂ g ∂ x n ∂ g ∂ x 1 ∂ 2 Λ ∂ x 1 2 ∂ 2 Λ ∂ x 1 ∂ x 2 ⋯ ∂ 2 Λ ∂ x 1 ∂ x n ∂ g ∂ x 2 ∂ 2 Λ ∂ x 2 ∂ x 1 ∂ 2 Λ ∂ x 2 2 ⋯ ∂ 2 Λ ∂ x 2 ∂ x n ⋮ ⋮ ⋮ ⋱ ⋮ ∂ g ∂ x n ∂ 2 Λ ∂ x n ∂ x 1 ∂ 2 Λ ∂ x n ∂ x 2 ⋯ ∂ 2 Λ ∂ x n 2 ] = [ 0 ∂ g ∂ x ( ∂ g ∂ x ) T ∂ 2 Λ ∂ x 2 ] {\displaystyle \mathbf {H} (\Lambda )={\begin{bmatrix}{\dfrac {\partial ^{2}\Lambda }{\partial \lambda ^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\\\left({\dfrac {\partial ^{2}\Lambda }{\partial \lambda \partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial g}{\partial x_{2}}}&\cdots &{\dfrac {\partial g}{\partial x_{n}}}\\[2.2ex]{\dfrac {\partial g}{\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}^{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{1}\,\partial x_{n}}}\\[2.2ex]{\dfrac {\partial g}{\partial x_{2}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}^{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{2}\,\partial x_{n}}}\\[2.2ex]\vdots &\vdots &\vdots &\ddots &\vdots \\[2.2ex]{\dfrac {\partial g}{\partial x_{n}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{1}}}&{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}\,\partial x_{2}}}&\cdots &{\dfrac {\partial ^{2}\Lambda }{\partial x_{n}^{2}}}\end{bmatrix}}={\begin{bmatrix}0&{\dfrac {\partial g}{\partial \mathbf {x} }}\\\left({\dfrac {\partial g}{\partial \mathbf {x} }}\right)^{\mathsf {T}}&{\dfrac {\partial ^{2}\Lambda }{\partial \mathbf {x} ^{2}}}\end{bmatrix}}} If there are, say, m {\displaystyle m} constraints then the zero in the upper-left corner is an m × m {\displaystyle m\times m} block of zeros, and there are m {\displaystyle m} border rows at the top and m {\displaystyle m} border columns at the left. The above rules stating that extrema are characterized (among critical points with a non-singular Hessian) by a positive-definite or negative-definite Hessian cannot apply here since a bordered Hessian can neither be negative-definite nor positive-definite, as z T H z = 0 {\displaystyle \mathbf {z} ^{\mathsf {T}}\mathbf {H} \mathbf {z} =0} if z {\displaystyle \mathbf {z} } is any vector whose sole non-zero entry is its first. The second derivative test consists here of sign restrictions of the determinants of a certain set of n − m {\displaystyle n-m} submatrices of the bordered Hessian. Intuitively, the m {\displaystyle m} constraints can be thought of as reducing the problem to one with n − m {\displaystyle n-m} free variables. (For example, the maximization of f ( x 1 , x 2 , x 3 ) {\displaystyle f\left(x_{1},x_{2},x_{3}\right)} subject to the constraint x 1 + x 2 + x 3 = 1 {\displaystyle x_{1}+x_{2}+x_{3}=1} can be reduced to the maximization of f ( x 1 , x 2 , 1 − x 1 − x 2 ) {\displaystyle f\left(x_{1},x_{2},1-x_{1}-x_{2}\right)} without constraint.) Specifically, sign conditions are imposed on the sequence of leading principal minors (determinants of upper-left-justified sub-matrices) of the bordered Hessian, for which the first 2 m {\displaystyle 2m} leading principal minors are neglected, the smallest minor consisting of the truncated first 2 m + 1 {\displaystyle 2m+1} rows and columns, the next consisting of the truncated first 2 m + 2 {\displaystyle 2m+2} rows and columns, and so on, with the last being the entire bordered Hessian; if 2 m + 1 {\displaystyle 2m+1} is larger than n + m , {\displaystyle n+m,} then the smallest leading principal minor is the Hessian itself. There are thus n − m {\displaystyle n-m} minors to consider, each evaluated at the specific point being considered as a candidate maximum or minimum. A sufficient condition for a local maximum is that these minors alternate in sign with the smallest one having the sign of ( − 1 ) m + 1 . {\displaystyle (-1)^{m+1}.} A sufficient condition for a local minimum is that all of these minors have the sign of ( − 1 ) m . {\displaystyle (-1)^{m}.} (In the unconstrained case of m = 0 {\displaystyle m=0} these conditions coincide with the conditions for the unbordered Hessian to be negative definite or positive definite respectively). === Vector-valued functions === If f {\displaystyle f} is instead a vector field f : R n → R m , {\displaystyle \mathbf {f} :\mathbb {R} ^{n}\to \mathbb {R} ^{m},} that is, f ( x ) = ( f 1 ( x ) , f 2 ( x ) , … , f m ( x ) ) , {\displaystyle \mathbf {f} (\mathbf {x} )=\left(f_{1}(\mathbf {x} ),f_{2}(\mathbf {x} ),\ldots ,f_{m}(\mathbf {x} )\right),} then the collection of second partial derivatives is not a n × n {\displaystyle n\times n} matrix, but rather a third-order tensor. This can be thought of as an array of m {\displaystyle m} Hessian matrices, one for each component of f {\displaystyle \mathbf {f} } : H ( f ) = ( H ( f 1 ) , H ( f 2 ) , … , H ( f m ) ) . {\displaystyle \mathbf {H} (\mathbf {f} )=\left(\mathbf {H} (f_{1}),\mathbf {H} (f_{2}),\ldots ,\mathbf {H} (f_{m})\right).} This tensor degenerates to the usual Hessian matrix when m = 1. {\displaystyle m=1.} === Generalization to the complex case === In the context of several complex variables, the Hessian may be generalized. Suppose f : C n → C , {\displaystyle f\colon \mathbb {C} ^{n}\to \mathbb {C} ,} and write f ( z 1 , … , z n ) . {\displaystyle f\left(z_{1},\ldots ,z_{n}\right).} Identifying C n {\displaystyle {\mathbb {C} }^{n}} with R 2 n {\displaystyle {\mathbb {R} }^{2n}} , the normal "real" Hessian is a 2 n × 2 n {\displaystyle 2n\times 2n} matrix. As the object of study in several complex variables are holomorphic functions, that is, solutions to the n-dimensional Cauchy–Riemann conditions, we usually look on the part of the Hessian that contains information invariant under holomorphic changes of coordinates. This "part" is the so-called complex Hessian, which is the matrix ( ∂ 2 f ∂ z j ∂ z ¯ k ) j , k . {\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial {\bar {z}}_{k}}}\right)_{j,k}.} Note that if f {\displaystyle f} is holomorphic, then its complex Hessian matrix is identically zero, so the complex Hessian is used to study smooth but not holomorphic functions, see for example Levi pseudoconvexity. When dealing with holomorphic functions, we could consider the Hessian matrix ( ∂ 2 f ∂ z j ∂ z k ) j , k . {\displaystyle \left({\frac {\partial ^{2}f}{\partial z_{j}\partial z_{k}}}\right)_{j,k}.} === Generalizations to Riemannian manifolds === Let ( M , g ) {\displaystyle (M,g)} be a Riemannian manifold and ∇ {\displaystyle \nabla } its Levi-Civita connection. Let f : M → R {\displaystyle f:M\to \mathbb {R} } be a smooth function. Define the Hessian tensor by Hess ( f ) ∈ Γ ( T ∗ M ⊗ T ∗ M ) by Hess ( f ) := ∇ ∇ f = ∇ d f , {\displaystyle \operatorname {Hess} (f)\in \Gamma \left(T^{*}M\otimes T^{*}M\right)\quad {\text{ by }}\quad \operatorname {Hess} (f):=\nabla \nabla f=\nabla df,} where this takes advantage of the fact that the first covariant derivative of a function is the same as its ordinary differential. Choosing local coordinates { x i } {\displaystyle \left\{x^{i}\right\}} gives a local expression for the Hessian as Hess ( f ) = ∇ i ∂ j f d x i ⊗ d x j = ( ∂ 2 f ∂ x i ∂ x j − Γ i j k ∂ f ∂ x k ) d x i ⊗ d x j {\displaystyle \operatorname {Hess} (f)=\nabla _{i}\,\partial _{j}f\ dx^{i}\!\otimes \!dx^{j}=\left({\frac {\partial ^{2}f}{\partial x^{i}\partial x^{j}}}-\Gamma _{ij}^{k}{\frac {\partial f}{\partial x^{k}}}\right)dx^{i}\otimes dx^{j}} where Γ i j k {\displaystyle \Gamma _{ij}^{k}} are the Christoffel symbols of the connection. Other equivalent forms for the Hessian are given by Hess ( f ) ( X , Y ) = ⟨ ∇ X grad f , Y ⟩ and Hess ( f ) ( X , Y ) = X ( Y f ) − d f ( ∇ X Y ) . {\displaystyle \operatorname {Hess} (f)(X,Y)=\langle \nabla _{X}\operatorname {grad} f,Y\rangle \quad {\text{ and }}\quad \operatorname {Hess} (f)(X,Y)=X(Yf)-df(\nabla _{X}Y).} == See also == The determinant of the Hessian matrix is a covariant; see Invariant of a binary form Polarization identity, useful for rapid calculations involving Hessians. Jacobian matrix – Matrix of partial derivatives of a vector-valued functionPages displaying short descriptions of redirect targets Hessian equation == References == == Further reading == Lewis, David W. (1991). Matrix Theory. Singapore: World Scientific. ISBN 978-981-02-0689-5. Magnus, Jan R.; Neudecker, Heinz (1999). "The Second Differential". Matrix Differential Calculus : With Applications in Statistics and Econometrics (Revised ed.). New York: Wiley. pp. 99–115. ISBN 0-471-98633-X. == External links == "Hessian of a function", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Weisstein, Eric W. "Hessian". MathWorld.
|
Wikipedia:Hester Bijl#0
|
Hester Bijl is a Dutch mathematician, professor, and the rector magnificus of Leiden University. She is Professor of Numerical Mathematics at the Mathematical Institute at Leidein University. == Life and career == Bijl received a PhD from the Delft University of Technology, along with a master's degree from Leiden University. In 1999, she started working the Delft University of Technology in the aerodynamics department. In 2013, she became dean of the faculty of aerospace engineering at the Delft University of Technology. In 2021, Bijl was appointed rector magnificus of Leiden University, succeeding Carel Stolker. In 2024, she was reappointed. She is the first female rector of the university. Bijl is on the supervisory board of the Netherlands Organisation for Applied Scientific Research and on the board of the Leiden Bio Science Park. She has been a visiting researcher at the NASA Langley Research Center. == References ==
|
Wikipedia:Hidden algebra#0
|
Hidden algebra provides a formal semantics for use in the field of software engineering, especially for concurrent distributed object systems. It supports correctness proofs. Hidden algebra was studied by Joseph Goguen. It handles features of large software-based systems, including concurrency, distribution, nondeterminism, and local states. It also handled object-oriented features like classes, subclasses (inheritance), attributes, and methods. Hidden algebra generalizes process algebra and transition system approaches. == References == == External links == Hidden Algebra Tutorial
|
Wikipedia:Hideyuki Matsumura#0
|
Hideyuki Matsumura (松村 英之, 1930–1995) was a Japanese mathematician particularly known for his textbooks in commutative algebra. He received his Ph.D. in 1958 from Kyoto University under the advisory of mathematician Yasuo Akizuki. == References == == External links == The Oberwolfach Photo Collection has photos of him.
|
Wikipedia:Hierarchical closeness#0
|
Hierarchical closeness (HC) is a structural centrality measure used in network theory or graph theory. It is extended from closeness centrality to rank how centrally located a node is in a directed network. While the original closeness centrality of a directed network considers the most important node to be that with the least total distance from all other nodes, hierarchical closeness evaluates the most important node as the one which reaches the most nodes by the shortest paths. The hierarchical closeness explicitly includes information about the range of other nodes that can be affected by the given node. In a directed network G ( V , A ) {\displaystyle G(V,A)} where V {\displaystyle V} is the set of nodes and A {\displaystyle A} is the set of interactions, hierarchical closeness of a node i {\displaystyle i} ∈ V {\displaystyle V} called C h c ( i ) {\displaystyle C_{hc}(i)} was proposed by Tran and Kwon as follows: C h c ( i ) = N R ( i ) + C ( c l o − i ) ( i ) {\displaystyle C_{hc}(i)=N_{R}(i)+C_{(clo-i)}(i)} where: N R ( i ) ∈ [ 0 , | V | − 1 ] {\displaystyle N_{R}(i)\in [0,|V|-1]} is the reachability of a node i {\displaystyle i} defined by N R ( i ) = | { j ∈ V : ∃ {\displaystyle N_{R}(i)=|\{j\in V:\exists } a path from i {\displaystyle i} to j } | {\displaystyle j\}|} , and C c l o ( i ) {\displaystyle C_{clo}(i)} is the normalized form of original closeness (Sabidussi, 1966). It can use a variant definition of closeness as follows: C c l o − i ( i ) = 1 | V | − 1 ∑ j ∈ V ∖ { i } 1 d ( i , j ) {\displaystyle C_{clo-i}(i)={\frac {1}{|V|-1}}\sum _{j\in V\setminus \{i\}}{\frac {1}{d(i,j)}}} where d ( i , j ) {\displaystyle d(i,j)} is the distance of the shortest path, if any, from i {\displaystyle i} to j {\displaystyle j} ; otherwise, d ( i , j ) {\displaystyle d(i,j)} is specified as an infinite value. In the formula, N R ( i ) {\displaystyle N_{R}(i)} represents the number of nodes in V {\displaystyle V} that can be reachable from i {\displaystyle i} . It can also represent the hierarchical position of a node in a directed network. It notes that if N R ( i ) = 0 {\displaystyle N_{R}(i)=0} , then C h c ( i ) = 0 {\displaystyle C_{hc}(i)=0} because C ( c l o − i ) ( i ) {\displaystyle C_{(clo-i)}(i)} is 0 {\displaystyle 0} . In cases where N R ( i ) > 0 {\displaystyle N_{R}(i)>0} , the reachability is a dominant factor because N R ( i ) ≥ 1 {\displaystyle N_{R}(i)\geq 1} but C ( c l o − i ) ( i ) < 1 {\displaystyle C_{(clo-i)}(i)<1} . In other words, the first term indicates the level of the global hierarchy and the second term presents the level of the local centrality. == Application == Hierarchical closeness can be used in biological networks to rank the risk of genes to carry diseases.[1] == References ==
|
Wikipedia:Hieronymus Georg Zeuthen#0
|
Hieronymus Georg Zeuthen (15 February 1839 – 6 January 1920) was a Danish mathematician. He is known for work on the enumerative geometry of conic sections, algebraic surfaces, and history of mathematics. == Biography == Zeuthen was born in Grimstrup near Varde where his father was a minister. In 1849, his father moved to a church in Sorø where Zeuthen began his secondary schooling. In 1857 he entered the University of Copenhagen to study mathematics and graduated with a master's degree in 1862. Following this he earned a scholarship to study abroad, and decided to visit Paris where he studied geometry with Michel Chasles. After returning to Copenhagen, Zeuthen submitted his doctoral dissertation on a new method to determine the characteristics of conic systems in 1865. Enumerative geometry remained his focus up until 1875. In 1871 he was appointed as an extraordinary professor at the University of Copenhagen, as well as becoming an editor of Matematisk Tidsskrift, a position he held for 18 years. For 39 years he served as secretary of the Royal Danish Academy of Sciences and Letters, during which he also lectured at the Polytechnic Institute. In 1886, he was promoted to ordinary professor at the University of Copenhagen, where he twice served as rector. After 1875 Zeuthen began to make contributions in other areas such as mechanics and algebraic geometry, as well as being recognised as an expert on the history of medieval and Greek mathematics. He wrote 40 papers and books on the history of mathematics, which covered many topics and several periods. He was an invited speaker at the International Congress of Mathematicians in 1897 at Zurich, in 1904 at Heidelberg, and in 1908 at Rome. == See also == Zeuthen–Segre invariant Ingeborg Hammer-Jensen, notable student and historian of science == Publications == Abriß einer elementar-geometrischen Kegelschnittlehre. Teubner 1882. Die Lehre von den Kegelschnitten im Altertum. Kopenhagen 1886 (Danish version 1885 in Forh.Vid.Selskab). Geschichte der Mathematik im Altertum und Mittelalter. Kopenhagen 1896 (Danish version 1893 publ. by Verlag A.F.Hoest). Histoire des Mathématiques dans l'Antiquité et le Moyen Age. Paris, Gauthier-Villars, 1902. Geschichte der Mathematik im XVI. und XVII. Jahrhundert. Teubner 1903, and as Heft 17 of Abhandlungen zur Geschichte der mathematischen Wissenschaften (ed. Moritz Cantor). The Danish version was published 1903 in Copenhagen. Die Mathematik im Altertum und im Mittelalter. Kopenhagen 1912. Lehrbuch der abzählenden Methoden der Geometrie. Teubner 1914. Hvorledes Mathematiken i tiden fra Platon til Euklid blev rationel Videnskab. Avec un résumé en francais. Forh.Dansk Vid.Selskab 1917, pp.199-369. == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Hieronymus Georg Zeuthen", MacTutor History of Mathematics Archive, University of St Andrews Hieronymus Georg Zeuthen at the Mathematics Genealogy Project Juel, Christian (1920). "Hieronymus Georg Zeuthen". Matematisk Tidsskrift A: 1–9. (in Danish)
|
Wikipedia:High-dimensional model representation#0
|
High-dimensional model representation is a finite expansion for a given multivariable function. The expansion was first described by Ilya M. Sobol as f ( x ) = f 0 + ∑ i = 1 n f i ( x i ) + ∑ i , j = 1 i < j n f i j ( x i , x j ) + ⋯ + f 12 … n ( x 1 , … , x n ) . {\displaystyle f(\mathbf {x} )=f_{0}+\sum _{i=1}^{n}f_{i}(x_{i})+\sum _{i,j=1 \atop i<j}^{n}f_{ij}(x_{i},x_{j})+\cdots +f_{12\ldots n}(x_{1},\ldots ,x_{n}).} The method, used to determine the right hand side functions, is given in Sobol's paper. A review can be found here: High Dimensional Model Representation (HDMR): Concepts and Applications. The underlying logic behind the HDMR is to express all variable interactions in a system in a hierarchical order. For instance f 0 {\displaystyle f_{0}} represents the mean response of the model f {\displaystyle f} . It can be considered as measuring what is left from the model after stripping down all variable effects. The uni-variate functions f i ( x i ) {\displaystyle f_{i}(x_{i})} , however represents the "individual" contributions of the variables. For instance, f 1 ( x 1 ) {\displaystyle f_{1}(x_{1})} is the portion of the model that can be controlled only by the variable x 1 {\displaystyle x_{1}} . For this reason, there can not be any constant in f 1 ( x 1 ) {\displaystyle f_{1}(x_{1})} because all constants are expressed in f 0 {\displaystyle f_{0}} . Going further into higher interactions,the next stop is bivariate functions f i j ( x i , x j ) {\displaystyle f_{ij}(x_{i},x_{j})} which represents the cooperative effect of variables x i {\displaystyle x_{i}} and x j {\displaystyle x_{j}} together. Similar logic applies here: the bivariate functions do not contain univarite functions nor constants as it violates the construction logic of HDMR. As we go into higher interactions, the number of interactions are increasing and at last we reach the residual term f 12 n ( x 1 , … , x n ) {\displaystyle f_{12n}(x_{1},\ldots ,x_{n})} representing the contribution only if all variable act together. == HDMR as an Approximation == The hierarchical representation model of HDMR brings an advantage if one needs to replace an existing model with a simpler one usually containing only univariate or bivariate terms. If the target model does not contain higher level of variable interactions, this approach can yield good approximations with the additional advantage of providing a clearer view of variable interactions. == See also == Variance-based sensitivity analysis Volterra series == References ==
|
Wikipedia:Higher-order compact finite difference scheme#0
|
High-order compact finite difference schemes are used for solving third-order differential equations created during the study of obstacle boundary value problems. They have been shown to be highly accurate and efficient. They are constructed by modifying the second-order scheme that was developed by Noor and Al-Said in 2002. The convergence rate of the high-order compact scheme is third order, the second-order scheme is fourth order. Differential equations are essential tools in mathematical modelling. Most physical systems are described in terms of mathematical models that include convective and diffusive transport of some variables. Finite difference methods are amongst the most popular methods that have been applied most frequently in solving such differential equations. A finite difference scheme is compact in the sense that the discretised formula comprises at most nine point stencils which includes a node in the middle about which differences are taken. In addition, greater order of accuracy (more than two) justifies the terminology 'higher-order compact finite difference scheme' (HOC). This can be achieved in several ways. The higher-order compact scheme considered here is by using the original differential equation to substitute for the leading truncation error terms in the finite difference equation. Overall, the scheme is found to be robust, efficient and accurate for most computational fluid dynamics (CFD) applications discussed here further. The simplest problem for the validation of the numerical algorithms is the Lid Driven cavity problem. Computed results in form of tables, graphs and figures for a fluid with Prandtl number = 0.71 with Rayleigh number (Ra) ranging from 103 to 107 are available in the literature. The efficacy of the scheme is proved when it very clearly captures the secondary and tertiary vortices at the sides of the cavity at high values of Ra. Another milestone was the development of these schemes for solving two dimensional steady/unsteady convection diffusion equations. A comprehensive study of flow past an impulsively started circular cylinder was made. The problem of flow past a circular cylinder has continued to generate tremendous interest amongst researchers working in CFD mainly because it displays almost all the fluid mechanical phenomena for incompressible, viscous flows in the simplest of geometrical settings. It was able to analyze and visualize the flow patterns more accurately for Reynold's number (Re) ranging from 10 to 9500 compared to the existing numerical results. This was followed by its extension to rotating counterpart of the cylinder surface for Re ranging from 200 to 1000. More complex phenomenon that involves a circular cylinder undergoing rotational oscillations while translating in a fluid is studied for Re as high as 500. Another benchmark in the history is its extension to multiphase flow phenomena. Natural processes such as gas bubble in oil, ice melting, wet steam are observed everywhere in nature. Such processes also play an important role with the practical applications in the area of biology, medicine, environmental remediation. The scheme has been successively implemented to solve one and two dimensional elliptic and parabolic equations with discontinuous coefficients and singular source terms. These type of problems hold importance numerically because they usually lead to non-smooth or discontinuous solutions across the interfaces. Expansion of this idea from fixed to moving interfaces with both regular and irregular geometries is currently going on. == References ==
|
Wikipedia:Higher-order operad#0
|
In algebra, a higher-order operad is a higher-dimensional generalization of an operad. == See also == Opetope == References == Heuts, Gijs; Hinich, Vladimir; Moerdijk, Ieke (2016). "On the equivalence between Lurie's model and the dendroidal model for infinity-operads". Advances in Mathematics. 302: 869–1043. arXiv:1305.3658. doi:10.1016/j.aim.2016.07.021. S2CID 119254588. == External links == https://ncatlab.org/nlab/show/(infinity%2C1)-operad
|
Wikipedia:Higuchi dimension#0
|
In fractal geometry, the Higuchi dimension (or Higuchi fractal dimension (HFD)) is an approximate value for the box-counting dimension of the graph of a real-valued function or time series. This value is obtained via an algorithmic approximation so one also talks about the Higuchi method. It has many applications in science and engineering and has been applied to subjects like characterizing primary waves in seismograms, clinical neurophysiology and analyzing changes in the electroencephalogram in Alzheimer's disease. == Formulation of the method == The original formulation of the method is due to T. Higuchi. Given a time series X : { 1 , … , N } → R {\displaystyle X:\{1,\dots ,N\}\to \mathbb {R} } consisting of N {\displaystyle N} data points and a parameter k m a x ≥ 2 {\displaystyle k_{\mathrm {max} }\geq 2} the Higuchi Fractal dimension (HFD) of X {\displaystyle X} is calculated in the following way: For each k ∈ { 1 , … , k m a x } {\displaystyle k\in \{1,\dots ,k_{\mathrm {max} }}\} and m ∈ { 1 , … , k } {\displaystyle m\in \{1,\dots ,k}\} define the length L m ( k ) {\displaystyle L_{m}(k)} by L m ( k ) = N − 1 ⌊ N − m k ⌋ k 2 ∑ i = 1 ⌊ N − m k ⌋ | X N ( m + i k ) − X N ( m + ( i − 1 ) k ) | . {\displaystyle L_{m}(k)={\frac {N-1}{\lfloor {\frac {N-m}{k}}\rfloor k^{2}}}\sum _{i=1}^{\lfloor {\frac {N-m}{k}}\rfloor }|X_{N}(m+ik)-X_{N}(m+(i-1)k)|.} The length L ( k ) {\displaystyle L(k)} is defined by the average value of the k {\displaystyle k} lengths L 1 ( k ) , … , L k ( k ) {\displaystyle L_{1}(k),\dots ,L_{k}(k)} , L ( k ) = 1 k ∑ m = 1 k L m ( k ) . {\displaystyle L(k)={\frac {1}{k}}\sum _{m=1}^{k}L_{m}(k).} The slope of the best-fitting linear function through the data points { ( log 1 k , log L ( k ) ) } {\displaystyle \left\{\left(\log {\frac {1}{k}},\log L(k)\right)\right\}} is defined to be the Higuchi fractal dimension of the time-series X {\displaystyle X} . == Application to functions == For a real-valued function f : [ 0 , 1 ] → R {\displaystyle f:[0,1]\to \mathbb {R} } one can partition the unit interval [ 0 , 1 ] {\displaystyle [0,1]} into N {\displaystyle N} equidistantly intervals [ t j , t j + 1 ) {\displaystyle [t_{j},t_{j+1})} and apply the Higuchi algorithm to the times series X ( j ) = f ( t j ) {\displaystyle X(j)=f(t_{j})} . This results into the Higuchi fractal dimension of the function f {\displaystyle f} . It was shown that in this case the Higuchi method yields an approximation for the box-counting dimension of the graph of f {\displaystyle f} as it follows a geometrical approach (see Liehr & Massopust 2020). == Robustness and stability == Applications to fractional Brownian functions and the Weierstrass function reveal that the Higuchi fractal dimension can be close to the box-dimension. On the other hand, the method can be unstable in the case where the data X ( 1 ) , … , X ( N ) {\displaystyle X(1),\dots ,X(N)} are periodic or if subsets of it lie on a horizontal line (see Liehr & Massopust 2020). == References ==
|
Wikipedia:Hilary Priestley#0
|
Hilary Ann Priestley is a British mathematician. She is a professor at the University of Oxford and a Fellow of St Anne's College, Oxford, where she has been Tutor in Mathematics since 1972. Hilary Priestley introduced ordered separable topological spaces; such topological spaces are now usually called Priestley spaces in her honour. The term "Priestley duality" is also used for her application of these spaces in the representation theory of distributive lattices. == Books == Priestley, Hilary A. (2003). Introduction to Complex Analysis (2nd ed.). Oxford University Press. ISBN 978-0-19-852562-2. Davey, Brian A.; Priestley, Hilary A. (2002). Introduction to Lattices and Order (2nd ed.). Cambridge University Press. ISBN 9780521784511. Priestley, Hilary A. (1997). Introduction to Integration. Oxford University Press. ISBN 978-0-19-850123-7. == References == == External links == Hilary Priestley home page Professor Hilary Priestley profile Archived 14 July 2014 at the Wayback Machine at the Mathematical Institute, University of Oxford Professor Hilary Ann Priestley profile at St Anne's College, Oxford Hilary Priestley on ResearchGate
|
Wikipedia:Hilary Shuard#0
|
Hilary Bertha Shuard CBE (14 November 1928 – 24 December 1992) was an expert on the teaching of mathematics in primary schools. She was a member of the Cockcroft Committee, and Deputy Principal of Homerton College, Cambridge for twenty years. == Life == Shuard was born in Chester on 14 November 1928. She was educated in mathematics at Oxford and Cambridge Universities, and also gained blues in hockey and cricket at Cambridge. Between 1953 and 1959, she taught at Christ's Hospital Hertford before joining the staff of the mathematics department at Homerton College Cambridge, where she remained until 1986. She was Deputy Principal of the college from 1966 until her retirement. Shuard was an "internationally known expert on mathematics in primary schools.” Shuard was president of the Mathematical Association for 1985–1986. From 1979 to 1989 she was President of the Cambridgeshire Women's Hockey Association. In 1970, Shuard published Primary Mathematics Today with Elizabeth Williams, which has since become a standard reference for primary school mathematics teachers. Shuard was a proponent of calculators in school, and was a member of the Cockcroft Committee, a government inquiry into mathematics education in schools. The Cockcroft Report was published in 1982. After this time Shuard set up the Prime project (Primary Initiatives in Mathematics Education), which was part of the National Curriculum Council, and included work on the acceptance of calculators in the primary curriculum, something for which Shuard was a strong advocate. Shuard died in Cambridge on 24 December 1992, aged 64. == Recognition == Shuard was awarded a CBE in 1987. == References ==
|
Wikipedia:Hilbert–Burch theorem#0
|
In mathematics, the Hilbert–Burch theorem describes the structure of some free resolutions of a quotient of a local or graded ring in the case that the quotient has projective dimension 2. Hilbert (1890) proved a version of this theorem for polynomial rings, and Burch (1968, p. 944) proved a more general version. Several other authors later rediscovered and published variations of this theorem. Eisenbud (1995, theorem 20.15) gives a statement and proof. == Statement == If R is a local ring with an ideal I and 0 → R m → f R n → R → R / I → 0 {\displaystyle 0\rightarrow R^{m}{\stackrel {f}{\rightarrow }}R^{n}\rightarrow R\rightarrow R/I\rightarrow 0} is a free resolution of the R-module R/I, then m = n – 1 and the ideal I is aJ where a is a regular element of R and J, a depth-2 ideal, is the first Fitting ideal Fitt 1 I {\displaystyle \operatorname {Fitt} _{1}I} of I, i.e., the ideal generated by the determinants of the minors of size m of the matrix of f. == References == Burch, Lindsay (1968), "On ideals of finite homological dimension in local rings", Proc. Cambridge Philos. Soc., 64 (4): 941–948, doi:10.1017/S0305004100043620, ISSN 0008-1981, MR 0229634, S2CID 123231429, Zbl 0172.32302 Eisenbud, David (1995), Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Berlin, New York: Springer-Verlag, ISBN 3-540-94268-8, MR 1322960, Zbl 0819.13001 Eisenbud, David (2005), The Geometry of Syzygies. A second course in commutative algebra and algebraic geometry, Graduate Texts in Mathematics, vol. 229, New York, NY: Springer-Verlag, ISBN 0-387-22215-4, Zbl 1066.14001 Hilbert, David (1890), "Ueber die Theorie der algebraischen Formen", Mathematische Annalen (in German), 36 (4): 473–534, doi:10.1007/BF01208503, ISSN 0025-5831, JFM 22.0133.01, S2CID 179177713
|
Wikipedia:Hilbert–Kunz function#0
|
In algebra, the Hilbert–Kunz function of a local ring (R, m) of prime characteristic p is the function f ( q ) = length R ( R / m [ q ] ) {\displaystyle f(q)=\operatorname {length} _{R}(R/m^{[q]})} where q is a power of p and m[q] is the ideal generated by the q-th powers of elements of the maximal ideal m. The notion was introduced by Ernst Kunz, who used it to characterize a regular ring as a Noetherian ring in which the Frobenius morphism is flat. If d is the dimension of the local ring, Monsky showed that f(q)/(q^d) is c+O(1/q) for some real constant c. This constant, the "Hilbert-Kunz multiplicity", is greater than or equal to 1. Watanabe and Yoshida strengthened some of Kunz's results, showing that in the unmixed case, the ring is regular precisely when c=1. Hilbert–Kunz functions and multiplicities have been studied for their own sake. Brenner and Trivedi have treated local rings coming from the homogeneous co-ordinate rings of smooth projective curves, using techniques from algebraic geometry. Han, Monsky, and Teixeira have treated diagonal hypersurfaces and various related hypersurfaces. But there is no known technique for determining the Hilbert–Kunz function or c in general. In particular the question of whether c is always rational wasn't settled until recently (by Brenner—it needn't be, and indeed can be transcendental). Hochster and Huneke related Hilbert-Kunz multiplicities to "tight closure" and Brenner and Monsky used Hilbert–Kunz functions to show that localization need not preserve tight closure. The question of how c behaves as the characteristic goes to infinity (say for a hypersurface defined by a polynomial with integer coefficients) has also received attention; once again open questions abound. A comprehensive overview is to be found in Craig Huneke's article "Hilbert-Kunz multiplicities and the F-signature" arXiv:1409.0467. This article is also found on pages 485-525 of the Springer volume "Commutative Algebra: Expository Papers Dedicated to David Eisenbud on the Occasion of His 65th Birthday", edited by Irena Peeva. == References == == Bibliography == E. Kunz, "On noetherian rings of characteristic p," Am. J. Math, 98, (1976), 999–1013. 1 Edward Miller, Lance; Swanson, Irena (2012). "Hilbert-Kunz functions of 2 x 2 determinantal rings". arXiv:1206.1015 [math.AC].
|
Wikipedia:Hilbert–Poincaré series#0
|
In mathematics, and in particular in the field of algebra, a Hilbert–Poincaré series (also known under the name Hilbert series), named after David Hilbert and Henri Poincaré, is an adaptation of the notion of dimension to the context of graded algebraic structures (where the dimension of the entire structure is often infinite). It is a formal power series in one indeterminate, say t {\displaystyle t} , where the coefficient of t n {\displaystyle t^{n}} gives the dimension (or rank) of the sub-structure of elements homogeneous of degree n {\displaystyle n} . It is closely related to the Hilbert polynomial in cases when the latter exists; however, the Hilbert–Poincaré series describes the rank in every degree, while the Hilbert polynomial describes it only in all but finitely many degrees, and therefore provides less information. In particular the Hilbert–Poincaré series cannot be deduced from the Hilbert polynomial even if the latter exists. In good cases, the Hilbert–Poincaré series can be expressed as a rational function of its argument t {\displaystyle t} . == Definition == Let K be a field, and let V = ⨁ i ∈ N V i {\displaystyle V=\textstyle \bigoplus _{i\in \mathbb {N} }V_{i}} be an N {\displaystyle \mathbb {N} } -graded vector space over K, where each subspace V i {\displaystyle V_{i}} of vectors of degree i is finite-dimensional. Then the Hilbert–Poincaré series of V is the formal power series ∑ i ∈ N dim K ( V i ) t i . {\displaystyle \sum _{i\in \mathbb {N} }\dim _{K}(V_{i})t^{i}.} A similar definition can be given for an N {\displaystyle \mathbb {N} } -graded R-module over any commutative ring R in which each submodule of elements homogeneous of a fixed degree n is free of finite rank; it suffices to replace the dimension by the rank. Often the graded vector space or module of which the Hilbert–Poincaré series is considered has additional structure, for instance, that of a ring, but the Hilbert–Poincaré series is independent of the multiplicative or other structure. Example: Since there are ( n + k k ) {\displaystyle \textstyle {\binom {n+k}{k}}} monomials of degree k in variables X 0 , … , X n {\displaystyle X_{0},\dots ,X_{n}} (by induction, say), one can deduce that the sum of the Hilbert–Poincaré series of K [ X 0 , … , X n ] {\displaystyle K[X_{0},\dots ,X_{n}]} is the rational function 1 / ( 1 − t ) n + 1 {\displaystyle 1/(1-t)^{n+1}} . == Hilbert–Serre theorem == Suppose M is a finitely generated graded module over A [ x 1 , … , x n ] , deg x i = d i {\displaystyle A[x_{1},\dots ,x_{n}],\deg x_{i}=d_{i}} with an Artinian ring (e.g., a field) A. Then the Poincaré series of M is a polynomial with integral coefficients divided by ∏ ( 1 − t d i ) {\displaystyle \prod (1-t^{d_{i}})} . The standard proof today is an induction on n. Hilbert's original proof made a use of Hilbert's syzygy theorem (a projective resolution of M), which gives more homological information. Here is a proof by induction on the number n of indeterminates. If n = 0 {\displaystyle n=0} , then, since M has finite length, M k = 0 {\displaystyle M_{k}=0} if k is large enough. Next, suppose the theorem is true for n − 1 {\displaystyle n-1} and consider the exact sequence of graded modules (exact degree-wise), with the notation N ( l ) k = N k + l {\displaystyle N(l)_{k}=N_{k+l}} , 0 → K ( − d n ) → M ( − d n ) → x n M → C → 0 {\displaystyle 0\to K(-d_{n})\to M(-d_{n}){\overset {x_{n}}{\to }}M\to C\to 0} . Since the length is additive, Poincaré series are also additive. Hence, we have: P ( M , t ) = − P ( K ( − d n ) , t ) + P ( M ( − d n ) , t ) − P ( C , t ) {\displaystyle P(M,t)=-P(K(-d_{n}),t)+P(M(-d_{n}),t)-P(C,t)} . We can write P ( M ( − d n ) , t ) = t d n P ( M , t ) {\displaystyle P(M(-d_{n}),t)=t^{d_{n}}P(M,t)} . Since K is killed by x n {\displaystyle x_{n}} , we can regard it as a graded module over A [ x 0 , … , x n − 1 ] {\displaystyle A[x_{0},\dots ,x_{n-1}]} ; the same is true for C. The theorem thus now follows from the inductive hypothesis. == Chain complex == An example of graded vector space is associated to a chain complex, or cochain complex C of vector spaces; the latter takes the form 0 → C 0 ⟶ d 0 C 1 ⟶ d 1 C 2 ⟶ d 2 ⋯ ⟶ d n − 1 C n ⟶ 0. {\displaystyle 0\to C^{0}{\stackrel {d_{0}}{\longrightarrow }}C^{1}{\stackrel {d_{1}}{\longrightarrow }}C^{2}{\stackrel {d_{2}}{\longrightarrow }}\cdots {\stackrel {d_{n-1}}{\longrightarrow }}C^{n}\longrightarrow 0.} The Hilbert–Poincaré series (here often called the Poincaré polynomial) of the graded vector space ⨁ i C i {\displaystyle \bigoplus _{i}C^{i}} for this complex is P C ( t ) = ∑ j = 0 n dim ( C j ) t j . {\displaystyle P_{C}(t)=\sum _{j=0}^{n}\dim(C^{j})t^{j}.} The Hilbert–Poincaré polynomial of the cohomology, with cohomology spaces Hj = Hj(C), is P H ( t ) = ∑ j = 0 n dim ( H j ) t j . {\displaystyle P_{H}(t)=\sum _{j=0}^{n}\dim(H^{j})t^{j}.} A famous relation between the two is that there is a polynomial Q ( t ) {\displaystyle Q(t)} with non-negative coefficients, such that P C ( t ) − P H ( t ) = ( 1 + t ) Q ( t ) . {\displaystyle P_{C}(t)-P_{H}(t)=(1+t)Q(t).} == References == Atiyah, Michael Francis; Macdonald, I.G. (1969). Introduction to Commutative Algebra. Westview Press. ISBN 978-0-201-40751-8.
