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Vertex separator
In graph theory, a vertex subset $S\subset V$ is a vertex separator (or vertex cut, separating set) for nonadjacent vertices a and b if the removal of S from the graph separates a and b into distinct connected components.
Relevant topics on
Graph connectivity
• Connectivity
• Algebraic connectivity
• Cycle rank
• Rank (graph theory)
• SPQR tree
• St-connectivity
• K-connectivity certificate
• Pixel connectivity
• Vertex separator
• Strongly connected component
• Biconnected graph
• Bridge
Not to be confused with cut vertex.
Examples
Consider a grid graph with r rows and c columns; the total number n of vertices is r × c. For instance, in the illustration, r = 5, c = 8, and n = 40. If r is odd, there is a single central row, and otherwise there are two rows equally close to the center; similarly, if c is odd, there is a single central column, and otherwise there are two columns equally close to the center. Choosing S to be any of these central rows or columns, and removing S from the graph, partitions the graph into two smaller connected subgraphs A and B, each of which has at most n⁄2 vertices. If r ≤ c (as in the illustration), then choosing a central column will give a separator S with $r\leq {\sqrt {n}}$ vertices, and similarly if c ≤ r then choosing a central row will give a separator with at most ${\sqrt {n}}$ vertices. Thus, every grid graph has a separator S of size at most ${\sqrt {n}},$ the removal of which partitions it into two connected components, each of size at most n⁄2.[1]
To give another class of examples, every free tree T has a separator S consisting of a single vertex, the removal of which partitions T into two or more connected components, each of size at most n⁄2. More precisely, there is always exactly one or exactly two vertices, which amount to such a separator, depending on whether the tree is centered or bicentered.[2]
As opposed to these examples, not all vertex separators are balanced, but that property is most useful for applications in computer science, such as the planar separator theorem.
Minimal separators
Let S be an (a,b)-separator, that is, a vertex subset that separates two nonadjacent vertices a and b. Then S is a minimal (a,b)-separator if no proper subset of S separates a and b. More generally, S is called a minimal separator if it is a minimal separator for some pair (a,b) of nonadjacent vertices. Notice that this is different from minimal separating set which says that no proper subset of S is a minimal (u,v)-separator for any pair of vertices (u,v). The following is a well-known result characterizing the minimal separators:[3]
Lemma. A vertex separator S in G is minimal if and only if the graph G – S, obtained by removing S from G, has two connected components C1 and C2 such that each vertex in S is both adjacent to some vertex in C1 and to some vertex in C2.
The minimal (a,b)-separators also form an algebraic structure: For two fixed vertices a and b of a given graph G, an (a,b)-separator S can be regarded as a predecessor of another (a,b)-separator T, if every path from a to b meets S before it meets T. More rigorously, the predecessor relation is defined as follows: Let S and T be two (a,b)-separators in G. Then S is a predecessor of T, in symbols $S\sqsubseteq _{a,b}^{G}T$, if for each x ∈ S \ T, every path connecting x to b meets T. It follows from the definition that the predecessor relation yields a preorder on the set of all (a,b)-separators. Furthermore, Escalante (1972) proved that the predecessor relation gives rise to a complete lattice when restricted to the set of minimal (a,b)-separators in G.
See also
• Chordal graph, a graph in which every minimal separator is a clique.
• k-vertex-connected graph
Notes
1. George (1973). Instead of using a row or column of a grid graph, George partitions the graph into four pieces by using the union of a row and a column as a separator.
2. Jordan (1869)
3. Golumbic (1980).
References
• Escalante, F. (1972). "Schnittverbände in Graphen". Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg. 38: 199–220. doi:10.1007/BF02996932.
• George, J. Alan (1973), "Nested dissection of a regular finite element mesh", SIAM Journal on Numerical Analysis, 10 (2): 345–363, doi:10.1137/0710032, JSTOR 2156361.
• Golumbic, Martin Charles (1980), Algorithmic Graph Theory and Perfect Graphs, Academic Press, ISBN 0-12-289260-7.
• Jordan, Camille (1869). "Sur les assemblages de lignes". Journal für die reine und angewandte Mathematik (in French). 70 (2): 185–190.
• Rosenberg, Arnold; Heath, Lenwood (2002). Graph Separators, with Applications. Springer. doi:10.1007/b115747.
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Vertex-transitive graph
In the mathematical field of graph theory, a vertex-transitive graph is a graph G in which, given any two vertices v1 and v2 of G, there is some automorphism
$f:G\to G\ $
Graph families defined by their automorphisms
distance-transitive → distance-regular ← strongly regular
↓
symmetric (arc-transitive) ← t-transitive, t ≥ 2 skew-symmetric
↓
(if connected)
vertex- and edge-transitive
→ edge-transitive and regular → edge-transitive
↓ ↓ ↓
vertex-transitive → regular → (if bipartite)
biregular
↑
Cayley graph ← zero-symmetric asymmetric
such that
$f(v_{1})=v_{2}.\ $
In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices.[1] A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical.
Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).
Finite examples
Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices.[2]
Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees.[3]
Properties
The edge-connectivity of a vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d + 1)/3.[1] If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.[4]
Infinite examples
Infinite vertex-transitive graphs include:
• infinite paths (infinite in both directions)
• infinite regular trees, e.g. the Cayley graph of the free group
• graphs of uniform tessellations (see a complete list of planar tessellations), including all tilings by regular polygons
• infinite Cayley graphs
• the Rado graph
Two countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture stated that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001.[5] In 2005, Eskin, Fisher, and Whyte confirmed the counterexample.[6]
See also
• Edge-transitive graph
• Lovász conjecture
• Semi-symmetric graph
• Zero-symmetric graph
References
1. Godsil, Chris; Royle, Gordon (2013) [2001], Algebraic Graph Theory, Graduate Texts in Mathematics, vol. 207, Springer, ISBN 978-1-4613-0163-9.
2. Potočnik P., Spiga P. & Verret G. (2013), "Cubic vertex-transitive graphs on up to 1280 vertices", Journal of Symbolic Computation, 50: 465–477, arXiv:1201.5317, doi:10.1016/j.jsc.2012.09.002, S2CID 26705221.
3. Lauri, Josef; Scapellato, Raffaele (2003), Topics in graph automorphisms and reconstruction, London Mathematical Society Student Texts, vol. 54, Cambridge University Press, p. 44, ISBN 0-521-82151-7, MR 1971819. Lauri and Scapelleto credit this construction to Mark Watkins.
4. Babai, L. (1996), Technical Report TR-94-10, University of Chicago, archived from the original on 2010-06-11
5. Diestel, Reinhard; Leader, Imre (2001), "A conjecture concerning a limit of non-Cayley graphs" (PDF), Journal of Algebraic Combinatorics, 14 (1): 17–25, doi:10.1023/A:1011257718029, S2CID 10927964.
6. Eskin, Alex; Fisher, David; Whyte, Kevin (2005). "Quasi-isometries and rigidity of solvable groups". arXiv:math.GR/0511647..
External links
• Weisstein, Eric W. "Vertex-transitive graph". MathWorld.
• A census of small connected cubic vertex-transitive graphs . Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012.
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Asymptote
In analytic geometry, an asymptote (/ˈæsɪmptoʊt/) of a curve is a line such that the distance between the curve and the line approaches zero as one or both of the x or y coordinates tends to infinity. In projective geometry and related contexts, an asymptote of a curve is a line which is tangent to the curve at a point at infinity.[1][2]
The word asymptote is derived from the Greek ἀσύμπτωτος (asumptōtos) which means "not falling together", from ἀ priv. + σύν "together" + πτωτ-ός "fallen".[3] The term was introduced by Apollonius of Perga in his work on conic sections, but in contrast to its modern meaning, he used it to mean any line that does not intersect the given curve.[4]
There are three kinds of asymptotes: horizontal, vertical and oblique. For curves given by the graph of a function y = ƒ(x), horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. Vertical asymptotes are vertical lines near which the function grows without bound. An oblique asymptote has a slope that is non-zero but finite, such that the graph of the function approaches it as x tends to +∞ or −∞.
More generally, one curve is a curvilinear asymptote of another (as opposed to a linear asymptote) if the distance between the two curves tends to zero as they tend to infinity, although the term asymptote by itself is usually reserved for linear asymptotes.
Asymptotes convey information about the behavior of curves in the large, and determining the asymptotes of a function is an important step in sketching its graph.[5] The study of asymptotes of functions, construed in a broad sense, forms a part of the subject of asymptotic analysis.
Introduction
The idea that a curve may come arbitrarily close to a line without actually becoming the same may seem to counter everyday experience. The representations of a line and a curve as marks on a piece of paper or as pixels on a computer screen have a positive width. So if they were to be extended far enough they would seem to merge, at least as far as the eye could discern. But these are physical representations of the corresponding mathematical entities; the line and the curve are idealized concepts whose width is 0 (see Line). Therefore, the understanding of the idea of an asymptote requires an effort of reason rather than experience.
Consider the graph of the function $f(x)={\frac {1}{x}}$ shown in this section. The coordinates of the points on the curve are of the form $\left(x,{\frac {1}{x}}\right)$ where x is a number other than 0. For example, the graph contains the points (1, 1), (2, 0.5), (5, 0.2), (10, 0.1), ... As the values of $x$ become larger and larger, say 100, 1,000, 10,000 ..., putting them far to the right of the illustration, the corresponding values of $y$, .01, .001, .0001, ..., become infinitesimal relative to the scale shown. But no matter how large $x$ becomes, its reciprocal ${\frac {1}{x}}$ is never 0, so the curve never actually touches the x-axis. Similarly, as the values of $x$ become smaller and smaller, say .01, .001, .0001, ..., making them infinitesimal relative to the scale shown, the corresponding values of $y$, 100, 1,000, 10,000 ..., become larger and larger. So the curve extends farther and farther upward as it comes closer and closer to the y-axis. Thus, both the x and y-axis are asymptotes of the curve. These ideas are part of the basis of concept of a limit in mathematics, and this connection is explained more fully below.[6]
Asymptotes of functions
The asymptotes most commonly encountered in the study of calculus are of curves of the form y = ƒ(x). These can be computed using limits and classified into horizontal, vertical and oblique asymptotes depending on their orientation. Horizontal asymptotes are horizontal lines that the graph of the function approaches as x tends to +∞ or −∞. As the name indicates they are parallel to the x-axis. Vertical asymptotes are vertical lines (perpendicular to the x-axis) near which the function grows without bound. Oblique asymptotes are diagonal lines such that the difference between the curve and the line approaches 0 as x tends to +∞ or −∞.
Vertical asymptotes
The line x = a is a vertical asymptote of the graph of the function y = ƒ(x) if at least one of the following statements is true:
1. $\lim _{x\to a^{-}}f(x)=\pm \infty ,$
2. $\lim _{x\to a^{+}}f(x)=\pm \infty ,$
where $\lim _{x\to a^{-}}$ is the limit as x approaches the value a from the left (from lesser values), and $\lim _{x\to a^{+}}$ is the limit as x approaches a from the right.
For example, if ƒ(x) = x/(x–1), the numerator approaches 1 and the denominator approaches 0 as x approaches 1. So
$\lim _{x\to 1^{+}}{\frac {x}{x-1}}=+\infty $
$\lim _{x\to 1^{-}}{\frac {x}{x-1}}=-\infty $
and the curve has a vertical asymptote x = 1.
The function ƒ(x) may or may not be defined at a, and its precise value at the point x = a does not affect the asymptote. For example, for the function
$f(x)={\begin{cases}{\frac {1}{x}}&{\text{if }}x>0,\\5&{\text{if }}x\leq 0.\end{cases}}$
has a limit of +∞ as x → 0+, ƒ(x) has the vertical asymptote x = 0, even though ƒ(0) = 5. The graph of this function does intersect the vertical asymptote once, at (0, 5). It is impossible for the graph of a function to intersect a vertical asymptote (or a vertical line in general) in more than one point. Moreover, if a function is continuous at each point where it is defined, it is impossible that its graph does intersect any vertical asymptote.
A common example of a vertical asymptote is the case of a rational function at a point x such that the denominator is zero and the numerator is non-zero.
If a function has a vertical asymptote, then it isn't necessarily true that the derivative of the function has a vertical asymptote at the same place. An example is
$f(x)={\tfrac {1}{x}}+\sin({\tfrac {1}{x}})\quad $ at $\quad x=0$.
This function has a vertical asymptote at $x=0,$ because
$\lim _{x\to 0^{+}}f(x)=\lim _{x\to 0^{+}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=+\infty ,$
and
$\lim _{x\to 0^{-}}f(x)=\lim _{x\to 0^{-}}\left({\tfrac {1}{x}}+\sin \left({\tfrac {1}{x}}\right)\right)=-\infty $.
The derivative of $f$ is the function
$f'(x)={\frac {-(\cos({\tfrac {1}{x}})+1)}{x^{2}}}$.
For the sequence of points
$x_{n}={\frac {(-1)^{n}}{(2n+1)\pi }},\quad $ for $\quad n=0,1,2,\ldots $
that approaches $x=0$ both from the left and from the right, the values $f'(x_{n})$ are constantly $0$. Therefore, both one-sided limits of $f'$ at $0$ can be neither $+\infty $ nor $-\infty $. Hence $f'(x)$ doesn't have a vertical asymptote at $x=0$.
Horizontal asymptotes
Horizontal asymptotes are horizontal lines that the graph of the function approaches as x → ±∞. The horizontal line y = c is a horizontal asymptote of the function y = ƒ(x) if
$\lim _{x\rightarrow -\infty }f(x)=c$ or $\lim _{x\rightarrow +\infty }f(x)=c$.
In the first case, ƒ(x) has y = c as asymptote when x tends to −∞, and in the second ƒ(x) has y = c as an asymptote as x tends to +∞.
For example, the arctangent function satisfies
$\lim _{x\rightarrow -\infty }\arctan(x)=-{\frac {\pi }{2}}$ and $\lim _{x\rightarrow +\infty }\arctan(x)={\frac {\pi }{2}}.$
So the line y = –π/2 is a horizontal asymptote for the arctangent when x tends to –∞, and y = π/2 is a horizontal asymptote for the arctangent when x tends to +∞.
Functions may lack horizontal asymptotes on either or both sides, or may have one horizontal asymptote that is the same in both directions. For example, the function ƒ(x) = 1/(x2+1) has a horizontal asymptote at y = 0 when x tends both to −∞ and +∞ because, respectively,
$\lim _{x\to -\infty }{\frac {1}{x^{2}+1}}=\lim _{x\to +\infty }{\frac {1}{x^{2}+1}}=0.$
Other common functions that have one or two horizontal asymptotes include x ↦ 1/x (that has an hyperbola as it graph), the Gaussian function $x\mapsto \exp(-x^{2}),$ the error function, and the logistic function.
Oblique asymptotes
When a linear asymptote is not parallel to the x- or y-axis, it is called an oblique asymptote or slant asymptote. A function ƒ(x) is asymptotic to the straight line y = mx + n (m ≠ 0) if
$\lim _{x\to +\infty }\left[f(x)-(mx+n)\right]=0\,{\mbox{ or }}\lim _{x\to -\infty }\left[f(x)-(mx+n)\right]=0.$
In the first case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to +∞, and in the second case the line y = mx + n is an oblique asymptote of ƒ(x) when x tends to −∞.
An example is ƒ(x) = x + 1/x, which has the oblique asymptote y = x (that is m = 1, n = 0) as seen in the limits
$\lim _{x\to \pm \infty }\left[f(x)-x\right]$
$=\lim _{x\to \pm \infty }\left[\left(x+{\frac {1}{x}}\right)-x\right]$
$=\lim _{x\to \pm \infty }{\frac {1}{x}}=0.$
Elementary methods for identifying asymptotes
The asymptotes of many elementary functions can be found without the explicit use of limits (although the derivations of such methods typically use limits).
General computation of oblique asymptotes for functions
The oblique asymptote, for the function f(x), will be given by the equation y = mx + n. The value for m is computed first and is given by
$m\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}f(x)/x$
where a is either $-\infty $ or $+\infty $ depending on the case being studied. It is good practice to treat the two cases separately. If this limit doesn't exist then there is no oblique asymptote in that direction.
Having m then the value for n can be computed by
$n\;{\stackrel {\text{def}}{=}}\,\lim _{x\rightarrow a}(f(x)-mx)$
where a should be the same value used before. If this limit fails to exist then there is no oblique asymptote in that direction, even should the limit defining m exist. Otherwise y = mx + n is the oblique asymptote of ƒ(x) as x tends to a.
For example, the function ƒ(x) = (2x2 + 3x + 1)/x has
$m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {2x^{2}+3x+1}{x^{2}}}=2$ and then
$n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\left({\frac {2x^{2}+3x+1}{x}}-2x\right)=3$
so that y = 2x + 3 is the asymptote of ƒ(x) when x tends to +∞.
The function ƒ(x) = ln x has
$m=\lim _{x\rightarrow +\infty }f(x)/x=\lim _{x\rightarrow +\infty }{\frac {\ln x}{x}}=0$ and then
$n=\lim _{x\rightarrow +\infty }(f(x)-mx)=\lim _{x\rightarrow +\infty }\ln x$, which does not exist.
So y = ln x does not have an asymptote when x tends to +∞.
Asymptotes for rational functions
A rational function has at most one horizontal asymptote or oblique (slant) asymptote, and possibly many vertical asymptotes.
The degree of the numerator and degree of the denominator determine whether or not there are any horizontal or oblique asymptotes. The cases are tabulated below, where deg(numerator) is the degree of the numerator, and deg(denominator) is the degree of the denominator.
The cases of horizontal and oblique asymptotes for rational functions
deg(numerator)−deg(denominator) Asymptotes in general Example Asymptote for example
< 0 $y=0$ $f(x)={\frac {1}{x^{2}+1}}$ $y=0$
= 0 y = the ratio of leading coefficients $f(x)={\frac {2x^{2}+7}{3x^{2}+x+12}}$ $y={\frac {2}{3}}$
= 1 y = the quotient of the Euclidean division of the numerator by the denominator $f(x)={\frac {2x^{2}+3x+5}{x}}=2x+3+{\frac {5}{x}}$ $y=2x+3$
> 1 none $f(x)={\frac {2x^{4}}{3x^{2}+1}}$ no linear asymptote, but a curvilinear asymptote exists
The vertical asymptotes occur only when the denominator is zero (If both the numerator and denominator are zero, the multiplicities of the zero are compared). For example, the following function has vertical asymptotes at x = 0, and x = 1, but not at x = 2.
$f(x)={\frac {x^{2}-5x+6}{x^{3}-3x^{2}+2x}}={\frac {(x-2)(x-3)}{x(x-1)(x-2)}}$
Oblique asymptotes of rational functions
When the numerator of a rational function has degree exactly one greater than the denominator, the function has an oblique (slant) asymptote. The asymptote is the polynomial term after dividing the numerator and denominator. This phenomenon occurs because when dividing the fraction, there will be a linear term, and a remainder. For example, consider the function
$f(x)={\frac {x^{2}+x+1}{x+1}}=x+{\frac {1}{x+1}}$
shown to the right. As the value of x increases, f approaches the asymptote y = x. This is because the other term, 1/(x+1), approaches 0.
If the degree of the numerator is more than 1 larger than the degree of the denominator, and the denominator does not divide the numerator, there will be a nonzero remainder that goes to zero as x increases, but the quotient will not be linear, and the function does not have an oblique asymptote.
Transformations of known functions
If a known function has an asymptote (such as y=0 for f(x)=ex), then the translations of it also have an asymptote.
• If x=a is a vertical asymptote of f(x), then x=a+h is a vertical asymptote of f(x-h)
• If y=c is a horizontal asymptote of f(x), then y=c+k is a horizontal asymptote of f(x)+k
If a known function has an asymptote, then the scaling of the function also have an asymptote.
• If y=ax+b is an asymptote of f(x), then y=cax+cb is an asymptote of cf(x)
For example, f(x)=ex-1+2 has horizontal asymptote y=0+2=2, and no vertical or oblique asymptotes.
General definition
Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)). Suppose that the curve tends to infinity, that is:
$\lim _{t\rightarrow b}(x^{2}(t)+y^{2}(t))=\infty .$
A line ℓ is an asymptote of A if the distance from the point A(t) to ℓ tends to zero as t → b.[7] From the definition, only open curves that have some infinite branch can have an asymptote. No closed curve can have an asymptote.
For example, the upper right branch of the curve y = 1/x can be defined parametrically as x = t, y = 1/t (where t > 0). First, x → ∞ as t → ∞ and the distance from the curve to the x-axis is 1/t which approaches 0 as t → ∞. Therefore, the x-axis is an asymptote of the curve. Also, y → ∞ as t → 0 from the right, and the distance between the curve and the y-axis is t which approaches 0 as t → 0. So the y-axis is also an asymptote. A similar argument shows that the lower left branch of the curve also has the same two lines as asymptotes.
Although the definition here uses a parameterization of the curve, the notion of asymptote does not depend on the parameterization. In fact, if the equation of the line is $ax+by+c=0$ then the distance from the point A(t) = (x(t),y(t)) to the line is given by
${\frac {|ax(t)+by(t)+c|}{\sqrt {a^{2}+b^{2}}}}$
if γ(t) is a change of parameterization then the distance becomes
${\frac {|ax(\gamma (t))+by(\gamma (t))+c|}{\sqrt {a^{2}+b^{2}}}}$
which tends to zero simultaneously as the previous expression.
An important case is when the curve is the graph of a real function (a function of one real variable and returning real values). The graph of the function y = ƒ(x) is the set of points of the plane with coordinates (x,ƒ(x)). For this, a parameterization is
$t\mapsto (t,f(t)).$
This parameterization is to be considered over the open intervals (a,b), where a can be −∞ and b can be +∞.
An asymptote can be either vertical or non-vertical (oblique or horizontal). In the first case its equation is x = c, for some real number c. The non-vertical case has equation y = mx + n, where m and $n$ are real numbers. All three types of asymptotes can be present at the same time in specific examples. Unlike asymptotes for curves that are graphs of functions, a general curve may have more than two non-vertical asymptotes, and may cross its vertical asymptotes more than once.
Curvilinear asymptotes
Let A : (a,b) → R2 be a parametric plane curve, in coordinates A(t) = (x(t),y(t)), and B be another (unparameterized) curve. Suppose, as before, that the curve A tends to infinity. The curve B is a curvilinear asymptote of A if the shortest distance from the point A(t) to a point on B tends to zero as t → b. Sometimes B is simply referred to as an asymptote of A, when there is no risk of confusion with linear asymptotes.[8]
For example, the function
$y={\frac {x^{3}+2x^{2}+3x+4}{x}}$
has a curvilinear asymptote y = x2 + 2x + 3, which is known as a parabolic asymptote because it is a parabola rather than a straight line.[9]
Asymptotes and curve sketching
Asymptotes are used in procedures of curve sketching. An asymptote serves as a guide line to show the behavior of the curve towards infinity.[10] In order to get better approximations of the curve, curvilinear asymptotes have also been used [11] although the term asymptotic curve seems to be preferred.[12]
Algebraic curves
The asymptotes of an algebraic curve in the affine plane are the lines that are tangent to the projectivized curve through a point at infinity.[13] For example, one may identify the asymptotes to the unit hyperbola in this manner. Asymptotes are often considered only for real curves,[14] although they also make sense when defined in this way for curves over an arbitrary field.[15]
A plane curve of degree n intersects its asymptote at most at n−2 other points, by Bézout's theorem, as the intersection at infinity is of multiplicity at least two. For a conic, there are a pair of lines that do not intersect the conic at any complex point: these are the two asymptotes of the conic.
A plane algebraic curve is defined by an equation of the form P(x,y) = 0 where P is a polynomial of degree n
$P(x,y)=P_{n}(x,y)+P_{n-1}(x,y)+\cdots +P_{1}(x,y)+P_{0}$
where Pk is homogeneous of degree k. Vanishing of the linear factors of the highest degree term Pn defines the asymptotes of the curve: setting Q = Pn, if Pn(x, y) = (ax − by) Qn−1(x, y), then the line
$Q'_{x}(b,a)x+Q'_{y}(b,a)y+P_{n-1}(b,a)=0$
is an asymptote if $Q'_{x}(b,a)$ and $Q'_{y}(b,a)$ are not both zero. If $Q'_{x}(b,a)=Q'_{y}(b,a)=0$ and $P_{n-1}(b,a)\neq 0$, there is no asymptote, but the curve has a branch that looks like a branch of parabola. Such a branch is called a parabolic branch, even when it does not have any parabola that is a curvilinear asymptote. If $Q'_{x}(b,a)=Q'_{y}(b,a)=P_{n-1}(b,a)=0,$ the curve has a singular point at infinity which may have several asymptotes or parabolic branches.
Over the complex numbers, Pn splits into linear factors, each of which defines an asymptote (or several for multiple factors). Over the reals, Pn splits in factors that are linear or quadratic factors. Only the linear factors correspond to infinite (real) branches of the curve, but if a linear factor has multiplicity greater than one, the curve may have several asymptotes or parabolic branches. It may also occur that such a multiple linear factor corresponds to two complex conjugate branches, and does not corresponds to any infinite branch of the real curve. For example, the curve x4 + y2 - 1 = 0 has no real points outside the square $|x|\leq 1,|y|\leq 1$, but its highest order term gives the linear factor x with multiplicity 4, leading to the unique asymptote x=0.
Asymptotic cone
The hyperbola
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=1$
has the two asymptotes
$y=\pm {\frac {b}{a}}x.$
The equation for the union of these two lines is
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}=0.$
Similarly, the hyperboloid
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=1$
is said to have the asymptotic cone[16][17]
${\frac {x^{2}}{a^{2}}}-{\frac {y^{2}}{b^{2}}}-{\frac {z^{2}}{c^{2}}}=0.$
The distance between the hyperboloid and cone approaches 0 as the distance from the origin approaches infinity.
More generally, consider a surface that has an implicit equation $P_{d}(x,y,z)+P_{d-2}(x,y,z)+\cdots P_{0}=0,$ where the $P_{i}$ are homogeneous polynomials of degree $i$ and $P_{d-1}=0$. Then the equation $P_{d}(x,y,z)=0$ defines a cone which is centered at the origin. It is called an asymptotic cone, because the distance to the cone of a point of the surface tends to zero when the point on the surface tends to infinity.
See also
• Big O notation
References
General references
• Kuptsov, L.P. (2001) [1994], "Asymptote", Encyclopedia of Mathematics, EMS Press
Specific references
1. Williamson, Benjamin (1899), "Asymptotes", An elementary treatise on the differential calculus
2. Nunemacher, Jeffrey (1999), "Asymptotes, Cubic Curves, and the Projective Plane", Mathematics Magazine, 72 (3): 183–192, CiteSeerX 10.1.1.502.72, doi:10.2307/2690881, JSTOR 2690881
3. Oxford English Dictionary, second edition, 1989.
4. D.E. Smith, History of Mathematics, vol 2 Dover (1958) p. 318
5. Apostol, Tom M. (1967), Calculus, Vol. 1: One-Variable Calculus with an Introduction to Linear Algebra (2nd ed.), New York: John Wiley & Sons, ISBN 978-0-471-00005-1, §4.18.
6. Reference for section: "Asymptote" The Penny Cyclopædia vol. 2, The Society for the Diffusion of Useful Knowledge (1841) Charles Knight and Co., London p. 541
7. Pogorelov, A. V. (1959), Differential geometry, Translated from the first Russian ed. by L. F. Boron, Groningen: P. Noordhoff N. V., MR 0114163, §8.
8. Fowler, R. H. (1920), The elementary differential geometry of plane curves, Cambridge, University Press, hdl:2027/uc1.b4073882, ISBN 0-486-44277-2, p. 89ff.
9. William Nicholson, The British enciclopaedia, or dictionary of arts and sciences; comprising an accurate and popular view of the present improved state of human knowledge, Vol. 5, 1809
10. Frost, P. An elementary treatise on curve tracing (1918) online
11. Fowler, R. H. The elementary differential geometry of plane curves Cambridge, University Press, 1920, pp 89ff.(online at archive.org)
12. Frost, P. An elementary treatise on curve tracing, 1918, page 5
13. C.G. Gibson (1998) Elementary Geometry of Algebraic Curves, § 12.6 Asymptotes, Cambridge University Press ISBN 0-521-64140-3,
14. Coolidge, Julian Lowell (1959), A treatise on algebraic plane curves, New York: Dover Publications, ISBN 0-486-49576-0, MR 0120551, pp. 40–44.
15. Kunz, Ernst (2005), Introduction to plane algebraic curves, Boston, MA: Birkhäuser Boston, ISBN 978-0-8176-4381-2, MR 2156630, p. 121.
16. L.P. Siceloff, G. Wentworth, D.E. Smith Analytic geometry (1922) p. 271
17. P. Frost Solid geometry (1875) This has a more general treatment of asymptotic surfaces.
External links
Wikimedia Commons has media related to Asymptotics.
• Asymptote at PlanetMath.
• Hyperboloid and Asymptotic Cone, string surface model, 1872 Archived 2012-02-15 at the Wayback Machine from the Science Museum
Authority control: National
• Japan
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Spatial gradient
A spatial gradient is a gradient whose components are spatial derivatives, i.e., rate of change of a given scalar physical quantity with respect to the position coordinates. Homogeneous regions have spatial gradient vector norm equal to zero. When evaluated over vertical position (altitude or depth), it is called vertical gradient; the remainder is called horizontal gradient, the vector projection of the full gradient onto the horizontal plane.
Examples:
Biology
• Concentration gradient, the ratio of solute concentration between two adjoining regions
• Potential gradient, the difference in electric charge between two adjoining regions
Fluid dynamics and earth science
• Density gradient
• Pressure gradient
• Temperature gradient
• Geothermal gradient
• Sound speed gradient
• Wind gradient
• Lapse rate
See also
• Grade (slope)
• Time derivative
• Material derivative
• Structure tensor
• Surface gradient
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Vertical and horizontal bundles
In mathematics, the vertical bundle and the horizontal bundle are vector bundles associated to a smooth fiber bundle. More precisely, given a smooth fiber bundle $\pi \colon E\to B$, the vertical bundle $VE$ and horizontal bundle $HE$ are subbundles of the tangent bundle $TE$ of $E$ whose Whitney sum satisfies $VE\oplus HE\cong TE$. This means that, over each point $e\in E$, the fibers $V_{e}E$ and $H_{e}E$ form complementary subspaces of the tangent space $T_{e}E$. The vertical bundle consists of all vectors that are tangent to the fibers, while the horizontal bundle requires some choice of complementary subbundle.
To make this precise, define the vertical space $V_{e}E$ at $e\in E$ to be $\ker(d\pi _{e})$. That is, the differential $d\pi _{e}\colon T_{e}E\to T_{b}B$ (where $b=\pi (e)$) is a linear surjection whose kernel has the same dimension as the fibers of $\pi $. If we write $F=\pi ^{-1}(b)$, then $V_{e}E$ consists of exactly the vectors in $T_{e}E$ which are also tangent to $F$. The name is motivated by low-dimensional examples like the trivial line bundle over a circle, which is sometimes depicted as a vertical cylinder projecting to a horizontal circle. A subspace $H_{e}E$ of $T_{e}E$ is called a horizontal space if $T_{e}E$ is the direct sum of $V_{e}E$ and $H_{e}E$.
The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE; this is the vertical bundle of E. Likewise, provided the horizontal spaces $H_{e}E$ vary smoothly with e, their disjoint union is a horizontal bundle. The use of the words "the" and "a" here is intentional: each vertical subspace is unique, defined explicitly by $\ker(d\pi _{e})$. Excluding trivial cases, there are an infinite number of horizontal subspaces at each point. Also note that arbitrary choices of horizontal space at each point will not, in general, form a smooth vector bundle; they must also vary in an appropriately smooth way.
The horizontal bundle is one way to formulate the notion of an Ehresmann connection on a fiber bundle. Thus, for example, if E is a principal G-bundle, then the horizontal bundle is usually required to be G-invariant: such a choice is equivalent to a connection on the principal bundle.[1] This notably occurs when E is the frame bundle associated to some vector bundle, which is a principal $\operatorname {GL} _{n}$ bundle.
Formal definition
Let π:E→B be a smooth fiber bundle over a smooth manifold B. The vertical bundle is the kernel VE := ker(dπ) of the tangent map dπ : TE → TB.[2]
Since dπe is surjective at each point e, it yields a regular subbundle of TE. Furthermore, the vertical bundle VE is also integrable.
An Ehresmann connection on E is a choice of a complementary subbundle HE to VE in TE, called the horizontal bundle of the connection. At each point e in E, the two subspaces form a direct sum, such that TeE = VeE ⊕ HeE.
Example
A simple example of a smooth fiber bundle is a Cartesian product of two manifolds. Consider the bundle B1 := (M × N, pr1) with bundle projection pr1 : M × N → M : (x, y) → x. Applying the definition in the paragraph above to find the vertical bundle, we consider first a point (m,n) in M × N. Then the image of this point under pr1 is m. The preimage of m under this same pr1 is {m} × N, so that T(m,n) ({m} × N) = {m} × TN. The vertical bundle is then VB1 = M × TN, which is a subbundle of T(M ×N). If we take the other projection pr2 : M × N → N : (x, y) → y to define the fiber bundle B2 := (M × N, pr2) then the vertical bundle will be VB2 = TM × N.
In both cases, the product structure gives a natural choice of horizontal bundle, and hence an Ehresmann connection: the horizontal bundle of B1 is the vertical bundle of B2 and vice versa.
Properties
Various important tensors and differential forms from differential geometry take on specific properties on the vertical and horizontal bundles, or even can be defined in terms of them. Some of these are:
• A vertical vector field is a vector field that is in the vertical bundle. That is, for each point e of E, one chooses a vector $v_{e}\in V_{e}E$ where $V_{e}E\subset T_{e}E=T_{e}(E_{\pi (e)})$ is the vertical vector space at e.[2]
• A differentiable r-form $\alpha $ on E is said to be a horizontal form if $\alpha (v_{1},...,v_{r})=0$ whenever at least one of the vectors $v_{1},...,v_{r}$ is vertical.
• The connection form vanishes on the horizontal bundle, and is non-zero only on the vertical bundle. In this way, the connection form can be used to define the horizontal bundle: The horizontal bundle is the kernel of the connection form.
• The solder form or tautological one-form vanishes on the vertical bundle and is non-zero only on the horizontal bundle. By definition, the solder form takes its values entirely in the horizontal bundle.
• For the case of a frame bundle, the torsion form vanishes on the vertical bundle, and can be used to define exactly that part that needs to be added to an arbitrary connection to turn it into a Levi-Civita connection, i.e. to make a connection be torsionless. Indeed, if one writes θ for the solder form, then the torsion tensor Θ is given by Θ = D θ (with D the exterior covariant derivative). For any given connection ω, there is a unique one-form σ on TE, called the contorsion tensor, that is vanishing in the vertical bundle, and is such that ω+σ is another connection 1-form that is torsion-free. The resulting one-form ω+σ is nothing other than the Levi-Civita connection. One can take this as a definition: since the torsion is given by $\Theta =D\theta =d\theta +\omega \wedge \theta $, the vanishing of the torsion is equivalent to having $d\theta =-(\omega +\sigma )\wedge \theta $, and it is not hard to show that σ must vanish on the vertical bundle, and that σ must be G-invariant on each fibre (more precisely, that σ transforms in the adjoint representation of G). Note that this defines the Levi-Civita connection without making any explicit reference to any metric tensor (although the metric tensor can be understood to be a special case of a solder form, as it establishes a mapping between the tangent and cotangent bundles of the base space, i.e. between the horizontal and vertical subspaces of the frame bundle).
• In the case where E is a principal bundle, then the fundamental vector field must necessarily live in the vertical bundle, and vanish in any horizontal bundle.
Notes
1. David Bleecker, Gauge Theory and Variational Principles (1981) Addison-Wesely Publishing Company ISBN 0-201-10096-7 (See theorem 1.2.4)
2. Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag (page 77)
References
• Choquet-Bruhat, Yvonne; DeWitt-Morette, Cécile (1977), Analysis, Manifolds and Physics, Amsterdam: Elsevier, ISBN 978-0-7204-0494-4
• Kobayashi, Shoshichi; Nomizu, Katsumi (1996). Foundations of Differential Geometry, Vol. 1 (New ed.). Wiley Interscience. ISBN 0-471-15733-3.
• Kolář, Ivan; Michor, Peter; Slovák, Jan (1993), Natural Operations in Differential Geometry (PDF), Springer-Verlag
• Krupka, Demeter; Janyška, Josef (1990), Lectures on differential invariants, Univerzita J. E. Purkyně V Brně, ISBN 80-210-0165-8
• Saunders, D.J. (1989), The geometry of jet bundles, Cambridge University Press, ISBN 0-521-36948-7
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Inverted snub dodecadodecahedron
In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60.[1] It is given a Schläfli symbol sr{5/3,5}.
Inverted snub dodecadodecahedron
TypeUniform star polyhedron
ElementsF = 84, E = 150
V = 60 (χ = −6)
Faces by sides60{3}+12{5}+12{5/2}
Coxeter diagram
Wythoff symbol| 5/3 2 5
Symmetry groupI, [5,3]+, 532
Index referencesU60, C76, W114
Dual polyhedronMedial inverted pentagonal hexecontahedron
Vertex figure
3.3.5.3.5/3
Bowers acronymIsdid
Cartesian coordinates
Cartesian coordinates for the vertices of an inverted snub dodecadodecahedron are all the even permutations of
(±2α, ±2, ±2β),
(±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
(±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
(±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
(±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
β = (α2/τ+τ)/(ατ−1/τ),
where τ = (1+√5)/2 is the golden mean and α is the negative real root of τα4−α3+2α2−α−1/τ, or approximately −0.3352090. Taking the odd permutations of the above coordinates with an odd number of plus signs gives another form, the enantiomorph of the other one.
Related polyhedra
Medial inverted pentagonal hexecontahedron
Medial inverted pentagonal hexecontahedron
TypeStar polyhedron
Face
ElementsF = 60, E = 150
V = 84 (χ = −6)
Symmetry groupI, [5,3]+, 532
Index referencesDU60
dual polyhedronInverted snub dodecadodecahedron
The medial inverted pentagonal hexecontahedron (or midly petaloid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the uniform inverted snub dodecadodecahedron. Its faces are irregular nonconvex pentagons, with one very acute angle.
Proportions
Denote the golden ratio by $\phi $, and let $\xi \approx -0.236\,993\,843\,45$ be the largest (least negative) real zero of the polynomial $P=8x^{4}-12x^{3}+5x+1$. Then each face has three equal angles of $\arccos(\xi )\approx 103.709\,182\,219\,53^{\circ }$, one of $\arccos(\phi ^{2}\xi +\phi )\approx 3.990\,130\,423\,41^{\circ }$ and one of $360^{\circ }-\arccos(\phi ^{-2}\xi -\phi ^{-1})\approx 224.882\,322\,917\,99^{\circ }$. Each face has one medium length edge, two short and two long ones. If the medium length is $2$, then the short edges have length
$1-{\sqrt {(1-\xi )/(\phi ^{3}-\xi )}}\approx 0.474\,126\,460\,54$,
and the long edges have length
$1+{\sqrt {(1-\xi )/(-\phi ^{-3}-\xi )}}\approx 37.551\,879\,448\,54$.
The dihedral angle equals $\arccos(\xi /(\xi +1))\approx 108.095\,719\,352\,34^{\circ }$. The other real zero of the polynomial $P$ plays a similar role for the medial pentagonal hexecontahedron.
See also
• List of uniform polyhedra
• Snub dodecadodecahedron
References
• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, ISBN 978-0-521-54325-5, MR 0730208 p. 124
1. Roman, Maeder. "60: inverted snub dodecadodecahedron". MathConsult.{{cite web}}: CS1 maint: url-status (link)
External links
• Weisstein, Eric W. "Medial inverted pentagonal hexecontahedron". MathWorld.
• Weisstein, Eric W. "Inverted snub dodecadodecahedron". MathWorld.
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Brownian excursion
In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In particular, a Brownian excursion process is a Wiener process conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive. BEPs are important because, among other reasons, they naturally arise as the limit process of a number of conditional functional central limit theorems.[1]
Definition
A Brownian excursion process, $e$, is a Wiener process (or Brownian motion) conditioned to be positive and to take the value 0 at time 1. Alternatively, it is a Brownian bridge process conditioned to be positive.
Another representation of a Brownian excursion $e$ in terms of a Brownian motion process W (due to Paul Lévy and noted by Kiyosi Itô and Henry P. McKean, Jr.[2]) is in terms of the last time $\tau _{-}$ that W hits zero before time 1 and the first time $\tau _{+}$ that Brownian motion $W$ hits zero after time 1:[2]
$\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{{\frac {|W((1-t)\tau _{-}+t\tau _{+})|}{\sqrt {\tau _{+}-\tau _{-}}}}:\ 0\leq t\leq 1\right\}.$
Let $\tau _{m}$ be the time that a Brownian bridge process $W_{0}$ achieves its minimum on [0, 1]. Vervaat (1979) shows that
$\{e(t):\ {0\leq t\leq 1}\}\ {\stackrel {d}{=}}\ \left\{W_{0}(\tau _{m}+t{\bmod {1}})-W_{0}(\tau _{m}):\ 0\leq t\leq 1\right\}.$
Properties
Vervaat's representation of a Brownian excursion has several consequences for various functions of $e$. In particular:
$M_{+}\equiv \sup _{0\leq t\leq 1}e(t)\ {\stackrel {d}{=}}\ \sup _{0\leq t\leq 1}W_{0}(t)-\inf _{0\leq t\leq 1}W_{0}(t),$
(this can also be derived by explicit calculations[3][4]) and
$\int _{0}^{1}e(t)\,dt\ {\stackrel {d}{=}}\ \int _{0}^{1}W_{0}(t)\,dt-\inf _{0\leq t\leq 1}W_{0}(t).$
The following result holds:[5]
$EM_{+}={\sqrt {\pi /2}}\approx 1.25331\ldots ,\,$
and the following values for the second moment and variance can be calculated by the exact form of the distribution and density:[5]
$EM_{+}^{2}\approx 1.64493\ldots \ ,\ \ \operatorname {Var} (M_{+})\approx 0.0741337\ldots .$
Groeneboom (1989), Lemma 4.2 gives an expression for the Laplace transform of (the density) of $\int _{0}^{1}e(t)\,dt$. A formula for a certain double transform of the distribution of this area integral is given by Louchard (1984).
Groeneboom (1983) and Pitman (1983) give decompositions of Brownian motion $W$ in terms of i.i.d Brownian excursions and the least concave majorant (or greatest convex minorant) of $W$.
For an introduction to Itô's general theory of Brownian excursions and the Itô Poisson process of excursions, see Revuz and Yor (1994), chapter XII.
Connections and applications
The Brownian excursion area
$A_{+}\equiv \int _{0}^{1}e(t)\,dt$
arises in connection with the enumeration of connected graphs, many other problems in combinatorial theory; see e.g.[6][7][8][9][10] and the limit distribution of the Betti numbers of certain varieties in cohomology theory.[11] Takacs (1991a) shows that $A_{+}$ has density
$f_{A_{+}}(x)={\frac {2{\sqrt {6}}}{x^{2}}}\sum _{j=1}^{\infty }v_{j}^{2/3}e^{-v_{j}}U\left(-{\frac {5}{6}},{\frac {4}{3}};v_{j}\right)\ \ {\text{ with }}\ \ v_{j}={\frac {2|a_{j}|^{3}}{27x^{2}}}$
where $a_{j}$ are the zeros of the Airy function and $U$ is the confluent hypergeometric function. Janson and Louchard (2007) show that
$f_{A_{+}}(x)\sim {\frac {72{\sqrt {6}}}{\sqrt {\pi }}}x^{2}e^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty ,$
and
$P(A_{+}>x)\sim {\frac {6{\sqrt {6}}}{\sqrt {\pi }}}xe^{-6x^{2}}\ \ {\text{ as }}\ \ x\rightarrow \infty .$
They also give higher-order expansions in both cases.
Janson (2007) gives moments of $A_{+}$ and many other area functionals. In particular,
$E(A_{+})={\frac {1}{2}}{\sqrt {\frac {\pi }{2}}},\ \ E(A_{+}^{2})={\frac {5}{12}}\approx 0.416666\ldots ,\ \ \operatorname {Var} (A_{+})={\frac {5}{12}}-{\frac {\pi }{8}}\approx .0239675\ldots \ .$
Brownian excursions also arise in connection with queuing problems,[12] railway traffic,[13][14] and the heights of random rooted binary trees.[15]
Related processes
• Brownian bridge
• Brownian meander
• reflected Brownian motion
• skew Brownian motion
Notes
1. Durrett, Iglehart: Functionals of Brownian Meander and Brownian Excursion, (1975)
2. Itô and McKean (1974, page 75)
3. Chung (1976)
4. Kennedy (1976)
5. Durrett and Iglehart (1977)
6. Wright, E. M. (1977). "The number of connected sparsely edged graphs". Journal of Graph Theory. 1 (4): 317–330. doi:10.1002/jgt.3190010407.
7. Wright, E. M. (1980). "The number of connected sparsely edged graphs. III. Asymptotic results". Journal of Graph Theory. 4 (4): 393–407. doi:10.1002/jgt.3190040409.
8. Spencer J (1997). "Enumerating graphs and Brownian motion". Communications on Pure and Applied Mathematics. 50 (3): 291–294. doi:10.1002/(sici)1097-0312(199703)50:3<291::aid-cpa4>3.0.co;2-6.
9. Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys. 4: 80–145. arXiv:0704.2289. Bibcode:2007arXiv0704.2289J. doi:10.1214/07-PS104. S2CID 14563292.
10. Flajolet, P.; Louchard, G. (2001). "Analytic variations on the Airy distribution". Algorithmica. 31 (3): 361–377. CiteSeerX 10.1.1.27.3450. doi:10.1007/s00453-001-0056-0. S2CID 6522038.
11. Reineke M (2005). "Cohomology of noncommutative Hilbert schemes". Algebras and Representation Theory. 8 (4): 541–561. arXiv:math/0306185. doi:10.1007/s10468-005-8762-y. S2CID 116587916.
12. Iglehart D. L. (1974). "Functional central limit theorems for random walks conditioned to stay positive". The Annals of Probability. 2 (4): 608–619. doi:10.1214/aop/1176996607.
13. Takacs L (1991a). "A Bernoulli excursion and its various applications". Advances in Applied Probability. 23 (3): 557–585. doi:10.1017/s0001867800023739.
14. Takacs L (1991b). "On a probability problem connected with railway traffic". Journal of Applied Mathematics and Stochastic Analysis. 4: 263–292. doi:10.1155/S1048953391000011.
15. Takacs L (1994). "On the Total Heights of Random Rooted Binary Trees". Journal of Combinatorial Theory, Series B. 61 (2): 155–166. doi:10.1006/jctb.1994.1041.
References
• Chung, K. L. (1975). "Maxima in Brownian excursions". Bulletin of the American Mathematical Society. 81 (4): 742–745. doi:10.1090/s0002-9904-1975-13852-3. MR 0373035.
• Chung, K. L. (1976). "Excursions in Brownian motion". Arkiv för Matematik. 14 (1): 155–177. Bibcode:1976ArM....14..155C. doi:10.1007/bf02385832. MR 0467948.
• Durrett, Richard T.; Iglehart, Donald L. (1977). "Functionals of Brownian meander and Brownian excursion". Annals of Probability. 5 (1): 130–135. doi:10.1214/aop/1176995896. JSTOR 2242808. MR 0436354.
• Groeneboom, Piet (1983). "The concave majorant of Brownian motion". Annals of Probability. 11 (4): 1016–1027. doi:10.1214/aop/1176993450. JSTOR 2243513. MR 0714964.
• Groeneboom, Piet (1989). "Brownian motion with a parabolic drift and Airy functions". Probability Theory and Related Fields. 81: 79–109. doi:10.1007/BF00343738. MR 0981568. S2CID 119980629.
• Itô, Kiyosi; McKean, Jr., Henry P. (2013) [1974]. Diffusion Processes and their Sample Paths. Classics in Mathematics (Second printing, corrected ed.). Springer-Verlag, Berlin. ISBN 978-3540606291. MR 0345224.
• Janson, Svante (2007). "Brownian excursion area, Wright's constants in graph enumeration, and other Brownian areas". Probability Surveys. 4: 80–145. arXiv:0704.2289. Bibcode:2007arXiv0704.2289J. doi:10.1214/07-ps104. MR 2318402. S2CID 14563292.
• Janson, Svante; Louchard, Guy (2007). "Tail estimates for the Brownian excursion area and other Brownian areas". Electronic Journal of Probability. 12: 1600–1632. arXiv:0707.0991. Bibcode:2007arXiv0707.0991J. doi:10.1214/ejp.v12-471. MR 2365879. S2CID 6281609.
• Kennedy, Douglas P. (1976). "The distribution of the maximum Brownian excursion". Journal of Applied Probability. 13 (2): 371–376. doi:10.2307/3212843. JSTOR 3212843. MR 0402955. S2CID 222386970.
• Lévy, Paul (1948). Processus Stochastiques et Mouvement Brownien. Gauthier-Villars, Paris. MR 0029120.
• Louchard, G. (1984). "Kac's formula, Levy's local time and Brownian excursion". Journal of Applied Probability. 21 (3): 479–499. doi:10.2307/3213611. JSTOR 3213611. MR 0752014. S2CID 123640749.
• Pitman, J. W. (1983). "Remarks on the convex minorant of Brownian motion". Seminar on Stochastic Processes, 1982. Progr. Probab. Statist. Vol. 5. Birkhauser, Boston. pp. 219–227. MR 0733673.