|
Wikipedia:Hilda Assiyatun#0
|
Hilda Assiyatun is an Indonesian mathematician, a professor in the Faculty of Mathematics and Natural Sciences of the Bandung Institute of Technology, the vice president for education and teaching of the Indonesian Mathematical Society (IndoMS), and the president of the Indonesian Combinatorial Society (InaCombS). Her research has concerned graph theory, with topics including graph Ramsey theory, the metric dimension of graphs, and graph coloring. == Education and career == Assiyatun received the equivalent of a bachelor's and master's degree from the Bandung Institute of Technology in 1992 and 1995 respectively. She completed a Ph.D. in Australia in 2002, at the University of Melbourne, with the dissertation Large Subgraphs of Regular Graphs supervised by Nick Wormald. Through her collaborations with Wormald, she has Erdős number 2. At the Bandung Institute of Technology, she was promoted to full professor in Ramsey theory in 2023, and gave her inaugural lecture as professor in 2024. == Society leadership == She has been vice president for education and teaching of the Indonesian Mathematical Society since 2020. After serving as vice president of the Indonesian Combinatorial Society, she was elected president in 2022, and is president for the 2024–2026 term. == References == == External links == Hilda Assiyatun publications indexed by Google Scholar
|
Wikipedia:Hilda Lyon#0
|
Hilda Margaret Lyon, MA, MSc, AFRAeS (31 May 1896 – 2 December 1946) was a British engineer who invented the "Lyon Shape", a streamlined design used for airships and submarines. == Early life and education == Lyon was born in 1896 in Market Weighton, Yorkshire. She was the youngest daughter of Thomas and Margaret Lyon (née Green); her father was a grocer. Hilda Lyon attended Beverley High School and then in 1915 went to Newnham College, Cambridge, from which she obtained a MA in mathematics. == Career in aviation == After graduating, Lyon took an Air Ministry course in aeroplane stress-analysis and then obtained a job as a technical assistant. She saw no prospect of promotion or more responsibility "for a woman mathematician" in this job, so she and her sister quit their jobs and went to Switzerland for six weeks. In 1918, Lyon worked as an Aircraft Technical Assistant for Siddeley-Deasy. She moved to George Parnall & Co. in 1920. Around 1922, Lyon was admitted as an Associate Fellow of the Royal Aeronautical Society. From 1925 onwards, she was a member of technical staff at the Royal Airship Works in Cardington, helping to develop the R101 rigid airship through her work on aerodynamics. In 1930, Lyon was awarded the R38 Memorial Prize by the Royal Aeronautical Society for her paper "The Strength of Transverse Frames of Rigid Airships". It was the first time that any prize of the society had been won by a woman. Lyon's work involved travel to America, Canada, and Germany. In 1930 she went to America on a Mary Ewart Travelling Scholarship and took up studies at Massachusetts Institute of Technology, where she was first permitted to use a wind tunnel. In 1932, she submitted a thesis on "The Effect of Turbulence on the Drag of Airship Models" to obtain her MSc. After submitting her thesis, Lyon went to Göttingen in Germany, and conducted research at the Kaiser Wilhelm Gesellschaft für Strömungsforschung with Ludwig Prandtl. After her return to Britain, Lyon spent time at home as a carer, keeping up with her research at the same time by using the libraries of the University of Hull and the University of Leeds and by visiting the National Physical Laboratory and the Royal Aircraft Establishment. During this time, she worked on aeroelastic flutter and elastic blades. From 1937, Lyon returned to full-time aerodynamic research as a Principal Scientific Officer at the Royal Aircraft Establishment in Farnborough. She first worked in wind tunnels on boundary layer suction, then joined the Stability Section. She later became head of this section, and also served on the Aeronautical Research Council. Lyon died on 2 December 1946 following an operation. After her death, her research and the "Lyon Shape" which she devised were incorporated into the American submarine USS Albacore, which had the prototype streamlined hull form for almost all subsequent US submarines. == Commemoration == Lyon's biography was published by the Oxford Dictionary of National Biography on 9 May 2019 as part of their support for the Women's Engineering Society's centenary, and a new book has been published about Hilda's life and work. A blue plaque was installed in Lyon's honour at the site of her father's grocers’ shop (now a Morrisons Daily) in Market Weighton in the East Riding of Yorkshire on Thursday, 27 June 2019. In 2021, St Mary's Church in Beverley, East Yorkshire announced their intention of installing a stone carving of Hilda Lyon as part of a programme of celebrating women in the restoration of the stonework of the medieval church. The other eight figures include fellow engineer and WES member Amy Johnson, Mary Wollstonecraft, Mary Seacole, Marie Curie, Rosalind Franklin, Helen Sharman and Ada Lovelace. == References == == External links == The Strength of Transverse Frames of Rigid Airships - R38 prize paper. The Effect of Turbulence on the Drag of Airship Models - MSc thesis.
|
Wikipedia:Hillel Furstenberg#0
|
Hillel "Harry" Furstenberg (Hebrew: הלל (הארי) פורסטנברג; born September 29, 1935) is a German-born American-Israeli mathematician and professor emeritus at the Hebrew University of Jerusalem. He is a member of the Israel Academy of Sciences and Humanities and U.S. National Academy of Sciences and a laureate of the Abel Prize and the Wolf Prize in Mathematics. He is known for his application of probability theory and ergodic theory methods to other areas of mathematics, including number theory and Lie groups. == Biography == Furstenberg was born to German Jews in Nazi Germany, in 1935 (originally named "Fürstenberg"). In 1939, shortly after Kristallnacht, his family escaped to the United States and settled in the Washington Heights neighborhood of New York City, escaping the Holocaust. He attended Marsha Stern Talmudical Academy and then Yeshiva University, where he concluded his BA and MSc studies at the age of 20 in 1955. Furstenberg published several papers as an undergraduate, including "Note on one type of indeterminate form" (1953) and "On the infinitude of primes" (1955). Both appeared in the American Mathematical Monthly, the latter provided a topological proof of Euclid's famous theorem that there are infinitely many primes. == Academic career == Furstenberg pursued his doctorate at Princeton University under the supervision of Salomon Bochner. In 1958 he received his PhD for his thesis, Prediction Theory. From 1959–1960, Furstenberg served as the C. L. E. Moore instructor at the Massachusetts Institute of Technology. Furstenberg got his first job as an assistant professor in 1961 at the University of Minnesota. Furstenberg was promoted to full professor at Minnesota but moved to Israel in 1965 to join at Hebrew University's Einstein Institute of Mathematics. He retired from Hebrew University in 2003. Furstenberg serves as an Advisory Committee member of The Center for Advanced Studies in Mathematics at Ben Gurion University of the Negev. In 2003, Hebrew University and Ben-Gurion University held a joint conference to celebrate Furstenberg's retirement. The four-day Conference on Probability in Mathematics was subtitled Furstenfest 2003 and included four days of lectures. In 1993, Furstenberg won the Israel Prize and in 2007, the Wolf Prize in mathematics. He is a member of the Israel Academy of Sciences and Humanities (elected 1974), the American Academy of Arts and Sciences (international honorary member since 1995), and the U.S. National Academy of Sciences (elected 1989). Furstenberg has taught generations of students, including Alexander Lubotzky, Yuval Peres, Tamar Ziegler, Shahar Mozes, and Vitaly Bergelson. == Research accomplishments == Furstenberg gained attention at an early stage in his career for producing an innovative topological proof of the infinitude of prime numbers in 1955. In a series of articles beginning in 1963 with A Poisson Formula for Semi-Simple Lie Groups, he continued to establish himself as a ground-breaking thinker. His work showing that the behavior of random walks on a group is intricately related to the structure of the group—which led to what is now called the Furstenberg boundary—has been hugely influential in the study of lattices and Lie groups. In his 1967 paper, Disjointness in ergodic theory, minimal sets, and a problem in Diophantine approximation, Furstenberg introduced the notion of 'disjointness,' a notion in ergodic systems that is analogous to coprimality for integers. The notion turned out to have applications in areas such as number theory, fractals, signal processing and electrical engineering. In 1977, he gave an ergodic theory reformulation, and subsequently proof, of Szemerédi's theorem. This is described in his 1977 paper, Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions. Furstenberg used methods from ergodic theory to prove a celebrated result by Endre Szemerédi, which states that any subset of integers with positive upper density contains arbitrarily large arithmetic progressions. His insights then led to later important results, such as the proof by Ben Green and Terence Tao that the sequence of prime numbers includes arbitrary large arithmetic progressions. He proved unique ergodicity of horocycle flows on compact hyperbolic Riemann surfaces in the early 1970s. The Furstenberg boundary and Furstenberg compactification of a locally symmetric space are named after him, as is the Furstenberg–Sárközy theorem in additive number theory. == Personal life == In 1958, Furstenberg married Rochelle (née) Cohen, a journalist and literary critic. Together they have five children and sixteen grandchildren. == Awards == 1977 – Rothschild Prize in Mathematics. 1993 – Furstenberg received the Israel Prize, for exact sciences. 1993 – Furstenberg received the Harvey Prize from Technion. 2006/7 – He received the Wolf Prize in Mathematics. 2006 – He delivered the Paul Turán Memorial Lectures. 2020 – He received the Abel Prize with Gregory Margulis "for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics". == Selected publications == Furstenberg, Harry, Stationary processes and prediction theory, Princeton, N.J., Princeton University Press, 1960. LCCN 60-12226 Furstenberg, Harry (March 1963). "A Poisson Formula for Semi-Simple Lie Groups". Annals of Mathematics. Second Series. 77 (2): 335–386. doi:10.2307/1970220. JSTOR 1970220. Furstenberg, Harry (1967). "Disjointness in ergodic theory, minimal sets, and a problem in diophantine approximation". Mathematical Systems Theory. 1: 1–49. doi:10.1007/BF01692494. S2CID 206801948. Furstenberg, Harry (1977). "Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions". Journal d'Analyse Mathématique. 31: 204–256. doi:10.1007/BF02813304. MR 0498471. S2CID 120917478. Furstenberg, Harry, Recurrence in ergodic theory and combinatorial number theory, Princeton, N.J., Princeton Univ. Press, 1981. Furstenberg, Harry (July 14, 2014). 2014 pbk edition. Princeton University Press. ISBN 978-0-691-08269-1. Furstenberg, Hillel (August 8, 2014). Ergodic Theory and Fractal Geometry. American Mathematical Society. ISBN 978-1-4704-1034-6. LCCN 2014010556. == See also == List of Israel Prize recipients == References == == External links == O'Connor, John J.; Robertson, Edmund F., "Hillel Furstenberg", MacTutor History of Mathematics Archive, University of St Andrews Mathematics Genealogy page Press release Archived March 3, 2016, at the Wayback Machine Israel Academy of Sciences and Humanities (Hebrew)
|
Wikipedia:Hiroshi Fujita#0
|
Hiroshi Fujita (Japanese: 藤田 宏, Hepburn: Fujita Hiroshi) (born 7 December 1928 in Osaka) is a retired Japanese mathematician who worked in partial differential equations. He obtained his Ph.D. at the University of Tokyo, under the supervision of Tosio Kato. == Mathematical contributions == His most widely cited paper, published in 1966, studied the partial differential equation ∂ u ∂ t = ∂ 2 u ∂ x 1 2 + ⋯ + ∂ 2 u ∂ x n 2 + u p , {\displaystyle {\frac {\partial u}{\partial t}}={\frac {\partial ^{2}u}{\partial x_{1}^{2}}}+\cdots +{\frac {\partial ^{2}u}{\partial x_{n}^{2}}}+u^{p},} and showed that there is a "threshold" value p0 > 1 for which p > p0 implies the existence of nonconstant solutions which exist for all positive t and all real values of the x variables. By contrast, if p is between 1 and p0 then such solutions cannot exist. This paper initiated the study of similar and analogous phenomena for various parabolic and hyperbolic partial differential equations. The impact of Fujita's paper is described by the well-known survey articles of Levine (1990) and Deng & Levine (2000). In collaboration with Kato, Fujita applied the semigroup approach in evolutionary partial differential equations to the Navier–Stokes equations of fluid mechanics. They found the existence of unique locally defined strong solutions under certain fractional derivative-based assumptions on the initial velocity. Their approach has been adopted by other influential works, such as Giga & Miyakawa (1985), to allow for different assumptions on the initial velocity. The full understanding of the smoothness and maximal extension of such solutions is currently considered as a major problem of partial differential equations and mathematical physics. == Selected publications == Tosio Kato and Hiroshi Fujita. On the nonstationary Navier-Stokes system. Rend. Sem. Mat. Univ. Padova 32 (1962), 243–260. Hiroshi Fujita and Tosio Kato. On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal. 16 (1964), 269–315. Hiroshi Fujita. On the blowing up of solutions of the Cauchy problem for ut = Δu + u1+α. J. Fac. Sci. Univ. Tokyo Sect. I 13 (1966), 109–124. Mathematical theory of sedimentation analysis (book) Functional-Analytic Methods for Partial Differential Equations (1990, Springer), Proceedings of a Conference and a Symposium held in Tokyo, Japan, July 3–9, 1989. Edited by Hiroshi Fujita, Teruo Ikebe and Shige T. Kuroda. Proceedings of the Ninth International Congress on Mathematical Education, Edited by Hiroshi Fujita et al. == References ==
|
Wikipedia:Hiroshi Haruki#0
|
Hiroshi Haruki (春木 博, Haruki Hiroshi, died September 13, 1997) was a Japanese mathematician. A world-renowned expert in functional equations, he is best known for discovering Haruki's theorem and Haruki's lemma in plane geometry. Some of his published work, such as: "On a Characteristic Property of Confocal Conic Sections" is available (open source) on Project Euclid. Haruki earned his MSc and PhD from Osaka University and taught there. He was a professor at the University of Waterloo in Canada from 1966 till his retirement in 1986. He was a founding member of the university's computer science department (1967). == See also == List of University of Waterloo people == References == News release, Department of Computer Science, University of Waterloo. == External links == Haruki's theorem on MathWorld Hiroshi Haruki's Lemma (Interactive Mathematics Miscellany and Puzzles) Hiroshi Haruki's Theorem (Interactive Mathematics Miscellany and Puzzles)
|
Wikipedia:Hiroshi Okamura#0
|
Hiroshi Okamura (岡村 博, Okamura Hiroshi, November 10, 1905 – September 3, 1948) was a Japanese mathematician who made contributions to analysis and the theory of differential equations. He was a professor at Kyoto University. He discovered the necessary and sufficient conditions on initial value problems of ordinary differential equations for the solution to be unique. He also refined the second mean value theorem of integration. == Works == Hiroshi Okamura (1941), "Sur l'unicité des solutions d'un système d'équations différentielles ordinaires", Mem. Coll. Sci., Kyoto Imperial Univ. (in French), 23: 225–231 Hiroshi Okamura (1942), "Condition nécessaire et suffisante remplie par les équations différentielles ordinaires sans points de Peano", Mem. Coll. Sci., Kyoto Imperial Univ. (in French), 24: 21–28 Hiroshi Okamura (1943), "Sur une sorte de distance relative à un système différentiel", Nippon Sugaku-Buturigakkwai Kizi Dai 3 Ki (in French), 25: 514–523 Hiroshi Okamura (1950), "On the surface integral and Gauss-Green's theorem", Memoirs of the College of Science, University of Kyoto, A: Mathematics, 26 (9): 5–14, doi:10.1215/kjm/1250778050 (posthumous) == References ==
|
Wikipedia:Hirsch–Plotkin radical#0
|
In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal locally nilpotent subgroups of the group. It was named by Gruenberg (1961) after Kurt Hirsch and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpotent; this fact is the key ingredient in its construction. The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such. In a finite group, the Hirsch–Plotkin radical coincides with the Fitting subgroup but for infinite groups the two subgroups can differ. The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case. == References ==
|
Wikipedia:Hisashi Terao#0
|
Hisashi Terao (寺尾 寿, Terao Hisashi) (1855-1923) was a Japanese astronomer and mathematician. He graduated from the Tokyo imperial University as well as from the University of Paris, and he was one of the founding members and the first principal of The Tokyo Academy of Physics (now Tokyo University of Science). Notable students that studied under him include Shin Hirayama, Hisashi Kimura, and Kiyotsugu Hirayama. He is also called one of the first astronomers of the Meiji era. == Biography == Terao was born as the eldest son to Kiheita Terao, a samurai of the Fukuoka Domain in Haruyoshi village, Naka District, Chikuzen Province, Japan (now Nakasu, Hakata-ku, Fukuoka, Fukuoka Prefecture). He studied at the Shuyukan Han school (currently Fukuoka Prefectural Shuyukan High School) and went on in 1873 to stay in Tokyo where he took a course in French language at the Tokyo School of Foreign Studies (currently, Tokyo University of Foreign Studies) which he completed before enrolling at a Kaisei school (which would later become Tokyo University) where he majored in physics. He would also go on to study astronomy under foreign government advisor Emile-Jean Lépissier before completing his physics course and graduating from the physics department of Tokyo University. In 1879, he went to study in France on a government-funded exchange program where he would continue to further his studies in astronomy and mathematics. On July 12, he began as an apprentice at the Montsouris Observatory in Paris before beginning his course of mathematics and celestial mechanics at the University of Paris on November 1. In December he would go on to study practical aspects of his course in the Paris Observatory. After completing his course, he was granted a bachelor's degree of mathematical science. In 1882, he joined a French government-organized observation of the transit of Venus on the island of Martinique in the Caribbean Sea, he also visited some observatories in the United States. He returned to Japan the following year in 1883. Following his return, he became an assistant chief of staff for the Ministry of Education, Culture, Sports, Science and Technology. He also worked on longitude and latitude determinate in Sendai, which saw the first use of a Meridian circle in latitude determination in Japanese history. He helped found The Tokyo Academy of Physics and became its first principal. In 1884, he became a professor of astronomy at the academy, and it was at this time he would go on to give lectures in universities as a mathematician on topics such as elliptic functions and theta functions. He was a proponent for the romanization of the Japanese language, and in January 1885 he along with Masakazu Tomoya, Ryōkichi Yatabe, Yamakawa Kenjirō, Naokichi Matsui, Arikata Kumamoto, and Jirō Kitao, ˆfounded the Rōmaji-kai (羅馬字会, "Romanization Group"). On the second of June 1886 he became head of the National Astronomical Observatory of Japan. In 1889 he attended an assembly of the International Association of Geodesy, where he bought back an international prototype meter to Japan. === Portrait === In 1883, 17-year-old Japanese artist Kuroda Seiki studied French under Hisashi Terao, and passed the entrance exam for the French course at the Tokyo School of Foreign Languages. In 1909, in celebration of the 25th anniversary of Terao's tenure at the Tokyo Imperial University, Kuroda Seiki drew a portrait of Terao in appreciation of him. == References ==
|
Wikipedia:History of Hindu Mathematics#0
|
History of Hindu Mathematics: A Source Book is a treatise on the history of Indian mathematics authored by Bibhutibhushan Datta and Awadhesh Narayan Singh and originally published in two parts in 1930's. The book has since been reissued in one volume by Asia Publishing House in 1962. The treatise has been a standard reference for the history of Indian mathematics for many years. == History of the book == Bibhutibhushan Datta, the senior author of the book, delivered a lecture titled "Contribution of the Ancient Hindus to Mathematics" on 20 December 1927 to the Allahabad University Mathematical Association. This address was published in the Bulletin of the Allahabad University Mathematical Association in two papers totalling 60 pages in length. Datta expanded this paper and wrote the treatise History of Hindu Mathematics in three volumes. Datta retired from academic life in 1933 and became an itinerant ascetic. At the time of retirement, the manuscript of the three-volume work was entrusted to his junior colleague Awadhesh Narayan Singh. Singh published the first two of these volumes as a joint publication. The first volume titled History of Hindu Mathematics. A Source Book (Part 1: Numerical notation and arithmetic) was published in 1935 and the second volume titled History of Hindu Mathematics. A Source Book (Part 2: Algebra) was published in 1938. The planned third volume was never published. == Contents of the book == The book is a source book and index. Under various topics are the collected translations of Sanskrit texts as found in Hindu mathematical texts. === Part 1 === Part 1 of the book is divided into chapters. Chapter 1 gives details of the various methods employed by the Hindus for denoting numbers. The chapter also contains details of the gradual evolution of the decimal place value notation in India. Chapter 2 deals with arithmetic in general and it contains the details of various methods for performing the arithmetical operations using a "board". The evolution of the operations of addition, subtraction, multiplication, division, squaring, cubing, and the extraction square root and cube root are all discussed in detail. === Part 2 === The whole of Part 2, running to about 307 pages, constitutes just one chapter numbered as Chapter 3 of the book. Some of the topics discussed in this chapter are linear equations with one unknown and with two unknowns, quadratic equations, linear indeterminate equations, solutions of equations of the form Nx2 + 1 = y2, indeterminate equations of higher degrees, and rational triangles. == From reviews == Reactions to the publication of the book were mixed: some were highly favorable and some were highly critical. For example, the reviewer in American Mathematical Monthly found the book "From the standpoint of authoritative subject matter and from that of book-making, it is a notable history", whereas the reviewer in Isis, a journal of the History of Science Society, found the book "... a mathematical panegyric on Hindu history. A history of Hindu mathematics still remains to be written". === From the review in American Mathematical Monthly === "Readers of the 'Monthly' have become familiar with the name of one of the authors of this work since articles by B Datta have been appearing over a period of ten years. Through these articles, he has won a place as a reliable research worker in the field of Hindu mathematics. It is gratifying that a work on the history of Hindu mathematics has now come from the hands of these two Hindu scholars; moreover that a complete history to appear in compact form is here begun with the promise of volumes to come. The work under consideration is the first part and deals with the history of the numeral notation and of arithmetic. The second part, we are told, is devoted to algebra and the third part contains the history of geometry, trigonometry, calculus and various other topics such as magic squares, theory of series and permutations and combinations. ... Datta and Singh's 'History of Hindu Mathematics' should be in every library which reaches standards covered by the word "approved." It should be owned by individuals who have any interest whatever in the history of the progress of science. From the standpoint of authoritative subject matter and from that of book-making, it is a notable history." === From the review in Isis === ". . . I would like to say that the book before us certainly contains an abundance of material for the history of Hindu mathematics, but the material presented has to be used with caution. The authors, as native Hindu scholars, are certainly possessed of a deep erudition in Hindu literature, but they display a lack of training in the modern methods of philological and historical criticism, which deficiency is still enhanced by a too perspicuous bias and a tendency towards exaggerating the achievements of the Hindu race. As an industrious collection of material and as a starting point for further critical investigation the present volume is very welcome indeed. But, on the whole, it impresses us as a mathematical panegyric on Hindu history. A history of Hindu mathematics still remains to be written." == Scan available == A scan of the book is available: History of Hindu Mathematics: A Source Book – via Internet Archive. == References ==
|
Wikipedia:History of algebra#0
|
Much of the history of Algeria has taken place on the fertile coastal plain of North Africa, which is often called the Maghreb. North Africa served as a transit region for people moving towards Europe or the Middle East, thus, the region's inhabitants have been influenced by populations from other areas, including the Carthaginians, Romans, and Vandals. The region was conquered by the Muslims in the early 8th century AD, but broke off from the Umayyad Caliphate after the Berber Revolt of 740. During the Ottoman period, Algeria became an important state in the Mediterranean sea which led to many naval conflicts. The last significant events in the country's recent history have been the Algerian War and Algerian Civil War. == Prehistory == Evidence of the early human occupation of Algeria is demonstrated by the discovery of 1.8 million year old Oldowan stone tools found at Ain Hanech in 1992. In 1954 fossilised Homo erectus bones were discovered by C. Arambourg at Ternefine that are 700,000 years old. Neolithic civilization (marked by animal domestication and subsistence agriculture) developed in the Saharan and Mediterranean Maghrib between 6000 and 2000 BC. This type of economy, richly depicted in the Tassili n'Ajjer cave paintings in southeastern Algeria, predominated in the Maghrib until the classical period. == Numidia == Numidia (Berber: Inumiden; 202–40 BC) was the ancient kingdom of the Numidians located in northwest Africa, initially comprising the territory that now makes up modern-day Algeria, but later expanding across what is today known as Tunisia, Libya, and some parts of Morocco. The polity was originally divided between the Massylii in the east and the Masaesyli in the west. During the Second Punic War (218–201 BC), Masinissa, king of the Massylii, defeated Syphax of the Masaesyli to unify Numidia into one kingdom. The kingdom began as a sovereign state and later alternated between being a Roman province and a Roman client state. Numidia, at its largest extent, was bordered by Mauretania to the west, at the Moulouya River, Africa to the east (also exercising control over Tripolitania), the Mediterranean Sea to the north, and the Sahara to the south. It was one of the first major states in the history of Algeria and the Berbers. === War With Rome === By 112 BC, Jugurtha resumed his war with Adherbal. He incurred the wrath of Rome in the process by killing some Roman businessmen who were aiding Adherbal. After a brief war with Rome, Jugurtha surrendered and received a highly favourable peace treaty, which raised suspicions of bribery once more. The local Roman commander was summoned to Rome to face corruption charges brought by his political rival Gaius Memmius. Jugurtha was also forced to come to Rome to testify against the Roman commander, where Jugurtha was completely discredited once his violent and ruthless past became widely known, and after he had been suspected of murdering a Numidian rival. War broke out between Numidia and the Roman Republic and several legions were dispatched to North Africa under the command of the Consul Quintus Caecilius Metellus Numidicus. The war dragged out into a long and seemingly endless campaign as the Romans tried to defeat Jugurtha decisively. Frustrated at the apparent lack of action, Metellus' lieutenant Gaius Marius returned to Rome to seek election as Consul. Marius was elected, and then returned to Numidia to take control of the war. He sent his Quaestor Sulla to neighbouring Mauretania in order to eliminate their support for Jugurtha. With the help of Bocchus I of Mauretania, Sulla captured Jugurtha and brought the war to a conclusive end. Jugurtha was brought to Rome in chains and was placed in the Tullianum. Jugurtha was executed by the Romans in 104 BC, after being paraded through the streets in Gaius Marius' Triumph. === Independence === The Greek historians referred to these peoples as "Νομάδες" (i.e. Nomads), which by Latin interpretation became "Numidae" (but cf. also the correct use of Nomades). Historian Gabriel Camps, however, disputes this claim, favoring instead an African origin for the term. The name appears first in Polybius (second century BC) to indicate the peoples and territory west of Carthage including the entire north of Algeria as far as the river Mulucha (Muluya), about 160 kilometres (100 mi) west of Oran. The Numidians were composed of two great tribal groups: the Massylii in eastern Numidia, and the Masaesyli in the west. During the first part of the Second Punic War, the eastern Massylii, under their king Gala, were allied with Carthage, while the western Masaesyli, under king Syphax, were allied with Rome. The Kingdom of Masaesyli under Syphax extended from the Moulouya river to Oued Rhumel. However, in 206 BC, the new king of the eastern Massylii, Masinissa, allied himself with Rome, and Syphax of the Masaesyli switched his allegiance to the Carthaginian side. At the end of the war, the victorious Romans gave all of Numidia to Masinissa of the Massylii. At the time of his death in 148 BC, Masinissa's territory extended from the Moulouya to the boundary of the Carthaginian territory, and also southeast as far as Cyrenaica to the gulf of Sirte, so that Numidia entirely surrounded Carthage (Appian, Punica, 106) except towards the sea. Furthermore, after the capture of Syphax the king in modern day Morocco with his capital based in Tingis, Bokkar, had become a vassal of Massinissa. Massinissa had also penetrated as far south beyond the Atlas to the Gaetuli and Fezzan was part of his domain. In 179 B.C. Masinissa had received a golden crown from the inhabitants of Delos as he had offered them a shipload of grain. A statue of Masinissa was set up in Delos in honour of him as well as an inscription dedicated to him in Delos by a native from Rhodes. His sons too had statues of them erected on the island of Delos and the King of Bithynia, Nicomedes, had also dedicated a statue to Masinissa. After the death of the long-lived Masinissa around 148 BC, he was succeeded by his son Micipsa. When Micipsa died in 118 BC, he was succeeded jointly by his two sons Hiempsal I and Adherbal and Masinissa's illegitimate grandson, Jugurtha, who was very popular among the Numidians. Hiempsal and Jugurtha quarrelled immediately after the death of Micipsa. Jugurtha had Hiempsal killed, which led to open war with Adherbal. Phoenician traders arrived on the North African coast around 900 BC and established Carthage (in present-day Tunisia) around 800 BC. During the classical period, Berber civilization was already at a stage in which agriculture, manufacturing, trade, and political organization supported several states. Trade links between Carthage and the Berbers in the interior grew, but territorial expansion also resulted in the enslavement or military recruitment of some Berbers and in the extraction of tribute from others. The Carthaginian state declined because of successive defeats by the Romans in the Punic Wars, and in 146 BC, the city of Carthage was destroyed. As Carthaginian power waned, the influence of Berber leaders in the hinterland grew. By the 2nd century BC, several large but loosely administered Berber kingdoms had emerged. After that, king Masinissa managed to unify Numidia under his rule. == Roman empire == Christianity arrived in the 2nd century. By the end of the 4th century, the settled areas had become Christianized, and some Berber tribes had converted en masse. After the fall of the Western Roman Empire, Algeria came under the control of the Vandal Kingdom. Later, the Eastern Roman Empire (also known as the Byzantine Empire) conquered Algeria from the Vandals, incorporating it into the Praetorian prefecture of Africa and later the Exarchate of Africa. == Medieval Muslim Algeria == From the 8th century Umayyad conquest of North Africa led by Musa bin Nusayr, Arab colonization started. The 11th century invasion of migrants from the Arabian peninsula brought oriental tribal customs. The introduction of Islam and Arabic had a profound impact on North Africa. The new religion and language introduced changes in social and economic relations, and established links with the Arab world through acculturation and assimilation. The second Arab military expeditions into the Maghreb, between 642 and 669, resulted in the spread of Islam. The Umayyads (a Muslim dynasty based in Damascus from 661 to 750) recognised that the strategic necessity of dominating the Mediterranean dictated a concerted military effort on the North African front. By 711 Umayyad forces helped by Berber converts to Islam had conquered all of North Africa. In 750 the Abbasids succeeded the Umayyads as Muslim rulers and moved the caliphate to Baghdad. Under the Abbasids, Berber Kharijites Sufri Banu Ifran were opposed to Umayyad and Abbasids. After, the Rustumids (761–909) actually ruled most of the central Maghrib from Tahirt, southwest of Algiers. The imams gained a reputation for honesty, piety, and justice, and the court of Tahirt was noted for its support of scholarship. The Rustumid imams failed, however, to organise a reliable standing army, which opened the way for Tahirt's demise under the assault of the Fatimid dynasty. The Fatimids left the rule of most of Algeria to the Zirids and Hammadid (972–1148), a Berber dynasty that centered significant local power in Algeria for the first time, but who were still at war with Banu Ifran (kingdom of Tlemcen) and Maghraoua (942-1068). This period was marked by constant conflict, political instability, and economic decline. Following a large incursion of Arab Bedouin from Egypt beginning in the first half of the 11th century, the use of Arabic spread to the countryside, and sedentary Berbers were gradually Arabised. The Almoravid ("those who have made a religious retreat") movement developed early in the 11th century among the Sanhaja Berbers of southern Morocco. The movement's initial impetus was religious, an attempt by a tribal leader to impose moral discipline and strict adherence to Islamic principles on followers. But the Almoravid movement shifted to engaging in military conquest after 1054. By 1106, the Almoravids had conquered the Maghreb as far east as Algiers and Morocco, and Spain up to the Ebro River. Like the Almoravids, the Almohads ("unitarians") found their inspiration in Islamic reform. The Almohads took control of Morocco by 1146, captured Algiers around 1151, and by 1160 had completed the conquest of the central Maghrib. The zenith of Almohad power occurred between 1163 and 1199. For the first time, the Maghrib was united under a local regime, but the continuing wars in Spain overtaxed the resources of the Almohads, and in the Maghrib their position was compromised by factional strife and a renewal of tribal warfare. In the central Maghrib, the Abdalwadid founded a dynasty that ruled the Kingdom of Tlemcen in Algeria. For more than 300 years, until the region came under Ottoman suzerainty in the 16th century, the Zayanids kept a tenuous hold in the central Maghrib. Many coastal cities asserted their autonomy as municipal republics governed by merchant oligarchies, tribal chieftains from the surrounding countryside, or the privateers who operated out of their ports. Nonetheless, Tlemcen, the "pearl of the Maghrib," prospered as a commercial center. === Berber dynasties === According to historians of the Middle Ages, the Berbers were divided into two branches, both going back to their ancestors Mazigh. The two branches, called Botr and Barnès were divided into tribes, and each Maghreb region is made up of several tribes. The large Berber tribes or peoples are Sanhaja, Houara, Zenata, Masmuda, Kutama, Awarba, Barghawata ... etc. Each tribe is divided into sub tribes. All these tribes had independent and territorial decisions. Several Berber dynasties emerged during the Middle Ages: - In North and West Africa, in Spain (al-Andalus), Sicily, Egypt, as well as in the southern part of the Sahara, in modern-day Mali, Niger, and Senegal. The medieval historian Ibn Khaldun described the follying Berber dynasties: Zirid, Banu Ifran, Maghrawa, Almoravid, Hammadid, Almohad Caliphate, Marinid, Zayyanid, Wattasid, Meknes, Hafsid dynasty, Fatimids. The invasion of the Banu Hilal Arab tribes in the 11th century sacked Kairouan, and the area under Zirid control was reduced to the coastal region, and the Arab conquests fragmented into petty Bedouin emirates. === Maghrawa Dynasty === The Maghrawa or Meghrawa (Arabic: المغراويون) were a large Zenata Berber tribal confederation whose cradle and seat of power was the territory located on the Chlef in the north-western part of today's Algeria, bounded by the Ouarsenis to the south, the Mediterranean Sea to the north and Tlemcen to the west. They ruled these areas on behalf of the Umayyad Caliphate of Cordoba at the end of the 10th century and during the first half of the 11th century. The Maghrawa confederation of zanata Berbers supposedly originated in the region of modern Algeria between Tlemcen and Tenes.The confederation of Maghrawa were the majority people of the central Maghreb among the Zenata (Gaetuli). Both nomadic and sedentary, the Maghrawa lived under the command of Maghrawa chiefs or Zenata. Algiers has been the territory of the Maghrawa since ancient times. The name Maghrawa was transcribed into Greek by historians. The great kingdom of the Maghrawa was located between Algiers, Cherchell, Ténès, Chlef, Miliana and Médéa. The Maghrawa imposed their domination in the Aurès.[when?] Chlef and its surroundings were populated by the Maghrawa according to Ibn Khaldun. The Maghrawa settled and extended their domination throughout the Dahra and beyond Miliana to the Tafna wadi near Tlemcen,[when?] and were found as far away as Mali.[citation needed] The Maghrawa were one of the first Berber tribes to submit to Islam in the 7th century. They supported Uqba ibn Nafi in his campaign to the Atlantic in 683. They defected from Sunni Islam and became Kharijite Muslims from the 8th century, and allied first with the Idrisids, and, from the 10th century on, with the Umayyads of Córdoba in Al-Andalus. As a result, they were caught up in the Umayyad-Fatimid conflict in Morocco and Algeria. Although they won a victory over the allies of the Fatimids in 924, they soon allied with them. When they switched back to the side of Córdoba, the Zirids briefly took control over most of Morocco, and ruled on behalf of the Fatimids. In 976/977 the Maghrawa conquered Sijilmasa from the Banu Midrar, and in 980 were able to drive the Miknasa out of Sijilmasa as well. The Maghrawa reached their peak under Ziri ibn Atiyya (to 1001), who achieved supremacy in Fez under Umayyad suzerainty, and expanded their territory at the expense of the Banu Ifran in the northern Maghreb – another Zenata tribe whose alliances had shifted often between the Fatimids and the Umayyads of Córdoba. Ziri ibn Atiyya conquered as much as he could of what is now northern Morocco and was able to achieve supremacy in Fez by 987. In 989 he defeated his enemy, Abu al-Bahār, which resulted in Ziri ruling from Zab to Sous Al-Aqsa, in 991 achieving supremacy in the western Maghreb. As a result of his victory he was invited to Córdoba by Ibn Abi 'Amir al-Mansur (also Latinized as Almanzor), the regent of Caliph Hisham II and de facto ruler of the Caliphate of Córdoba. Ziri brought many gifts and Al-Mansur housed him in a lavish palace, but Ziri soon returned to North Africa. The Banu Ifran took advantage of his absence and, under Yaddū, managed to capture Fez.[full citation needed] After a bloody struggle, Ziri reconquered Fez in 993 and displayed Yaddū's severed head on its walls.[citation needed] A period of peace followed, in which Ziri founded the city of Oujda in 994 and made it his capital. However, Ziri was loyal to the Umayyad caliphs in Cordoba and increasingly resented the way that Ibn Abi 'Amir was holding Hisham II captive while progressively usurping his power. In 997 Ziri rejected Ibn Abi 'Amir's authority and declared himself a direct supporter of Caliph Hisham II. Ibn Abi 'Amir sent an invasion force to Morocco. After three unsuccessful months, Ibn Abi 'Amir's army was forced to retreat to the safety of Tangiers, so Ibn Abi 'Amir sent a powerful reinforcements under his son Abd al-Malik.[citation needed] The armies clashed near Tangiers, and in this battle, Ziri was stabbed by an African soldier who reported to Abd al-Malik that he had seriously wounded the Zenata leader. Abd al-Malik pressed home the advantage, and the wounded Ziri fled, hotly pursued by the Caliph's army. The inhabitants of Fez would not let him enter the city, but opened the gates to Abd al-Malik on 13 October 998. Ziri fled to the Sahara, where he rallied the Zenata tribes and overthrew the unpopular remnants of the Idrisid dynasty at Tiaret. He was able to expand his territory to include Tlemcen and other parts of western Algeria, this time under Fatimid protection. Ziri died in 1001 of the after-effects of the stab wounds. He was succeeded by his son Al-Mu'izz, who made peace with Al-Mansur, and regained possession of all his father's former territories.[citation needed] A revolt against the Andalusian Umayyads was put down by Ibn Abi 'Amir, although the Maghrawa were able to regain power in Fez. Under the succeeding rulers al-Muizz (1001–1026), Hamman (1026–1039) and Dunas (1039), they consolidated their rule in northern and central Morocco.[citation needed] Internal power struggles after 1060 enabled the Almoravid dynasty to conquer the Maghrawa realm in 1070 and put an end to their rule. In the mid 11th century the Maghrawa still controlled most of Morocco, notably most of the Sous and Draa River area as well as Aghmat, Fez and Sijilmasa. Later, Zenata power declined. The Maghrawa and Banu Ifran began oppressing their subjects, shedding their blood, violating their women, breaking into homes to seize food and depriving traders of their goods. Anyone who tried to ward them off was killed. === Zirid Dynasty === The Zirid dynasty (Arabic: الزيريون, romanized: az-zīriyyūn), Banu Ziri (Arabic: بنو زيري, romanized: banū zīrī), or the Zirid state (Arabic: الدولة الزيرية, romanized: ad-dawla az-zīriyya) was a Sanhaja Berber dynasty from modern-day Algeria which ruled the central Maghreb from 972 to 1014 and Ifriqiya (eastern Maghreb) from 972 to 1148. Descendants of Ziri ibn Manad, a military leader of the Fatimid Caliphate and the eponymous founder of the dynasty, the Zirids were emirs who ruled in the name of the Fatimids. The Zirids gradually established their autonomy in Ifriqiya through military conquest until officially breaking with the Fatimids in the mid-11th century. The rule of the Zirid emirs opened the way to a period in North African history where political power was held by Berber dynasties such as the Almoravid dynasty, Almohad Caliphate, Zayyanid dynasty, Marinid Sultanate and Hafsid dynasty. Under Buluggin ibn Ziri the Zirids extended their control westwards and briefly occupied Fez and much of present-day Morocco after 980, but encountered resistance from the local Zenata Berbers who gave their allegiance to the Caliphate of Cordoba. To the east, Zirid control was extended over Tripolitania after 978 and as far as Ajdabiya (in present-day Libya). One member of the dynastic family, Zawi ibn Ziri, revolted and fled to al-Andalus, eventually founding the Taifa of Granada in 1013, after the collapse of the Caliphate of Cordoba. Another branch of the Zirids, the Hammadids, broke away from the main branch after various internal disputes and took control of the territories of the central Maghreb after 1015. The Zirids proper were then designated as Badicides and occupied only Ifriqiya between 1048 and 1148. They were based in Kairouan until 1057, when they moved the capital to Mahdia on the coast. The Zirids of Ifriqiya also intervened in Sicily during the 11th century, as the Kalbids, the dynasty who governed the island on behalf of the Fatimids, fell into disorder. The Zirids of Granada surrendered to the Almoravids in 1090, but the Badicides and the Hammadids remained independent during this time. Sometime between 1041 and 1051 the Zirid ruler al-Mu'izz ibn Badis renounced the Fatimid Caliphs and recognized the Sunni Muslim Abbasid Caliphate. In retaliation, the Fatimids instigated the migration of the Banu Hilal tribe to the Maghreb, dealing a serious blow to Zirid power in Ifriqiya. In the 12th century, the Hilalian invasions combined with the attacks of the Normans of Sicily along the coast further weakened Zirid power. The last Zirid ruler, al-Hasan, surrendered Mahdia to the Normans in 1148, thus ending independent Zirid rule. The Almohad Caliphate conquered the central Maghreb and Ifriqiya by 1160, ending the Hammadid dynasty in turn and finally unifying the whole of the Maghreb. ==== Origins and establishment ==== The Zirids were Sanhaja Berbers, from the sedentary Talkata tribe, originating from the area of modern Algeria. In the 10th century this tribe served as vassals of the Fatimid Caliphate, an Isma'ili Shi'a state that challenged the authority of the Sunni Abbasid caliphs. The progenitor of the Zirid dynasty, Ziri ibn Manad (r. 935–971) was installed as governor of the central Maghreb (roughly north-eastern Algeria today) on behalf of the Fatimids, guarding the western frontier of the Fatimid Caliphate. With Fatimid support Ziri founded his own capital and palace at 'Ashir, south-east of Algiers, in 936. He proved his worth as a key ally in 945, during the Kharijite rebellion of Abu Yazid, when he helped break Abu Yazid's siege of the Fatimid capital, Mahdia. After playing this valuable role, he expanded 'Ashir with a new palace circa 947. In 959 he aided Jawhar al-Siqili on a Fatimid military expedition which successfully conquered Fez and Sijilmasa in present-day Morocco. On their return home to the Fatimid capital they paraded the emir of Fez and the “Caliph” Ibn Wasul of Sijilmasa in cages in a humiliating manner. After this success, Ziri was also given Tahart to govern on behalf of the Fatimids. He was eventually killed in battle against the Zanata in 971. When the Fatimids moved their capital to Egypt in 972, Ziri's son Buluggin ibn Ziri (r. 971–984) was appointed viceroy of Ifriqiya. He soon led a new expedition west and by 980 he had conquered Fez and most of Morocco, which had previously been retaken by the Umayyads of Cordoba in 973. He also led a successful expedition to Barghawata territory, from which he brought back a large number of slaves to Ifriqiya. In 978 the Fatimids also granted Buluggin overlordship of Tripolitania (in present-day Libya), allowing him to appoint his own governor in Tripoli. In 984 Buluggin died in Sijilmasa from an illness and his successor decided to abandon Morocco in 985. ==== Buluggin's successors and the first divisions ==== After Buluggin's death, rule of the Zirid state passed to his son, Al-Mansur ibn Buluggin (r. 984–996), and continued through his descendants. However, this alienated the other sons of Ziri ibn Manad who now found themselves excluded from power. In 999 many of these brothers launched a rebellion in 'Ashir against Badis ibn al-Mansur (r. 996–1016), Buluggin's grandson, marking the first serious break in the unity of the Zirids. The rebels were defeated in battle by Hammad ibn Buluggin, Badis' uncle, and most of the brothers were killed. The only remaining brother of stature, Zawi ibn Ziri, led the remaining rebels westwards and sought new opportunity in al-Andalus under the Umayyads Caliphs of Cordoba, the former enemies of the Fatimids and Zirids. He and his followers eventually founded an independent kingdom in al-Andalus, the Taifa of Granada, in 1013. After 1001 Tripolitania broke away under the leadership of Fulful ibn Sa'id ibn Khazrun, a Maghrawa leader who founded the Banu Khazrun dynasty, which endured until 1147. Fulful fought a protracted war against Badis ibn al-Mansur and sought outside help from the Fatimids and even from the Umayyads of Cordoba, but after his death in 1009 the Zirids were able to retake Tripoli for a time. The region nonetheless remained effectively under control of the Banu Khazrun, who fluctuated between practical autonomy and full independence, often playing the Fatimids and the Zirids against each other. The Zirids finally lost Tripoli to them in 1022. Badis appointed Hammad ibn Buluggin as governor of 'Ashir and the western Zirid territories in 997. He gave Hammad a great deal of autonomy, allowing him to campaign against the Zanata and control any new territories he conquered. Hammad constructed his own capital, the Qal'at Bani Hammad, in 1008, and in 1015 he rebelled against Badis and declared himself independent altogether, while also recognizing the Abbasids instead of the Fatimids as caliphs. Badis besieged Hammad's capital and nearly subdued him, but died in 1016 shortly before this could be accomplished. His son and successor, al-Mu'izz ibn Badis (r. 1016–1062), defeated Hammad in 1017, which forced the negotiation of a peace agreement between them. Hammad resumed his recognition of the Fatimids as caliphs but remained independent, forging a new Hammadid state which controlled a large part of present-day Algeria thereafter. ==== Apogee in Ifriqiya ==== The Zirid period of Ifriqiya is considered a high point in its history, with agriculture, industry, trade and learning, both religious and secular, all flourishing, especially in their capital, Qayrawan (Kairouan). The early reign of al-Mu'izz ibn Badis (r. 1016–1062) was particularly prosperous and marked the height of their power in Ifriqiya. In the eleventh century, when the question of Berber origin became a concern, the dynasty of al-Mu'izz started, as part of the Zirids' propaganda, to emphasize its supposed links to the Himyarite kings as a title to nobility, a theme that was taken the by court historians of the period. Management of the area by later Zirid rulers was neglectful as the agricultural economy declined, prompting an increase in banditry among the rural population. The relationship between the Zirids their Fatimid overlords varied - in 1016 thousands of Shiites died in rebellions in Ifriqiya, and the Fatimids encouraged the defection of Tripolitania from the Zirids, but nevertheless the relationship remained close. In 1049 the Zirids broke away completely by adopting Sunni Islam and recognizing the Abbasids of Baghdad as rightful Caliphs, a move which was popular with the urban Arabs of Kairouan. In Sicily the Kalbids continued to govern on behalf of the Fatimids but the island descended into political disarray during the 11th century, inciting the Zirids to intervene on the island. In 1025 (or 1021), al-Mu'izz ibn Badis sent a fleet of 400 ships to the island in response to the Byzantines reconquering Calabria (in southern Italy) from the Muslims, but the fleet was lost in a powerful storm off the coast of Pantelleria. In 1036, the Muslim population of the island request aid from al-Mu'izz to overthrow the Kalbid emir Ahmad ibn Yusuf al-Akhal, whose rule they considered flawed and unjust. The request also contained a pledge to recognize al-Mu'izz as their ruler. Al-Mu'izz, eager to expand his influence after the fragmentation of Zirid North Africa, accepted and sent his son, 'Abdallah, to the island with a large army. Al-Akhal, who had been in negotiations with the Byzantines, requested help from them. A Byzantine army intervened and defeated the Zirid army on the island, but it then withdrew to Calabria, allowing 'Abdallah to finish off al-Akhal. Al-Akhal was besieged in Palermo and killed in 1038. 'Abdallah was subsequently forced to withdraw from the island, either due to the ever-divided Sicilians turning against him or due to another Byzantine invasion in 1038, led by George Maniakes. Another Kalbid amir, al-Hasan al-Samsam, was elected to govern Sicily, but Muslim rule there disintegrated into various petty factions leading up to the Norman conquest of the island in the second half of the 11th century. ==== Hilalian invasions and withdrawal to Mahdia ==== The Zirids renounced the Fatimids and recognized the Abbasid Caliphs in 1048-49, or sometime between 1041 and 1051. In retaliation, the Fatimids sent the Arab tribes of the Banu Hilal and the Banu Sulaym to the Maghreb. The Banu Sulaym settled first in Cyrenaica, but the Banu Hilal continued towards Ifriqiya. The Zirids attempted to stop their advance towards Ifriqiya, they sent 30,000 Sanhaja cavalry to meet the 3,000 Arab cavalry of Banu Hilal in the Battle of Haydaran of 14 April 1052. Nevertheless, the Zirids were decisively defeated and were forced to retreat, opening the road to Kairouan for the Hilalian Arab cavalry. The resulting anarchy devastated the previously flourishing agriculture, and the coastal towns assumed a new importance as conduits for maritime trade and bases for piracy against Christian shipping, as well as being the last holdout of the Zirids. The Banu Hilal invasions eventually forced al-Mu'izz ibn Badis to abandon Kairouan in 1057 and move his capital to Mahdia, while the Banu Hilal largely roamed and pillaged the interior of the former Zirid territories. As a result of the Zirid withdrawal, various local principalities emerged in different areas. In Tunis, the shaykhs of the city elected Abd al-Haqq ibn Abd al-Aziz ibn Khurasan (r. 1059-1095) as local ruler. He founded the local Banu Khurasan dynasty that governed the city thereafter, alternately recognizing the Hammadids or the Zirids as overlords depending on the circumstances. In Qabis (Gabès), the Zirid governor, al-Mu'izz ibn Muhammad ibn Walmiya remained loyal until 1062 when, outraged by the expulsion of his two brothers from Mahdia by al-Mu'izz ibn Badis, he declared his independence and placed himself under the protection of Mu'nis ibn Yahya, a chief of Banu Hilal. Sfaqus (Sfax) was declared independent by the Zirid governor, Mansur al-Barghawati, who was murdered and succeeded by his cousin Hammu ibn Malil al-Barghawati. Al-Mui'zz ibn Badis was succeeded by his son, Tamim ibn al-Mu'izz (r. 1062-1108), who spent much of his reign attempting to restore Zirid power in the region. In 1063 he repelled a siege of Mahdia by the independent ruler of Sfax while also capturing the important port of Sus (Sousse). Meanwhile, the Hammadid ruler al-Nasir ibn 'Alannas (r. 1062-1088) began to intervene in Ifriqiya around this time, having his sovereignty recognized in Sfax, Tunis, and Kairouan. Tamim organized a coalition with some of the Banu Hilal and Banu Sulaym tribes and succeeded in inflicting a heavy defeat on al-Nasir at the Battle of Sabiba in 1065. The war between the Zirids and Hammadids continued until 1077, when a truce was negotiated, sealed by a marriage between Tamim and one of al-Nasir's daughters. In 1074 Tamim sent a naval expedition to Calabria where they ravaged the Italian coasts, plundered Nicotera and enslaved many of its inhabitants. The next year (1075) another Zirid raid resulted in the capture of Mazara in Sicily; however, the Zirid emir rethought his involvement in Sicily and decided to withdraw, abandoning what they had briefly held. In 1087, the Zirid capital, Mahdia, was sacked by the Pisans. According to Ettinghausen, Grabar, and Jenkins-Madina, the Pisa Griffin is believed to have been part of the spoils taken during the sack. In 1083 Mahdia was besieged by a chief of the Banu Hilal, Malik ibn 'Alawi. Unable to take the city, Malik instead turned to Kairouan and captured that city, but Tamim marched out with his entire army and defeated the Banu Hilal forces, at which point he also brought Kairouan back under Zirid control. He went on to capture Gabès in 1097 and Sfax in 1100. Gabès, however, soon declared itself independent again under the leadership of the Banu Jami', a family from the Riyahi branch of the Banu Hilal. Tamim's son and successor, Yahya ibn Tamim (r. 1108-1116), formally recognized the Fatimid caliphs again and received an emissary from Cairo in 1111. He captured an important fortress near Carthage called Iqlibiya and his fleet launched raids against Sardinia and Genoa, bringing back many captives. He was assassinated in 1116 and succeeded by his son, 'Ali ibn Yahya (r. 1116-1121). 'Ali continued to recognize the Fatimids, receiving another embassy from Cairo in 1118. He imposed his authority on Tunis, but failed to recapture Gabès from its local ruler, Rafi' ibn Jami', whose counterattack he then had to repel from Mahdia. He was succeeded by his son al-Hasan in 1121, the last Zirid ruler. ==== End of Zirid rule ==== During the 1130s and 1140s the Normans of Sicily began to capture cities and islands along the coast of Ifriqiya. Jerba was captured in 1135 and Tripoli was captured in 1146. In 1148, the Normans captured Sfax, Gabès, and Mahdia. In Mahdia, the population was weakened by years of famine and the bulk of the Zirid army was away on another campaign when the Norman fleet, commanded by George of Antioch, arrived off the coast. Al-Hasan decided to abandon the city, leaving it to be occupied, which effectively ended the Zirid dynasty's rule. Al-Hasan fled to the citadel of al-Mu'allaqa near Carthage and stayed there for a several months. He planned to flee to the Fatimid court in Egypt but the Norman fleet blocked his way, so instead he headed west, making for the Almohad court of 'Abd al-Mu'min in Marrakesh. He obtained permission from Yahya ibn al-'Aziz, the Hammadid ruler, to cross his territory, but after entering Hammadid territory he was detained and placed under house arrest in Algiers. When 'Abd al-Mu'min captured Algiers in 1151, he freed al-Hasan, who accompanied him back to Marrakesh. Later, when 'Abd al-Mu'min conquered Mahdia in 1160, placing all of Ifriqiya under Almohad rule, al-Hasan was with him. 'Abd al-Mu'min appointed him governor of Mahdia, where he remained, residing in the suburb of Zawila, until 'Abd al-Mu'min's death in 1163. The new Almohad caliph, Abu Ya'qub Yusuf, subsequently ordered him to come back to Marrakesh, but al-Hasan died along the way in Tamasna in 1167. === Hammadid Dynasty === The Hammadid dynasty (Arabic: الحمّاديون) was a branch of the Sanhaja Berber dynasty that ruled an area roughly corresponding to north-eastern modern Algeria between 1008 and 1152. The state reached its peak under Nasir ibn Alnas during which it was briefly the most important state in Northwest Africa. The Hammadid dynasty's first capital was at Qalaat Beni Hammad. It was founded in 1007, and is now a UNESCO World Heritage Site. When the area was sacked by the Banu Hilal tribe, the Hammadids moved their capital to Béjaïa in 1090. === Almohad Caliphate === The Almohad Caliphate (IPA: ; Arabic: خِلَافَةُ ٱلْمُوَحِّدِينَ or دَوْلَةُ ٱلْمُوَحِّدِينَ or ٱلدَّوْلَةُ ٱلْمُوَحِّدِيَّةُ from Arabic: ٱلْمُوَحِّدُونَ, romanized: al-Muwaḥḥidūn, lit. 'those who profess the unity of God': 246 ) was a North African Berber Muslim empire founded in the 12th century. At its height, it controlled much of the Iberian Peninsula (Al Andalus) and North Africa (the Maghreb). The Almohad docrtine was founded by Ibn Tumart among the Berber Masmuda tribes, but the Almohad caliphate and its ruling dynasty were founded after his death by Abd al-Mu'min al-Gumi, which was born in the Hammadid region of Tlemcen, Algeria. Around 1120, Ibn Tumart first established a Berber state in Tinmel in the Atlas Mountains. Under Abd al-Mu'min (r. 1130–1163) they succeeded in overthrowing the ruling Almoravid dynasty governing Morocco in 1147, when he conquered Marrakesh and declared himself caliph. They then extended their power over all of the Maghreb by 1159. Al-Andalus soon followed, and all of Muslim Iberia was under Almohad rule by 1172. The turning point of their presence in the Iberian Peninsula came in 1212, when Muhammad III, "al-Nasir" (1199–1214) was defeated at the Battle of Las Navas de Tolosa in the Sierra Morena by an alliance of the Christian forces from Castile, Aragon and Navarre. Much of the remaining territories of al-Andalus were lost in the ensuing decades, with the cities of Córdoba and Seville falling to the Christians in 1236 and 1248 respectively. The Almohads continued to rule in Africa until the piecemeal loss of territory through the revolt of tribes and districts enabled the rise of their most effective enemies, the Marinids, from northern Morocco in 1215. The last representative of the line, Idris al-Wathiq, was reduced to the possession of Marrakesh, where he was murdered by a slave in 1269; the Marinids seized Marrakesh, ending the Almohad domination of the Western Maghreb. ==== Origins ==== The Almohad movement originated with Ibn Tumart, a member of the Masmuda, a Berber tribal confederation of the Atlas Mountains of southern Morocco. At the time, Morocco, western Algeria and Spain (al-Andalus), were under the rule of the Almoravids, a Sanhaja Berber dynasty. Early in his life, Ibn Tumart went to Spain to pursue his studies, and thereafter to Baghdad to deepen them. In Baghdad, Ibn Tumart attached himself to the theological school of al-Ash'ari, and came under the influence of the teacher al-Ghazali. He soon developed his own system, combining the doctrines of various masters. Ibn Tumart's main principle was a strict unitarianism (tawhid), which denied the independent existence of the attributes of God as being incompatible with His unity, and therefore a polytheistic idea. Ibn Tumart represented a revolt against what he perceived as anthropomorphism in Muslim orthodoxy. His followers would become known as the al-Muwaḥḥidūn ("Almohads"), meaning those who affirm the unity of God. After his return to the Maghreb c. 1117, Ibn Tumart spent some time in various Ifriqiyan cities, preaching and agitating, heading riotous attacks on wine-shops and on other manifestations of laxity. He laid the blame for the latitude on the ruling dynasty of the Almoravids, whom he accused of obscurantism and impiety. He also opposed their sponsorship of the Maliki school of jurisprudence, which drew upon consensus (ijma) and other sources beyond the Qur'an and Sunnah in their reasoning, an anathema to the stricter Zahirism favored by Ibn Tumart. His antics and fiery preaching led fed-up authorities to move him along from town to town. After being expelled from Bejaia, Ibn Tumart set up camp in Mellala, in the outskirts of the city, where he received his first disciples – notably, al-Bashir (who would become his chief strategist) and Abd al-Mu'min (a Zenata Berber, who would later become his successor). In 1120, Ibn Tumart and his small band of followers proceeded to Morocco, stopping first in Fez, where he briefly engaged the Maliki scholars of the city in debate. He even went so far as to assault the sister of the Almoravid emir ʿAli ibn Yusuf, in the streets of Fez, because she was going about unveiled, after the manner of Berber women. After being expelled from Fez, he went to Marrakesh, where he successfully tracked down the Almoravid emir Ali ibn Yusuf at a local mosque, and challenged the emir, and the leading scholars of the area, to a doctrinal debate. After the debate, the scholars concluded that Ibn Tumart's views were blasphemous and the man dangerous, and urged him to be put to death or imprisoned. But the emir decided merely to expel him from the city. Ibn Tumart took refuge among his own people, the Hargha, in his home village of Igiliz (exact location uncertain), in the Sous valley. He retreated to a nearby cave, and lived out an ascetic lifestyle, coming out only to preach his program of puritan reform, attracting greater and greater crowds. At length, towards the end of Ramadan in late 1121, after a particularly moving sermon, reviewing his failure to persuade the Almoravids to reform by argument, Ibn Tumart 'revealed' himself as the true Mahdi, a divinely guided judge and lawgiver, and was recognized as such by his audience. This was effectively a declaration of war on the Almoravid state. On the advice of one of his followers, Omar Hintati, a prominent chieftain of the Hintata, Ibn Tumart abandoned his cave in 1122 and went up into the High Atlas, to organize the Almohad movement among the highland Masmuda tribes. Besides his own tribe, the Hargha, Ibn Tumart secured the adherence of the Ganfisa, the Gadmiwa, the Hintata, the Haskura, and the Hazraja to the Almohad cause. Around 1124, Ibn Tumart erected the ribat of Tinmel, in the valley of the Nfis in the High Atlas, an impregnable fortified complex, which would serve both as the spiritual center and military headquarters of the Almohad movement. For the first eight years, the Almohad rebellion was limited to a guerilla war along the peaks and ravines of the High Atlas. Their principal damage was in rendering insecure (or altogether impassable) the roads and mountain passes south of Marrakesh – threatening the route to all-important Sijilmassa, the gateway of the trans-Saharan trade. Unable to send enough manpower through the narrow passes to dislodge the Almohad rebels from their easily defended mountain strong points, the Almoravid authorities reconciled themselves to setting up strongholds to confine them there (most famously the fortress of Tasghîmût that protected the approach to Aghmat, which was conquered by the Almohads in 1132), while exploring alternative routes through more easterly passes. Ibn Tumart organized the Almohads as a commune, with a minutely detailed structure. At the core was the Ahl ad-dār ("House of the Mahdi:), composed of Ibn Tumart's family. This was supplemented by two councils: an inner Council of Ten, the Mahdi's privy council, composed of his earliest and closest companions; and the consultative Council of Fifty, composed of the leading sheikhs of the Masmuda tribes. The early preachers and missionaries (ṭalaba and huffāẓ) also had their representatives. Militarily, there was a strict hierarchy of units. The Hargha tribe coming first (although not strictly ethnic; it included many "honorary" or "adopted" tribesmen from other ethnicities, e.g. Abd al-Mu'min himself). This was followed by the men of Tinmel, then the other Masmuda tribes in order, and rounded off by the black fighters, the ʻabīd. Each unit had a strict internal hierarchy, headed by a mohtasib, and divided into two factions: one for the early adherents, another for the late adherents, each headed by a mizwar (or amzwaru); then came the sakkakin (treasurers), effectively the money-minters, tax-collectors, and bursars, then came the regular army (jund), then the religious corps – the muezzins, the hafidh and the hizb – followed by the archers, the conscripts, and the slaves. Ibn Tumart's closest companion and chief strategist, al-Bashir, took upon himself the role of "political commissar", enforcing doctrinal discipline among the Masmuda tribesmen, often with a heavy hand. In early 1130, the Almohads finally descended from the mountains for their first sizeable attack in the lowlands. It was a disaster. The Almohads swept aside an Almoravid column that had come out to meet them before Aghmat, and then chased their remnant all the way to Marrakesh. They laid siege to Marrakesh for forty days until, in April (or May) 1130, the Almoravids sallied from the city and crushed the Almohads in the bloody Battle of al-Buhayra (named after a large garden east of the city). The Almohads were thoroughly routed, with huge losses. Half their leadership was killed in action, and the survivors only just managed to scramble back to the mountains. Ibn Tumart died shortly after, in August 1130. That the Almohad movement did not immediately collapse after such a devastating defeat and the death of their charismatic Mahdi, is likely due to the skills of his successor, Abd al-Mu'min.: 70 Ibn Tumart's death was kept a secret for three years, a period which Almohad chroniclers described as a ghayba or "occultation". This period likely gave Abd al-Mu'min time to secure his position as successor to the political leadership of the movement.: 70 Although a Zenata Berber from Tagra (Algeria), and thus an alien among the Masmuda of southern Morocco, Abd al-Mu'min nonetheless saw off his principal rivals and hammered wavering tribes back to the fold. In an ostentatious gesture of defiance, in 1132, if only to remind the emir that the Almohads were not finished, Abd al-Mu'min led an audacious night operation that seized Tasghîmût fortress and dismantled it thoroughly, carting off its great gates back to Tinmel. Three years after Ibn Tumart's death he was officially proclaimed "Caliph". In order to neutralise the Masmudas, to whom he was a stranger, Abd al-Mumin relied on his tribe of origin, the Kumiyas (a Berber tribe from Orania), which he integrated massively into the army and within the Almohad power. He thus appointed his son as his successor and his other children as governors of the provinces of the Caliphate. The Kumiyas would later form the bodyguard of Abd al Mumin and his successor. In addition, he also relied on Arabs, representatives of the great Hilalian families, whom he deported to Morocco to weaken the influence of the Masmuda sheikhs. These moves have the effect of advancing the Arabisation of the future Morocco. ==== Al-Andalus ==== Abd al-Mu'min then came forward as the lieutenant of the Mahdi Ibn Tumart. Between 1130 and his death in 1163, Abd al-Mu'min not only rooted out the Almoravids, but extended his power over all northern Africa as far as Egypt, becoming amir of Marrakesh in 1147. Al-Andalus followed the fate of Africa. Between 1146 and 1173, the Almohads gradually wrested control from the Almoravids over the Moorish principalities in Iberia. The Almohads transferred the capital of Muslim Iberia from Córdoba to Seville. They founded a great mosque there; its tower, the Giralda, was erected in 1184 to mark the accession of Ya'qub I. The Almohads also built a palace there called Al-Muwarak on the site of the modern day Alcázar of Seville. The Almohad princes had a longer and more distinguished career than the Almoravids. The successors of Abd al-Mumin, Abu Yaqub Yusuf (Yusuf I, ruled 1163–1184) and Abu Yusuf Yaqub al-Mansur (Yaʻqūb I, ruled 1184–1199), were both able men. Initially their government drove many Jewish and Christian subjects to take refuge in the growing Christian states of Portugal, Castile, and Aragon. Ultimately they became less fanatical than the Almoravids, and Ya'qub al-Mansur was a highly accomplished man who wrote a good Arabic style and protected the philosopher Averroes. In 1190–1191, he campaigned in southern Portugal and won back territory lost in 1189. His title of "al-Manṣūr" ("the Victorious") was earned by his victory over Alfonso VIII of Castile in the Battle of Alarcos (1195). From the time of Yusuf II, however, the Almohads governed their co-religionists in Iberia and central North Africa through lieutenants, their dominions outside Morocco being treated as provinces. When Almohad emirs crossed the Straits it was to lead a jihad against the Christians and then return to Morocco. ==== Holding years ==== In 1212, the Almohad Caliph Muhammad 'al-Nasir' (1199–1214), the successor of al-Mansur, after an initially successful advance north, was defeated by an alliance of the four Christian kings of Castile, Aragón, Navarre, and Portugal, at the Battle of Las Navas de Tolosa in the Sierra Morena. The battle broke the Almohad advance, but the Christian powers remained too disorganized to profit from it immediately. Before his death in 1213, al-Nasir appointed his young ten-year-old son as the next caliph Yusuf II "al-Mustansir". The Almohads passed through a period of effective regency for the young caliph, with power exercised by an oligarchy of elder family members, palace bureaucrats and leading nobles. The Almohad ministers were careful to negotiate a series of truces with the Christian kingdoms, which remained more-or-less in place for next fifteen years (the loss of Alcácer do Sal to the Kingdom of Portugal in 1217 was an exception). In early 1224, the youthful caliph died in an accident, without any heirs. The palace bureaucrats in Marrakesh, led by the wazir Uthman ibn Jam'i, quickly engineered the election of his elderly grand-uncle, Abd al-Wahid I 'al-Makhlu', as the new Almohad caliph. But the rapid appointment upset other branches of the family, notably the brothers of the late al-Nasir, who governed in al-Andalus. The challenge was immediately raised by one of them, then governor in Murcia, who declared himself Caliph Abdallah al-Adil. With the help of his brothers, he quickly seized control of al-Andalus. His chief advisor, the shadowy Abu Zayd ibn Yujjan, tapped into his contacts in Marrakesh, and secured the deposition and assassination of Abd al-Wahid I, and the expulsion of the al-Jami'i clan. This coup has been characterized as the pebble that finally broke al-Andalus. It was the first internal coup among the Almohads. The Almohad clan, despite occasional disagreements, had always remained tightly knit and loyally behind dynastic precedence. Caliph al-Adil's murderous breach of dynastic and constitutional propriety marred his acceptability to other Almohad sheikhs. One of the recusants was his cousin, Abd Allah al-Bayyasi ("the Baezan"), the Almohad governor of Jaén, who took a handful of followers and decamped for the hills around Baeza. He set up a rebel camp and forged an alliance with the hitherto quiet Ferdinand III of Castile. Sensing his greater priority was Marrakesh, where recusant Almohad sheikhs had rallied behind Yahya, another son of al-Nasir, al-Adil paid little attention to this little band of misfits. === Zayyanid Dynasty === The Kingdom of Tlemcen or Zayyanid Kingdom of Tlemcen (Arabic: الزيانيون) was a Berber kingdom in what is now the northwest of Algeria. Its territory stretched from Tlemcen to the Chelif bend and Algiers, and at its zenith reached Sijilmasa and the Moulouya River in the west, Tuat to the south and the Soummam in the east. The Tlemcen Kingdom was established after the demise of the Almohad Caliphate in 1236, and later fell under Ottoman rule in 1554. It was ruled by sultans of the Zayyanid dynasty. The capital of the Tlemcen kingdom centred on Tlemcen, which lay on the primary east–west route between Morocco and Ifriqiya. The kingdom was situated between the realm of the Marinids the west, centred on Fez, and the Hafsids to the east, centred on Tunis. Tlemcen was a hub for the north–south trade route from Oran on the Mediterranean coast to the Western Sudan. As a prosperous trading centre, it attracted its more powerful neighbours. At different times the kingdom was invaded and occupied by the Marinids from the west, by the Hafsids from the east, and by Aragonese from the north. At other times, they were able to take advantage of turmoil among their neighbours: during the reign of Abu Tashfin I (r. 1318–1337) the Zayyanids occupied Tunis and in 1423, under the reign of Abu Malek, they briefly captured Fez.: 287 In the south the Zayyanid realm included Tuat, Tamentit and the Draa region which was governed by Abdallah Ibn Moslem ez Zerdali, a sheikh of the Zayyanids. ==== Rise to power (13th century) ==== The Bānu ʿabd āl-Wād, also called the Bānu Ziyān or Zayyanids after Yaghmurasen Ibn Zyan, the founder of the dynasty, were leaders of a Berber group who had long been settled in the Central Maghreb. Although contemporary chroniclers asserted that they had a noble Arab origin, he reportedly spoke in Zenati dialect and denied the lineage that genealogists had attributed to him. The town of Tlemcen, called Pomaria by the Romans, is about 806m above sea level in fertile, well-watered country. Tlemcen was an important centre under the Almoravid dynasty and its successors the Almohad Caliphate, who began a new wall around the town in 1161. Yaghmurasen