• Revuz, Daniel; Yor, Marc (2004). Continuous Martingales and Brownian Motion. Grundlehren der mathematischen Wissenschaften. Vol. 293. Springer-Verlag, Berlin. doi:10.1007/978-3-662-06400-9. ISBN 978-3-642-08400-3. MR 1725357.
• Vervaat, W. (1979). "A relation between Brownian bridge and Brownian excursion". Annals of Probability. 7 (1): 143–149. doi:10.1214/aop/1176995155. JSTOR 2242845. MR 0515820.
Stochastic processes
Discrete time
• Bernoulli process
• Branching process
• Chinese restaurant process
• Galton–Watson process
• Independent and identically distributed random variables
• Markov chain
• Moran process
• Random walk
• Loop-erased
• Self-avoiding
• Biased
• Maximal entropy
Continuous time
• Additive process
• Bessel process
• Birth–death process
• pure birth
• Brownian motion
• Bridge
• Excursion
• Fractional
• Geometric
• Meander
• Cauchy process
• Contact process
• Continuous-time random walk
• Cox process
• Diffusion process
• Empirical process
• Feller process
• Fleming–Viot process
• Gamma process
• Geometric process
• Hawkes process
• Hunt process
• Interacting particle systems
• Itô diffusion
• Itô process
• Jump diffusion
• Jump process
• Lévy process
• Local time
• Markov additive process
• McKean–Vlasov process
• Ornstein–Uhlenbeck process
• Poisson process
• Compound
• Non-homogeneous
• Schramm–Loewner evolution
• Semimartingale
• Sigma-martingale
• Stable process
• Superprocess
• Telegraph process
• Variance gamma process
• Wiener process
• Wiener sausage
Both
• Branching process
• Galves–Löcherbach model
• Gaussian process
• Hidden Markov model (HMM)
• Markov process
• Martingale
• Differences
• Local
• Sub-
• Super-
• Random dynamical system
• Regenerative process
• Renewal process
• Stochastic chains with memory of variable length
• White noise
Fields and other
• Dirichlet process
• Gaussian random field
• Gibbs measure
• Hopfield model
• Ising model
• Potts model
• Boolean network
• Markov random field
• Percolation
• Pitman–Yor process
• Point process
• Cox
• Poisson
• Random field
• Random graph
Time series models
• Autoregressive conditional heteroskedasticity (ARCH) model
• Autoregressive integrated moving average (ARIMA) model
• Autoregressive (AR) model
• Autoregressive–moving-average (ARMA) model
• Generalized autoregressive conditional heteroskedasticity (GARCH) model
• Moving-average (MA) model
Financial models
• Binomial options pricing model
• Black–Derman–Toy
• Black–Karasinski
• Black–Scholes
• Chan–Karolyi–Longstaff–Sanders (CKLS)
• Chen
• Constant elasticity of variance (CEV)
• Cox–Ingersoll–Ross (CIR)
• Garman–Kohlhagen
• Heath–Jarrow–Morton (HJM)
• Heston
• Ho–Lee
• Hull–White
• LIBOR market
• Rendleman–Bartter
• SABR volatility
• Vašíček
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Ample line bundle
In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of an ample line bundle, although there are several related classes of line bundles. Roughly speaking, positivity properties of a line bundle are related to having many global sections. Understanding the ample line bundles on a given variety X amounts to understanding the different ways of mapping X into projective space. In view of the correspondence between line bundles and divisors (built from codimension-1 subvarieties), there is an equivalent notion of an ample divisor.
In more detail, a line bundle is called basepoint-free if it has enough sections to give a morphism to projective space. A line bundle is semi-ample if some positive power of it is basepoint-free; semi-ampleness is a kind of "nonnegativity". More strongly, a line bundle on a complete variety X is very ample if it has enough sections to give a closed immersion (or "embedding") of X into projective space. A line bundle is ample if some positive power is very ample.
An ample line bundle on a projective variety X has positive degree on every curve in X. The converse is not quite true, but there are corrected versions of the converse, the Nakai–Moishezon and Kleiman criteria for ampleness.
Introduction
Pullback of a line bundle and hyperplane divisors
Given a morphism $f\colon X\to Y$ of schemes, a vector bundle E on Y (or more generally a coherent sheaf on Y) has a pullback to X, $f^{*}E$ (see Sheaf of modules#Operations). The pullback of a vector bundle is a vector bundle of the same rank. In particular, the pullback of a line bundle is a line bundle. (Briefly, the fiber of $f^{*}E$ at a point x in X is the fiber of E at f(x).)
The notions described in this article are related to this construction in the case of a morphism to projective space
$f\colon X\to \mathbb {P} ^{n},$
with E = O(1) the line bundle on projective space whose global sections are the homogeneous polynomials of degree 1 (that is, linear functions) in variables $x_{0},\ldots ,x_{n}$. The line bundle O(1) can also be described as the line bundle associated to a hyperplane in $\mathbb {P} ^{n}$ (because the zero set of a section of O(1) is a hyperplane). If f is a closed immersion, for example, it follows that the pullback $f^{*}O(1)$ is the line bundle on X associated to a hyperplane section (the intersection of X with a hyperplane in $\mathbb {P} ^{n}$).
Basepoint-free line bundles
Let X be a scheme over a field k (for example, an algebraic variety) with a line bundle L. (A line bundle may also be called an invertible sheaf.) Let $a_{0},...,a_{n}$ be elements of the k-vector space $H^{0}(X,L)$ of global sections of L. The zero set of each section is a closed subset of X; let U be the open subset of points at which at least one of $a_{0},\ldots ,a_{n}$ is not zero. Then these sections define a morphism
$f\colon U\to \mathbb {P} _{k}^{n},\ x\mapsto [a_{0}(x),\ldots ,a_{n}(x)].$
In more detail: for each point x of U, the fiber of L over x is a 1-dimensional vector space over the residue field k(x). Choosing a basis for this fiber makes $a_{0}(x),\ldots ,a_{n}(x)$ into a sequence of n+1 numbers, not all zero, and hence a point in projective space. Changing the choice of basis scales all the numbers by the same nonzero constant, and so the point in projective space is independent of the choice.
Moreover, this morphism has the property that the restriction of L to U is isomorphic to the pullback $f^{*}O(1)$.[1]
The base locus of a line bundle L on a scheme X is the intersection of the zero sets of all global sections of L. A line bundle L is called basepoint-free if its base locus is empty. That is, for every point x of X there is a global section of L which is nonzero at x. If X is proper over a field k, then the vector space $H^{0}(X,L)$ of global sections has finite dimension; the dimension is called $h^{0}(X,L)$.[2] So a basepoint-free line bundle L determines a morphism $f\colon X\to \mathbb {P} ^{n}$ over k, where $n=h^{0}(X,L)-1$, given by choosing a basis for $H^{0}(X,L)$. Without making a choice, this can be described as the morphism
$f\colon X\to \mathbb {P} (H^{0}(X,L))$
from X to the space of hyperplanes in $H^{0}(X,L)$, canonically associated to the basepoint-free line bundle L. This morphism has the property that L is the pullback $f^{*}O(1)$.
Conversely, for any morphism f from a scheme X to projective space $\mathbb {P} ^{n}$ over k, the pullback line bundle $f^{*}O(1)$ is basepoint-free. Indeed, O(1) is basepoint-free on $\mathbb {P} ^{n}$, because for every point y in $\mathbb {P} ^{n}$ there is a hyperplane not containing y. Therefore, for every point x in X, there is a section s of O(1) over $\mathbb {P} ^{n}$ that is not zero at f(x), and the pullback of s is a global section of $f^{*}O(1)$ that is not zero at x. In short, basepoint-free line bundles are exactly those that can be expressed as the pullback of O(1) by some morphism to projective space.
Nef, globally generated, semi-ample
The degree of a line bundle L on a proper curve C over k is defined as the degree of the divisor (s) of any nonzero rational section s of L. The coefficients of this divisor are positive at points where s vanishes and negative where s has a pole. Therefore, any line bundle L on a curve C such that $H^{0}(C,L)\neq 0$ has nonnegative degree (because sections of L over C, as opposed to rational sections, have no poles).[3] In particular, every basepoint-free line bundle on a curve has nonnegative degree. As a result, a basepoint-free line bundle L on any proper scheme X over a field is nef, meaning that L has nonnegative degree on every (irreducible) curve in X.[4]
More generally, a sheaf F of $O_{X}$-modules on a scheme X is said to be globally generated if there is a set I of global sections $s_{i}\in H^{0}(X,F)$ such that the corresponding morphism
$\bigoplus _{i\in I}O_{X}\to F$
of sheaves is surjective.[5] A line bundle is globally generated if and only if it is basepoint-free.
For example, every quasi-coherent sheaf on an affine scheme is globally generated.[6] Analogously, in complex geometry, Cartan's theorem A says that every coherent sheaf on a Stein manifold is globally generated.
A line bundle L on a proper scheme over a field is semi-ample if there is a positive integer r such that the tensor power $L^{\otimes r}$ is basepoint-free. A semi-ample line bundle is nef (by the corresponding fact for basepoint-free line bundles).[7]
Very ample line bundles
A line bundle L on a proper scheme X over a field k is said to be very ample if it is basepoint-free and the associated morphism
$f\colon X\to \mathbb {P} _{k}^{n}$
is a closed immersion. Here $n=h^{0}(X,L)-1$. Equivalently, L is very ample if X can be embedded into projective space of some dimension over k in such a way that L is the restriction of the line bundle O(1) to X.[8] The latter definition is used to define very ampleness for a line bundle on a proper scheme over any commutative ring.[9]
The name "very ample" was introduced by Alexander Grothendieck in 1961.[10] Various names had been used earlier in the context of linear systems of divisors.
For a very ample line bundle L on a proper scheme X over a field with associated morphism f, the degree of L on a curve C in X is the degree of f(C) as a curve in $\mathbb {P} ^{n}$. So L has positive degree on every curve in X (because every subvariety of projective space has positive degree).[11]
Definitions
Ample invertible sheaves on quasi-compact schemes
Ample line bundles are used most often on proper schemes, but they can be defined in much wider generality.
Let X be a scheme, and let ${\mathcal {L}}$ be an invertible sheaf on X. For each $x\in X$, let ${\mathfrak {m}}_{x}$ denote the ideal sheaf of the reduced subscheme supported only at x. For $s\in \Gamma (X,{\mathcal {L}})$, define
$X_{s}=\{x\in X\colon s_{x}\not \in {\mathfrak {m}}_{x}{\mathcal {L}}_{x}\}.$
Equivalently, if $\kappa (x)$ denotes the residue field at x (considered as a skyscraper sheaf supported at x), then
$X_{s}=\{x\in X\colon {\bar {s}}_{x}\neq 0\in \kappa (x)\otimes {\mathcal {L}}_{x}\},$
where ${\bar {s}}_{x}$ is the image of s in the tensor product.
Fix $s\in \Gamma (X,{\mathcal {L}})$. For every s, the restriction ${\mathcal {L}}|_{X_{s}}$ is a free ${\mathcal {O}}_{X}$-module trivialized by the restriction of s, meaning the multiplication-by-s morphism ${\mathcal {O}}_{X_{s}}\to {\mathcal {L}}|_{X_{s}}$ is an isomorphism. The set $X_{s}$ is always open, and the inclusion morphism $X_{s}\to X$ is an affine morphism. Despite this, $X_{s}$ need not be an affine scheme. For example, if $s=1\in \Gamma (X,{\mathcal {O}}_{X})$, then $X_{s}=X$ is open in itself and affine over itself but generally not affine.
Assume X is quasi-compact. Then ${\mathcal {L}}$ is ample if, for every $x\in X$, there exists an $n\geq 1$ and an $s\in \Gamma (X,{\mathcal {L}}^{\otimes n})$ such that $x\in X_{s}$ and $X_{s}$ is an affine scheme.[12] For example, the trivial line bundle ${\mathcal {O}}_{X}$ is ample if and only if X is quasi-affine.[13]
In general, it is not true that every $X_{s}$ is affine. For example, if $X=\mathbf {P} ^{2}\setminus \{O\}$ for some point O, and if ${\mathcal {L}}$ is the restriction of ${\mathcal {O}}_{\mathbf {P} ^{2}}(1)$ to X, then ${\mathcal {L}}$ and ${\mathcal {O}}_{\mathbf {P} ^{2}}(1)$ have the same global sections, and the non-vanishing locus of a section of ${\mathcal {L}}$ is affine if and only if the corresponding section of ${\mathcal {O}}_{\mathbf {P} ^{2}}(1)$ contains O.
It is necessary to allow powers of ${\mathcal {L}}$ in the definition. In fact, for every N, it is possible that $X_{s}$ is non-affine for every $s\in \Gamma (X,{\mathcal {L}}^{\otimes n})$ with $n\leq N$. Indeed, suppose Z is a finite set of points in $\mathbf {P} ^{2}$, $X=\mathbf {P} ^{2}\setminus Z$, and ${\mathcal {L}}={\mathcal {O}}_{\mathbf {P} ^{2}}(1)|_{X}$. The vanishing loci of the sections of ${\mathcal {L}}^{\otimes N}$ are plane curves of degree N. By taking Z to be a sufficiently large set of points in general position, we may ensure that no plane curve of degree N (and hence any lower degree) contains all the points of Z. In particular their non-vanishing loci are all non-affine.
Define $\textstyle S=\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {L}}^{\otimes n})$. Let $p\colon X\to \operatorname {Spec} \mathbf {Z} $ denote the structural morphism. There is a natural isomorphism between ${\mathcal {O}}_{X}$-algebra homomorphisms $\textstyle p^{*}({\tilde {S}})\to \bigoplus _{n\geq 0}{\mathcal {L}}^{\otimes n}$ and endomorphisms of the graded ring S. The identity endomorphism of S corresponds to a homomorphism $\varepsilon $. Applying the $\operatorname {Proj} $ functor produces a morphism from an open subscheme of X, denoted $G(\varepsilon )$, to $\operatorname {Proj} S$.
The basic characterization of ample invertible sheaves states that if X is a quasi-compact quasi-separated scheme and ${\mathcal {L}}$ is an invertible sheaf on X, then the following assertions are equivalent:[14]
1. ${\mathcal {L}}$ is ample.
2. The open sets $X_{s}$, where $s\in \Gamma (X,{\mathcal {L}}^{\otimes n})$ and $n\geq 0$, form a basis for the topology of X.
3. The open sets $X_{s}$ with the property of being affine, where $s\in \Gamma (X,{\mathcal {L}}^{\otimes n})$ and $n\geq 0$, form a basis for the topology of X.
4. $G(\varepsilon )=X$ and the morphism $G(\varepsilon )\to \operatorname {Proj} S$ is a dominant open immersion.
5. $G(\varepsilon )=X$ and the morphism $G(\varepsilon )\to \operatorname {Proj} S$ is a homeomorphism of the underlying topological space of X with its image.
6. For every quasi-coherent sheaf ${\mathcal {F}}$ on X, the canonical map $\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {F}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {F}}$ is surjective.
7. For every quasi-coherent sheaf of ideals ${\mathcal {J}}$ on X, the canonical map $\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}$ is surjective.
8. For every quasi-coherent sheaf of ideals ${\mathcal {J}}$ on X, the canonical map $\bigoplus _{n\geq 0}\Gamma (X,{\mathcal {J}}\otimes _{{\mathcal {O}}_{X}}{\mathcal {L}}^{\otimes n})\otimes _{\mathbf {Z} }{\mathcal {L}}^{\otimes {-n}}\to {\mathcal {J}}$ is surjective.
9. For every quasi-coherent sheaf ${\mathcal {F}}$ of finite type on X, there exists an integer $n_{0}$ such that for $n\geq n_{0}$, ${\mathcal {F}}\otimes {\mathcal {L}}^{\otimes n}$ is generated by its global sections.
10. For every quasi-coherent sheaf ${\mathcal {F}}$ of finite type on X, there exists integers $n>0$ and $k>0$ such that ${\mathcal {F}}$ is isomorphic to a quotient of ${\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}$.
11. For every quasi-coherent sheaf of ideals ${\mathcal {J}}$ of finite type on X, there exists integers $n>0$ and $k>0$ such that ${\mathcal {J}}$ is isomorphic to a quotient of ${\mathcal {L}}^{\otimes (-n)}\otimes {\mathcal {O}}_{X}^{k}$.
On proper schemes
When X is separated and finite type over an affine scheme, an invertible sheaf ${\mathcal {L}}$ is ample if and only if there exists a positive integer r such that the tensor power ${\mathcal {L}}^{\otimes r}$ is very ample.[15][16] In particular, a proper scheme over R has an ample line bundle if and only if it is projective over R. Often, this characterization is taken as the definition of ampleness.
The rest of this article will concentrate on ampleness on proper schemes over a field, as this is the most important case. An ample line bundle on a proper scheme X over a field has positive degree on every curve in X, by the corresponding statement for very ample line bundles.
A Cartier divisor D on a proper scheme X over a field k is said to be ample if the corresponding line bundle O(D) is ample. (For example, if X is smooth over k, then a Cartier divisor can be identified with a finite linear combination of closed codimension-1 subvarieties of X with integer coefficients.)
Weakening the notion of "very ample" to "ample" gives a flexible concept with a wide variety of different characterizations. A first point is that tensoring high powers of an ample line bundle with any coherent sheaf whatsoever gives a sheaf with many global sections. More precisely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that the sheaf $F\otimes L^{\otimes r}$ is globally generated for all $r\geq s$. Here s may depend on F.[17][18]
Another characterization of ampleness, known as the Cartan–Serre–Grothendieck theorem, is in terms of coherent sheaf cohomology. Namely, a line bundle L on a proper scheme X over a field (or more generally over a Noetherian ring) is ample if and only if for every coherent sheaf F on X, there is an integer s such that
$H^{i}(X,F\otimes L^{\otimes r})=0$
for all $i>0$ and all $r\geq s$.[19][18] In particular, high powers of an ample line bundle kill cohomology in positive degrees. This implication is called the Serre vanishing theorem, proved by Jean-Pierre Serre in his 1955 paper Faisceaux algébriques cohérents.
Examples/Non-examples
• The trivial line bundle $O_{X}$ on a projective variety X of positive dimension is basepoint-free but not ample. More generally, for any morphism f from a projective variety X to some projective space $\mathbb {P} ^{n}$ over a field, the pullback line bundle $L=f^{*}O(1)$ is always basepoint-free, whereas L is ample if and only if the morphism f is finite (that is, all fibers of f have dimension 0 or are empty).[20]
• For an integer d, the space of sections of the line bundle O(d) over $\mathbb {P} _{\mathbb {C} }^{1}$ is the complex vector space of homogeneous polynomials of degree d in variables x,y. In particular, this space is zero for d < 0. For $d\geq 0$, the morphism to projective space given by O(d) is
$\mathbb {P} ^{1}\to \mathbb {P} ^{d}$
by
$[x,y]\mapsto [x^{d},x^{d-1}y,\ldots ,y^{d}].$
This is a closed immersion for $d\geq 1$, with image a rational normal curve of degree d in $\mathbb {P} ^{d}$. Therefore, O(d) is basepoint-free if and only if $d\geq 0$, and very ample if and only if $d\geq 1$. It follows that O(d) is ample if and only if $d\geq 1$.
• For an example where "ample" and "very ample" are different, let X be a smooth projective curve of genus 1 (an elliptic curve) over C, and let p be a complex point of X. Let O(p) be the associated line bundle of degree 1 on X. Then the complex vector space of global sections of O(p) has dimension 1, spanned by a section that vanishes at p.[21] So the base locus of O(p) is equal to p. On the other hand, O(2p) is basepoint-free, and O(dp) is very ample for $d\geq 3$ (giving an embedding of X as an elliptic curve of degree d in $\mathbb {P} ^{d-1}$). Therefore, O(p) is ample but not very ample. Also, O(2p) is ample and basepoint-free but not very ample; the associated morphism to projective space is a ramified double cover $X\to \mathbb {P} ^{1}$.
• On curves of higher genus, there are ample line bundles L for which every global section is zero. (But high multiples of L have many sections, by definition.) For example, let X be a smooth plane quartic curve (of degree 4 in $\mathbb {P} ^{2}$) over C, and let p and q be distinct complex points of X. Then the line bundle $L=O(2p-q)$ is ample but has $H^{0}(X,L)=0$.[22]
Criteria for ampleness of line bundles
Intersection theory
Further information: intersection theory § Intersection theory in algebraic geometry
To determine whether a given line bundle on a projective variety X is ample, the following numerical criteria (in terms of intersection numbers) are often the most useful. It is equivalent to ask when a Cartier divisor D on X is ample, meaning that the associated line bundle O(D) is ample. The intersection number $D\cdot C$ can be defined as the degree of the line bundle O(D) restricted to C. In the other direction, for a line bundle L on a projective variety, the first Chern class $c_{1}(L)$ means the associated Cartier divisor (defined up to linear equivalence), the divisor of any nonzero rational section of L.
On a smooth projective curve X over an algebraically closed field k, a line bundle L is very ample if and only if $h^{0}(X,L\otimes O(-x-y))=h^{0}(X,L)-2$ for all k-rational points x,y in X.[23] Let g be the genus of X. By the Riemann–Roch theorem, every line bundle of degree at least 2g + 1 satisfies this condition and hence is very ample. As a result, a line bundle on a curve is ample if and only if it has positive degree.[24]
For example, the canonical bundle $K_{X}$ of a curve X has degree 2g − 2, and so it is ample if and only if $g\geq 2$. The curves with ample canonical bundle form an important class; for example, over the complex numbers, these are the curves with a metric of negative curvature. The canonical bundle is very ample if and only if $g\geq 2$ and the curve is not hyperelliptic.[25]
The Nakai–Moishezon criterion (named for Yoshikazu Nakai (1963) and Boris Moishezon (1964)) states that a line bundle L on a proper scheme X over a field is ample if and only if $\int _{Y}c_{1}(L)^{{\text{dim}}(Y)}>0$ for every (irreducible) closed subvariety Y of X (Y is not allowed to be a point).[26] In terms of divisors, a Cartier divisor D is ample if and only if $D^{{\text{dim}}(Y)}\cdot Y>0$ for every (nonzero-dimensional) subvariety Y of X. For X a curve, this says that a divisor is ample if and only if it has positive degree. For X a surface, the criterion says that a divisor D is ample if and only if its self-intersection number $D^{2}$ is positive and every curve C on X has $D\cdot C>0$.
Kleiman's criterion
To state Kleiman's criterion (1966), let X be a projective scheme over a field. Let $N_{1}(X)$ be the real vector space of 1-cycles (real linear combinations of curves in X) modulo numerical equivalence, meaning that two 1-cycles A and B are equal in $N_{1}(X)$ if and only if every line bundle has the same degree on A and on B. By the Néron–Severi theorem, the real vector space $N_{1}(X)$ has finite dimension. Kleiman's criterion states that a line bundle L on X is ample if and only if L has positive degree on every nonzero element C of the closure of the cone of curves NE(X) in $N_{1}(X)$. (This is slightly stronger than saying that L has positive degree on every curve.) Equivalently, a line bundle is ample if and only if its class in the dual vector space $N^{1}(X)$ is in the interior of the nef cone.[27]
Kleiman's criterion fails in general for proper (rather than projective) schemes X over a field, although it holds if X is smooth or more generally Q-factorial.[28]
A line bundle on a projective variety is called strictly nef if it has positive degree on every curve Nagata (1959). and David Mumford constructed line bundles on smooth projective surfaces that are strictly nef but not ample. This shows that the condition $c_{1}(L)^{2}>0$ cannot be omitted in the Nakai–Moishezon criterion, and it is necessary to use the closure of NE(X) rather than NE(X) in Kleiman's criterion.[29] Every nef line bundle on a surface has $c_{1}(L)^{2}\geq 0$, and Nagata and Mumford's examples have $c_{1}(L)^{2}=0$.
C. S. Seshadri showed that a line bundle L on a proper scheme over an algebraically closed field is ample if and only if there is a positive real number ε such that deg(L|C) ≥ εm(C) for all (irreducible) curves C in X, where m(C) is the maximum of the multiplicities at the points of C.[30]
Several characterizations of ampleness hold more generally for line bundles on a proper algebraic space over a field k. In particular, the Nakai-Moishezon criterion is valid in that generality.[31] The Cartan-Serre-Grothendieck criterion holds even more generally, for a proper algebraic space over a Noetherian ring R.[32] (If a proper algebraic space over R has an ample line bundle, then it is in fact a projective scheme over R.) Kleiman's criterion fails for proper algebraic spaces X over a field, even if X is smooth.[33]
Openness of ampleness
On a projective scheme X over a field, Kleiman's criterion implies that ampleness is an open condition on the class of an R-divisor (an R-linear combination of Cartier divisors) in $N^{1}(X)$, with its topology based on the topology of the real numbers. (An R-divisor is defined to be ample if it can be written as a positive linear combination of ample Cartier divisors.[34]) An elementary special case is: for an ample divisor H and any divisor E, there is a positive real number b such that $H+aE$ is ample for all real numbers a of absolute value less than b. In terms of divisors with integer coefficients (or line bundles), this means that nH + E is ample for all sufficiently large positive integers n.
Ampleness is also an open condition in a quite different sense, when the variety or line bundle is varied in an algebraic family. Namely, let $f\colon X\to Y$ be a proper morphism of schemes, and let L be a line bundle on X. Then the set of points y in Y such that L is ample on the fiber $X_{y}$ is open (in the Zariski topology). More strongly, if L is ample on one fiber $X_{y}$, then there is an affine open neighborhood U of y such that L is ample on $f^{-1}(U)$ over U.[35]
Kleiman's other characterizations of ampleness
Kleiman also proved the following characterizations of ampleness, which can be viewed as intermediate steps between the definition of ampleness and numerical criteria. Namely, for a line bundle L on a proper scheme X over a field, the following are equivalent:[36]
• L is ample.
• For every (irreducible) subvariety $Y\subset X$ of positive dimension, there is a positive integer r and a section $s\in H^{0}(Y,{\mathcal {L}}^{\otimes r})$ which is not identically zero but vanishes at some point of Y.
• For every (irreducible) subvariety $Y\subset X$ of positive dimension, the holomorphic Euler characteristics of powers of L on Y go to infinity:
$\chi (Y,{\mathcal {L}}^{\otimes r})\to \infty $ as $r\to \infty $.
Generalizations
Ample vector bundles
Robin Hartshorne defined a vector bundle F on a projective scheme X over a field to be ample if the line bundle ${\mathcal {O}}(1)$ on the space $\mathbb {P} (F)$ of hyperplanes in F is ample.[37]
Several properties of ample line bundles extend to ample vector bundles. For example, a vector bundle F is ample if and only if high symmetric powers of F kill the cohomology $H^{i}$ of coherent sheaves for all $i>0$.[38] Also, the Chern class $c_{r}(F)$ of an ample vector bundle has positive degree on every r-dimensional subvariety of X, for $1\leq r\leq {\text{rank}}(F)$.[39]
Big line bundles
Main article: Iitaka dimension
A useful weakening of ampleness, notably in birational geometry, is the notion of a big line bundle. A line bundle L on a projective variety X of dimension n over a field is said to be big if there is a positive real number a and a positive integer $j_{0}$ such that $h^{0}(X,L^{\otimes j})\geq aj^{n}$ for all $j\geq j_{0}$. This is the maximum possible growth rate for the spaces of sections of powers of L, in the sense that for every line bundle L on X there is a positive number b with $h^{0}(X,L^{\otimes j})\leq bj^{n}$ for all j > 0.[40]
There are several other characterizations of big line bundles. First, a line bundle is big if and only if there is a positive integer r such that the rational map from X to $\mathbb {P} (H^{0}(X,L^{\otimes r}))$ given by the sections of $L^{\otimes r}$ is birational onto its image.[41] Also, a line bundle L is big if and only if it has a positive tensor power which is the tensor product of an ample line bundle A and an effective line bundle B (meaning that $H^{0}(X,B)\neq 0$).[42] Finally, a line bundle is big if and only if its class in $N^{1}(X)$ is in the interior of the cone of effective divisors.[43]
Bigness can be viewed as a birationally invariant analog of ampleness. For example, if $f\colon X\to Y$ is a dominant rational map between smooth projective varieties of the same dimension, then the pullback of a big line bundle on Y is big on X. (At first sight, the pullback is only a line bundle on the open subset of X where f is a morphism, but this extends uniquely to a line bundle on all of X.) For ample line bundles, one can only say that the pullback of an ample line bundle by a finite morphism is ample.[20]
Example: Let X be the blow-up of the projective plane $\mathbb {P} ^{2}$ at a point over the complex numbers. Let H be the pullback to X of a line on $\mathbb {P} ^{2}$, and let E be the exceptional curve of the blow-up $\pi \colon X\to \mathbb {P} ^{2}$. Then the divisor H + E is big but not ample (or even nef) on X, because
$(H+E)\cdot E=E^{2}=-1<0.$
This negativity also implies that the base locus of H + E (or of any positive multiple) contains the curve E. In fact, this base locus is equal to E.
Relative ampleness
Given a quasi-compact morphism of schemes $f:X\to S$, an invertible sheaf L on X is said to be ample relative to f or f-ample if the following equivalent conditions are met:[44][45]
1. For each open affine subset $U\subset S$, the restriction of L to $f^{-1}(U)$ is ample (in the usual sense).
2. f is quasi-separated and there is an open immersion $X\hookrightarrow \operatorname {Proj} _{S}({\mathcal {R}}),\,{\mathcal {R}}:=f_{*}\left(\bigoplus _{0}^{\infty }L^{\otimes n}\right)$ induced by the adjunction map:
$f^{*}{\mathcal {R}}\to \bigoplus _{0}^{\infty }L^{\otimes n}$.
3. The condition 2. without "open".
The condition 2 says (roughly) that X can be openly compactified to a projective scheme with ${\mathcal {O}}(1)=L$ (not just to a proper scheme).
See also
General algebraic geometry
• Algebraic geometry of projective spaces
• Fano variety: a variety whose canonical bundle is anti-ample
• Matsusaka's big theorem
• Divisorial scheme: a scheme admitting an ample family of line bundles
Ampleness in complex geometry
• Holomorphic vector bundle
• Kodaira embedding theorem: on a compact complex manifold, ampleness and positivity coincide.
• Kodaira vanishing theorem
• Lefschetz hyperplane theorem: an ample divisor in a complex projective variety X is topologically similar to X.
Notes
1. Hartshorne (1977), Theorem II.7.1.
2. Hartshorne (1977), Theorem III.5.2; (tag 02O6).
3. Hartshorne (1977), Lemma IV.1.2.
4. Lazarsfeld (2004), Example 1.4.5.
5. tag 01AM.
6. Hartshorne (1977), Example II.5.16.2.
7. Lazarsfeld (2004), Definition 2.1.26.
8. Hartshorne (1977), section II.5.
9. tag 02NP.
10. Grothendieck, EGA II, Definition 4.2.2.
11. Hartshorne (1977), Proposition I.7.6 and Example IV.3.3.2.
12. tag 01PS.
13. tag 01QE.
14. EGA II, Théorème 4.5.2 and Proposition 4.5.5.
15. EGA II, Proposition 4.5.10.
16. tag 01VU.
17. Hartshorne (1977), Theorem II.7.6
18. Lazarsfeld (2004), Theorem 1.2.6.
19. Hartshorne (1977), Proposition III.5.3
20. Lazarsfeld (2004), Theorem 1.2.13.
21. Hartshorne (1977), Example II.7.6.3.
22. Hartshorne (1977), Exercise IV.3.2(b).
23. Hartshorne (1977), Proposition IV.3.1.
24. Hartshorne (1977), Corollary IV.3.3.
25. Hartshorne (1977), Proposition IV.5.2.
26. Lazarsfeld (2004), Theorem 1.2.23, Remark 1.2.29; Kleiman (1966), Theorem III.1.
27. Lazarsfeld (2004), Theorems 1.4.23 and 1.4.29; Kleiman (1966), Theorem IV.1.
28. Fujino (2005), Corollary 3.3; Lazarsfeld (2004), Remark 1.4.24.
29. Lazarsfeld (2004), Example 1.5.2.
30. Lazarsfeld (2004), Theorem 1.4.13; Hartshorne (1970), Theorem I.7.1.
31. Kollár (1990), Theorem 3.11.
32. tag 0D38.
33. Kollár (1996), Chapter VI, Appendix, Exercise 2.19.3.
34. Lazarsfeld (2004), Definition 1.3.11.
35. Lazarsfeld (2004), Theorem 1.2.17 and its proof.
36. Lazarsfeld (2004), Example 1.2.32; Kleiman (1966), Theorem III.1.
37. Lazarsfeld (2004), Definition 6.1.1.
38. Lazarsfeld (2004), Theorem 6.1.10.
39. Lazarsfeld (2004), Theorem 8.2.2.
40. Lazarsfeld (2004), Corollary 2.1.38.
41. Lazarsfeld (2004), section 2.2.A.
42. Lazarsfeld (2004), Corollary 2.2.7.
43. Lazarsfeld (2004), Theorem 2.2.26.
44. tag 01VG.
45. Grothendieck & Dieudonné 1961, Proposition 4.6.3.
Sources
• Fujino, Osamu (2005), "On the Kleiman-Mori cone", Proceedings of the Japan Academy, Series A, Mathematical Sciences, 81 (5): 80–84, arXiv:math/0501055, Bibcode:2005math......1055F, doi:10.3792/pjaa.81.80, MR 2143547
• Grothendieck, Alexandre; Dieudonné, Jean (1961). "Éléments de géométrie algébrique: II. Étude globale élémentaire de quelques classes de morphismes". Publications Mathématiques de l'IHÉS. 8. doi:10.1007/bf02699291. MR 0217084.
• Hartshorne, Robin (1970), Ample Subvarieties of Algebraic Varieties, Lecture Notes in Mathematics, vol. 156, Berlin, Heidelberg: Springer-Verlag, doi:10.1007/BFb0067839, ISBN 978-3-540-05184-8, MR 0282977
• Hartshorne, Robin (1977), Algebraic Geometry, Berlin, New York: Springer-Verlag, ISBN 978-0-387-90244-9, MR 0463157
• Kleiman, Steven L. (1966), "Toward a numerical theory of ampleness", Annals of Mathematics, Second Series, 84 (3): 293–344, doi:10.2307/1970447, ISSN 0003-486X, JSTOR 1970447, MR 0206009
• Kollár, János (1990), "Projectivity of complete moduli", Journal of Differential Geometry, 32, doi:10.4310/jdg/1214445046, MR 1064874
• Kollár, János (1996), Rational curves on algebraic varieties, Berlin, Heidelberg: Springer-Verlag, doi:10.1007/978-3-662-03276-3, ISBN 978-3-642-08219-1, MR 1440180
• Lazarsfeld, Robert (2004), Positivity in algebraic geometry (2 vols.), Berlin: Springer-Verlag, doi:10.1007/978-3-642-18808-4, ISBN 3-540-22533-1, MR 2095471
• Nagata, Masayoshi (1959), "On the 14th problem of Hilbert", American Journal of Mathematics, 81 (3): 766–772, doi:10.2307/2372927, JSTOR 2372927, MR 0154867
• "Section 29.37 (01VG): Relatively ample sheaves—The Stacks project".
• Stacks Project, Tag 01AM.
• Stacks Project, Tag 01PS.
• Stacks Project, Tag 01QE.
• Stacks Project, Tag 01VU.
• Stacks Project, Tag 02NP.
• Stacks Project, Tag 02O6
• Stacks Project, Tag 0D38.
External links
• The Stacks Project
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Very large-scale neighborhood search
In mathematical optimization, neighborhood search is a technique that tries to find good or near-optimal solutions to a combinatorial optimisation problem by repeatedly transforming a current solution into a different solution in the neighborhood of the current solution. The neighborhood of a solution is a set of similar solutions obtained by relatively simple modifications to the original solution. For a very large-scale neighborhood search, the neighborhood is large and possibly exponentially sized.
The resulting algorithms can outperform algorithms using small neighborhoods because the local improvements are larger. If neighborhood searched is limited to just one or a very small number of changes from the current solution, then it can be difficult to escape from local minima, even with additional meta-heuristic techniques such as Simulated Annealing or Tabu search. In large neighborhood search techniques, the possible changes from one solution to its neighbor may allow tens or hundreds of values to change, and this means that the size of the neighborhood may itself be sufficient to allow the search process to avoid or escape local minima, though additional meta-heuristic techniques can still improve performance.
References
• Ahuja, Ravindra K.; Orlin, James B.; Sharma, Dushyant (2000), "Very large-scale neighborhood search" (PDF), International Transactions in Operational Research, 7 (4–5): 301–317, doi:10.1111/j.1475-3995.2000.tb00201.x
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Verónica Martínez de la Vega
Verónica Martínez de la Vega y Mansilla is a Mexican mathematician whose research involves topology and hypertopology. She is a researcher in the Institute of Mathematics at the National Autonomous University of Mexico (UNAM).[1]
Education and career
Martínez de la Vega was born in Mexico City, on January 5, 1971. Her family worked as lawyers, and discouraged her from going into science, but nevertheless she ended up studying mathematics at UNAM, and wrote an undergraduate thesis in topology that she published as a journal paper in Topology and its Applications.[2] Continuing to graduate study in topology at UNAM, she completed her PhD in 2002 with the dissertation Estudio sobre dendroides y compactaciones supervised by Polish topologist Janusz J. Charatonik, becoming his only female doctoral student.[2][3]
After postgraduate research at UAM Iztapalapa and California State University, Sacramento, she joined the Institute of Mathematics as a researcher in 2005.[4]
Recognition
Martínez de la Vega is a member of the Mexican Academy of Sciences.[5] In 2017 UNAM gave her their "Reconocimiento Sor Juana Inés de la Cruz" award.[2]
References
1. "Dra. Verónica Martínez de la Vega y Mansilla", Directory, UNAM Faculty of Sciences, retrieved 2022-11-26
2. "Verónica Martínez de la Vega y Mansilla recibe el "Reconocimiento Sor Juana Inés de la Cruz"", Noticias del IM, UNAM Institute of Mathematics, retrieved 2022-11-26
3. Verónica Martínez de la Vega at the Mathematics Genealogy Project
4. Verónica Martínez de la Vega (Investigadora), UNAM Institute of Mathematics, retrieved 2022-11-26
5. Mathematics section members (PDF), Mexican Academy of Sciences, 2021, retrieved 2022-11-26
Authority control: Academics
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Edoardo Vesentini
Edoardo Vesentini (31 May 1928 – 28 March 2020) was an Italian mathematician and politician who introduced the Andreotti–Vesentini theorem. He was awarded the Caccioppoli Prize in 1962.
Vasentini was born in Rome, and died on 28 March 2020, aged 91.[1]
References
• Vesentini, Edoardo (2005), "Beniamino Segre and Italian geometry" (PDF), Rendiconti di Matematica e delle sue Applicazioni, 25 (2): 185–193, MR 2197882, Zbl 1093.01009.
• Edoardo Vesentini at the Mathematics Genealogy Project
• Premio Caccioppoli 1962 a Edoardo Vesentini
1. "E' morto Edoardo Vesentini, direttore emerito della Scuola Normale Superiore di Pisa". LaNazione.it.
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Linear algebraic group
In mathematics, a linear algebraic group is a subgroup of the group of invertible $n\times n$ matrices (under matrix multiplication) that is defined by polynomial equations. An example is the orthogonal group, defined by the relation $M^{T}M=I_{n}$ where $M^{T}$ is the transpose of $M$.
Algebraic structure → Group theory
Group theory
Basic notions
• Subgroup
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Group homomorphisms
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• Glossary of group theory
• List of group theory topics
Finite groups
• Cyclic group Zn
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Classification of finite simple groups
• cyclic
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Modular groups
• PSL(2, $\mathbb {Z} $)
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Topological and Lie groups
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Algebraic groups
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Many Lie groups can be viewed as linear algebraic groups over the field of real or complex numbers. (For example, every compact Lie group can be regarded as a linear algebraic group over R (necessarily R-anisotropic and reductive), as can many noncompact groups such as the simple Lie group SL(n,R).) The simple Lie groups were classified by Wilhelm Killing and Élie Cartan in the 1880s and 1890s. At that time, no special use was made of the fact that the group structure can be defined by polynomials, that is, that these are algebraic groups. The founders of the theory of algebraic groups include Maurer, Chevalley, and Kolchin (1948). In the 1950s, Armand Borel constructed much of the theory of algebraic groups as it exists today.
One of the first uses for the theory was to define the Chevalley groups.
Examples
For a positive integer $n$, the general linear group $GL(n)$ over a field $k$, consisting of all invertible $n\times n$ matrices, is a linear algebraic group over $k$. It contains the subgroups
$U\subset B\subset GL(n)$
consisting of matrices of the form, resp.,
$\left({\begin{array}{cccc}1&*&\dots &*\\0&1&\ddots &\vdots \\\vdots &\ddots &\ddots &*\\0&\dots &0&1\end{array}}\right)$ and $\left({\begin{array}{cccc}*&*&\dots &*\\0&*&\ddots &\vdots \\\vdots &\ddots &\ddots &*\\0&\dots &0&*\end{array}}\right)$.
The group $U$ is an example of a unipotent linear algebraic group, the group $B$ is an example of a solvable algebraic group called the Borel subgroup of $GL(n)$. It is a consequence of the Lie-Kolchin theorem that any connected solvable subgroup of $\mathrm {GL} (n)$ is conjugated into $B$. Any unipotent subgroup can be conjugated into $U$.
Another algebraic subgroup of $\mathrm {GL} (n)$ is the special linear group $\mathrm {SL} (n)$ of matrices with determinant 1.
The group $\mathrm {GL} (1)$ is called the multiplicative group, usually denoted by $\mathbf {G} _{\mathrm {m} }$. The group of $k$-points $\mathbf {G} _{\mathrm {m} }(k)$ is the multiplicative group $k^{*}$ of nonzero elements of the field $k$. The additive group $\mathbf {G} _{\mathrm {a} }$, whose $k$-points are isomorphic to the additive group of $k$, can also be expressed as a matrix group, for example as the subgroup $U$ in $\mathrm {GL} (2)$ :
${\begin{pmatrix}1&*\\0&1\end{pmatrix}}.$
These two basic examples of commutative linear algebraic groups, the multiplicative and additive groups, behave very differently in terms of their linear representations (as algebraic groups). Every representation of the multiplicative group $\mathbf {G} _{\mathrm {m} }$ is a direct sum of irreducible representations. (Its irreducible representations all have dimension 1, of the form $x\mapsto x^{n}$ for an integer $n$.) By contrast, the only irreducible representation of the additive group $\mathbf {G} _{\mathrm {a} }$ is the trivial representation. So every representation of $\mathbf {G} _{\mathrm {a} }$ (such as the 2-dimensional representation above) is an iterated extension of trivial representations, not a direct sum (unless the representation is trivial). The structure theory of linear algebraic groups analyzes any linear algebraic group in terms of these two basic groups and their generalizations, tori and unipotent groups, as discussed below.
Definitions
For an algebraically closed field k, much of the structure of an algebraic variety X over k is encoded in its set X(k) of k-rational points, which allows an elementary definition of a linear algebraic group. First, define a function from the abstract group GL(n,k) to k to be regular if it can be written as a polynomial in the entries of an n×n matrix A and in 1/det(A), where det is the determinant. Then a linear algebraic group G over an algebraically closed field k is a subgroup G(k) of the abstract group GL(n,k) for some natural number n such that G(k) is defined by the vanishing of some set of regular functions.
For an arbitrary field k, algebraic varieties over k are defined as a special case of schemes over k. In that language, a linear algebraic group G over a field k is a smooth closed subgroup scheme of GL(n) over k for some natural number n. In particular, G is defined by the vanishing of some set of regular functions on GL(n) over k, and these functions must have the property that for every commutative k-algebra R, G(R) is a subgroup of the abstract group GL(n,R). (Thus an algebraic group G over k is not just the abstract group G(k), but rather the whole family of groups G(R) for commutative k-algebras R; this is the philosophy of describing a scheme by its functor of points.)
In either language, one has the notion of a homomorphism of linear algebraic groups. For example, when k is algebraically closed, a homomorphism from G ⊂ GL(m) to H ⊂ GL(n) is a homomorphism of abstract groups G(k) → H(k) which is defined by regular functions on G. This makes the linear algebraic groups over k into a category. In particular, this defines what it means for two linear algebraic groups to be isomorphic.
In the language of schemes, a linear algebraic group G over a field k is in particular a group scheme over k, meaning a scheme over k together with a k-point 1 ∈ G(k) and morphisms
$m\colon G\times _{k}G\to G,\;i\colon G\to G$
over k which satisfy the usual axioms for the multiplication and inverse maps in a group (associativity, identity, inverses). A linear algebraic group is also smooth and of finite type over k, and it is affine (as a scheme). Conversely, every affine group scheme G of finite type over a field k has a faithful representation into GL(n) over k for some n.[1] An example is the embedding of the additive group Ga into GL(2), as mentioned above. As a result, one can think of linear algebraic groups either as matrix groups or, more abstractly, as smooth affine group schemes over a field. (Some authors use "linear algebraic group" to mean any affine group scheme of finite type over a field.)
For a full understanding of linear algebraic groups, one has to consider more general (non-smooth) group schemes. For example, let k be an algebraically closed field of characteristic p > 0. Then the homomorphism f: Gm → Gm defined by x ↦ xp induces an isomorphism of abstract groups k* → k*, but f is not an isomorphism of algebraic groups (because x1/p is not a regular function). In the language of group schemes, there is a clearer reason why f is not an isomorphism: f is surjective, but it has nontrivial kernel, namely the group scheme μp of pth roots of unity. This issue does not arise in characteristic zero. Indeed, every group scheme of finite type over a field k of characteristic zero is smooth over k.[2] A group scheme of finite type over any field k is smooth over k if and only if it is geometrically reduced, meaning that the base change $G_{\overline {k}}$ is reduced, where ${\overline {k}}$ is an algebraic closure of k.[3]
Since an affine scheme X is determined by its ring O(X) of regular functions, an affine group scheme G over a field k is determined by the ring O(G) with its structure of a Hopf algebra (coming from the multiplication and inverse maps on G). This gives an equivalence of categories (reversing arrows) between affine group schemes over k and commutative Hopf algebras over k. For example, the Hopf algebra corresponding to the multiplicative group Gm = GL(1) is the Laurent polynomial ring k[x, x−1], with comultiplication given by
$x\mapsto x\otimes x.$
Basic notions
For a linear algebraic group G over a field k, the identity component Go (the connected component containing the point 1) is a normal subgroup of finite index. So there is a group extension
$1\to G^{\circ }\to G\to F\to 1,$
where F is a finite algebraic group. (For k algebraically closed, F can be identified with an abstract finite group.) Because of this, the study of algebraic groups mostly focuses on connected groups.
Various notions from abstract group theory can be extended to linear algebraic groups. It is straightforward to define what it means for a linear algebraic group to be commutative, nilpotent, or solvable, by analogy with the definitions in abstract group theory. For example, a linear algebraic group is solvable if it has a composition series of linear algebraic subgroups such that the quotient groups are commutative. Also, the normalizer, the center, and the centralizer of a closed subgroup H of a linear algebraic group G are naturally viewed as closed subgroup schemes of G. If they are smooth over k, then they are linear algebraic groups as defined above.
One may ask to what extent the properties of a connected linear algebraic group G over a field k are determined by the abstract group G(k). A useful result in this direction is that if the field k is perfect (for example, of characteristic zero), or if G is reductive (as defined below), then G is unirational over k. Therefore, if in addition k is infinite, the group G(k) is Zariski dense in G.[4] For example, under the assumptions mentioned, G is commutative, nilpotent, or solvable if and only if G(k) has the corresponding property.
The assumption of connectedness cannot be omitted in these results. For example, let G be the group μ3 ⊂ GL(1) of cube roots of unity over the rational numbers Q. Then G is a linear algebraic group over Q for which G(Q) = 1 is not Zariski dense in G, because $G({\overline {\mathbf {Q} }})$ is a group of order 3.
Over an algebraically closed field, there is a stronger result about algebraic groups as algebraic varieties: every connected linear algebraic group over an algebraically closed field is a rational variety.[5]
The Lie algebra of an algebraic group
The Lie algebra ${\mathfrak {g}}$ of an algebraic group G can be defined in several equivalent ways: as the tangent space T1(G) at the identity element 1 ∈ G(k), or as the space of left-invariant derivations. If k is algebraically closed, a derivation D: O(G) → O(G) over k of the coordinate ring of G is left-invariant if
$D\lambda _{x}=\lambda _{x}D$
for every x in G(k), where λx: O(G) → O(G) is induced by left multiplication by x. For an arbitrary field k, left invariance of a derivation is defined as an analogous equality of two linear maps O(G) → O(G) ⊗O(G).[6] The Lie bracket of two derivations is defined by [D1, D2] =D1D2 − D2D1.
The passage from G to ${\mathfrak {g}}$ is thus a process of differentiation. For an element x ∈ G(k), the derivative at 1 ∈ G(k) of the conjugation map G → G, g ↦ xgx−1, is an automorphism of ${\mathfrak {g}}$, giving the adjoint representation:
$\operatorname {Ad} \colon G\to \operatorname {Aut} ({\mathfrak {g}}).$
Over a field of characteristic zero, a connected subgroup H of a linear algebraic group G is uniquely determined by its Lie algebra ${\mathfrak {h}}\subset {\mathfrak {g}}$.[7] But not every Lie subalgebra of ${\mathfrak {g}}$ corresponds to an algebraic subgroup of G, as one sees in the example of the torus G = (Gm)2 over C. In positive characteristic, there can be many different connected subgroups of a group G with the same Lie algebra (again, the torus G = (Gm)2 provides examples). For these reasons, although the Lie algebra of an algebraic group is important, the structure theory of algebraic groups requires more global tools.