ibn Zayyan (1235–83) of the Bānu ʿabd āl-Wād was governor of Tlemcen under the Almohads. He inherited leadership of the family from his brother in 1235. When the Almohad empire began to fall apart, in 1235, Yaghmurasen declared his independence. The city of Tlemcen became the capital of one of three successor states, ruled for centuries by successive Ziyyanid sultans. Its flag was a white crescent pointing upwards on a blue field. The kingdom covered the less fertile regions of the Tell Atlas. Its people included a minority of settled farmers and villagers, and a majority of nomadic herders. Yaghmurasen was able to maintain control over the rival Berber groups, and when faced with the outside threat of the Marinid dynasty, he formed an alliance with the Emir of Granada and the King of Castile, Alfonso X. According to Ibn Khaldun, "he was the bravest, most dreaded and honourable man of the 'Abd-la-Wadid family. No one looked after the interest of his people, maintained the influence of the kingdom and managed the state administration better than he did." In 1248 he defeated the Almohad Caliph in the Battle of Oujda during which the Almohad Caliph was killed. In 1264 he managed to conquer Sijilmasa, therefore bringing Sijilmasa and Tlemcen, the two most important outlets for trans-Saharan trade under one authority. Sijilmasa remained under his control for 11 years. Before his death he instructed his son and heir Uthman to remain on the defensive with the Marinid kingdom, but to expand into Hafsid territory if possible. ==== 14th century ==== For most of its history the kingdom was on the defensive, threatened by stronger states to the east and the west. The nomadic Arabs to the south also took advantage of the frequent periods of weakness to raid the centre and take control of pastures in the south. The city of Tlemcen was several times attacked or besieged by the Marinids, and large parts of the kingdom were occupied by them for several decades in the fourteenth century. The Marinid Abu Yaqub Yusuf an-Nasr besieged Tlemcen from 1299 to 1307. During the siege he built a new town, al-Mansura, diverting most of the trade to this town. The new city was fortified and had a mosque, baths and palaces. The siege was raised when Abu Yakub was murdered in his sleep by one of his eunuchs. When the Marinids left in 1307, the Zayyanids promptly destroyed al-Mansura. The Zayyanid king Abu Zayyan I died in 1308 and was succeeded by Abu Hammu I (r. 1308–1318). Abu Hammu was later killed in a conspiracy instigated by his son and heir Abu Tashufin I (r. 1318–1337). The reigns of Abu Hammu I and Abu Tashufin I marked the second apogee of the Zayyanids, a period during which they consolidated their hegemony in the central Maghreb. Tlemcen recovered its trade and its population grew, reaching about 100,000 by around the 1330s. Abu Tashufin initiated hostilities against Ifriqiya while the Marinids were distracted by their internal struggles. He besieged Béjaïa and sent an army into Tunisia that defeated the Hafsid king Abu Yahya Abu Bakr II, who fled to Constantine while the Zayyanids occupied Tunis in 1325. The Marinid sultan Abu al-Hasan (r. 1331–1348) cemented an alliance with Hafsids by marrying a Hafsid princess. Upon being attacked by the Zayyanids again, the Hafsids appealed to Abu al-Hasan for help, providing him with an excuse to invade his neighbour. The Marinid sultan initiated a siege of Tlemcen in 1335 and the city fell in 1337. Abu Tashufin died during the fighting. Abu al-Hasan received delegates from Egypt, Granada, Tunis and Mali congratulating him on his victory, by which he had gained complete control of the trans-Saharan trade. In 1346 the Hafsid Sultan, Abu Bakr, died and a dispute over the succession ensued. In 1347 Abu al-Hasan annexed Ifriqiya, briefly reuniting the Maghrib territories as they had been under the Almohads. However, Abu al-Hasan went too far in attempting to impose more authority over the Arab tribes, who revolted and in April 1348 defeated his army near Kairouan. His son, Abu Inan Faris, who had been serving as governor of Tlemcen, returned to Fez and declared that he was sultan. Tlemcen and the central Maghreb revolted. The Zayyanid Abu Thabit I (1348-1352) was proclaimed king of Tlemcen. Abu al-Hasan had to return from Ifriqiya by sea. After failing to retake Tlemcen and being defeated by his son, Abu al-Hasan died in May 1351. In 1352 Abu Inan Faris recaptured Tlemcen. He also reconquered the central Maghreb. He took Béjaïa in 1353 and Tunis in 1357, becoming master of Ifriqiya. In 1358 he was forced to return to Fez due to Arab opposition, where he fell sick and was killed. The Zayyanid king Abu Hammu Musa II (r. 1359–1389) next took the throne of Tlemcen. He pursued an expansionist policy, pushing towards Fez in the west and into the Chelif valley and Béjaïa in the east. He had a long reign punctuated by fighting against the Marinids or various rebel groups. The Marinids reoccupied Tlemcen in 1360 and in 1370. In both cases, the Marinids found they were unable to hold the region against local resistance. Abu Hammu attacked the Hafsids in Béjaïa again in 1366, but this resulted in Hafsid intervention in the kingdom's affairs. The Hafsid sultan released Abu Hammu's cousin, Abu Zayyan, and helped him in laying claim to the Zayyanid throne. This provoked an internecine war between the two Zayyanids until 1378, when Abu Hammu finally captured Abu Zayyan in Algiers. The historian Ibn Khaldun lived in Tlemcen for a period during the generally prosperous reign of Abu Hammu Musa II, and helped him in negotiations with the nomadic Arabs. He said of this period, "Here [in Tlemcen] science and arts developed with success; here were born scholars and outstanding men, whose glory penetrated into other countries." Abu Hammu was deposed by his son, Abu Tashfin II (1389–94), and the state went into decline. ==== Decline (late 14th and 15th centuries) ==== In the late 14th century and the 15th century, the state was increasingly weak and became intermittently a vassal of Hafsid Ifriqiya, Marinid Morocco or the Crown of Aragon. In 1386 Abu Hammu moved his capital to Algiers, which he judged less vulnerable, but a year later his son, Abu Tashufin, overthrew him and took him prisoner. Abu Hammu was sent on a ship towards Alexandria but he escaped along the way when the ship stopped in Tunis. In 1388 he recaptured Tlemcen, forcing his son to flee. Abu Tashufin sought refuge in Fez and enlisted the aid of the Marinids, who sent an army to occupy Tlemcen and reinstall him on the throne. As a result, Abu Tashufin and his successors recognized the suzerainty of the Marinids and paid them an annual tribute. During the reign of the Marinid sultan Abu Sa'id, the Zayyanids rebelled on several occasions and Abu Sa'id had to reassert his authority.: 33–39 After Abu Sa'id's death in 1420 the Marinids were plunged into political turmoil. The Zayyanid emir, Abu Malek, used this opportunity to throw off Marinid authority and captured Fez in 1423. Abu Malek installed Muhammad, a Marinid prince, as a Zayyanid vassal in Fez.: 287 : 47–49 The Wattasids, a family related to the Marinids, continued to govern from Salé, where they proclaimed Abd al-Haqq II, an infant, as the successor to the Marinid throne, with Abu Zakariyya al-Wattasi as regent. The Hafsid sultan, Abd al-Aziz II, reacted to Abu Malek's rising influence by sending military expeditions westward, installing his own Zayyanid client king (Abu Abdallah II) in Tlemcen and pursuing Abu Malek to Fez. Abu Malek's Marinid puppet, Muhammad, was deposed and the Wattasids returned with Abd al-Haqq II to Fez, acknowledging Hafsid suzerainty.: 287 : 47–49 The Zayyanids remained vassals of the Hafsids until the end of the 15th century, when the Spanish expansion along the coast weakened the rule of both dynasties. By the end of the 15th century the Kingdom of Aragon had gained effective political control, intervening in the dynastic disputes of the amirs of Tlemcen, whose authority had shrunk to the town and its immediate neighbourship. When the Spanish took the city of Oran from the kingdom in 1509, continuous pressure from the Berbers prompted the Spanish to attempt a counterattack against the city of Tlemcen (1543), which was deemed by the Papacy to be a crusade. The Spanish under Martin of Angulo had also suffered a prior defeat in 1535 when they attempted to install a client ruler in Tlemcen. The Spanish failed to take the city in the first attack, but the strategic vulnerability of Tlemcen caused the kingdom's weight to shift toward the safer and more heavily fortified corsair base at Algiers. Tlemcen was captured in 1551 by the Ottoman Empire under Hassan Pasha. The last Zayyanid sultan's son escaped to Oran, then a Spanish possession. He was baptized and lived a quiet life as Don Carlos at the court of Philip II of Spain. Under the Ottoman Empire Tlemcen quickly lost its former importance, becoming a sleepy provincial town. The failure of the kingdom to become a powerful state can be explained by the lack of geographical or cultural unity, the constant internal disputes and the reliance on irregular Arab-Berber nomads for the military. === Kingdom of Beni Abbas === === Kingdom of Kuku === === Christian conquest of Spain === The final triumph of the 700-year Christian conquest of Spain was marked by the fall of Granada in 1492. Christian Spain imposed its influence on the Maghrib coast by constructing fortified outposts and collecting tribute. But Spain never sought to extend its North African conquests much beyond a few modest enclaves. Privateering was an age-old practice in the Mediterranean, and North African rulers engaged in it increasingly in the late 16th and early 17th centuries because it was so lucrative. Until the 17th century the Barbary pirates used galleys, but a Dutch renegade of the name of Zymen Danseker taught them the advantage of using sailing ships. Algeria became the privateering city-state par excellence, and two privateer brothers were instrumental in extending Ottoman influence in Algeria. At about the time Spain was establishing its presidios in the Maghrib, the Muslim privateer brothers Aruj and Khair ad Din—the latter known to Europeans as Barbarossa, or Red Beard—were operating successfully off Tunisia. In 1516 Aruj moved his base of operations to Algiers but was killed in 1518. Khair ad Din succeeded him as military commander of Algiers, and the Ottoman sultan gave him the title of beglerbey (provincial governor). === Spanish enclaves === The Spanish expansionist policy in North Africa began with the Catholic Monarchs and the regent Cisneros, once the Reconquista in the Iberian Peninsula was finished. That way, several towns and outposts in the Algerian coast were conquered and occupied: Mers El Kébir (1505), Oran (1509), Algiers (1510) and Bugia (1510). The Spanish conquest of Oran was won with much bloodshed: 4,000 Algerians were massacred, and up to 8,000 were taken prisoner. For about 200 years, Oran's inhabitants were virtually held captive in their fortress walls, ravaged by famine and plague; Spanish soldiers, too, were irregularly fed and paid. The Spaniards left Algiers in 1529, Bujia in 1554, Mers El Kébir and Oran in 1708. The Spanish returned in 1732 when the armada of the Duke of Montemar was victorious in the Battle of Aïn-el-Turk and retook Oran and Mers El Kébir; the Spanish massacred many Muslim soldiers. In 1751, a Spanish adventurer, named John Gascon, obtained permission, and vessels and fireworks, to go against Algiers, and set fire, at night, to the Algerian fleet. The plan, however, miscarried. In 1775, Charles III of Spain sent a large force to attack Algiers, under the command of Alejandro O'Reilly (who had led Spanish forces in crushing French rebellion in Louisiana), resulting in a disastrous defeat. The Algerians suffered 5,000 casualties. The Spanish navy bombarded Algiers in 1784; over 20,000 cannonballs were fired, much of the city and its fortifications were destroyed and most of the Algerian fleet was sunk. Oran and Mers El Kébir were held until 1792, when they were sold by the king Charles IV to the Bey of Algiers. == Regency of Algiers == The Regency of Algiers (Arabic: دولة الجزائر, romanized: Dawlat al-Jaza'ir) was a state in North Africa lasting from 1516 to 1830, until it was conquered by the French. Situated between the regency of Tunis in the east, the Sultanate of Morocco (from 1553) in the west and Tuat as well as the country south of In Salah in the south (and the Spanish and Portuguese possessions of North Africa), the Regency originally extended its borders from La Calle in the east to Trara in the west and from Algiers to Biskra, and afterwards spread to the present eastern and western borders of Algeria. It had various degrees of autonomy throughout its existence, in some cases reaching complete independence, recognized even by the Ottoman sultan. The country was initially governed by governors appointed by the Ottoman sultan (1518–1659), rulers appointed by the Odjak of Algiers (1659–1710), and then Deys elected by the Divan of Algiers from (1710-1830). === Establishment === From 1496, the Spanish conquered numerous possessions on the North African coast: Melilla (1496), Mers El Kébir (1505), Oran (1509), Bougie (1510), Tripoli (1510), Algiers, Shershell, Dellys, and Tenes. The Spaniards later led unsuccessful expeditions to take Algiers in the Algiers expedition in 1516, 1519 and another failed expedition in 1541. Around the same time, the Ottoman privateer brothers Oruç and Hayreddin—both known to Europeans as Barbarossa, or "Red Beard"—were operating successfully off Tunisia under the Hafsids. In 1516, Oruç moved his base of operations to Algiers. He asked for the protection of the Ottoman Empire in 1517, but was killed in 1518 during his invasion of the Zayyanid Kingdom of Tlemcen. Hayreddin succeeded him as military commander of Algiers. In 1551 Hasan Pasha, the son of Hayreddin defeated the Spanish-Moroccan armies during a campaign to recapture Tlemcen, thus cementing Ottoman control in western and central Algeria. After that, the conquest of Algeria sped up. In 1552 Salah Rais, with the help of some Kabyle kingdoms, conquered Touggourt, and established a foothold in the Sahara. In the 1560s eastern Algeria was centralized, and the power struggle which had been present ever since the Emirate of Béjaïa collapsed came to an end. During the 16th, 17th, and early 18th century, the Kabyle Kingdoms of Kuku and Ait Abbas managed to maintain their independence repelling Ottoman attacks several times, notably in the First Battle of Kalaa of the Beni Abbes. This was mainly thanks to their ideal position deep inside the Kabylia Mountains and their great organisation, and the fact that unlike in the West and East where collapsing kingdoms such as Tlemcen or Béjaïa were present, Kabylia had two new and energetic emirates. === Base in the war against Spain === Hayreddin Barbarossa established the military basis of the regency. The Ottomans provided a supporting garrison of 2,000 Turkish troops with artillery. He left Hasan Agha in command as his deputy when he had to leave for Constantinople in 1533. The son of Barbarossa, Hasan Pashan was in 1544 when his father retired, the first governor of the Regency to be directly appointed by the Ottoman Empire. He took the title of beylerbey. Algiers became a base in the war against Spain, and also in the Ottoman conflicts with Morocco. Beylerbeys continued to be nominated for unlimited tenures until 1587. After Spain had sent an embassy to Constantinople in 1578 to negotiate a truce, leading to a formal peace in August 1580, the Regency of Algiers was a formal Ottoman territory, rather than just a military base in the war against Spain. At this time, the Ottoman Empire set up a regular Ottoman administration in Algiers and its dependencies, headed by Pashas, with 3-year terms to help considate Ottoman power in the Maghreb. === Mediterranean privateers === Despite the end of formal hostilities with Spain in 1580, attacks on Christian and especially Catholic shipping, with slavery for the captured, became prevalent in Algiers and were actually the main industry and source of revenues of the Regency. In the early 17th century, Algiers also became, along with other North African ports such as Tunis, one of the bases for Anglo-Turkish piracy. There were as many as 8,000 renegades in the city in 1634. (Renegades were former Christians, sometimes fleeing the law, who voluntarily moved to Muslim territory and converted to Islam.) Hayreddin Barbarossa is credited with tearing down the Peñón of Algiers and using the stone to build the inner harbor. A contemporary letter states: "The infinity of goods, merchandise jewels and treasure taken by our English pirates daily from Christians and carried to Algire and Tunis to the great enriching of Mores and Turks and impoverishing of Christians" Privateers and slavery of Christians originating from Algiers were a major problem throughout the centuries, leading to regular punitive expeditions by European powers. Spain (1567, 1775, 1783), Denmark (1770), France (1661, 1665, 1682, 1683, 1688), England (1622, 1655, 1672), all led naval bombardments against Algiers. Abraham Duquesne fought the Barbary pirates in 1681 and bombarded Algiers between 1682 and 1683, to help Christian captives. === Political Turmoil (1659-1713) === ==== The Agha period ==== In 1659 the Janissaries of the Odjak of Algiers took over the country, and removed the local Pasha with the blessing of the Ottoman Sultan. From there on a system of dual leaders was in place. There was first and foremost the Agha, elected by the Odjak, and the Pasha appointed by the Ottoman Sublime Porte, whom was a major cause of unrest. Of course, this duality was not stable. All of the Aghas were assassinated, without an exception. Even the first Agha was killed after only 1 year of rule. Thanks to this the Pashas from Constantinople were able to increase the power, and reaffirm Turkish control over the region. In 1671, the Rais, the pirate captains, elected a new leader, Mohamed Trik. The Janissaries also supported him, and started calling him the Dey, which means Uncle in Turkish. ==== Early Dey period (1671-1710) ==== In the early Dey period the country worked similarly to before, with the Pasha still holding considerable powers, but instead of the Janissaries electing their own leaders freely, other factions such as the Taifa of Rais also wanted to elect the deys. Mohammed Trik, taking over during a time instability was faced with heavy issues. Not only were the Janissaries on a rampage, removing any leaders for even the smallest mistakes (even if those leaders were elected by them), but the native populace was also restless. The conflicts with European powers didn't help this either. In 1677, following an explosion in Algiers and several attempts at his life, Mohammed escaped to Tripoli leaving Algiers to Baba Hassan. Just 4 years into his rule he was already at war with one of the most powerful countries in Europe, the Kingdom of France. In 1682 France bombarded Algiers for the first time. The Bombardment was inconclusive, and the leader of the fleet Abraham Duquesne failed to secure the submission of Algiers. The next year, Algiers was bombarded again, this time liberating a few slaves. Before a peace treaty could be signed though, Baba Hassan was deposed and killed by a Rais called Mezzo Morto Hüseyin. Continuing the war against France he was defeated in a naval battle in 1685, near Cherchell, and at last a French Bombardment in 1688 brought an end to his reign, and the war. His successor, Hadj Chabane was elected by the Raïs. He defeated Morocco in the Battle of Moulouya and defeated Tunis as well. He went back to Algiers, but he was assassinated in 1695 by the Janissaries whom once again took over the country. From there on Algiers was in turmoil once again. Leaders were assassinated, despite not even ruling for a year, and the Pasha was still a cause of unrest. The only notable event during this time of unrest was the recapture of Oran and Mers-el-Kébir from the Spanish. ==== Coup of Baba Ali Chaouche, and independence ==== Baba Ali Chaouche, also written as Chaouch, took over the country, ending the rule of the Janissaries. The Pasha attempted to resist him, but instead he was sent home, and told to never come back, and if he did he will be executed. He also sent a letter to the Ottoman sultan declaring that Algiers will from then on act as an independent state, and will not be an Ottoman vassal, but an ally at best. The Sublime Porte, enraged, tried to send another Pasha to Algiers, whom was then sent back to Constantinople by the Algerians. This marked the de facto independence of Algiers from the Ottoman Empire. === Danish–Algerian War === In the mid-1700s Dano-Norwegian trade in the Mediterranean expanded. In order to protect the lucrative business against piracy, Denmark–Norway had secured a peace deal with the states of Barbary Coast. It involved paying an annual tribute to the individual rulers and additionally to the States. In 1766, Algiers had a new ruler, dey Baba Mohammed ben-Osman. He demanded that the annual payment made by Denmark-Norway should be increased, and he should receive new gifts. Denmark–Norway refused the demands. Shortly after, Algerian pirates hijacked three Dano-Norwegian ships and allowed the crew to be sold as slaves. They threatened to bombard the Algerian capital if the Algerians did not agree to a new peace deal on Danish terms. Algiers was not intimidated by the fleet, the fleet was of 2 frigates, 2 bomb galiot and 4 ship of the line. === Algerian-Sharifian War === In the west, the Algerian-Cherifian conflicts shaped the western border of Algeria. There were numerous battles between the Regency of Algiers and the Sharifian Empires for example: the campaign of Tlemcen in 1551, the campaign of Tlemcen in 1557, the Battle of Moulouya and the Battle of Chelif. The independent Kabyle Kingdoms also had some involvement, the Kingdom of Beni Abbes participated in the campaign of Tlemcen in 1551 and the Kingdom of Kuku provided Zwawa troops for the capture of Fez in 1576 in which Abd al-Malik was installed as an Ottoman vassal ruler over the Saadi Dynasty. The Kingdom of Kuku also participated in the capture of Fez in 1554 in which Salih Rais defeated the Moroccan army and conquered Morocco up until Fez, adding these territories to the Ottoman crown and placing Ali Abu Hassun as the ruler and vassal to the Ottoman sultan. In 1792 the Regency of Algiers managed to take possession of the Moroccan Rif and Oujda, which they then abandoned in 1795 for unknown reasons. === Barbary Wars === During the early 19th century, Algiers again resorted to widespread piracy against shipping from Europe and the young United States of America, mainly due to internal fiscal difficulties, and the damage caused by the Napoleonic Wars. This in turn led to the First Barbary War and Second Barbary War, which culminated in August 1816 when Lord Exmouth executed a naval bombardment of Algiers, the biggest, and most successful one. The Barbary Wars resulted in a major victory for the American, British, and Dutch Navy. === Political status === ==== 1516-1567 ==== In between 1516 and 1567, the rulers of the Regency were chosen by the Ottoman sultan. During the first few decades, Algiers was completely aligned with the Ottoman Empire, although it later gained a certain level of autonomy as it was the westernmost province of the Ottoman Empire, and administering it directly would have been problematic. ==== 1567-1710 ==== During this period a form of dual leadership was in place, with the Aghas sharing power and influence with a Pasha appointed by the Ottoman sultan from Constantinople. After 1567, the Deys became the main leaders of the country, although the Pashas still retained some power. ==== 1710-1830 ==== After a coup by Baba Ali Chaouch, the political situation of Algiers became complicated. ==== Relation with the Ottoman Empire ==== Some sources describe it as completely independent from the Ottomans, albeit the state was still nominally part of the Ottoman Empire. Cur Abdy, dey of Algiers shouted at an Ottoman envoy for claiming that the Ottoman Padishah was the king of Algiers ("King of Algiers? King of Algiers? If he is the King of Algiers then who am I?"). Despite the Ottomans having no influence in Algiers, and the Algerians often ignoring orders from the Ottoman sultan, such as in 1784. In some cases Algiers also participated in the Ottoman Empire's wars, such as the Russo-Turkish War (1787–1792), albeit this was not common, and in 1798 for example Algiers sold wheat to the French Empire campaigning in Egypt against the Ottomans through two Jewish traders. In some cases, Algiers was declared to be a country rebelling against the holy law of Islam by the Ottoman Caliph. This usually meant a declaration of war by the Ottomans against the Deylik of Algiers. This could happen due to many reasons. For example, under the rule of Haji Ali Dey, Algerian pirates regularly attacked Ottoman shipments, and Algiers waged war against the Beylik of Tunis, despite several protests by the Ottoman Porte, which resulted in a declaration of war. It can be thus said that the relationship between the Ottoman Empire and Algiers mainly depended on what the Dey at the time wanted. While in some cases, if the relationship between the two was favorable, Algiers did participate in Ottoman wars, Algiers otherwise remained completely autonomous from the rest of the Empire similar to the other Barbary States. == French rule == === 19th century colonialism === North African boundaries have shifted during various stages of the conquests. The borders of modern Algeria were expanded by the French, whose colonization began in 1830 (French invasion began on July 5), though it was not fully conquered and pacified until 1903. To benefit French colonists (many of whom were not in fact of French origin but Italian, Maltese, and Spanish) and nearly the entirety of whom lived in urban areas, northern Algeria was eventually organized into overseas departments of France, with representatives in the French National Assembly. France controlled the entire country, but the traditional Muslim population in the rural areas remained separated from the modern economic infrastructure of the European community. As a result of what the French considered an insult to the French consul in Algiers by the Day in 1827, France blockaded Algiers for three years. In 1830, France invaded and occupied the coastal areas of Algeria, citing a diplomatic incident as casus belli. Hussein Dey went into exile. French colonization then gradually penetrated southwards, and came to have a profound impact on the area and its populations. The European conquest, initially accepted in the Algiers region, was soon met by a rebellion, led by Abdel Kadir, which took roughly a decade for the French troops to put down. By 1848 nearly all of northern Algeria was under French control, and the new government of the French Second Republic declared the occupied lands an integral part of France. Three "civil territories"—Algiers, Oran, and Constantine—were organized as French départements (local administrative units) under a civilian government. During the "Pacification of Algeria", which lasted until 1903, the French perpetrated atrocities which included mass executions of civilians and prisoners and the use of concentration camps; many estimates indicates that the native Algerian population fell by one-third in the years between the French invasion and the end of fighting in the mid-1870s due to warfare, disease and starvation. Various governments and scholars consider France's actions in Algeria as constituting a genocide. Napoleon III set up a Project of Arab kingdom in Algeria between 1860 and 1870. His goal was to take Algeria out of legal limbo and make it a kingdom associated with France before his project was abandoned by the Third Republic. In addition to enduring the affront of being ruled by a foreign, non-Muslim power, many Algerians lost their lands to the new government or to colonists. Traditional leaders were eliminated, coopted, or made irrelevant, and the traditional educational system was largely dismantled; social structures were stressed to the breaking point. From 1856, native Muslims and Jews were viewed as French subjects not citizens. However, in 1865, Napoleon III allowed them to apply for full French citizenship, a measure that few took, since it involved renouncing the right to be governed by sharia law in personal matters, and was considered a kind of apostasy; in 1870, the Crémieux Decree made French citizenship automatic for Jewish natives, a move which largely angered many Muslims, which resulted in the Jews being seen as the accomplices of the colonial power by anti-colonial Algerians. Nonetheless, this period saw progress in health, some infrastructures, and the overall expansion of the economy of Algeria, as well as the formation of new social classes, which, after exposure to ideas of equality and political liberty, would help propel the country to independence. During the colonization France focused on eradicating the local culture by destroying hundreds years old palaces and important buildings. It is estimated that around half of Algiers, a city founded in the 10th century, was destroyed. Many segregatory laws were levied against the Algerians and their culture. === Rise of Algerian nationalism and French resistance === A new generation of Islamic leadership emerged in Algeria at the time of World War I and grew to maturity during the 1920s and 1930s. Various groups were formed in opposition to French rule, most notable the National Liberation Front (FLN) and the National Algerian Movement. Colons (colonists), or, more popularly, pieds noirs (literally, black feet) dominated the government and controlled the bulk of Algeria's wealth. Throughout the colonial era, they continued to block or delay all attempts to implement even the most modest reforms. But from 1933 to 1936, mounting social, political, and economic crises in Algeria induced the indigenous population to engage in numerous acts of political protest. The government responded with more restrictive laws governing public order and security. Algerian Muslims rallied to the French side at the start of World War II as they had done in World War I. But the colons were generally sympathetic to the collaborationist Vichy regime established following France's defeat by Nazi Germany. After the fall of the Vichy regime in Algeria (November 11, 1942) as a result of Operation Torch, the Free French commander in chief in North Africa slowly rescinded repressive Vichy laws, despite opposition by colon extremists. In March 1943, Muslim leader Ferhat Abbas presented the French administration with the Manifesto of the Algerian People, signed by 56 Algerian nationalist and international leaders. The manifesto demanded an Algerian constitution that would guarantee immediate and effective political participation and legal equality for Muslims. Instead, the French administration in 1944 instituted a reform package, based on the 1936 Viollette Plan, that granted full French citizenship only to certain categories of "meritorious" Algerian Muslims, who numbered about 60,000. In April 1945 the French had arrested the Algerian nationalist leader Messali Hadj. On May 1 the followers of his Parti du Peuple Algérien (PPA) participated in demonstrations which were violently put down by the police. Several Algerians were killed. The tensions between the Muslim and colon communities exploded on May 8, 1945, V-E Day, causing the Sétif and Guelma massacre. When a Muslim march was met with violence, marchers rampaged. The army and police responded by conducting a prolonged and systematic ratissage (literally, raking over) of suspected centers of dissidence. According to official French figures, 1,500 Muslims died as a result of these countermeasures. Other estimates vary from 6,000 to as high as 45,000 killed. Many nationalists drew the conclusion that independence could not be won by peaceful means, and so started organizing for violent rebellion. In August 1947, the French National Assembly approved the government-proposed Organic Statute of Algeria. This law called for the creation of an Algerian Assembly with one house representing Europeans and "meritorious" Muslims and the other representing the remaining 8 million or more Muslims. Muslim and colon deputies alike abstained or voted against the statute but for diametrically opposed reasons: the Muslims because it fell short of their expectations and the colons because it went too far. === Algerian War of Independence (1954–1962) === The Algerian War of Independence (1954–1962), brutal and long, was the most recent major turning point in the country's history. Although often fratricidal, it ultimately united Algerians and seared the value of independence and the philosophy of anticolonialism into the national consciousness. In the early morning hours of November 1, 1954, the National Liberation Front (Front de Libération Nationale—FLN) launched attacks throughout Algeria in the opening salvo of a war of independence. An important watershed in this war was the massacre of Pieds-Noirs civilians by the FLN near the town of Philippeville in August 1955. Which prompted Jacques Soustelle into calling for more repressive measures against the rebels. The French authorities claimed that 1,273 "guerrillas" died in what Soustelle admitted were "severe" reprisals. The FLN subsequently, giving names and addresses, claimed that 12,000 Muslims were killed. After Philippeville, all-out war began in Algeria. The FLN fought largely using guerrilla tactics whilst the French counter-insurgency tactics often included severe reprisals and repression. Eventually, protracted negotiations led to a cease-fire signed by France and the FLN on March 18, 1962, at Evian, France. The Evian accords also provided for continuing economic, financial, technical, and cultural relations, along with interim administrative arrangements until a referendum on self-determination could be held. The Evian accords guaranteed the religious and property rights of French settlers, but the perception that they would not be respected led to the exodus of one million pieds-noirs and harkis. Abusive tactics of the French Army remains a controversial subject in France to this day. Deliberate illegal methods were used, such as beatings, mutilations, hanging by the feet or hands, torture by electroshock, waterboarding, sleep deprivation and sexual assaults, among others. French war crimes against Algerian civilians were also committed, including indiscriminate shootings of civilians, bombings of villages suspected of helping the ALN, rape, disembowelment of pregnant women, imprisonment without food in small cells (some of which were small enough to impede lying down), throwing prisoners out of helicopters to their death or into the sea with concrete on their feet, and burying people alive. The FLN also committed many atrocities, both against French pieds-noirs and against fellow Algerians whom they deemed as supporting the French. These crimes included killing unarmed men, women and children, rape and disembowelment or decapitation of women and murdering children by slitting their throats or banging their heads against walls. Between 350,000 and 1 million Algerians are estimated to have died during the war, and more than 2 million, out of a total Muslim population of 9 or 10 million, were made into refugees or forcibly relocated into government-controlled camps. Much of the countryside and agriculture was devastated, along with the modern economy, which had been dominated by urban European settlers (the pied-noirs). French sources estimated that at least 70,000 Muslim civilians were killed or abducted and presumed killed, by the FLN during the Algerian War. Nearly one million people of mostly French, Spanish and Italian descent left the country at independence due to the privileges that they lost as settlers and their unwillingness to be on equal footing with indigenous Algerians along with them left most Algerians of Jewish descent and those Muslim Algerians who had supported a French Algeria (harkis). 30–150,000 pro-French Muslims were also killed in Algeria by FLN in post-war reprisals. == Independent Algeria == === Ben Bella presidency (1962–65) === The Algerian independence referendum was held in French Algeria on 1 July 1962, passing with 99.72% of the vote. As a result, France declared Algeria independent on 3 July. On 8 September 1963, the first Algerian constitution was adopted by nationwide referendum under close supervision by the National Liberation Front (FLN). Later that month, Ahmed Ben Bella was formally elected the first president of Algeria for a five-year term after receiving support from the FLN and the military, led by Colonel Houari Boumédiène. However, the war for independence and its aftermath had severely disrupted Algeria's society and economy. In addition to the destruction of much of Algeria's infrastructure, an exodus of the upper-class French and European colons from Algeria deprived the country of most of its managers, civil servants, engineers, teachers, physicians, and skilled workers. The homeless and displaced numbered in the hundreds of thousands, many suffering from illness, and some 70 percent of the workforce was unemployed. The months immediately following independence witnessed the pell-mell rush of Algerians and government officials to claim the property and jobs left behind by the European colons. For example in the 1963 March Decrees, President Ben Bella declared all agricultural, industrial, and commercial properties previously owned and operated by Europeans vacant, thereby legalizing confiscation by the state. The military played an important role in Ben Bella's administration. Since the president recognized the role that the military played in bringing him to power, he appointed senior military officers as ministers and other important positions within the new state, including naming Colonel Boumédiène as defence minister. These military officials played a core role into implementing the country's security and foreign policy. Under the new constitution, Ben Bella's presidency combined the functions of chief of state and head of government with those of supreme commander of the armed forces. He formed his government without needing legislative approval and was responsible for the definition and direction of its policies. There was no effective institutional check on the president's powers. As a result, opposition leader Hocine Aït-Ahmed quit the National Assembly in 1963 to protest the increasingly dictatorial tendencies of the regime and formed a clandestine resistance movement, the Socialist Forces Front (Front des Forces Socialistes—FFS), dedicated to overthrowing the Ben Bella regime by force. Late summer 1963 saw sporadic incidents attributed to the FFS, but more serious fighting broke out a year later, and the army moved quickly and in force to crush a rebellion. Minister of Defense Boumédiène had no qualms about sending the army to put down regional uprisings because he felt they posed a threat to the state. However, President Ben Bella attempted to co-opt allies from among these regional leaders in order to undermine the ability of military commanders to influence foreign and security policy. Tensions consequently built between Boumédiène and Ben Bella, and in 1965 the military removed Ben Bella in a coup d'état, replacing him with Boumédiène as head of state. === The 1965 coup and the Boumédienne military regime === On 19 June 1965, Houari Boumédiène deposed Ahmed Ben Bella in a military coup d'état that was both swift and bloodless. Ben Bella "disappeared", and would not be seen again until he was released from house arrest in 1980 by Boumédiène's successor, Colonel Chadli Bendjedid. Boumédiène immediately dissolved the National Assembly and suspended the 1963 constitution. Political power resided in the Nation Council of the Algerian Revolution (Conseil National de la Révolution Algérienne—CNRA), a predominantly military body intended to foster cooperation among various factions in the army and the party. Houari Boumédiène's position as head of government and of state was initially insecure, partly because of his lack of a significant power base outside of the armed forces. He relied strongly on a network of former associates known as the Oujda group, named after Boumédiène's posting as National Liberation Army (Armée de Libération Nationale—ALN) leader in the Moroccan border town of Oujda during the war years, but he could not fully dominate his fractious regime. This situation may have accounted for his deference to collegial rule. Over Boumédiène's 11-year reign as Chairman of the CNRA, the council introduced two formal mechanisms: the People's Municipal Assembly (Assemblée Populaires Communales) and the People's Provincial Assembly (Assemblée Populaires de Wilaya) for popular participation in politics. Under Boumédiène's rule, leftist and socialist concepts were merged with Islam. Boumédiène also used Islam to opportunistically consolidate his power. On one hand, he made token concessions and cosmetic changes to the government to appear more Islamic, such as putting Islamist Ahmed Taleb Ibrahimi in charge of national education in 1965 and adopting policies criminalizing gambling, establishing Friday as the national holiday, and dropping plans to introduce birth control to paint an Islamic image of the new government. But on the other hand, Boumédiène's government also progressively repressed Islamic groups, such as by ordering the dissolution of Al Qiyam. Following attempted coups—most notably that of chief-of-staff Col. Tahar Zbiri in December 1967—and a failed assassination attempt on 25 April 1968, Boumédiène consolidated power and forced military and political factions to submit. He took a systematic, authoritarian approach to state building, arguing that Algeria needed stability and an economic base before building any political institutions. Eleven years after Boumédiène took power, after much public debate, a long-promised new constitution was promulgated in November 1976. The constitution restored the National Assembly and gave it legislative, consent, and oversight functions. Boumédiène was later elected president with 95 percent of the cast votes. === Bendjedid rule (1978–92), the 1992 Coup d'État and the rise of the civil war === Boumédiène's death on 27 December 1978 set off a struggle within the FLN to choose a successor. A deadlock occurred between two candidates was broken when Colonel Chadli Bendjedid, a moderate who had collaborated with Boumédiène in deposing Ahmed Ben Bella, was sworn in on February 9, 1979. He was re-elected in 1984 and 1988. After the violent 1988 October Riots, a new constitution was adopted in 1989 that eradicated the Algerian one-party state by allowing the formation of political associations in addition to the FLN. It also removed the armed forces, which had run the government since the days of Boumédiène, from a role in the operation of the government. Among the scores of parties that sprang up under the new constitution, the militant Islamic Salvation Front (Front Islamique du Salut—FIS) was the most successful, winning a majority of votes in the June 1990 municipal elections, as well as the first stage of the December national legislative elections. The surprising first round of success for the fundamentalist FIS party in the December 1991 balloting caused the army to discuss options to intervene in the election. Officers feared that an Islamist government would interfere with their positions and core interests in economic, national security, and foreign policy, since the FIS has promised to make a fundamental re-haul of the social, political, and economic structure to achieve a radical Islamist agenda. Senior military figures, such as Defence Minister Khaled Nezzar, Chief of the General Staff Abdelmalek Guenaizia, and other leaders of the navy, Gendarmerie, and security services, all agreed that the FIS should be stopped from gaining power at the polling box. They also agreed that Bendjedid would need to be removed from office due to his determination to uphold the country's new constitution by continuing with the second round of ballots. On 11 January 1992, Bendjedid announced his resignation on national television, saying it was necessary to "protect the unity of the people and the security of the country". Later that same day, the High Council of State (Haut Comité d'Etat—HCE), which was composed of five people (including Khaled Nezzar, Tedjini Haddam, Ali Kafi, Mohamed Boudiaf and Ali Haroun), was appointed to carry out the duties of the president. The new government, led by Sid Ahmed Ghozali, banned all political activity at mosques and began stopping people from attending prayers at popular mosques. The FIS was legally dissolved by Interior Minister Larbi Belkheir on 9 February for attempting "insurrections against the state". A state of emergency was also declared and extraordinary powers, such as curtailing the right to associate, were granted to the regime. Between January and March, a growing number of FIS militants were arrested by the military, including Abdelkader Hachani and his successors, Othman Aissani and Rabah Kebir. Following the announcement to dissolve the FIS and implement a state of emergency on 9 February, the Algerian security forces used their new emergency powers to conduct large scale arrests of FIS members and housed them in 5 "detention centers" in the Sahara. Between 5,000 (official number) and 30,000 (FIS number) people were detained. This crackdown led to a fundamental Islamic insurgency, resulting in the continuous and brutal 10 year-long Algerian Civil War. During the civil war, the secular state apparatus nonetheless allowed elections featuring pro-government and moderate religious-based parties. The civil war lasted from 1991 to 2002. === Civil War and Bouteflika (1992–2019) === After Chadli Bendjedid resigned from the presidency in the military coup of 1992, a series of figureheads were selected by the military to assume the presidency, as officers were reluctant to assume public political power even though they had manifested control over the government. Additionally, the military's senior leaders felt a need to give a civilian face to the new political regime they had hastily constructed in the aftermath of Benjedid's ousting and the termination of elections, preferring a friendlier non-military face to front the regime. The first such head of state was Mohamed Boudiaf, who was appointed president of the High Council of State (HCE) in February 1992 after a 27-year exile in Morocco. However, Boudiaf quickly came to odds with the military when attempts by Boudiaf to appoint his own staff or form a political party were viewed with suspicion by officers. Boudiaf also launched political initiatives, such as a rigorous anti-corruption campaign in April 1992 and the sacking of Khaled Nezzar from his post as Defence Minister, which were seen by the military as an attempt to remove their influence in the government. The former of these initiatives was especially hazardous to the many senior military officials who had benefited massively and illegally from the political system for years. In the end, Boudiaf was assassinated in June 1992 by one of his bodyguards with Islamist sympathies. Ali Kafi briefly assumed the HCE presidency after Boudiaf's death, before Liamine Zéroual was appointed as a long-term replacement in 1994. However, Zéroual only remained in office for four years before he announced his retirement, as he quickly became embroiled in a clan warfare within the upper classes of the military and fell out with groups of the more senior generals. After this Abdelaziz Bouteflika, Boumédiène's foreign minister, succeeded as the president. As the Algerian civil war wound to a close, presidential elections were held again in April 1999. Although seven candidates qualified for election, all but Abdelaziz Bouteflika, who had the support of the military as well as the National Liberation Front (FLN), withdrew on the eve of the election amid charges of electoral fraud and interference from the military. Bouteflika went on to win with 70 percent of the cast votes. Despite the purportedly democratic elections, the civilian government immediately after the 1999 elections only acted as a sort of 'hijab' over the true government, mostly running day-to-day businesses, while the military still largely ran the country behind the scenes. For example, ministerial mandates to individuals were only granted with the military's approval, and different factions of the military invested in various political parties and the press, using them as pawns to gain influence. However, the military's influence over politics decreased gradually, leaving Bouteflika with more authority on deciding policy. One reason for this was that the senior commanders who had dominated the political scene during the 1960s and 1970s started to retire. Bouteflika's former experience as Boumédiène's foreign minister earned him connections that rejuvenated Algeria's international reputation, which had been tarnished in the early 1990s due to the civil war. On the domestic front, Bouteflika's policy of "national reconciliation" to bring a close to civilian violence earned him a popular mandate that helped him to win further presidential terms in 2004, 2009 and 2014. In 2010, journalists gathered to demonstrate for press freedom and against Bouteflika's self-appointed role as editor-in-chief of Algeria's state television station. In February 2011, the government rescinded the state of emergency that had been in place since 1992 but still banned all protest gatherings and demonstrations. However, in April 2011, over 2,000 protesters defied the official ban and took to the streets of Algiers, clashing with police forces. These protests can be seen as a part of the Arab Spring, with protesters noting that they were inspired by the recent Egyptian revolution, and that Algeria was a police state that was "corrupt to the bone". In 2019, after 20 years in office, Bouteflika announced in February that he would seek a fifth term of office. This sparked widespread discontent around Algeria and protests in Algiers. Despite later attempts at saying he would resign after his term finished in late April, Bouteflika resigned on 2 April, after the chief of the army, Ahmed Gaid Salah, made a declaration that he was "unfit for office". Despite Gaid Salah being loyal to Bouteflika, many in the military identified with civilians, as nearly 70 percent of the army are civilian conscripts who are required to serve for 18 months. Also, since demonstrators demanded a change to the whole governmental system, many army officers aligned themselves with demonstrators in the hopes of surviving an anticipated revolution and retaining their positions. === After Bouteflika (2019-) === After the resignation of Abdelaziz Bouteflika on 9 April 2019, the President of the Council of the Nation Abdelkader Bensalah became acting president of Algeria. Following the presidential election on 12 December 2019, Abdelmadjid Tebboune was elected president after taking 58% of the votes, beating the candidates from both main parties, the National Liberation Front and the Democratic National Rally. On the eve of the first anniversary of the Hirak Movement, which led to the resignation of former president Bouteflika, President Abdelmadjid Tebboune announced in a statement to the Algerian national media that 22 February would be declared the Algerian "National Day of Fraternity and Cohesion between the People and Its Army for Democracy." In the same statement, Tebboune spoke in favor of the Hirak Movement, saying that "the blessed Hirak has preserved the country from a total collapse", and that he had "made a personal commitment to carry out all of the [movement's] demands." On 21 and 22 February 2020, masses of demonstrators (with turnout comparable to well-established Algerian holidays like the Algerian Day of Independence) gathered to honor the anniversary of the Hirak Movement and the newly established national day. In an effort to contain the COVID-19 pandemic, Tebboune announced on 17 March 2020 that "marches and rallies, whatever their motives" would be prohibited. But after protesters and journalists were arrested for participating in such marches, Tebboune faced accusations of attempting to "silence Algerians." Notably, the government's actions were condemned by Amnesty International, which said in a statement that "when all eyes [...] are on the management of the COVID-19 pandemic, the Algerian authorities are devoting time to speeding up the prosecution and trial of activists, journalists, and supporters of the Hirak movement." The National Committee for the Liberation of Detainees (Comité national pour la libération des détenus—CNLD) estimated that around 70 prisoners of conscience were imprisoned by 2 July 2020 and that several of the imprisoned were arrested for Facebook posts. On 28 December 2019, the then-recently inaugurated President Tebboune met with Ahmed Benbitour, the former Algerian Head of Government, with whom he discussed the "foundations of the new Republic." On 8 January 2020, Tebboune established a "commission of experts" composed of 17 members (a majority of which were professors of constitutional law) responsible for examining the previous constitution and making any necessary revisions. Led by Ahmed Laraba, the commission was required to submit its proposals to Tebboune directly within the following two months. In a letter to Laraba on the same day, Tebboune outlined seven axes around which the commission should focus its discussion. These areas of focus included strengthening citizens' rights, combating corruption, consolidating the balance of powers in the Algerian government, increasing the oversight powers of parliament, promoting the independence of the judiciary, furthering citizens' equality under the law, and constitutionalizing elections. Tebboune's letter also included a call for an "immutable and intangible" two-term limit to anyone serving as president — a major point of contention in the initial Hirak Movement protests, which were spurred by former president Abdelaziz Bouteflika's announcement to run for a fifth term. The preliminary draft revision of the constitution was publicly published on 7 May 2020, but the Laraba Commission (as the "commission of experts" came to be known) was open to additional proposals from the public until 20 June. By 3 June, the commission had received an estimated 1,200 additional public proposals. After all revisions were considered by the Laraba Commission, the draft was introduced to the Cabinet of Algeria (Council of Ministers). The revised constitution was adopted in the Council of Ministers on 6 September, in the People's National Assembly on 10 September, and in the Council of the Nation on 12 September. The constitutional changes were approved in the 1 November 2020 referendum, with 66.68% of voters participating in favour of the changes. On 16 February 2021, mass protests and a wave of nationwide rallies and peaceful demonstrations against the government of Abdelmadjid Tebboune began. In May 2021, Algeria prohibited any protests that do not have prior approval by authorities. In September 2024, President Tebboune won a second term with a landslide 84.3 percent of the vote, although his opponents called the results fraud. == See also == Culture of Algeria Colonial heads of Algeria List of heads of government of Algeria History of Africa History of North Africa List of presidents of Algeria Politics of Algeria Prime Minister of Algeria History of cities in Algeria: Algiers history and timeline Oran history and timeline == References == === Notes === === References === === Sources === Abun-Nasr, Jamil M. (1987). A History of the Maghrib in the Islamic Period. Cambridge: Cambridge University Press. ISBN 978-0-521-33767-0. (118) (94) Brett, Michael (2017). The Fatimid Empire. Edinburgh: Edinburgh University Press. ISBN 9781474421522. Despois, J.; Marçais, G.; Colombe, M.; Emerit, M.; Despois, J.; Marçais, PH. (1986) [1960]. "Algeria". In Bearman, P.; Bianquis, Th.; Bosworth, C.E.; van Donzel, E.; Heinrichs, W.P. (eds.). Encyclopaedia of Islam. Vol. I (2nd ed.). Leiden, Netherlands: Brill Publishers. ISBN 9004081143. Ettinghausen, Richard; Grabar, Oleg; Jenkins-Madina, Marilyn (2001). Islamic Art and Architecture: 650–1250. Yale University Press. ISBN 978-0-300-08869-4. Retrieved 2013-03-17. Fage, J. D.; Oliver, Roland Anthony (1975). The Cambridge History of Africa: From c. 500 B.C. to A.D. 1050. Cambridge University Press. ISBN 978-0-521-20981-6. (p.15) (p. 344) Ghalem, Mohamed; Ramaoun, Hassan (2000). L'Algérie: histoire, société et culture (in French). Alger: Casbah Éditions. ISBN 9961-64-189-2. BnF 39208583s. Horne, Alistair (1977). A Savage War of Peace: Algeria 1954-1962. New York Review (published 2006). ISBN 978-1-59017-218-6. Horne, Alistair (1978). A Savage War of Peace: Algeria 1954–1962. New York Review of Books. ISBN 978-1-59017-218-6. Hrbek, I. (1997). "The disintegration of political unity in the Maghrib". In Joseph Ki-Zerbo; Djibril T. Niane (eds.). General History of Africa, vol. IV: Africa from the Twelfth to the Sixteenth Century. UNESCO, James Curry Ltd., and Univ. Calif. Press. Julien, Charles-André (1994). Histoire de l'Afrique du Nord: des origines à 1830 (in French). Payot. ISBN 9782228887892. Kaddache, Mahfoud (1998). L'Algérie durant la période ottomane (in French). Office des publications universitaires. ISBN 978-9961-0-0099-1. Retrieved 2016-09-08. Kaddache, Mahfoud (2011). L'Algérie des Algériens. ISBN 978-9961-9662-1-1. Koulakssis, Ahmed; Meynier, Gilbert (1987). L'emir Khaled: premier zaʼîm? Identité algérienne et colonialisme français. L'Harmattan. ISBN 978-2-85802-859-7. Meynier, Gilbert (2010). L'Algérie, coeur du Maghreb classique: de l'ouverture islamo-arabe au repli (698-1518) (in French). La Découverte. ISBN 978-2-7071-5231-2. Murray, John (1874). A handbook for travellers in Algeria. Retrieved 2013-05-15. Niane, Djibril Tamsir (1984). Africa from the Twelfth to the Sixteenth Century: 4. University of California Press. ISBN 978-0-435-94810-8. Retrieved 2013-05-15. Panzac, Daniel (1995). Histoire économique et sociale de l'Empire ottoman et de la Turquie (1326-1960): actes du sixième congrès international tenu à Aix-en-Provence du 1er au 4 juillet 1992. Peeters Publishers. ISBN 978-9-068-31799-2. Ruano, Delfina S. (2006). "Hafsids". In Josef W Meri (ed.). Medieval Islamic Civilization: an Encyclopedia. Routledge. Tarabulsi, Hasna (2006). "The Zayyanids of Tlemcen and the Hatsids of Tunis". IBN JALDUN: STUDIES. Fundación El legado andalusì. ISBN 978-84-96556-34-8. Retrieved 2013-05-15. Tassy, Mr Laugier de (1725). Histoire du royaume d'Alger: avec l'etat présent de son gouvernement, de ses forces de terre & de mer, de ses revenus, police, justice, politique & commerce (in French). Amsterdam: Henri du Sauzet. Tibi, Amin (2002). "Zīrids". In Bearman, P. J.; Bianquis, Th.; Bosworth, C. E.; van Donzel, E. & Heinrichs, W. P. (eds.). The Encyclopaedia of Islam, Second Edition. Volume XI: W–Z. Leiden: E. J. Brill. pp. 513–516. ISBN 978-90-04-12756-2. Wingfield, Lewis (1868). Under the Palms in Algeria and Tunis ... Hurst and Blackett. Retrieved 2013-05-15. == Further reading == Ageron, Charles Robert, and Michael Brett. Modern Algeria: A History from 1830 to the Present (1992) Bennoune, Mahfoud (1988). The Making of Contemporary Algeria – Colonial Upheavals and Post-Independence Development, 1830–1987. Cambridge: Cambridge University Press. ISBN 978-0-521-30150-3. Derradji, Abder-Rahmane. The Algerian Guerrilla Campaign, Strategy & Tactics (Lewiston, New York: Edwin Mellen Press, 1997). Derradji, Abder-Rahmane. A Concise History of Political Violence in Algeria: Brothers in Faith Enemies in Arms (2 vol. Lewiston, New York: Edwin Mellen Press, 2002), Horne, Alistair. A Savage War of Peace: Algeria 1954-1962 (2006) Laouisset, Djamel (2009). A Retrospective Study of the Algerian Iron and Steel Industry. New York City: Nova Publishers. ISBN 978-1-61761-190-2. Le Sueur, James D. (2010). Algeria since 1989: between terror and democracy. Global history of the present. Halifax [N.S.] : London; New York: Fernwood Pub.; Zed Books. ISBN 9781552662564. McDougall, James. (2017) A history of Algeria (Cambridge UP, 2017). Roberts, Hugh (2003). The Battlefield – Algeria, 1988–2002. Studies in a Broken Polity. London: Verso Books. ISBN 978-1-85984-684-1. Ruedy, John (1992). Modern Algeria – The Origins and Development of a Nation. Bloomington: Indiana University Press. ISBN 978-0-253-34998-9. Sessions, Jennifer E. By Sword and Plow: France and the Conquest of Algeria (Cornell University Press; 2011) 352 pages Stora, Benjamin (2004). Algeria, 1830-2000 : a short history. Ithaca: Cornell University Press. ISBN 9780801489167. Sidaoui, Riadh (2009). "Islamic Politics and the Military – Algeria 1962–2008". Religion and Politics – Islam and Muslim Civilisation. Farnham: Ashgate Publishing. ISBN 0-7546-7418-5. "Algeria : Country Studies - Federal Research Division, Library of Congress". Lcweb2.loc.gov. Archived from the original on 2013-01-15. Retrieved 2012-12-25. === Historiography and memory === Branche, Raphaëlle. "The martyr's torch: memory and power in Algeria." Journal of North African Studies 16.3 (2011): 431–443. Cohen, William B. "Pied-Noir memory, history, and the Algerian War." in Europe's Invisible Migrants (2003): 129-145 online. Hannoum, Abdelmajid. "The historiographic state: how Algeria once became French." History and Anthropology 19.2 (2008): 91-114. online Hassett, Dónal. Mobilizing Memory: The Great War and the Language of Politics in Colonial Algeria, 1918-1939 (Oxford UP, 2019). House, Jim. "Memory and the Creation of Solidarity during the Decolonization of Algeria." Yale French Studies 118/119 (2010): 15-38 online. Johnson, Douglas. "Algeria: some problems of modern history." Journal of African history (1964): 221–242. Lorcin, Patricia M.E., ed. Algeria and France, 1800-2000: identity, memory, nostalgia (Syracuse UP, 2006). McDougall, James. History and the Culture of Nationalism in Algeria (Cambridge UP, 2006) excerpt. Vince, Natalya. Our fighting sisters: Nation, memory and gender in Algeria, 1954–2012 (Manchester UP, 2072115). == External links == "Algeria". State.gov. 2012-08-17. Retrieved 2012-12-25. "Countries Ab-Am". Rulers.org. Retrieved 2012-12-25. List of rulers for Algeria
|
Wikipedia:History of the function concept#0
|
The mathematical concept of a function dates from the 17th century in connection with the development of calculus; for example, the slope d y / d x {\displaystyle dy/dx} of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme. Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Karl Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Georg Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another. == Functions before the 17th century == In the 12th century, mathematician Sharaf al-Din al-Tusi analyzed the equation x3 + d = b ⋅ x2 in the form x2 ⋅ (b – x) = d, stating that the left hand side must at least equal the value of d for the equation to have a solution. He then determined the maximum value of this expression. It is arguable that the isolation of this expression is an early approach to the notion of a "function". A value less than d means no positive solution; a value equal to d corresponds to one solution, while a value greater than d corresponds to two solutions. Sharaf al-Din's analysis of this equation was a notable development in Islamic mathematics, but his work was not pursued any further at that time, neither in the Muslim world nor in Europe. According to Jean Dieudonné and Ponte, the concept of a function emerged in the 17th century as a result of the development of analytic geometry and the infinitesimal calculus. Nevertheless, Medvedev suggests that the implicit concept of a function is one with an ancient lineage. Ponte also sees more explicit approaches to the concept in the Middle Ages: Historically, some mathematicians can be regarded as having foreseen and come close to a modern formulation of the concept of function. Among them is [Nicole] Oresme (1323–1382) . . . In his theory, some general ideas about independent and dependent variable quantities seem to be present. The development of analytical geometry around 1640 allowed mathematicians to go between geometric problems about curves and algebraic relations between "variable coordinates x and y." Calculus was developed using the notion of variables, with their associated geometric meaning, which persisted well into the eighteenth century. However, the terminology of "function" came to be used in interactions between Leibniz and Bernoulli towards the end of the 17th century. == Notion of function in analysis == The term "function" was literally introduced by Gottfried Leibniz, in a 1673 letter, to describe a quantity related to points of a curve, such as a coordinate or curve's slope. Johann Bernoulli started calling expressions made of a single variable "functions." In 1698, he agreed with Leibniz that any quantity formed "in an algebraic and transcendental manner" may be called a function of x. By 1718, he came to regard as a function "any expression made up of a variable and some constants." Alexis Claude Clairaut (in approximately 1734) and Leonhard Euler introduced the familiar notation f ( x ) {\displaystyle {f(x)}} for the value of a function. The functions considered in those times are called today differentiable functions. For this type of function, one can talk about limits and derivatives; both are measurements of the output or the change in the output as it depends on the input or the change in the input. Such functions are the basis of calculus. === Euler === In the first volume of his fundamental text Introductio in analysin infinitorum, published in 1748, Euler gave essentially the same definition of a function as his teacher Bernoulli, as an expression or formula involving variables and constants e.g., x 2 + 3 x + 2 {\displaystyle {x^{2}+3x+2}} . Euler's own definition reads: A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities. Euler also allowed multi-valued functions whose values are determined by an implicit equation. In 1755, however, in his Institutiones calculi differentialis, Euler gave a more general concept of a function: When certain quantities depend on others in such a way that they undergo a change when the latter change, then the first are called functions of the second. This name has an extremely broad character; it encompasses all the ways in which one quantity can be determined in terms of others. Medvedev considers that "In essence this is the definition that became known as Dirichlet's definition." Edwards also credits Euler with a general concept of a function and says further that The relations among these quantities are not thought of as being given by formulas, but on the other hand they are surely not thought of as being the sort of general set-theoretic, anything-goes subsets of product spaces that modern mathematicians mean when they use the word "function". === Fourier === In his Théorie Analytique de la Chaleur, Joseph Fourier claimed that an arbitrary function could be represented by a Fourier series. Fourier had a general conception of a function, which included functions that were neither continuous nor defined by an analytical expression. Related questions on the nature and representation of functions, arising from the solution of the wave equation for a vibrating string, had already been the subject of dispute between Jean le Rond d'Alembert and Euler, and they had a significant impact in generalizing the notion of a function. Luzin observes that: The modern understanding of function and its definition, which seems correct to us, could arise only after Fourier's discovery. His discovery showed clearly that most of the misunderstandings that arose in the debate about the vibrating string were the result of confusing two seemingly identical but actually vastly different concepts, namely that of function and that of its analytic representation. Indeed, prior to Fourier's discovery no distinction was drawn between the concepts of "function" and of "analytic representation," and it was this discovery that brought about their disconnection. === Cauchy === During the 19th century, mathematicians started to formalize all the different branches of mathematics. One of the first to do so was Augustin-Louis Cauchy; his somewhat imprecise results were later made completely rigorous by Weierstrass, who advocated building calculus on arithmetic rather than on geometry, which favoured Euler's definition over Leibniz's (see arithmetization of analysis). According to Smithies, Cauchy thought of functions as being defined by equations involving real or complex numbers, and tacitly assumed they were continuous: Cauchy makes some general remarks about functions in Chapter I, Section 1 of his Analyse algébrique (1821). From what he says there, it is clear that he normally regards a function as being defined by an analytic expression (if it is explicit) or by an equation or a system of equations (if it is implicit); where he differs from his predecessors is that he is prepared to consider the possibility that a function may be defined only for a restricted range of the independent variable. === Lobachevsky and Dirichlet === Nikolai Lobachevsky and Peter Gustav Lejeune Dirichlet are traditionally credited with independently giving the modern "formal" definition of a function as a relation in which every first element has a unique second element. Lobachevsky (1834) writes that The general concept of a function requires that a function of x be defined as a number given for each x and varying gradually with x. The value of the function can be given either by an analytic expression, or by a condition that provides a means of examining all numbers and choosing one of them; or finally the dependence may exist but remain unknown. while Dirichlet (1837) writes If now a unique finite y corresponding to each x, and moreover in such a way that when x ranges continuously over the interval from a to b, y = f ( x ) {\displaystyle {y=f(x)}} also varies continuously, then y is called a continuous function of x for this interval. It is not at all necessary here that y be given in terms of x by one and the same law throughout the entire interval, and it is not necessary that it be regarded as a dependence expressed using mathematical operations. Eves asserts that "the student of mathematics usually meets the Dirichlet definition of function in his introductory course in calculus. Dirichlet's claim to this formalization has been disputed by Imre Lakatos: There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837] paper for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values: ... However, Gardiner says "...it seems to me that Lakatos goes too far, for example, when he asserts that 'there is ample evidence that [Dirichlet] had no idea of [the modern function] concept'." Moreover, as noted above, Dirichlet's paper does appear to include a definition along the lines of what is usually ascribed to him, even though (like Lobachevsky) he states it only for continuous functions of a real variable. Similarly, Lavine observes that: It is a matter of some dispute how much credit Dirichlet deserves for the modern definition of a function, in part because he restricted his definition to continuous functions....I believe Dirichlet defined the notion of continuous function to make it clear that no rule or law is required even in the case of continuous functions, not just in general. This would have deserved special emphasis because of Euler's definition of a continuous function as one given by single expression-or law. But I also doubt there is sufficient evidence to settle the dispute. Because Lobachevsky and Dirichlet have been credited as among the first to introduce the notion of an arbitrary correspondence, this notion is sometimes referred to as the Dirichlet or Lobachevsky-Dirichlet definition of a function. A general version of this definition was later used by Bourbaki (1939), and some in the education community refer to it as the "Dirichlet–Bourbaki" definition of a function. === Dedekind === Dieudonné, who was one of the founding members of the Bourbaki group, credits a precise and general modern definition of a function to Dedekind in his work Was sind und was sollen die Zahlen, which appeared in 1888 but had already been drafted in 1878. Dieudonné observes that instead of confining himself, as in previous conceptions, to real (or complex) functions, Dedekind defines a function as a single-valued mapping between any two sets: What was new and what was to be essential for the whole of mathematics was the entirely general conception of a function. === Hardy === Hardy 1908, pp. 26–28 defined a function as a relation between two variables x and y such that "to some values of x at any rate correspond values of y." He neither required the function to be defined for all values of x nor to associate each value of x to a single value of y. This broad definition of a function encompasses more relations than are ordinarily considered functions in contemporary mathematics. For example, Hardy's definition includes multivalued functions and what in computability theory are called partial functions. == Logicians' function == === Prior to 1850 === Logicians of this time were primarily involved with analyzing syllogisms (the 2000-year-old Aristotelian forms and otherwise), or as Augustus De Morgan (1847) stated it: "the examination of that part of reasoning which depends upon the manner in which inferences are formed, and the investigation of general maxims and rules for constructing arguments". At this time the notion of (logical) "function" is not explicit, but at least in the work of De Morgan and George Boole it is implied: we see abstraction of the argument forms, the introduction of variables, the introduction of a symbolic algebra with respect to these variables, and some of the notions of set theory. De Morgan's 1847 "FORMAL LOGIC OR, The Calculus of Inference, Necessary and Probable" observes that "[a] logical truth depends upon the structure of the statement, and not upon the particular matters spoken of"; he wastes no time (preface page i) abstracting: "In the form of the proposition, the copula is made as abstract as the terms". He immediately (p. 1) casts what he calls "the proposition" (present-day propositional function or relation) into a form such as "X is Y", where the symbols X, "is", and Y represent, respectively, the subject, copula, and predicate. While the word "function" does not appear, the notion of "abstraction" is there, "variables" are there, the notion of inclusion in his symbolism "all of the Δ is in the О" (p. 9) is there, and lastly a new symbolism for logical analysis of the notion of "relation" (he uses the word with respect to this example " X)Y " (p. 75) ) is there: " A1 X)Y To take an X it is necessary to take a Y" [or To be an X it is necessary to be a Y] " A1 Y)X To take a Y it is sufficient to take a X" [or To be a Y it is sufficient to be an X], etc. In his 1848 The Nature of Logic Boole asserts that "logic . . . is in a more especial sense the science of reasoning by signs", and he briefly discusses the notions of "belonging to" and "class": "An individual may possess a great variety of attributes and thus belonging to a great variety of different classes". Like De Morgan he uses the notion of "variable" drawn from analysis; he gives an example of "represent[ing] the class oxen by x and that of horses by y and the conjunction and by the sign + . . . we might represent the aggregate class oxen and horses by x + y". In the context of "the Differential Calculus" Boole defined (circa 1849) the notion of a function as follows: "That quantity whose variation is uniform . . . is called the independent variable. That quantity whose variation is referred to the variation of the former is said to be a function of it. The Differential calculus enables us in every case to pass from the function to the limit. This it does by a certain Operation. But in the very Idea of an Operation is . . . the idea of an inverse operation. To effect that inverse operation in the present instance is the business of the Int[egral] Calculus." == Logicians' function 1850–1950 == Eves observes "that logicians have endeavored to push down further the starting level of the definitional development of mathematics and to derive the theory of sets, or classes, from a foundation in the logic of propositions and propositional functions". But by the late 19th century the logicians' research into the foundations of mathematics was undergoing a major split. The direction of the first group, the Logicists, can probably be summed up best by Bertrand Russell 1903 – "to fulfil two objects, first, to show that all mathematics follows from symbolic logic, and secondly to discover, as far as possible, what are the principles of symbolic logic itself." The second group of logicians, the set-theorists, emerged with Georg Cantor's "set theory" (1870–1890) but were driven forward partly as a result of Russell's discovery of a paradox that could be derived from Frege's conception of "function", but also as a