Semisimple and unipotent elements
Main article: Jordan–Chevalley decomposition
For an algebraically closed field k, a matrix g in GL(n,k) is called semisimple if it is diagonalizable, and unipotent if the matrix g − 1 is nilpotent. Equivalently, g is unipotent if all eigenvalues of g are equal to 1. The Jordan canonical form for matrices implies that every element g of GL(n,k) can be written uniquely as a product g = gssgu such that gss is semisimple, gu is unipotent, and gss and gu commute with each other.
For any field k, an element g of GL(n,k) is said to be semisimple if it becomes diagonalizable over the algebraic closure of k. If the field k is perfect, then the semisimple and unipotent parts of g also lie in GL(n,k). Finally, for any linear algebraic group G ⊂ GL(n) over a field k, define a k-point of G to be semisimple or unipotent if it is semisimple or unipotent in GL(n,k). (These properties are in fact independent of the choice of a faithful representation of G.) If the field k is perfect, then the semisimple and unipotent parts of a k-point of G are automatically in G. That is (the Jordan decomposition): every element g of G(k) can be written uniquely as a product g = gssgu in G(k) such that gss is semisimple, gu is unipotent, and gss and gu commute with each other.[8] This reduces the problem of describing the conjugacy classes in G(k) to the semisimple and unipotent cases.
Tori
Main article: Algebraic torus
A torus over an algebraically closed field k means a group isomorphic to (Gm)n, the product of n copies of the multiplicative group over k, for some natural number n. For a linear algebraic group G, a maximal torus in G means a torus in G that is not contained in any bigger torus. For example, the group of diagonal matrices in GL(n) over k is a maximal torus in GL(n), isomorphic to (Gm)n. A basic result of the theory is that any two maximal tori in a group G over an algebraically closed field k are conjugate by some element of G(k).[9] The rank of G means the dimension of any maximal torus.
For an arbitrary field k, a torus T over k means a linear algebraic group over k whose base change $T_{\overline {k}}$ to the algebraic closure of k is isomorphic to (Gm)n over ${\overline {k}}$, for some natural number n. A split torus over k means a group isomorphic to (Gm)n over k for some n. An example of a non-split torus over the real numbers R is
$T=\{(x,y)\in A_{\mathbf {R} }^{2}:x^{2}+y^{2}=1\},$
with group structure given by the formula for multiplying complex numbers x+iy. Here T is a torus of dimension 1 over R. It is not split, because its group of real points T(R) is the circle group, which is not isomorphic even as an abstract group to Gm(R) = R*.
Every point of a torus over a field k is semisimple. Conversely, if G is a connected linear algebraic group such that every element of $G({\overline {k}})$ is semisimple, then G is a torus.[10]
For a linear algebraic group G over a general field k, one cannot expect all maximal tori in G over k to be conjugate by elements of G(k). For example, both the multiplicative group Gm and the circle group T above occur as maximal tori in SL(2) over R. However, it is always true that any two maximal split tori in G over k (meaning split tori in G that are not contained in a bigger split torus) are conjugate by some element of G(k).[11] As a result, it makes sense to define the k-rank or split rank of a group G over k as the dimension of any maximal split torus in G over k.
For any maximal torus T in a linear algebraic group G over a field k, Grothendieck showed that $T_{\overline {k}}$ is a maximal torus in $G_{\overline {k}}$.[12] It follows that any two maximal tori in G over a field k have the same dimension, although they need not be isomorphic.
Unipotent groups
Let Un be the group of upper-triangular matrices in GL(n) with diagonal entries equal to 1, over a field k. A group scheme over a field k (for example, a linear algebraic group) is called unipotent if it is isomorphic to a closed subgroup scheme of Un for some n. It is straightforward to check that the group Un is nilpotent. As a result, every unipotent group scheme is nilpotent.
A linear algebraic group G over a field k is unipotent if and only if every element of $G({\overline {k}})$ is unipotent.[13]
The group Bn of upper-triangular matrices in GL(n) is a semidirect product
$B_{n}=T_{n}\ltimes U_{n},$
where Tn is the diagonal torus (Gm)n. More generally, every connected solvable linear algebraic group is a semidirect product of a torus with a unipotent group, T ⋉ U.[14]
A smooth connected unipotent group over a perfect field k (for example, an algebraically closed field) has a composition series with all quotient groups isomorphic to the additive group Ga.[15]
Borel subgroups
The Borel subgroups are important for the structure theory of linear algebraic groups. For a linear algebraic group G over an algebraically closed field k, a Borel subgroup of G means a maximal smooth connected solvable subgroup. For example, one Borel subgroup of GL(n) is the subgroup B of upper-triangular matrices (all entries below the diagonal are zero).
A basic result of the theory is that any two Borel subgroups of a connected group G over an algebraically closed field k are conjugate by some element of G(k).[16] (A standard proof uses the Borel fixed-point theorem: for a connected solvable group G acting on a proper variety X over an algebraically closed field k, there is a k-point in X which is fixed by the action of G.) The conjugacy of Borel subgroups in GL(n) amounts to the Lie–Kolchin theorem: every smooth connected solvable subgroup of GL(n) is conjugate to a subgroup of the upper-triangular subgroup in GL(n).
For an arbitrary field k, a Borel subgroup B of G is defined to be a subgroup over k such that, over an algebraic closure ${\overline {k}}$ of k, $B_{\overline {k}}$is a Borel subgroup of $G_{\overline {k}}$. Thus G may or may not have a Borel subgroup over k.
For a closed subgroup scheme H of G, the quotient space G/H is a smooth quasi-projective scheme over k.[17] A smooth subgroup P of a connected group G is called parabolic if G/P is projective over k (or equivalently, proper over k). An important property of Borel subgroups B is that G/B is a projective variety, called the flag variety of G. That is, Borel subgroups are parabolic subgroups. More precisely, for k algebraically closed, the Borel subgroups are exactly the minimal parabolic subgroups of G; conversely, every subgroup containing a Borel subgroup is parabolic.[18] So one can list all parabolic subgroups of G (up to conjugation by G(k)) by listing all the linear algebraic subgroups of G that contain a fixed Borel subgroup. For example, the subgroups P ⊂ GL(3) over k that contain the Borel subgroup B of upper-triangular matrices are B itself, the whole group GL(3), and the intermediate subgroups
$\left\{{\begin{bmatrix}*&*&*\\0&*&*\\0&*&*\end{bmatrix}}\right\}$ and $\left\{{\begin{bmatrix}*&*&*\\*&*&*\\0&0&*\end{bmatrix}}\right\}.$
The corresponding projective homogeneous varieties GL(3)/P are (respectively): the flag manifold of all chains of linear subspaces
$0\subset V_{1}\subset V_{2}\subset A_{k}^{3}$
with Vi of dimension i; a point; the projective space P2 of lines (1-dimensional linear subspaces) in A3; and the dual projective space P2 of planes in A3.
Semisimple and reductive groups
Main article: Reductive group
A connected linear algebraic group G over an algebraically closed field is called semisimple if every smooth connected solvable normal subgroup of G is trivial. More generally, a connected linear algebraic group G over an algebraically closed field is called reductive if every smooth connected unipotent normal subgroup of G is trivial.[19] (Some authors do not require reductive groups to be connected.) A semisimple group is reductive. A group G over an arbitrary field k is called semisimple or reductive if $G_{\overline {k}}$ is semisimple or reductive. For example, the group SL(n) of n × n matrices with determinant 1 over any field k is semisimple, whereas a nontrivial torus is reductive but not semisimple. Likewise, GL(n) is reductive but not semisimple (because its center Gm is a nontrivial smooth connected solvable normal subgroup).
Every compact connected Lie group has a complexification, which is a complex reductive algebraic group. In fact, this construction gives a one-to-one correspondence between compact connected Lie groups and complex reductive groups, up to isomorphism.[20]
A linear algebraic group G over a field k is called simple (or k-simple) if it is semisimple, nontrivial, and every smooth connected normal subgroup of G over k is trivial or equal to G.[21] (Some authors call this property "almost simple".) This differs slightly from the terminology for abstract groups, in that a simple algebraic group may have nontrivial center (although the center must be finite). For example, for any integer n at least 2 and any field k, the group SL(n) over k is simple, and its center is the group scheme μn of nth roots of unity.
Every connected linear algebraic group G over a perfect field k is (in a unique way) an extension of a reductive group R by a smooth connected unipotent group U, called the unipotent radical of G:
$1\to U\to G\to R\to 1.$
If k has characteristic zero, then one has the more precise Levi decomposition: every connected linear algebraic group G over k is a semidirect product $R\ltimes U$ of a reductive group by a unipotent group.[22]
Classification of reductive groups
Main article: Reductive group
Reductive groups include the most important linear algebraic groups in practice, such as the classical groups: GL(n), SL(n), the orthogonal groups SO(n) and the symplectic groups Sp(2n). On the other hand, the definition of reductive groups is quite "negative", and it is not clear that one can expect to say much about them. Remarkably, Claude Chevalley gave a complete classification of the reductive groups over an algebraically closed field: they are determined by root data.[23] In particular, simple groups over an algebraically closed field k are classified (up to quotients by finite central subgroup schemes) by their Dynkin diagrams. It is striking that this classification is independent of the characteristic of k. For example, the exceptional Lie groups G2, F4, E6, E7, and E8 can be defined in any characteristic (and even as group schemes over Z). The classification of finite simple groups says that most finite simple groups arise as the group of k-points of a simple algebraic group over a finite field k, or as minor variants of that construction.
Every reductive group over a field is the quotient by a finite central subgroup scheme of the product of a torus and some simple groups. For example,
$GL(n)\cong (G_{m}\times SL(n))/\mu _{n}.$
For an arbitrary field k, a reductive group G is called split if it contains a split maximal torus over k (that is, a split torus in G which remains maximal over an algebraic closure of k). For example, GL(n) is a split reductive group over any field k. Chevalley showed that the classification of split reductive groups is the same over any field. By contrast, the classification of arbitrary reductive groups can be hard, depending on the base field. For example, every nondegenerate quadratic form q over a field k determines a reductive group SO(q), and every central simple algebra A over k determines a reductive group SL1(A). As a result, the problem of classifying reductive groups over k essentially includes the problem of classifying all quadratic forms over k or all central simple algebras over k. These problems are easy for k algebraically closed, and they are understood for some other fields such as number fields, but for arbitrary fields there are many open questions.
Applications
Representation theory
One reason for the importance of reductive groups comes from representation theory. Every irreducible representation of a unipotent group is trivial. More generally, for any linear algebraic group G written as an extension
$1\to U\to G\to R\to 1$
with U unipotent and R reductive, every irreducible representation of G factors through R.[24] This focuses attention on the representation theory of reductive groups. (To be clear, the representations considered here are representations of G as an algebraic group. Thus, for a group G over a field k, the representations are on k-vector spaces, and the action of G is given by regular functions. It is an important but different problem to classify continuous representations of the group G(R) for a real reductive group G, or similar problems over other fields.)
Chevalley showed that the irreducible representations of a split reductive group over a field k are finite-dimensional, and they are indexed by dominant weights.[25] This is the same as what happens in the representation theory of compact connected Lie groups, or the finite-dimensional representation theory of complex semisimple Lie algebras. For k of characteristic zero, all these theories are essentially equivalent. In particular, every representation of a reductive group G over a field of characteristic zero is a direct sum of irreducible representations, and if G is split, the characters of the irreducible representations are given by the Weyl character formula. The Borel–Weil theorem gives a geometric construction of the irreducible representations of a reductive group G in characteristic zero, as spaces of sections of line bundles over the flag manifold G/B.
The representation theory of reductive groups (other than tori) over a field of positive characteristic p is less well understood. In this situation, a representation need not be a direct sum of irreducible representations. And although irreducible representations are indexed by dominant weights, the dimensions and characters of the irreducible representations are known only in some cases. Andersen, Jantzen and Soergel (1994) determined these characters (proving Lusztig's conjecture) when the characteristic p is sufficiently large compared to the Coxeter number of the group. For small primes p, there is not even a precise conjecture.
Group actions and geometric invariant theory
An action of a linear algebraic group G on a variety (or scheme) X over a field k is a morphism
$G\times _{k}X\to X$
that satisfies the axioms of a group action. As in other types of group theory, it is important to study group actions, since groups arise naturally as symmetries of geometric objects.
Part of the theory of group actions is geometric invariant theory, which aims to construct a quotient variety X/G, describing the set of orbits of a linear algebraic group G on X as an algebraic variety. Various complications arise. For example, if X is an affine variety, then one can try to construct X/G as Spec of the ring of invariants O(X)G. However, Masayoshi Nagata showed that the ring of invariants need not be finitely generated as a k-algebra (and so Spec of the ring is a scheme but not a variety), a negative answer to Hilbert's 14th problem. In the positive direction, the ring of invariants is finitely generated if G is reductive, by Haboush's theorem, proved in characteristic zero by Hilbert and Nagata.
Geometric invariant theory involves further subtleties when a reductive group G acts on a projective variety X. In particular, the theory defines open subsets of "stable" and "semistable" points in X, with the quotient morphism only defined on the set of semistable points.
Related notions
Linear algebraic groups admit variants in several directions. Dropping the existence of the inverse map $i\colon G\to G$, one obtains the notion of a linear algebraic monoid.[26]
Lie groups
For a linear algebraic group G over the real numbers R, the group of real points G(R) is a Lie group, essentially because real polynomials, which describe the multiplication on G, are smooth functions. Likewise, for a linear algebraic group G over C, G(C) is a complex Lie group. Much of the theory of algebraic groups was developed by analogy with Lie groups.
There are several reasons why a Lie group may not have the structure of a linear algebraic group over R.
• A Lie group with an infinite group of components G/Go cannot be realized as a linear algebraic group.
• An algebraic group G over R may be connected as an algebraic group while the Lie group G(R) is not connected, and likewise for simply connected groups. For example, the algebraic group SL(2) is simply connected over any field, whereas the Lie group SL(2,R) has fundamental group isomorphic to the integers Z. The double cover H of SL(2,R), known as the metaplectic group, is a Lie group that cannot be viewed as a linear algebraic group over R. More strongly, H has no faithful finite-dimensional representation.
• Anatoly Maltsev showed that every simply connected nilpotent Lie group can be viewed as a unipotent algebraic group G over R in a unique way.[27] (As a variety, G is isomorphic to affine space of some dimension over R.) By contrast, there are simply connected solvable Lie groups that cannot be viewed as real algebraic groups. For example, the universal cover H of the semidirect product S1 ⋉ R2 has center isomorphic to Z, which is not a linear algebraic group, and so H cannot be viewed as a linear algebraic group over R.
Abelian varieties
Algebraic groups which are not affine behave very differently. In particular, a smooth connected group scheme which is a projective variety over a field is called an abelian variety. In contrast to linear algebraic groups, every abelian variety is commutative. Nonetheless, abelian varieties have a rich theory. Even the case of elliptic curves (abelian varieties of dimension 1) is central to number theory, with applications including the proof of Fermat's Last Theorem.
Tannakian categories
The finite-dimensional representations of an algebraic group G, together with the tensor product of representations, form a tannakian category RepG. In fact, tannakian categories with a "fiber functor" over a field are equivalent to affine group schemes. (Every affine group scheme over a field k is pro-algebraic in the sense that it is an inverse limit of affine group schemes of finite type over k.[28]) For example, the Mumford–Tate group and the motivic Galois group are constructed using this formalism. Certain properties of a (pro-)algebraic group G can be read from its category of representations. For example, over a field of characteristic zero, RepG is a semisimple category if and only if the identity component of G is pro-reductive.[29]
See also
• The groups of Lie type are the finite simple groups constructed from simple algebraic groups over finite fields.
• Lang's theorem
• Generalized flag variety, Bruhat decomposition, BN pair, Weyl group, Cartan subgroup, group of adjoint type, parabolic induction
• Real form (Lie theory), Satake diagram
• Adelic algebraic group, Weil's conjecture on Tamagawa numbers
• Langlands classification, Langlands program, geometric Langlands program
• Torsor, nonabelian cohomology, special group, cohomological invariant, essential dimension, Kneser–Tits conjecture, Serre's conjecture II
• Pseudo-reductive group
• Differential Galois theory
• Distribution on a linear algebraic group
Notes
1. Milne (2017), Corollary 4.10.
2. Milne (2017), Corollary 8.39.
3. Milne (2017), Proposition 1.26(b).
4. Borel (1991), Theorem 18.2 and Corollary 18.4.
5. Borel (1991), Remark 14.14.
6. Milne (2017), section 10.e.
7. Borel (1991), section 7.1.
8. Milne (2017), Theorem 9.18.
9. Borel (1991), Corollary 11.3.
10. Milne (2017), Corollary 17.25
11. Springer (1998), Theorem 15.2.6.
12. Borel (1991), 18.2(i).
13. Milne (2017), Corollary 14.12.
14. Borel (1991), Theorem 10.6.
15. Borel (1991), Theorem 15.4(iii).
16. Borel (1991), Theorem 11.1.
17. Milne (2017), Theorems 7.18 and 8.43.
18. Borel (1991), Corollary 11.2.
19. Milne (2017), Definition 6.46.
20. Bröcker & tom Dieck (1985), section III.8; Conrad (2014), section D.3.
21. Conrad (2014), after Proposition 5.1.17.
22. Conrad (2014), Proposition 5.4.1.
23. Springer (1998), 9.6.2 and 10.1.1.
24. Milne (2017), Lemma 19.16.
25. Milne (2017), Theorem 22.2.
26. Renner, Lex (2006), Linear Algebraic Monoids, Springer.
27. Milne (2017), Theorem 14.37.
28. Deligne & Milne (1982), Corollary II.2.7.
29. Deligne & Milne (1982), Remark II.2.28.
References
• Andersen, H. H.; Jantzen, J. C.; Soergel, W. (1994), Representations of Quantum Groups at a pth Root of Unity and of Semisimple Groups in Characteristic p: Independence of p, Astérisque, vol. 220, Société Mathématique de France, ISSN 0303-1179, MR 1272539
• Borel, Armand (1991) [1969], Linear Algebraic Groups (2nd ed.), New York: Springer-Verlag, ISBN 0-387-97370-2, MR 1102012
• Bröcker, Theodor; tom Dieck, Tammo (1985), Representations of Compact Lie Groups, Springer Nature, ISBN 0-387-13678-9, MR 0781344
• Conrad, Brian (2014), "Reductive group schemes" (PDF), Autour des schémas en groupes, vol. 1, Paris: Société Mathématique de France, pp. 93–444, ISBN 978-2-85629-794-0, MR 3309122
• Deligne, Pierre; Milne, J. S. (1982), "Tannakian categories", Hodge Cycles, Motives, and Shimura Varieties, Lecture Notes in Mathematics, vol. 900, Springer Nature, pp. 101–228, ISBN 3-540-11174-3, MR 0654325
• De Medts, Tom (2019), Linear Algebraic Groups (course notes) (PDF), Ghent University
• Humphreys, James E. (1975), Linear Algebraic Groups, Springer, ISBN 0-387-90108-6, MR 0396773
• Kolchin, E. R. (1948), "Algebraic matric groups and the Picard–Vessiot theory of homogeneous linear ordinary differential equations", Annals of Mathematics, Second Series, 49 (1): 1–42, doi:10.2307/1969111, ISSN 0003-486X, JSTOR 1969111, MR 0024884
• Milne, J. S. (2017), Algebraic Groups: The Theory of Group Schemes of Finite Type over a Field, Cambridge University Press, ISBN 978-1107167483, MR 3729270
• Springer, Tonny A. (1998) [1981], Linear Algebraic Groups (2nd ed.), New York: Birkhäuser, ISBN 0-8176-4021-5, MR 1642713
External links
• "Linear algebraic group", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
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Vexillary permutation
In mathematics, a vexillary permutation is a permutation μ of the positive integers containing no subpermutation isomorphic to the permutation (2143); in other words, there do not exist four numbers i < j < k < l with μ(j) < μ(i) < μ(l) < μ(k). They were introduced by Lascoux and Schützenberger (1982, 1985). The word "vexillary" means flag-like, and comes from the fact that vexillary permutations are related to flags of modules.
Guibert, Pergola & Pinzani (2001) showed that vexillary involutions are enumerated by Motzkin numbers.
See also
• Riffle shuffle permutation, a subclass of the vexillary permutations
References
• Guibert, O.; Pergola, E.; Pinzani, R. (2001), "Vexillary involutions are enumerated by Motzkin numbers", Annals of Combinatorics, 5 (2): 153–174, doi:10.1007/PL00001297, ISSN 0218-0006, MR 1904383
• Lascoux, Alain; Schützenberger, Marcel-Paul (1982), "Polynômes de Schubert", Comptes Rendus de l'Académie des Sciences, Série I, 294 (13): 447–450, ISSN 0249-6291, MR 0660739
• Lascoux, Alain; Schützenberger, Marcel-Paul (1985), "Schubert polynomials and the Littlewood–Richardson rule", Letters in Mathematical Physics. A Journal for the Rapid Dissemination of Short Contributions in the Field of Mathematical Physics, 10 (2): 111–124, doi:10.1007/BF00398147, ISSN 0377-9017, MR 0815233
• Macdonald, I.G. (1991b), Notes on Schubert polynomials, Publications du Laboratoire de combinatoire et d'informatique mathématique, vol. 6, Laboratoire de combinatoire et d'informatique mathématique (LACIM), Université du Québec a Montréal, ISBN 978-2-89276-086-6
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Vickrey auction
A Vickrey auction or sealed-bid second-price auction (SBSPA) is a type of sealed-bid auction. Bidders submit written bids without knowing the bid of the other people in the auction. The highest bidder wins but the price paid is the second-highest bid. This type of auction is strategically similar to an English auction and gives bidders an incentive to bid their true value. The auction was first described academically by Columbia University professor William Vickrey in 1961[1] though it had been used by stamp collectors since 1893.[2] In 1797 Johann Wolfgang von Goethe sold a manuscript using a sealed-bid, second-price auction.[3]
Part of a series on
Auctions
Types
• All-pay
• Amsterdam
• Anglo-Dutch
• Barter double
• Best/not best
• Brazilian
• Calcutta
• Candle
• Chinese
• Click-box bidding
• Combinatorial
• Common value
• Deferred-acceptance
• Discriminatory price
• Double
• Dutch
• English
• Forward
• French
• Generalized first-price
• Generalized second-price
• Japanese
• Knapsack
• Multi-attribute
• Multiunit
• No-reserve
• Rank
• Reverse
• Scottish
• Sealed first-price
• Simultaneous ascending
• Single-price
• Traffic light
• Uniform price
• Unique bid
• Value of revenues
• Vickrey
• Vickrey–Clarke–Groves
• Walrasian
• Yankee
Bidding
• Shading
• Calor licitantis
• Cancellation hunt
• Jump
• Rigging
• Sniping
• Suicide
• Tacit collusion
Contexts
• Algorithms
• Autos
• Art
• Charity
• Children
• Cricket players
• Domain names
• Flowers
• Loans
• Scam
• Slaves
• Spectrum
• Stamps
• Virginity
• Wine
• Wives
Theory
• Digital goods
• Price of anarchy
• Revenue equivalence
• Winner's curse
Online
• Ebidding
• Private electronic market
• Software
Vickrey's original paper mainly considered auctions where only a single, indivisible good is being sold. The terms Vickrey auction and second-price sealed-bid auction are, in this case only, equivalent and used interchangeably. In the case of multiple identical goods, the bidders submit inverse demand curves and pay the opportunity cost.[4]
Vickrey auctions are much studied in economic literature but uncommon in practice. Generalized variants of the Vickrey auction for multiunit auctions exist, such as the generalized second-price auction used in Google's and Yahoo!'s online advertisement programmes[5][6] (not incentive compatible) and the Vickrey–Clarke–Groves auction (incentive compatible).
Properties
Self-revelation and incentive compatibility
In a Vickrey auction with private values each bidder maximizes their expected utility by bidding (revealing) their valuation of the item for sale. These type of auctions are sometimes used for specified pool trading in the agency mortgage-backed securities (MBS) market.
Ex-post efficiency
A Vickrey auction is decision efficient (the winner is the bidder with the highest valuation) under the most general circumstances; it thus provides a baseline model against which the efficiency properties of other types of auctions can be posited. It is only ex-post efficient (sum of transfers equal to zero) if the seller is included as "player zero," whose transfer equals the negative of the sum of the other players' transfers (i.e. the bids).
Weaknesses
• It does not allow for price discovery, that is, discovery of the market price if the buyers are unsure of their own valuations, without sequential auctions.
• Sellers may use shill bids to increase profit.
Proof of dominance of truthful bidding
The dominant strategy in a Vickrey auction with a single, indivisible item is for each bidder to bid their true value of the item.[7]
Let $v_{i}$ be bidder i's value for the item. Let $b_{i}$ be bidder i's bid for the item. The payoff for bidder i is ${\begin{cases}v_{i}-\max _{j\neq i}b_{j}&{\text{if }}b_{i}>\max _{j\neq i}b_{j}\\0&{\text{otherwise}}\end{cases}}$
The strategy of overbidding is dominated by bidding truthfully. Assume that bidder i bids $b_{i}>v_{i}$.
If $\max _{j\neq i}b_{j}<v_{i}$ then the bidder would win the item with a truthful bid as well as an overbid. The bid's amount does not change the payoff so the two strategies have equal payoffs in this case.
If $\max _{j\neq i}b_{j}>b_{i}$ then the bidder would lose the item either way so the strategies have equal payoffs in this case.
If $v_{i}<\max _{j\neq i}b_{j}<b_{i}$ then only the strategy of overbidding would win the auction. The payoff would be negative for the strategy of overbidding because they paid more than their value of the item, while the payoff for a truthful bid would be zero. Thus the strategy of bidding higher than one's true valuation is dominated by the strategy of truthfully bidding.
The strategy of underbidding is dominated by bidding truthfully. Assume that bidder i bids $b_{i}<v_{i}$.
If $\max _{j\neq i}b_{j}>v_{i}$ then the bidder would lose the item with a truthful bid as well as an underbid, so the strategies have equal payoffs for this case.
If $\max _{j\neq i}b_{j}<b_{i}$ then the bidder would win the item either way so the strategies have equal payoffs in this case.
If $b_{i}<\max _{j\neq i}b_{j}<v_{i}$ then only the strategy of truthfully bidding would win the auction. The payoff for the truthful strategy would be positive as they paid less than their value of the item, while the payoff for an underbid bid would be zero. Thus the strategy of underbidding is dominated by the strategy of truthfully bidding.
Truthful bidding dominates the other possible strategies (underbidding and overbidding) so it is an optimal strategy.
Revenue equivalence of the Vickrey auction and sealed first price auction
The two most common auctions are the sealed first price (or high-bid) auction and the open ascending price (or English) auction. In the former each buyer submits a sealed bid. The high bidder is awarded the item and pays his or her bid. In the latter, the auctioneer announces successively higher asking prices and continues until no one is willing to accept a higher price. Suppose that a buyer's valuation is $v$ and the current asking price is $b$. If $v<b$, then the buyer loses by raising his hand. If $v>b$ and the buyer is not the current high bidder, it is more profitable to bid than to let someone else be the winner. Thus it is a dominant strategy for a buyer to drop out of the bidding when the asking price reaches his or her valuation. Thus, just as in the Vickrey sealed second price auction, the price paid by the buyer with the highest valuation is equal to the second highest value.
Consider then the expected payment in the sealed second-price auction. Vickrey considered the case of two buyers and assumed that each buyer's value was an independent draw from a uniform distribution with support $[0,1]$. With buyers bidding according to their dominant strategies, a buyer with valuation $v$ wins if his opponent's value $x<v$. Suppose that $v$ is the high value. Then the winning payment is uniformly distributed on the interval $[0,v]$ and so the expected payment of the winner is
$e(v)={\tfrac {1}{2}}v$.
We now argue that in the sealed first price auction the equilibrium bid of a buyer with valuation $v$ is
$B(v)=e(v)={\tfrac {1}{2}}v$.
That is, the payment of the winner in the sealed first-price auction is equal to the expected revenue in the sealed second-price auction.
Proof of revenue equivalence
Suppose that buyer 2 bids according to the strategy $B(v)=v/2$, where $B(v)$ is the buyer's bid for a valuation $v$. We need to show that buyer 1's best response is to use the same strategy.
Note first that if buyer 2 uses the strategy $B(v)=v/2$, then buyer 2's maximum bid is $B(1)=1/2$ and so buyer 1 wins with probability 1 with any bid of 1/2 or more. Consider then a bid $b$ on the interval $[0,1/2]$. Let buyer 2's value be $x$. Then buyer 1 wins if $B(x)=x/2<b$, that is, if $x<2b$. Under Vickrey's assumption of uniformly distributed values, the win probability is $w(b)=2b$. Buyer 1's expected payoff is therefore
$U(b)=w(b)(v-b)=2b(v-b)={\tfrac {1}{2}}[{{v}^{2}}-{{(v-2b)}^{2}}]$
Note that $U(b)$ takes on its maximum at $b=v/2=B(v)$.
Use in network routing
In network routing, VCG mechanisms are a family of payment schemes based on the added value concept. The basic idea of a VCG mechanism in network routing is to pay the owner of each link or node (depending on the network model) that is part of the solution, its declared cost plus its added value. In many routing problems, this mechanism is not only strategyproof, but also the minimum among all strategyproof mechanisms.
In the case of network flows, unicast or multicast, a minimum cost flow (MCF) in graph G is calculated based on the declared costs dk of each of the links and payment is calculated as follows:
Each link (or node) $\scriptstyle e_{k}$ in the MCF is paid
$p_{k}=d_{k}+MCF(G-e_{k})-MCF(G)$,
where MCF(G) indicates the cost of the minimum cost flow in graph G and G − ek indicates graph G without the link ek. Links not in the MCF are paid nothing. This routing problem is one of the cases for which VCG is strategyproof and minimum.
In 2004, it was shown that the expected VCG overpayment of an Erdős–Rényi random graph with n nodes and edge probability p, $\scriptstyle G\in G(n,p)$ approaches
${\frac {p}{2-p}}$
as n, approaches $\scriptstyle \infty $, for $np=\omega ({\sqrt {n\log n}})$. Prior to this result, it was known that VCG overpayment in G(n, p) is
$\Omega \left({\frac {1}{np}}\right)$
and
$O(1)\,$
with high probability given
$np=\omega (\log n).\,$
Generalizations
The most obvious generalization to multiple or divisible goods is to have all winning bidders pay the amount of the highest non-winning bid. This is known as a uniform price auction. The uniform-price auction does not, however, result in bidders bidding their true valuations as they do in a second-price auction unless each bidder has demand for only a single unit. A generalization of the Vickrey auction that maintains the incentive to bid truthfully is known as the Vickrey–Clarke–Groves (VCG) mechanism. The idea in VCG is that items are assigned to maximize the sum of utilities; then each bidder pays the "opportunity cost" that their presence introduces to all the other players. This opportunity cost for a bidder is defined as the total bids of all the other bidders that would have won if the first bidder had not bid, minus the total bids of all the other actual winning bidders.
A different kind of generalization is to set a reservation price—a minimum price below which the item is not sold at all. In some cases, setting a reservation price can substantially increase the revenue of the auctioneer. This is an example of Bayesian-optimal mechanism design.
In mechanism design, the revelation principle can be viewed as a generalization of the Vickrey auction.
See also
• Auction theory
• First-price sealed-bid auction
• VCG auction
References
• Vijay Krishna, Auction Theory, Academic Press, 2002.
• Peter Cramton, Yoav Shoham, Richard Steinberg (Eds), Combinatorial Auctions, MIT Press, 2006, Chapter 1. ISBN 0-262-03342-9.
• Paul Milgrom, Putting Auction Theory to Work, Cambridge University Press, 2004.
• Teck Ho, "Consumption and Production" UC Berkeley, Haas Class of 2010.
Notes
1. Vickrey, William (1961). "Counterspeculation, Auctions, and Competitive Sealed Tenders". The Journal of Finance. 16 (1): 8–37. doi:10.1111/j.1540-6261.1961.tb02789.x.
2. Lucking-Reiley, David (2000). "Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce". Journal of Economic Perspectives. 14 (3): 183–192. doi:10.1257/jep.14.3.183.
3. Benny Moldovanu and Manfred Tietzel (1998). "Goethe's Second-Price Auction". The Journal of Political Economy. 106 (4): 854–859. CiteSeerX 10.1.1.560.8278. doi:10.1086/250032. JSTOR 2990730. S2CID 53490333.
4. Jones, Derek (2003). "Auction Theory for the New Economy". New Economy Handbook. Emerald Publishing Ltd. ISBN 978-0123891723.
5. Benjamin Edelman, Michael Ostrovsky, and Michael Schwarz: "Internet Advertising and the Generalized Second-Price Auction: Selling Billions of Dollars Worth of Keywords". American Economic Review 97(1), 2007 pp 242–259.
6. Hal R. Varian: "Position Auctions". International Journal of Industrial Organization, 2006, doi:10.1016/j.ijindorg.2006.10.002 .
7. von Ahn, Luis (30 September 2008). "Auctions" (PDF). 15–396: Science of the Web Course Notes. Carnegie Mellon University. Archived from the original (PDF) on 8 October 2008. Retrieved 6 November 2008.
Authority control: National
• Germany
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Vicky Neale
Victoria Neale (1984 – 3 May 2023) was a British mathematician and writer. She was Whitehead Lecturer at Oxford's Mathematical Institute and Supernumerary Fellow at Balliol College.[2][3] Her research specialty was number theory. The author of the 2017 book Closing the Gap: The Quest to Understand Prime Numbers,[4][5] she was interviewed on several BBC radio programs as a mathematics expert.[6][7] In addition, she wrote for The Conversation and The Guardian.[8][9] Her other educational and outreach activities included lecturing at the PROMYS Europe high-school program[10] and helping to organize the European Girls' Mathematical Olympiad.[11]
Vicky Neale
Born
Victoria Neale[1]
1984 (1984)
Died3 May 2023 (aged 39)
CitizenshipUnited Kingdom
Alma mater
• University of Cambridge (BA)
• University of Cambridge (PhD)
Scientific career
Fields
• Number theory
Institutions
• University of Cambridge
• University of Oxford
ThesisBracket quadratics as asymptotic bases for the natural numbers (2011)
Doctoral advisorBen Green
Websitepeople.maths.ox.ac.uk/neale/
Neale was born in 1984.[12] She obtained her PhD in 2011 from the University of Cambridge. Her thesis work, supervised by Ben Joseph Green, concerned Waring's problem.[2][1] She then taught at Cambridge while being Director of Studies in mathematics at Murray Edwards College,[11][13] before moving to Oxford in the summer of 2014.[14]
Neale died on 3 May 2023, at the age of 39.[15] She had been diagnosed with a rare type of cancer in 2021.[16]
References
1. Vicky Neale at the Mathematics Genealogy Project
2. Neale, Vicky (3 August 2018). "Homepage". Mathematical Institute, University of Oxford. Retrieved 7 August 2018.
3. "Speakers and Panellists - ACME". Advisory Committee on Mathematics Education. Retrieved 7 August 2018.
"BCME 9 Plenary Speakers". British Congress of Mathematics Education. 2018. Retrieved 10 August 2018.
4. Neale, Vicky (2017). Closing the Gap: The Quest to Understand Prime Numbers. Oxford University Press. ISBN 9780198788287. OCLC 1030559953.
5. Reviews of Closing the Gap include the following:
• Hunacek, Mark (12 February 2018). "Closing the Gap | Mathematical Association of America". Mathematical Association of America. Retrieved 7 August 2018.
• Freiberger, Marianne (12 December 2017). "'Closing the gap'". Plus Magazine. Retrieved 7 August 2018.
• Bultheel, Adhemar (February 2018). "Review: Closing the Gap". European Mathematical Society. Retrieved 10 September 2018.
• Kalaydzhieva, Nikoleta; Porritt, Sam (28 June 2018). "Closing the Gap". Chalkdust. Retrieved 7 August 2018.
• Fried, Michael N. (3 July 2018). "Prime Numbers, Mathematical Pencils, and Massive Collaboration". Mathematical Thinking and Learning. 20 (3): 248–250. doi:10.1080/10986065.2018.1483932. ISSN 1098-6065.
6. Among her appearances are the following:
• "Fermat's Last Theorem, In Our Time - BBC Radio 4". BBC. Retrieved 7 August 2018.
• "Numbers Numbers Everywhere, Series 10, The Infinite Monkey Cage - BBC Radio 4". BBC. Retrieved 7 August 2018.
• "e, In Our Time - BBC Radio 4". BBC. Retrieved 7 August 2018.
• "Vicky Neale on the Mathematics of Beauty, A History of Ideas - BBC Radio 4". BBC. Retrieved 7 August 2018.
• "Maths: Alex Bellos, Neil deGrasse Tyson, Serafina Cuomo, Vicky Neale, Free Thinking - BBC Radio 3". BBC. Retrieved 7 August 2018.
7. She is also quoted as a mathematics expert in, for example,
• Flyn, Cal (10 July 2017). "What Makes Maths Beautiful?". New Humanist. Retrieved 10 August 2018.
• Sample, Ian (21 November 2016). "Magic numbers: can maths equations be beautiful?". The Guardian. Retrieved 10 August 2018.
8. Neale, Vicky (17 February 2017). "Mathematics is beautiful (no, really)". The Conversation. Retrieved 7 August 2018.
9. Neale, Vicky (26 November 2015). "Solving for Xmas: how to make mathematical Christmas cards". The Guardian. Retrieved 7 August 2018.
10. "Annual Report 2016" (PDF). Clay Mathematics Institute. 26 June 2017. Retrieved 7 August 2018.
11. "Principal Faculty | PROMYS-Europe: Program in Mathematics for Young Scientists". promys-europe.org. Retrieved 7 August 2018.
12. Vicky Neale 1984–2023, Balliol College
13. Gowers, Timothy (11 January 2014). "Introduction to Cambridge IA Analysis I 2014". Gowers' Weblog. Retrieved 10 August 2018.
14. "Balliol Maths: a plurality of women". Floreat Domus 2015. Balliol College. Retrieved 10 August 2018.
15. "Vicky Neale | Mathematical Institute". Mathematical Institute, University of Oxford. 4 May 2023. Archived from the original on 4 May 2023. Retrieved 4 May 2023.
16. Dr Vicky Neale (1984-2023), London Mathematical Society
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Victor Puiseux
Victor Alexandre Puiseux (French: [pɥizø]; 16 April 1820 – 9 September 1883) was a French mathematician and astronomer. Puiseux series are named after him, as is in part the Bertrand–Diquet–Puiseux theorem. His work on algebraic functions and uniformization makes him a direct precursor of Bernhard Riemann, for what concerns the latter's work on this subject and his introduction of Riemann surfaces.[1] He was also an accomplished amateur mountaineer. A peak in the French alps, which he climbed in 1848, is named after him.
A species of Israeli gecko, Ptyodactylus puiseuxi, is named in his honor.[2]
Life
He was born in 1820 in Argenteuil, Val-d'Oise. He occupied the chair of celestial mechanics at the Sorbonne. Excelling in mathematical analysis, he introduced new methods in his account of algebraic functions, and by his contributions to celestial mechanics advanced knowledge in that direction. In 1871, he was unanimously elected to the French Academy.
One of his sons, Pierre Henri Puiseux, was a famous astronomer.
He died in 1883 in Frontenay, France.
References
1. Athanase Papadopoulos, « Cauchy and Puiseux: Two precursors of Riemann », In: From Riemann to differential geometry and relativity (L. Ji, A. Papadopoulos and S. Yamada, ed.) Berlin: Springer., 2017, p. 209-235.
2. Beolens, Bo; Watkins, Michael; Grayson, Michael (2011). The Eponym Dictionary of Reptiles. Baltimore: Johns Hopkins University Press. xiii + 296 pp. ISBN 978-1-4214-0135-5. ("Puiseux", p. 212).
• O'Connor, John J.; Robertson, Edmund F., "Victor Alexandre Puiseux", MacTutor History of Mathematics Archive, University of St Andrews
• Victor Puiseux at the Mathematics Genealogy Project
This article incorporates text from a publication now in the public domain: Herbermann, Charles, ed. (1913). "Victor-Alexandre Puiseux". Catholic Encyclopedia. New York: Robert Appleton Company.
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Victor Ginzburg
Victor Ginzburg (born 1957) is a Russian American mathematician who works in representation theory and in noncommutative geometry. He is known for his contributions to geometric representation theory, especially, for his works on representations of quantum groups and Hecke algebras, and on the geometric Langlands program (Satake equivalence of categories). He is currently a Professor of Mathematics at the University of Chicago.[1][2]
Victor Ginzburg
2012 in Oberwolfach
Born1957 (age 65–66)
Moscow, Russia
NationalityAmerican
Alma materMoscow State University
Known forGinzburg dg algebra
Koszul duality
Scientific career
FieldsMathematics
InstitutionsUniversity of Chicago
Doctoral advisorAlexandre Kirillov
Israel Gelfand
Career
Ginzburg received his Ph.D. at Moscow State University in 1985, under the direction of Alexandre Kirillov and Israel Gelfand.
Ginzburg wrote a textbook Representation theory and complex geometry with Neil Chriss on geometric representation theory.
A paper by Alexander Beilinson, Ginzburg, and Wolfgang Soergel introduced the concept of Koszul duality (cf. Koszul algebra) and the technique of "mixed categories" to representation theory. Furthermore, Ginzburg and Mikhail Kapranov developed Koszul duality theory for operads.
In noncommutative geometry, Ginzburg defined, following earlier ideas of Maxim Kontsevich, the notion of Calabi–Yau algebra. An important role in the theory of motivic Donaldson–Thomas invariants is played by the so-called "Ginzburg dg algebra", a Calabi-Yau (dg)-algebra of dimension 3 associated with any cyclic potential on the path algebra of a quiver.
Selected publications
• Beilinson, Alexander; Ginzburg, Victor; Soergel, Wolfgang (1996), "Koszul duality patterns in representation theory" (PDF), Journal of the American Mathematical Society, 9 (2): 473–527, doi:10.1090/S0894-0347-96-00192-0, MR 1322847
• Chriss, Neil; Ginzburg, Victor (1997), Representation theory and complex geometry, Boston, MA: Birkhäuser, MR 1433132
• Etingof, Pavel; Ginzburg, Victor (2002), "Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism", Inventiones Mathematicae, 147 (2): 243–348, arXiv:math/0011114, Bibcode:2002InMat.147..243E, doi:10.1007/s002220100171, MR 1881922, S2CID 119708574
• Ginzburg, Victor (2005). "Lectures on Noncommutative Geometry". arXiv:math/0506603.
• Ginzburg, Victor (2006). "Calabi-Yau Algebras". arXiv:math/0612139.
• Ginzburg, Victor; Kapranov, Mikhail (1994), "Koszul duality for operads", Duke Mathematical Journal, 76 (1): 203–272, doi:10.1215/S0012-7094-94-07608-4, MR 1301191, S2CID 115166937
References
1. Koppes, Steve (June 8, 2006), "Victor Ginzburg, Professor in Mathematics and the College", The University of Chicago Chronicle.
2. "MMJ: Vol.7 (2007), N.4. - Victor Ginzburg".
External links
Victor Ginzburg at the Mathematics Genealogy Project
Wikimedia Commons has media related to Victor Ginzburg.
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Victor Moll
Victor Hugo Moll (born 1956) is a Chilean American mathematician specializing in calculus.
Moll studied at the Universidad Santa Maria and at the New York University with a master's degree in 1982 and a doctorate in 1984 with Henry P. McKean (Stability in the Large for Solitary Wave Solutions to McKean's Nerve Conduction Caricature).[1] He was a post-doctoral student at Temple University and became an assistant professor in 1986 and an associate professor in 1992 and in 2001 Professor at Tulane University.
In 1990–1991, he was a visiting professor at the University of Utah, in 1999 at the Universidad Técnica Federico Santa María in Valparaíso, and in 1995 a visiting scientist at the Courant Institute of Mathematical Sciences of New York University.
He deals with classical analysis, symbolic arithmetic and experimental mathematics, special functions and number theory.
Projects
Inspired by a 1988 paper in which Ilan Vardi proved several integrals in Table of Integrals, Series, and Products,[2] a well-known comprehensive table of integrals originally compiled by the Russian mathematicians Iosif Moiseevich Ryzhik (Russian: Иосиф Моисеевич Рыжик) and Izrail Solomonovich Gradshteyn (Израиль Соломонович Градштейн) in 1943 and subsequently expanded and translated into several languages, Victor Moll and George Boros started a project to prove all integrals listed in Gradshteyn and Ryzhik and add additional commentary and references.[3] In the foreword of the book Irresistible Integrals (2004), they wrote:[4]
It took a short time to realize that this task was monumental.
Nevertheless, the efforts have resulted in about 900 entries from Gradshteyn and Ryzhik discussed in a series of more than 30 articles of which papers 1 to 28 have been published in issues 14 to 26 of Scientia, Universidad Técnica Federico Santa María (UTFSM), between 2007 and 2015[5] and compiled into a two-volume book series Special Integrals of Gradshteyn and Ryzhik: the Proofs (2014–2015).[6][7] Moll also assisted Daniel Zwillinger editing the eight English edition of Gradshteyn and Ryzhik in 2014.[8]
Moll also took on the task to revise and expand the classical landmark work "A Course of Modern Analysis" by Whittaker and Watson, which was originally published in 1902 and last revised in 1927, to publish a new edition in 2021.
Publications
• The evaluation of integrals, a personal story, Notices AMS, 2002, No. 3
• with Henry McKean Elliptic Curves: function theory, geometry, arithmetic, Cambridge University Press, 1997
• Numbers and functions: from a classical-experimental mathematician’s point of view, AMS, 2012
• Editor with Tewodros Amdeberhan Tapas in experimental mathematics, AMS Special Session on Experimental Mathematics, 5 January 2007, New Orleans, Louisiana, AMS, 2008
• Editor with Tewodros Amdeberhan, Luis A. Medina Gems in experimental mathematics, AMS Special Session, Experimental Mathematics, 5 January 2009, Washington, DC, AMS, 2010
• with George Boros Irresistible Integrals: Symbolics, Analysis and Experiments in the Evaluation of Integrals, Cambridge University Press, 2004.
• A Course of Modern Analysis, 2021
See also
• Gradshteyn and Ryzhik (GR)
• Whittaker and Watson
References
1. Victor Moll at the Mathematics Genealogy Project
2. Vardi, Ilan (April 1988). "Integrals: An Introduction to Analytic Number Theory" (PDF). American Mathematical Monthly. 95 (4): 308–315. doi:10.2307/2323562. JSTOR 2323562. Archived (PDF) from the original on 2016-03-15. Retrieved 2016-03-14.
3. Moll, Victor Hugo (April 2010) [2009-08-30]. "Seized Opportunities" (PDF). Notices of the American Mathematical Society. 57 (4): 476–484. Archived (PDF) from the original on 2016-04-08. Retrieved 2016-04-08.
4. Boros, George; Moll, Victor Hugo (2006) [September 2004]. Irresistible Integrals. Symbolics, Analysis and Experiments in the Evaluation of Integrals (reprinted 1st ed.). Cambridge University Press (CUP). p. xi. ISBN 978-0-521-79186-1. Retrieved 2016-02-22. (NB. This edition contains many typographical errors.)
5. Moll, Victor Hugo (2012). "Index of the papers in Revista Scientia with formulas from GR". Retrieved 2016-02-17.
6. Moll, Victor Hugo (2014-10-01). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume I. ISBN 978-1-4822-5651-2. Retrieved 2016-02-12. {{cite book}}: |work= ignored (help)
7. Moll, Victor Hugo (2015-08-24). Special Integrals of Gradshteyn and Ryzhik: the Proofs – Volume II. ISBN 978-1-4822-5653-6. Retrieved 2016-02-12. {{cite book}}: |work= ignored (help)
8. Gradshteyn, Izrail Solomonovich; Ryzhik, Iosif Moiseevich; Geronimus, Yuri Veniaminovich; Tseytlin, Michail Yulyevich; Jeffrey, Alan (2015) [October 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Table of Integrals, Series, and Products. Translated by Scripta Technica, Inc. (8 ed.). Academic Press, Inc. ISBN 978-0-12-384933-5. GR:12. Retrieved 2016-02-21.
External links
• Moll, Victor Hugo. "Victor Hugo Moll". Archived from the original on 2021-12-21. Retrieved 2022-01-22.
• Testimonios: Dr. Victor H. Moll (October 15, 2022)
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Victor Kac
Victor Gershevich (Grigorievich) Kac (Russian: Виктор Гершевич (Григорьевич) Кац; born 19 December 1943) is a Soviet and American mathematician at MIT, known for his work in representation theory. He co-discovered[2] Kac–Moody algebras, and used the Weyl–Kac character formula for them to reprove the Macdonald identities. He classified the finite-dimensional simple Lie superalgebras, and found the Kac determinant formula for the Virasoro algebra. He is also known for the Kac–Weisfeiler conjectures with Boris Weisfeiler.
Victor Gershevich Kac
Born (1943-12-19) December 19, 1943
Buguruslan, Orenburg Oblast, Russian SFSR
Alma materMoscow State University (MS)
Moscow State University (PhD)
Known for
• Kac–Moody algebra
• Weyl–Kac character formula
• Classification of Lie superalgebras
• Kac–Weisfeiler conjectures
• Kac determinant formula for Virasoro algebra
Awards
• Sloan Research Fellowship (1981)
• Medal of the Collège de France (1981)
• Guggenheim Fellowship (1986)
• Wigner Medal (1996)
• American Academy of Arts and Sciences (2007)
• National Academy of Sciences (2013)
• Steele Prize (2015)
Scientific career
FieldsMathematics
InstitutionsMIT
ThesisSimple Irreducible Graded Lie Algebras of Finite Growth (1968)
Doctoral advisorÈrnest Borisovich Vinberg[1]
Biography
Kac studied mathematics at Moscow State University, receiving his MS in 1965 and his PhD in 1968.[3] From 1968 to 1976, he held a teaching position at the Moscow Institute of Electronic Machine Building (MIEM). He left the Soviet Union in 1977, becoming an associate professor of mathematics at MIT. In 1981, he was promoted to full professor. Kac received a Sloan Fellowship and the Medal of the Collège de France, both in 1981, and a Guggenheim Fellowship in 1986. He received the Wigner Medal (1996) "in recognition of work on affine Lie algebras that has had wide influence in theoretical physics". In 1978 he was an Invited Speaker (Highest weight representations of infinite dimensional Lie algebras) at the International Congress of Mathematicians (ICM) in Helsinki. Kac was a plenary speaker at the 1988 American Mathematical Society centennial conference. In 2002 he gave a plenary lecture, Classification of Supersymmetries, at the ICM in Beijing.
Kac is a Fellow of the American Mathematical Society,[4] an Honorary member of the Moscow Mathematical Society, Fellow of the American Academy of Arts and Sciences and a Member of the National Academy of Sciences.
The research of Victor Kac primarily concerns representation theory and mathematical physics. His work appears in mathematics and physics and in the development of quantum field theory, string theory and the theory of integrable systems.
Kac has published 13 books and over 200 articles in mathematics and physics journals and is listed as an ISI highly cited researcher.[5] Victor Kac was awarded the 2015 AMS Leroy P. Steele Prize for Lifetime Achievement.[6]
He was married with Michèle Vergne[7] and they have a daughter, Marianne Kac-Vergne, who is a professor of American civilization at the university of Picardie. His brother Boris Katz is a principal research scientist at MIT.[8]
Kac–Moody algebra
"Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A.J. Coleman[9]
Bibliography
• Kac, Victor G. (1994) [1985]. Infinite-Dimensional Lie Algebras (3rd ed.). Cambridge University Press. ISBN 0-521-46693-8.
• Kac, V. (1985). Infinite Dimensional Groups with Applications. New York: Springer. ISBN 9781461211044. OCLC 840277997.
• Seligman, George B. (1987). "Review: Infinite-dimensional Lie algebras, by Victor G. Kac, 2nd edition" (PDF). Bull. Amer. Math. Soc. (N.S.). 16: 144–149. doi:10.1090/S0273-0979-1987-15492-9.
• Kac, Victor G.; Raina, A. K. (1987). Bombay lectures on highest weight representations of infinite-dimensional Lie algebras. Singapore: World Scientific. ISBN 9971503956. OCLC 18475755.
• Kac, Victor (1997). Vertex Algebras for Beginners (University Lecture Series, No 10). American Mathematical Society. ISBN 0-8218-0643-2.
• Kac, Victor G.; Cheung, Pokman (2002). Quantum calculus. New York: Springer. ISBN 0387953418. OCLC 47243954.
• Kac, Victor G.; Raina, A. K. (2013). Bombay Lectures on Highest Weight Representations of Infinite Dimensional Lie Algebras. Advanced Series in Mathematical Physics. Vol. 29 (2nd ed.). World Scientific Publishing. doi:10.1142/8882. ISBN 978-981-4522-18-2.
References
1. Mathematics Genealogy Project: https://www.genealogy.math.ndsu.nodak.edu/id.php?id=37054
2. Stephen Berman, Karen Parshall "Victor Kac and Robert Moody — their paths to Kac–Moody-Algebras", Mathematical Intelligencer, 2002, Nr.1
3. Victor Kac, A Biographical Interview: http://dynkincollection.library.cornell.edu/sites/default/files/Victor%20Kac%20%28RI-ED%29.pdf
4. List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
5. "List of ISI highly cited researchers".
6. 2015 AMS Steele Prizes
7. La Gazette des Mathématiciens 165, retrieved 2021-04-22.
8. Negri, Gloria (4 October 2006). "Clara Katz; Soviet émigré saved ailing granddaughter". The Boston Globe.
9. Coleman, A. John, "The Greatest Mathematical Paper of All Time", The Mathematical Intelligencer, vol. 11, no. 3, pp. 29–38.
External links
• Victor Kac's home page at MIT
• Victor Kac at the Mathematics Genealogy Project
• Victor Kac, A Biographical Interview,
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Victor J. Katz
Victor Joseph Katz (born 31 December 1942, Philadelphia)[1] is an American mathematician, historian of mathematics, and teacher known for using the history of mathematics in teaching mathematics.
Victor Joseph Katz
Born31 December 1942
Philadelphia, Pennsylvania, USA
NationalityAmerican
OccupationMathematician
Known forUsing the history of mathematics to teach the subject
Academic background
Alma materPrinceton University
Academic work
Notable worksHistory of Mathematics: An Introduction (1993)
Biography
Katz received in 1963 from Princeton University a bachelor's degree and in 1968 from Brandeis University a Ph.D. in mathematics under Maurice Auslander with thesis The Brauer group of a regular local ring.[2] He became at Federal City College an assistant professor and then in 1973 an associate professor and, after the merger of Federal City College into the University of the District of Columbia in 1977, a full professor there in 1980. He retired there as professor emeritus in 2005.
As a mathematician Katz specializes in algebra, but he is mainly known for his work on the history of mathematics and its uses in teaching. He wrote a textbook History of Mathematics: An Introduction (1993), for which he won in 1995 the Watson Davis and Helen Miles Davis Prize. He organized workshops and congresses for the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics. The MAA published a collection of teaching materials by Katz as a compact disk with the title Historical Modules for the Teaching and Learning of Mathematics. With Frank Swetz, he was a founding editor of a free online journal on the history of mathematics under the aegis of the MAA; the journal is called Convergence: Where Mathematics, History, and Teaching Interact.[3] In the journal Convergence, Katz and Swetz published a series Mathematical Treasures.[4][5] For a study of the possibilities for using mathematical history in schools, Katz received a grant from the National Science Foundation.
Personal
He has been married since 1969 to Phyllis Katz (née Friedman), a science educator who developed and directed the U.S. national nonprofit organization Hands On Science Outreach, Inc. (HOSO). The couple have three children.
Selected publications
As author
• History of Mathematics: An Introduction, New York: Harper Collins, 1993, 3rd edition Pearson 2008 (a shortened edition was published in 2003 by Pearson)
• with Karen Hunger Parshall: Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century, Princeton University Press 2014[6][7]
• with John B. Fraleigh: A first course in abstract algebra, Addison-Wesley 2003
As editor
• The Mathematics of Egypt, Mesopotamia, China, India and Islam: A Sourcebook, Princeton University Press 2007[8]
• with Bengt Johansson, Frank Swetz, Otto Bekken, John Fauvel: Learn from the Masters, MAA 1994 (contribution by Katz: Historical ideas in teaching linear algebra, Napier's logarithms adapted for today's classroom)
• Using History to Teach Mathematics: An International Perspective, MAA 2000, MAA Notes (No. 51)[9][10]
• with Marlow Anderson, Robin Wilson: Sherlock Holmes in Babylon and other Tales of Mathematical History, (collection of reprints from the journal Mathematics Magazine of MAA; contribution by Katz: Ideas of calculus in Islam and India), MAA 2004[11]
• with Marlow Anderson, Robin Wilson: Who gave you the epsilon? and other tales of mathematical history, MAA 2009 (continuation of the collection of essays on the history of mathematics from MAA journal; contribution by Katz: The history of Stokes' theorem)[12]
• with Constantinos Tzanakis: Recent Developments on Introducing a Historical Dimension in Mathematics Education, MAA 2011
References
1. biographical information from American Men and Women of Science, Thomson Gale 2004
2. Victor J. Katz at the Mathematics Genealogy Project
3. MAA, Convergence
4. Katz, Swetz, Mathematical Treasures, Omar Khayyam's Algebra
5. Katz, V. J.; Swetz, F. (March 2011). "Mathematical Treasures" (PDF). HPM Newskletter. No. 76. pp. 2–4.
6. Jongsma, Calvin (26 February 2015). "Review of Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century by Victor J. Katz and Karen Hunger Parshall". MAA Reviews, Mathematical Association of America.
7. Chen, Jiang-Ping Jeff (March 2015). "Review of Taming the Unknown: A History of Algebra from Antiquity to the Early Twentieth Century by Victor J. Katz and Karen Hunger Parshall". The College Mathematics Journal. 46 (2): 149–152. doi:10.4169/college.math.j.46.2.149. S2CID 218544510.
8. Montelle, Clemency (2015). "Review of The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook ed. by Victor J. Katz". Aestimatio: Critical Reviews in the History of Science. 4: 179–191. doi:10.33137/aestimatio.v4i0.25818.
9. Sandifer, Ed (3 January 2001). "Review of Using History to Teach Mathematics: An International Perspective by Victor J. Katz". MAA Reviews, Mathematical Association of America.
10. Deakin, Michael A. B. (2001). "Review of Using History to Teach Mathematics: An International Perspective ed. by Victor J. Katz" (PDF). Zentralblatt für Didaktik der Mathematik. 33 (5): 137–138. doi:10.1007/BF02656618.
11. Gouvêa, Fernando Q. (2015). "Review of Sherlock Holmes in Babylon and Other Tales of Mathematical History ed. by Marlow Anderson, Victor Katz, and Robin Wilson". Aestimatio: Critical Reviews in the History of Science. 2: 67–79. doi:10.33137/aestimatio.v2i0.25743.
12. Davis, Philip J. (18 October 2009). "Review of Who Gave You the Epsilon? & Other Tales of Mathematical History ed. by Marlow Anderson, Victor Katz, and Robin Wilson". SIAM News, Society for Industrial and Applied Mathematics.
External links
• Biography from the MAA
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Victor Kolyvagin
Victor Alexandrovich Kolyvagin (Russian: Виктор Александрович Колывагин, born 11 March, 1955) is a Russian mathematician who wrote a series of papers on Euler systems, leading to breakthroughs on the Birch and Swinnerton-Dyer conjecture, and Iwasawa's conjecture for cyclotomic fields.[1] His work also influenced Andrew Wiles's work on Fermat's Last Theorem.[2][3]
Victor Kolyvagin
NationalityRussian
Alma materMoscow State University
Scientific career
FieldsMathematics
InstitutionsJohns Hopkins University, CUNY
Doctoral advisorYuri Manin
Career
Kolyvagin received his Ph.D. in Mathematics in 1981 from Moscow State University,[4] where his advisor was Yuri I. Manin. He then worked at Steklov Institute of Mathematics in Moscow[2] until 1994. Since 1994 he has been a professor of mathematics in the United States. He was a professor at Johns Hopkins University until 2002 when he became the first person to hold the Mina Rees Chair in mathematics at the Graduate Center Faculty at The City University of New York.[5][4]
Awards
In 1990 he received the Chebyshev Prize of the USSR Academy of Sciences.[4]
References
1. Rubin, Karl (2000). Euler Systems. ISBN 0-691-05075-9. {{cite book}}: |journal= ignored (help)
2. Cipra, Barry (1993). "Fermat Proof Hits a Stumbling Block". Science. American Association for the Advancement of Science. 263 (5142): 1967–8. Bibcode:1993Sci...262.1967C. doi:10.1126/science.262.5142.1967-a. JSTOR 2882956.
3. Cipra, Barry A. (January 6, 1989). "Getting a Grip on Elliptic Curves Author". Science. New Series. American Association for the Advancement of Science Stable. 243 (4887): 30–31. doi:10.1126/science.243.4887.30. JSTOR 1703169. PMID 17780417. Archived from the original on 2022-11-04.
4. Arenson, Karen W. (August 7, 2002). "Benefactor's Chair Filled at CUNY". The New York Times.
5. Targeted News Service (2009-12-22). "NSF Invests a Million Dollars in Number Theory at the CUNY Graduate Center".
Link
• Victor Kolyvagin at the Mathematics Genealogy Project
• Kolyvagin's Biography
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Victor Lomonosov
Victor Lomonosov (7 February 1946 – 29 March 2018) was a Russian-American mathematician known for his work in functional analysis. In operator theory, he is best known for his work in 1973 on the invariant subspace problem, which was described by Walter Rudin in his classical book on Functional Analysis as "Lomonosov's spectacular invariant subspace theorem".[1] The Theorem Lomonosov gives a very short proof, using the Schauder fixed point theorem, that if the bounded linear operator T on a Banach space commutes with a non-zero compact operator then T has a non-trivial invariant subspace.[2] Lomonosov has also published on the Bishop–Phelps theorem[3] and Burnside's Theorem [4]
Lomonosov received his master's degree from the Moscow State University in 1969 and his Ph.D. from National University of Kharkiv in 1974 (adviser Vladimir Matsaev). He was appointed at the rank of Associate Professor at Kent State University in the fall of1991, becoming Professor at the same university in 1999.
References
1. Rudin, Walter (1991) [1973]. Functional Analysis (2nd ed.). New York: McGraw-Hill. ISBN 0-07-100944-2.
2. Lomonosov, V. I. (1973). "Invariant subspaces of the family of operators that commute with a completely continuous operator". Akademija Nauk SSSR. Funkcional' Nyi Analiz I Ego Prilozenija. 7 (3): 55–56. doi:10.1007/BF01080698. MR 0420305. S2CID 121421267.
3. Lomonosov, Victor (2000). "A counterexample to the Bishop-Phelps theorem in complex spaces". Israel Journal of Mathematics. 115: 25–28. doi:10.1007/bf02810578. S2CID 53646715.
4. Lomonosov, Victor (1991). "An extension of Burnside's theorem to infinite-dimensional spaces". Israel Journal of Mathematics. 75 (2–3): 329–339. doi:10.1007/bf02776031. S2CID 120120695.
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Victor Mazurov
Victor Danilovich Mazurov (Russian: Виктор Данилович Мазуров; born January 31, 1943) is a Russian mathematician. He is well known for his works in group theory and is the founder of the Novosibirsk school of finite groups. Mazurov is a Corresponding Member of the Russian Academy of Sciences.
Mazurov's parents Daniil Petrovich and Evstolia Ivanovna were teachers.[1] Victor went to elementary school in a village of Kuvashi and finished high school with highest honors in Zlatoust. He then moved to Sverdlovsk (now Yekaterinburg) to study mathematics in Ural State University. His advisers in Sverdlovsk were Victor Busarkin and Albert Starostin.[2] In 1963 Mazurov married his university classmate Nadezhda Khomenko. After graduating in 1965, they moved to Novosibirsk where Mazurov joined the research staff of the Sobolev Institute of Mathematics (Russian: Институт математики СО РАН).
Mazurov is an editor (with Evgenyj Khukhro) of the "Kourovka Notebook",[3] a periodically updated collection of over 1,000 open problems in Group Theory.
Mazurov obtained several results that contributed to the proof of the classification of finite simple groups, also known as the Enormous Theorem[4] and considered one of the greatest achievements in mathematics of the 20th century.
He is one of the initial group of fellows of the American Mathematical Society.[5]
References
1. Sib Math Journal, Vol. 54(1), 2013 http://math.nsc.ru/LBRT/a4/Mazurov/rus/bio_ru.htm
2. Victor Mazurov at the Mathematics Genealogy Project
3. Khukhro, E. I.; Mazurov, V. D. (2014). "Unsolved Problems in Group Theory. The Kourovka Notebook". arXiv:1401.0300 [math.GR].
4. "Enormous Theorem".
5. Jackson, Allyn (May 2013), "Fellows of the AMS: Inaugural Class" (PDF), Notices of the American Mathematical Society, 60 (5): 631–637
External links
• Victor D. Mazurov Personal webpage at the Sobolev Institute of Mathematics
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Victor Săhleanu
Victor Aurelian Săhleanu (Romanian: [ˈviktor a.ureliˈan səhˈle̯anu]; 19 January 1924 – 26 August 1997) was a Romanian physician and anthropologist. He was a leading figure in anthropology in his country from the late 1960s until his death.
Biography
Education and early career
Săhleanu was born in Gura Humorului, in the Bukovina region of the Kingdom of Romania. After finishing secondary school at the Aron Pumnul High School in Cernăuți, he entered the medical faculty of the University of Bucharest, from which he graduated in 1948. At that point, with the onset of the Communist regime, the institution became the Carol Davila University of Medicine and Pharmacy. Between 1944 and 1946, he took part-time courses at the literature and philosophy faculty as well, but did not earn a degree. In 1949, he became a doctor of medicine and surgery, with a thesis on "Considerations regarding Field Medicine". Ștefan-Marius Milcu presided over the doctoral committee, which also included Constantin Ion Parhon. He began working in hospitals while still a student, and during 1946, was a junior teaching assistant in the anatomical pathology department.[1]
After graduation in 1948, he won a competition to become an intern at the Parhon-led endocrinology institute, where he was also a researcher from 1954 to 1961. He worked in endocrinology for a total of seventeen years, during which he founded the institute's morphopathology laboratory. From 1950 to 1952, he was a peer reviewer at the Milcu-led anthropology collective, a section of the endocrinology institute that was essentially a continuation of the Francisc Rainer-founded anthropology institute. In 1954, he signed up for part-time classes at the physics and mathematics faculty in Bucharest, graduating in 1961.[1]
Involvement in anthropology and legacy
In 1963, he became a primary care endocrinologist and, at the request of Eugen A. Pora, began teaching courses in biophysics and biomathematics at Babeș-Bolyai University in Cluj. In 1965, he earned the title of Doctor of Science. The same year, he was transferred from the endocrinology to the geriatrics institute. In 1969, he was transferred from Babeș-Bolyai and named adjunct scientific director at Bucharest's center for anthropological research;[1] from that point until his death, he was at the forefront of anthropology in Romania.[2] In 1974, when the center became a laboratory within the Victor Babeș institute, Săhleanu became its director, serving for eight years.[1]
In 1982, due to the so-called "Transcendental Meditation Affair", he was excluded from scientific life. His works were withdrawn from libraries, his name could no longer appear in books or publications, and he was transferred to work as a doctor at a hospital in the Titan neighborhood. In 1984, aged 60, he retired upon his request. Between 1982 and 1984, he taught postgraduate courses in anthropology at a United Nations demographic center in Bucharest.[1] As an anthropologist, he developed an interdisciplinary approach to the field that combined biology and culture, exploring the relationship between anatomical features and their behavioral, symbolic and cultural significance. In 1980, he was editor-in-chief of Romania's first atlas of biological anthropology.[2]
In February 1990, after the fall of the regime, he was restored as head of the Romanian Academy's anthropological research center, by government decree. He died in 1997, following complications from a cerebral hemorrhage.[3] He and his wife Zoe, a pediatrician, had two sons: Adrian George, who became a philologist and psychoanalyst; and Valentin, later an architect.[2]
Săhleanu published over 2000 articles and 60 books, in fields that included methodology, medical psychology and psychoanalysis, ethics, aesthetics and the history of medicine and science.[2] He kept a diary, from age 17 until his final days, that reached over 25,000 pages.[4] He was also an essayist and poet, publishing volumes in 1961, 1972, 1977 and 1997; and was among the founders of the Romanian Society of Writer and Journalist Physicians.[5]
Alexandru Ofrim stated that Săhleanu wrote communist propaganda against erotic pleasure.[6][7]
• With his mother at a family reunion in 1935
• As a high school student
• At his wedding in 1948
• In his office at the anthropology institute
Notes
1. Ciuhuța, p.9
2. Kozma, p.34
3. Ciuhuța, p.10
4. Kozma, p.35
5. Ciuhuța, p.12
6. Ofrim, Alexandru (18 September 2008). "Tot ce trebuia să știm despre sex - din cărți". Dilema veche (in Romanian). Retrieved 15 February 2019.
7. "Private Life and Social Practices during the Golden Age". Muzeul Municipiului București – Site Oficial. Retrieved 1 December 2020.
References
• (in Romanian) Andrei Kozma, Cristiana Glavce, Constantin Bălăceanu-Stolnici (eds.), Antropologie și mediu. Editura Niculescu, Bucharest, 2014, ISBN 978-973-748-859-6
• Mircea Ștefan Ciuhuța, "Victor Săhleanu, personalitate de prim rang în antropologia românească", p. 9-13
• Andrei Kozma, "Victor Aurelian Săhleanu, poetul om de știință", p. 34-9
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Victor Thébault
Victor Michael Jean-Marie Thébault (1882–1960) was a French mathematician best known for propounding three problems in geometry. The name Thébault's theorem is used by some authors to refer to the first of these problems and by others to refer to the third.
Thébault was born on March 6, 1882, in Ambrières-les-Grand (today a part of Ambrières-les-Vallées, Mayenne) in the northwest of France. He got his education at a teacher's college in Laval, where he studied from 1898 to 1901. After his graduation he taught for three years at Pré-en-Pail until he got a professorship at technical school in Ernée. In 1909 he placed first in a competitive exams, which yielded him a certificate to work as a science professor at teachers' colleges. Thébault however found a professor's salary insufficient to support his large family and hence he left teaching to become a factory superintendent at Ernée from 1910 to 1923. In 1924 he became a chief insurance inspector in Le Mans, a position he held until his retirement in 1940. During his retirement he lived in Tennie. He died on March 19, 1960, shortly after a severe stroke and was survived by his wife, five sons and a daughter.[1]
Despite leaving teaching Thébault stayed active in mathematics with number theory and geometry being his main areas of interest. He published a large number of articles in math journals all over the world and aside from regular articles he also contributed many original problems and solutions to their problem sections. He published over 1000 original problems in various mathematical magazines[2] and his contributions to the problem section of the American Mathematical Monthly alone comprise over 600 problems and solutions. In recognition of his contributions the French government bestowed two titles on him. In 1932 he became an Officier de L'Instruction Publique and in a 1935 a Chevalier de l'Order de Couronne de Belgium.[1]
Notes
1. C. W. Trigg: Victor Thebault 1882-1960. Mathematics Magazine, Vol. 33, No. 5 (May - Jun., 1960) (JSTOR)
2. Alexander Ostermann, Gerhard Wanner: Geometry by Its History. Springer, 2012, p. 181
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Victor Batyrev
Victor Vadimovich Batyrev (Виктор Вадимович Батырев, born 31 August 1961, Moscow)[1] is a Russian mathematician, specializing in algebraic and arithmetic geometry and its applications to mathematical physics. He is a professor at the University of Tübingen.
Biography
Batyrev studied mathematics from 1978 to 1985 at Moscow State University. From 1991 he was at the University of Essen, where he earned his habilitation in 1993. Since 1996 he has been a professor at the University of Tübingen.[2]
He received in 1994 the Gottschalk-Diederich-Baedeker Prize. In 1995 he received the Heinz Maier-Leibnitz Prize for his habilitation thesis Hodge Theory of Hypersurfaces in Toric Varieties and Recent Developments in Quantum Physics. In 1998 he was an invited speaker at the International Congress of Mathematicians in Berlin and gave a talk Mirror Symmetry and Toric Geometry.[3] In 2003 he was elected a member of the Heidelberger Akademie der Wissenschaften.[4]
Selected publications
• Batyrev, V. V.; Manin, Yu. I. (1990). "Sur le nombre des points rationnels de hauté borné des variétés algébriques". Mathematische Annalen. 286 (1–3): 27–43. doi:10.1007/bf01453564. S2CID 119945673.
• Batyrev, Victor V. (1993). "Quantum cohomology rings of toric manifolds". arXiv:alg-geom/9310004. Bibcode:1993alg.geom.10004B. {{cite journal}}: Cite journal requires |journal= (help)
• Batyrev, Victor V. (1994). "Dual polyhedra and mirror symmetry for Calabi–Yau hypersurfaces in toric varieties". Journal of Algebraic Geometry: 493–535. arXiv:alg-geom/9310003. Bibcode:1993alg.geom.10003B.
• Batyrev, Victor V.; Borisov, Lev A. (1996). "Mirror duality and string theoretic Hodge numbers". Inventiones Mathematicae. 126 (1): 183–203. arXiv:alg-geom/9509009. Bibcode:1996InMat.126..183B. doi:10.1007/s002220050093. S2CID 55227866.
• Batyrev, Victor V.; Dais, Dimitrios I. (1996). "Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry". Topology. 35 (4): 901–929. arXiv:alg-geom/9410001. doi:10.1016/0040-9383(95)00051-8. S2CID 15604511.
• Batyrev, Victor V. (1997). "Stringy Hodge numbers of varieties with Gorenstein canonical singularities". arXiv:alg-geom/9711008. Bibcode:1997alg.geom.11008B. {{cite journal}}: Cite journal requires |journal= (help)
• Batyrev, Victor V.; Tschinkel, Yuri (1998). "Manin's conjecture for toric manifolds". Journal of Algebraic Geometry. 7: 15–53. arXiv:alg-geom/9510014. Bibcode:1995alg.geom.10014B.
References
1. Jahrbuch der Heidelberger Akademie der Wissenschaften 2010
2. homepage of Victor Batyrev at the University of Tübingen
3. Batyrev, Victor V. (1998). "Mirror symmetry and toric geometry". Doc. Math. (Bielefeld) Extra Vol. ICM Berlin, 1998, vol. II. pp. 239–248.
4. page for Batyrev at homepage of Heidelberger Akademie der Wissenschaften
External links
• mathnet.ru
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Victor W. Marek
Victor Witold Marek, formerly Wiktor Witold Marek known as Witek Marek (born 22 March 1943) is a Polish mathematician and computer scientist working in the fields of theoretical computer science and mathematical logic.
Biography
Victor Witold Marek studied mathematics at the Faculty of Mathematics and Physics of the University of Warsaw. Supervised by Andrzej Mostowski, he received both a magister degree in mathematics in 1964 and a doctoral degree in mathematics in 1968. He completed habilitation in mathematics in 1972.
In 1970–1971, Marek was a postdoctoral researcher at Utrecht University, the Netherlands, where he worked under Dirk van Dalen. In 1967–1968 as well as in 1973–1975, he was a researcher at the Institute of Mathematics of the Polish Academy of Sciences, Warsaw, Poland. In 1979–1980 and 1982–1983 he worked at the Venezuelan Institute of Scientific Research. In 1976, he was appointed an Assistant Professor of Mathematics at the University of Warsaw.
In 1983, he was appointed a professor of computer science at the University of Kentucky. In 1989–1990, he was a Visiting Professor of Mathematics at Cornell University, Ithaca, New York. In 2001–2002, he was a visitor at the Department of Mathematics of the University of California, San Diego.
In 2013, Professor Marek was the Chair of the Program Committee of the scientific conference commemorating Andrzej Mostowski's Centennial.
Legacy
Teaching
He has supervised a number of graduate theses and projects. He was an advisor of 16 doctoral candidates both in mathematics and computer science. In particular, he advised dissertations in mathematics by Małgorzata Dubiel-Lachlan, Roman Kossak, Adam Krawczyk, Tadeusz Kreid, Roman Murawski, Andrzej Pelc, Zygmunt Ratajczyk, Marian Srebrny, and Zygmunt Vetulani. In computer science his students were V. K. Cody Bumgardner, Waldemar W. Koczkodaj, Witold Lipski, Joseph Oldham, Inna Pivkina, Michał Sobolewski , Paweł Traczyk, and Zygmunt Vetulani. These individuals have worked in various institutions of higher education in Canada, France, Poland, and the United States.
Mathematics
He investigated a number of areas in the foundations of mathematics, for instance infinitary combinatorics (large cardinals), metamathematics of set theory, the hierarchy of constructible sets,[1] models of second-order arithmetic,[2] the impredicative theory of Kelley–Morse classes.[3] He proved that the so-called Fraïssé conjecture (second-order theories of countable ordinals are all different) is entailed by Gödel's axiom of constructibility. Together with Marian Srebrny, he investigated properties of gaps in a constructible universe.
Computer science
He studied logical foundations of computer science. In the early 1970s, in collaboration with Zdzisław Pawlak,[4][5] he investigated Pawlak's information storage and retrieval systems,[6] which then was a widely studied concept, especially in Eastern Europe. These systems were essentially single-table relational databases, but unlike Codd's relational databases were bags rather than sets of records. These investigations, in turn, led Pawlak to the concept of rough set,[5] studied by Marek and Pawlak in 1981.[7] The concept of rough set, in computer science, statistics, topology, universal algebra, combinatorics, and modal logic, turned out to be an expressive language for describing, and especially manipulating an incomplete information.
Logic
In the area of nonmonotonic logics, a group of logics related to artificial intelligence, he focused on investigations of Reiter's default logic,[8] and autoepistemic logic of R. Moore. These investigations led to a form of logic programming called answer set programming[9] a computational knowledge representation formalism, studied both in Europe and in the United States. Together with Mirosław Truszczynski, he proved that the problem of existence of stable models of logic programs is NP-complete. In a stronger formalism admitting function symbols, along with Nerode and Remmel he showed that the analogous problem is Σ1
1
-complete.
Publications
V. W. Marek is an author of over 180 scientific papers in the area of foundations of mathematics and of computer science. He was also an editor of numerous proceedings of scientific meetings. Additionally, he authored or coauthored several books. These include:
• Logika i Podstawy Matematyki w Zadaniach (jointly with Janusz Onyszkiewicz)
• Logic and Foundations of Mathematics in problems (jointly with Janusz Onyszkiewicz)
• Analiza Kombinatoryczna (jointly with W. Lipski),
• Nonmonotonic Logic – Context-dependent Reasoning (jointly with M. Truszczyński),
• Introduction to Mathematics of Satisfiability.
References
1. W. Marek and M. Srebrny, Gaps in constructible universe, Annals of Mathematical Logic, 6:359–394, 1974.
2. K.R. Apt and W. Marek, Second order arithmetic and related topics, Annals of Mathematical Logic, 6:177–229, 1974
3. W. Marek, On the metamathematics of impredicative set theory. Dissertationes Mathematicae 98, 45 pages, 1973
4. Z. Pawlak, Mathematical foundations of information retrieval. Institute of Computer Sciences, Polish Academy of Sciences, Technical Report 101, 8 pages, 1973
5. Z. Pawlak, Rough sets. Institute of Computer Science, Polish Academy of Sciences, Technical Report 431, 12 pages, 1981
6. W. Marek and Z. Pawlak On the foundations of information retrieval. Bull. Acad. Pol. Sci. 22:447–452, 1974
7. W. Marek and Z. Pawlak. Rough sets and information systems, Institute of Computer Science, Technical Report 441, Polish Academy of Sciences, 15 pages, 1981
8. M.Denecker, V.W. Marek and M. Truszczynski, Uniform semantic treatment of default and autoepistemic logics. Artificial Intelligence. 143:79–122, 2003
9. V.W. Marek and M. Truszczynski, Stable logic programming – an alternative logic programming paradigm. In: 25 years of Logic Programming Paradigm, pages 375–398, Springer-Verlag, 1999
External links
• Personal page of Dr. V.W. Marek at the University of Kentucky
• Papers online
• Slides and other scientific materials
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Victor Pan
Victor Yakovlevich Pan (Russian: Пан Виктор Яковлевич) is a Soviet and American mathematician and computer scientist, known for his research on algorithms for polynomials and matrix multiplication.
Education and career
Pan earned his Ph.D. at Moscow University in 1964, under the supervision of Anatoli Georgievich Vitushkin,[1] and continued his work at the Soviet Academy of Sciences. During that time, he published a number of significant papers and became known informally as "polynomial Pan" for his pioneering work in the area of polynomial computations. In late 1970s, he immigrated to the United States and held positions at several institutions including IBM Research. Since 1988, he has taught at Lehman College of the City University of New York.[2]
Contributions
Victor Pan is an expert in computational complexity and has developed a number of new algorithms. One of his notable early results is a proof that the number of multiplications in Horner's method is optimal.[CVP]
In the theory of matrix multiplication algorithms, Pan in 1978 published an algorithm with running time $O(n^{2.795})$. This was the first improvement over the Strassen algorithm after nearly a decade, and kicked off a long line of improvements in fast matrix multiplication that later included the Coppersmith–Winograd algorithm and subsequent developments.[SNO] He wrote the text How to Multiply Matrices Faster (Springer, 1984) surveying early developments in this area.[3][HMM] His 1982 algorithm[P82] still held the record in 2020 for the fastest "practically useful" matrix multiplication algorithm (i.e., with a small base size and manageable hidden constants).[4] In 1998, with his student Xiaohan Huang, Pan showed that matrix multiplication algorithms can take advantage of rectangular matrices with unbalanced aspect ratios, multiplying them more quickly than the time bounds one would obtain using square matrix multiplication algorithms.[FRM]
Since that work, Pan has returned to symbolic and numeric computation and to an earlier theme of his research, computations with polynomials. He developed fast algorithms for the numerical computation of polynomial roots,[UP] and, with Bernard Mourrain, algorithms for multivariate polynomials based on their relations to structured matrices.[5][MPD] He also authored or co-authored several more books, on matrix and polynomial computation,[6][PMC] structured matrices,[7][SMP] and on numerical root-finding procedures.[8][NMR]
Recognition
Pan was appointed Distinguished Professor at Lehman College in 2000.[2]
In 2013 he became a fellow of the American Mathematical Society, for "contributions to the mathematical theory of computation".[9]
Selected publications
Research papers
CVP.
Pan, V. Ja. (1966), "On means of calculating values of polynomials", Russian Math. Surveys, 21: 105–136, doi:10.1070/rm1966v021n01abeh004147, MR 0207178, S2CID 250869179
SNO.
Pan, V. Ya. (October 1978), "Strassen's algorithm is not optimal: Trilinear technique of aggregating, uniting and canceling for constructing fast algorithms for matrix operations", Proceedings of the 19th Annual Symposium on Foundations of Computer Science (FOCS 1978), IEEE, doi:10.1109/sfcs.1978.34, S2CID 14348408
P82.
Pan, Victor Y. (1982), "Trilinear aggregating with implicit canceling for a new acceleration of matrix multiplication", Computers and Mathematics with Applications, 8: 23–34, doi:10.1016/0898-1221(82)90037-2, MR 0644547
FRM.
Huang, Xiaohan; Pan, Victor Y. (1998), "Fast rectangular matrix multiplication and applications", Journal of Complexity, 14 (2): 257–299, doi:10.1006/jcom.1998.0476, MR 1629113
MPD.
Mourrain, Bernard; Pan, Victor Y. (2000), "Multivariate polynomials, duality, and structured matrices" (PDF), Journal of Complexity, 16 (1): 110–180, doi:10.1006/jcom.1999.0530, MR 1762401 (winner, J. Complexity best paper award)[5]
UP.
Pan, Victor Y. (2002), "Univariate polynomials: nearly optimal algorithms for numerical factorization and root-finding", Journal of Symbolic Computation, 33 (5): 701–733, doi:10.1006/jsco.2002.0531, MR 1919911
Books
HMM.
Pan, Victor (1984), How to Multiply Matrices Faster, Lecture Notes in Computer Science, vol. 179, Berlin: Springer-Verlag, doi:10.1007/3-540-13866-8, ISBN 3-540-13866-8, S2CID 5280107[3]
PMC.
Bini, Dario; Pan, Victor Y. (1994), Polynomial and Matrix Computations, Vol. I: Fundamental Algorithms, Progress in Theoretical Computer Science, Boston, MA: Birkhäuser, doi:10.1007/978-1-4612-0265-3, ISBN 0-8176-3786-9, S2CID 30728536[6]
SMP.
Pan, Victor Y. (2001), Structured Matrices and Polynomials: Unified Superfast Algorithms, New York: Springer-Verlag, doi:10.1007/978-1-4612-0129-8, ISBN 0-8176-4240-4[7]
NMR.
McNamee, J. M.; Pan, V. Y. (2013), Numerical Methods for Roots of Polynomials, Part II, Studies in Computational Mathematics, vol. 16, Amsterdam: Elsevier/Academic Press, ISBN 978-0-444-52730-1[8]
References
1. Victor Pan at the Mathematics Genealogy Project
2. Victor Pan of Lehman mathematics faculty selected as Distinguished Professor, Lehman College, archived from the original on 2018-02-14
3. Reviews of How to Multiply Matrices Faster:
• Gladwell, Ian (1986), Mathematical Reviews, Lecture Notes in Computer Science, 179, doi:10.1007/3-540-13866-8, ISBN 978-3-540-13866-2, MR 0765701, S2CID 5280107{{citation}}: CS1 maint: untitled periodical (link)
• Coppersmith, Don (July 1986), SIAM Review, 28 (2): 250–252, doi:10.1137/1028072, JSTOR 2030488{{citation}}: CS1 maint: untitled periodical (link)
• Probert, Robert L. (November–December 1986), American Scientist, 74 (6): 682, JSTOR 27854420{{citation}}: CS1 maint: untitled periodical (link)
4. Karstadt, Elaye; Schwartz, Oded (2020), "Matrix multiplication, a little faster", Journal of the ACM, 67 (1): 1–31, doi:10.1145/3364504, MR 4061328, S2CID 211041916
5. "Best paper awards", Journal of Complexity, retrieved 2018-10-16
6. Reviews of Polynomial and Matrix Computations:
• Gupta, Murli M. (1995), Mathematical Reviews, doi:10.1007/978-1-4612-0265-3, ISBN 978-1-4612-6686-0, MR 1289412, S2CID 30728536{{citation}}: CS1 maint: untitled periodical (link)
• Tate, Stephen R. (June 1995), ACM SIGACT News, 26 (2): 26–27, doi:10.1145/202840.606473, S2CID 4740448{{citation}}: CS1 maint: untitled periodical (link)
• Eberly, Wayne (March 1996), SIAM Review, 38 (1): 161–165, doi:10.1137/1038020, JSTOR 2132983{{citation}}: CS1 maint: untitled periodical (link)
• Higham, Nicholas J. (April 1996), Mathematics of Computation, 65 (214): 888–889, JSTOR 2153629{{citation}}: CS1 maint: untitled periodical (link)
• Emiris, I. Z.; Galligo, A. (September 1996), ACM SIGSAM Bulletin, 30 (3): 21–23, doi:10.1145/240065.570109, S2CID 14598227{{citation}}: CS1 maint: untitled periodical (link)
7. Review of Structured Matrices and Polynomials:
• Melman, Aaron (2002), Mathematical Reviews, doi:10.1007/978-1-4612-0129-8, ISBN 978-1-4612-6625-9, MR 1843842{{citation}}: CS1 maint: untitled periodical (link)
8. Review of Numerical Methods for Roots of Polynomials, Part II:
• Proinov, Petko D., Mathematical Reviews, MR 3293902{{citation}}: CS1 maint: untitled periodical (link)
9. "List of Fellows of the American Mathematical Society", American Mathematical Society, retrieved 22 May 2015
External links
• Victor Pan publications indexed by Google Scholar
• Profile in American Scientist
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Victoria Howle
Victoria E. Howle is an American applied mathematician specializing in numerical linear algebra and known as one of the developers of the Trilinos open-source software library for scientific computing. She is an associate professor in the Department of Mathematics and Statistics at Texas Tech University.
Education and career
Howle graduated from Rutgers University in 1988 with a bachelor's degree in English literature.[1] She earned her Ph.D. in 2001 from Cornell University. Her dissertation, Efficient Iterative Methods for Ill-Conditioned Linear and Nonlinear Network Problems, was supervised by Stephen Vavasis.[2]
After working as a researcher at Sandia National Laboratories from 2000 to 2007, she took a faculty position at Texas Tech in 2007.[1]
Service and recognition
Howle was one of the inaugural winners of the AWM Service Award of the Association for Women in Mathematics, in 2013.[3][4] The award honored her service to the association, including founding its annual essay contest in which students write biographies of women mathematicians.[1][4]
References
1. Curriculum vitae (PDF), September 13, 2019, retrieved 2020-05-13
2. Victoria Howle at the Mathematics Genealogy Project
3. "AWM Service Award" (PDF), AWM Awards Given in San Diego, Notices of the American Mathematical Society, 60 (5): 616–617, May 2013
4. Association for Women in Mathematics Service Award 2013, Association for Women in Mathematics, retrieved 2020-05-13
External links
• Home page
• Victoria Howle publications indexed by Google Scholar
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Vicumpriya Perera
Vicumpriya Perera (Sinhala: විකුම්ප්රිය පෙරේරා) is a Sri Lankan born mathematician, lyricist, poet and music producer.[1][2] He has published three books of Sinhala poetry, Mekunu Satahan (Sinhala: මැකුනු සටහන්) in 2001,[3] Paa Satahan (Sinhala: පා සටහන්) in 2013, [4][5] and Mawbime Suwandha (Sinhala: මව්බිමේ සුවඳ) in 2023.[6] He has written over 200 songs and has produced eleven Sinhala song albums. He currently works as a mathematics professor in Ohio, US.
Vicumpriya Perera
Born19 February ????
Sri Lanka
NationalitySri Lankan
EducationAnanda College, St. Anthony's College, Wattala
Occupation(s)lyricist, poet, mathematician
Life and career
Vicumpriya Perera is originally from Wattala, Sri Lanka. He is a graduate of St. Anthony's College, Wattala and Ananda College, Maradana, Sri Lanka. He received a Bachelor of Science degree in Mathematics with first class honors from University of Colombo, Sri Lanka and continued his graduate studies at Indiana University - Purdue University at Indianapolis. He obtained a doctorate degree from Purdue University in Pure Mathematics with research concentrating on operator algebras and functional analysis in 1993.[7]
Vicumpriya Perera lives in Ohio, US, where he has worked as a mathematics professor at Kent State University (Trumbull campus) since 1998.[8][9] He works in operator algebra, which is an area of pure mathematics.
List of albums
The following is a list of the songs albums that Vicumpriya Perera has produced. Vicumpriya Perera was the sole composer of the lyrics of all of them.
TitleSinhalaYearNo. of songsSinger(s)Music directorsCitation(s)
Paa Satahanපා සටහන්200820Bhadraji Mahinda Jayatilaka, Praneeth Mash, Sumith Vanniarachchi, and Indeevari Abeywardena.Bhadraji Mahinda Jayatilaka, Shantha Gunaratne, and Lassana Jayasekara.[10]
Weli Aetayakවැලි ඇටයක්200918Nalin Jayawardena, with duet singers Santhuri Waidyasekera, Sangeeth Wickramasinghe, Ananda Waidyasekera, Nijamali Jayawardena, and Sanduni Rashmika.Sangeeth Wickramasinghe, Ananda Waidyasekera, Rohan Jayawardena, Rukshan Karunanayake, Jayanga Dedigama and Sanuka Wickramasinghe.[11]
Ukusu Esඋකුසු ඇස්201018Bhadraji Mahinda Jayatilaka, Nalin Jayawardena, Chimes of the Seventies, Vidarshana Kodagoda, Dammika Tissarachchi, Isuru Roshan, Anura Dias, Channa and Ravindra Kasturisinghe, Mahesh Fernando, Indeevari Abeywardena, and Praneeth Mash.Rohan Jayawardena, Sangeeth Wickramasinghe, Ananda Waidyasekera, Rukshan Karunanayake, and Shantha Gunaratne.[12]
Niwaadu Kaleනිවාඩු කාලේ201116Sanduni Rashmika, with duet singers Nalin Jayawardena, Santhuri Waidyasekera, and Jayanga Dedigama.Title song composed by Bhadraji Mahinda Jayatilaka and directed by Sangeeth Wickramasinghe. Other songs directed by Ananda Waidyasekera.[13]
Viduli Eliyakවිදුලි එලියක්201216Nilupuli Dilhara, with duet singers Keerthi Pasquel, Nimal Gunasekera, Nalin Jayawardena, and Sanduni Rashmika.Ananda Waidyasekera. Bhadraji Mahinda Jayatilaka composed music for one song in the album as well.[14]
Mal Renuwakමල් රේණුවක්201316Nalin Jayawardena, with duet singers Amilaa Nadeeshani (second runner-up of Sirasa Superstar season 2 (2007)), Rupa Indumathi, Bhadraji Mahinda Jayatilaka, Walter Fernando, and Thilini Athukorala.Rohana Weerasinghe, Navaratne Gamage, Sarath De Alwis, H M Jayawardena, Nimal Mendis, Ernest Soysa, Bhadraji Mahinda Jayatilaka, Rohan Jayawardena, Mervin Priyantha, Ananda Waidyasekera, Sangeeth Wickramasinghe and Rukshan Karunanayake.[15][16][17]
Siththaruwananiසිත්තරුවාණනි201416Nalin Jayawardena, with duet singer Nimanthi Chamodini (Sri Lankan reality musical show star)Sangeeth Wickramasinghe[18][19][20]
Indikalaa Pem Medurakඉඳිකලා පෙම් මැදුරක්201516Devananda Waidyasekera, Chandrakumar Kandanarachchi, Thyaga N Edward, Walter Fernando, Ajith Ariyarathna, Nalin Jayawardena, Athula Sri Gamage, Srilal Fonseka, and Praneeth Shiwanka PereraAnanda Waidyasekera[21]
Ithiri Giyaada Aadareඉතිරී ගියාද ආදරේ201616Nalin Jayawardena, Rohan Jayawardena, Dhammika Tissaarachchi, Minali Gamage, Amanda Perera, Sangeeth Wickramasinghe and Renuka WickramasingheRohan Jayawardena, Nimal Mendis, Bhadraji Mahinda Jayatilaka, Ananda Waidyasekera, Sangeeth Wickramasinghe, Rukshan Karunanayake, and Jayanga Dedigama[22]
Mawbime Suwandhaමව්බිමේ සුවඳ201814Sangeeth Nipun Professor Sanath Nandasiri, Visharada Edward Jayakody, Anuradha Nandasiri, Dayan Witharana, Swarnalatha Kaveeshwara, Visharada Charitha Priyadarshani Peiris, Visharada Sarath Peiris, Nirasha Ratnayake, Nadeesha Dayaratne, Visharada Harshana Disanayake, and Upendra PiyasenaVisharada Sarath Peiris[23]
Minpasu Aayeමින්පසු ආයේ202015Nalin Jayawardena, Shashika Srimali and Dhammika EdussooriyaDhammika Edussooriya[24]
Siththaruwanani included songs from the sinhala classical musical genre (sarala gee). Instrumentalists for this album consisted of Sri Lankan musicians Mahendra Pasquel, Sarath Fernando, Dhananjaya Somasiri, Janaka Bogoda, Susil Amarasinghe, Rohana Dharmakeerthi, Shelton Wijesekera, and Dilusha Ravindranath.[20]
Other productions
In 2005, Vicumpriya Perera (along with Nalin Jayawardena, and Jaanaka Wimaladharma) produced a compact disc set, Dhammapadaya (Sinhala: ධම්මපදය), under the Lanka Heritage label.[25] The set contained four discs, and consisted of complete the Dhammapada stanzas in the original Pali language followed by the Sinhala translations chanted by venerable Beruwala Siri Sobhitha Thero of the Sri Lanka Buddhist Vihara in Perth, Australia.[lower-alpha 1] In 2006, this disc set had an English release called Dhammapada.[25] This version had the original Dhammapada stanzas (again in Pali) followed by the English translations written and rendered by Dr. Gil Fronsdal,[27] director and resident teacher Insight Meditation Center, Redwood City, California, US.
In 2012 Vicumpriya Perera (in collaboration with Nalin Jayawardena) produced a Sinhala Audiobook called Kulageyin Kulageyata (Sinhala: කුලගෙයින් කුලගෙයට) under the Lanka Heritage, LLC. The book was written in 2009 by Bhadraji Mahinda Jayatilaka, who provided most of the voice work . The audiobook has a total length of five compact discs, and was published by Sarasavi Publishers,[28] Nugegoda, Sri Lanka.
Notes
1. Dhammapada is a widely read Buddhist scripture containing 423 pali verses spanning into 26 chapters called varga.[26]
References
1. "Digital legacy of Sinhala songs". Sunday Observer. Retrieved 5 December 2017.
2. "Vicumpriya Perera – mathematician, translator and electronic recorder". Sunday Island. Retrieved 2 December 2013.
3. "Mekunu Satahan ebook". Retrieved 9 December 2013.
4. Perera, Vicumpriya (2013). Paa Satahan. Nugegoda, Sri Lanka: Sarasavi Publishers. ISBN 9789556717921.
5. "A homeland in cyberspace". Sunday Observer. Retrieved 5 December 2017.
6. "Mawbime Suwandha". Retrieved 9 April 2023.
7. Perera, Vicumpriya (1993). Real Valued Spectral Flow in a Type II-[infinity] Factor. Purdue University.
8. "Vic Perera". Retrieved 5 December 2017.
9. "Vicumpriya Perera". Retrieved 6 December 2017.
10. Paa Satahan – Bhadraji Mahinda Jayatilaka
11. "Weli Aetayak': Nalin and Vicumpriya's joint venture". Sunday Times. Retrieved 2 December 2013.
12. Ukusu Es
13. Niwadu Kaale – Sanduni Rashmika
14. Viduli Eliyak – Nilupuli Dilhara
15. Mal Renuwak – Nalin Jayawardena
16. Thilakarathne, Indeewara. "Depicting life through songs". Ceylon Today. Retrieved 2 December 2013.
17. Madugalle, Dushyantha. "Promoting Lankan culture in and out of diaspora". Sunday Observer. Retrieved 20 December 2013.
18. Siththaruwanani – Nalin Jayawardena
19. Withanachchi, Thinani. "Nalin in Sri Lanka to release 11th CD". Sarasaviya. Retrieved 12 September 2014.
20. "Voice from Australia, Lyrics from America and Music from Sri Lanka". Sarasaviya. Retrieved 12 September 2014.
21. Vicumpriya Perera Lyrics, Vol. 8: Indikalaa Pem Medurak
22. Vicumpriya Perera Lyrics 9: Ithiri Giyaada Aadare
23. Lyrical Compositions of Dr. Vicumpriya Perera, Vol.10: Fragrances Of The Motherland
24. Minpasu Aaye - Lyrical Compositions of Vicumpriya Perera 11
25. "Dhammapada at Lanka Heritage website". Retrieved 2 December 2013.