reaction against Russell's proposed solution. Ernst Zermelo's set-theoretic response was his 1908 Investigations in the foundations of set theory I – the first axiomatic set theory; here too the notion of "propositional function" plays a role. === George Boole's The Laws of Thought 1854; John Venn's Symbolic Logic 1881 === In his An Investigation into the laws of thought Boole now defined a function in terms of a symbol x as follows: "8. Definition. – Any algebraic expression involving symbol x is termed a function of x, and may be represented by the abbreviated form f(x)" Boole then used algebraic expressions to define both algebraic and logical notions, e.g., 1 − x is logical NOT(x), xy is the logical AND(x,y), x + y is the logical OR(x, y), x(x + y) is xx + xy, and "the special law" xx = x2 = x. In his 1881 Symbolic Logic Venn was using the words "logical function" and the contemporary symbolism (x = f(y), y = f −1(x), cf page xxi) plus the circle-diagrams historically associated with Venn to describe "class relations", the notions "'quantifying' our predicate", "propositions in respect of their extension", "the relation of inclusion and exclusion of two classes to one another", and "propositional function" (all on p. 10), the bar over a variable to indicate not-x (page 43), etc. Indeed he equated unequivocally the notion of "logical function" with "class" [modern "set"]: "... on the view adopted in this book, f(x) never stands for anything but a logical class. It may be a compound class aggregated of many simple classes; it may be a class indicated by certain inverse logical operations, it may be composed of two groups of classes equal to one another, or what is the same thing, their difference declared equal to zero, that is, a logical equation. But however composed or derived, f(x) with us will never be anything else than a general expression for such logical classes of things as may fairly find a place in ordinary Logic". === Frege's Begriffsschrift 1879 === Gottlob Frege's Begriffsschrift (1879) preceded Giuseppe Peano (1889), but Peano had no knowledge of Frege 1879 until after he had published his 1889. Both writers strongly influenced Russell (1903). Russell in turn influenced much of 20th-century mathematics and logic through his Principia Mathematica (1913) jointly authored with Alfred North Whitehead. At the outset Frege abandons the traditional "concepts subject and predicate", replacing them with argument and function respectively, which he believes "will stand the test of time. It is easy to see how regarding a content as a function of an argument leads to the formation of concepts. Furthermore, the demonstration of the connection between the meanings of the words if, and, not, or, there is, some, all, and so forth, deserves attention". Frege begins his discussion of "function" with an example: Begin with the expression "Hydrogen is lighter than carbon dioxide". Now remove the sign for hydrogen (i.e., the word "hydrogen") and replace it with the sign for oxygen (i.e., the word "oxygen"); this makes a second statement. Do this again (using either statement) and substitute the sign for nitrogen (i.e., the word "nitrogen") and note that "This changes the meaning in such a way that "oxygen" or "nitrogen" enters into the relations in which "hydrogen" stood before". There are three statements: "Hydrogen is lighter than carbon dioxide." "Oxygen is lighter than carbon dioxide." "Nitrogen is lighter than carbon dioxide." Now observe in all three a "stable component, representing the totality of [the] relations"; call this the function, i.e., "... is lighter than carbon dioxide", is the function. Frege calls the argument of the function "[t]he sign [e.g., hydrogen, oxygen, or nitrogen], regarded as replaceable by others that denotes the object standing in these relations". He notes that we could have derived the function as "Hydrogen is lighter than . . .." as well, with an argument position on the right; the exact observation is made by Peano (see more below). Finally, Frege allows for the case of two (or more) arguments. For example, remove "carbon dioxide" to yield the invariant part (the function) as: "... is lighter than ... " The one-argument function Frege generalizes into the form Φ(A) where A is the argument and Φ( ) represents the function, whereas the two-argument function he symbolizes as Ψ(A, B) with A and B the arguments and Ψ( , ) the function and cautions that "in general Ψ(A, B) differs from Ψ(B, A)". Using his unique symbolism he translates for the reader the following symbolism: "We can read |--- Φ(A) as "A has the property Φ. |--- Ψ(A, B) can be translated by "B stands in the relation Ψ to A" or "B is a result of an application of the procedure Ψ to the object A". === Peano's The Principles of Arithmetic 1889 === Peano defined the notion of "function" in a manner somewhat similar to Frege, but without the precision. First Peano defines the sign "K means class, or aggregate of objects", the objects of which satisfy three simple equality-conditions, a = a, (a = b) = (b = a), IF ((a = b) AND (b = c)) THEN (a = c). He then introduces φ, "a sign or an aggregate of signs such that if x is an object of the class s, the expression φx denotes a new object". Peano adds two conditions on these new objects: First, that the three equality-conditions hold for the objects φx; secondly, that "if x and y are objects of class s and if x = y, we assume it is possible to deduce φx = φy". Given all these conditions are met, φ is a "function presign". Likewise he identifies a "function postsign". For example if φ is the function presign a+, then φx yields a+x, or if φ is the function postsign +a then xφ yields x+a. === Bertrand Russell's The Principles of Mathematics 1903 === While the influence of Cantor and Peano was paramount, in Appendix A "The Logical and Arithmetical Doctrines of Frege" of The Principles of Mathematics, Russell arrives at a discussion of Frege's notion of function, "...a point in which Frege's work is very important, and requires careful examination". In response to his 1902 exchange of letters with Frege about the contradiction he discovered in Frege's Begriffsschrift Russell tacked this section on at the last moment. For Russell the bedeviling notion is that of variable: "6. Mathematical propositions are not only characterized by the fact that they assert implications, but also by the fact that they contain variables. The notion of the variable is one of the most difficult with which logic has to deal. For the present, I openly wish to make it plain that there are variables in all mathematical propositions, even where at first sight they might seem to be absent. . . . We shall find always, in all mathematical propositions, that the words any or some occur; and these words are the marks of a variable and a formal implication". As expressed by Russell "the process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition ... So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of mathematics to do it"; these generalizations Russell named propositional functions. Indeed he cites and quotes from Frege's Begriffsschrift and presents a vivid example from Frege's 1891 Function und Begriff: That "the essence of the arithmetical function 2x3 + x is what is left when the x is taken away, i.e., in the above instance 2( )3 + ( ). The argument x does not belong to the function but the two taken together make the whole". Russell agreed with Frege's notion of "function" in one sense: "He regards functions – and in this I agree with him – as more fundamental than predicates and relations" but Russell rejected Frege's "theory of subject and assertion", in particular "he thinks that, if a term a occurs in a proposition, the proposition can always be analysed into a and an assertion about a". === Evolution of Russell's notion of "function" 1908–1913 === Russell would carry his ideas forward in his 1908 Mathematical logical as based on the theory of types and into his and Whitehead's 1910–1913 Principia Mathematica. By the time of Principia Mathematica Russell, like Frege, considered the propositional function fundamental: "Propositional functions are the fundamental kind from which the more usual kinds of function, such as "sin x" or log x or "the father of x" are derived. These derivative functions . . . are called "descriptive functions". The functions of propositions . . . are a particular case of propositional functions". Propositional functions: Because his terminology is different from the contemporary, the reader may be confused by Russell's "propositional function". An example may help. Russell writes a propositional function in its raw form, e.g., as φŷ: "ŷ is hurt". (Observe the circumflex or "hat" over the variable y). For our example, we will assign just 4 values to the variable ŷ: "Bob", "This bird", "Emily the rabbit", and "y". Substitution of one of these values for variable ŷ yields a proposition; this proposition is called a "value" of the propositional function. In our example there are four values of the propositional function, e.g., "Bob is hurt", "This bird is hurt", "Emily the rabbit is hurt" and "y is hurt." A proposition, if it is significant—i.e., if its truth is determinate—has a truth-value of truth or falsity. If a proposition's truth value is "truth" then the variable's value is said to satisfy the propositional function. Finally, per Russell's definition, "a class [set] is all objects satisfying some propositional function" (p. 23). Note the word "all" – this is how the contemporary notions of "For all ∀" and "there exists at least one instance ∃" enter the treatment (p. 15). To continue the example: Suppose (from outside the mathematics/logic) one determines that the propositions "Bob is hurt" has a truth value of "falsity", "This bird is hurt" has a truth value of "truth", "Emily the rabbit is hurt" has an indeterminate truth value because "Emily the rabbit" doesn't exist, and "y is hurt" is ambiguous as to its truth value because the argument y itself is ambiguous. While the two propositions "Bob is hurt" and "This bird is hurt" are significant (both have truth values), only the value "This bird" of the variable ŷ satisfies the propositional function φŷ: "ŷ is hurt". When one goes to form the class α: φŷ: "ŷ is hurt", only "This bird" is included, given the four values "Bob", "This bird", "Emily the rabbit" and "y" for variable ŷ and their respective truth-values: falsity, truth, indeterminate, ambiguous. Russell defines functions of propositions with arguments, and truth-functions f(p). For example, suppose one were to form the "function of propositions with arguments" p1: "NOT(p) AND q" and assign its variables the values of p: "Bob is hurt" and q: "This bird is hurt". (We are restricted to the logical linkages NOT, AND, OR and IMPLIES, and we can only assign "significant" propositions to the variables p and q). Then the "function of propositions with arguments" is p1: NOT("Bob is hurt") AND "This bird is hurt". To determine the truth value of this "function of propositions with arguments" we submit it to a "truth function", e.g., f(p1): f( NOT("Bob is hurt") AND "This bird is hurt" ), which yields a truth value of "truth". The notion of a "many-one" functional relation": Russell first discusses the notion of "identity", then defines a descriptive function (pages 30ff) as the unique value ιx that satisfies the (2-variable) propositional function (i.e., "relation") φŷ. N.B. The reader should be warned here that the order of the variables are reversed! y is the independent variable and x is the dependent variable, e.g., x = sin(y). Russell symbolizes the descriptive function as "the object standing in relation to y": R'y =DEF (ιx)(x R y). Russell repeats that "R'y is a function of y, but not a propositional function [sic]; we shall call it a descriptive function. All the ordinary functions of mathematics are of this kind. Thus in our notation "sin y" would be written " sin 'y ", and "sin" would stand for the relation sin 'y has to y". == Formalist's function: David Hilbert's axiomatization of mathematics (1904–1927) == David Hilbert set himself the goal of "formalizing" classical mathematics "as a formal axiomatic theory, and this theory shall be proved to be consistent, i.e., free from contradiction". In Hilbert 1927 The Foundations of Mathematics he frames the notion of function in terms of the existence of an "object": 13. A(a) --> A(ε(A)) Here ε(A) stands for an object of which the proposition A(a) certainly holds if it holds of any object at all; let us call ε the logical ε-function". [The arrow indicates "implies".] Hilbert then illustrates the three ways how the ε-function is to be used, firstly as the "for all" and "there exists" notions, secondly to represent the "object of which [a proposition] holds", and lastly how to cast it into the choice function. Recursion theory and computability: But the unexpected outcome of Hilbert's and his student Bernays's effort was failure; see Gödel's incompleteness theorems of 1931. At about the same time, in an effort to solve Hilbert's Entscheidungsproblem, mathematicians set about to define what was meant by an "effectively calculable function" (Alonzo Church 1936), i.e., "effective method" or "algorithm", that is, an explicit, step-by-step procedure that would succeed in computing a function. Various models for algorithms appeared, in rapid succession, including Church's lambda calculus (1936), Stephen Kleene's μ-recursive functions(1936) and Alan Turing's (1936–7) notion of replacing human "computers" with utterly-mechanical "computing machines" (see Turing machines). It was shown that all of these models could compute the same class of computable functions. Church's thesis holds that this class of functions exhausts all the number-theoretic functions that can be calculated by an algorithm. The outcomes of these efforts were vivid demonstrations that, in Turing's words, "there can be no general process for determining whether a given formula U of the functional calculus K [Principia Mathematica] is provable"; see more at Independence (mathematical logic) and Computability theory. == Development of the set-theoretic definition == Set theory began with the work of the logicians with the notion of "class" (modern "set") for example De Morgan (1847), Jevons (1880), Venn (1881), Frege (1879) and Peano (1889). It was given a push by Georg Cantor's attempt to define the infinite in set-theoretic treatment (1870–1890) and a subsequent discovery of an antinomy (contradiction, paradox) in this treatment (Cantor's paradox), by Russell's discovery (1902) of an antinomy in Frege's 1879 (Russell's paradox), by the discovery of more antinomies in the early 20th century (e.g., the 1897 Burali-Forti paradox and the 1905 Richard paradox), and by resistance to Russell's complex treatment of logic and dislike of his axiom of reducibility (1908, 1910–1913) that he proposed as a means to evade the antinomies. === Russell's paradox 1902 === In 1902 Russell sent a letter to Frege pointing out that Frege's 1879 Begriffsschrift allowed a function to be an argument of itself: "On the other hand, it may also be that the argument is determinate and the function indeterminate . . .." From this unconstrained situation Russell was able to form a paradox: "You state ... that a function, too, can act as the indeterminate element. This I formerly believed, but now this view seems doubtful to me because of the following contradiction. Let w be the predicate: to be a predicate that cannot be predicated of itself. Can w be predicated of itself?" Frege responded promptly that "Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build arithmetic". From this point forward development of the foundations of mathematics became an exercise in how to dodge "Russell's paradox", framed as it was in "the bare [set-theoretic] notions of set and element". === Zermelo's set theory (1908) modified by Skolem (1922) === The notion of "function" appears as Zermelo's axiom III—the Axiom of Separation (Axiom der Aussonderung). This axiom constrains us to use a propositional function Φ(x) to "separate" a subset MΦ from a previously formed set M: "AXIOM III. (Axiom of separation). Whenever the propositional function Φ(x) is definite for all elements of a set M, M possesses a subset MΦ containing as elements precisely those elements x of M for which Φ(x) is true". As there is no universal set — sets originate by way of Axiom II from elements of (non-set) domain B – "...this disposes of the Russell antinomy so far as we are concerned". But Zermelo's "definite criterion" is imprecise, and is fixed by Weyl, Fraenkel, Skolem, and von Neumann. In fact Skolem in his 1922 referred to this "definite criterion" or "property" as a "definite proposition": "... a finite expression constructed from elementary propositions of the form a ε b or a = b by means of the five operations [logical conjunction, disjunction, negation, universal quantification, and existential quantification]. van Heijenoort summarizes: "A property is definite in Skolem's sense if it is expressed . . . by a well-formed formula in the simple predicate calculus of first order in which the sole predicate constants are ε and possibly, =. ... Today an axiomatization of set theory is usually embedded in a logical calculus, and it is Weyl's and Skolem's approach to the formulation of the axiom of separation that is generally adopted. In this quote the reader may observe a shift in terminology: nowhere is mentioned the notion of "propositional function", but rather one sees the words "formula", "predicate calculus", "predicate", and "logical calculus." This shift in terminology is discussed more in the section that covers "function" in contemporary set theory. === Wiener–Hausdorff–Kuratowski ordered pair definition 1914–1921 === The history of the notion of "ordered pair" is not clear. As noted above, Frege (1879) proposed an intuitive ordering in his definition of a two-argument function Ψ(A, B). Norbert Wiener in his 1914 (see below) observes that his own treatment essentially "revert(s) to Schröder's treatment of a relation as a class of ordered couples". Russell (1903) considered the definition of a relation (such as Ψ(A, B)) as a "class of couples" but rejected it: "There is a temptation to regard a relation as definable in extension as a class of couples. This is the formal advantage that it avoids the necessity for the primitive proposition asserting that every couple has a relation holding between no other pairs of terms. But it is necessary to give sense to the couple, to distinguish the referent [domain] from the relatum [converse domain]: thus a couple becomes essentially distinct from a class of two terms, and must itself be introduced as a primitive idea. . . . It seems therefore more correct to take an intensional view of relations, and to identify them rather with class-concepts than with classes." By 1910–1913 and Principia Mathematica Russell had given up on the requirement for an intensional definition of a relation, stating that "mathematics is always concerned with extensions rather than intensions" and "Relations, like classes, are to be taken in extension". To demonstrate the notion of a relation in extension Russell now embraced the notion of ordered couple: "We may regard a relation ... as a class of couples ... the relation determined by φ(x, y) is the class of couples (x, y) for which φ(x, y) is true". In a footnote he clarified his notion and arrived at this definition: "Such a couple has a sense, i.e., the couple (x, y) is different from the couple (y, x) unless x = y. We shall call it a "couple with sense," ... it may also be called an ordered couple. But he goes on to say that he would not introduce the ordered couples further into his "symbolic treatment"; he proposes his "matrix" and his unpopular axiom of reducibility in their place. An attempt to solve the problem of the antinomies led Russell to propose his "doctrine of types" in an appendix B of his 1903 The Principles of Mathematics. In a few years he would refine this notion and propose in his 1908 The Theory of Types two axioms of reducibility, the purpose of which were to reduce (single-variable) propositional functions and (dual-variable) relations to a "lower" form (and ultimately into a completely extensional form); he and Alfred North Whitehead would carry this treatment over to Principia Mathematica 1910–1913 with a further refinement called "a matrix". The first axiom is *12.1; the second is *12.11. To quote Wiener the second axiom *12.11 "is involved only in the theory of relations". Both axioms, however, were met with skepticism and resistance; see more at Axiom of reducibility. By 1914 Norbert Wiener, using Whitehead and Russell's symbolism, eliminated axiom *12.11 (the "two-variable" (relational) version of the axiom of reducibility) by expressing a relation as an ordered pair using the null set. At approximately the same time, Hausdorff (1914, p. 32) gave the definition of the ordered pair (a, b) as {{a,1}, {b, 2}}. A few years later Kuratowski (1921) offered a definition that has been widely used ever since, namely {{a, b}, {a}}". As noted by Suppes (1960) "This definition . . . was historically important in reducing the theory of relations to the theory of sets. Observe that while Wiener "reduced" the relational *12.11 form of the axiom of reducibility he did not reduce nor otherwise change the propositional-function form *12.1; indeed he declared this "essential to the treatment of identity, descriptions, classes and relations". === Schönfinkel's notion of function as a many-one correspondence 1924 === Where exactly the general notion of "function" as a many-one correspondence derives from is unclear. Russell in his 1920 Introduction to Mathematical Philosophy states that "It should be observed that all mathematical functions result form one-many [sic – contemporary usage is many-one] relations . . . Functions in this sense are descriptive functions". A reasonable possibility is the Principia Mathematica notion of "descriptive function" – R 'y =DEF (ιx)(x R y): "the singular object that has a relation R to y". Whatever the case, by 1924, Moses Schönfinkel expressed the notion, claiming it to be "well known": "As is well known, by function we mean in the simplest case a correspondence between the elements of some domain of quantities, the argument domain, and those of a domain of function values ... such that to each argument value there corresponds at most one function value". According to Willard Quine, Schönfinkel 1924 "provide[s] for ... the whole sweep of abstract set theory. The crux of the matter is that Schönfinkel lets functions stand as arguments. For Schönfinkel, substantially as for Frege, classes are special sorts of functions. They are propositional functions, functions whose values are truth values. All functions, propositional and otherwise, are for Schönfinkel one-place functions". Remarkably, Schönfinkel reduces all mathematics to an extremely compact functional calculus consisting of only three functions: Constancy, fusion (i.e., composition), and mutual exclusivity. Quine notes that Haskell Curry (1958) carried this work forward "under the head of combinatory logic". === Von Neumann's set theory 1925 === By 1925 Abraham Fraenkel (1922) and Thoralf Skolem (1922) had amended Zermelo's set theory of 1908. But von Neumann was not convinced that this axiomatization could not lead to the antinomies. So he proposed his own theory, his 1925 An axiomatization of set theory. It explicitly contains a "contemporary", set-theoretic version of the notion of "function": "[Unlike Zermelo's set theory] [w]e prefer, however, to axiomatize not "set" but "function". The latter notion certainly includes the former. (More precisely, the two notions are completely equivalent, since a function can be regarded as a set of pairs, and a set as a function that can take two values.)". At the outset he begins with I-objects and II-objects, two objects A and B that are I-objects (first axiom), and two types of "operations" that assume ordering as a structural property obtained of the resulting objects [x, y] and (x, y). The two "domains of objects" are called "arguments" (I-objects) and "functions" (II-objects); where they overlap are the "argument functions" (he calls them I-II objects). He introduces two "universal two-variable operations" – (i) the operation [x, y]: ". . . read 'the value of the function x for the argument y . . . it itself is a type I object", and (ii) the operation (x, y): ". . . (read 'the ordered pair x, y') whose variables x and y must both be arguments and that itself produces an argument (x, y). Its most important property is that x1 = x2 and y1 = y2 follow from (x1 = y2) = (x2 = y2)". To clarify the function pair he notes that "Instead of f(x) we write [f,x] to indicate that f, just like x, is to be regarded as a variable in this procedure". To avoid the "antinomies of naive set theory, in Russell's first of all . . . we must forgo treating certain functions as arguments". He adopts a notion from Zermelo to restrict these "certain functions". Suppes observes that von Neumann's axiomatization was modified by Bernays "in order to remain nearer to the original Zermelo system . . . He introduced two membership relations: one between sets, and one between sets and classes". Then Gödel [1940] further modified the theory: "his primitive notions are those of set, class and membership (although membership alone is sufficient)". This axiomatization is now known as von Neumann–Bernays–Gödel set theory. === Bourbaki 1939 === In 1939, the collaboration NIcolas Bourbaki, in addition to giving the well-known ordered pair definition of a function as a certain subset of the cartesian product E × F, gave the following: "Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E and a variable element y of F is called a functional relation in y if, for all x ∈ E, there exists a unique y ∈ F which is in the given relation with x. We give the name of function to the operation which in this way associates with every element x ∈ E the element y ∈ F which is in the given relation with x, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function." == Since 1950 == === In contemporary set theory === Both axiomatic and naive forms of Zermelo's set theory as modified by Fraenkel (1922) and Skolem (1922) define "function" as a relation, define a relation as a set of ordered pairs, and define an ordered pair as a set of two "dissymetric" sets. While the reader of Suppes (1960) Axiomatic Set Theory or Halmos (1970) Naive Set Theory observes the use of function-symbolism in the axiom of separation, e.g., φ(x) (in Suppes) and S(x) (in Halmos), they will see no mention of "proposition" or even "first order predicate calculus". In their place are "expressions of the object language", "atomic formulae", "primitive formulae", and "atomic sentences". Kleene (1952) defines the words as follows: "In word languages, a proposition is expressed by a sentence. Then a 'predicate' is expressed by an incomplete sentence or sentence skeleton containing an open place. For example, "___ is a man" expresses a predicate ... The predicate is a propositional function of one variable. Predicates are often called 'properties' ... The predicate calculus will treat of the logic of predicates in this general sense of 'predicate', i.e., as propositional function". In 1954, Bourbaki, on p. 76 in Chapitre II of Theorie des Ensembles (theory of sets), gave a definition of a function as a triple f = (F, A, B). Here F is a functional graph, meaning a set of pairs where no two pairs have the same first member. On p. 77 (op. cit.) Bourbaki states (literal translation): "Often we shall use, in the remainder of this Treatise, the word function instead of functional graph." Suppes (1960) in Axiomatic Set Theory, formally defines a relation (p. 57) as a set of pairs, and a function (p. 86) as a relation where no two pairs have the same first member. === Relational form of a function === The reason for the disappearance of the words "propositional function" e.g., in Suppes (1960), and Halmos (1970), is explained by Tarski (1946) together with further explanation of the terminology: "An expression such as x is an integer, which contains variables and, on replacement of these variables by constants becomes a sentence, is called a SENTENTIAL [i.e., propositional cf his index] FUNCTION. But mathematicians, by the way, are not very fond of this expression, because they use the term "function" with a different meaning. ... sentential functions and sentences composed entirely of mathematical symbols (and not words of everyday language), such as: x + y = 5 are usually referred to by mathematicians as FORMULAE. In place of "sentential function" we shall sometimes simply say "sentence" – but only in cases where there is no danger of any misunderstanding". For his part Tarski calls the relational form of function a "FUNCTIONAL RELATION or simply a FUNCTION". After a discussion of this "functional relation" he asserts that: "The concept of a function which we are considering now differs essentially from the concepts of a sentential [propositional] and of a designatory function .... Strictly speaking ... [these] do not belong to the domain of logic or mathematics; they denote certain categories of expressions which serve to compose logical and mathematical statements, but they do not denote things treated of in those statements... . The term "function" in its new sense, on the other hand, is an expression of a purely logical character; it designates a certain type of things dealt with in logic and mathematics." See more about "truth under an interpretation" at Alfred Tarski. == Notes == == References == Boole, George (1854). An Investigation into the Laws of Thought on which are founded the Laws of Thought and Probabilities. Walton and Marberly. De Morgan, Augustus (1847). Formal Logic, or The Calculus of Inference, Necessary and Probable. Walton and Marberly. Dedekind, Richard; Pogorzelski, H.; Ryan, W.; Snyder, W. (1995). What are Numbers and What Should They Be?. Research Institute for Mathematics. Dieudonné, Jean (1992). Mathematics-The Music of Reason. Springer-Verlag. Dirichlet, G. P. Lejeune (1889). Gesammelte Werke, Bd. I. Berlin. ISBN 9780828402255. {{cite book}}: ISBN / Date incompatibility (help)CS1 maint: location missing publisher (link) Edwards, Harold M. (2007). "Euler's definition of the derivative". Bulletin of the American Mathematical Society. 44 (4): 575–580. doi:10.1090/s0273-0979-07-01174-3. MR 2338366. Euler, Leonhard (1988). Introduction to Analysis of the Infinite. Book I. Springer-Verlag. Euler, Leonhard (2000). Foundations of Differential Calculus. Springer-Verlag. Eves, Howard (1990). Foundations and Fundamental Concepts of Mathematics (3rd ed.). Dover. ISBN 0-486-69609-X. Fourier, Joseph (1822). Théorie analytique de la chaleur. Paris: Firmin Didot Père et Fils. Grattan-Guinness, Ivor; Bornet, Gérard (1997). George Boole: Selected Manuscripts on Logic and its Philosophy. Springer-Verlag. ISBN 3-7643-5456-9. Halmos, Paul (1970). Naive Set Theory. New York, Springer-Verlag. ISBN 9780387900926. Hardy, Godfrey Harold (1908). A Course of Pure Mathematics. Cambridge University Press (published 1993). ISBN 978-0-521-09227-2. {{cite book}}: ISBN / Date incompatibility (help) Kleene, Stephen Cole (1952). Introduction to Metamathematics. North-Holland (published 1971). ISBN 978-0-7204-2103-3. {{cite book}}: ISBN / Date incompatibility (help) Lavine, Shaughan (1994). Understanding the Infinite. Harvard University Press. Lobachevsky, Nikolai (1951). Works. Moscow-Leningrad.{{cite book}}: CS1 maint: location missing publisher (link) Luzin, N. (1998). "Function: Part II". The American Mathematical Monthly. 105 (3): 263–270. doi:10.2307/2589085. JSTOR 2589085. Medvedev, Fyodor A. (1991). Scenes from the History of Real Functions. Birkhauser. ISBN 9780817625726. Ponte, João Pedro (1992). "The history of the concept of function and some educational implications". The Mathematics Educator. 3 (2): 3–8. Russell, Bertrand (1903). The Principles of Mathematics. Cambridge University Press. Russell, Bertrand (1920). Introduction to Mathematical Philosophy (2nd ed.). Dover. ISBN 0-486-27724-0. {{cite book}}: ISBN / Date incompatibility (help) Smithies, Frank (1997). Cauchy and the Creation of Complex Function Theory. Cambridge University Press. Suppes, Patrick (1960). Axiomatic Set Theory (1972 ed.). Dover. ISBN 0-486-61630-4. {{cite book}}: ISBN / Date incompatibility (help) cf. his Chapter 1 Introduction. Tarski, Alfred (1946). Introduction to Logic and to the Methodology of Deductive Sciences (1995 ed.). Courier Dover. ISBN 0-486-28462-X. {{cite book}}: ISBN / Date incompatibility (help) Venn, John (1881). Symbolic Logic. Macmillan. van Heijenoort, Jean (1976) [1967]. From Frege to Godel: A Source Book in Mathematical Logic, 1879–1931 (3rd printing ed.). Harvard University Press. ISBN 0-674-32449-8. ——; Frege, Gottlob (1967) [1879]. "Frege (1879) Begriffsschrift, a formula language, modeled upon that of arithmetic, for pure thought". ibid. pp. 1–82. With commentary by van Heijenoort. ——; Peano, Giuseppe (1967) [1889]. "Peano (1889) The principles of arithmetic, presented by a new method". ibid. pp. 83–97. With commentary by van Heijenoort. ——; Russell, Bertrand (1967) [1902]. "Russell (1902) Letter to Frege". ibid. pp. 124–125. With commentary by van Heijenoort. Wherein Russell announces his discovery of a "paradox" in Frege's work. ——; Frege, Gottlob (1967) [1902]. "Frege (1902) Letter to Russell". ibid. pp. 126–128. With commentary by van Heijenoort. ——; Hilbert, David (1967) [1904]. "Hilbert (1904) On the foundations of logic and arithmetic". ibid. pp. 129–138. With commentary by van Heijenoort. ——; Richard, Jules (1967) [1905]. "Richard (1905) The principles of mathematics and the problem of sets". ibid. pp. 142–144. With commentary by van Heijenoort. The Richard paradox. ——; Russell, Bertrand (1967) [1908a]. "Russell (1908a) Mathematical logic as based on the theory of types". ibid. pp. 150–182. With commentary by Willard Quine. ——; Zermelo, Ernst (1967) [1908]. "Zermelo (1908) A new proof of the possibility of a well-ordering". ibid. pp. 183–198. With commentary by van Heijenoort. Wherein Zermelo rails against Poincaré's (and therefore Russell's) notion of impredicative definition. ——; Zermelo, Ernst (1967) [1908a]. "Zermelo (1908a) Investigations in the foundations of set theory I". ibid. pp. 199–215. With commentary by van Heijenoort. Wherein Zermelo attempts to solve Russell's paradox by structuring his axioms to restrict the universal domain B (from which objects and sets are pulled by definite properties) so that it itself cannot be a set, i.e., his axioms disallow a universal set. ——; Whitehead, Alfred North; Russell, Bertrand (1967) [1910]. "Whitehead and Russell (1910) Incomplete symbols: Descriptions". ibid. pp. 216–223. With commentary by W. V. Quine. ——; Wiener, Norbert (1967) [1914]. "Wiener (1914) A simplification of the logic of relations". ibid. pp. 224–227. With commentary by van Heijenoort. ——; Skolem, Thoralf (1967) [1922]. "Skolem (1922) Some remarks on axiomatized set theory". ibid. pp. 290–301. With commentary by van Heijenoort. Wherein Skolem defines Zermelo's vague "definite property". ——; Schönfinkel, Moses (1967) [1924]. "Schönfinkel (1924) On the building blocks of mathematical logic". ibid. pp. 355–366. With commentary by Willard Quine. The start of combinatory logic. ——; von Neumann, John (1967) [1925]. "von Neumann (1925) An axiomatization of set theory". ibid. pp. 393–413. With commentary by van Heijenoort. Wherein von Neumann creates "classes" as distinct from "sets" (the "classes" are Zermelo's "definite properties"), and now there is a universal set, etc. ——; Hilbert, David (1967) [1927]. "Hilbert(1927) The foundations of mathematics". ibid. pp. 464–479. With commentary by van Heijenoort. Whitehead, Alfred North; Russell, Bertrand (1913). Principia Mathematica to *56 (1962 ed.). Cambridge University Press. ISBN 978-0-521-62606-4. {{cite book}}: ISBN / Date incompatibility (help) == Further reading == Dubinsky, Ed; Harel, Guershon (1992). The Concept of Function: Aspects of Epistemology and Pedagogy. Mathematical Association of America. ISBN 0-88385-081-8. Frege, Gottlob (1879). Begriffsschrift: eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle. Kleiner, Israel (1989). "Evolution of the Function Concept: A Brief Survey". The College Mathematics Journal. 20 (4). Mathematical Association of America: 282–300. doi:10.2307/2686848. JSTOR 2686848. Lützen, Jesper (2003). "Between rigor and applications: Developments in the concept of function in mathematical analysis". In Roy Porter (ed.). The Cambridge History of Science: The modern physical and mathematical sciences. Cambridge University Press. ISBN 0521571995. An approachable and diverting historical presentation. Malik, M. A. (1980). "Historical and pedagogical aspects of the definition of function". International Journal of Mathematical Education in Science and Technology. 11 (4): 489–492. doi:10.1080/0020739800110404. Monna, A. F. (1972). "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue". Archive for History of Exact Sciences. 9 (1): 57–84. doi:10.1007/BF00348540. S2CID 120506760. Reichenbach, Hans (1947) Elements of Symbolic Logic, Dover Publishing Inc., New York NY, ISBN 0-486-24004-5. Ruthing, D. (1984). "Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.". Mathematical Intelligencer. 6 (4): 72–77. doi:10.1007/BF03026743. S2CID 189883712. Youschkevitch, A. P. (1976). "The concept of function up to the middle of the 19th century". Archive for History of Exact Sciences. 16 (1): 37–85. doi:10.1007/BF00348305. S2CID 121038818. == External links == Functions from cut-the-knot.
|
Wikipedia:Hitoshi Kumano-Go#0
|
Hitoshi Kumano-Go (4 October 1935 – 24 August 1982) was a Japanese mathematician who specialized in partial differential equations. He is especially recognized for his work on pseudo-differential operators and Fourier integral operators. == Life == Hitoshi Kumano-go was born on 4 October 1935 in Arita, Wakayama Prefecture. After finishing high school in 1954, he studied mathematics and graduated from Osaka University in 1958. He started a doctoral work at the same university under the supervision of Mitio Nagumo. He received his PhD in 1963. He was promoted to associate professor in 1967 and to full professorship in 1971. From 1967 to 1969, Kumano-go had been a visiting member at the Courant Institute of Mathematical Sciences of New York University. In May 1981 he entered Osaka University Hospital where a brain tumor was discovered. Kumano-go died on 24 August 1982 in Osaka at the age of 46. == Work == Kumano-Go first studied the local and global uniqueness of the solutions of the Cauchy problem for partial differential equations. While at the Courant Institute of Mathematical Sciences, Kumano-Go collaborated with Kurt Friedrichs, Peter Lax and Louis Nirenberg among others. He made major contributions to the theory of pseudo-differential operators. Its work contributed to the construction of the fundamental solution of a first order hyperbolic partial differential equation. His treatise on pseudo differential operators was first published in Japanese in 1974 and translated into English in 1981. Kumano-Go also made important contribution to the study of Fourier integral operators. == References ==