26. Müller, F. Max (1881). The Dhammapada (Sacred Books of the East, Vol. X). Oxford University Press.
27. Gil Fronsdal, and Jack Kornfield (foreword) (2005). The Dhammapada: A New Translation of the Buddhist Classic with Annotations, Boston: Shambhala. ISBN 1-59030-211-7.
28. "Sarasavi Prakashakayo". Retrieved 2 December 2013.
External links
• Lyrics of Vicumpriya Perera
• Vicumpriya Perera music on Google Play
• Home Page of Insight Meditation Center of Redwood City, CA
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
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Vidyadhar P. Godambe
Vidyadhar Prabhakar Godambe FRSC (1 June 1926 – 9 June 2016) was an Indian statistician. He was a Distinguished Professor Emeritus at the University of Waterloo. Godambe was known for formulating and developing a theory of estimating equations.
Vidyadhar P. Godambe
Born(1926-06-01)1 June 1926
Pune, India
Died9 June 2016(2016-06-09) (aged 90)
Known forsurvey sampling
estimating equations
Academic background
EducationB.Sc., Fergusson College
M.Sc., Statistics, 1950, Bombay University
PhD., 1958, University of London
ThesisRobust and non-parametric inference and other general criteria for statistical decisions (1958)
Doctoral advisorGeorge Alfred Barnard
Academic work
InstitutionsDominion Bureau of Statistics
Rashtrasant Tukadoji Maharaj Nagpur University
Bombay University
Johns Hopkins University
University of Michigan
University of Waterloo
Early life and education
Godambe was born on 1 June 1926, in Pune, India as the second oldest of four children.[1] He was frail and sickly growing up so he attended the local school from age five to 10.[2] Godambe later attended Nutan Marathi Vidyalaya in Pune and Fergusson College for his Bachelor of Science in mathematics.[1]
After earning his Master's degree, Godambe accepted a position in the Bureau of Economics and Statistics with the Government of Bombay. While there, he submitted papers for publication in the Journal of the Royal Statistical Society and Bulletin of the Bureau of Economics and Statistics, Bombay.[2] Godambe shortly thereafter left Bombay to pursue a PhD at the University of London, and accepted a fellowship at the University of California, Berkeley. Upon his return, and completion of his thesis, Godambe was appointed a Senior Research Fellow at the Indian Statistical Institute and Professor and Head of the Statistics Department in Nagpur.[1]
Career
Godambe eventually left Nagpur and accepted a position at Bombay University as a professor for one year. He then moved to North America and worked at the Dominion Bureau of Statistics, alongside Ivan Fellegi, then taught at Johns Hopkins University, University of Michigan, and finally the University of Waterloo.[3] Godambe began his career at the University of Waterloo in July 1967 as a visiting professor in Statistics and Actuarial Sciences but was granted tenure as Professor in July 1969.[4] A few years later, in 1971, Godambe and Mary Thompson read a paper to the Royal Statistical Society entitled ‘Bayes, fiducial, and frequency aspects of statistical inference in survey sampling.[3]
From there, "Godambe’s paradox" was invented. Based on the paper published in the Journal of the Royal Statistical Society, he demonstrated that the likelihood principle implies that inference should be independent of the sampling design in general, which led to the development of model theory in survey sampling.[3] His method of estimating equations argued that all statistical inferences should adhere to his "ancillarity principle."[5]
Awards and honours
In 1987, Godambe was honoured with the Statistical Society of Canada (SSC) Gold Medal[6] and was later named an honorary member.[7] In 1991, he was appointed a Distinguished Professor Emeritus at the University of Waterloo.[8] In 2002, Godambe was elected a Fellow of the Royal Society of Canada.[9][10]
References
1. "OBITUARY:V.P. Godambe, 1926–2016". bulletin.imstat.org. Retrieved 2 December 2019.
2. Thompson, Mary E. (November 2002). "A Conversation with V. P. Godambe". Statistical Science. 17 (4): 458–466. doi:10.1214/ss/1049993204. JSTOR 3182767.
3. "A Conversation with V.P. Godambe". ssc.ca. 12 March 2010. Retrieved 2 December 2019.
4. "Remembering Vidyadhar Godambe; other notes". uwaterloo.ca. 5 July 2016. Retrieved 2 December 2019.
5. Christian Genest; Mark J. Schervish (December 1985). "Resolution of Godambe's Paradox". The Canadian Journal of Statistics. 13 (4): 293–297. doi:10.2307/3314949. JSTOR 3314949.
6. "Vidyadhar Prabhakar Godambe, SSC Gold Medalist 1987". ssc.ca. Retrieved 2 December 2019.
7. "Vidyadhar Prabhakar Godambe, Honorary Member 2001". ssc.ca. Retrieved 2 December 2019.
8. "1960 - 1999". uwaterloo.ca. Retrieved 2 December 2019.
9. "Notices of the American Mathematical Society" (PDF). ams.org. December 2002. p. 1399. Retrieved 2 December 2019.
10. "Two named to Royal Society of Canada". bulletin.uwaterloo.ca. 28 June 2002. Retrieved 2 December 2019.
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Viennot's geometric construction
In mathematics, Viennot's geometric construction (named after Xavier Gérard Viennot) gives a diagrammatic interpretation of the Robinson–Schensted correspondence in terms of shadow lines. It has a generalization to the Robinson–Schensted–Knuth correspondence, which is known as the matrix-ball construction.
The construction
Starting with a permutation $\sigma \in S_{n}$, written in two-line notation, say:
$\sigma ={\begin{pmatrix}1&2&\cdots &n\\\sigma _{1}&\sigma _{2}&\cdots &\sigma _{n}\end{pmatrix}},$
one can apply the Robinson–Schensted correspondence to this permutation, yielding two standard Young tableaux of the same shape, P and Q. P is obtained by performing a sequence of insertions, and Q is the recording tableau, indicating in which order the boxes were filled.
Viennot's construction starts by plotting the points $(i,\sigma _{i})$ in the plane, and imagining there is a light that shines from the origin, casting shadows straight up and to the right. This allows consideration of the points which are not shadowed by any other point; the boundary of their shadows then forms the first shadow line. Removing these points and repeating the procedure, one obtains all the shadow lines for this permutation. Viennot's insight is then that these shadow lines read off the first rows of P and Q (in fact, even more than that; these shadow lines form a "timeline", indicating which elements formed the first rows of P and Q after the successive insertions). One can then repeat the construction, using as new points the previous unlabelled corners, which allows to read off the other rows of P and Q.
Animation
For example consider the permutation
$\sigma ={\begin{pmatrix}1&2&3&4&5&6&7&8\\3&8&1&2&4&7&5&6\end{pmatrix}}.$
Then Viennot's construction goes as follows:
Applications
One can use Viennot's geometric construction to prove that if $\sigma $ corresponds to the pair of tableaux P,Q under the Robinson–Schensted correspondence, then $\sigma ^{-1}$ corresponds to the switched pair Q,P. Indeed, taking $\sigma $ to $\sigma ^{-1}$ reflects Viennot's construction in the $y=x$-axis, and this precisely switches the roles of P and Q.
See also
• Plactic monoid
• Jeu de taquin
References
• Bruce E. Sagan. The Symmetric Group. Springer, 2001.
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Viète's formula
In mathematics, Viète's formula is the following infinite product of nested radicals representing twice the reciprocal of the mathematical constant π:
${\frac {2}{\pi }}={\frac {\sqrt {2}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2}}}}{2}}\cdot {\frac {\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}{2}}\cdots $
This article is about a formula for π. For formulas for symmetric functions of the roots, see Vieta's formulas.
It can also be represented as:
${\frac {2}{\pi }}=\prod _{n=1}^{\infty }\cos {\frac {\pi }{2^{n+1}}}$
The formula is named after François Viète, who published it in 1593.[1] As the first formula of European mathematics to represent an infinite process,[2] it can be given a rigorous meaning as a limit expression,[3] and marks the beginning of mathematical analysis. It has linear convergence, and can be used for calculations of π,[4] but other methods before and since have led to greater accuracy. It has also been used in calculations of the behavior of systems of springs and masses,[5] and as a motivating example for the concept of statistical independence.
The formula can be derived as a telescoping product of either the areas or perimeters of nested polygons converging to a circle. Alternatively, repeated use of the half-angle formula from trigonometry leads to a generalized formula, discovered by Leonhard Euler, that has Viète's formula as a special case. Many similar formulas involving nested roots or infinite products are now known.
Significance
François Viète (1540–1603) was a French lawyer, privy councillor to two French kings, and amateur mathematician. He published this formula in 1593 in his work Variorum de rebus mathematicis responsorum, liber VIII. At this time, methods for approximating π to (in principle) arbitrary accuracy had long been known. Viète's own method can be interpreted as a variation of an idea of Archimedes of approximating the circumference of a circle by the perimeter of a many-sided polygon,[1] used by Archimedes to find the approximation[6]
${\frac {223}{71}}<\pi <{\frac {22}{7}}.$
By publishing his method as a mathematical formula, Viète formulated the first instance of an infinite product known in mathematics,[7][8] and the first example of an explicit formula for the exact value of π.[9][10] As the first representation in European mathematics of a number as the result of an infinite process rather than of a finite calculation,[11] Eli Maor highlights Viète's formula as marking the beginning of mathematical analysis[2] and Jonathan Borwein calls its appearance "the dawn of modern mathematics".[12]
Using his formula, Viète calculated π to an accuracy of nine decimal digits.[4] However, this was not the most accurate approximation to π known at the time, as the Persian mathematician Jamshīd al-Kāshī had calculated π to an accuracy of nine sexagesimal digits and 16 decimal digits in 1424.[12] Not long after Viète published his formula, Ludolph van Ceulen used a method closely related to Viète's to calculate 35 digits of π, which were published only after van Ceulen's death in 1610.[12]
Beyond its mathematical and historical significance, Viète's formula can be used to explain the different speeds of waves of different frequencies in an infinite chain of springs and masses, and the appearance of π in the limiting behavior of these speeds.[5] Additionally, a derivation of this formula as a product of integrals involving the Rademacher system, equal to the integral of products of the same functions, provides a motivating example for the concept of statistical independence.[13]
Interpretation and convergence
Viète's formula may be rewritten and understood as a limit expression[3]
$\lim _{n\rightarrow \infty }\prod _{i=1}^{n}{\frac {a_{i}}{2}}={\frac {2}{\pi }}$
where
${\begin{aligned}a_{1}&={\sqrt {2}}\\a_{n}&={\sqrt {2+a_{n-1}}}.\end{aligned}}$
For each choice of $n$, the expression in the limit is a finite product, and as $n$ gets arbitrarily large these finite products have values that approach the value of Viète's formula arbitrarily closely. Viète did his work long before the concepts of limits and rigorous proofs of convergence were developed in mathematics; the first proof that this limit exists was not given until the work of Ferdinand Rudio in 1891.[1][14]
The rate of convergence of a limit governs the number of terms of the expression needed to achieve a given number of digits of accuracy. In Viète's formula, the numbers of terms and digits are proportional to each other: the product of the first n terms in the limit gives an expression for π that is accurate to approximately 0.6n digits.[4][15] This convergence rate compares very favorably with the Wallis product, a later infinite product formula for π. Although Viète himself used his formula to calculate π only with nine-digit accuracy, an accelerated version of his formula has been used to calculate π to hundreds of thousands of digits.[4]
Related formulas
Viète's formula may be obtained as a special case of a formula for the sinc function that has often been attributed to Leonhard Euler[16], more than a century later:[1]
${\frac {\sin x}{x}}=\cos {\frac {x}{2}}\cdot \cos {\frac {x}{4}}\cdot \cos {\frac {x}{8}}\cdots $
Substituting x = π/2 in this formula yields:[17]
${\frac {2}{\pi }}=\cos {\frac {\pi }{4}}\cdot \cos {\frac {\pi }{8}}\cdot \cos {\frac {\pi }{16}}\cdots $
Then, expressing each term of the product on the right as a function of earlier terms using the half-angle formula:
$\cos {\frac {x}{2}}={\sqrt {\frac {1+\cos x}{2}}}$
gives Viète's formula.[9]
It is also possible to derive from Viète's formula a related formula for π that still involves nested square roots of two, but uses only one multiplication:[18]
$\pi =\lim _{k\to \infty }2^{k}\underbrace {\sqrt {2-{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2+\cdots +{\sqrt {2}}}}}}}}}}}} _{k{\text{ square roots}}},$
which can be rewritten compactly as
${\begin{aligned}\pi &=\lim _{k\to \infty }2^{k}{\sqrt {2-a_{k}}}\\[5px]a_{1}&=0\\a_{k}&={\sqrt {2+a_{k-1}}}.\end{aligned}}$
Many formulae for π and other constants such as the golden ratio are now known, similar to Viète's in their use of either nested radicals or infinite products of trigonometric functions.[8][18][19][20][21][22][23][24]
Derivation
Viète obtained his formula by comparing the areas of regular polygons with 2n and 2n + 1 sides inscribed in a circle.[1][2] The first term in the product, √2/2, is the ratio of areas of a square and an octagon, the second term is the ratio of areas of an octagon and a hexadecagon, etc. Thus, the product telescopes to give the ratio of areas of a square (the initial polygon in the sequence) to a circle (the limiting case of a 2n-gon). Alternatively, the terms in the product may be instead interpreted as ratios of perimeters of the same sequence of polygons, starting with the ratio of perimeters of a digon (the diameter of the circle, counted twice) and a square, the ratio of perimeters of a square and an octagon, etc.[25]
Another derivation is possible based on trigonometric identities and Euler's formula. Repeatedly applying the double-angle formula
$\sin x=2\sin {\frac {x}{2}}\cos {\frac {x}{2}},$
leads to a proof by mathematical induction that, for all positive integers n,
$\sin x=2^{n}\sin {\frac {x}{2^{n}}}\left(\prod _{i=1}^{n}\cos {\frac {x}{2^{i}}}\right).$
The term 2n sin x/2n goes to x in the limit as n goes to infinity, from which Euler's formula follows. Viète's formula may be obtained from this formula by the substitution x = π/2.[9][13]
References
1. Beckmann, Petr (1971). A History of π (2nd ed.). Boulder, Colorado: The Golem Press. pp. 94–95. ISBN 978-0-88029-418-8. MR 0449960.
2. Maor, Eli (2011). Trigonometric Delights. Princeton, New Jersey: Princeton University Press. pp. 50, 140. ISBN 978-1-4008-4282-7.
3. Eymard, Pierre; Lafon, Jean Pierre (2004). "2.1 Viète's infinite product". The Number pi. Translated by Wilson, Stephen S. Providence, Rhode Island: American Mathematical Society. pp. 44–46. ISBN 978-0-8218-3246-2. MR 2036595.
4. Kreminski, Rick (2008). "π to thousands of digits from Vieta's formula". Mathematics Magazine. 81 (3): 201–207. doi:10.1080/0025570X.2008.11953549. JSTOR 27643107. S2CID 125362227.
5. Cullerne, J. P.; Goekjian, M. C. Dunn (December 2011). "Teaching wave propagation and the emergence of Viète's formula". Physics Education. 47 (1): 87–91. doi:10.1088/0031-9120/47/1/87. S2CID 122368450.
6. Beckmann 1971, p. 67.
7. De Smith, Michael J. (2006). Maths for the Mystified: An Exploration of the History of Mathematics and Its Relationship to Modern-day Science and Computing. Leicester: Matador. p. 165. ISBN 978-1905237-81-4.
8. Moreno, Samuel G.; García-Caballero, Esther M. (2013). "On Viète-like formulas". Journal of Approximation Theory. 174: 90–112. doi:10.1016/j.jat.2013.06.006. MR 3090772.
9. Morrison, Kent E. (1995). "Cosine products, Fourier transforms, and random sums". The American Mathematical Monthly. 102 (8): 716–724. arXiv:math/0411380. doi:10.2307/2974641. JSTOR 2974641. MR 1357488.
10. Oldham, Keith B.; Myland, Jan C.; Spanier, Jerome (2010). An Atlas of Functions: with Equator, the Atlas Function Calculator. New York: Springer. p. 15. doi:10.1007/978-0-387-48807-3. ISBN 978-0-387-48807-3.
11. Very similar infinite trigonometric series for $\pi $ appeared earlier in Indian mathematics, in the work of Madhava of Sangamagrama (c. 1340 – 1425), but were not known in Europe until much later. See: Plofker, Kim (2009). "7.3.1 Mādhava on the circumference and arcs of the circle". Mathematics in India. Princeton, New Jersey: Princeton University Press. pp. 221–234. ISBN 978-0-691-12067-6.
12. Borwein, Jonathan M. (2013). "The life of Pi: From Archimedes to ENIAC and beyond" (PDF). In Sidoli, Nathan; Van Brummelen, Glen (eds.). From Alexandria, Through Baghdad: Surveys and Studies in the Ancient Greek and Medieval Islamic Mathematical Sciences in Honor of J. L. Berggren. Berlin & Heidelberg: Springer. pp. 531–561. doi:10.1007/978-3-642-36736-6_24. ISBN 978-3-642-36735-9.
13. Kac, Mark (1959). "Chapter 1: From Vieta to the notion of statistical independence". Statistical Independence in Probability, Analysis and Number Theory. Carus Mathematical Monographs. Vol. 12. New York: John Wiley & Sons for the Mathematical Association of America. pp. 1–12. MR 0110114.
14. Rudio, F. (1891). "Ueber die Convergenz einer von Vieta herrührenden eigentümlichen Produktentwicklung" [On the convergence of a special product expansion due to Vieta]. Historisch-litterarische Abteilung der Zeitschrift für Mathematik und Physik (in German). 36: 139–140. JFM 23.0263.02.
15. Osler, Thomas J. (2007). "A simple geometric method of estimating the error in using Vieta's product for π". International Journal of Mathematical Education in Science and Technology. 38 (1): 136–142. doi:10.1080/00207390601002799. S2CID 120145020.
16. Euler, Leonhard (1738). "De variis modis circuli quadraturam numeris proxime exprimendi" [On various methods for expressing the quadrature of a circle with verging numbers]. Commentarii Academiae Scientiarum Petropolitanae (in Latin). 9: 222–236. Translated into English by Thomas W. Polaski. See final formula. The same formula is also in Euler, Leonhard (1783). "Variae observationes circa angulos in progressione geometrica progredientes" [Various observations about angles proceeding in geometric progression]. Opuscula Analytica (in Latin). 1: 345–352. Translated into English by Jordan Bell, arXiv:1009.1439. See the formula in numbered paragraph 3.
17. Wilson, Robin J. (2018). Euler's pioneering equation: the most beautiful theorem in mathematics (PDF) (First ed.). Oxford, United Kingdom. pp. 57–58. ISBN 9780198794929.{{cite book}}: CS1 maint: location missing publisher (link)
18. Servi, L. D. (2003). "Nested square roots of 2". The American Mathematical Monthly. 110 (4): 326–330. doi:10.2307/3647881. JSTOR 3647881. MR 1984573.
19. Nyblom, M. A. (2012). "Some closed-form evaluations of infinite products involving nested radicals". Rocky Mountain Journal of Mathematics. 42 (2): 751–758. doi:10.1216/RMJ-2012-42-2-751. MR 2915517.
20. Levin, Aaron (2006). "A geometric interpretation of an infinite product for the lemniscate constant". The American Mathematical Monthly. 113 (6): 510–520. doi:10.2307/27641976. JSTOR 27641976. MR 2231136.
21. Levin, Aaron (2005). "A new class of infinite products generalizing Viète's product formula for π". The Ramanujan Journal. 10 (3): 305–324. doi:10.1007/s11139-005-4852-z. MR 2193382. S2CID 123023282.
22. Osler, Thomas J. (2007). "Vieta-like products of nested radicals with Fibonacci and Lucas numbers". Fibonacci Quarterly. 45 (3): 202–204. MR 2437033.
23. Stolarsky, Kenneth B. (1980). "Mapping properties, growth, and uniqueness of Vieta (infinite cosine) products". Pacific Journal of Mathematics. 89 (1): 209–227. doi:10.2140/pjm.1980.89.209. MR 0596932.
24. Allen, Edward J. (1985). "Continued radicals". The Mathematical Gazette. 69 (450): 261–263. doi:10.2307/3617569. JSTOR 3617569. S2CID 250441699.
25. Rummler, Hansklaus (1993). "Squaring the circle with holes". The American Mathematical Monthly. 100 (9): 858–860. doi:10.2307/2324662. JSTOR 2324662. MR 1247533.
External links
• Viète's Variorum de rebus mathematicis responsorum, liber VIII (1593) on Google Books. The formula is on the second half of p. 30.
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Vieta's formulas
In mathematics, Vieta's formulas relate the coefficients of a polynomial to sums and products of its roots. They are named after François Viète (more commonly referred to by the Latinised form of his name, "Franciscus Vieta").
For a method for computing π, see Viète's formula.
Basic formulas
Any general polynomial of degree n
$P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}$
(with the coefficients being real or complex numbers and an ≠ 0) has n (not necessarily distinct) complex roots r1, r2, ..., rn by the fundamental theorem of algebra. Vieta's formulas relate the polynomial's coefficients to signed sums of products of the roots r1, r2, ..., rn as follows:
${\begin{cases}r_{1}+r_{2}+\dots +r_{n-1}+r_{n}=-{\dfrac {a_{n-1}}{a_{n}}}\\[1ex](r_{1}r_{2}+r_{1}r_{3}+\cdots +r_{1}r_{n})+(r_{2}r_{3}+r_{2}r_{4}+\cdots +r_{2}r_{n})+\cdots +r_{n-1}r_{n}={\dfrac {a_{n-2}}{a_{n}}}\\[1ex]{}\quad \vdots \\[1ex]r_{1}r_{2}\cdots r_{n}=(-1)^{n}{\dfrac {a_{0}}{a_{n}}}.\end{cases}}$
(*)
Vieta's formulas can equivalently be written as
$\sum _{1\leq i_{1}<i_{2}<\cdots <i_{k}\leq n}\left(\prod _{j=1}^{k}r_{i_{j}}\right)=(-1)^{k}{\frac {a_{n-k}}{a_{n}}}$
for k = 1, 2, ..., n (the indices ik are sorted in increasing order to ensure each product of k roots is used exactly once).
The left-hand sides of Vieta's formulas are the elementary symmetric polynomials of the roots.
Vieta's system (*) can be solved by Newton's method through an explicit simple iterative formula, the Durand-Kerner method.
Generalization to rings
Vieta's formulas are frequently used with polynomials with coefficients in any integral domain R. Then, the quotients $a_{i}/a_{n}$ belong to the field of fractions of R (and possibly are in R itself if $a_{n}$ happens to be invertible in R) and the roots $r_{i}$ are taken in an algebraically closed extension. Typically, R is the ring of the integers, the field of fractions is the field of the rational numbers and the algebraically closed field is the field of the complex numbers.
Vieta's formulas are then useful because they provide relations between the roots without having to compute them.
For polynomials over a commutative ring that is not an integral domain, Vieta's formulas are only valid when $a_{n}$ is not a zero-divisor and $P(x)$ factors as $a_{n}(x-r_{1})(x-r_{2})\dots (x-r_{n})$. For example, in the ring of the integers modulo 8, the quadratic polynomial $P(x)=x^{2}-1$ has four roots: 1, 3, 5, and 7. Vieta's formulas are not true if, say, $r_{1}=1$ and $r_{2}=3$, because $P(x)\neq (x-1)(x-3)$. However, $P(x)$ does factor as $(x-1)(x-7)$ and also as $(x-3)(x-5)$, and Vieta's formulas hold if we set either $r_{1}=1$ and $r_{2}=7$ or $r_{1}=3$ and $r_{2}=5$.
Example
Vieta's formulas applied to quadratic and cubic polynomials:
The roots $r_{1},r_{2}$ of the quadratic polynomial $P(x)=ax^{2}+bx+c$ satisfy
$r_{1}+r_{2}=-{\frac {b}{a}},\quad r_{1}r_{2}={\frac {c}{a}}.$
The first of these equations can be used to find the minimum (or maximum) of P; see Quadratic equation § Vieta's formulas.
The roots $r_{1},r_{2},r_{3}$ of the cubic polynomial $P(x)=ax^{3}+bx^{2}+cx+d$ satisfy
$r_{1}+r_{2}+r_{3}=-{\frac {b}{a}},\quad r_{1}r_{2}+r_{1}r_{3}+r_{2}r_{3}={\frac {c}{a}},\quad r_{1}r_{2}r_{3}=-{\frac {d}{a}}.$
Proof
Vieta's formulas can be proved by expanding the equality
$a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots +a_{1}x+a_{0}=a_{n}(x-r_{1})(x-r_{2})\cdots (x-r_{n})$
(which is true since $r_{1},r_{2},\dots ,r_{n}$ are all the roots of this polynomial), multiplying the factors on the right-hand side, and identifying the coefficients of each power of $x.$
Formally, if one expands $(x-r_{1})(x-r_{2})\cdots (x-r_{n}),$ the terms are precisely $(-1)^{n-k}r_{1}^{b_{1}}\cdots r_{n}^{b_{n}}x^{k},$ where $b_{i}$ is either 0 or 1, accordingly as whether $r_{i}$ is included in the product or not, and k is the number of $r_{i}$ that are included, so the total number of factors in the product is n (counting $x^{k}$ with multiplicity k) – as there are n binary choices (include $r_{i}$ or x), there are $2^{n}$ terms – geometrically, these can be understood as the vertices of a hypercube. Grouping these terms by degree yields the elementary symmetric polynomials in $r_{i}$ – for xk, all distinct k-fold products of $r_{i}.$
As an example, consider the quadratic
$f(x)=a_{2}x^{2}+a_{1}x+a_{0}=a_{2}(x-r_{1})(x-r_{2})=a_{2}(x^{2}-x(r_{1}+r_{2})+r_{1}r_{2}).$
Comparing identical powers of $x$, we find $a_{2}=a_{2}$, $a_{1}=-a_{2}(r_{1}+r_{2})$ and $a_{0}=a_{2}(r_{1}r_{2})$, with which we can for example identify $r_{1}+r_{2}=-a_{1}/a_{2}$ and $r_{1}r_{2}=a_{0}/a_{2}$, which are Vieta's formula's for $n=2$.
History
As reflected in the name, the formulas were discovered by the 16th-century French mathematician François Viète, for the case of positive roots.
In the opinion of the 18th-century British mathematician Charles Hutton, as quoted by Funkhouser,[1] the general principle (not restricted to positive real roots) was first understood by the 17th-century French mathematician Albert Girard:
...[Girard was] the first person who understood the general doctrine of the formation of the coefficients of the powers from the sum of the roots and their products. He was the first who discovered the rules for summing the powers of the roots of any equation.
See also
• Content (algebra)
• Descartes' rule of signs
• Newton's identities
• Gauss–Lucas theorem
• Properties of polynomial roots
• Rational root theorem
• Symmetric polynomial and elementary symmetric polynomial
References
1. (Funkhouser 1930)
• "Viète theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Funkhouser, H. Gray (1930), "A short account of the history of symmetric functions of roots of equations", American Mathematical Monthly, Mathematical Association of America, 37 (7): 357–365, doi:10.2307/2299273, JSTOR 2299273
• Vinberg, E. B. (2003), A course in algebra, American Mathematical Society, Providence, R.I, ISBN 0-8218-3413-4
• Djukić, Dušan; et al. (2006), The IMO compendium: a collection of problems suggested for the International Mathematical Olympiads, 1959–2004, Springer, New York, NY, ISBN 0-387-24299-6
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François Viète
François Viète, Seigneur de la Bigotière (Latin: Franciscus Vieta; 1540 – 23 February 1603), commonly known by his mononym, Vieta, was a French mathematician whose work on new algebra was an important step towards modern algebra, due to his innovative use of letters as parameters in equations. He was a lawyer by trade, and served as a privy councillor to both Henry III and Henry IV of France.
François Viète
Born1540
Fontenay-le-Comte, Kingdom of France
Died23 February 1603 (aged 62–63)
Paris, Kingdom of France
NationalityFrench
Other namesFranciscus Vieta
EducationUniversity of Poitiers
(LL.B., 1559)
Known forNew algebra (the first symbolic algebra)
Vieta's formulas
Viète's formula
Scientific career
FieldsAstronomy, mathematics (algebra and trigonometry)
Notable studentsAlexander Anderson
InfluencesPeter Ramus
Gerolamo Cardano[1]
InfluencedPierre de Fermat
René Descartes[2]
Signature
Biography
Early life and education
Viète was born at Fontenay-le-Comte in present-day Vendée. His grandfather was a merchant from La Rochelle. His father, Etienne Viète, was an attorney in Fontenay-le-Comte and a notary in Le Busseau. His mother was the aunt of Barnabé Brisson, a magistrate and the first president of parliament during the ascendancy of the Catholic League of France.
Viète went to a Franciscan school and in 1558 studied law at Poitiers, graduating as a Bachelor of Laws in 1559. A year later, he began his career as an attorney in his native town.[3] From the outset, he was entrusted with some major cases, including the settlement of rent in Poitou for the widow of King Francis I of France and looking after the interests of Mary, Queen of Scots.
Serving Parthenay
In 1564, Viète entered the service of Antoinette d'Aubeterre, Lady Soubise, wife of Jean V de Parthenay-Soubise, one of the main Huguenot military leaders and accompanied him to Lyon to collect documents about his heroic defence of that city against the troops of Jacques of Savoy, 2nd Duke of Nemours just the year before.
The same year, at Parc-Soubise, in the commune of Mouchamps in present-day Vendée, Viète became the tutor of Catherine de Parthenay, Soubise's twelve-year-old daughter. He taught her science and mathematics and wrote for her numerous treatises on astronomy and trigonometry, some of which have survived. In these treatises, Viète used decimal numbers (twenty years before Stevin's paper) and he also noted the elliptic orbit of the planets,[4] forty years before Kepler and twenty years before Giordano Bruno's death.
John V de Parthenay presented him to King Charles IX of France. Viète wrote a genealogy of the Parthenay family and following the death of Jean V de Parthenay-Soubise in 1566 his biography.
In 1568, Antoinette, Lady Soubise, married her daughter Catherine to Baron Charles de Quellenec and Viète went with Lady Soubise to La Rochelle, where he mixed with the highest Calvinist aristocracy, leaders like Coligny and Condé and Queen Jeanne d’Albret of Navarre and her son, Henry of Navarre, the future Henry IV of France.
In 1570, he refused to represent the Soubise ladies in their infamous lawsuit against the Baron De Quellenec, where they claimed the Baron was unable (or unwilling) to provide an heir.
First steps in Paris
In 1571, he enrolled as an attorney in Paris, and continued to visit his student Catherine. He regularly lived in Fontenay-le-Comte, where he took on some municipal functions. He began publishing his Universalium inspectionum ad Canonem mathematicum liber singularis and wrote new mathematical research by night or during periods of leisure. He was known to dwell on any one question for up to three days, his elbow on the desk, feeding himself without changing position (according to his friend, Jacques de Thou).[5]
In 1572, Viète was in Paris during the St. Bartholomew's Day massacre. That night, Baron De Quellenec was killed after having tried to save Admiral Coligny the previous night. The same year, Viète met Françoise de Rohan, Lady of Garnache, and became her adviser against Jacques, Duke of Nemours.
In 1573, he became a councillor of the Parliament of Brittany, at Rennes, and two years later, he obtained the agreement of Antoinette d'Aubeterre for the marriage of Catherine of Parthenay to Duke René de Rohan, Françoise's brother.
In 1576, Henri, duc de Rohan took him under his special protection, recommending him in 1580 as "maître des requêtes". In 1579, Viète finished the printing of his Universalium inspectionum (Mettayer publisher), published as an appendix to a book of two trigonometric tables (Canon mathematicus, seu ad triangula, the "canon" referred to by the title of his Universalium inspectionum, and Canonion triangulorum laterum rationalium). A year later, he was appointed maître des requêtes to the parliament of Paris, committed to serving the king. That same year, his success in the trial between the Duke of Nemours and Françoise de Rohan, to the benefit of the latter, earned him the resentment of the tenacious Catholic League.
Exile in Fontenay
Between 1583 and 1585, the League persuaded Henry III to release Viète, Viète having been accused of sympathy with the Protestant cause. Henry of Navarre, at Rohan's instigation, addressed two letters to King Henry III of France on March 3 and April 26, 1585, in an attempt to obtain Viète's restoration to his former office, but he failed.[3]
Viète retired to Fontenay and Beauvoir-sur-Mer, with François de Rohan. He spent four years devoted to mathematics, writing his New Algebra (1591).
Code-breaker to two kings
In 1589, Henry III took refuge in Blois. He commanded the royal officials to be at Tours before 15 April 1589. Viète was one of the first who came back to Tours. He deciphered the secret letters of the Catholic League and other enemies of the king. Later, he had arguments with the classical scholar Joseph Juste Scaliger. Viète triumphed against him in 1590.
After the death of Henry III, Viète became a privy councillor to Henry of Navarre, now Henry IV.[6]: 75–77 He was appreciated by the king, who admired his mathematical talents. Viète was given the position of councillor of the parlement at Tours. In 1590, Viète discovered the key to a Spanish cipher, consisting of more than 500 characters, and this meant that all dispatches in that language which fell into the hands of the French could be easily read.[7]
Henry IV published a letter from Commander Moreo to the King of Spain. The contents of this letter, read by Viète, revealed that the head of the League in France, Charles, Duke of Mayenne, planned to become king in place of Henry IV. This publication led to the settlement of the Wars of Religion. The King of Spain accused Viète of having used magical powers. In 1593, Viète published his arguments against Scaliger. Beginning in 1594, he was appointed exclusively deciphering the enemy's secret codes.
Gregorian calendar
In 1582, Pope Gregory XIII published his bull Inter gravissimas and ordered Catholic kings to comply with the change from the Julian calendar, based on the calculations of the Calabrian doctor Aloysius Lilius, aka Luigi Lilio or Luigi Giglio. His work was resumed, after his death, by the scientific adviser to the Pope, Christopher Clavius.
Viète accused Clavius, in a series of pamphlets (1600), of introducing corrections and intermediate days in an arbitrary manner, and misunderstanding the meaning of the works of his predecessor, particularly in the calculation of the lunar cycle. Viète gave a new timetable, which Clavius cleverly refuted,[8] after Viète's death, in his Explicatio (1603).
It is said that Viète was wrong. Without doubt, he believed himself to be a kind of "King of Times" as the historian of mathematics, Dhombres, claimed.[9] It is true that Viète held Clavius in low esteem, as evidenced by De Thou:
He said that Clavius was very clever to explain the principles of mathematics, that he heard with great clarity what the authors had invented, and wrote various treatises compiling what had been written before him without quoting its references. So, his works were in a better order which was scattered and confused in early writings.
The Adriaan van Roomen problem
In 1596, Scaliger resumed his attacks from the University of Leyden. Viète replied definitively the following year. In March that same year, Adriaan van Roomen sought the resolution, by any of Europe's top mathematicians, to a polynomial equation of degree 45. King Henri IV received a snub from the Dutch ambassador, who claimed that there was no mathematician in France. He said it was simply because some Dutch mathematician, Adriaan van Roomen, had not asked any Frenchman to solve his problem.
Viète came, saw the problem, and, after leaning on a window for a few minutes, solved it. It was the equation between sin(x) and sin(x/45). He resolved this at once, and said he was able to give at the same time (actually the next day) the solution to the other 22 problems to the ambassador. "Ut legit, ut solvit," he later said. Further, he sent a new problem back to Van Roomen, for resolution by Euclidean tools (rule and compass) of the lost answer to the problem first set by Apollonius of Perga. Van Roomen could not overcome that problem without resorting to a trick (see detail below).
Final years
In 1598, Viète was granted special leave. Henry IV, however, charged him to end the revolt of the Notaries, whom the King had ordered to pay back their fees. Sick and exhausted by work, he left the King's service in December 1602 and received 20,000 écu, which were found at his bedside after his death.
A few weeks before his death, he wrote a final thesis on issues of cryptography, whose memory made obsolete all encryption methods of the time. He died on 23 February 1603, as De Thou wrote,[10] leaving two daughters, Jeanne, whose mother was Barbe Cottereau, and Suzanne, whose mother was Julienne Leclerc. Jeanne, the eldest, died in 1628, having married Jean Gabriau, a councillor of the parliament of Brittany. Suzanne died in January 1618 in Paris.
The cause of Viète's death is unknown. Alexander Anderson, student of Viète and publisher of his scientific writings, speaks of a "praeceps et immaturum autoris fatum." (meeting an untimely end).[7][11]
Work and thought
Background
At the end of the 16th century, mathematics was placed under the dual aegis of Greek geometry and the Arabic procedures for resolution. At the time of Viète, algebra therefore oscillated between arithmetic, which gave the appearance of a list of rules; and geometry, which seemed more rigorous. Meanwhile, Italian mathematicians Luca Pacioli, Scipione del Ferro, Niccolò Fontana Tartaglia, Gerolamo Cardano, Lodovico Ferrari, and especially Raphael Bombelli (1560) all developed techniques for solving equations of the third degree, which heralded a new era.
On the other hand, from the German school of Coss, the Welsh mathematician Robert Recorde (1550) and the Dutchman Simon Stevin (1581) brought an early algebraic notation: the use of decimals and exponents. However, complex numbers remained at best a philosophical way of thinking. Descartes, almost a century after their invention, used them as imaginary numbers. Only positive solutions were considered and using geometrical proof was common.
The mathematician's task was in fact twofold. It was necessary to produce algebra in a more geometrical way (i.e. to give it a rigorous foundation), and it was also necessary to make geometry more algebraic, allowing for analytical calculation in the plane. Viète and Descartes solved this dual task in a double revolution.
Viète's symbolic algebra
Firstly, Viète gave algebra a foundation as strong as that of geometry. He then ended the algebra of procedures (al-Jabr and al-Muqabala), creating the first symbolic algebra, and claiming that with it, all problems could be solved (nullum non problema solvere).[12][13]
In his dedication of the Isagoge to Catherine de Parthenay, Viète wrote:
"These things which are new are wont in the beginning to be set forth rudely and formlessly and must then be polished and perfected in succeeding centuries. Behold, the art which I present is new, but in truth so old, so spoiled and defiled by the barbarians, that I considered it necessary, in order to introduce an entirely new form into it, to think out and publish a new vocabulary, having gotten rid of all its pseudo-technical terms..."[14]
Viète did not know "multiplied" notation (given by William Oughtred in 1631) or the symbol of equality, =, an absence which is more striking because Robert Recorde had used the present symbol for this purpose since 1557, and Guilielmus Xylander had used parallel vertical lines since 1575.[7] Note also the use of a 'u' like symbol with a number above it for an unknown to a given power by Rafael Bombelli in 1572.[15]
Viète had neither much time, nor students able to brilliantly illustrate his method. He took years in publishing his work (he was very meticulous), and most importantly, he made a very specific choice to separate the unknown variables, using consonants for parameters and vowels for unknowns. In this notation he perhaps followed some older contemporaries, such as Petrus Ramus, who designated the points in geometrical figures by vowels, making use of consonants, R, S, T, etc., only when these were exhausted.[7] This choice proved unpopular with future mathematicians and Descartes, among others, preferred the first letters of the alphabet to designate the parameters and the latter for the unknowns.
Viète also remained a prisoner of his time in several respects. First, he was heir of Ramus and did not address the lengths as numbers. His writing kept track of homogeneity, which did not simplify their reading. He failed to recognize the complex numbers of Bombelli and needed to double-check his algebraic answers through geometrical construction. Although he was fully aware that his new algebra was sufficient to give a solution, this concession tainted his reputation.
However, Viète created many innovations: the binomial formula, which would be taken by Pascal and Newton, and the coefficients of a polynomial to sums and products of its roots, called Viète's formula.
Geometric algebra
Viète was well skilled in most modern artifices, aiming at the simplification of equations by the substitution of new quantities having a certain connection with the primitive unknown quantities. Another of his works, Recensio canonica effectionum geometricarum, bears a modern stamp, being what was later called an algebraic geometry—a collection of precepts how to construct algebraic expressions with the use of ruler and compass only. While these writings were generally intelligible, and therefore of the greatest didactic importance, the principle of homogeneity, first enunciated by Viète, was so far in advance of his times that most readers seem to have passed it over. That principle had been made use of by the Greek authors of the classic age; but of later mathematicians only Hero, Diophantus, etc., ventured to regard lines and surfaces as mere numbers that could be joined to give a new number, their sum.[7]
The study of such sums, found in the works of Diophantus, may have prompted Viète to lay down the principle that quantities occurring in an equation ought to be homogeneous, all of them lines, or surfaces, or solids, or supersolids — an equation between mere numbers being inadmissible. During the centuries that have elapsed between Viète's day and the present, several changes of opinion have taken place on this subject. Modern mathematicians like to make homogeneous such equations as are not so from the beginning, in order to get values of a symmetrical shape. Viète himself did not see that far; nevertheless, he indirectly suggested the thought. He also conceived methods for the general resolution of equations of the second, third and fourth degrees different from those of Scipione dal Ferro and Lodovico Ferrari, with which he had not been acquainted. He devised an approximate numerical solution of equations of the second and third degrees, wherein Leonardo of Pisa must have preceded him, but by a method which was completely lost.[7]
Above all, Viète was the first mathematician who introduced notations for the problem (and not just for the unknowns).[12] As a result, his algebra was no longer limited to the statement of rules, but relied on an efficient computational algebra, in which the operations act on the letters and the results can be obtained at the end of the calculations by a simple replacement. This approach, which is the heart of contemporary algebraic method, was a fundamental step in the development of mathematics.[16] With this, Viète marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period.
The logic of species
Being wealthy, Viète began to publish at his own expense, for a few friends and scholars in almost every country of Europe, the systematic presentation of his mathematic theory, which he called "species logistic" (from species: symbol) or art of calculation on symbols (1591).[17]
He described in three stages how to proceed for solving a problem:
• As a first step, he summarized the problem in the form of an equation. Viète called this stage the Zetetic. It denotes the known quantities by consonants (B, D, etc.) and the unknown quantities by the vowels (A, E, etc.)
• In a second step, he made an analysis. He called this stage the Poristic. Here mathematicians must discuss the equation and solve it. It gives the characteristic of the problem, porisma (corrollary), from which we can move to the next step.
• In the last step, the exegetical analysis, he returned to the initial problem which presents a solution through a geometrical or numerical construction based on porisma.
Among the problems addressed by Viète with this method is the complete resolution of the quadratic equations of the form $X^{2}+Xb=c$ and third-degree equations of the form $X^{3}+aX=b$ (Viète reduced it to quadratic equations). He knew the connection between the positive roots of an equation (which, in his day, were alone thought of as roots) and the coefficients of the different powers of the unknown quantity (see Viète's formulas and their application on quadratic equations). He discovered the formula for deriving the sine of a multiple angle, knowing that of the simple angle with due regard to the periodicity of sines. This formula must have been known to Viète in 1593.[7]
Viète's formula
Main article: Viète's formula
In 1593, based on geometrical considerations and through trigonometric calculations perfectly mastered, he discovered the first infinite product in the history of mathematics by giving an expression of π, now known as Viète's formula:[18]
$\pi =2\times {\frac {2}{\sqrt {2}}}\times {\frac {2}{\sqrt {2+{\sqrt {2}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}\times {\frac {2}{\sqrt {2+{\sqrt {2+{\sqrt {2+{\sqrt {2}}}}}}}}}\times \cdots $
He provides 10 decimal places of π by applying the Archimedes method to a polygon with 6 × 216 = 393,216 sides.
Adriaan van Roomen's problem
This famous controversy is told by Tallemant des Réaux in these terms (46th story from the first volume of Les Historiettes. Mémoires pour servir à l’histoire du XVIIe siècle):
"In the times of Henri the fourth, a Dutchman called Adrianus Romanus, a learned mathematician, but not so good as he believed, published a treatise in which he proposed a question to all the mathematicians of Europe, but did not ask any Frenchman. Shortly after, a state ambassador came to the King at Fontainebleau. The King took pleasure in showing him all the sights, and he said people there were excellent in every profession in his kingdom. 'But, Sire,' said the ambassador, 'you have no mathematician, according to Adrianus Romanus, who didn't mention any in his catalog.' 'Yes, we have,' said the King. 'I have an excellent man. Go and seek Monsieur Viette,' he ordered. Vieta, who was at Fontainebleau, came at once. The ambassador sent for the book from Adrianus Romanus and showed the proposal to Vieta, who had arrived in the gallery, and before the King came out, he had already written two solutions with a pencil. By the evening he had sent many other solutions to the ambassador."
This suggests that the Adrien van Roomen problem is an equation of 45°, which Viète recognized immediately as a chord of an arc of 8° (${\tfrac {1}{45}}$ turn). It was then easy to determine the following 22 positive alternatives, the only valid ones at the time.
When, in 1595, Viète published his response to the problem set by Adriaan van Roomen, he proposed finding the resolution of the old problem of Apollonius, namely to find a circle tangent to three given circles. Van Roomen proposed a solution using a hyperbola, with which Viète did not agree, as he was hoping for a solution using Euclidean tools.
Viète published his own solution in 1600 in his work Apollonius Gallus. In this paper, Viète made use of the center of similitude of two circles.[7] His friend De Thou said that Adriaan van Roomen immediately left the University of Würzburg, saddled his horse and went to Fontenay-le-Comte, where Viète lived. According to De Thou, he stayed a month with him, and learned the methods of the new algebra. The two men became friends and Viète paid all van Roomen's expenses before his return to Würzburg.
This resolution had an almost immediate impact in Europe and Viète earned the admiration of many mathematicians over the centuries. Viète did not deal with cases (circles together, these tangents, etc.), but recognized that the number of solutions depends on the relative position of the three circles and outlined the ten resulting situations. Descartes completed (in 1643) the theorem of the three circles of Apollonius, leading to a quadratic equation in 87 terms, each of which is a product of six factors (which, with this method, makes the actual construction humanly impossible).[19]
Religious and political beliefs
Viète was accused of Protestantism by the Catholic League, but he was not a Huguenot. His father was, according to Dhombres.[20] Indifferent in religious matters, he did not adopt the Calvinist faith of Parthenay, nor that of his other protectors, the Rohan family. His call to the parliament of Rennes proved the opposite. At the reception as a member of the court of Brittany, on 6 April 1574, he read in public a statement of Catholic faith.[20]
Nevertheless, Viète defended and protected Protestants his whole life, and suffered, in turn, the wrath of the League. It seems that for him, the stability of the state was to be preserved and that under this requirement, the King's religion did not matter. At that time, such people were called "Politicals."
Furthermore, at his death, he did not want to confess his sins. A friend had to convince him that his own daughter would not find a husband, were he to refuse the sacraments of the Catholic Church. Whether Viète was an atheist or not is a matter of debate.[20]
Publications
Chronological list
• Between 1564 and 1568, Viète prepared for his student, Catherine de Parthenay, some textbooks of astronomy and trigonometry and a treatise that was never published: Harmonicon coeleste.
• In 1579, the trigonometric tables Canon mathematicus, seu ad triangula, published together with a table of rational-sided triangles Canonion triangulorum laterum rationalium, and a book of trigonometry Universalium inspectionum ad canonem mathematicum – which he published at his own expense and with great printing difficulties. This text contains many formulas on the sine and cosine and is unusual in using decimal numbers. The trigonometric tables here exceeded those of Regiomontanus (Triangulate Omnimodis, 1533) and Rheticus (1543, annexed to De revolutionibus of Copernicus). (Alternative scan of a 1589 reprint)
• In 1589, Deschiffrement d'une lettre escripte par le Commandeur Moreo au Roy d'Espaigne son maître.
• In 1590, Deschiffrement description of a letter by the Commander Moreo at Roy Espaigne of his master, Tours: Mettayer.
• In 1591:
• In artem analyticem isagoge (Introduction to the art of analysis), also known as Algebra Nova (New Algebra) Tours: Mettayer, in 9 folio; the first edition of the Isagoge.
• Zeteticorum libri quinque. Tours: Mettayer, in 24 folio; which are the five books of Zetetics, a collection of problems from Diophantus solved using the analytical art.
• Between 1591 and 1593, Effectionum geometricarum canonica recensio. Tours: Mettayer, in 7 folio.
• In 1593:
• Vietae Supplementum geometriae. Tours: Francisci, in 21 folio.
• Francisci Vietae Variorum de rebus responsorum mathematics liber VIII. Tours: Mettaye, in 49 folio; about the challenges of Scaliger.
• Variorum de rebus mathematicis responsorum liber VIII; the "Eighth Book of Varied Responses" in which he talks about the problems of the trisection of the angle (which he acknowledges that it is bound to an equation of third degree) of squaring the circle, building the regular heptagon, etc.
• In 1594, Munimen adversus nova cyclometrica. Paris: Mettayer, in quarto, 8 folio; again, a response against Scaliger.
• In 1595, Ad problema quod omnibus mathematicis totius orbis construendum proposuit Adrianus Romanus, Francisci Vietae responsum. Paris: Mettayer, in quarto, 16 folio; about the Adriaan van Roomen problem.
• In 1600:
• De numerosa potestatum ad exegesim resolutione. Paris: Le Clerc, in 36 folio; work that provided the means for extracting roots and solutions of equations of degree at most 6.
• Francisci Vietae Apollonius Gallus. Paris: Le Clerc, in quarto, 13 folio; where he referred to himself as the French Apollonius.
• Between 1600 and 1602:
• Fontenaeensis libellorum supplicum in Regia magistri relatio Kalendarii vere Gregoriani ad ecclesiasticos doctores exhibita Pontifici Maximi Clementi VIII. Paris: Mettayer, in quarto, 40 folio.
• Francisci Vietae adversus Christophorum Clavium expostulatio. Paris: Mettayer, in quarto, 8 folio; his theses against Clavius.
Posthumous publications
• 1612:
• Supplementum Apollonii Galli edited by Marin Ghetaldi.
• Supplementum Apollonii Redivivi sive analysis problematis bactenus desiderati ad Apollonii Pergaei doctrinam a Marino Ghetaldo Patritio Regusino hujusque non ita pridem institutam edited by Alexander Anderson.
• 1615:
• Ad Angularum Sectionem Analytica Theoremata F. Vieta primum excogitata at absque ulla demonstratione ad nos transmissa, iam tandem demonstrationibus confirmata edited by Alexander Anderson.
• Pro Zetetico Apolloniani problematis a se jam pridem edito in supplemento Apollonii Redivivi Zetetico Apolloniani problematis a se jam pridem edito; in qua ad ea quae obiter inibi perstrinxit Ghetaldus respondetur edited by Alexander Anderson
• Francisci Vietae Fontenaeensis, De aequationum — recognitione et emendatione tractatus duo per Alexandrum Andersonum edited by Alexander Anderson
• 1617: Animadversionis in Franciscum Vietam, a Clemente Cyriaco nuper editae brevis diakrisis edited by Alexander Anderson
• 1619: Exercitationum Mathematicarum Decas Prima edited by Alexander Anderson
• 1631: In artem analyticem isagoge. Eiusdem ad logisticem speciosam notae priores, nunc primum in lucem editae. Paris: Baudry, in 12 folio; the second edition of the Isagoge, including the posthumously published Ad logisticem speciosam notae priores.
Reception and influence
During the ascendancy of the Catholic League, Viète's secretary was Nathaniel Tarporley, perhaps one of the more interesting and enigmatic mathematicians of 16th-century England. When he returned to London, Tarporley became one of the trusted friends of Thomas Harriot.
Apart from Catherine de Parthenay, Viète's other notable students were: French mathematician Jacques Aleaume, from Orleans, Marino Ghetaldi of Ragusa, Jean de Beaugrand and the Scottish mathematician Alexander Anderson. They illustrated his theories by publishing his works and continuing his methods. At his death, his heirs gave his manuscripts to Peter Aleaume.[21] We give here the most important posthumous editions:
• In 1612: Supplementum Apollonii Galli of Marino Ghetaldi.
• From 1615 to 1619: Animadversionis in Franciscum Vietam, Clemente a Cyriaco nuper by Alexander Anderson
• Francisci Vietae Fontenaeensis ab aequationum recognitione et emendatione Tractatus duo Alexandrum per Andersonum. Paris, Laquehay, 1615, in 4, 135 p. The death of Alexander Anderson unfortunately halted the publication.
• In 1630, an Introduction en l'art analytic ou nouvelle algèbre ('Introduction to the analytic art or modern algebra),[22] translated into French and commentary by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin.
• The Five Books of François Viette's Zetetic (Les cinq livres des zététiques de François Viette), put into French, and commented increased by mathematician J. L. Sieur de Vaulezard. Paris, Jacquin, p. 219.
The same year, there appeared an Isagoge by Antoine Vasset (a pseudonym of Claude Hardy), and the following year, a translation into Latin of Beaugrand, which Descartes would have received.
In 1648, the corpus of mathematical works printed by Frans van Schooten, professor at Leiden University (Elzevirs presses). He was assisted by Jacques Golius and Mersenne.
The English mathematicians Thomas Harriot and Isaac Newton, and the Dutch physicist Willebrord Snellius, the French mathematicians Pierre de Fermat and Blaise Pascal all used Viète's symbolism.
About 1770, the Italian mathematician Targioni Tozzetti, found in Florence Viète's Harmonicon coeleste. Viète had written in it: Describat Planeta Ellipsim ad motum anomaliae ad Terram. (That shows he adopted Copernicus' system and understood before Kepler the elliptic form of the orbits of planets.)[23]
In 1841, the French mathematician Michel Chasles was one of the first to reevaluate his role in the development of modern algebra.
In 1847, a letter from François Arago, perpetual secretary of the Academy of Sciences (Paris), announced his intention to write a biography of François Viète.
Between 1880 and 1890, the polytechnician Fréderic Ritter, based in Fontenay-le-Comte, was the first translator of the works of François Viète and his first contemporary biographer with Benjamin Fillon.
Descartes' views on Viète
Thirty-four years after the death of Viète, the philosopher René Descartes published his method and a book of geometry that changed the landscape of algebra and built on Viète's work, applying it to the geometry by removing its requirements of homogeneity. Descartes, accused by Jean Baptiste Chauveau, a former classmate of La Flèche, explained in a letter to Mersenne (1639 February) that he never read those works.[24] Descartes accepted the Viète's view of mathematics for which the study shall stress the self-evidence of the results that Descartes implemented translating the symbolic algebra in geometric reasoning.[25] The locution mathesis universalis was derived from van Roomen's works.[25]
"I have no knowledge of this surveyor and I wonder what he said, that we studied Viète's work together in Paris, because it is a book which I cannot remember having seen the cover, while I was in France."
Elsewhere, Descartes said that Viète's notations were confusing and used unnecessary geometric justifications. In some letters, he showed he understands the program of the Artem Analyticem Isagoge; in others, he shamelessly caricatured Viète's proposals. One of his biographers, Charles Adam,[26] noted this contradiction:
"These words are surprising, by the way, for he (Descartes) had just said a few lines earlier that he had tried to put in his geometry only what he believed "was known neither by Vieta nor by anyone else". So he was informed of what Viète knew; and he must have read his works previously."
Current research has not shown the extent of the direct influence of the works of Viète on Descartes. This influence could have been formed through the works of Adriaan van Roomen or Jacques Aleaume at the Hague, or through the book by Jean de Beaugrand.[27]
In his letters to Mersenne, Descartes consciously minimized the originality and depth of the work of his predecessors. "I began," he says, "where Vieta finished". His views emerged in the 17th century and mathematicians won a clear algebraic language without the requirements of homogeneity. Many contemporary studies have restored the work of Parthenay's mathematician, showing he had the double merit of introducing the first elements of literal calculation and building a first axiomatic for algebra.[28]
Although Viète was not the first to propose notation of unknown quantities by letters - Jordanus Nemorarius had done this in the past - we can reasonably estimate that it would be simplistic to summarize his innovations for that discovery and place him at the junction of algebraic transformations made during the late sixteenth – early 17th century.
See also
• Michael Stifel
• Rafael Bombelli
Notes
1. Jacqueline A. Stedall, From Cardano's Great Art to Lagrange's Reflections: Filling a Gap in the History of Algebra, European Mathematical Society, 2011, p. 20.
2. H. Ben-Yami, Descartes' Philosophical Revolution: A Reassessment, Palgrave Macmillan, 2015, p. 179: "[Descartes'] work in mathematics was apparently influenced by Vieta's, despite his denial of any acquaintance with the latter’s work."
3. Cantor 1911, p. 57.
4. Goldstein, Bernard R. (1998), "What's new in Kepler's new astronomy?", in Earman, John; Norton, John D. (eds.), The Cosmos of Science: Essays of Exploration, Pittsburgh-Konstanz series in the philosophy and history of science, University of Pittsburgh Press, pp. 3–23, ISBN 9780822972013. See in particular p. 21: "an unpublished manuscript by Viète includes a mathematical discussion of an ellipse in a planetary model".
5. Kinser, Sam. The works of Jacques-Auguste de Thou. Google Books
6. Bashmakova, I. G., & Smirnova, G. S., The Beginnings and Evolution of Algebra (Washington, D.C.: Mathematical Association of America, 2000), pp. 75–77
7. Cantor 1911, p. 58.
8. Clavius, Christophorus. Operum mathematicorum tomus quintus continens Romani Christophorus Clavius, published by Anton Hierat, Johann Volmar, place Royale Paris, in 1612
9. Otte, Michael; Panza, Marco. Analysis and synthesis in mathematics. Google Books
10. De thou (from University of Saint Andrews) Archived 2008-07-08 at the Wayback Machine
11. Ball, Walter William Rouse. A short account of the history of mathematics. Google Books
12. H. J. M. Bos : Redefining geometrical exactness: Descartes' transformation Google Books
13. Jacob Klein: Greek mathematical thought and the origin of algebra, Google Books
14. Hadden, Richard W. (1994), On the Shoulders of Merchants: Exchange and the Mathematical Conception of Nature in Early Modern Europe, New York: State University of New York Press, ISBN 0-585-04483-X.
15. Stedall, Jacqueline Anne (2000). A large discourse concerning algebra: John Wallis's 1685 Treatise of algebra (Thesis). The Open University Press.
16. Helena M. Pycior : Symbols, Impossible Numbers, and Geometric Entanglements: British Algebra... Google books
17. Peter Murphy, Peter Murphy (LL. B.) : Evidence, proof, and facts: a book of sources, Google Books
18. Variorum de rebus Mathèmaticis Reíponíorum Liber VIII
19. Henk J.M. Bos: Descartes, Elisabeth and Apollonius’ Problem. In The Correspondence of René Descartes 1643, Quæstiones Infinitæ, pages 202–212. Zeno Institute of Philosophy, Utrecht, Theo Verbeek edition, Erik-Jan Bos and Jeroen van de Ven, 2003
20. Dhombres, Jean. François Viète et la Réforme. Available at cc-parthenay.fr Archived 2007-09-11 at the Wayback Machine (in French)
21. De Thou, Jacques-Auguste available at L'histoire universelle (fr) and at Universal History (en) Archived 2008-07-08 at the Wayback Machine
22. Viète, François (1983). The Analytic Art, translated by T. Richard Witmer. Kent, Ohio: The Kent State University Press.
23. Article about Harmonicon coeleste: Adsabs.harvard.edu "The Planetary Theory of François Viète, Part 1".
24. Letter from Descartes to Mersenne. (PDF) Pagesperso-orange.fr, February 20, 1639 (in French)
25. Bullynck, Maarten (2018). The 'Everyday' in Mathematics On the usability of mathematical practices for doing history. pp. 11, 10. Archived from the original on July 9, 2020. {{cite book}}: |website= ignored (help)
26. Archive.org, Charles Adam, Vie et Oeuvre de Descartes Paris, L Cerf, 1910, p 215.
27. Chikara Sasaki. Descartes' mathematical thought p.259
28. For example: Hairer, E (2008). Analysis by its history. New York: Springer. p. 6. ISBN 9780387770314.
Bibliography
• Bailey Ogilvie, Marilyn; Harvey, Joy Dorothy. The Biographical Dictionary of Women in Science: L–Z. Google Books. p 985.
• Bachmakova, Izabella G., Slavutin, E.I. “ Genesis Triangulorum de François Viète et ses recherches dans l’analyse indéterminée ”, Archives for History of Exact Science, 16 (4), 1977, 289-306.
• Bashmakova, Izabella Grigorievna; Smirnova Galina S; Shenitzer, Abe. The Beginnings and Evolution of Algebra. Google Books. pp. 75–.
• Biard, Joel; Rāshid, Rushdī. Descartes et le Moyen Age. Paris: Vrin, 1998. Google Books (in French)
• Burton, David M (1985). The History of Mathematics: An Introduction. Newton, Massachusetts: Allyn and Bacon, Inc.
• Cajori, F. (1919). A History of Mathematics. pp. 152 and onward.
• Calinger, Ronald (ed.) (1995). Classics of Mathematics. Englewood Cliffs, New Jersey: Prentice–Hall, Inc.
• Calinger, Ronald. Vita mathematica. Mathematical Association of America. Google Books
• Chabert, Jean-Luc; Barbin, Évelyne; Weeks, Chris. A History of Algorithms. Google Books
• Derbyshire, John (2006). Unknown Quantity a Real and Imaginary History of Algebra. Scribd.com Archived 2009-12-21 at the Wayback Machine
• Eves, Howard (1980). Great Moments in Mathematics (Before 1650). The Mathematical Association of America. Google Books
• Grisard, J. (1968) François Viète, mathématicien de la fin du seizième siècle: essai bio-bibliographique (Thèse de doctorat de 3ème cycle) École Pratique des Hautes Études, Centre de Recherche d'Histoire des Sciences et des Techniques, Paris. (in French)
• Godard, Gaston. François Viète (1540–1603), Father of Modern Algebra. Université de Paris-VII, France, Recherches vendéennes. ISSN 1257-7979 (in French)
• W. Hadd, Richard. On the shoulders of merchants. Google Books
• Hofmann, Joseph E (1957). The History of Mathematics, translated by F. Graynor and H. O. Midonick. New York, New York: The Philosophical Library.
• Joseph, Anthony. Round tables. European Congress of Mathematics. Google Books
• Michael Sean Mahoney (1994). The mathematical career of Pierre de Fermat (1601–1665). Google Books
• Jacob Klein. Die griechische Logistik und die Entstehung der Algebra in: Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abteilung B: Studien, Band 3, Erstes Heft, Berlin 1934, p. 18–105 and Zweites Heft, Berlin 1936, p. 122–235; translated in English by Eva Brann as: Greek Mathematical Thought and the Origin of Algebra. Cambridge, Mass. 1968, ISBN 0-486-27289-3
• Mazur, Joseph (2014). Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers. Princeton, New Jersey: Princeton University Press.
• Nadine Bednarz, Carolyn Kieran, Lesley Lee. Approaches to algebra. Google Books
• Otte, Michael; Panza, Marco. Analysis and Synthesis in Mathematics. Google Books
• Pycior, Helena M. Symbols, Impossible Numbers, and Geometric Entanglements. Google Books
• Francisci Vietae Opera Mathematica, collected by F. Van Schooten. Leyde, Elzévir, 1646, p. 554 Hildesheim-New-York: Georg Olms Verlag (1970). (in Latin)
• The intégral corpus (excluding Harmonicon) was published by Frans van Schooten, professor at Leyde as Francisci Vietæ. Opera mathematica, in unum volumen congesta ac recognita, opera atque studio Francisci a Schooten, Officine de Bonaventure et Abraham Elzevier, Leyde, 1646. Gallica.bnf.fr (pdf). (in Latin)
• Stillwell, John. Mathematics and its history. Google Books
• Varadarajan, V. S. (1998). Algebra in Ancient and Modern Times The American Mathematical Society. Google Books
Attribution
• This article incorporates text from a publication now in the public domain: Cantor, Moritz (1911). "Vieta, François". In Chisholm, Hugh (ed.). Encyclopædia Britannica. Vol. 28 (11th ed.). Cambridge University Press. pp. 57–58.
External links
• Literature by and about François Viète in the German National Library catalogue
• François Viète at Library of Congress
• O'Connor, John J.; Robertson, Edmund F., "François Viète", MacTutor History of Mathematics Archive, University of St Andrews
• New Algebra (1591) online
• Francois Viète: Father of Modern Algebraic Notation
• The Lawyer and the Gambler
• About Tarporley
• Site de Jean-Paul Guichard (in French)
• L'algèbre nouvelle (in French)
• "About the Harmonicon" (PDF). Archived from the original (PDF) on 2011-08-07. Retrieved 2009-06-18. (200 KB). (in French)
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Vieta jumping
In number theory, Vieta jumping, also known as root flipping, is a proof technique. It is most often used for problems in which a relation between two integers is given, along with a statement to prove about its solutions. In particular, it can be used to produce new solutions of a quadratic Diophantine equation from known ones. There exist multiple variations of Vieta jumping, all of which involve the common theme of infinite descent by finding new solutions to an equation using Vieta's formulas.
History
Vieta jumping is a classical method in the theory of quadratic Diophantine equations and binary quadratic forms. For example, it was used in the analysis of Markov equation back in 1879 and in the 1953 paper of Mills.[1]
In 1988, the method came to the attention to mathematical olympiad problems in the light of the first olympiad problem to use it in a solution that was proposed for the International Mathematics Olympiad and assumed to be the most difficult problem on the contest:[2][3]
Let a and b be positive integers such that ab + 1 divides a2 + b2. Show that ${\frac {a^{2}+b^{2}}{ab+1}}$ is the square of an integer.[4]
Arthur Engel wrote the following about the problem's difficulty:
Nobody of the six members of the Australian problem committee could solve it. Two of the members were husband and wife George and Esther Szekeres, both famous problem solvers and problem creators. Since it was a number theoretic problem it was sent to the four most renowned Australian number theorists. They were asked to work on it for six hours. None of them could solve it in this time. The problem committee submitted it to the jury of the XXIX IMO marked with a double asterisk, which meant a superhard problem, possibly too hard to pose. After a long discussion, the jury finally had the courage to choose it as the last problem of the competition. Eleven students gave perfect solutions.
Among the eleven students receiving the maximum score for solving this problem were Ngô Bảo Châu, Ravi Vakil, Zvezdelina Stankova, and Nicușor Dan.[5] Emanouil Atanassov (from Bulgaria) solved the problem in a paragraph and received a special prize.[6]
Standard Vieta jumping
The concept of standard Vieta jumping is a proof by contradiction, and consists of the following four steps:[7]
1. Assume toward a contradiction that some solution (a1, a2, ...) exists that violates the given requirements.
2. Take the minimal such solution according to some definition of minimality.
3. Replace some ai by a variable x in the formulas, and obtain an equation for which ai is a solution.
4. Using Vieta's formulas, show that this implies the existence of a smaller solution, hence a contradiction.
Example
Problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square.[8][9]
1. Fix some value k that is a non-square positive integer. Assume there exist positive integers (a, b) for which k = a2 + b2/ab + 1.
2. Let (A, B) be positive integers for which k = A2 + B2/AB + 1 and such that A + B is minimized, and without loss of generality assume A ≥ B.
3. Fixing B, replace A with the variable x to yield x2 – (kB)x + (B2 – k) = 0. We know that one root of this equation is x1 = A. By standard properties of quadratic equations, we know that the other root satisfies x2 = kB – A and x2 = B2 – k/A.
4. The first expression for x2 shows that x2 is an integer, while the second expression implies that x2 ≠ 0 since k is not a perfect square. From x22 + B2/x2B + 1 = k > 0 it further follows that x2B > −1, and hence x2 is a positive integer. Finally, A ≥ B implies that x2 = B2 − k/A < B2/A ≤ A, hence x2 < A, and thus x2 + B < A + B, which contradicts the minimality of A + B.
Constant descent Vieta jumping
The method of constant descent Vieta jumping is used when we wish to prove a statement regarding a constant k having something to do with the relation between a and b. Unlike standard Vieta jumping, constant descent is not a proof by contradiction, and it consists of the following four steps:[10]
1. The equality case is proven so that it may be assumed that a > b.
2. b and k are fixed and the expression relating a, b, and k is rearranged to form a quadratic with coefficients in terms of b and k, one of whose roots is a. The other root, x2 is determined using Vieta's formulas.
3. For all (a, b) above a certain base case, show that 0 < x2 < b < a and that x2 is an integer. Thus, while maintaining the same k, we may replace (a, b) with (b, x2) and repeat this process until we arrive at the base case.
4. Prove the statement for the base case, and as k has remained constant through this process, this is sufficient to prove the statement for all ordered pairs.
Example
Let a and b be positive integers such that ab divides a2 + b2 + 1. Prove that 3ab = a2 + b2 + 1.[11]
1. If a = b, a2 dividing 2a2 + 1 implies that a2 divides 1, and hence the positive integers a = b = 1, and 3(1)(1) = 12 + 12 + 1. So, without loss of generality, assume that a > b.
2. For any (a, b) satisfying the given condition, let k = a2 + b2 + 1/ab and rearrange and substitute to get x2 − (kb) x + (b2 + 1) = 0. One root to this quadratic is a, so by Vieta's formulas the other root may be written as follows: x2 = kb − a = b2 + 1/a.
3. The first equation shows that x2 is an integer and the second that it is positive. Because a > b and they are both integers, a ≥ b + 1, and hence ab ≥ b2 + b; As long as b > 1, we always have ab > b2 + 1, and therefore x2 = b2 + 1/a < b. Thus, while maintaining the same k, we may replace (a, b) with (b, x2) and repeat this process until we arrive at the base case.
4. The base case we arrive at is the case where b = 1. For (a, 1) to satisfy the given condition, a must divide a2 + 2, which implies that a divides 2, making a either 1 or 2. The first case is eliminated because a = b. In the second case, k = a2 + b2 + 1/ab = 6/2 = 3. As k has remained constant throughout this process of Vieta jumping, this is sufficient to show that for any (a, b) satisfying the given condition, k will always equal 3.
Geometric interpretation
Vieta jumping can be described in terms of lattice points on hyperbolas in the first quadrant.[2] The same process of finding smaller roots is used instead to find lower lattice points on a hyperbola while remaining in the first quadrant. The procedure is as follows:
1. From the given condition we obtain the equation of a family of hyperbolas that are unchanged by switching x and y so that they are symmetric about the line y = x.
2. Prove the desired statement for the intersections of the hyperbolas and the line y = x.
3. Assume there is some lattice point (x, y) on some hyperbola and without loss of generality x < y. Then by Vieta's formulas, there is a corresponding lattice point with the same x-coordinate on the other branch of the hyperbola, and by reflection through y = x a new point on the original branch of the hyperbola is obtained.
4. It is shown that this process produces lower points on the same branch and can be repeated until some condition (such as x = 0) is achieved. Then by substitution of this condition into the equation of the hyperbola, the desired conclusion will be proven.
Example
This method can be applied to problem #6 at IMO 1988: Let a and b be positive integers such that ab + 1 divides a2 + b2. Prove that a2 + b2/ab + 1 is a perfect square.
1. Let a2 + b2/ab + 1 = q and fix the value of q. If q = 1, q is a perfect square as desired. If q = 2, then (a-b)2 = 2 and there is no integral solution a, b. When q > 2, the equation x2 + y2 − qxy − q = 0 defines a hyperbola H and (a,b) represents an integral lattice point on H.
2. If (x,x) is an integral lattice point on H with x > 0, then (since q is integral) one can see that x = 1). This proposition's statement is then true for the point (x,x).
3. Now let P = (x, y) be a lattice point on a branch H with x, y > 0 and x ≠ y (as the previous remark covers the case x = y). By symmetry, we can assume that x < y and that P is on the higher branch of H. By applying Vieta's Formulas, (x, qx − y) is a lattice point on the lower branch of H. Let y′ = qx − y. From the equation for H, one sees that 1 + x y′ > 0. Since x > 0, it follows that y′ ≥ 0. Hence the point (x, y′) is in the first quadrant. By reflection, the point (y′, x) is also a point in the first quadrant on H. Moreover from Vieta's formulas, yy′ = x2 - q, and y′ = x2 - q/ y. Combining this equation with x < y, one can show that y′ < x. The new constructed point Q = (y′, x) is then in the first quadrant, on the higher branch of H, and with smaller x,y-coordinates than the point P we started with.
4. The process in the previous step can be repeated whenever the point Q has a positive x-coordinate. However, since the x-coordinates of these points will form a decreasing sequence of non-negative integers, the process can only be repeated finitely many times before it produces a point Q = (0, y) on the upper branch of H; by substitution, q = y2 is a square as required.
See also
• Vieta's formulas
• Proof by contradiction
• Infinite descent
• Markov number
• Apollonian gasket
Notes
1. Mills 1953.
2. Arthur Engel (1998). Problem Solving Strategies. Problem Books in Mathematics. Springer. p. 127. doi:10.1007/b97682. ISBN 978-0-387-98219-9.
3. "The Return of the Legend of Question Six". Numberphile. August 16, 2016. Archived from the original on 2021-12-20 – via YouTube.
4. "International Mathematical Olympiad". www.imo-official.org. Retrieved 29 September 2020.
5. "Results of the 1988 International Mathematical Olympiad". Imo-official.org. Retrieved 2013-03-03.
6. "Individual ranking of Emanouil Atanassov". International Mathematical Olympiad.
7. Yimin Ge (2007). "The Method of Vieta Jumping" (PDF). Mathematical Reflections. 5.
8. "AoPS Forum – One of my favourites problems, yeah!". Artofproblemsolving.com. Retrieved 2023-01-08.
9. K. S. Brown. "N = (x^2 + y^2)/(1+xy) is a Square". MathPages.com. Retrieved 2016-09-26.
10. "AoPS Forum — Lemur Numbers". Artofproblemsolving.com. Retrieved 2023-01-08.
11. "AoPS Forum – x*y | x^2+y^2+1". Artofproblemsolving.com. 2005-06-07. Retrieved 2023-01-08.
External links
• Vieta Root Jumping at Brilliant.org
• Mills, W. H. (1953). "A system of quadratic Diophantine equations". Pacific J. Math. 3 (1): 209–220.
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Vietnamese numerals
Historically Vietnamese has two sets of numbers: one is etymologically native Vietnamese; the other uses Sino-Vietnamese vocabulary. In the modern language the native Vietnamese vocabulary is used for both everyday counting and mathematical purposes. The Sino-Vietnamese vocabulary is used only in fixed expressions or in Sino-Vietnamese words, in a similar way that Latin and Greek numerals are used in modern English (e.g., the bi- prefix in bicycle).
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List of numeral systems
For numbers up to one million, native Vietnamese terms is often used the most, whilst mixed Sino-Vietnamese origin words and native Vietnamese words are used for units of one million or above.
Concept
For non-official purposes prior to the 20th century, Vietnamese had a writing system known as Hán-Nôm. Sino-Vietnamese numbers were written in Chữ Hán and native vocabulary was written in Chữ Nôm. Hence, there are two concurrent system in Vietnamese nowadays in the romanized script, one for native Vietnamese and one for Sino-Vietnamese.
In the modern Vietnamese writing system, numbers are written as Arabic numerals or in the romanized script Chữ Quốc ngữ (một, hai, ba), which had a Chữ Nôm character. Less common for numbers under one million are the numbers of Sino-Vietnamese origin (nhất [1], nhị [2], tam [3]), using Chữ Hán (classical Chinese characters). Chữ Hán and Chữ Nôm has all but become obsolete in the Vietnamese language, with the Latin-style of reading, writing, and pronouncing native Vietnamese and Sino-Vietnamese being wide spread instead, when France occupied Vietnam. Chữ Hán can still be seen in traditional temples or traditional literature or in cultural artefacts. The Hán-Nôm Institute resides in Hanoi, Vietnam.
Basic figures
The following table is an overview of the basic Vietnamese numeric figures, provided in both native and Sino-Vietnamese counting systems. The form that is highlighted in green is the most widely used in all purposes whilst the ones highlighted in blue are seen as archaic but may still be in use. There are slight differences between the Hanoi and Saigon dialects of Vietnamese, readings between each are differentiated below.
Number Native Vietnamese Sino-Vietnamese Notes
Chữ quốc ngữ Chữ Nôm Chữ quốc ngữ Hán tự
0 không 空 linh 空 • 〇（零） The foreign-language borrowed word "zêrô (zêro, dê-rô)" is often used in physics-related publications, or colloquially.
1 một 𠬠 nhất 一（壹）
2 hai 𠄩 nhị 二（貳）
3 ba 𠀧 tam 三（叄）
4 bốn 𦊚 tứ 四（肆） In the ordinal number system, the Sino-Vietnamese "tư／四" is more systematic; as the digit 4 appears after the number 20 when counting upwards, the Sino-Vietnamese "tư／四" is more commonly used.
5 năm 𠄼 ngũ 五（伍） In numbers above ten that end in five (such as 115, 25, 1055), five is alternatively pronounced as "lăm／𠄻" to avoid possible confusion with "năm／𢆥", a homonym of năm, meaning "year". Exceptions to this rule are numbers ending in 05 (such as 605, 9405).
6 sáu 𦒹 lục 六（陸）
7 bảy 𦉱 thất 七（柒） In some Vietnamese dialects, it is also read as "bẩy".
8 tám 𠔭 bát 八（捌）
9 chín 𠃩 cửu 九（玖）
10 mười • một chục 𨒒 thập 十（拾） Chục is used colloquially. "Ten eggs" may be called một chục quả trứng rather than mười quả trứng. It's also used in compounds like mươi instead of mười (e.g.: hai mươi/chục "twenty").
100 trăm • một trăm 𤾓 • 𠬠𤾓 bách (bá) 百（佰） The Sino-Vietnamese "bách／百" is commonly used as a morpheme (in compound words), and is rarely used in the field of mathematics as a digit. Example: "bách phát bách trúng／百發百中".
1,000 nghìn (ngàn) • một nghìn (ngàn) 𠦳 • 𠬠𠦳 thiên 千（仟） The Sino-Vietnamese "thiên／千" is commonly used as a morpheme, but rarely used in a mathematical sense, however only in counting bricks, it is used. Example: "thiên kim／千金". "nghìn" is the standard word in Northern Vietnam, whilst "ngàn" is the word used in the South.
10,000 mười nghìn (ngàn) 𨒒𠦳 vạn • một vạn 萬 • 𠬠萬 The "một／𠬠" within "một vạn／𠬠萬" is a native Vietnamese (intrinsic term) morpheme. This was officially used in Vietnamese in the past, however, this unit has become less common after 1945, but in counting bricks, it is still widely used. The borrowed native pronunciation muôn for 萬 is still used in slogans such as "muôn năm" (ten thousand years/endless).
100,000 trăm nghìn (ngàn) • một trăm nghìn (ngàn) 𤾓𠦳 • 𠬠𤾓𠦳 ức • một ức • mười vạn[1] 億 • 𠬠億 • 𨒒萬 The "mười／𨒒" and "một／𠬠" within "mười vạn／𨒒萬" and "một ức／𠬠億" are native Vietnamese (intrinsic term) morphemes.
1,000,000 (none) (none) triệu • một triệu • một trăm vạn[2] 兆 • 𠬠兆 • 𠬠𤾓萬 The "một／𠬠" and "trăm／𤾓" within "một triệu／𠬠兆" and "một trăm vạn／𠬠𤾓萬" are native Vietnamese (intrinsic term) morphemes.
10,000,000 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) mười triệu 𨒒兆 The "mười／𨒒" within "mười triệu／𨒒兆" is a native Vietnamese (intrinsic term) morpheme.
100,000,000 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) trăm triệu 𤾓兆 The "trăm／𤾓" within "trăm triệu／𤾓兆" is a native Vietnamese (intrinsic term) morpheme.
1,000,000,000 (none) (none) tỷ 秭[3]
1012 (mixed usage of Sino-Vietnamese and native Vietnamese systems) (mixed usage of Sino-Vietnamese and native Vietnamese systems) nghìn (ngàn) tỷ 𠦳秭
1015 (none) (none) triệu tỷ 兆秭
1018 (none) (none) tỷ tỷ 秭秭
Some other features of Vietnamese numerals include the following:
• Outside of fixed Sino-Vietnamese expressions, Sino-Vietnamese words are usually used in combination with native Vietnamese words. For instance, "mười triệu" combines native "mười" and Sino-Vietnamese "triệu".
• Modern Vietnamese separates place values in thousands instead of myriads. For example, "123123123" is recorded in Vietnamese as "một trăm hai mươi ba triệu một trăm hai mươi ba nghìn (ngàn) một trăm hai mươi ba, or '123 million, 123 thousand and 123'.[4] Meanwhile, in Chinese, Japanese & Korean, the same number is rendered as "1億2312萬3123" (1 hundred-million, 2312 ten-thousand and 3123).
• Sino-Vietnamese numbers are not in frequent use in modern Vietnamese. Sino-Vietnamese numbers such as "vạn／萬" 'ten thousand', "ức／億" 'hundred-thousand' and "triệu／兆" 'million' are used for figures exceeding one thousand, but with the exception of "triệu" are becoming less commonly used. Number values for these words are used for each numeral increasing tenfold in digit value, 億 being the number for 105, 兆 for 106, et cetera. However, Triệu in Vietnamese and 兆 in Modern Chinese now have different values.
Other figures
NumberChữ quốc ngữHán-NômNotes
11 mười một𨒒𠬠
12 mười hai • một tá𨒒𠄩 • 𠬠打"một tá／𠬠打" is often used within mathematics-related occasions, to which "tá" represents the foreign loanword "dozen".
14 mười bốn • mười tư𨒒𦊚 • 𨒒四"mười tư／𨒒四" is often used within literature-related occasions, to which "tư／四" forms part of the Sino-Vietnamese vocabulary.
15 mười lăm𨒒𠄻Here, five is pronounced "lăm／𠄻", or also "nhăm／𠄶" by some speakers in the north.
19 mười chín𨒒𠃩
20 hai mươi • hai chục𠄩𨒒 • 𠄩𨔿
21 hai mươi mốt𠄩𨒒𠬠For numbers which include the digit 1 from 21 to 91, the number 1 is pronounced "mốt".
24 hai mươi tư𠄩𨒒四When the digit 4 appears in numbers after 20 as the last digit of a 3-digit group, it is more common to use "tư／四".
25 hai mươi lăm𠄩𨒒𠄻Here, five is pronounced "lăm".
50 năm mươi • năm chục𠄼𨒒 • 𠄼𨔿When "𨒒" (10) appears after the number 20, the pronunciation changes to "mươi".
101 một trăm linh một • một trăm lẻ một𠬠𤾓零𠬠 • 𠬠𤾓𥘶𠬠"Một trăm linh một／𠬠𤾓零𠬠" is the Northern form, where "linh／零" forms part of the Sino-Vietnamese vocabulary; "một trăm lẻ một／𠬠𤾓𥘶𠬠" is commonly used in the Southern and Central dialect groups of Vietnam.
1001 một nghìn (ngàn) không trăm linh một • một nghìn (ngàn) không trăm lẻ một𠬠𠦳空𤾓零𠬠 • 𠬠𠦳空𤾓𥘶𠬠When the hundreds digit is occupied by a zero, these are expressed using "không trăm／空𤾓".
10055 mười nghìn (ngàn) không trăm năm mươi lăm𨒒𠦳空𤾓𠄼𨒒𠄻
• When the number 1 appears after 20 in the unit digit, the pronunciation changes to "mốt".
• When the number 4 appears after 20 in the unit digit, it is more common to use Sino-Vietnamese "tư／四".
• When the number 5 appears after 10 in the unit digit, the pronunciation changes to "lăm／𠄻".
• When "mười" appears after 20, the pronunciation changes to "mươi".
Ordinal numbers
Vietnamese ordinal numbers are generally preceded by the prefix "thứ-", which is a Sino-Vietnamese word which corresponds to "次-". For the ordinal numbers of one and four, the Sino-Vietnamese readings "nhất／一" and "tư／四" are more commonly used; two is occasionally rendered using the Sino-Vietnamese "nhị／二". In all other cases, the native Vietnamese number is used.
In formal cases, the ordinal number with the structure "đệ (第) + Sino-Vietnamese numbers" is used, especially in calling the generation of monarches, with an example being Nữ vương Elizabeth đệ nhị/女王 Elizabeth 第二 (Queen Elizabeth II).
Ordinal numberChữ quốc ngữHán-Nôm
1stthứ nhất次一
2ndthứ hai • thứ nhì次𠄩 • 次二
3rdthứ ba次𠀧
4ththứ tư次四
5ththứ năm次𠄼
nththứ "n"次「n」
Footnotes
1. Tu dien Han Viet Thieu Chuu：「(1): ức, mười vạn là một ức.」
2. Tu dien Han Viet Thieu Chuu：「(3): triệu, một trăm vạn.」
3. Hán Việt Từ Điển Trích Dẫn 漢越辭典摘引：「Một ngàn lần một triệu là một tỉ 秭 (*). Tức là 1.000.000.000. § Ghi chú: Ngày xưa, mười vạn 萬 là một ức 億, một vạn ức là một tỉ 秭.」
4. Triệu means one million in Vietnamese, but the Chinese number that is the source of the Vietnamese word, "兆" (Mandarin zhào), is officially rendered as 1012 in Taiwan, and commonly designated as 106 in the People's Republic of China (See various scale systems).
See also
• Japanese numerals, Korean numerals, Chinese numerals
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Vietoris–Rips complex
In topology, the Vietoris–Rips complex, also called the Vietoris complex or Rips complex, is a way of forming a topological space from distances in a set of points. It is an abstract simplicial complex that can be defined from any metric space M and distance δ by forming a simplex for every finite set of points that has diameter at most δ. That is, it is a family of finite subsets of M, in which we think of a subset of k points as forming a (k − 1)-dimensional simplex (an edge for two points, a triangle for three points, a tetrahedron for four points, etc.); if a finite set S has the property that the distance between every pair of points in S is at most δ, then we include S as a simplex in the complex.
History
The Vietoris–Rips complex was originally called the Vietoris complex, for Leopold Vietoris, who introduced it as a means of extending homology theory from simplicial complexes to metric spaces.[1] After Eliyahu Rips applied the same complex to the study of hyperbolic groups, its use was popularized by Mikhail Gromov (1987), who called it the Rips complex.[2] The name "Vietoris–Rips complex" is due to Jean-Claude Hausmann (1995).[3]
Relation to Čech complex
The Vietoris–Rips complex is closely related to the Čech complex (or nerve) of a set of balls, which has a simplex for every finite subset of balls with nonempty intersection. In a geodesically convex space Y, the Vietoris–Rips complex of any subspace X ⊂ Y for distance δ has the same points and edges as the Čech complex of the set of balls of radius δ/2 in Y that are centered at the points of X. However, unlike the Čech complex, the Vietoris–Rips complex of X depends only on the intrinsic geometry of X, and not on any embedding of X into some larger space.
As an example, consider the uniform metric space M3 consisting of three points, each at unit distance from each other. The Vietoris–Rips complex of M3, for δ = 1, includes a simplex for every subset of points in M3, including a triangle for M3 itself. If we embed M3 as an equilateral triangle in the Euclidean plane, then the Čech complex of the radius-1/2 balls centered at the points of M3 would contain all other simplexes of the Vietoris–Rips complex but would not contain this triangle, as there is no point of the plane contained in all three balls. However, if M3 is instead embedded into a metric space that contains a fourth point at distance 1/2 from each of the three points of M3, the Čech complex of the radius-1/2 balls in this space would contain the triangle. Thus, the Čech complex of fixed-radius balls centered at M3 differs depending on which larger space M3 might be embedded into, while the Vietoris–Rips complex remains unchanged.
If any metric space X is embedded in an injective metric space Y, the Vietoris–Rips complex for distance δ and X coincides with the Čech complex of the balls of radius δ/2 centered at the points of X in Y. Thus, the Vietoris–Rips complex of any metric space M equals the Čech complex of a system of balls in the tight span of M.
Relation to unit disk graphs and clique complexes
The Vietoris–Rips complex for δ = 1 contains an edge for every pair of points that are at unit distance or less in the given metric space. As such, its 1-skeleton is the unit disk graph of its points. It contains a simplex for every clique in the unit disk graph, so it is the clique complex or flag complex of the unit disk graph.[4] More generally, the clique complex of any graph G is a Vietoris–Rips complex for the metric space having as points the vertices of G and having as its distances the lengths of the shortest paths in G.
Other results
If M is a closed Riemannian manifold, then for sufficiently small values of δ the Vietoris–Rips complex of M, or of spaces sufficiently close to M, is homotopy equivalent to M itself.[5]
Chambers, Erickson & Worah (2008) describe efficient algorithms for determining whether a given cycle is contractible in the Rips complex of any finite point set in the Euclidean plane.
Applications
As with unit disk graphs, the Vietoris–Rips complex has been applied in computer science to model the topology of ad hoc wireless communication networks. One advantage of the Vietoris–Rips complex in this application is that it can be determined only from the distances between the communication nodes, without having to infer their exact physical locations. A disadvantage is that, unlike the Čech complex, the Vietoris–Rips complex does not directly provide information about gaps in communication coverage, but this flaw can be ameliorated by sandwiching the Čech complex between two Vietoris–Rips complexes for different values of δ.[6] An implementation of Vietoris-Rips complexes can be found in the TDAstats R package.[7]
Vietoris–Rips complexes have also been applied for feature-extraction in digital image data; in this application, the complex is built from a high-dimensional metric space in which the points represent low-level image features.[8]
The collection of all Vietoris-Rips complexes is a commonly applied construction in persistent homology and topological data analysis, and is known as the Rips filtration.[9]
Notes
1. Vietoris (1927); Lefschetz (1942); Hausmann (1995); Reitberger (2002).
2. Hausmann (1995); Reitberger (2002).
3. Reitberger (2002).
4. Chambers, Erickson & Worah (2008).
5. Hausmann (1995), Latschev (2001).
6. de Silva & Ghrist (2006), Muhammad & Jadbabaie (2007).
7. Wadhwa et al. 2018.
8. Carlsson, Carlsson & de Silva (2006).
9. Dey, Tamal K.; Shi, Dayu; Wang, Yusu (2019-01-30). "SimBa: An Efficient Tool for Approximating Rips-filtration Persistence via Simplicial Batch Collapse". ACM Journal of Experimental Algorithmics. 24: 1.5:1–1.5:16. doi:10.1145/3284360. ISSN 1084-6654. S2CID 216028146.
References
• Carlsson, Erik; Carlsson, Gunnar; de Silva, Vin (2006), "An algebraic topological method for feature identification" (PDF), International Journal of Computational Geometry and Applications, 16 (4): 291–314, doi:10.1142/S021819590600204X, S2CID 5831809, archived from the original (PDF) on 2019-03-04.
• Chambers, Erin W.; Erickson, Jeff; Worah, Pratik (2008), "Testing contractibility in planar Rips complexes", Proceedings of the 24th Annual ACM Symposium on Computational Geometry, pp. 251–259, CiteSeerX 10.1.1.296.6424, doi:10.1145/1377676.1377721, S2CID 8072058.
• Chazal, Frédéric; Oudot, Steve (2008), "Towards persistence-based reconstruction in euclidean spaces", Proceedings of the twenty-fourth annual symposium on Computational geometry, pp. 232–241, arXiv:0712.2638, doi:10.1145/1377676.1377719, ISBN 978-1-60558-071-5, S2CID 1020710{{citation}}: CS1 maint: date and year (link).
• de Silva, Vin; Ghrist, Robert (2006), "Coordinate-free coverage in sensor networks with controlled boundaries via homology", The International Journal of Robotics Research, 25 (12): 1205–1222, doi:10.1177/0278364906072252, S2CID 10210836.
• Gromov, Mikhail (1987), "Hyperbolic groups", Essays in group theory, Mathematical Sciences Research Institute Publications, vol. 8, Springer-Verlag, pp. 75–263.
• Hausmann, Jean-Claude (1995), "On the Vietoris–Rips complexes and a cohomology theory for metric spaces", Prospects in Topology: Proceedings of a conference in honour of William Browder, Annals of Mathematics Studies, vol. 138, Princeton University Press, pp. 175–188, MR 1368659.
• Latschev, Janko (2001), "Vietoris-Rips complexes of metric spaces near a closed Riemannian manifold", Archiv der Mathematik, 77 (6): 522–528, doi:10.1007/PL00000526, MR 1879057, S2CID 119878137.
• Lefschetz, Solomon (1942), Algebraic Topology, New York: Amer. Math. Soc., p. 271, MR 0007093.
• Muhammad, A.; Jadbabaie, A. (2007), "Dynamic coverage verification in mobile sensor networks via switched higher order Laplacians" (PDF), in Broch, Oliver (ed.), Robotics: Science and Systems, MIT Press.
• Reitberger, Heinrich (2002), "Leopold Vietoris (1891–2002)" (PDF), Notices of the American Mathematical Society, 49 (20).
• Vietoris, Leopold (1927), "Über den höheren Zusammenhang kompakter Räume und eine Klasse von zusammenhangstreuen Abbildungen", Mathematische Annalen, 97 (1): 454–472, doi:10.1007/BF01447877, S2CID 121172198.
• Wadhwa, Raoul; Williamson, Drew; Dhawan, Andrew; Scott, Jacob (2018), "TDAstats: R pipeline for computing persistent homology in topological data analysis", Journal of Open Source Software, 3 (28): 860, Bibcode:2018JOSS....3..860R, doi:10.21105/joss.00860, PMC 7771879, PMID 33381678
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Leopold Vietoris
Leopold Vietoris (/viːˈtɔːrɪs/; German: [viːˈtoːʀɪs]; 4 June 1891 – 9 April 2002) was an Austrian mathematician, World War I veteran and supercentenarian. He was born in Radkersburg and died in Innsbruck.
Leopold Vietoris
Leopold Vietoris on his 110th birthday
Born(1891-06-04)4 June 1891
Bad Radkersburg, Styria
Austria-Hungary
Died(2002-04-09)9 April 2002
(aged 110 years, 309 days)
Innsbruck, Tyrol
Austria
NationalityAustrian
Alma materTU Wien
University of Vienna
Known forContributions to topology
Being a supercentenarian
Spouse(s)Klara Riccabona (m. 1928–1935) (her death)
Maria Josefa Vincentia Vietoris, born von Riccabona zu Reichenfels (m. 1936–2002) (her death)
Children6
Scientific career
FieldsMathematics
InstitutionsUniversity of Innsbruck
Doctoral advisorsGustav Ritter von Escherich
Wilhelm Wirtinger
He was known for his contributions to topology—notably the Mayer–Vietoris sequence—and other fields of mathematics, his interest in mathematical history and for being a keen alpinist.
Biography
Vietoris studied mathematics and geometry at the Vienna University of Technology.[1] He was drafted in 1914 in World War I and was wounded in September that same year.[1] On 4 November 1918, one week before the Armistice of Villa Giusti, he became an Italian prisoner of war.[1] After returning to Austria, he attended the University of Vienna, where he earned his PhD in 1920, with a thesis written under the supervision of Gustav von Escherich and Wilhelm Wirtinger.[1][2]
In autumn 1928 he married his first wife Klara Riccabona, who later died while giving birth to their sixth daughter.[1] In 1936 he married Klara's sister, Maria Riccabona.[1]
Vietoris was survived by his six daughters, 17 grandchildren, and 30 great-grandchildren.[3]
He lends his name to a few mathematical concepts:
• Vietoris topology (see topological space)
• Vietoris homology (see homology theory)
• Mayer–Vietoris sequence
• Vietoris–Begle mapping theorem
• Vietoris–Rips complex
Vietoris remained scientifically active in his later years, even writing one paper on trigonometric sums at the age of 103.[4]
Vietoris lived to be 110 years and 309 days old, and became the oldest verified Austrian man ever.[5]
Decorations and awards
• Austrian Decoration for Science and Art (1973)
• Grand Gold Decoration for Services to the Republic of Austria (1981)
• Honorary member of the German Mathematical Society (1992)
Notes
1. Reitberger, Heinrich (November 2002). "Leopold Vietoris (1891–2002)" (PDF). American Mathematical Society. Retrieved 5 September 2003.
2. Leopold Vietoris at the Mathematics Genealogy Project
3. "Professor Dr. Leopold Vietoris" (PDF). Geo Imagining. Retrieved 11 October 2009.
4. Reitberger, Heinrich (November 2002). "Leopold Vietoris (1891–2002)" (PDF). Notices of the American Mathematical Society. 49 (10): 1235.
5. "Verified Supercentenarians (Ranked By Age) Gerontology Research Group". 1 January 2014. Retrieved 28 February 2019.
References
• Weibel, Peter, ed. (2005). Beyond Art: A Third Culture: A Comparative Study in Cultures, Art and Science in 20th Century Austria and Hungary. Springer Science & Business Media. p. 260. ISBN 978-3-211-24562-0.
External links
• O'Connor, John J.; Robertson, Edmund F., "Leopold Vietoris", MacTutor History of Mathematics Archive, University of St Andrews
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Vietoris–Begle mapping theorem
The Vietoris–Begle mapping theorem is a result in the mathematical field of algebraic topology. It is named for Leopold Vietoris and Edward G. Begle. The statement of the theorem, below, is as formulated by Stephen Smale.
Theorem
Let $X$ and $Y$ be compact metric spaces, and let $f:X\to Y$ be surjective and continuous. Suppose that the fibers of $f$ are acyclic, so that
${\tilde {H}}_{r}(f^{-1}(y))=0,$ for all $0\leq r\leq n-1$ and all $y\in Y$,
with ${\tilde {H}}_{r}$ denoting the $r$th reduced Vietoris homology group. Then, the induced homomorphism
$f_{*}:{\tilde {H}}_{r}(X)\to {\tilde {H}}_{r}(Y)$
is an isomorphism for $r\leq n-1$ and a surjection for $r=n$.
Note that as stated the theorem doesn't hold for homology theories like singular homology. For example, Vietoris homology groups of the closed topologist's sine curve and of a segment are isomorphic (since the first projects onto the second with acyclic fibers). But the singular homology differs, since the segment is path connected and the topologist's sine curve is not.