|
Wikipedia:HoDoMS#0
|
HoDoMS (Heads of Departments of Mathematical Sciences) is an educational company that acts as a body to represent the heads of United Kingdom higher education departments of mathematical sciences. It aims to discuss and promote the interests of higher education mathematics in the UK and to facilitate dialogue between departments. == Governance == HoDoMS is operated by a committee including four officer roles which are listed below with incumbents. The committee includes observers from the Institute of Mathematics and its Applications, The OR Society, the Royal Statistical Society, the Council for the Mathematical Sciences and the Edinburgh Mathematical Society. == Activities == The main activity of HoDoMS is to run an annual conference bringing members together for briefings and discussion on current issues. For example, the 2020 conference heard briefings on policy issues such as research funding, the Research Excellence Framework 2021, the Teaching Excellence Framework as well as practicalities such as online marking, knowledge exchange, teaching as a career for mathematics undergraduates, and academics and mental health. HoDoMS also collaborates with other organisations, for example with the London Mathematical Society on an 'Education Day' in 2019 and with the Institute of Mathematics and its Applications and the Isaac Newton Institute on an 'Induction Course for New Lecturers in the Mathematical Sciences' in 2021 == History == The first meeting of HoDoMS took place on 14 September 1995 at University College, London under its first chair, Graham Wilks. On 14 August 2018, HoDoMS was incorporated as a Private company limited by guarantee. == Affiliations == HoDoMS is a member of the Joint Mathematical Council of the United Kingdom (JMC). == See also == Engineering Professors' Council == References == == External links == HoDoMS Web site
|
Wikipedia:Hobby–Rice theorem#0
|
In mathematics, and in particular the necklace splitting problem, the Hobby–Rice theorem is a result that is useful in establishing the existence of certain solutions. It was proved in 1965 by Charles R. Hobby and John R. Rice; a simplified proof was given in 1976 by A. Pinkus. == The theorem == Define a partition of the interval [0,1] as a division of the interval into n + 1 {\displaystyle n+1} subintervals by as an increasing sequence of n {\displaystyle n} numbers: 0 = z 0 < z 1 < ⋯ < z n ⏟ < z n + 1 = 1 {\displaystyle 0=z_{0}<\underbrace {z_{1}<\dotsb <z_{n}} <z_{n+1}=1} Define a signed partition as a partition in which each subinterval i {\displaystyle i} has an associated sign δ i {\displaystyle \delta _{i}} : δ 1 , … , δ k + 1 ∈ { + 1 , − 1 } {\displaystyle \delta _{1},\dotsc ,\delta _{k+1}\in \left\{+1,-1\right\}} The Hobby–Rice theorem says that for every n continuously integrable functions: g 1 , … , g n : [ 0 , 1 ] ⟶ R {\displaystyle g_{1},\dotsc ,g_{n}\colon [0,1]\longrightarrow \mathbb {R} } there exists a signed partition of [0,1] such that: ∑ i = 1 n + 1 δ i ∫ z i − 1 z i g j ( z ) d z = 0 for 1 ≤ j ≤ n . {\displaystyle \sum _{i=1}^{n+1}\delta _{i}\!\int _{z_{i-1}}^{z_{i}}g_{j}(z)\,dz=0{\text{ for }}1\leq j\leq n.} (in other words: for each of the n functions, its integral over the positive subintervals equals its integral over the negative subintervals). == Application to fair division == The theorem was used by Noga Alon in the context of necklace splitting in 1987. Suppose the interval [0,1] is a cake. There are n partners and each of the n functions is a value-density function of one partner. We want to divide the cake into two parts such that all partners agree that the parts have the same value. This fair-division challenge is sometimes referred to as the consensus-halving problem. The Hobby–Rice theorem implies that this can be done with n cuts. == References ==
|
Wikipedia:Hochster–Roberts theorem#0
|
In algebra, the Hochster–Roberts theorem, introduced by Melvin Hochster and Joel L. Roberts in 1974, states that rings of invariants of linearly reductive groups acting on regular rings are Cohen–Macaulay. In other words, if V is a rational representation of a linearly reductive group G over a field k, then there exist algebraically independent invariant homogeneous polynomials f 1 , ⋯ , f d {\displaystyle f_{1},\cdots ,f_{d}} such that k [ V ] G {\displaystyle k[V]^{G}} is a free finite graded module over k [ f 1 , ⋯ , f d ] {\displaystyle k[f_{1},\cdots ,f_{d}]} . In 1987, Jean-François Boutot proved that if a variety over a field of characteristic 0 has rational singularities then so does its quotient by the action of a reductive group; this implies the Hochster–Roberts theorem in characteristic 0 as rational singularities are Cohen–Macaulay. In characteristic p>0, there are examples of groups that are reductive (or even finite) acting on polynomial rings whose rings of invariants are not Cohen–Macaulay. == References ==
|
Wikipedia:Hodge star operator#0
|
In mathematics, the Hodge star operator or Hodge star is a linear map defined on the exterior algebra of a finite-dimensional oriented vector space endowed with a nondegenerate symmetric bilinear form. Applying the operator to an element of the algebra produces the Hodge dual of the element. This map was introduced by W. V. D. Hodge. For example, in an oriented 3-dimensional Euclidean space, an oriented plane can be represented by the exterior product of two basis vectors, and its Hodge dual is the normal vector given by their cross product; conversely, any vector is dual to the oriented plane perpendicular to it, endowed with a suitable bivector. Generalizing this to an n-dimensional vector space, the Hodge star is a one-to-one mapping of k-vectors to (n – k)-vectors; the dimensions of these spaces are the binomial coefficients ( n k ) = ( n n − k ) {\displaystyle {\tbinom {n}{k}}={\tbinom {n}{n-k}}} . The naturalness of the star operator means it can play a role in differential geometry when applied to the cotangent bundle of a pseudo-Riemannian manifold, and hence to differential k-forms. This allows the definition of the codifferential as the Hodge adjoint of the exterior derivative, leading to the Laplace–de Rham operator. This generalizes the case of 3-dimensional Euclidean space, in which divergence of a vector field may be realized as the codifferential opposite to the gradient operator, and the Laplace operator on a function is the divergence of its gradient. An important application is the Hodge decomposition of differential forms on a closed Riemannian manifold. == Formal definition for k-vectors == Let V be an n-dimensional oriented vector space with a nondegenerate symmetric bilinear form ⟨ ⋅ , ⋅ ⟩ {\displaystyle \langle \cdot ,\cdot \rangle } , referred to here as a scalar product. (In more general contexts such as pseudo-Riemannian manifolds and Minkowski space, the bilinear form may not be positive-definite.) This induces a scalar product on k-vectors α , β ∈ ⋀ k V {\textstyle \alpha ,\beta \in \bigwedge ^{\!k}V} , for 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} , by defining it on simple k-vectors α = α 1 ∧ ⋯ ∧ α k {\displaystyle \alpha =\alpha _{1}\wedge \cdots \wedge \alpha _{k}} and β = β 1 ∧ ⋯ ∧ β k {\displaystyle \beta =\beta _{1}\wedge \cdots \wedge \beta _{k}} to equal the Gram determinant: 14 ⟨ α , β ⟩ = det ( ⟨ α i , β j ⟩ i , j = 1 k ) {\displaystyle \langle \alpha ,\beta \rangle =\det \left(\left\langle \alpha _{i},\beta _{j}\right\rangle _{i,j=1}^{k}\right)} extended to ⋀ k V {\textstyle \bigwedge ^{\!k}V} through linearity. The unit n-vector ω ∈ ⋀ n V {\displaystyle \omega \in {\textstyle \bigwedge }^{\!n}V} is defined in terms of an oriented orthonormal basis { e 1 , … , e n } {\displaystyle \{e_{1},\ldots ,e_{n}\}} of V as: ω := e 1 ∧ ⋯ ∧ e n . {\displaystyle \omega :=e_{1}\wedge \cdots \wedge e_{n}.} (Note: In the general pseudo-Riemannian case, orthonormality means ⟨ e i , e j ⟩ ∈ { δ i j , − δ i j } {\displaystyle \langle e_{i},e_{j}\rangle \in \{\delta _{ij},-\delta _{ij}\}} for all pairs of basis vectors.) The Hodge star operator is a linear operator on the exterior algebra of V, mapping k-vectors to (n – k)-vectors, for 0 ≤ k ≤ n {\displaystyle 0\leq k\leq n} . It has the following property, which defines it completely:: 15 α ∧ ( ⋆ β ) = ⟨ α , β ⟩ ω {\displaystyle \alpha \wedge ({\star }\beta )=\langle \alpha ,\beta \rangle \,\omega } for all k-vectors α , β ∈ ⋀ k V . {\displaystyle \alpha ,\beta \in {\textstyle \bigwedge }^{\!k}V.} Dually, in the space ⋀ n V ∗ {\displaystyle {\textstyle \bigwedge }^{\!n}V^{*}} of n-forms (alternating n-multilinear functions on V n {\displaystyle V^{n}} ), the dual to ω {\displaystyle \omega } is the volume form det {\displaystyle \det } , the function whose value on v 1 ∧ ⋯ ∧ v n {\displaystyle v_{1}\wedge \cdots \wedge v_{n}} is the determinant of the n × n {\displaystyle n\times n} matrix assembled from the column vectors of v j {\displaystyle v_{j}} in e i {\displaystyle e_{i}} -coordinates. Applying det {\displaystyle \det } to the above equation, we obtain the dual definition: det ( α ∧ ⋆ β ) = ⟨ α , β ⟩ {\displaystyle \det(\alpha \wedge {\star }\beta )=\langle \alpha ,\beta \rangle } for all k-vectors α , β ∈ ⋀ k V . {\displaystyle \alpha ,\beta \in {\textstyle \bigwedge }^{\!k}V.} Equivalently, taking α = α 1 ∧ ⋯ ∧ α k {\displaystyle \alpha =\alpha _{1}\wedge \cdots \wedge \alpha _{k}} , β = β 1 ∧ ⋯ ∧ β k {\displaystyle \beta =\beta _{1}\wedge \cdots \wedge \beta _{k}} , and ⋆ β = β 1 ⋆ ∧ ⋯ ∧ β n − k ⋆ {\displaystyle {\star }\beta =\beta _{1}^{\star }\wedge \cdots \wedge \beta _{n-k}^{\star }} : det ( α 1 ∧ ⋯ ∧ α k ∧ β 1 ⋆ ∧ ⋯ ∧ β n − k ⋆ ) = det ( ⟨ α i , β j ⟩ ) . {\displaystyle \det \left(\alpha _{1}\wedge \cdots \wedge \alpha _{k}\wedge \beta _{1}^{\star }\wedge \cdots \wedge \beta _{n-k}^{\star }\right)\ =\ \det \left(\langle \alpha _{i},\beta _{j}\rangle \right).} This means that, writing an orthonormal basis of k-vectors as e I = e i 1 ∧ ⋯ ∧ e i k {\displaystyle e_{I}\ =\ e_{i_{1}}\wedge \cdots \wedge e_{i_{k}}} over all subsets I = { i 1 < ⋯ < i k } {\displaystyle I=\{i_{1}<\cdots <i_{k}\}} of [ n ] = { 1 , … , n } {\displaystyle [n]=\{1,\ldots ,n\}} , the Hodge dual is the (n – k)-vector corresponding to the complementary set I ¯ = [ n ] ∖ I = { i ¯ 1 < ⋯ < i ¯ n − k } {\displaystyle {\bar {I}}=[n]\smallsetminus I=\left\{{\bar {i}}_{1}<\cdots <{\bar {i}}_{n-k}\right\}} : ⋆ e I = s ⋅ t ⋅ e I ¯ , {\displaystyle {\star }e_{I}=s\cdot t\cdot e_{\bar {I}},} where s ∈ { 1 , − 1 } {\displaystyle s\in \{1,-1\}} is the sign of the permutation i 1 ⋯ i k i ¯ 1 ⋯ i ¯ n − k {\displaystyle i_{1}\cdots i_{k}{\bar {i}}_{1}\cdots {\bar {i}}_{n-k}} and t ∈ { 1 , − 1 } {\displaystyle t\in \{1,-1\}} is the product ⟨ e i 1 , e i 1 ⟩ ⋯ ⟨ e i k , e i k ⟩ {\displaystyle \langle e_{i_{1}},e_{i_{1}}\rangle \cdots \langle e_{i_{k}},e_{i_{k}}\rangle } . In the Riemannian case, t = 1 {\displaystyle t=1} . Since Hodge star takes an orthonormal basis to an orthonormal basis, it is an isometry on the exterior algebra ⋀ V {\textstyle \bigwedge V} . == Geometric explanation == The Hodge star is motivated by the correspondence between a subspace W of V and its orthogonal subspace (with respect to the scalar product), where each space is endowed with an orientation and a numerical scaling factor. Specifically, a non-zero decomposable k-vector w 1 ∧ ⋯ ∧ w k ∈ ⋀ k V {\displaystyle w_{1}\wedge \cdots \wedge w_{k}\in \textstyle \bigwedge ^{\!k}V} corresponds by the Plücker embedding to the subspace W {\displaystyle W} with oriented basis w 1 , … , w k {\displaystyle w_{1},\ldots ,w_{k}} , endowed with a scaling factor equal to the k-dimensional volume of the parallelepiped spanned by this basis (equal to the Gramian, the determinant of the matrix of scalar products ⟨ w i , w j ⟩ {\displaystyle \langle w_{i},w_{j}\rangle } ). The Hodge star acting on a decomposable vector can be written as a decomposable (n − k)-vector: ⋆ ( w 1 ∧ ⋯ ∧ w k ) = u 1 ∧ ⋯ ∧ u n − k , {\displaystyle {\star }(w_{1}\wedge \cdots \wedge w_{k})\,=\,u_{1}\wedge \cdots \wedge u_{n-k},} where u 1 , … , u n − k {\displaystyle u_{1},\ldots ,u_{n-k}} form an oriented basis of the orthogonal space U = W ⊥ {\displaystyle U=W^{\perp }\!} . Furthermore, the (n − k)-volume of the u i {\displaystyle u_{i}} -parallelepiped must equal the k-volume of the w i {\displaystyle w_{i}} -parallelepiped, and w 1 , … , w k , u 1 , … , u n − k {\displaystyle w_{1},\ldots ,w_{k},u_{1},\ldots ,u_{n-k}} must form an oriented basis of V {\displaystyle V} . A general k-vector is a linear combination of decomposable k-vectors, and the definition of Hodge star is extended to general k-vectors by defining it as being linear. == Examples == === Two dimensions === In two dimensions with the normalized Euclidean metric and orientation given by the ordering (x, y), the Hodge star on k-forms is given by ⋆ 1 = d x ∧ d y ⋆ d x = d y ⋆ d y = − d x ⋆ ( d x ∧ d y ) = 1. {\displaystyle {\begin{aligned}{\star }\,1&=dx\wedge dy\\{\star }\,dx&=dy\\{\star }\,dy&=-dx\\{\star }(dx\wedge dy)&=1.\end{aligned}}} === Three dimensions === A common example of the Hodge star operator is the case n = 3, when it can be taken as the correspondence between vectors and bivectors. Specifically, for Euclidean R3 with the basis d x , d y , d z {\displaystyle dx,dy,dz} of one-forms often used in vector calculus, one finds that ⋆ d x = d y ∧ d z ⋆ d y = d z ∧ d x ⋆ d z = d x ∧ d y . {\displaystyle {\begin{aligned}{\star }\,dx&=dy\wedge dz\\{\star }\,dy&=dz\wedge dx\\{\star }\,dz&=dx\wedge dy.\end{aligned}}} The Hodge star relates the exterior and cross product in three dimensions: ⋆ ( u ∧ v ) = u × v ⋆ ( u × v ) = u ∧ v . {\displaystyle {\star }(\mathbf {u} \wedge \mathbf {v} )=\mathbf {u} \times \mathbf {v} \qquad {\star }(\mathbf {u} \times \mathbf {v} )=\mathbf {u} \wedge \mathbf {v} .} Applied to three dimensions, the Hodge star provides an isomorphism between axial vectors and bivectors, so each axial vector a is associated with a bivector A and vice versa, that is: A = ⋆ a , a = ⋆ A {\displaystyle \mathbf {A} ={\star }\mathbf {a} ,\ \ \mathbf {a} ={\star }\mathbf {A} } . The Hodge star can also be interpreted as a form of the geometric correspondence between an axis of rotation and an infinitesimal rotation (see also: 3D rotation group#Lie algebra) around the axis, with speed equal to the length of the axis of rotation. A scalar product on a vector space V {\displaystyle V} gives an isomorphism V ≅ V ∗ {\displaystyle V\cong V^{*}\!} identifying V {\displaystyle V} with its dual space, and the vector space L ( V , V ) {\displaystyle L(V,V)} is naturally isomorphic to the tensor product V ∗ ⊗ V ≅ V ⊗ V {\displaystyle V^{*}\!\!\otimes V\cong V\otimes V} . Thus for V = R 3 {\displaystyle V=\mathbb {R} ^{3}} , the star mapping ⋆ : V → ⋀ 2 V ⊂ V ⊗ V {\textstyle \textstyle {\star }:V\to \bigwedge ^{\!2}\!V\subset V\otimes V} takes each vector v {\displaystyle \mathbf {v} } to a bivector ⋆ v ∈ V ⊗ V {\displaystyle {\star }\mathbf {v} \in V\otimes V} , which corresponds to a linear operator L v : V → V {\displaystyle L_{\mathbf {v} }:V\to V} . Specifically, L v {\displaystyle L_{\mathbf {v} }} is a skew-symmetric operator, which corresponds to an infinitesimal rotation: that is, the macroscopic rotations around the axis v {\displaystyle \mathbb {v} } are given by the matrix exponential exp ( t L v ) {\displaystyle \exp(tL_{\mathbf {v} })} . With respect to the basis d x , d y , d z {\displaystyle dx,dy,dz} of R 3 {\displaystyle \mathbb {R} ^{3}} , the tensor d x ⊗ d y {\displaystyle dx\otimes dy} corresponds to a coordinate matrix with 1 in the d x {\displaystyle dx} row and d y {\displaystyle dy} column, etc., and the wedge d x ∧ d y = d x ⊗ d y − d y ⊗ d x {\displaystyle dx\wedge dy\,=\,dx\otimes dy-dy\otimes dx} is the skew-symmetric matrix [ 0 1 0 − 1 0 0 0 0 0 ] {\displaystyle \scriptscriptstyle \left[{\begin{array}{rrr}\,0\!\!&\!\!1&\!\!\!\!0\!\!\!\!\!\!\\[-.5em]\,\!-1\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!\\[-.5em]\,0\!\!&\!\!0\!\!&\!\!\!\!0\!\!\!\!\!\!\end{array}}\!\!\!\right]} , etc. That is, we may interpret the star operator as: v = a d x + b d y + c d z ⟶ ⋆ v ≅ L v = [ 0 c − b − c 0 a b − a 0 ] . {\displaystyle \mathbf {v} =a\,dx+b\,dy+c\,dz\quad \longrightarrow \quad {\star }{\mathbf {v} }\ \cong \ L_{\mathbf {v} }\ =\left[{\begin{array}{rrr}0&c&-b\\-c&0&a\\b&-a&0\end{array}}\right].} Under this correspondence, cross product of vectors corresponds to the commutator Lie bracket of linear operators: L u × v = L v L u − L u L v = − [ L u , L v ] {\displaystyle L_{\mathbf {u} \times \mathbf {v} }=L_{\mathbf {v} }L_{\mathbf {u} }-L_{\mathbf {u} }L_{\mathbf {v} }=-\left[L_{\mathbf {u} },L_{\mathbf {v} }\right]} . === Four dimensions === In case n = 4 {\displaystyle n=4} , the Hodge star acts as an endomorphism of the second exterior power (i.e. it maps 2-forms to 2-forms, since 4 − 2 = 2). If the signature of the metric tensor is all positive, i.e. on a Riemannian manifold, then the Hodge star is an involution. If the signature is mixed, i.e., pseudo-Riemannian, then applying the operator twice will return the argument up to a sign – see § Duality below. This particular endomorphism property of 2-forms in four dimensions makes self-dual and anti-self-dual two-forms natural geometric objects to study. That is, one can describe the space of 2-forms in four dimensions with a basis that "diagonalizes" the Hodge star operator with eigenvalues ± 1 {\displaystyle \pm 1} (or ± i {\displaystyle \pm i} , depending on the signature). For concreteness, we discuss the Hodge star operator in Minkowski spacetime where n = 4 {\displaystyle n=4} with metric signature (− + + +) and coordinates ( t , x , y , z ) {\displaystyle (t,x,y,z)} . The volume form is oriented as ε 0123 = 1 {\displaystyle \varepsilon _{0123}=1} . For one-forms, ⋆ d t = − d x ∧ d y ∧ d z , ⋆ d x = − d t ∧ d y ∧ d z , ⋆ d y = − d t ∧ d z ∧ d x , ⋆ d z = − d t ∧ d x ∧ d y , {\displaystyle {\begin{aligned}{\star }dt&=-dx\wedge dy\wedge dz\,,\\{\star }dx&=-dt\wedge dy\wedge dz\,,\\{\star }dy&=-dt\wedge dz\wedge dx\,,\\{\star }dz&=-dt\wedge dx\wedge dy\,,\end{aligned}}} while for 2-forms, ⋆ ( d t ∧ d x ) = − d y ∧ d z , ⋆ ( d t ∧ d y ) = − d z ∧ d x , ⋆ ( d t ∧ d z ) = − d x ∧ d y , ⋆ ( d x ∧ d y ) = d t ∧ d z , ⋆ ( d z ∧ d x ) = d t ∧ d y , ⋆ ( d y ∧ d z ) = d t ∧ d x . {\displaystyle {\begin{aligned}{\star }(dt\wedge dx)&=-dy\wedge dz\,,\\{\star }(dt\wedge dy)&=-dz\wedge dx\,,\\{\star }(dt\wedge dz)&=-dx\wedge dy\,,\\{\star }(dx\wedge dy)&=dt\wedge dz\,,\\{\star }(dz\wedge dx)&=dt\wedge dy\,,\\{\star }(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d x μ ∧ d x ν ) = η μ κ η ν λ ε κ λ ρ σ 1 2 ! d x ρ ∧ d x σ . {\displaystyle {\begin{aligned}{\star }(dx^{\mu })&=\eta ^{\mu \lambda }\varepsilon _{\lambda \nu \rho \sigma }{\frac {1}{3!}}dx^{\nu }\wedge dx^{\rho }\wedge dx^{\sigma }\,,\\{\star }(dx^{\mu }\wedge dx^{\nu })&=\eta ^{\mu \kappa }\eta ^{\nu \lambda }\varepsilon _{\kappa \lambda \rho \sigma }{\frac {1}{2!}}dx^{\rho }\wedge dx^{\sigma }\,.\end{aligned}}} Hodge dual of three- and four-forms can be easily deduced from the fact that, in the Lorentzian signature, ⋆ 2 = 1 {\displaystyle {\star }^{2}=1} for odd-rank forms and ⋆ 2 = − 1 {\displaystyle {\star }^{2}=-1} for even-rank forms. An easy rule to remember for these Hodge operations is that given a form α {\displaystyle \alpha } , its Hodge dual ⋆ α {\displaystyle {\star }\alpha } may be obtained by writing the components not involved in α {\displaystyle \alpha } in an order such that α ∧ ( ⋆ α ) = d t ∧ d x ∧ d y ∧ d z {\displaystyle \alpha \wedge ({\star }\alpha )=dt\wedge dx\wedge dy\wedge dz} . An extra minus sign will enter only if α {\displaystyle \alpha } contains d t {\displaystyle dt} . (For (+ − − −), one puts in a minus sign only if α {\displaystyle \alpha } involves an odd number of the space-associated forms d x {\displaystyle dx} , d y {\displaystyle dy} and d z {\displaystyle dz} .) Note that the combinations ( d x μ ∧ d x ν ) ± := 1 2 ( d x μ ∧ d x ν ∓ i ⋆ ( d x μ ∧ d x ν ) ) {\displaystyle (dx^{\mu }\wedge dx^{\nu })^{\pm }:={\frac {1}{2}}{\big (}dx^{\mu }\wedge dx^{\nu }\mp i{\star }(dx^{\mu }\wedge dx^{\nu }){\big )}} take ± i {\displaystyle \pm i} as the eigenvalue for Hodge star operator, i.e., ⋆ ( d x μ ∧ d x ν ) ± = ± i ( d x μ ∧ d x ν ) ± , {\displaystyle {\star }(dx^{\mu }\wedge dx^{\nu })^{\pm }=\pm i(dx^{\mu }\wedge dx^{\nu })^{\pm },} and hence deserve the name self-dual and anti-self-dual two-forms. Understanding the geometry, or kinematics, of Minkowski spacetime in self-dual and anti-self-dual sectors turns out to be insightful in both mathematical and physical perspectives, making contacts to the use of the two-spinor language in modern physics such as spinor-helicity formalism or twistor theory. === Conformal invariance === The Hodge star is conformally invariant on n-forms on a 2n-dimensional vector space V {\displaystyle V} , i.e. if g {\displaystyle g} is a metric on V {\displaystyle V} and λ > 0 {\displaystyle \lambda >0} , then the induced Hodge stars ⋆ g , ⋆ λ g : Λ n V → Λ n V {\displaystyle {\star }_{g},{\star }_{\lambda g}:\Lambda ^{n}V\to \Lambda ^{n}V} are the same. === Example: Derivatives in three dimensions === The combination of the ⋆ {\displaystyle {\star }} operator and the exterior derivative d generates the classical operators grad, curl, and div on vector fields in three-dimensional Euclidean space. This works out as follows: d takes a 0-form (a function) to a 1-form, a 1-form to a 2-form, and a 2-form to a 3-form (and takes a 3-form to zero). For a 0-form f = f ( x , y , z ) {\displaystyle f=f(x,y,z)} , the first case written out in components gives: d f = ∂ f ∂ x d x + ∂ f ∂ y d y + ∂ f ∂ z d z . {\displaystyle df={\frac {\partial f}{\partial x}}\,dx+{\frac {\partial f}{\partial y}}\,dy+{\frac {\partial f}{\partial z}}\,dz.} The scalar product identifies 1-forms with vector fields as d x ↦ ( 1 , 0 , 0 ) {\displaystyle dx\mapsto (1,0,0)} , etc., so that d f {\displaystyle df} becomes grad f = ( ∂ f ∂ x , ∂ f ∂ y , ∂ f ∂ z ) {\textstyle \operatorname {grad} f=\left({\frac {\partial f}{\partial x}},{\frac {\partial f}{\partial y}},{\frac {\partial f}{\partial z}}\right)} . In the second case, a vector field F = ( A , B , C ) {\displaystyle \mathbf {F} =(A,B,C)} corresponds to the 1-form φ = A d x + B d y + C d z {\displaystyle \varphi =A\,dx+B\,dy+C\,dz} , which has exterior derivative: d φ = ( ∂ C ∂ y − ∂ B ∂ z ) d y ∧ d z + ( ∂ C ∂ x − ∂ A ∂ z ) d x ∧ d z + ( ∂ B ∂ x − ∂ A ∂ y ) d x ∧ d y . {\displaystyle d\varphi =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}}\right)dy\wedge dz+\left({\frac {\partial C}{\partial x}}-{\frac {\partial A}{\partial z}}\right)dx\wedge dz+\left({\partial B \over \partial x}-{\frac {\partial A}{\partial y}}\right)dx\wedge dy.} Applying the Hodge star gives the 1-form: ⋆ d φ = ( ∂ C ∂ y − ∂ B ∂ z ) d x − ( ∂ C ∂ x − ∂ A ∂ z ) d y + ( ∂ B ∂ x − ∂ A ∂ y ) d z , {\displaystyle {\star }d\varphi =\left({\partial C \over \partial y}-{\partial B \over \partial z}\right)\,dx-\left({\partial C \over \partial x}-{\partial A \over \partial z}\right)\,dy+\left({\partial B \over \partial x}-{\partial A \over \partial y}\right)\,dz,} which becomes the vector field curl F = ( ∂ C ∂ y − ∂ B ∂ z , − ∂ C ∂ x + ∂ A ∂ z , ∂ B ∂ x − ∂ A ∂ y ) {\textstyle \operatorname {curl} \mathbf {F} =\left({\frac {\partial C}{\partial y}}-{\frac {\partial B}{\partial z}},\,-{\frac {\partial C}{\partial x}}+{\frac {\partial A}{\partial z}},\,{\frac {\partial B}{\partial x}}-{\frac {\partial A}{\partial y}}\right)} . In the third case, F = ( A , B , C ) {\displaystyle \mathbf {F} =(A,B,C)} again corresponds to φ = A d x + B d y + C d z {\displaystyle \varphi =A\,dx+B\,dy+C\,dz} . Applying Hodge star, exterior derivative, and Hodge star again: ⋆ φ = A d y ∧ d z − B d x ∧ d z + C d x ∧ d y , d ⋆ φ = ( ∂ A ∂ x + ∂ B ∂ y + ∂ C ∂ z ) d x ∧ d y ∧ d z , ⋆ d ⋆ φ = ∂ A ∂ x + ∂ B ∂ y + ∂ C ∂ z = div F . {\displaystyle {\begin{aligned}{\star }\varphi &=A\,dy\wedge dz-B\,dx\wedge dz+C\,dx\wedge dy,\\d{\star \varphi }&=\left({\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}\right)dx\wedge dy\wedge dz,\\{\star }d{\star }\varphi &={\frac {\partial A}{\partial x}}+{\frac {\partial B}{\partial y}}+{\frac {\partial C}{\partial z}}=\operatorname {div} \mathbf {F} .\end{aligned}}} One advantage of this expression is that the identity d2 = 0, which is true in all cases, has as special cases two other identities: (1) curl grad f = 0, and (2) div curl F = 0. In particular, Maxwell's equations take on a particularly simple and elegant form, when expressed in terms of the exterior derivative and the Hodge star. The expression ⋆ d ⋆ {\displaystyle {\star }d{\star }} (multiplied by an appropriate power of −1) is called the codifferential; it is defined in full generality, for any dimension, further in the article below. One can also obtain the Laplacian Δf = div grad f in terms of the above operations: Δ f = ⋆ d ⋆ d f = ∂ 2 f ∂ x 2 + ∂ 2 f ∂ y 2 + ∂ 2 f ∂ z 2 . {\displaystyle \Delta f={\star }d{\star }df={\frac {\partial ^{2}f}{\partial x^{2}}}+{\frac {\partial ^{2}f}{\partial y^{2}}}+{\frac {\partial ^{2}f}{\partial z^{2}}}.} The Laplacian can also be seen as a special case of the more general Laplace–deRham operator Δ = d δ + δ d {\displaystyle \Delta =d\delta +\delta d} where in three dimensions, δ = ( − 1 ) k ⋆ d ⋆ {\displaystyle \delta =(-1)^{k}{\star }d{\star }} is the codifferential for k {\displaystyle k} -forms. Any function f {\displaystyle f} is a 0-form, and δ f = 0 {\displaystyle \delta f=0} and so this reduces to the ordinary Laplacian. For the 1-form φ {\displaystyle \varphi } above, the codifferential is δ = − ⋆ d ⋆ {\displaystyle \delta =-{\star }d{\star }} and after some straightforward calculations one obtains the Laplacian acting on φ {\displaystyle \varphi } . == Duality == Applying the Hodge star twice leaves a k-vector unchanged up to a sign: for η ∈ ⋀ k V {\displaystyle \eta \in {\textstyle \bigwedge }^{k}V} in an n-dimensional space V, one has ⋆ ⋆ η = ( − 1 ) k ( n − k ) s η , {\displaystyle {\star }{\star }\eta =(-1)^{k(n-k)}s\,\eta ,} where s is the parity of the signature of the scalar product on V, that is, the sign of the determinant of the matrix of the scalar product with respect to any basis. For example, if n = 4 and the signature of the scalar product is either (+ − − −) or (− + + +) then s = −1. For Riemannian manifolds (including Euclidean spaces), we always have s = 1. The above identity implies that the inverse of ⋆ {\displaystyle {\star }} can be given as ⋆ − 1 : ⋀ k V → ⋀ n − k V η ↦ ( − 1 ) k ( n − k ) s ⋆ η {\displaystyle {\begin{aligned}{\star }^{-1}:~{\textstyle \bigwedge }^{\!k}V&\to {\textstyle \bigwedge }^{\!n-k}V\\\eta &\mapsto (-1)^{k(n-k)}\!s\,{\star }\eta \end{aligned}}} If n is odd then k(n − k) is even for any k, whereas if n is even then k(n − k) has the parity of k. Therefore: ⋆ − 1 = { s ⋆ n is odd ( − 1 ) k s ⋆ n is even {\displaystyle {\star }^{-1}={\begin{cases}s\,{\star }&n{\text{ is odd}}\\(-1)^{k}s\,{\star }&n{\text{ is even}}\end{cases}}} where k is the degree of the element operated on. == On manifolds == For an n-dimensional oriented pseudo-Riemannian manifold M, we apply the construction above to each cotangent space T p ∗ M {\displaystyle {\text{T}}_{p}^{*}M} and its exterior powers ⋀ k T p ∗ M {\textstyle \bigwedge ^{k}{\text{T}}_{p}^{*}M} , and hence to the differential k-forms ζ ∈ Ω k ( M ) = Γ ( ⋀ k T ∗ M ) {\textstyle \zeta \in \Omega ^{k}(M)=\Gamma \left(\bigwedge ^{k}{\text{T}}^{*}\!M\right)} , the global sections of the bundle ⋀ k T ∗ M → M {\textstyle \bigwedge ^{k}\mathrm {T} ^{*}\!M\to M} . The Riemannian metric induces a scalar product on ⋀ k T p ∗ M {\textstyle \bigwedge ^{k}{\text{T}}_{p}^{*}M} at each point p ∈ M {\displaystyle p\in M} . We define the Hodge dual of a k-form ζ {\displaystyle \zeta } , defining ⋆ ζ {\displaystyle {\star }\zeta } as the unique (n – k)-form satisfying η ∧ ⋆ ζ = ⟨ η , ζ ⟩ ω {\displaystyle \eta \wedge {\star }\zeta \ =\ \langle \eta ,\zeta \rangle \,\omega } for every k-form η {\displaystyle \eta } , where ⟨ η , ζ ⟩ {\displaystyle \langle \eta ,\zeta \rangle } is a real-valued function on M {\displaystyle M} , and the volume form ω {\displaystyle \omega } is induced by the pseudo-Riemannian metric. Integrating this equation over M {\displaystyle M} , the right side becomes the L 2 {\displaystyle L^{2}} (square-integrable) scalar product on k-forms, and we obtain: ∫ M η ∧ ⋆ ζ = ∫ M ⟨ η , ζ ⟩ ω . {\displaystyle \int _{M}\eta \wedge {\star }\zeta \ =\ \int _{M}\langle \eta ,\zeta \rangle \ \omega .} More generally, if M {\displaystyle M} is non-orientable, one can define the Hodge star of a k-form as a (n – k)-pseudo differential form; that is, a differential form with values in the canonical line bundle. === Computation in index notation === We compute in terms of tensor index notation with respect to a (not necessarily orthonormal) basis { ∂ ∂ x 1 , … , ∂ ∂ x n } {\textstyle \left\{{\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right\}} in a tangent space V = T p M {\displaystyle V=T_{p}M} and its dual basis { d x 1 , … , d x n } {\displaystyle \{dx_{1},\ldots ,dx_{n}\}} in V ∗ = T p ∗ M {\displaystyle V^{*}=T_{p}^{*}M} , having the metric matrix ( g i j ) = ( ⟨ ∂ ∂ x i , ∂ ∂ x j ⟩ ) {\textstyle (g_{ij})=\left(\left\langle {\frac {\partial }{\partial x_{i}}},{\frac {\partial }{\partial x_{j}}}\right\rangle \right)} and its inverse matrix ( g i j ) = ( ⟨ d x i , d x j ⟩ ) {\displaystyle (g^{ij})=(\langle dx^{i},dx^{j}\rangle )} . The Hodge dual of a decomposable k-form is: ⋆ ( d x i 1 ∧ ⋯ ∧ d x i k ) = | det [ g i j ] | ( n − k ) ! g i 1 j 1 ⋯ g i k j k ε j 1 … j n d x j k + 1 ∧ ⋯ ∧ d x j n . {\displaystyle {\star }\left(dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}\right)\ =\ {\frac {\sqrt {\left|\det[g_{ij}]\right|}}{(n-k)!}}g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}\varepsilon _{j_{1}\dots j_{n}}dx^{j_{k+1}}\wedge \dots \wedge dx^{j_{n}}.} Here ε j 1 … j n {\displaystyle \varepsilon _{j_{1}\dots j_{n}}} is the Levi-Civita symbol with ε 1 … n = 1 {\displaystyle \varepsilon _{1\dots n}=1} , and we implicitly take the sum over all values of the repeated indices j 1 , … , j n {\displaystyle j_{1},\ldots ,j_{n}} . The factorial ( n − k ) ! {\displaystyle (n-k)!} accounts for double counting, and is not present if the summation indices are restricted so that j k + 1 < ⋯ < j n {\displaystyle j_{k+1}<\dots <j_{n}} . The absolute value of the determinant is necessary since it may be negative, as for tangent spaces to Lorentzian manifolds. An arbitrary differential form can be written as follows: α = 1 k ! α i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k = ∑ i 1 < ⋯ < i k α i 1 , … , i k d x i 1 ∧ ⋯ ∧ d x i k . {\displaystyle \alpha \ =\ {\frac {1}{k!}}\alpha _{i_{1},\dots ,i_{k}}dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}\ =\ \sum _{i_{1}<\dots <i_{k}}\alpha _{i_{1},\dots ,i_{k}}dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}.} The factorial k ! {\displaystyle k!} is again included to account for double counting when we allow non-increasing indices. We would like to define the dual of the component α i 1 , … , i k {\displaystyle \alpha _{i_{1},\dots ,i_{k}}} so that the Hodge dual of the form is given by ⋆ α = 1 ( n − k ) ! ( ⋆ α ) i k + 1 , … , i n d x i k + 1 ∧ ⋯ ∧ d x i n . {\displaystyle {\star }\alpha ={\frac {1}{(n-k)!}}({\star }\alpha )_{i_{k+1},\dots ,i_{n}}dx^{i_{k+1}}\wedge \dots \wedge dx^{i_{n}}.} Using the above expression for the Hodge dual of d x i 1 ∧ ⋯ ∧ d x i k {\displaystyle dx^{i_{1}}\wedge \dots \wedge dx^{i_{k}}} , we find: ( ⋆ α ) j k + 1 , … , j n = | det [ g a b ] | k ! α i 1 , … , i k g i 1 j 1 ⋯ g i k j k ε j 1 , … , j n . {\displaystyle ({\star }\alpha )_{j_{k+1},\dots ,j_{n}}={\frac {\sqrt {\left|\det[g_{ab}]\right|}}{k!}}\alpha _{i_{1},\dots ,i_{k}}\,g^{i_{1}j_{1}}\cdots g^{i_{k}j_{k}}\,\varepsilon _{j_{1},\dots ,j_{n}}\,.} Although one can apply this expression to any tensor α {\displaystyle \alpha } , the result is antisymmetric, since contraction with the completely anti-symmetric Levi-Civita symbol cancels all but the totally antisymmetric part of the tensor. It is thus equivalent to antisymmetrization followed by applying the Hodge star. The unit volume form ω = ⋆ 1 ∈ ⋀ n V ∗ {\textstyle \omega ={\star }1\in \bigwedge ^{n}V^{*}} is given by: ω = | det [ g i j ] | d x 1 ∧ ⋯ ∧ d x n . {\displaystyle \omega ={\sqrt {\left|\det[g_{ij}]\right|}}\;dx^{1}\wedge \cdots \wedge dx^{n}.} === Codifferential === The most important application of the Hodge star on manifolds is to define the codifferential δ {\displaystyle \delta } on k {\displaystyle k} -forms. Let δ = ( − 1 ) n ( k + 1 ) + 1 s ⋆ d ⋆ = ( − 1 ) k ⋆ − 1 d ⋆ {\displaystyle \delta =(-1)^{n(k+1)+1}s\ {\star }d{\star }=(-1)^{k}\,{\star }^{-1}d{\star }} where d {\displaystyle d} is the exterior derivative or differential, and s = 1 {\displaystyle s=1} for Riemannian manifolds. Then d : Ω k ( M ) → Ω k + 1 ( M ) {\displaystyle d:\Omega ^{k}(M)\to \Omega ^{k+1}(M)} while δ : Ω k ( M ) → Ω k − 1 ( M ) . {\displaystyle \delta :\Omega ^{k}(M)\to \Omega ^{k-1}(M).} The codifferential is not an antiderivation on the exterior algebra, in contrast to the exterior derivative. The codifferential is the adjoint of the exterior derivative with respect to the square-integrable scalar product: ⟨ ⟨ η , δ ζ ⟩ ⟩ = ⟨ ⟨ d η , ζ ⟩ ⟩ , {\displaystyle \langle \!\langle \eta ,\delta \zeta \rangle \!\rangle \ =\ \langle \!\langle d\eta ,\zeta \rangle \!\rangle ,} where ζ {\displaystyle \zeta } is a k {\displaystyle k} -form and η {\displaystyle \eta } a ( k − 1 ) {\displaystyle (k\!-\!1)} -form. This property is useful as it can be used to define the codifferential even when the manifold is non-orientable (and the Hodge star operator not defined). The identity can be proved from Stokes' theorem for smooth forms: 0 = ∫ M d ( η ∧ ⋆ ζ ) = ∫ M ( d η ∧ ⋆ ζ + ( − 1 ) k − 1 η ∧ ⋆ ⋆ − 1 d ⋆ ζ ) = ⟨ ⟨ d η , ζ ⟩ ⟩ − ⟨ ⟨ η , δ ζ ⟩ ⟩ , {\displaystyle 0\ =\ \int _{M}d(\eta \wedge {\star }\zeta )\ =\ \int _{M}\left(d\eta \wedge {\star }\zeta +(-1)^{k-1}\eta \wedge {\star }\,{\star }^{-1}d\,{\star }\zeta \right)\ =\ \langle \!\langle d\eta ,\zeta \rangle \!\rangle -\langle \!\langle \eta ,\delta \zeta \rangle \!\rangle ,} provided M {\displaystyle M} has empty boundary, or η {\displaystyle \eta } or ⋆ ζ {\displaystyle {\star }\zeta } has zero boundary values. (The proper definition of the above requires specifying a topological vector space that is closed and complete on the space of smooth forms. The Sobolev space is conventionally used; it allows the convergent sequence of forms ζ i → ζ {\displaystyle \zeta _{i}\to \zeta } (as i → ∞ {\displaystyle i\to \infty } ) to be interchanged with the combined differential and integral operations, so that ⟨ ⟨ η , δ ζ i ⟩ ⟩ → ⟨ ⟨ η , δ ζ ⟩ ⟩ {\displaystyle \langle \!\langle \eta ,\delta \zeta _{i}\rangle \!\rangle \to \langle \!\langle \eta ,\delta \zeta \rangle \!\rangle } and likewise for sequences converging to η {\displaystyle \eta } .) Since the differential satisfies d 2 = 0 {\displaystyle d^{2}=0} , the codifferential has the corresponding property δ 2 = ( − 1 ) n s 2 ⋆ d ⋆ ⋆ d ⋆ = ( − 1 ) n k + k + 1 s 3 ⋆ d 2 ⋆ = 0. {\displaystyle \delta ^{2}=(-1)^{n}s^{2}{\star }d{\star }{\star }d{\star }=(-1)^{nk+k+1}s^{3}{\star }d^{2}{\star }=0.} The Laplace–deRham operator is given by Δ = ( δ + d ) 2 = δ d + d δ {\displaystyle \Delta =(\delta +d)^{2}=\delta d+d\delta } and lies at the heart of Hodge theory. It is symmetric: ⟨ ⟨ Δ ζ , η ⟩ ⟩ = ⟨ ⟨ ζ , Δ η ⟩ ⟩ {\displaystyle \langle \!\langle \Delta \zeta ,\eta \rangle \!\rangle =\langle \!\langle \zeta ,\Delta \eta \rangle \!\rangle } and non-negative: ⟨ ⟨ Δ η , η ⟩ ⟩ ≥ 0. {\displaystyle \langle \!\langle \Delta \eta ,\eta \rangle \!\rangle \geq 0.} The Hodge star sends harmonic forms to harmonic forms. As a consequence of Hodge theory, the de Rham cohomology is naturally isomorphic to the space of harmonic k-forms, and so the Hodge star induces an isomorphism of cohomology groups ⋆ : H Δ k ( M ) → H Δ n − k ( M ) , {\displaystyle {\star }:H_{\Delta }^{k}(M)\to H_{\Delta }^{n-k}(M),} which in turn gives canonical identifications via Poincaré duality of H k(M) with its dual space. In coordinates, with notation as above, the codifferential of the form α {\displaystyle \alpha } may be written as δ α = − 1 k ! g m l ( ∂ ∂ x l α m , i 1 , … , i k − 1 − Γ m l j α j , i 1 , … , i k − 1 ) d x i 1 ∧ ⋯ ∧ d x i k − 1 , {\displaystyle \delta \alpha =\ -{\frac {1}{k!}}g^{ml}\left({\frac {\partial }{\partial x_{l}}}\alpha _{m,i_{1},\dots ,i_{k-1}}-\Gamma _{ml}^{j}\alpha _{j,i_{1},\dots ,i_{k-1}}\right)dx^{i_{1}}\wedge \dots \wedge dx^{i_{k-1}},} where here Γ m l j {\displaystyle \Gamma _{ml}^{j}} denotes the Christoffel symbols of { ∂ ∂ x 1 , … , ∂ ∂ x n } {\textstyle \left\{{\frac {\partial }{\partial x_{1}}},\ldots ,{\frac {\partial }{\partial x_{n}}}\right\}} . ==== Poincare lemma for codifferential ==== In analogy to the Poincare lemma for exterior derivative, one can define its version for codifferential, which reads If δ ω = 0 {\displaystyle \delta \omega =0} for ω ∈ Λ k ( U ) {\displaystyle \omega \in \Lambda ^{k}(U)} , where U {\displaystyle U} is a star domain on a manifold, then there is α ∈ Λ k + 1 ( U ) {\displaystyle \alpha \in \Lambda ^{k+1}(U)} such that ω = δ α {\displaystyle \omega =\delta \alpha } . A practical way of finding α {\displaystyle \alpha } is to use cohomotopy operator h {\displaystyle h} , that is a local inverse of δ {\displaystyle \delta } . One has to define a homotopy operator H β = ∫ 0 1 K ⌟ β | F ( t , x ) t k d t , {\displaystyle H\beta =\int _{0}^{1}{\mathcal {K}}\lrcorner \beta |_{F(t,x)}t^{k}dt,} where F ( t , x ) = x 0 + t ( x − x 0 ) {\displaystyle F(t,x)=x_{0}+t(x-x_{0})} is the linear homotopy between its center x 0 ∈ U {\displaystyle x_{0}\in U} and a point x ∈ U {\displaystyle x\in U} , and the (Euler) vector K = ∑ i = 1 n ( x − x 0 ) i ∂ x i {\displaystyle {\mathcal {K}}=\sum _{i=1}^{n}(x-x_{0})^{i}\partial _{x^{i}}} for n = dim ( U ) {\displaystyle n=\dim(U)} is inserted into the form β ∈ Λ ∗ ( U ) {\displaystyle \beta \in \Lambda ^{*}(U)} . We can then define cohomotopy operator as h : Λ ( U ) → Λ ( U ) , h := η ⋆ − 1 H ⋆ {\displaystyle h:\Lambda (U)\rightarrow \Lambda (U),\quad h:=\eta {\star }^{-1}H\star } , where η β = ( − 1 ) k β {\displaystyle \eta \beta =(-1)^{k}\beta } for β ∈ Λ k ( U ) {\displaystyle \beta \in \Lambda ^{k}(U)} . The cohomotopy operator fulfills (co)homotopy invariance formula δ h + h δ = I − S x 0 , {\displaystyle \delta h+h\delta =I-S_{x_{0}},} where S x 0 = ⋆ − 1 s x 0 ∗ ⋆ {\displaystyle S_{x_{0}}={\star }^{-1}s_{x_{0}}^{*}{\star }} , and s x 0 ∗ {\displaystyle s_{x_{0}}^{*}} is the pullback along the constant map s x 0 : x → x 0 {\displaystyle s_{x_{0}}:x\rightarrow x_{0}} . Therefore, if we want to solve the equation δ ω = 0 {\displaystyle \delta \omega =0} , applying cohomotopy invariance formula we get ω = δ h ω + S x 0 ω , {\displaystyle \omega =\delta h\omega +S_{x_{0}}\omega ,} where h ω ∈ Λ k + 1 ( U ) {\displaystyle h\omega \in \Lambda ^{k+1}(U)} is a differential form we are looking for, and "constant of integration" S x 0 ω {\displaystyle S_{x_{0}}\omega } vanishes unless ω {\displaystyle \omega } is a top form. Cohomotopy operator fulfills the following properties: h 2 = 0 , δ h δ = δ , h δ h = h {\displaystyle h^{2}=0,\quad \delta h\delta =\delta ,\quad h\delta h=h} . They make it possible to use it to define anticoexact forms on U {\displaystyle U} by Y ( U ) = { ω ∈ Λ ( U ) | ω = h δ ω } {\displaystyle {\mathcal {Y}}(U)=\{\omega \in \Lambda (U)|\omega =h\delta \omega \}} , which together with exact forms C ( U ) = { ω ∈ Λ ( U ) | ω = δ h ω } {\displaystyle {\mathcal {C}}(U)=\{\omega \in \Lambda (U)|\omega =\delta h\omega \}} make a direct sum decomposition Λ ( U ) = C ( U ) ⊕ Y ( U ) {\displaystyle \Lambda (U)={\mathcal {C}}(U)\oplus {\mathcal {Y}}(U)} . This direct sum is another way of saying that the cohomotopy invariance formula is a decomposition of unity, and the projector operators on the summands fulfills idempotence formulas: ( h δ ) 2 = h δ , ( δ h ) 2 = δ h {\displaystyle (h\delta )^{2}=h\delta ,\quad (\delta h)^{2}=\delta h} . These results are extension of similar results for exterior derivative. == Citations == == References ==