References
• "Leopold Vietoris (1891–2002)", Notices of the American Mathematical Society, vol. 49, no. 10 (November 2002) by Heinrich Reitberger
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Viewpoints: Mathematical Perspective and Fractal Geometry in Art
Viewpoints: Mathematical Perspective and Fractal Geometry in Art is a textbook on mathematics and art. It was written by mathematicians Marc Frantz and Annalisa Crannell, and published in 2011 by the Princeton University Press (ISBN 9780691125923). The Basic Library List Committee of the Mathematical Association of America has recommended it for inclusion in undergraduate mathematics libraries.[1]
Topics
The first seven chapters of the book concern perspectivity, while its final two concern fractals and their geometry.[1][2] Topics covered within the chapters on perspectivity include coordinate systems for the plane and for Euclidean space, similarity, angles, and orthocenters, one-point and multi-point perspective, and anamorphic art.[1][3] In the fractal chapters, the topics include self-similarity, exponentiation, and logarithms, and fractal dimension. Beyond this mathematical material, the book also describes methods for artists to depict scenes in perspective, and for viewers of art to understand the perspectives in the artworks they see,[1] for instance by finding the optimal point from which to view an artwork.[2] The chapters are ordered by difficulty, and begin with experiments that the students can perform on their own to motivate the material in each chapter.[3]
The book is heavily illustrated by artworks and photography (such as the landscapes of Ansel Adams) and includes a series of essays or interviews by contemporary artists on the mathematical content of their artworks.[1][3] An appendix contains suggestions aimed at teachers of this material.[3]
Audience and reception
Viewpoints is intended as a textbook for mathematics classes aimed at undergraduate liberal arts students,[1][2][4] as a way to show these students how geometry can be used in their everyday life.[2] However, it could even be used for high school art students,[2][3] and reviewer Paul Kelley writes that "it will be of value to anyone interested in an elementary introduction to the mathematics and practice of perspective drawing".[2] It differs from many other liberal arts mathematics textbooks in its relatively narrow focus on geometry and perspective, and its avoidance of more well-covered ground in mathematics and the arts such as symmetry and the geometry of polyhedra.[2]
Although reviewer Blake Mellor complains that the connection between the material on perspective and on fractal geometry "feels forced", he concludes that "this is an excellent text".[4] Reviewer Paul Kelley writes that the book's "step-by-step progression" through its topics makes it "readable [and] easy-to-follow", and that "Students can learn a great deal from this book."[2] Reviewer Alexander Bogomolny calls it "an elegant fusion of mathematical ideas and practical aspects of fine art".[1]
References
1. Bogomolny, Alexander (September 2011), "Review of Viewpoints", MAA Reviews, Mathematical Association of America
2. Kelley, Paul (December 2012 – January 2013), "Review of Viewpoints", The Mathematics Teacher, 106 (5): 399, doi:10.5951/mathteacher.106.5.0398, JSTOR 10.5951/mathteacher.106.5.0398
3. Marchetti, Elena (February 2015), "Review of Viewpoints", Nexus Network Journal, 17 (2): 685–687, doi:10.1007/s00004-015-0237-9
4. Mellor, Blake (December 2011), "Review of Viewpoints", Journal of Mathematics and the Arts, 5 (4): 221–222, doi:10.1080/17513472.2011.624443
Mathematics and art
Concepts
• Algorithm
• Catenary
• Fractal
• Golden ratio
• Hyperboloid structure
• Minimal surface
• Paraboloid
• Perspective
• Camera lucida
• Camera obscura
• Plastic number
• Projective geometry
• Proportion
• Architecture
• Human
• Symmetry
• Tessellation
• Wallpaper group
Forms
• Algorithmic art
• Anamorphic art
• Architecture
• Geodesic dome
• Islamic
• Mughal
• Pyramid
• Vastu shastra
• Computer art
• Fiber arts
• 4D art
• Fractal art
• Islamic geometric patterns
• Girih
• Jali
• Muqarnas
• Zellij
• Knotting
• Celtic knot
• Croatian interlace
• Interlace
• Music
• Origami
• Sculpture
• String art
• Tiling
Artworks
• List of works designed with the golden ratio
• Continuum
• Mathemalchemy
• Mathematica: A World of Numbers... and Beyond
• Octacube
• Pi
• Pi in the Sky
Buildings
• Cathedral of Saint Mary of the Assumption
• Hagia Sophia
• Pantheon
• Parthenon
• Pyramid of Khufu
• Sagrada Família
• Sydney Opera House
• Taj Mahal
Artists
Renaissance
• Paolo Uccello
• Piero della Francesca
• Leonardo da Vinci
• Vitruvian Man
• Albrecht Dürer
• Parmigianino
• Self-portrait in a Convex Mirror
19th–20th
Century
• William Blake
• The Ancient of Days
• Newton
• Jean Metzinger
• Danseuse au café
• L'Oiseau bleu
• Giorgio de Chirico
• Man Ray
• M. C. Escher
• Circle Limit III
• Print Gallery
• Relativity
• Reptiles
• Waterfall
• René Magritte
• La condition humaine
• Salvador Dalí
• Crucifixion
• The Swallow's Tail
• Crockett Johnson
Contemporary
• Max Bill
• Martin and Erik Demaine
• Scott Draves
• Jan Dibbets
• John Ernest
• Helaman Ferguson
• Peter Forakis
• Susan Goldstine
• Bathsheba Grossman
• George W. Hart
• Desmond Paul Henry
• Anthony Hill
• Charles Jencks
• Garden of Cosmic Speculation
• Andy Lomas
• Robert Longhurst
• Jeanette McLeod
• Hamid Naderi Yeganeh
• István Orosz
• Hinke Osinga
• Antoine Pevsner
• Tony Robbin
• Alba Rojo Cama
• Reza Sarhangi
• Oliver Sin
• Hiroshi Sugimoto
• Daina Taimiņa
• Roman Verostko
• Margaret Wertheim
Theorists
Ancient
• Polykleitos
• Canon
• Vitruvius
• De architectura
Renaissance
• Filippo Brunelleschi
• Leon Battista Alberti
• De pictura
• De re aedificatoria
• Piero della Francesca
• De prospectiva pingendi
• Luca Pacioli
• De divina proportione
• Leonardo da Vinci
• A Treatise on Painting
• Albrecht Dürer
• Vier Bücher von Menschlicher Proportion
• Sebastiano Serlio
• Regole generali d'architettura
• Andrea Palladio
• I quattro libri dell'architettura
Romantic
• Samuel Colman
• Nature's Harmonic Unity
• Frederik Macody Lund
• Ad Quadratum
• Jay Hambidge
• The Greek Vase
Modern
• Owen Jones
• The Grammar of Ornament
• Ernest Hanbury Hankin
• The Drawing of Geometric Patterns in Saracenic Art
• G. H. Hardy
• A Mathematician's Apology
• George David Birkhoff
• Aesthetic Measure
• Douglas Hofstadter
• Gödel, Escher, Bach
• Nikos Salingaros
• The 'Life' of a Carpet
Publications
• Journal of Mathematics and the Arts
• Lumen Naturae
• Making Mathematics with Needlework
• Rhythm of Structure
• Viewpoints: Mathematical Perspective and Fractal Geometry in Art
Organizations
• Ars Mathematica
• The Bridges Organization
• European Society for Mathematics and the Arts
• Goudreau Museum of Mathematics in Art and Science
• Institute For Figuring
• Mathemalchemy
• National Museum of Mathematics
Related
• Droste effect
• Mathematical beauty
• Patterns in nature
• Sacred geometry
• Category
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Viggo Brun
Viggo Brun (13 October 1885 – 15 August 1978) was a Norwegian professor, mathematician and number theorist. [1]
Viggo Brun
Born13 October 1885
Lier, Norway
Died15 August 1978
Drøbak, Norway
CitizenshipNorway
Known forBrun's Theorem, Brun Sieve
Scientific career
FieldsNumber Theory
Contributions
In 1915, he introduced a new method, based on Legendre's version of the sieve of Eratosthenes, now known as the Brun sieve, which addresses additive problems such as Goldbach's conjecture and the twin prime conjecture. He used it to prove that there exist infinitely many integers n such that n and n+2 have at most nine prime factors, and that all large even integers are the sum of two numbers with at most nine prime factors.[2]
He also showed that the sum of the reciprocals of twin primes converges to a finite value, now called Brun's constant: by contrast, the sum of the reciprocals of all primes is divergent. He developed a multi-dimensional continued fraction algorithm in 1919–1920 and applied this to problems in musical theory. He also served as praeses of the Royal Norwegian Society of Sciences and Letters in 1946.[3]
Biography
Brun was born at Lier in Buskerud, Norway. He studied at the University of Oslo and began research at the University of Göttingen in 1910. In 1923, Brun became a professor at the
Technical University in Trondheim and in 1946 a professor at the University of Oslo.[4]
He retired in 1955 at the age of 70 and died in 1978 (at 92 years-old) at Drøbak in Akershus, Norway.[5]
See also
• Brun's theorem
• Brun-Titchmarsh theorem
• Brun sieve
• Sieve theory
References
1. "Viggo Brun". numbertheory.org. 18 June 2003. Retrieved January 1, 2017.
2. J J O'Connor; E F Robertson. "Viggo Brun". School of Mathematics and Statistics, University of St Andrews, Scotland. Retrieved January 1, 2017.
3. Bratberg, Terje (1996). "Vitenskapsselskapet". In Arntzen, Jon Gunnar (ed.). Trondheim byleksikon. Oslo: Kunnskapsforlaget. pp. 599–600. ISBN 82-573-0642-8.
4. "Viggo Brun". Store norske leksikon. Retrieved January 1, 2017.
5. Bent Birkeland. "Viggo Brun". Norsk biografisk leksikon. Retrieved January 1, 2017.
Other sources
• H. Halberstam and H. E. Richert, Sieve methods, Academic Press (1974) ISBN 0-12-318250-6. Gives an account of Brun's sieve.
• C.J. Scriba, Viggo Brun, Historia Mathematica 7 (1980) 1–6.
• C.J. Scriba, Zur Erinnerung an Viggo Brun, Mitt. Math. Ges. Hamburg 11 (1985) 271-290
External links
• Brun's Constant
• Brun's Pure Sieve
• Viggo Brun personal archive exists at NTN University Library Dorabiblioteket
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Vi Hart
Victoria Hart (born 1988),[2] commonly known as Vi Hart (/ˈvaɪ hɑːrt, ˈviː hɑːrt/),[3] is an American mathematician and YouTuber. They describe themself as a "recreational mathemusician" and are well-known for creating mathematical videos on YouTube[4][5][6] and popularizing mathematics.[7][8] Hart founded the virtual reality research group eleVR and has co-authored several research papers on computational geometry and the mathematics of paper folding.[9][10]
Vi Hart
Hart in 2012, sitting on top of a finished project
Born
Victoria Hart
1988 (age 34–35)
NationalityAmerican
Occupation(s)YouTube personality, educator, inventor
Known forMathematical/musical YouTube videos
YouTube information
Channel
• Vihart
Years active2009–present
Genres
• Education
• Music
Subscribers1.44 million[1]
Total views149 million[1]
Creator Awards
100,000 subscribers
1,000,000 subscribers
Last updated: 16 Jul 2023
Together with another YouTube mathematics popularizer, Matt Parker, Hart won the 2018 Communications Award of the Joint Policy Board for Mathematics for "entertaining, thought-provoking mathematics and music videos on YouTube that explain mathematical concepts through doodles".[11]
Early life and education
Hart is the child of mathematical sculptor George W. Hart, and received a degree in music at Stony Brook University.[4] Hart identifies as "gender agnostic";[12] in a video released in 2015, they spoke about their lack of gender identity—including lacking non-binary identities such as agender—and their attitude to gendered terms such as pronouns as a "linguistic game" that they were not interested in playing. They indicated that they have no preference and do not care which pronouns they are called by.[13]
Career
Hart's career as a mathematics popularizer began in 2010 with a video series about "doodling in math class". After these recreational mathematics videos — which introduced topics like fractal dimensions — grew popular, they were featured in The New York Times and on National Public Radio,[4][14] eventually gaining the support of the Khan Academy and making videos for the educational site as their "Resident Mathemusician".[15][16] Many of Hart's videos combined mathematics and music, such as "Twelve tones", which was called "deliriously and delightfully profound" by Salon.[17]
Together with Henry Segerman, Hart wrote "The Quaternion Group as a Symmetry Group", which was included in the anthology The Best Writing on Mathematics 2015.[18]
In 2014, Hart founded a research group called eleVR, with Emily Eifler and Andrea Hawksley, to research virtual reality (VR). The group created VR videos, and had also collaborated on educational computer games.[19][20][21][3][22] They created the game Hypernom, where the player has to eat part of 4 dimensional polytopes which are stereographically projected into 3D and viewed using a virtual reality headset.[23][24] In June, eleVR released an open source web video player that worked with the Oculus Rift.[25] In the same year Hart created the playable blog post Parable of the Polygons with Nicky Case. The game was based on economist Thomas Schelling's Dynamic Models of Segregation.[20][26] In May 2016, eleVR joined Y Combinator Research (YCR) as part of the Human Advancement Research Community (HARC) project[27], in which Hart was listed as a Principal Investigator.[28]
Hart is a Senior Research Project Manager at Microsoft.[29]
References
1. "About Vihart". YouTube.
2. "Khan Academy's mathemusician Vi Hart brings dull lessons to life". Wired. Retrieved January 27, 2016.
3. "FAQ". Vi Hart.com. Archived from the original on December 13, 2014. Retrieved December 12, 2014.
4. Chang, Kenneth (January 17, 2011), "Bending and Stretching Classroom Lessons to Make Math Inspire", The New York Times.
5. Bell, Melissa (December 17, 2010), "Making math magic: Vi Hart doodles her lessons", The Washington Post.
6. Krulwich, Robert (December 16, 2010), I Hate Math! (Not After This, You Won't), NPR
7. "Weird geometry: Art enters the hyperbolic realm". New Scientist. Retrieved January 4, 2023.
8. "Parable of the Polygons". Parable of the Polygons. Retrieved January 4, 2023.
9. Vi Hart at DBLP Bibliography Server . Retrieved March 29, 2014.
10. "Reshaping the Universe: VR Landscapes Explore Mind-Bending Geometry". Live Science. March 29, 2017.
11. "Vi Hart and Matt Parker to Receive 2018 JPBM Communications Awards", News, Events and Announcements, American Mathematical Society, December 8, 2017
12. Hart, Vi [@vihartvihart] (April 30, 2014). "Fun fact: I consider myself gender agnostic. "Person," not "Woman," please. I respect your religion, but don't like having it pushed on me" (Tweet). Archived from the original on March 5, 2016 – via Twitter.
13. Hart, Vi (June 8, 2015). On Gender (Online video). YouTube.
14. "I Hate Math! (Not After This, You Won't)". NPR.org. Retrieved November 12, 2016.
15. Khan Academy (January 3, 2012), Announcement, retrieved January 7, 2018
16. Gans, Joshua (January 24, 2012). "Learning on Speed". Harvard Business Review. Retrieved January 8, 2018.
17. Leonard, Andrew (June 28, 2013). "The mad genius of Vi Hart". Salon. Retrieved January 8, 2018.
18. Hart, Vi; Segerman, Henry (January 12, 2016). "The Quaternion Group as a Symmetry Group". In Pitici, Mircea (ed.). The Best Writing on Mathematics 2015. Princeton University Press. pp. 141–153. arXiv:1404.6596. Bibcode:2014arXiv1404.6596H. ISBN 9781400873371.
19. "About Us". eleVR. Retrieved December 12, 2014.
20. Case, Nicky; Hart, Vi. "Parable of the Polygons". Retrieved December 12, 2014.
21. Bhatia, Aatish (December 8, 2014). "Empirical Zeal How Small Biases Lead to a Divided World: An Interactive Exploration of Racial Segregation". Wired.
22. "Introducing eleVR – Vi Hart". vihart.com. Archived from the original on May 23, 2022. Retrieved November 28, 2017.
23. Lawson-Perfect, Christian (July 31, 2015). "Hypernom". The Aperiodical. Retrieved April 5, 2016.
24. Hart, Vi; Hawksley, Andrea; Segerman, Henry; Bosch, Marc ten (July 21, 2015). "Hypernom: Mapping VR Headset Orientation to S^3". Proceedings of Bridges 2015: Mathematics, Music, Art, Architecture, Culture. pp. 387–390. arXiv:1507.05707. Bibcode:2015arXiv150705707H.
25. "eleVR: the first web video player for virtual reality".
26. Farokhmanesh, Megan (December 11, 2014). "A visual guide to bias, as explained by adorable shapes". Polygon.
27. "eleVR leaving YCR – elevr". elevr.com.
28. Altman, Sam (May 11, 2016). "HARC". Y Combinator Blog. Retrieved June 20, 2016.
29. Allen, Danielle (April 21, 2020). "Roadmap to Pandemic Resilience" (PDF). Edmond J. Safra Center for Ethics. Harvard University. Retrieved April 21, 2020.
External links
Wikimedia Commons has media related to Vi Hart.
• Vi Hart's channel on YouTube
• Vi Hart's second's channel on YouTube
• "Vi Hart". Khan Academy. Archived from the original on January 28, 2016.{{cite web}}: CS1 maint: unfit URL (link)
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Vikram Bhagvandas Mehta
Vikram Bhagvandas Mehta (August 15, 1946 – June 4, 2014) was an Indian mathematician who worked on algebraic geometry and vector bundles. Together with Annamalai Ramanathan he introduced the notion of Frobenius split varieties, which led to the solution of several problems about Schubert varieties.[1] He is also known to have worked, from the 2000s onward, on the fundamental group scheme. It was precisely in the year 2002 when he and Subramanian published a proof of a conjecture by Madhav V. Nori[2] that brought back into the limelight the theory of an object that until then had met with little success.[3]
Awards
The Council of Scientific and Industrial Research awarded him the Shanti Swarup Bhatnagar Prize for Science and Technology in 1991 for his work in algebraic geometry.[4]
References
1. Bhagvandas Mehta, Vikram; Ramanathan, Annamalai (July 1985). "Frobenius splitting and cohomology vanishing for Schubert varieties". Annals of Mathematics. 122 (1): 27–40. doi:10.2307/1971368. ISSN 0003-486X. JSTOR 1971368 – via JSTOR.
2. M. V. Nori On the Representations of the Fundamental Group, Compositio Mathematica, Vol. 33, Fasc. 1, (1976), p. 29-42
3. V. B. Mehta, S. Subramanian On the Fundamental Group Scheme, Inventiones mathematicae, 148, 143-150 (2002)
4. "Awardee Details: Shanti Swarup Bhatnagar Prize". ssbprize.gov.in. Retrieved 19 October 2020.
External links
• Vikram Bhagvandas Mehta citation
Recipients of Shanti Swarup Bhatnagar Prize for Science and Technology in Mathematical Science
1950s–70s
• K. S. Chandrasekharan & C. R. Rao (1959)
• K. G. Ramanathan (1965)
• A. S. Gupta & C. S. Seshadri (1972)
• P. C. Jain & M. S. Narasimhan (1975)
• K. R. Parthasarathy & S. K. Trehan (1976)
• M. S. Raghunathan (1977)
• E. M. V. Krishnamurthy (1978)
• S. Raghavan & S. Ramanan (1979)
1980s
• R. Sridharan (1980)
• J. K. Ghosh (1981)
• B. L. S. Prakasa Rao & J. B. Shukla (1982)
• I. B. S. Passi & Phoolan Prasad (1983)
• S. K. Malik & R. Parthasarathy (1985)
• T. Parthasarathy & U. B. Tewari (1986)
• Raman Parimala & T. N. Shorey (1987)
• M. B. Banerjee & K. B. Sinha (1988)
• Gopal Prasad (1989)
1990s
• R. Balasubramanian & S. G. Dani (1990)
• V. B. Mehta & A. Ramanathan (1991)
• Maithili Sharan (1992)
• Karmeshu & Navin M. Singhi (1993)
• N. Mohan Kumar (1994)
• Rajendra Bhatia (1995)
• V. S. Sunder (1996)
• Subhashis Nag & T. R. Ramadas (1998)
• Rajeeva Laxman Karandikar (1999)
2000s
• Rahul Mukerjee (2000)
• Gadadhar Misra & T. N. Venkataramana (2001)
• Dipendra Prasad & S. Thangavelu (2002)
• Manindra Agrawal & V. Srinivas (2003)
• Arup Bose & Sujatha Ramdorai (2004)
• Probal Chaudhuri & K. H. Paranjape (2005)
• Vikraman Balaji & Indranil Biswas (2006)
• B. V. Rajarama Bhat (2007)
• Rama Govindarajan (2007)
• Jaikumar Radhakrishnan (2008)
• Suresh Venapally (2009)
2010s
• Mahan Mitra & Palash Sarkar (2011)
• Siva Athreya & Debashish Goswami (2012)
• Eknath Prabhakar Ghate (2013)
• Kaushal Kumar Verma (2014)
• K Sandeep & Ritabrata Munshi (2015)
• Amalendu Krishna (2016)
• Naveen Garg (2016)
• (Not awarded) (2017)
• Amit Kumar & Nitin Saxena (2018)
• Neena Gupta & Dishant Mayurbhai Pancholi (2019)
2020s
• Rajat Subhra Hazra (2020)
• U. K. Anandavardhanan (2020)
• Anish Ghosh (2021)
• Saket Saurabh (2021)
Authority control: Academics
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Viktor Maslov (mathematician)
Viktor Pavlovich Maslov (Russian: Виктор Павлович Маслов; 15 June 1930 – 3 August 2023) was a Russian mathematical physicist. He was a member of the Russian Academy of Sciences. He obtained his doctorate in physico-mathematical sciences in 1957.[2] His main fields of interest were quantum theory, idempotent analysis, non-commutative analysis, superfluidity, superconductivity, and phase transitions. He was editor-in-chief of Mathematical Notes and Russian Journal of Mathematical Physics.
Viktor Maslov
Виктор Маслов
Born
Viktor Pavlovich Maslov
(1930-06-15)15 June 1930
Moscow, Russian SFSR, USSR
Died3 August 2023(2023-08-03) (aged 93)
NationalityRussian
CitizenshipRussian
Alma materLomonosov Moscow State University
Known forMaslov index
Spouse
Lê Vũ Anh
(m. 1975–1981)
AwardsState Prize of the USSR (1978); A.M.Lyapunov Gold Medal (USSR Academy of Science, 1982); Lenin Prize (1985); State Prize of the Russian Federation (1997); Demidov prize (2000); Independent Russian Triumph Prize (2002); State Prize of the Russian Federation (2013)[1]
Scientific career
Fieldsphysico-mathematics
InstitutionsLomonosov Moscow State University
Doctoral advisorSergei Fomin[2]
The Maslov index is named after him. He also introduced the concept of Lagrangian submanifold.[2]
Early life and career
Viktor Pavlovich Maslov was born in Moscow on 15 June 1930. He was the son of statistician Pavel Maslov and researcher Izolda Lukomskaya, and the grandson of the economist and agriculturalist Petr Maslov. At the beginning of World War II, he was evacuated to Kazan with his mother, grandmother and other members of his mother's family.[3]
In 1953 he graduated from the Physics Department of the Moscow State University and taught at the university. In 1957 he defended his Ph.D. thesis and in 1966, his doctoral dissertation. In 1984, he was elected an academician within Department of Mathematics of the Academy of Sciences of the USSR.[4]
From 1968 to 1998, he headed the Department of Applied Mathematics at the Moscow Institute of Electronics and Mathematics. From 1992 to 2016, he was in charge of the Department of Quantum Statistics and Field Theory of the Physics Faculty of Moscow State University.[4]
Maslov headed the laboratory of the mechanics of natural disasters at the Institute for Problems in Mechanics of the Russian Academy of Sciences. He was a research professor at the Department of Applied Mathematics at Moscow Institute of Electronics and Mathematics of Higher School of Economics.[4]
Scientific acitivity
Maslov was known as a prominent specialist in the field of mathematical physics, differential equations, functional analysis, mechanics and quantum physics. He developed asymptotic methods that are widely applied to equations arising in quantum mechanics, field theory, statistical physics and abstract mathematics, that bear his name.[5]
Maslov's asymptotic methods are closely related to such problems as the theory of a self-consistent field in quantum and classical statistics, superfluidity and superconductivity, quantization of solitons, quantum field theory in strong external fields and in curved space-time, the method of expansion in the inverse number of particle types. In 1983, he attended the International Congress of Mathematicians in Warsaw, where he presented a plenary report "Non-standard characteristics of asymptotic problems".[6]
Maslov dealt with the problems of liquid and gas, carried out fundamental research on the problems of magnetohydrodynamics related to the dynamo problem. He also made calculations for the emergency unit of the Chernobyl nuclear power plant during the 1986 disaster. In 1991, he made model and forecasts of the economic situation in Russia.[6]
From the early 1990s, he worked on the use of equations of mathematical physics in economics and financial analysis. In particular, he managed to predict the 1998 Russian financial crisis, and even earlier, the collapse of the economic and, as a consequence, the collapse of the political system of the USSR.[6]
In 2008, Maslov in his own words predicted a global recession in the late 2000s. He calculated the critical number of U.S. debt, and found out that a crisis should break out in the near future. In the calculations, he used equations similar to the equations of phase transition in physics. In the mid-1980s, Maslov introduced the term tropical mathematics, in which the operations of the conditional optimization problem were considered.[7]
Personal life
In the early 1970s, he met Lê Vũ Anh, the daughter of Lê Duẩn, then General Secretary of the Communist Party of Vietnam, when she was a student at the Faculty of Physics in Moscow State University. The romance was considered scandalous because Vietnamese students studying abroad were not allowed to have romantic relationships with foreigners and anyone caught would have to be disciplined and may be sent back to Vietnam. In order to avoid trouble, she returned home to marry a Vietnamese student from the same university and wanted to stay in Vietnam to forget her love affair with Maslov. However, she was forced by her father to return to USSR to complete her studies.[8]
When she and her husband returned to Moscow, Anh realized that she did not love her husband and could not forget her former lover. She decided to live separately from her husband and secretly went back and forth with Maslov. After being pregnant for the second time, after having a miscarriage for the first time, Anh had enough energy to ask her husband for a divorce in order to be able to marry Maslov. In 1975, she and Maslov married. She gave birth to a daughter on 31 October 1977 named Lena. Meeting her father by chance when he went to USSR for a state visit, Anh confessed all her love affairs. Lê Duẩn did not accept it and tried to lure her back to the country. However, Anh gradually reconciled with her family.[8]
After giving birth to her second daughter Tania, Anh gave birth to her son, Anton, on 1981. Anh died shortly after giving birth to her son, due to hemorrhage.[9]
Immediately after Anh died, a dispute over custody of his three children with his wife's family occurred. An official from the Vietnamese Communist Party's Central Committee took over the communication between Maslov and Anh's family. Both sides proposed a compromise solution, Maslov kept his daughters and son would be returned to Lê Duẩn. Maslov only allowed his son to go to Vietnam for two years. But after the deadline, his son never returned to him. Maslov had to fight for two more years before Lê Duẩn accepted to bring his grandson to meet his father.[10]
However, the son that Maslov met was no longer Anton Maslov as before, but a Vietnamese citizen with the new name Nguyễn An Hoàn and he was unable to speak Russian. According to Maslov, Lê Duẩn did not intend to return the child, but also hoped to bring back his daughters. Fearing the loss of his children, Maslov contacted the son of the President of the Supreme Soviet of the Soviet Union Andrei Gromyko, a close friend of Soviet leader Mikhail Gorbachev. He was advised to write to Gorbachev and was promised to convince Gorbachev to read it. After a massive legal struggle, Lê Duẩn gave up the idea of taking him and his children back.[9]
His children later resided in England and the Netherlands, where they were highly successful in their respective professions.[9]
Maslov later re-married a woman named Irina, who was at the same age as his ex-wife Anh. Irina is a linguist and she received the title of Associate Doctor of Science in 1991. For the last three decades, he lived in Troitsk.[9]
Viktor Maslow died on 3 August 2023, at the age of 93.[11]
Selected books
• Maslov, V. P. (1972). Théorie des perturbations et méthodes asymptotiques. Dunod; 384 pages{{cite book}}: CS1 maint: postscript (link)[12]
• Karasëv, M. V.; Maslov, V. P.: Nonlinear Poisson brackets. Geometry and quantization. Translated from the Russian by A. Sossinsky [A. B. Sosinskiĭ] and M. Shishkova. Translations of Mathematical Monographs, 119. American Mathematical Society, Providence, RI, 1993.[13]
• Kolokoltsov, Vassili N.; Maslov, Victor P.: Idempotent analysis and its applications. Translation of Idempotent analysis and its application in optimal control (Russian), "Nauka" Moscow, 1994. Translated by V. E. Nazaikinskii. With an appendix by Pierre Del Moral. Mathematics and its Applications, 401. Kluwer Academic Publishers Group, Dordrecht, 1997.
• Maslov, V. P.; Fedoriuk, M. V.: Semi-classical approximation in quantum mechanics. Translated from the Russian by J. Niederle and J. Tolar. Mathematical Physics and Applied Mathematics, 7. Contemporary Mathematics, 5. D. Reidel Publishing Co., Dordrecht-Boston, Mass., 1981.[14]
This book was cited over 700 times at Google Scholar in 2011.
• Maslov, V. P. Operational methods. Translated from the Russian by V. Golo, N. Kulman and G. Voropaeva. Mir Publishers, Moscow, 1976.
References
1. Viktor Maslov, HSE
2. "Fiftieth Anniversary of research and teaching by Viktor Pavlovich Maslov" (PDF).
3. "The Central Database of Shoah Victims' Names". www.yvng.yadvashem.org. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
4. "Маслов, Виктор Павлович". www.tass.ru. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
5. Proceedings of the International Congress of Mathematicians. August 16-24, 1983, Warszawa
6. "Академику Маслову Виктору Павловичу - 90 лет!". www.ras.ru. 2020-06-15. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
7. Medvedev, Yuri (2009-03-12). "Он рассчитал катастрофy". www.rg.ru. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
8. "Về câu chuyện tình của con gái Tổng Bí thư Lê Duẩn với viện sĩ khoa học Nga". www.cand.com.vn. 2016-08-26. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
9. "Hồi ký của VS Maslov về mối tình với Lê Vũ Anh". www.nguoivietodessa.com. 2016-08-28. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
10. "Câu chuyện tình buồn bí mật của Lê Vũ Anh con gái ông Lê Duẩn lấy chồng người Nga". www.ttx.vanganh.org. Retrieved 2021-07-29.{{cite web}}: CS1 maint: url-status (link)
11. Умер Виктор Павлович Маслов (in Russian)
12. Streater, R. F. (1975). "Review of Théorie des perturbations et méthodes asymptotiques by V. P. Maslov". Bulletin of the London Mathematical Society. 7 (3): 334. doi:10.1112/blms/7.3.334. ISSN 0024-6093.
13. Libermann, P. (1996). "Book Review: Nonlinear Poisson brackets, geometry and quantization". Bulletin of the American Mathematical Society. 33: 101–106. doi:10.1090/S0273-0979-96-00619-2.
14. Blattner, Robert J.; Ralston, James (1983). "joint review of Lagrangian analysis and quantum mechanics, a mathematical structure related to asymptotic expansions and the Maslow index by Jean Leray; Semi-classical approximation in quantum mechanics by V. P. Maslow and M. V. Fedoriuk". Bulletin of the American Mathematical Society. 9 (3): 387–397. doi:10.1090/S0273-0979-1983-15224-2.
External links
• Viktor Maslov at the Mathematics Genealogy Project
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Viktor Valentinovich Novozhilov
Viktor Valentinovich Novozhilov (Russian: Виктор Валентинович Новожилов) (27 October [O.S. 15 October] 1892 – 15 August 1970) was a Soviet economist and mathematician, known for his development of techniques for the mathematical analysis of economic phenomena. He was awarded the Lenin Prize (1965) and served as head of the Laboratory for Economic Assessment Systems at the Leningrad office of the Central Economic Mathematical Institute.
Viktor Valentinovich Novozhilov
Born(1892-10-27)27 October 1892
Kharkov, Kharkov Governorate, Russian Empire (now Ukraine)
Died15 August 1970(1970-08-15) (aged 77)
Leningrad, Soviet Union
Alma materSt. Volodymyr Kyiv University
Scientific career
FieldsEconomics
Biography
Novozhilov graduated from high school with a gold medal in 1911 and entered the St. Volodymyr Kyiv University, which he completed in 1915, becoming an assistant professor in political economics and statistics. In 1922, he moved to the Leningrad Polytechnic Institute, where he served as the head of the department of Auto Industry Economics from 1938 to 1951. From 1951 to 1966, Novozhilov was the head of the Statistics Department at the Leningrad Engineering and Economics Institute.
He was part of the government-sponsored team engaged in economic reform analysis in the 1920s in the Soviet Union. He performed extensive research in the field of economic analysis for agriculture and made specific recommendations regarding optimal investment levels in a socialist agricultural setting.
External links
• Paper on Novozhilov's contribution
• Alternative encyclopedic article
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William Feller
William "Vilim" Feller (July 7, 1906 – January 14, 1970), born Vilibald Srećko Feller, was a Croatian–American mathematician specializing in probability theory.
William Feller
Born
Vilibald Srećko Feller
July 7, 1906 (1906-07-07)
Zagreb, Austro-Hungarian Monarchy (now Croatia)
DiedJanuary 14, 1970 (1970-01-15) (aged 63)
New York City, US
NationalityCroatian–American
Alma materUniversity of Zagreb
University of Göttingen
Known forFeller process
Feller's coin-tossing constants
Feller-continuous process
Feller's paradox
Feller's theorem
Feller–Pareto distribution
Feller–Tornier constant
Feller–Miyadera–Phillips theorem
Proof by intimidation
Stars and bars
AwardsNational Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1969)
Scientific career
FieldsMathematician
InstitutionsUniversity of Kiel
University of Copenhagen
University of Stockholm
University of Lund
Brown University
Cornell University
Princeton University
Doctoral advisorRichard Courant
Doctoral studentsPatrick Billingsley
George Forsythe
Robert Kurtz
Henry McKean
Lawrence Shepp
Hale Trotter
Benjamin Weiss
David A. Freedman
InfluencesStanko Vlögel
Signature
Early life and education
Feller was born in Zagreb to Ida Oemichen-Perc, a Croatian–Austrian Catholic, and Eugen Viktor Feller, son of a Polish–Jewish father (David Feller) and an Austrian mother (Elsa Holzer).[1]
Eugen Feller was a famous chemist and created Elsa fluid named after his mother. According to Gian-Carlo Rota, Eugen Feller's surname was a "Slavic tongue twister", which William changed at the age of twenty.[2] This claim appears to be false. His forename, Vilibald, was chosen by his Catholic mother for the saint day of his birthday.[3]
Work
Feller held a docent position at the University of Kiel beginning in 1928. Because he refused to sign a Nazi oath,[4] he fled the Nazis and went to Copenhagen, Denmark in 1933. He also lectured in Sweden (Stockholm and Lund).[5] As a refugee in Sweden, Feller reported being troubled by increasing fascism at the universities. He reported that the mathematician Torsten Carleman would offer his opinion that Jews and foreigners should be executed.[6]
Finally, in 1939 he arrived in the U.S., where he became a citizen in 1944 and was on the faculty at Brown and Cornell. In 1950 he became a professor at Princeton University.
The works of Feller are contained in 104 papers and two books on a variety of topics such as mathematical analysis, theory of measurement, functional analysis, geometry, and differential equations in addition to his work in mathematical statistics and probability.
Feller was one of the greatest probabilists of the twentieth century. He is remembered for his championing of probability theory as a branch of mathematical analysis in Sweden and the United States. In the middle of the 20th century, probability theory was popular in France and Russia, while mathematical statistics was more popular in the United Kingdom and the United States, according to the Swedish statistician, Harald Cramér.[7] His two-volume textbook on probability theory and its applications was called "the most successful treatise on probability ever written" by Gian-Carlo Rota.[8] By stimulating his colleagues and students in Sweden and then in the United States, Feller helped establish research groups studying the analytic theory of probability. In his research, Feller contributed to the study of the relationship between Markov chains and differential equations, where his theory of generators of one-parameter semigroups of stochastic processes gave rise to the theory of "Feller operators".
Results
Numerous topics relating to probability are named after him, including Feller processes, Feller's explosion test, Feller–Brown movement, and the Lindeberg–Feller theorem. Feller made fundamental contributions to renewal theory, Tauberian theorems, random walks, diffusion processes, and the law of the iterated logarithm. Feller was among those early editors who launched the journal Mathematical Reviews.
Notable books
• An Introduction to Probability Theory and its Applications, Volume I, 3rd edition (1968); 1st edn. (1950);[9] 2nd edn. (1957)[10]
• An Introduction to Probability Theory and its Applications, Volume II, 2nd edition (1971)
Recognition
In 1949, Feller was named a Fellow of the American Statistical Association.[11] He was elected to the American Academy of Arts and Sciences in 1958, the United States National Academy of Sciences in 1960, and the American Philosophical Society in 1966.[12][13][14] Feller won the National Medal of Science in 1969. He was president of the Institute of Mathematical Statistics.
See also
• Feller condition
• Beta distribution
• Compound Poisson distribution
• Gillespie algorithm
• Kolmogorov equations
• Poisson point process
• Stability (probability)
• St. Petersburg paradox
• Stochastic process
References
1. Zubrinic, Darko (2006). "William Feller (1906-1970)". Croatianhistory.net. Accessed 3 July 2018.
2. Rota, Gian-Carlo (1996). Indiscrete Thoughts. Birkhäuser. ISBN 0-8176-3866-0.
3. O'Connor, John J.; Robertson, Edmund F., "William Feller", MacTutor History of Mathematics Archive, University of St Andrews
4. "Biography of William Feller". History of William Feller. Retrieved 2006-06-27.
5. Siegmund-Schultze, Reinhard (2009). Mathematicians fleeing from Nazi Germany: Individual fates and global impact. Princeton, New Jersey: Princeton University Press. pp. xxviii+471. ISBN 978-0-691-14041-4. MR 2522825.
6. (Siegmund-Schultze 2009, p. 135)
7. Preface to his Mathematical Methods of Statistics.
8. Page 199: Indiscrete Thoughts.
9. Wolfowitz, J. (1951). "Review: An introduction to probability theory and its applications, Vol. I, 1st ed., by W. Feller" (PDF). Bull. Amer. Math. Soc. 57 (2): 156–159. doi:10.1090/s0002-9904-1951-09491-4.
10. "Review: An introduction to probability theory and its applications, Vol. I, 2nd ed., by W. Feller" (PDF). Bull. Amer. Math. Soc. 64 (6): 393. 1958. doi:10.1090/s0002-9904-1958-10252-9.
11. "View/Search Fellows of the ASA". American Statistical Association. Retrieved 2016-07-22.
12. "William Feller". American Academy of Arts & Sciences. Retrieved 2022-09-29.
13. "William Feller". www.nasonline.org. Retrieved 2022-09-29.
14. "APS Member History". search.amphilsoc.org. Retrieved 2022-09-29.
External links
Wikiquote has quotations related to William Feller.
• William Feller at the Mathematics Genealogy Project
• A biographical memoir by Murray Rosenblatt
• Croatian Giants of Science - in Croatian
• O'Connor, John J.; Robertson, Edmund F., "William Feller", MacTutor History of Mathematics Archive, University of St Andrews
• "Fine Hall in its golden age: Remembrances of Princeton in the early fifties" by Gian-Carlo Rota. Contains a section on Feller at Princeton.
• Feller Matriculation Form giving personal details
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Villarceau circles
In geometry, Villarceau circles (/viːlɑːrˈsoʊ/) are a pair of circles produced by cutting a torus obliquely through the center at a special angle.
Given an arbitrary point on a torus, four circles can be drawn through it. One is in a plane parallel to the equatorial plane of the torus and another perpendicular to that plane (these are analogous to lines of latitude and longitude on the Earth). The other two are Villarceau circles. They are obtained as the intersection of the torus with a plane that passes through the center of the torus and touches it tangentially at two antipodal points. If one considers all these planes, one obtains two families of circles on the torus. Each of these families consists of disjoint circles that cover each point of the torus exactly once and thus forms a 1-dimensional foliation of the torus.
The Villarceau circles are named after the French astronomer and mathematician Yvon Villarceau (1813–1883) who wrote about them in 1848.
Mannheim (1903) showed that the Villarceau circles meet all of the parallel circular cross-sections of the torus at the same angle, a result that he said a Colonel Schoelcher had presented at a congress in 1891.
Example
Consider a horizontal torus in xyz space, centered at the origin and with major radius 5 and minor radius 3. That means that the torus is the locus of some vertical circles of radius three whose centers are on a circle of radius five in the horizontal xy plane. Points on this torus satisfy this equation:
$0=(x^{2}+y^{2}+z^{2}+16)^{2}-100(x^{2}+y^{2}).\,\!$
Slicing with the z = 0 plane produces two concentric circles, x2 + y2 = 22 and x2 + y2 = 82, the outer and inner equator. Slicing with the x = 0 plane produces two side-by-side circles, (y − 5)2 + z2 = 32 and (y + 5)2 + z2 = 32.
Two example Villarceau circles can be produced by slicing with the plane 3x = 4z. One is centered at (0, +3, 0) and the other at (0, −3, 0); both have radius five. They can be written in parametric form as
$(x,y,z)=(4\cos \vartheta ,+3+5\sin \vartheta ,3\cos \vartheta )\,\!$
and
$(x,y,z)=(4\cos \vartheta ,-3+5\sin \vartheta ,3\cos \vartheta ).\,\!$
The slicing plane is chosen to be tangent to the torus at two points while passing through its center. It is tangent at (16⁄5, 0, 12⁄5) and at (−16⁄5, 0, −12⁄5). The angle of slicing is uniquely determined by the dimensions of the chosen torus. Rotating any one such plane around the z-axis gives all of the Villarceau circles for that torus.
Existence and equations
A proof of the circles’ existence can be constructed from the fact that the slicing plane is tangent to the torus at two points. One characterization of a torus is that it is a surface of revolution. Without loss of generality, choose a coordinate system so that the axis of revolution is the z axis. Begin with a circle of radius r in the xz plane, centered at (R, 0, 0).
$0=(x-R)^{2}+z^{2}-r^{2}\,\!$
Sweeping replaces x by (x2 + y2)1/2, and clearing the square root produces a quartic equation.
$0=(x^{2}+y^{2}+z^{2}+R^{2}-r^{2})^{2}-4R^{2}(x^{2}+y^{2}).\,\!$
The cross-section of the swept surface in the xz plane now includes a second circle.
$0=(x+R)^{2}+z^{2}-r^{2}\,\!$
This pair of circles has two common internal tangent lines, with slope at the origin found from the right triangle with hypotenuse R and opposite side r (which has its right angle at the point of tangency). Thus z/x equals ±r / (R2 − r2)1/2, and choosing the plus sign produces the equation of a plane bitangent to the torus.
$0=xr-z{\sqrt {R^{2}-r^{2}}}\,\!$
By symmetry, rotations of this plane around the z axis give all the bitangent planes through the center. (There are also horizontal planes tangent to the top and bottom of the torus, each of which gives a “double circle”, but not Villarceau circles.)
$0=xr\cos \varphi +yr\sin \varphi -z{\sqrt {R^{2}-r^{2}}}\,\!$
We can calculate the intersection of the plane(s) with the torus analytically, and thus show that the result is a symmetric pair of circles, one of which is a circle of radius R centered at
$(-r\sin \varphi ,r\cos \varphi ,0).\,\!$
A treatment along these lines can be found in Coxeter (1969).
A more abstract — and more flexible — approach was described by Hirsch (2002), using algebraic geometry in a projective setting. In the homogeneous quartic equation for the torus,
$0=(x^{2}+y^{2}+z^{2}+R^{2}w^{2}-r^{2}w^{2})^{2}-4R^{2}w^{2}(x^{2}+y^{2}),\,\!$
setting w to zero gives the intersection with the “plane at infinity”, and reduces the equation to
$0=(x^{2}+y^{2}+z^{2})^{2}.\,\!$
This intersection is a double point, in fact a double point counted twice. Furthermore, it is included in every bitangent plane. The two points of tangency are also double points. Thus the intersection curve, which theory says must be a quartic, contains four double points. But we also know that a quartic with more than three double points must factor (it cannot be irreducible), and by symmetry the factors must be two congruent conics. Hirsch extends this argument to any surface of revolution generated by a conic, and shows that intersection with a bitangent plane must produce two conics of the same type as the generator when the intersection curve is real.
Filling space
The torus plays a central role in the Hopf fibration of the 3-sphere, S3, over the ordinary sphere, S2, which has circles, S1, as fibers. When the 3-sphere is mapped to Euclidean 3-space by stereographic projection, the inverse image of a circle of latitude on S2 under the fiber map is a torus, and the fibers themselves are Villarceau circles.[1] Banchoff has explored such a torus with computer graphics imagery. One of the unusual facts about the circles is that each links through all the others, not just in its own torus but in the collection filling all of space; Berger has a discussion and drawing.[2]
See also
• Toric section
• Vesica piscis
Citations
1. Dorst 2019, §6. Hopf Fibration and Stereographic Projection from 4D.
2. Berger 1987, pp. 304–305, §18.9: Villarceau circles and parataxy.
References
• Banchoff, Thomas F. (1990). Beyond the Third Dimension. Scientific American Library. ISBN 978-0-7167-5025-3.
• Berger, Marcel (1987). Geometry II. Springer. ISBN 978-3-540-17015-0.
• Coxeter, H. S. M. (1969). Introduction to Geometry (2/e ed.). Wiley. pp. 132–133. ISBN 978-0-471-50458-0.
• Hirsch, Anton (2002). "Extension of the 'Villarceau-Section' to Surfaces of Revolution with a Generating Conic". Journal for Geometry and Graphics. Lemgo, Germany: Heldermann Verlag. 6 (2): 121–132. ISSN 1433-8157.
• Mannheim, M. A. (1903). "Sur le théorème de Schoelcher". Nouvelles Annales de Mathématiques. Paris: Carilian-Gœury et Vor. Dalmont. 4th series, volume 3: 105–107.
• Stachel, Hellmuth (2002). "Remarks on A. Hirsch's Paper concerning Villarceau Sections". Journal for Geometry and Graphics. Lemgo, Germany: Heldermann Verlag. 6 (2): 133–139. ISSN 1433-8157.
• Yvon Villarceau, Antoine Joseph François (1848). "Théorème sur le tore". Nouvelles Annales de Mathématiques. Série 1. Paris: Gauthier-Villars. 7: 345–347. OCLC: 2449182.
• Dorst, Leo (2019). "Conformal Villarceau Rotors". Advances in Applied Clifford Algebras. 29 (44). doi:10.1007/s00006-019-0960-5. S2CID 253592159.
External links
Wikimedia Commons has media related to Villarceau circles.
• Weisstein, Eric W. "Villarceau Circles". MathWorld.
• Flat Torus in the Three-Sphere
• (in French) The circles of the torus (Les cercles du tore)
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Ville's inequality
In probability theory, Ville's inequality provides an upper bound on the probability that a supermartingale exceeds a certain value. The inequality is named after Jean Ville, who proved it in 1939.[1][2][3][4] The inequality has applications in statistical testing.
Statement
Let $X_{0},X_{1},X_{2},\dots $ be a non-negative supermartingale. Then, for any real number $a>0,$
$\operatorname {P} \left[\sup _{n\geq 0}X_{n}\geq a\right]\leq {\frac {\operatorname {E} [X_{0}]}{a}}\ .$
The inequality is a generalization of Markov's inequality.
References
1. Ville, Jean (1939). Etude Critique de la Notion de Collectif (PDF) (Thesis).
2. Durrett, Rick (2019). Probability Theory and Examples (Fifth ed.). Exercise 4.8.2: Cambridge University Press.{{cite book}}: CS1 maint: location (link)
3. Howard, Steven R. (2019). Sequential and Adaptive Inference Based on Martingale Concentration (Thesis).
4. Choi, K. P. (1988). "Some sharp inequalities for Martingale transforms". Transactions of the American Mathematical Society. 307 (1): 279–300. doi:10.1090/S0002-9947-1988-0936817-3. S2CID 121892687.
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Vilma Mesa
Vilma María Mesa Narváez (born 1963)[1] is a Colombian-American mathematics educator whose research topics have included secondary-school curriculum development, college-level calculus instruction, mathematics in community colleges, international perspectives in mathematics education, and inquiry-based learning.[2] She is a professor of education and mathematics at the University of Michigan, where she is affiliated with the Center for the Study of Higher and Post-secondary Education.[3]
Education and career
Mesa earned her bachelor's degrees in computer science and mathematics at the University of Los Andes (Colombia) in 1986 and 1987, respectively,[4] and became a computer programmer for the Colombian government and in the private sector.[3] From 1988 to 1995 she worked as a researcher at the University of Los Andes,[4] working in mathematics education and authoring textbooks on mathematics and statistics for applications including engineering and social sciences.[3]
In 1996, she began graduate study in mathematics education at the University of Georgia, where she earned her master's degree in 1996 and completed her Ph.D. in 2000.[4] Her dissertation, Conceptions of Function Promoted by Seventh- and Eighth-Grade Textbooks from Eighteen Countries, was jointly advised by Jeremy Kilpatrick and Edward Arthur Azoff.[5]
After postdoctoral research at the University of Michigan, she stayed on at the University of Michigan as a coordinator for the master's program in curriculum development and as an instructional consultant until she was hired in 2005 as an assistant professor of mathematics education in the School of Education. She was tenured in 2014 and added a joint appointment in the university's mathematics department in 2015.[4]
In 2016, she visited the University of Santiago, Chile as a Fulbright Scholar.[4][3]
Recognition
Mesa is the 2022 winner of the Louise Hay Award for Contributions to Mathematics Education, where she was recognised "for her distinguished contributions to mathematics education research at the collegiate level, for her teaching and mentorship, and as an advocate for access to mathematics for women and members of underprivileged populations."[2]
References
1. Birth year from WorldCat identities, retrieved 2022-02-02
2. "Louise Hay Award, 2022 Winner: Vilma Mesa", Awards, Association for Women in Mathematics, 1 February 2022, retrieved 2022-02-02
3. "Vilma Mesa", Calendar 2019, Lathisms, retrieved 2022-02-02
4. Curriculum vitae, retrieved 2022-02-02
5. Vilma Mesa at the Mathematics Genealogy Project
External links
• Home page
• Vilma Mesa publications indexed by Google Scholar
Authority control: Academics
• Mathematics Genealogy Project
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Vincent's theorem
In mathematics, Vincent's theorem—named after Alexandre Joseph Hidulphe Vincent—is a theorem that isolates the real roots of polynomials with rational coefficients.
Even though Vincent's theorem is the basis of the fastest method for the isolation of the real roots of polynomials, it was almost totally forgotten, having been overshadowed by Sturm's theorem; consequently, it does not appear in any of the classical books on the theory of equations (of the 20th century), except for Uspensky's book. Two variants of this theorem are presented, along with several (continued fractions and bisection) real root isolation methods derived from them.