|
Wikipedia:Hofstadter's butterfly#0
|
In condensed matter physics, Hofstadter's butterfly is a graph of the spectral properties of non-interacting two-dimensional electrons in a perpendicular magnetic field in a lattice. The fractal, self-similar nature of the spectrum was discovered in the 1976 Ph.D. work of Douglas Hofstadter and is one of the early examples of modern scientific data visualization. The name reflects the fact that, as Hofstadter wrote, "the large gaps [in the graph] form a very striking pattern somewhat resembling a butterfly." The Hofstadter butterfly plays an important role in the theory of the integer quantum Hall effect and the theory of topological quantum numbers. == History == The first mathematical description of electrons on a 2D lattice, acted on by a perpendicular homogeneous magnetic field, was studied by Rudolf Peierls and his student R. G. Harper in the 1950s. Hofstadter first described the structure in 1976 in an article on the energy levels of Bloch electrons in perpendicular magnetic fields. It gives a graphical representation of the spectrum of Harper's equation at different frequencies. One key aspect of the mathematical structure of this spectrum – the splitting of energy bands for a specific value of the magnetic field, along a single dimension (energy) – had been previously mentioned in passing by Soviet physicist Mark Azbel in 1964 (in a paper cited by Hofstadter), but Hofstadter greatly expanded upon that work by plotting all values of the magnetic field against all energy values, creating the two-dimensional plot that first revealed the spectrum's uniquely recursive geometric properties. Written while Hofstadter was at the University of Oregon, his paper was influential in directing further research. It predicted on theoretical grounds that the allowed energy level values of an electron in a two-dimensional square lattice, as a function of a magnetic field applied perpendicularly to the system, formed what is now known as a fractal set. That is, the distribution of energy levels for small-scale changes in the applied magnetic field recursively repeats patterns seen in the large-scale structure. "Gplot", as Hofstadter called the figure, was described as a recursive structure in his 1976 article in Physical Review B, written before Benoit Mandelbrot's newly coined word "fractal" was introduced in an English text. Hofstadter also discusses the figure in his 1979 book Gödel, Escher, Bach. The structure became generally known as "Hofstadter's butterfly". David J. Thouless and his team discovered that the butterfly's wings are characterized by Chern integers, which provide a way to calculate the Hall conductance in Hofstadter's model. === Confirmation === In 1997 the Hofstadter butterfly was reproduced in experiments with a microwave guide equipped with an array of scatterers. The similarity between the mathematical description of the microwave guide with scatterers and Bloch's waves in the magnetic field allowed the reproduction of the Hofstadter butterfly for periodic sequences of the scatterers. In 2001, Christian Albrecht, Klaus von Klitzing, and coworkers realized an experimental setup to test Thouless et al.'s predictions about Hofstadter's butterfly with a two-dimensional electron gas in a superlattice potential. In 2013, three separate groups of researchers independently reported evidence of the Hofstadter butterfly spectrum in graphene devices fabricated on hexagonal boron nitride substrates. In this instance the butterfly spectrum results from the interplay between the applied magnetic field and the large-scale moiré pattern that develops when the graphene lattice is oriented with near zero-angle mismatch to the boron nitride. In September 2017, John Martinis's group at Google, in collaboration with the Angelakis group at CQT Singapore, published results from a simulation of 2D electrons in a perpendicular magnetic field using interacting photons in 9 superconducting qubits. The simulation recovered Hofstadter's butterfly, as expected. In 2021 the butterfly was observed in twisted bilayer graphene at the second magic angle. == Theoretical model == In his original paper, Hofstadter considers the following derivation: a charged quantum particle in a two-dimensional square lattice, with a lattice spacing a {\displaystyle a} , is described by a periodic Schrödinger equation, under a perpendicular static homogeneous magnetic field restricted to a single Bloch band. For a 2D square lattice, the tight binding energy dispersion relation is W ( k ) = E 0 ( cos k x a + cos k y a ) = E 0 2 ( e i k x a + e − i k x a + e i k y a + e − i k y a ) {\displaystyle W(\mathbf {k} )=E_{0}(\cos k_{x}a+\cos k_{y}a)={\frac {E_{0}}{2}}(e^{ik_{x}a}+e^{-ik_{x}a}+e^{ik_{y}a}+e^{-ik_{y}a})} , where W ( k ) {\displaystyle W(\mathbf {k} )} is the energy function, k = ( k x , k y ) {\displaystyle \mathbf {k} =(k_{x},k_{y})} is the crystal momentum, and E 0 {\displaystyle E_{0}} is an empirical parameter. The magnetic field B = ∇ × A {\displaystyle \mathbf {B} =\nabla \times \mathbf {A} } , where A {\displaystyle \mathbf {A} } the magnetic vector potential, can be taken into account by using Peierls substitution, replacing the crystal momentum with the canonical momentum ℏ k → p − q A {\displaystyle \hbar \mathbf {k} \to \mathbf {p} -q\mathbf {A} } , where p = ( p x , p y ) {\displaystyle \mathbf {p} =(p_{x},p_{y})} is the particle momentum operator and q {\displaystyle q} is the charge of the particle ( q = − e {\displaystyle q=-e} for the electron, e {\displaystyle e} is the elementary charge). For convenience we choose the gauge A = ( 0 , B x , 0 ) {\displaystyle \mathbf {A} =(0,Bx,0)} . Using that e i p j a {\displaystyle e^{ip_{j}a}} is the translation operator, so that e i p j a ψ ( x , y ) = ψ ( x + a , y ) {\displaystyle e^{ip_{j}a}\psi (x,y)=\psi (x+a,y)} , where j = x , y , z {\displaystyle j=x,y,z} and ψ ( r ) = ψ ( x , y ) {\displaystyle \psi (\mathbf {r} )=\psi (x,y)} is the particle's two-dimensional wave function. One can use W ( p − q A ) {\displaystyle W(\mathbf {p} -q\mathbf {A} )} as an effective Hamiltonian to obtain the following time-independent Schrödinger equation: E ψ ( x , y ) = E 0 2 [ ψ ( x + a , y ) + ψ ( x − a , y ) + ψ ( x , y + a ) e − i q B x a / ℏ + ψ ( x , y − a ) e + i q B x a / ℏ ] . {\displaystyle E\psi (x,y)={\frac {E_{0}}{2}}\left[\psi (x+a,y)+\psi (x-a,y)+\psi (x,y+a)e^{-iqBxa/\hbar }+\psi (x,y-a)e^{+iqBxa/\hbar }\right].} Considering that the particle can only hop between points in the lattice, we write x = n a , y = m a {\displaystyle x=na,y=ma} , where n , m {\displaystyle n,m} are integers. Hofstadter makes the following ansatz: ψ ( x , y ) = g n e i ν m {\displaystyle \psi (x,y)=g_{n}e^{i\nu m}} , where ν {\displaystyle \nu } depends on the energy, in order to obtain Harper's equation (also known as almost Mathieu operator for λ = 1 {\displaystyle \lambda =1} ): g n + 1 + g n − 1 + 2 cos ( 2 π n α − ν ) g n = ϵ g n , {\displaystyle g_{n+1}+g_{n-1}+2\cos(2\pi n\alpha -\nu )g_{n}=\epsilon g_{n},} where ϵ = 2 E / E 0 {\displaystyle \epsilon =2E/E_{0}} and α = ϕ ( B ) / ϕ 0 {\displaystyle \alpha =\phi (B)/\phi _{0}} , ϕ ( B ) = B a 2 {\displaystyle \phi (B)=Ba^{2}} is proportional to the magnetic flux through a lattice cell and ϕ 0 = 2 π ℏ / q {\displaystyle \phi _{0}=2\pi \hbar /q} is the magnetic flux quantum. The flux ratio α {\displaystyle \alpha } can also be expressed in terms of the magnetic length l m = ℏ / e B {\textstyle l_{\rm {m}}={\sqrt {\hbar /eB}}} , such that α = ( 2 π ) − 1 ( a / l m ) 2 {\textstyle \alpha =(2\pi )^{-1}(a/l_{\rm {m}})^{2}} . Hofstadter's butterfly is the resulting plot of ϵ α {\displaystyle \epsilon _{\alpha }} as a function of the flux ratio α {\displaystyle \alpha } , where ϵ α {\displaystyle \epsilon _{\alpha }} is the set of all possible ϵ {\displaystyle \epsilon } that are a solution to Harper's equation. === Solutions to Harper's equation and Wannier treatment === Due to the cosine function's properties, the pattern is periodic on α {\displaystyle \alpha } with period 1 (it repeats for each quantum flux per unit cell). The graph in the region of α {\displaystyle \alpha } between 0 and 1 has reflection symmetry in the lines α = 1 2 {\displaystyle \alpha ={\frac {1}{2}}} and ϵ = 0 {\displaystyle \epsilon =0} . Note that ϵ {\displaystyle \epsilon } is necessarily bounded between -4 and 4. Harper's equation has the particular property that the solutions depend on the rationality of α {\displaystyle \alpha } . By imposing periodicity over n {\displaystyle n} , one can show that if α = P / Q {\displaystyle \alpha =P/Q} (a rational number), where P {\displaystyle P} and Q {\displaystyle Q} are distinct prime numbers, there are exactly Q {\displaystyle Q} energy bands. For large Q ≫ P {\displaystyle Q\gg P} , the energy bands converge to thin energy bands corresponding to the Landau levels. Gregory Wannier showed that by taking into account the density of states, one can obtain a Diophantine equation that describes the system, as n n 0 = S + T α {\displaystyle {\frac {n}{n_{0}}}=S+T\alpha } where n = ∫ − 4 ϵ F ρ ( ϵ ) d ϵ ; n 0 = ∫ − 4 4 ρ ( ϵ ) d ϵ {\displaystyle n=\int _{-4}^{\epsilon _{\rm {F}}}\rho (\epsilon )\mathrm {d} \epsilon \;;\;n_{0}=\int _{-4}^{4}\rho (\epsilon )\mathrm {d} \epsilon } where S {\displaystyle S} and T {\displaystyle T} are integers, and ρ ( ϵ ) {\displaystyle \rho (\epsilon )} is the density of states at a given α {\displaystyle \alpha } . Here n {\displaystyle n} counts the number of states up to the Fermi energy, and n 0 {\displaystyle n_{0}} corresponds to the levels of the completely filled band (from ϵ = − 4 {\displaystyle \epsilon =-4} to ϵ = 4 {\displaystyle \epsilon =4} ). This equation characterizes all the solutions of Harper's equation. Most importantly, one can derive that when α {\displaystyle \alpha } is an irrational number, there are infinitely many solution for ϵ α {\displaystyle \epsilon _{\alpha }} . The union of all ϵ α {\displaystyle \epsilon _{\alpha }} forms a self-similar fractal that is discontinuous between rational and irrational values of α {\displaystyle \alpha } . This discontinuity is nonphysical, and continuity is recovered for a finite uncertainty in B {\displaystyle B} or for lattices of finite size. The scale at which the butterfly can be resolved in a real experiment depends on the system's specific conditions. === Phase diagram, conductance and topology === The phase diagram of electrons in a two-dimensional square lattice, as a function of a perpendicular magnetic field, chemical potential and temperature, has infinitely many phases. Thouless and coworkers showed that each phase is characterized by an integral Hall conductance, where all integer values are allowed. These integers are known as Chern numbers. == See also == Aubry–André model Schrodinger’s Cat == References ==
|
Wikipedia:Holmgren's uniqueness theorem#0
|
In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients. == Simple form of Holmgren's theorem == We will use the multi-index notation: Let α = { α 1 , … , α n } ∈ N 0 n , {\displaystyle \alpha =\{\alpha _{1},\dots ,\alpha _{n}\}\in \mathbb {N} _{0}^{n},} , with N 0 {\displaystyle \mathbb {N} _{0}} standing for the nonnegative integers; denote | α | = α 1 + ⋯ + α n {\displaystyle |\alpha |=\alpha _{1}+\cdots +\alpha _{n}} and ∂ x α = ( ∂ ∂ x 1 ) α 1 ⋯ ( ∂ ∂ x n ) α n {\displaystyle \partial _{x}^{\alpha }=\left({\frac {\partial }{\partial x_{1}}}\right)^{\alpha _{1}}\cdots \left({\frac {\partial }{\partial x_{n}}}\right)^{\alpha _{n}}} . Holmgren's theorem in its simpler form could be stated as follows: Assume that P = ∑|α| ≤m Aα(x)∂αx is an elliptic partial differential operator with real-analytic coefficients. If Pu is real-analytic in a connected open neighborhood Ω ⊂ Rn, then u is also real-analytic. This statement, with "analytic" replaced by "smooth", is Hermann Weyl's classical lemma on elliptic regularity: If P is an elliptic differential operator and Pu is smooth in Ω, then u is also smooth in Ω. This statement can be proved using Sobolev spaces. == Classical form == Let Ω {\displaystyle \Omega } be a connected open neighborhood in R n {\displaystyle \mathbb {R} ^{n}} , and let Σ {\displaystyle \Sigma } be an analytic hypersurface in Ω {\displaystyle \Omega } , such that there are two open subsets Ω + {\displaystyle \Omega _{+}} and Ω − {\displaystyle \Omega _{-}} in Ω {\displaystyle \Omega } , nonempty and connected, not intersecting Σ {\displaystyle \Sigma } nor each other, such that Ω = Ω − ∪ Σ ∪ Ω + {\displaystyle \Omega =\Omega _{-}\cup \Sigma \cup \Omega _{+}} . Let P = ∑ | α | ≤ m A α ( x ) ∂ x α {\displaystyle P=\sum _{|\alpha |\leq m}A_{\alpha }(x)\partial _{x}^{\alpha }} be a differential operator with real-analytic coefficients. Assume that the hypersurface Σ {\displaystyle \Sigma } is noncharacteristic with respect to P {\displaystyle P} at every one of its points: C h a r P ∩ N ∗ Σ = ∅ {\displaystyle \mathop {\rm {Char}} P\cap N^{*}\Sigma =\emptyset } . Above, C h a r P = { ( x , ξ ) ⊂ T ∗ R n ∖ 0 : σ p ( P ) ( x , ξ ) = 0 } , with σ p ( x , ξ ) = ∑ | α | = m i | α | A α ( x ) ξ α {\displaystyle \mathop {\rm {Char}} P=\{(x,\xi )\subset T^{*}\mathbb {R} ^{n}\backslash 0:\sigma _{p}(P)(x,\xi )=0\},{\text{ with }}\sigma _{p}(x,\xi )=\sum _{|\alpha |=m}i^{|\alpha |}A_{\alpha }(x)\xi ^{\alpha }} the principal symbol of P {\displaystyle P} . N ∗ Σ {\displaystyle N^{*}\Sigma } is a conormal bundle to Σ {\displaystyle \Sigma } , defined as N ∗ Σ = { ( x , ξ ) ∈ T ∗ R n : x ∈ Σ , ξ | T x Σ = 0 } {\displaystyle N^{*}\Sigma =\{(x,\xi )\in T^{*}\mathbb {R} ^{n}:x\in \Sigma ,\,\xi |_{T_{x}\Sigma }=0\}} . The classical formulation of Holmgren's theorem is as follows: Holmgren's theorem Let u {\displaystyle u} be a distribution in Ω {\displaystyle \Omega } such that P u = 0 {\displaystyle Pu=0} in Ω {\displaystyle \Omega } . If u {\displaystyle u} vanishes in Ω − {\displaystyle \Omega _{-}} , then it vanishes in an open neighborhood of Σ {\displaystyle \Sigma } . == Relation to the Cauchy–Kowalevski theorem == Consider the problem ∂ t m u = F ( t , x , ∂ x α ∂ t k u ) , α ∈ N 0 n , k ∈ N 0 , | α | + k ≤ m , k ≤ m − 1 , {\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u),\quad \alpha \in \mathbb {N} _{0}^{n},\quad k\in \mathbb {N} _{0},\quad |\alpha |+k\leq m,\quad k\leq m-1,} with the Cauchy data ∂ t k u | t = 0 = ϕ k ( x ) , 0 ≤ k ≤ m − 1 , {\displaystyle \partial _{t}^{k}u|_{t=0}=\phi _{k}(x),\qquad 0\leq k\leq m-1,} Assume that F ( t , x , z ) {\displaystyle F(t,x,z)} is real-analytic with respect to all its arguments in the neighborhood of t = 0 , x = 0 , z = 0 {\displaystyle t=0,x=0,z=0} and that ϕ k ( x ) {\displaystyle \phi _{k}(x)} are real-analytic in the neighborhood of x = 0 {\displaystyle x=0} . Theorem (Cauchy–Kowalevski) There is a unique real-analytic solution u ( t , x ) {\displaystyle u(t,x)} in the neighborhood of ( t , x ) = ( 0 , 0 ) ∈ ( R × R n ) {\displaystyle (t,x)=(0,0)\in (\mathbb {R} \times \mathbb {R} ^{n})} . Note that the Cauchy–Kowalevski theorem does not exclude the existence of solutions which are not real-analytic. On the other hand, in the case when F ( t , x , z ) {\displaystyle F(t,x,z)} is polynomial of order one in z {\displaystyle z} , so that ∂ t m u = F ( t , x , ∂ x α ∂ t k u ) = ∑ α ∈ N 0 n , 0 ≤ k ≤ m − 1 , | α | + k ≤ m A α , k ( t , x ) ∂ x α ∂ t k u , {\displaystyle \partial _{t}^{m}u=F(t,x,\partial _{x}^{\alpha }\,\partial _{t}^{k}u)=\sum _{\alpha \in \mathbb {N} _{0}^{n},0\leq k\leq m-1,|\alpha |+k\leq m}A_{\alpha ,k}(t,x)\,\partial _{x}^{\alpha }\,\partial _{t}^{k}u,} Holmgren's theorem states that the solution u {\displaystyle u} is real-analytic and hence, by the Cauchy–Kowalevski theorem, is unique. == See also == Cauchy–Kowalevski theorem FBI transform == References ==
|
Wikipedia:Homeomorphism#0
|
In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word homomorphism comes from the Ancient Greek language: ὁμός (homos) meaning "same" and μορφή (morphe) meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German ähnlich meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925). Homomorphisms of vector spaces are also called linear maps, and their study is the subject of linear algebra. The concept of homomorphism has been generalized, under the name of morphism, to many other structures that either do not have an underlying set, or are not algebraic. This generalization is the starting point of category theory. A homomorphism may also be an isomorphism, an endomorphism, an automorphism, etc. (see below). Each of those can be defined in a way that may be generalized to any class of morphisms. == Definition == A homomorphism is a map between two algebraic structures of the same type (e.g. two groups, two fields, two vector spaces), that preserves the operations of the structures. This means a map f : A → B {\displaystyle f:A\to B} between two sets A {\displaystyle A} , B {\displaystyle B} equipped with the same structure such that, if ⋅ {\displaystyle \cdot } is an operation of the structure (supposed here, for simplification, to be a binary operation), then f ( x ⋅ y ) = f ( x ) ⋅ f ( y ) {\displaystyle f(x\cdot y)=f(x)\cdot f(y)} for every pair x {\displaystyle x} , y {\displaystyle y} of elements of A {\displaystyle A} . One says often that f {\displaystyle f} preserves the operation or is compatible with the operation. Formally, a map f : A → B {\displaystyle f:A\to B} preserves an operation μ {\displaystyle \mu } of arity k {\displaystyle k} , defined on both A {\displaystyle A} and B {\displaystyle B} if f ( μ A ( a 1 , … , a k ) ) = μ B ( f ( a 1 ) , … , f ( a k ) ) , {\displaystyle f(\mu _{A}(a_{1},\ldots ,a_{k}))=\mu _{B}(f(a_{1}),\ldots ,f(a_{k})),} for all elements a 1 , . . . , a k {\displaystyle a_{1},...,a_{k}} in A {\displaystyle A} . The operations that must be preserved by a homomorphism include 0-ary operations, that is the constants. In particular, when an identity element is required by the type of structure, the identity element of the first structure must be mapped to the corresponding identity element of the second structure. For example: A semigroup homomorphism is a map between semigroups that preserves the semigroup operation. A monoid homomorphism is a map between monoids that preserves the monoid operation and maps the identity element of the first monoid to that of the second monoid (the identity element is a 0-ary operation). A group homomorphism is a map between groups that preserves the group operation. This implies that the group homomorphism maps the identity element of the first group to the identity element of the second group, and maps the inverse of an element of the first group to the inverse of the image of this element. Thus a semigroup homomorphism between groups is necessarily a group homomorphism. A ring homomorphism is a map between rings that preserves the ring addition, the ring multiplication, and the multiplicative identity. Whether the multiplicative identity is to be preserved depends upon the definition of ring in use. If the multiplicative identity is not preserved, one has a rng homomorphism. A linear map is a homomorphism of vector spaces; that is, a group homomorphism between vector spaces that preserves the abelian group structure and scalar multiplication. A module homomorphism, also called a linear map between modules, is defined similarly. An algebra homomorphism is a map that preserves the algebra operations. An algebraic structure may have more than one operation, and a homomorphism is required to preserve each operation. Thus a map that preserves only some of the operations is not a homomorphism of the structure, but only a homomorphism of the substructure obtained by considering only the preserved operations. For example, a map between monoids that preserves the monoid operation and not the identity element, is not a monoid homomorphism, but only a semigroup homomorphism. The notation for the operations does not need to be the same in the source and the target of a homomorphism. For example, the real numbers form a group for addition, and the positive real numbers form a group for multiplication. The exponential function x ↦ e x {\displaystyle x\mapsto e^{x}} satisfies e x + y = e x e y , {\displaystyle e^{x+y}=e^{x}e^{y},} and is thus a homomorphism between these two groups. It is even an isomorphism (see below), as its inverse function, the natural logarithm, satisfies ln ( x y ) = ln ( x ) + ln ( y ) , {\displaystyle \ln(xy)=\ln(x)+\ln(y),} and is also a group homomorphism. == Examples == The real numbers are a ring, having both addition and multiplication. The set of all 2×2 matrices is also a ring, under matrix addition and matrix multiplication. If we define a function between these rings as follows: f ( r ) = ( r 0 0 r ) {\displaystyle f(r)={\begin{pmatrix}r&0\\0&r\end{pmatrix}}} where r is a real number, then f is a homomorphism of rings, since f preserves both addition: f ( r + s ) = ( r + s 0 0 r + s ) = ( r 0 0 r ) + ( s 0 0 s ) = f ( r ) + f ( s ) {\displaystyle f(r+s)={\begin{pmatrix}r+s&0\\0&r+s\end{pmatrix}}={\begin{pmatrix}r&0\\0&r\end{pmatrix}}+{\begin{pmatrix}s&0\\0&s\end{pmatrix}}=f(r)+f(s)} and multiplication: f ( r s ) = ( r s 0 0 r s ) = ( r 0 0 r ) ( s 0 0 s ) = f ( r ) f ( s ) . {\displaystyle f(rs)={\begin{pmatrix}rs&0\\0&rs\end{pmatrix}}={\begin{pmatrix}r&0\\0&r\end{pmatrix}}{\begin{pmatrix}s&0\\0&s\end{pmatrix}}=f(r)\,f(s).} For another example, the nonzero complex numbers form a group under the operation of multiplication, as do the nonzero real numbers. (Zero must be excluded from both groups since it does not have a multiplicative inverse, which is required for elements of a group.) Define a function f {\displaystyle f} from the nonzero complex numbers to the nonzero real numbers by f ( z ) = | z | . {\displaystyle f(z)=|z|.} That is, f {\displaystyle f} is the absolute value (or modulus) of the complex number z {\displaystyle z} . Then f {\displaystyle f} is a homomorphism of groups, since it preserves multiplication: f ( z 1 z 2 ) = | z 1 z 2 | = | z 1 | | z 2 | = f ( z 1 ) f ( z 2 ) . {\displaystyle f(z_{1}z_{2})=|z_{1}z_{2}|=|z_{1}||z_{2}|=f(z_{1})f(z_{2}).} Note that f cannot be extended to a homomorphism of rings (from the complex numbers to the real numbers), since it does not preserve addition: | z 1 + z 2 | ≠ | z 1 | + | z 2 | . {\displaystyle |z_{1}+z_{2}|\neq |z_{1}|+|z_{2}|.} As another example, the diagram shows a monoid homomorphism f {\displaystyle f} from the monoid ( N , + , 0 ) {\displaystyle (\mathbb {N} ,+,0)} to the monoid ( N , × , 1 ) {\displaystyle (\mathbb {N} ,\times ,1)} . Due to the different names of corresponding operations, the structure preservation properties satisfied by f {\displaystyle f} amount to f ( x + y ) = f ( x ) × f ( y ) {\displaystyle f(x+y)=f(x)\times f(y)} and f ( 0 ) = 1 {\displaystyle f(0)=1} . A composition algebra A {\displaystyle A} over a field F {\displaystyle F} has a quadratic form, called a norm, N : A → F {\displaystyle N:A\to F} , which is a group homomorphism from the multiplicative group of A {\displaystyle A} to the multiplicative group of F {\displaystyle F} . == Special homomorphisms == Several kinds of homomorphisms have a specific name, which is also defined for general morphisms. === Isomorphism === An isomorphism between algebraic structures of the same type is commonly defined as a bijective homomorphism.: 134 : 28 In the more general context of category theory, an isomorphism is defined as a morphism that has an inverse that is also a morphism. In the specific case of algebraic structures, the two definitions are equivalent, although they may differ for non-algebraic structures, which have an underlying set. More precisely, if f : A → B {\displaystyle f:A\to B} is a (homo)morphism, it has an inverse if there exists a homomorphism g : B → A {\displaystyle g:B\to A} such that f ∘ g = Id B and g ∘ f = Id A . {\displaystyle f\circ g=\operatorname {Id} _{B}\qquad {\text{and}}\qquad g\circ f=\operatorname {Id} _{A}.} If A {\displaystyle A} and B {\displaystyle B} have underlying sets, and f : A → B {\displaystyle f:A\to B} has an inverse g {\displaystyle g} , then f {\displaystyle f} is bijective. In fact, f {\displaystyle f} is injective, as f ( x ) = f ( y ) {\displaystyle f(x)=f(y)} implies x = g ( f ( x ) ) = g ( f ( y ) ) = y {\displaystyle x=g(f(x))=g(f(y))=y} , and f {\displaystyle f} is surjective, as, for any x {\displaystyle x} in B {\displaystyle B} , one has x = f ( g ( x ) ) {\displaystyle x=f(g(x))} , and x {\displaystyle x} is the image of an element of A {\displaystyle A} . Conversely, if f : A → B {\displaystyle f:A\to B} is a bijective homomorphism between algebraic structures, let g : B → A {\displaystyle g:B\to A} be the map such that g ( y ) {\displaystyle g(y)} is the unique element x {\displaystyle x} of A {\displaystyle A} such that f ( x ) = y {\displaystyle f(x)=y} . One has f ∘ g = Id B and g ∘ f = Id A , {\displaystyle f\circ g=\operatorname {Id} _{B}{\text{ and }}g\circ f=\operatorname {Id} _{A},} and it remains only to show that g is a homomorphism. If ∗ {\displaystyle *} is a binary operation of the structure, for every pair x {\displaystyle x} , y {\displaystyle y} of elements of B {\displaystyle B} , one has g ( x ∗ B y ) = g ( f ( g ( x ) ) ∗ B f ( g ( y ) ) ) = g ( f ( g ( x ) ∗ A g ( y ) ) ) = g ( x ) ∗ A g ( y ) , {\displaystyle g(x*_{B}y)=g(f(g(x))*_{B}f(g(y)))=g(f(g(x)*_{A}g(y)))=g(x)*_{A}g(y),} and g {\displaystyle g} is thus compatible with ∗ . {\displaystyle *.} As the proof is similar for any arity, this shows that g {\displaystyle g} is a homomorphism. This proof does not work for non-algebraic structures. For example, for topological spaces, a morphism is a continuous map, and the inverse of a bijective continuous map is not necessarily continuous. An isomorphism of topological spaces, called homeomorphism or bicontinuous map, is thus a bijective continuous map, whose inverse is also continuous. === Endomorphism === An endomorphism is a homomorphism whose domain equals the codomain, or, more generally, a morphism whose source is equal to its target.: 135 The endomorphisms of an algebraic structure, or of an object of a category, form a monoid under composition. The endomorphisms of a vector space or of a module form a ring. In the case of a vector space or a free module of finite dimension, the choice of a basis induces a ring isomorphism between the ring of endomorphisms and the ring of square matrices of the same dimension. === Automorphism === An automorphism is an endomorphism that is also an isomorphism.: 135 The automorphisms of an algebraic structure or of an object of a category form a group under composition, which is called the automorphism group of the structure. Many groups that have received a name are automorphism groups of some algebraic structure. For example, the general linear group GL n ( k ) {\displaystyle \operatorname {GL} _{n}(k)} is the automorphism group of a vector space of dimension n {\displaystyle n} over a field k {\displaystyle k} . The automorphism groups of fields were introduced by Évariste Galois for studying the roots of polynomials, and are the basis of Galois theory. === Monomorphism === For algebraic structures, monomorphisms are commonly defined as injective homomorphisms.: 134 : 29 In the more general context of category theory, a monomorphism is defined as a morphism that is left cancelable. This means that a (homo)morphism f : A → B {\displaystyle f:A\to B} is a monomorphism if, for any pair g {\displaystyle g} , h {\displaystyle h} of morphisms from any other object C {\displaystyle C} to A {\displaystyle A} , then f ∘ g = f ∘ h {\displaystyle f\circ g=f\circ h} implies g = h {\displaystyle g=h} . These two definitions of monomorphism are equivalent for all common algebraic structures. More precisely, they are equivalent for fields, for which every homomorphism is a monomorphism, and for varieties of universal algebra, that is algebraic structures for which operations and axioms (identities) are defined without any restriction (the fields do not form a variety, as the multiplicative inverse is defined either as a unary operation or as a property of the multiplication, which are, in both cases, defined only for nonzero elements). In particular, the two definitions of a monomorphism are equivalent for sets, magmas, semigroups, monoids, groups, rings, fields, vector spaces and modules. A split monomorphism is a homomorphism that has a left inverse and thus it is itself a right inverse of that other homomorphism. That is, a homomorphism f : A → B {\displaystyle f\colon A\to B} is a split monomorphism if there exists a homomorphism g : B → A {\displaystyle g\colon B\to A} such that g ∘ f = Id A . {\displaystyle g\circ f=\operatorname {Id} _{A}.} A split monomorphism is always a monomorphism, for both meanings of monomorphism. For sets and vector spaces, every monomorphism is a split monomorphism, but this property does not hold for most common algebraic structures. === Epimorphism === In algebra, epimorphisms are often defined as surjective homomorphisms.: 134 : 43 On the other hand, in category theory, epimorphisms are defined as right cancelable morphisms. This means that a (homo)morphism f : A → B {\displaystyle f:A\to B} is an epimorphism if, for any pair g {\displaystyle g} , h {\displaystyle h} of morphisms from B {\displaystyle B} to any other object C {\displaystyle C} , the equality g ∘ f = h ∘ f {\displaystyle g\circ f=h\circ f} implies g = h {\displaystyle g=h} . A surjective homomorphism is always right cancelable, but the converse is not always true for algebraic structures. However, the two definitions of epimorphism are equivalent for sets, vector spaces, abelian groups, modules (see below for a proof), and groups. The importance of these structures in all mathematics, especially in linear algebra and homological algebra, may explain the coexistence of two non-equivalent definitions. Algebraic structures for which there exist non-surjective epimorphisms include semigroups and rings. The most basic example is the inclusion of integers into rational numbers, which is a homomorphism of rings and of multiplicative semigroups. For both structures it is a monomorphism and a non-surjective epimorphism, but not an isomorphism. A wide generalization of this example is the localization of a ring by a multiplicative set. Every localization is a ring epimorphism, which is not, in general, surjective. As localizations are fundamental in commutative algebra and algebraic geometry, this may explain why in these areas, the definition of epimorphisms as right cancelable homomorphisms is generally preferred. A split epimorphism is a homomorphism that has a right inverse and thus it is itself a left inverse of that other homomorphism. That is, a homomorphism f : A → B {\displaystyle f\colon A\to B} is a split epimorphism if there exists a homomorphism g : B → A {\displaystyle g\colon B\to A} such that f ∘ g = Id B . {\displaystyle f\circ g=\operatorname {Id} _{B}.} A split epimorphism is always an epimorphism, for both meanings of epimorphism. For sets and vector spaces, every epimorphism is a split epimorphism, but this property does not hold for most common algebraic structures. In summary, one has split epimorphism ⟹ epimorphism (surjective) ⟹ epimorphism (right cancelable) ; {\displaystyle {\text{split epimorphism}}\implies {\text{epimorphism (surjective)}}\implies {\text{epimorphism (right cancelable)}};} the last implication is an equivalence for sets, vector spaces, modules, abelian groups, and groups; the first implication is an equivalence for sets and vector spaces. == Kernel == Any homomorphism f : X → Y {\displaystyle f:X\to Y} defines an equivalence relation ∼ {\displaystyle \sim } on X {\displaystyle X} by a ∼ b {\displaystyle a\sim b} if and only if f ( a ) = f ( b ) {\displaystyle f(a)=f(b)} . The relation ∼ {\displaystyle \sim } is called the kernel of f {\displaystyle f} . It is a congruence relation on X {\displaystyle X} . The quotient set X / ∼ {\displaystyle X/{\sim }} can then be given a structure of the same type as X {\displaystyle X} , in a natural way, by defining the operations of the quotient set by [ x ] ∗ [ y ] = [ x ∗ y ] {\displaystyle [x]\ast [y]=[x\ast y]} , for each operation ∗ {\displaystyle \ast } of X {\displaystyle X} . In that case the image of X {\displaystyle X} in Y {\displaystyle Y} under the homomorphism f {\displaystyle f} is necessarily isomorphic to X / ∼ {\displaystyle X/\!\sim } ; this fact is one of the isomorphism theorems. When the algebraic structure is a group for some operation, the equivalence class K {\displaystyle K} of the identity element of this operation suffices to characterize the equivalence relation. In this case, the quotient by the equivalence relation is denoted by X / K {\displaystyle X/K} (usually read as " X {\displaystyle X} mod K {\displaystyle K} "). Also in this case, it is K {\displaystyle K} , rather than ∼ {\displaystyle \sim } , that is called the kernel of f {\displaystyle f} . The kernels of homomorphisms of a given type of algebraic structure are naturally equipped with some structure. This structure type of the kernels is the same as the considered structure, in the case of abelian groups, vector spaces and modules, but is different and has received a specific name in other cases, such as normal subgroup for kernels of group homomorphisms and ideals for kernels of ring homomorphisms (in the case of non-commutative rings, the kernels are the two-sided ideals). == Relational structures == In model theory, the notion of an algebraic structure is generalized to structures involving both operations and relations. Let L be a signature consisting of function and relation symbols, and A, B be two L-structures. Then a homomorphism from A to B is a mapping h from the domain of A to the domain of B such that h(FA(a1,...,an)) = FB(h(a1),...,h(an)) for each n-ary function symbol F in L, RA(a1,...,an) implies RB(h(a1),...,h(an)) for each n-ary relation symbol R in L. In the special case with just one binary relation, we obtain the notion of a graph homomorphism. == Formal language theory == Homomorphisms are also used in the study of formal languages and are often briefly referred to as morphisms. Given alphabets Σ 1 {\displaystyle \Sigma _{1}} and Σ 2 {\displaystyle \Sigma _{2}} , a function h : Σ 1 ∗ → Σ 2 ∗ {\displaystyle h\colon \Sigma _{1}^{*}\to \Sigma _{2}^{*}} such that h ( u v ) = h ( u ) h ( v ) {\displaystyle h(uv)=h(u)h(v)} for all u , v ∈ Σ 1 {\displaystyle u,v\in \Sigma _{1}} is called a homomorphism on Σ 1 ∗ {\displaystyle \Sigma _{1}^{*}} . If h {\displaystyle h} is a homomorphism on Σ 1 ∗ {\displaystyle \Sigma _{1}^{*}} and ε {\displaystyle \varepsilon } denotes the empty string, then h {\displaystyle h} is called an ε {\displaystyle \varepsilon } -free homomorphism when h ( x ) ≠ ε {\displaystyle h(x)\neq \varepsilon } for all x ≠ ε {\displaystyle x\neq \varepsilon } in Σ 1 ∗ {\displaystyle \Sigma _{1}^{*}} . A homomorphism h : Σ 1 ∗ → Σ 2 ∗ {\displaystyle h\colon \Sigma _{1}^{*}\to \Sigma _{2}^{*}} on Σ 1 ∗ {\displaystyle \Sigma _{1}^{*}} that satisfies | h ( a ) | = k {\displaystyle |h(a)|=k} for all a ∈ Σ 1 {\displaystyle a\in \Sigma _{1}} is called a k {\displaystyle k} -uniform homomorphism. If | h ( a ) | = 1 {\displaystyle |h(a)|=1} for all a ∈ Σ 1 {\displaystyle a\in \Sigma _{1}} (that is, h {\displaystyle h} is 1-uniform), then h {\displaystyle h} is also called a coding or a projection. The set Σ ∗ {\displaystyle \Sigma ^{*}} of words formed from the alphabet Σ {\displaystyle \Sigma } may be thought of as the free monoid generated by Σ {\displaystyle \Sigma } . Here the monoid operation is concatenation and the identity element is the empty word. From this perspective, a language homomorphism is precisely a monoid homomorphism. == See also == Diffeomorphism Homomorphic encryption Homomorphic secret sharing – a simplistic decentralized voting protocol Morphism Quasimorphism == Notes == == Citations == == References == Krieger, Dalia (2006). "On critical exponents in fixed points of non-erasing morphisms". In Ibarra, Oscar H.; Dang, Zhe (eds.). Developments in language theory : 10th international conference, DLT 2006, Santa Barbara, CA, USA, June 26-29, 2006 : proceedings. Berlin: Springer. pp. 280–291. ISBN 978-3-540-35430-7. OCLC 262693179. Stanley N. Burris; H.P. Sankappanavar (2012). A Course in Universal Algebra (PDF). S. Burris and H.P. Sankappanavar. ISBN 978-0-9880552-0-9. Mac Lane, Saunders (1971), Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, Springer, ISBN 0-387-90036-5, Zbl 0232.18001 Fraleigh, John B.; Katz, Victor J. (2003), A First Course in Abstract Algebra, Addison-Wesley, ISBN 978-1-292-02496-7
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.