Sign variation
Let c0, c1, c2, ... be a finite or infinite sequence of real numbers. Suppose l < r and the following conditions hold:
1. If r = l+1 the numbers cl and cr have opposite signs.
2. If r ≥ l+2 the numbers cl+1, ..., cr−1 are all zero and the numbers cl and cr have opposite signs.
This is called a sign variation or sign change between the numbers cl and cr.
When dealing with the polynomial p(x) in one variable, one defines the number of sign variations of p(x) as the number of sign variations in the sequence of its coefficients.
Two versions of this theorem are presented: the continued fractions version due to Vincent,[1][2][3] and the bisection version due to Alesina and Galuzzi.[4][5]
Vincent's theorem: Continued fractions version (1834 and 1836)
If in a polynomial equation with rational coefficients and without multiple roots, one makes successive transformations of the form
$x=a_{1}+{\frac {1}{x'}},\quad x'=a_{2}+{\frac {1}{x''}},\quad x''=a_{3}+{\frac {1}{x'''}},\ldots $
where $a_{1},a_{2},a_{3},\ldots $ are any positive numbers greater than or equal to one, then after a number of such transformations, the resulting transformed equation either has zero sign variations or it has a single sign variation. In the first case there is no root, whereas in the second case there is a single positive real root. Furthermore, the corresponding root of the proposed equation is approximated by the finite continued fraction:[1][2][3]
$a_{1}+{\cfrac {1}{a_{2}+{\cfrac {1}{a_{3}+{\cfrac {1}{\ddots }}}}}}$
Moreover, if infinitely many numbers $a_{1},a_{2},a_{3},\ldots $ satisfying this property can be found, then the root is represented by the (infinite) corresponding continued fraction.
The above statement is an exact translation of the theorem found in Vincent's original papers;[1][2][3] however, the following remarks are needed for a clearer understanding:
• If $f_{n}(x)$ denotes the polynomial obtained after n substitutions (and after removing the denominator), then there exists N such that for all $n\geq N$ either $f_{n}(x)$ has no sign variation or it has one sign variation. In the latter case $f_{n}(x)$ has a single positive real root for all $n\geq N$.
• The continued fraction represents a positive root of the original equation, and the original equation may have more than one positive root. Moreover, assuming $a_{1}\geq 1$, we can only obtain a root of the original equation that is > 1. To obtain an arbitrary positive root we need to assume that $a_{1}\geq 0$.
• Negative roots are obtained by replacing x by −x, in which case the negative roots become positive.
Vincent's theorem: Bisection version (Alesina and Galuzzi 2000)
Let p(x) be a real polynomial of degree deg(p) that has only simple roots. It is possible to determine a positive quantity δ so that for every pair of positive real numbers a, b with $|b-a|<\delta $, every transformed polynomial of the form
$f(x)=(1+x)^{\deg(p)}p\left({\frac {a+bx}{1+x}}\right)$
(1)
has exactly 0 or 1 sign variations. The second case is possible if and only if p(x) has a single root within (a, b).
The Alesina–Galuzzi "a_b roots test"
From equation (1) the following criterion is obtained for determining whether a polynomial has any roots in the interval (a, b):
Perform on p(x) the substitution
$x\leftarrow {\frac {a+bx}{1+x}}$
and count the number of sign variations in the sequence of coefficients of the transformed polynomial; this number gives an upper bound on the number of real roots p(x) has inside the open interval (a, b). More precisely, the number ρab(p) of real roots in the open interval (a, b)—multiplicities counted—of the polynomial p(x) in R[x], of degree deg(p), is bounded above by the number of sign variations varab(p), where
$\operatorname {var} _{ab}(p)=\operatorname {var} \left((1+x)^{\deg(p)}p\left({\frac {a+bx}{1+x}}\right)\right),$
$\operatorname {var} _{ab}(p)=\operatorname {var} _{ba}(p)\geq \rho _{ab}(p).$
As in the case of Descartes' rule of signs if varab(p) = 0 it follows that ρab(p) = 0 and if varab(p) = 1 it follows that ρab(p) = 1.
A special case of the Alesina–Galuzzi "a_b roots test" is Budan's "0_1 roots test".
Sketch of a proof
A detailed discussion of Vincent's theorem, its extension, the geometrical interpretation of the transformations involved and three different proofs can be found in the work by Alesina and Galuzzi.[4][5] A fourth proof is due to Ostrowski[6] who rediscovered a special case of a theorem stated by Obreschkoff,[7] p. 81, in 1920–1923.
To prove (both versions of) Vincent's theorem Alesina and Galuzzi show that after a series of transformations mentioned in the theorem, a polynomial with one positive root eventually has one sign variation. To show this, they use the following corollary to the theorem by Obreschkoff of 1920–1923 mentioned earlier; that is, the following corollary gives the necessary conditions under which a polynomial with one positive root has exactly one sign variation in the sequence of its coefficients; see also the corresponding figure.
Corollary. (Obreschkoff's cone or sector theorem, 1920–1923[7] p. 81): If a real polynomial has one simple root x0, and all other (possibly multiple) roots lie in the sector
$S_{\sqrt {3}}=\left\{x=-\alpha +i\beta \ :\ |\beta |\leq {\sqrt {3}}|\alpha |,\alpha >0\right\}$ :\ |\beta |\leq {\sqrt {3}}|\alpha |,\alpha >0\right\}}
then the sequence of its coefficients has exactly one sign variation.
Consider now the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}},\qquad a,b,c,d\in \mathbb {Z} _{>0}$
and the three circles shown in the corresponding figure; assume that a/c < b/d.
• The (yellow) circle
$\left|x-{\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)\right|={\tfrac {1}{2}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
whose diameter lies on the real axis, with endpoints a/c and b/d, is mapped by the inverse Möbius transformation
$M^{-1}(x)={\frac {dx-b}{-cx+a}}$
onto the imaginary axis. For example the point
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)+{\tfrac {i}{2}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
gets mapped onto the point −i d/c. The exterior points get mapped onto the half-plane with Re(x) < 0.
• The two circles (only their blue crescents are visible) with center
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)\pm {\tfrac {i}{2{\sqrt {3}}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
and radius
${\tfrac {1}{\sqrt {3}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
are mapped by the inverse Möbius transformation
$M^{-1}(x)={\frac {dx-b}{-cx+a}}$
onto the lines Im(x) = ±√3 Re(x). For example the point
${\tfrac {1}{2}}\left({\tfrac {a}{c}}+{\tfrac {b}{d}}\right)-{\tfrac {3i}{2{\sqrt {3}}}}\left({\tfrac {b}{d}}-{\tfrac {a}{c}}\right)$
gets mapped to the point
${\tfrac {-d}{2c}}\left(1-i{\sqrt {3}}\right).$
The exterior points (those outside the eight-shaped figure) get mapped onto the $S_{\sqrt {3}}$ sector.
From the above it becomes obvious that if a polynomial has a single positive root inside the eight-shaped figure and all other roots are outside of it, it presents one sign variation in the sequence of its coefficients. This also guarantees the termination of the process.
Historical background
Early applications of Vincent's theorem
In his fundamental papers,[1][2][3] Vincent presented examples that show precisely how to use his theorem to isolate real roots of polynomials with continued fractions. However the resulting method had exponential computing time, a fact that mathematicians must have realized then, as was realized by Uspensky[8] p. 136, a century later.
The exponential nature of Vincent's algorithm is due to the way the partial quotients ai (in Vincent's theorem) are computed. That is, to compute each partial quotient ai (that is, to locate where the roots lie on the x-axis) Vincent uses Budan's theorem as a "no roots test"; in other words, to find the integer part of a root Vincent performs successive substitutions of the form x ← x+1 and stops only when the polynomials p(x) and p(x+1) differ in the number of sign variations in the sequence of their coefficients (i.e. when the number of sign variations of p(x+1) is decreased).
See the corresponding diagram where the root lies in the interval (5, 6). It can be easily inferred that, if the root is far away from the origin, it takes a lot of time to find its integer part this way, hence the exponential nature of Vincent's method. Below there is an explanation of how this drawback is overcome.
Disappearance of Vincent's theorem
Vincent was the last author in the 19th century to use his theorem for the isolation of the real roots of a polynomial.
The reason for that was the appearance of Sturm's theorem in 1827, which solved the real root isolation problem in polynomial time, by defining the precise number of real roots a polynomial has in a real open interval (a, b). The resulting (Sturm's) method for computing the real roots of polynomials has been the only one widely known and used ever since—up to about 1980, when it was replaced (in almost all computer algebra systems) by methods derived from Vincent's theorem, the fastest one being the Vincent–Akritas–Strzeboński (VAS) method.[9]
Serret included in his Algebra,[10] pp 363–368, Vincent's theorem along with its proof and directed all interested readers to Vincent's papers for examples on how it is used. Serret was the last author to mention Vincent's theorem in the 19th century.
Comeback of Vincent's theorem
In the 20th century Vincent's theorem cannot be found in any of the theory of equations books; the only exceptions are the books by Uspensky[8] and Obreschkoff,[7] where in the second there is just the statement of the theorem.
It was in Uspensky's book[8] that Akritas found Vincent's theorem and made it the topic of his Ph.D. Thesis "Vincent's Theorem in Algebraic Manipulation", North Carolina State University, USA, 1978. A major achievement at the time was getting hold of Vincent's original paper of 1836, something that had eluded Uspensky—resulting thus in a great misunderstanding. Vincent's original paper of 1836 was made available to Akritas through the commendable efforts (interlibrary loan) of a librarian in the Library of the University of Wisconsin–Madison, USA.
Real root isolation methods derived from Vincent's theorem
Isolation of the real roots of a polynomial is the process of finding open disjoint intervals such that each contains exactly one real root and every real root is contained in some interval. According to the French school of mathematics of the 19th century, this is the first step in computing the real roots, the second being their approximation to any degree of accuracy; moreover, the focus is on the positive roots, because to isolate the negative roots of the polynomial p(x) replace x by −x (x ← −x) and repeat the process.
The continued fractions version of Vincent's theorem can be used to isolate the positive roots of a given polynomial p(x) of degree deg(p). To see this, represent by the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}},\qquad a,b,c,d\in \mathbb {N} $
the continued fraction that leads to a transformed polynomial
$f(x)=(cx+d)^{\deg(p)}p\left({\frac {ax+b}{cx+d}}\right)$
(2)
with one sign variation in the sequence of its coefficients. Then, the single positive root of f(x) (in the interval (0, ∞)) corresponds to that positive root of p(x) that is in the open interval with endpoints ${\frac {b}{d}}$ and ${\frac {a}{c}}$. These endpoints are not ordered and correspond to M(0) and M(∞) respectively.
Therefore, to isolate the positive roots of a polynomial, all that must be done is to compute—for each root—the variables a, b, c, d of the corresponding Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}$
that leads to a transformed polynomial as in equation (2), with one sign variation in the sequence of its coefficients.
Crucial Observation: The variables a, b, c, d of a Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}$
(in Vincent's theorem) leading to a transformed polynomial—as in equation (2)—with one sign variation in the sequence of its coefficients can be computed:
• either by continued fractions, leading to the Vincent–Akritas–Strzebonski (VAS) continued fractions method,[9]
• or by bisection, leading to (among others) the Vincent–Collins–Akritas (VCA) bisection method.[11]
The "bisection part" of this all important observation appeared as a special theorem in the papers by Alesina and Galuzzi.[4][5]
All methods described below (see the article on Budan's theorem for their historical background) need to compute (once) an upper bound, ub, on the values of the positive roots of the polynomial under consideration. Exception is the VAS method where additionally lower bounds, lb, must be computed at almost every cycle of the main loop. To compute the lower bound lb of the polynomial p(x) compute the upper bound ub of the polynomial $x^{\deg(p)}p\left({\frac {1}{x}}\right)$ and set $lb={\frac {1}{ub}}$.
Excellent (upper and lower) bounds on the values of just the positive roots of polynomials have been developed by Akritas, Strzeboński and Vigklas based on previous work by Doru Stefanescu. They are described in P. S. Vigklas' Ph.D. Thesis[12] and elsewhere.[13] These bounds have already been implemented in the computer algebra systems Mathematica, SageMath, SymPy, Xcas etc.
All three methods described below follow the excellent presentation of François Boulier,[14] p. 24.
Continued fractions method
Only one continued fractions method derives from Vincent's theorem. As stated above, it started in the 1830s when Vincent presented, in the papers[1][2][3] several examples that show how to use his theorem to isolate the real roots of polynomials with continued fractions. However the resulting method had exponential computing time. Below is an explanation of how this method evolved.
Vincent–Akritas–Strzeboński (VAS, 2005)
This is the second method (after VCA) developed to handle the exponential behavior of Vincent's method.
The VAS continued fractions method is a direct implementation of Vincent's theorem. It was originally presented by Vincent from 1834 to 1938 in the papers [1][2][3] in a exponential form; namely, Vincent computed each partial quotient ai by a series of unit increments ai ← ai + 1, which are equivalent to substitutions of the form x ← x + 1.
Vincent's method was converted into its polynomial complexity form by Akritas, who in his 1978 Ph.D. Thesis (Vincent's theorem in algebraic manipulation, North Carolina State University, USA) computed each partial quotient ai as the lower bound, lb, on the values of the positive roots of a polynomial. This is called the ideal positive lower root bound that computes the integer part of the smallest positive root (see the corresponding figure). To wit, now set ai ← lb or, equivalently, perform the substitution x ← x + lb, which takes about the same time as the substitution x ← x + 1.
Finally, since the ideal positive lower root bound does not exist, Strzeboński[15] introduced in 2005 the substitution $x\leftarrow lb_{computed}*x$, whenever $lb_{computed}>16$; in general $lb>lb_{computed}$ and the value 16 was determined experimentally. Moreover, it has been shown[15] that the VAS (continued fractions) method is faster than the fastest implementation of the VCA (bisection) method,[16] a fact that was confirmed[17] independently; more precisely, for the Mignotte polynomials of high degree VAS is about 50,000 times faster than the fastest implementation of VCA.
In 2007, Sharma[18] removed the hypothesis of the ideal positive lower bound and proved that VAS is still polynomial in time.
VAS is the default algorithm for root isolation in Mathematica, SageMath, SymPy, Xcas.
For a comparison between Sturm's method and VAS use the functions realroot(poly) and time(realroot(poly)) of Xcas. By default, to isolate the real roots of poly realroot uses the VAS method; to use Sturm's method write realroot(sturm, poly). See also the External links for an application by A. Berkakis for Android devices that does the same thing.
Here is how VAS(p, M) works, where for simplicity Strzeboński's contribution is not included:
• Let p(x) be a polynomial of degree deg(p) such that p(0) ≠ 0. To isolate its positive roots, associate with p(x) the Möbius transformation M(x) = x and repeat the following steps while there are pairs {p(x), M(x)} to be processed.
• Use Descartes' rule of signs on p(x) to compute, if possible, (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, ∞). If there are no roots return the empty set, ∅ whereas if there is one root return the interval (a, b), where a = min(M(0), M(∞)), and b = max(M(0), M(∞)); if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x).[12][13]
• If there are two or more sign variations Descartes' rule of signs implies that there may be zero, one or more real roots inside the interval (0, ∞); in this case consider separately the roots of p(x) that lie inside the interval (0, 1) from those inside the interval (1, ∞). A special test must be made for 1.
• To guarantee that there are roots inside the interval (0, 1) the ideal lower bound, lb is used; that is the integer part of the smallest positive root is computed with the help of the lower bound,[12][13] $lb_{computed}$, on the values of the positive roots of p(x). If $lb_{computed}>1$, the substitution $x\leftarrow x+lb_{computed}$ is performed to p(x) and M(x), whereas if $lb_{computed}\leq 1$ use substitution(s) x ← x+1 to find the integer part of the root(s).
• To compute the roots inside the interval (0, 1) perform the substitution $x\leftarrow {\frac {1}{1+x}}$ to p(x) and M(x) and process the pair
$\left\{(1+x)^{\deg(p)}p\left({\tfrac {1}{1+x}}\right),M({\tfrac {1}{1+x}})\right\},$
whereas to compute the roots in the interval (1, ∞) perform the substitution x ← x + 1 to p(x) and M(x) and process the pair {p(1 + x), M(1 + x)}. It may well turn out that 1 is a root of p(x), in which case, M(1) is a root of the original polynomial and the isolation interval reduces to a point.
Below is a recursive presentation of VAS(p, M).
VAS(p, M):
Input: A univariate, square-free polynomial $p(x)\in \mathbb {Z} [x],p(0)\neq 0$, of degree deg(p), and the Möbius transformation
$M(x)={\frac {ax+b}{cx+d}}=x,\qquad a,b,c,d\in \mathbb {N} .$
Output: A list of isolating intervals of the positive roots of p(x).
1 var ← the number of sign variations of p(x) // Descartes' rule of signs;
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)} // a = min(M(0), M(∞)), b = max(M(0), M(∞)), but if b = ∞ set b = ub, where ub is an upper bound on the values of the positive roots of p(x);
4 lb ← the ideal lower bound on the positive roots of p(x);
5 if lb ≥ 1 then p ← p(x + lb), M ← M(x + lb);
6 p01 ← (x + 1)deg(p) p(1/x + 1), M01 ← M(1/x + 1) // Look for real roots in (0, 1);
7 m ← M(1) // Is 1 a root?
8 p1∞ ← p(x + 1), M1∞ ← M(x + 1) // Look for real roots in (1, ∞);
9 if p(1) ≠ 0 then
10 RETURN VAS(p01, M01) ∪ VAS(p1∞, M1∞)
11 else
12 RETURN VAS(p01, M01) ∪ {[m, m]} ∪ VAS(p1∞, M1∞)
13 end
Remarks
• For simplicity Strzeboński's contribution is not included.
• In the above algorithm with each polynomial there is associated a Möbius transformation M(x).
• In line 1 Descartes' rule of signs is applied.
• If lines 4 and 5 are removed from VAS(p, M) the resulting algorithm is Vincent's exponential one.
• Any substitution performed on the polynomial p(x) is also performed on the associated Möbius transformation M(x) (lines 5 6 and 8).
• The isolating intervals are computed from the Möbius transformation in line 3, except for integer roots computed in line 7 (also 12).
Example of VAS(p, M)
We apply the VAS method to p(x) = x3 − 7x + 7 (note that: M(x) = x).
Iteration 1
VAS(x3 − 7x + 7, x)
1 var ← 2 // the number of sign variations in the sequence of coefficients of p(x) = x3 − 7x + 7
4 lb ← 1 // the ideal lower bound—found using lbcomputed and substitution(s) x ← x + 1
5 p ← x3 + 3x2 − 4x + 1, M ← x + 1
6 p01 ← x3 − x2 − 2x + 1, M01 ← x + 2/x + 1
7 m ← 1
8 p1∞ ← x3 + 6x2 + 5x + 1, M1∞ ← x + 2
10 RETURN VAS(x3 − x2 − 2x + 1, x + 2/x + 1) ∪ VAS(x3 + 6x2 + 5x + 1, x + 2)
List of isolation intervals: { }.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}-x^{2}-2x+1,{\tfrac {x+2}{x+1}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 2
VAS(x3 − x2 − 2x + 1, x + 2/x + 1)
1 var ← 2 // the number of sign variations in the sequence of coefficients of p(x) = x3 − x2 − 2x + 1
4 lb ← 0 // the ideal lower bound—found using lbcomputed and substitution(s) x ← x + 1
6 p01 ← x3 + x2 − 2x − 1, M01 ← 2x + 3/x + 1
7 m ← 3/2
8 p1∞ ← x3 + 2x2 − x − 1, M1∞ ← x + 3/x + 2
10 RETURN VAS(x3 + x2 − 2x − 1, 2x + 3/x + 2) ∪ VAS(x3 + 2x2 − x − 1, x + 3/x + 2)
List of isolation intervals: { }.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+x^{2}-2x-1,{\tfrac {2x+3}{x+2}}\right\},\left\{x^{3}+2x^{2}-x-1,{\tfrac {x+3}{x+2}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 3
VAS(x3 + x2 − 2x − 1, 2x + 3/x + 2)
1 var ← 1 // the number of sign variations in the sequence of coefficients of p(x) = x3 + x2 − 2x − 1
3 RETURN {(3/2, 2)}
List of isolation intervals: {(3/2, 2)}.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+2x^{2}-x-1,{\tfrac {x+3}{x+2}}\right\},\{x^{3}+6x^{2}+5x+1,x+2\}\right\}.$
Remove the first and process it.
Iteration 4
VAS(x3 + 2x2 − x − 1, x + 3/x + 2)
1 var ← 1 // the number of sign variations in the sequence of coefficients of p(x) = x3 + 2x2 − x − 1
3 RETURN {(1, 3/2)}
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, M} to be processed:
$\left\{\left\{x^{3}+6x^{2}+5x+1,x+2\right\}\right\}.$
Remove the first and process it.
Iteration 5
VAS(x3 + 6x2 + 5x + 1, x + 2)
1 var ← 0 // the number of sign variations in the sequence of coefficients of p(x) = x3 + 6x2 + 5x + 1
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, M} to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
Bisection methods
There are various bisection methods derived from Vincent's theorem; they are all presented and compared elsewhere.[19] Here the two most important of them are described, namely, the Vincent–Collins–Akritas (VCA) method and the Vincent–Alesina–Galuzzi (VAG) method.
The Vincent–Alesina–Galuzzi (VAG) method is the simplest of all methods derived from Vincent's theorem but has the most time consuming test (in line 1) to determine if a polynomial has roots in the interval of interest; this makes it the slowest of the methods presented in this article.
By contrast, the Vincent–Collins–Akritas (VCA) method is more complex but uses a simpler test (in line 1) than VAG. This along with certain improvements[16] have made VCA the fastest bisection method.
Vincent–Collins–Akritas (VCA, 1976)
This was the first method developed to overcome the exponential nature of Vincent's original approach, and has had quite an interesting history as far as its name is concerned. This method, which isolates the real roots, using Descartes' rule of signs and Vincent's theorem, had been originally called modified Uspensky's algorithm by its inventors Collins and Akritas.[11] After going through names like "Collins–Akritas method" and "Descartes' method" (too confusing if ones considers Fourier's article[20]), it was finally François Boulier, of Lille University, who gave it the name Vincent–Collins–Akritas (VCA) method,[14] p. 24, based on the fact that "Uspensky's method" does not exist[21] and neither does "Descartes' method".[22] The best implementation of this method is due to Rouillier and Zimmerman,[16] and to this date, it is the fastest bisection method. It has the same worst case complexity as Sturm's algorithm, but is almost always much faster. It has been implemented in Maple's RootFinding package.
Here is how VCA(p, (a, b)) works:
• Given a polynomial porig(x) of degree deg(p), such that porig(0) ≠ 0, whose positive roots must be isolated, first compute an upper bound,[12][13] ub on the values of these positive roots and set p(x) = porig(ub * x) and (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (0, 1) and there is a bijection between them and the roots of porig(x), which all lie in the interval (a, b) = (0, ub) (see the corresponding figure); this bijection is expressed by α(a,b) = a +α(0,1)(b − a). Likewise, there is a bijection between the intervals (0, 1) and (0, ub).
• Repeat the following steps while there are pairs {p(x), (a, b)} to be processed.
• Use Budan's "0_1 roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (0, 1). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
• If there are two or more sign variations Budan's "0_1 roots test" implies that there may be zero, one, two or more real roots inside the interval (0, 1). In this case cut it in half and consider separately the roots of p(x) inside the interval (0, 1/2)—and that correspond to the roots of porig(x) inside the interval (a, 1/2(a + b)) from those inside the interval (1/2, 1) and correspond to the roots of porig(x) inside the interval (1/2(a + b), b); that is, process, respectively, the pairs
$\left\{2^{\deg(p)}p({\tfrac {x}{2}}),(a,{\tfrac {1}{2}}(a+b))\right\},\quad \left\{2^{\deg(p)}p({\tfrac {1}{2}}(x+1)),({\tfrac {1}{2}}(a+b),b)\right\}$
(see the corresponding figure). It may well turn out that 1/2 is a root of p(x), in which case 1/2(a + b) is a root of porig(x) and the isolation interval reduces to a point.
Below is a recursive presentation of the original algorithm VCA(p, (a, b)).
VCA(p, (a, b))
Input: A univariate, square-free polynomial p(ub * x) ∈ Z[x], p(0) ≠ 0 of degree deg(p), and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x). (The positive roots of p(ub * x) are all in the open interval (0, 1)).
Output: A list of isolating intervals of the positive roots of p(x)
1 var ← the number of sign variations of (x + 1)deg(p)p(1/x + 1) // Budan's "0_1 roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 p01/2 ← 2deg(p)p(x/2) // Look for real roots in (0, 1/2);
5 m ← 1/2(a + b) // Is 1/2 a root?
6 p1/21 ← 2deg(p)p(x + 1/2) // Look for real roots in (1/2, 1);
7 if p(1/2) ≠ 0 then
8 RETURN VCA (p01/2, (a, m)) ∪ VCA (p1/21, (m, b))
9 else
10 RETURN VCA (p01/2, (a, m)) ∪ {[m, m]} ∪ VCA (p1/21, (m, b))
11 end
Remark
• In the above algorithm with each polynomial there is associated an interval (a, b). As shown elsewhere,[22] p. 11, a Möbius transformation can also be associated with each polynomial in which case VCA looks more like VAS.
• In line 1 Budan's "0_1 roots test" is applied.
Example of VCA(p, (a,b))
Given the polynomial porig(x) = x3 − 7x + 7 and considering as an upper bound[12][13] on the values of the positive roots ub = 4 the arguments of the VCA method are: p(x) = 64x3 − 28x + 7 and (a, b) = (0, 4).
Iteration 1
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 7x3 − 7x2 − 35x + 43
4 p01/2 ← 64x3 − 112x + 56
5 m ← 2
6 p1/21 ← 64x3 + 192x2 + 80x + 8
7 p(1/2) = 1
8 RETURN VCA(64x3 − 112x + 56, (0, 2)) ∪ VCA(64x3 + 192x2 + 80x + 8, (2, 4))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}-112x+56,(0,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 2
VCA(64x3 − 112x + 56, (0, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 56x3 + 56x2 − 56x + 8
4 p01/2 ← 64x3 − 448x + 448
5 m ← 1
6 p1/21 ← 64x3 + 192x2 − 256x + 64
7 p(1/2) = 8
8 RETURN VCA(64x3 − 448x + 448, (0, 1)) ∪ VCA(64x3 + 192x2 − 256x + 64, (1, 2))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}-448x+448,(0,1)\right\},\left\{64x^{3}+192x^{2}-256x+64,(1,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 3
VCA(64x3 − 448x + 448, (0, 1))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 448x3 + 896x2 + 448x + 64
2 RETURN ∅
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+192x^{2}-256x+64,(1,2)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 4
VCA(64x3 + 192x2 − 256x + 64, (1, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 64x3 − 64x2 − 128x + 64
4 p01/2 ← 64x3 + 384x2 − 1024x + 512
5 m ← 3/2
6 p1/21 ← 64x3 + 576x2 − 64x + 64
7 p(1/2) = −8
8 RETURN VCA(64x3 + 384x2 − 1024x + 512, (1, 3/2)) ∪ VCA(64x3 + 576x2 − 64x − 64, (3/2, 2))
List of isolation intervals: { }.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+384x^{2}-1024x+512,\left(1,{\tfrac {3}{2}}\right)\right\},\left\{64x^{3}+576x^{2}-64x-64,\left({\tfrac {3}{2}},2\right)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 5
VCA(64x3 + 384x2 − 1024x + 512, (1, 3/2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 512x3 + 512x2 − 128x − 64
3 RETURN {(1, 3/2)}
List of isolation intervals: {(1, 3/2)}.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+576x^{2}-64x-64,\left({\tfrac {3}{2}},2\right)\right\},\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 6
VCA(64x3 + 576x2 − 64x − 64, (3/2, 2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = −64x3 − 256x2 + 256x + 512
3 RETURN {(3/2, 2)}
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, I} to be processed:
$\left\{\left\{64x^{3}+192x^{2}+80x+8,(2,4)\right\}\right\}.$
Remove the first and process it.
Iteration 7
VCA(64x3 + 192x2 + 80x + 8, (2, 4))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(1/x + 1) = 8x3 + 104x2 + 376x + 344
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of pairs {p, I} to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
Vincent–Alesina–Galuzzi (VAG, 2000)
This was developed last and is the simplest real root isolation method derived from Vincent's theorem.
Here is how VAG(p, (a, b)) works:
• Given a polynomial p(x) of degree deg(p), such that p(0) ≠ 0, whose positive roots must be isolated, first compute an upper bound,[12][13] ub on the values of these positive roots and set (a, b) = (0, ub). The positive roots of p(x) all lie in the interval (a, b).
• Repeat the following steps while there are intervals (a, b) to be processed; in this case the polynomial p(x) stays the same.
• Use the Alesina–Galuzzi "a_b roots test" on p(x) to compute (using the number var of sign variations in the sequence of its coefficients) the number of its roots inside the interval (a, b). If there are no roots return the empty set, ∅ and if there is one root return the interval (a, b).
• If there are two or more sign variations the Alesina–Galuzzi "a_b roots test" implies that there may be zero, one, two or more real roots inside the interval (a, b). In this case cut it in half and consider separately the roots of p(x) inside the interval (a, 1/2(a + b)) from those inside the interval (1/2(a + b), b); that is, process, respectively, the intervals (a, 1/2(a + b)) and (1/2(a + b), b). It may well turn out that 1/2(a + b) is a root of p(x), in which case the isolation interval reduces to a point.
Below is a recursive presentation of VAG(p, (a, b)).
VAG(p, (a, b))
Input: A univariate, square-free polynomial p(x) ∈ Z[x], p(0) ≠ 0 of degree deg(p) and the open interval (a, b) = (0, ub), where ub is an upper bound on the values of the positive roots of p(x).
Output: A list of isolating intervals of the positive roots of p(x).
1 var ← the number of sign variations of (x + 1)deg(p) p(a + bx/1 + x) // The Alesina–Galuzzi "a_b roots test";
2 if var = 0 then RETURN ∅;
3 if var = 1 then RETURN {(a, b)};
4 m ← 1/2(a + b) // Subdivide the interval (a, b) in two equal parts;
5 if p(m) ≠ 0 then
6 RETURN VAG(p, (a, m)) ∪ VAG(p, (m, b))
7 else
8 RETURN VAG(p, (a, m)) ∪ {[m, m]} ∪ VAG(p, (m, b))
9 end
Remarks
• Compared to VCA the above algorithm is extremely simple; by contrast, VAG uses the time consuming "a_b roots test" and that makes it much slower than VCA.[19]
• As Alesina and Galuzzi point out,[5] p. 189, there is a variant of this algorithm due to Donato Saeli. Saeli suggested that the mediant of the endpoints be used instead of their midpoint 1/2(a + b). However, it has been shown[19] that using the mediant of the endpoints is in general much slower than the "mid-point" version.
Example of VAG(p, (a,b))
Given the polynomial p(x) = x3 − 7x + 7 and considering as an upper bound[12][13] on the values of the positive roots ub = 4 the arguments of VAG are: p(x) = x3 − 7x + 7 and (a, b) = (0, 4).
Iteration 1
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(4x/x + 1) = 43x3 − 35x2 − 7x + 7
4 m ← 1/2(0 + 4) = 2
5 p(m) = 1
8 RETURN VAG(x3 − 7x + 7, (0, 2)) ∪ VAG(x3 − 7x + 7, (2, 4)
List of isolation intervals: {}.
List of intervals to be processed: {(0, 2), (2, 4)}.
Remove the first and process it.
Iteration 2
VAG(x3 − 7x + 7, (0, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(2x/x + 1) = x3 − 7x2 + 7x + 7
4 m ← 1/2(0 + 2) = 1
5 p(m) = 1
8 RETURN VAG(x3 − 7x + 7, (0, 1)) ∪ VAG(x3 − 7x + 7, (1, 2)
List of isolation intervals: {}.
List of intervals to be processed: {(0, 1), (1, 2), (2, 4)}.
Remove the first and process it.
Iteration 3
VAG(x3 − 7x + 7, (0, 1))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(x/x + 1) = x3 + 7x2 + 14x + 7
2 RETURN ∅
List of isolation intervals: {}.
List of intervals to be processed: {(1, 2), (2, 4)}.
Remove the first and process it.
Iteration 4
VAG(x3 − 7x + 7, (1, 2))
1 var ← 2 // the number of sign variations in the sequence of coefficients of (x + 1)3p(2x + 1/x + 1) = x3 − 2x2 − x + 1
4 m ← 1/2(1 + 2) = 3/2
5 p(m) = −1/8
8 RETURN VAG(x3 − 7x + 7, (1, 3/2)) ∪ VAG(x3 − 7x + 7, (3/2, 2))
List of isolation intervals: {}.
List of intervals to be processed: {(1, 3/2), (3/2, 2), (2, 4)}.
Remove the first and process it.
Iteration 5
VAG(x3 − 7x + 7, (1, 3/2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of 23(x + 1)3p(3/2x + 1/x + 1) = x3 + 2x2 − 8x − 8
3 RETURN (1, 3/2)
List of isolation intervals: {(1, 3/2)}.
List of intervals to be processed: {(3/2, 2), (2, 4)}.
Remove the first and process it.
Iteration 6
VAG(x3 − 7x + 7, (3/2, 2))
1 var ← 1 // the number of sign variations in the sequence of coefficients of 23(x + 1)3p(2x + 3/2/x + 1) = 8x3 + 4x2 − 4x − 1
3 RETURN (3/2, 2)
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of intervals to be processed: {(2, 4)}.
Remove the first and process it.
Iteration 7
VAG(x3 − 7x + 7, (2, 4))
1 var ← 0 // the number of sign variations in the sequence of coefficients of (x + 1)3p(4x + 2/x + 1) = 344x3 + 376x2 + 104x + 8
2 RETURN ∅
List of isolation intervals: {(1, 3/2), (3/2, 2)}.
List of intervals to be processed: ∅.
Finished.
Conclusion
Therefore, the two positive roots of the polynomial p(x) = x3 − 7x + 7 lie inside the isolation intervals (1, 3/2) and (3/2, 2)}. Each root can be approximated by (for example) bisecting the isolation interval it lies in until the difference of the endpoints is smaller than 10−6; following this approach, the roots turn out to be ρ1 = 1.3569 and ρ2 = 1.69202.
See also
• Properties of polynomial roots
• Root-finding algorithm
• Vieta's formulas
• Newton's method
References
1. Vincent, Alexandre Joseph Hidulphe (1834). "Mémoire sur la résolution des équations numériques". Mémoires de la Société Royale des Sciences, de l' Agriculture et des Arts, de Lille: 1–34.
2. Vincent, Alexandre Joseph Hidulphe (1836). "Note sur la résolution des équations numériques" (PDF). Journal de Mathématiques Pures et Appliquées. 1: 341–372.
3. Vincent, Alexandre Joseph Hidulphe (1838). "Addition à une précédente note relative à la résolution des équations numériques" (PDF). Journal de Mathématiques Pures et Appliquées. 3: 235–243.
4. Alesina, Alberto; Massimo Galuzzi (1998). "A new proof of Vincent's theorem". L'Enseignement Mathématique. 44 (3–4): 219–256.
5. Alesina, Alberto; Massimo Galuzzi (2000). "Vincent's Theorem from a Modern Point of View" (PDF). Categorical Studies in Italy 2000, Rendiconti del Circolo Matematico di Palermo, Serie II, N. 64: 179–191.
6. Ostrowski, A. M. (1950). "Note on Vincent's theorem". Annals of Mathematics. Second Series. 52 (3): 702–707. doi:10.2307/1969443. JSTOR 1969443.
7. Obreschkoff, Nikola (1963). Verteilung und Berechnung der Nullstellen reeller Polynome. Berlin: VEB Deutscher Verlag der Wissenschaften.
8. Uspensky, James Victor (1948). Theory of Equations. New York: McGraw–Hill Book Company.
9. Akritas, Alkiviadis G.; A.W. Strzeboński; P.S. Vigklas (2008). "Improving the performance of the continued fractions method using new bounds of positive roots" (PDF). Nonlinear Analysis: Modelling and Control. 13 (3): 265–279. doi:10.15388/NA.2008.13.3.14557.
10. Serret, Joseph A. (1877). Cours d'algèbre supérieure. Tome I. Gauthier-Villars.
11. Collins, George E.; Alkiviadis G. Akritas (1976). "Polynomial real root isolation using Descarte's rule of signs". Polynomial Real Root Isolation Using Descartes' Rule of Signs. SYMSAC '76, Proceedings of the third ACM symposium on Symbolic and algebraic computation. Yorktown Heights, NY, USA: ACM. pp. 272–275. doi:10.1145/800205.806346. ISBN 9781450377904. S2CID 17003369.
12. Vigklas, Panagiotis, S. (2010). Upper bounds on the values of the positive roots of polynomials (PDF). Ph. D. Thesis, University of Thessaly, Greece.{{cite book}}: CS1 maint: multiple names: authors list (link)
13. Akritas, Alkiviadis, G. (2009). "Linear and Quadratic Complexity Bounds on the Values of the Positive Roots of Polynomials". Journal of Universal Computer Science. 15 (3): 523–537.{{cite journal}}: CS1 maint: multiple names: authors list (link)
14. Boulier, François (2010). Systèmes polynomiaux : que signifie " résoudre " ? (PDF). Université Lille 1.
15. Akritas, Alkiviadis G.; Adam W. Strzeboński (2005). "A Comparative Study of Two Real Root Isolation Methods" (PDF). Nonlinear Analysis: Modelling and Control. 10 (4): 297–304. doi:10.15388/NA.2005.10.4.15110.
16. Rouillier, F.; P. Zimmerman (2004). "Efficient isolation of polynomial's real roots". Journal of Computational and Applied Mathematics. 162: 33–50. doi:10.1016/j.cam.2003.08.015.
17. Tsigaridas, Elias P.; Emiris, Ioannis Z. (2006). "Univariate polynomial real root isolation: Continued fractions revisited". In Azar, Yossi; Erlebach, Thomas (eds.). Algorithms – ESA 2006, 14th Annual European Symposium, Zurich, Switzerland, September 11–13, 2006, Proceedings. Lecture Notes in Computer Science. Vol. 4168. Springer. pp. 817–828. arXiv:cs/0604066. doi:10.1007/11841036_72.
18. Sharma, Vikram (2007). Complexity Analysis of Algorithms in Algebraic Computation (PDF). Ph.D. Thesis, Courant Institute of Mathematical Sciences, New York University,USA.
19. Akritas, Alkiviadis G.; Adam W. Strzeboński; Panagiotis S. Vigklas (2008). "On the Various Bisection Methods Derived from Vincent's Theorem". Serdica Journal of Computing. 2 (1): 89–104. doi:10.55630/sjc.2008.2.89-104. hdl:10525/376. S2CID 126142131.
20. Fourier, Jean Baptiste Joseph (1820). "Sur l'usage du théorème de Descartes dans la recherche des limites des racines". Bulletin des Sciences, par la Société Philomatique de Paris: 156–165.
21. Akritas, Alkiviadis G. (1986). "There is no "Uspensky's method."". There's no "Uspensky's Method". In: Proceedings of the fifth ACM Symposium on Symbolic and Algebraic Computation (SYMSAC '86, Waterloo, Ontario, Canada), pp. 88–90. pp. 88–90. doi:10.1145/32439.32457. ISBN 0897911997. S2CID 15446040.
22. Akritas, Alkiviadis G. (2008). There is no "Descartes' method". In: M.J.Wester and M. Beaudin (Eds), Computer Algebra in Education, AullonaPress, USA, pp. 19–35. ISBN 9780975454190.
External links
• Berkakis, Antonis: RealRoots, a free App for Android devices to compare Sturm's method and VAS
• https://play.google.com/store/apps/details?id=org.kde.necessitas.berkakis.realroots
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Vincent Léotaud
Vincent Léotaud (1595 – 1672) was a French Jesuit mathematician.[1][2]
Vincent Léotaud
Born1595
Vallouise
Died1672 (aged 76–77)
Embrun
OccupationMathematician
In his work Examen circuli quadraturae he affirmed the impossibility of squaring the circle, against the opinion of Grégoire de Saint-Vincent.[3][4]
Works
• Examen circuli quadraturae (in Latin). Vol. 1. Lyon: Guillaume Barbier. 1654.
• Examen circuli quadraturae (in Latin). Vol. 2. Lyon: Guillaume Barbier. 1654.
• Institutionum arithmeticarum libri quatuor (in Latin). Lyon: Guillaume Barbier. 1660.
• Cyclomathia seu Multiplex circuli contemplatio, tribus libris comprehensa (in Latin). Lyon: Benoit Coral. 1663.
• Magnetologia; in qua exponitur noua de magneticis philosophia (in Latin). Lyon: Laurent Anisson. 1668.
References
1. Léotaud, Vincent (1595-1672) (in French). Bibliothèque nationale de France.
2. "Léotaud, Vincent". Consortium of European Research Libraries.
3. Robson, Eleanor; Stedall, Jacqueline, eds. (2009). The Oxford Handbook of the History of Mathematics. Oxford University Press. p. 554. ISBN 9780199213122.
4. Zupanov, Ines G., ed. (2019). The Oxford Handbook of the Jesuits. Oxford University Press. p. 650. ISBN 9780190639631.
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Vincentio Reinieri
Vincentio (Vincenzio, Vincenzo) Reinieri (Renieri, Reiner) (30 March 1606 – 5 November 1647) was an Italian mathematician and astronomer. He was a friend and disciple of Galileo Galilei.
Vincentio Reinieri
Born(1606-03-30)30 March 1606
Died5 November 1642(1642-11-05) (aged 36)
NationalityItalian
Occupation(s)Mathematician, astronomer
Biography
Born at Genoa, he was a member of the Olivetan order. His order sent him to Rome in 1623. He met Galileo at Siena in 1633. Galileo had Reinieri update and attempt to improve his astronomical tables of the motions of Jupiter's moons, revising these tables for prediction of the positions of these satellites.
Reinieri's work led him to Arcetri, where he befriended Vincenzo Viviani. Reinieri enjoyed the same spirit of inquiry and love of debate as his mentor. On 5 February 1641 Reinieri wrote to Galileo from Pisa: "Not infrequently I am in some battle with the Peripatetic gentlemen, particularly when I note that those fattest with ignorance least appreciate your worth, and I have just given the head of one of those a good scrubbing." (Drake, p. 413-4)
Reinieri became professor of mathematics at the University of Pisa on the death of Dino Peri. He also taught Greek there. His astronomical work consisted of adding new observations of Jupiter's moons to Galileo's. To some degree, Reinieri improved the Galilean tables on the motions of these satellites. Before his death, Galileo decided to place all of the papers containing his observations and calculations in the hands of Reinieri. Reinieri was to finish and revise them.
Reinieri's observations of Jupiter's moons remained unpublished at the time of his premature death at Pisa in 1647. He was succeeded to the chair of mathematics by Famiano Michelini (c. 1600-1666).
Legacy
On Reinieri's death, papers concerning longitude entrusted to him by Galileo are said to have been stolen by a man named Giuseppe Agostini (Fahie, p. 374). However, scholars such as Antonio Favaro doubt whether this theft actually occurred (see Antonio Favaro, Documenti inediti per la Storia dei MSS. Galileiani, Rome, 1886, pp. 8–14).
The crater Reiner on the Moon is named after him.
Latin works
• Expugnata Hierusalem, poema, Publisher: Maceratae, Apud Petrum Salvionum (1628)
• Tabulae mediceae secundorum mobilium universales quibus per unicum prosthaphaereseon orbis canonem planetarum calculus exhibetur. Non solum tychonicè iuxta Rudolphinas Danicas & Lansbergianas, sed etiam iuxta Prutenicas Alphonsinas & Ptolemaicas, Publisher: Florentiae, typis nouis Amatoris Massae & Laurentij de Landis (1639)
• Tabulae Mediceae secundorum mobilium uniuersales (in Latin). Firenze: Amadore Massi & Lorenzo Landi. 1639.
• Tabulæ motuum cælestium universales : serenissimi magni ducis etruriæ Ferdinandi II. auspicijs primo editæ, & Mediceæ nuncupati, nunc vero auctæ, recognitæ, atque... Bernardini Fernandez de Velasco... iussu, ac sumptibus recusæ...Publisher: Florentiæ : typis Amatoris Massæ Foroliuien., 1647
Sources
• Drake, Stillman, Galileo at Work: His Scientific Biography (Chicago: University of Chicago Press, 1978), 464. ISBN 0-226-16226-5
• Fahie, J.J., Galileo: His Life and Work (London: John Murray, 1903), 374-5. - Google Books
Further reading
• A Selection from Italian Prose Writers: with a double translation: for the use of students of the Italian language on the Hamiltonian system, London, Hunt and Clark, 1828 - Google Books. Letters of Galileo to Renieri: pp. 142–147 (no images for remainder of letter), and pp. 242–253 (no images pp, 246-250).
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Vincenzo Brunacci
Vincenzo Brunacci (3 March 1768 – 16 June 1818) was an Italian mathematician born in Florence.[1] He was professor of Matematica sublime (infinitesimal calculus) in Pavia. He transmitted Lagrange's ideas to his pupils, including Ottaviano Fabrizio Mossotti, Antonio Bordoni and Gabrio Piola.
Vincenzo Brunacci
Born(1768-03-03)3 March 1768
Florence, Italy
Died16 June 1818(1818-06-16) (aged 50)
Pavia, Italy
NationalityItalian
Alma materUniversity of Pisa
Known forContributions to infinitesimal calculus
Scientific career
FieldsMathematics
Doctoral advisorPietro Paoli
Other academic advisorsSebastiano Canovai
Doctoral studentsOttaviano Fabrizio Mossotti
Antonio Bordoni
Gabrio Piola
Biography
He studied medicine, astronomy and mathematics at the University of Pisa. In 1788 he earned his laurea and the same year he started teaching mathematics at the Naval Institute of Livorno. In 1796, when Napoleon entered Italy, he endorsed the new order. Upon the reinstatement of the Austrian rule, he moved to France between 1799 and 1800. On returning he attained a chair at the University of Pisa. In 1801 he moved to the University of Pavia with the office of professor of infinitesimal calculus and become its dean.
Brunacci believed that Lagrange's approach, developed in the "Théorie des fonctions analytiques", was the correct one and that the infinitesimal concept was to be banned from analysis and mechanics. In Brunacci's university teaching infinitesimal calculus differently from Lagrange's principles was even prohibited as a rule. Brunacci passed his idea of analysis on to his students, among which Fabrizio Ottaviano Mossotti, Gabrio Piola and Antonio Bordoni.
He cooperated with the public administration, in 1805 he was in the Committee for the Naviglio Pavese (Pavia Canal) project and the following year as inspector of Waters and Roads.
In 1809 he joined the Committee for the new measurements and weights system and from 1811 he was inspector general of Public Education for the entire Italian Kingdom.
He died in Pavia in 1818.
Writings
• Opuscolo analitico, (1792).
• Calcolo integrale delle equazioni lineari, (1798).
• Corso di matematica sublime, in four volumes, Firenze, (1804–1807).
• Elementi di algebra e di geometria, in two volumes, Firenze, (1809).
• Trattato dell'ariete idraulico, (1810).
• Quale tra le pratiche usate in Italia per la dispensa delle acque è la più convenevole, e quali precauzioni ed artifizi dovrebbero aggiungersi per intieramente perfezionarla riducendo le antiche alle nuove misure metriche (in Italian). Verona: Mainardi. 1814.
• Trattato di navigazione (in Italian). Vol. 2. Milano: Stamperia reale. 1817.
• Trattato di navigazione, 1817
Notes
1. An Italian short biography Vincenzo Brunacci in Edizione Nazionale Mathematica Italiana online.
External links
• Vincenzo Brunacci at the Mathematics Genealogy Project
• An Italian short biography Vincenzo Brunacci in Edizione Nazionale Mathematica Italiana online.
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Vincenzo Flauti
Vincenzo Flauti (1782–1863) was an Italian mathematician.
Vincenzo Flauti
Born(1782-04-04)4 April 1782
Naples, Kingdom of Naples, today Italy
Died20 June 1863(1863-06-20) (aged 81)
Naples, Italy
Alma materUniversity of Naples
Scientific career
FieldsMathematics
InstitutionsUniversity of Naples
InfluencedNicola Fergola
Life and work
Flauti studied at the Liceo del Salvatore, the school led by Nicola Fergola. Although he began medical studies, he changed them to mathematics influenced by his master Fergola. He taught at the University of Naples from 1803 to 1860, succeeding Fergola in his chair in 1812.
In 1860, when the Kingdom of the Two Sicilies was conquered by Giuseppe Garibaldi and was incorporated into the Kingdom of Italy, Flauti was excluded from the Academy of Sciences of Naples and from his docent duties, because he had been a supporter of the Bourbon monarchy.
Flauti was the leader of the synthetic school of mathematics founded by Fergola.[1] In 1807, jointly with Felice Giannattasio, he was entrusted by the Bourbon government to write a mathematics textbook for all schoolchildren in the kingdom.[2]
References
1. Mazzotti 2002, p. 141.
2. Ferraro 2008, p. 108.
Bibliography
• Ferraro, Giovanni (2008). "Manuali di geometria elementare nella Napoli preunitaria (1806–1860)" (PDF). History of Education & Children's Literature (in Italian). 3 (2): 103–139. ISSN 1971-1093.
• Ferraro, Giovanni (2012). "Excellens in arte non debet mori". HAL (in Italian): 1–16.
• Mazzotti, Massimo (1998). "The Geometers of God: Mathematics and Reaction in the Kingdom of Naples" (PDF). Isis. 89 (4): 674–701. doi:10.1086/384160. hdl:10036/31212. ISSN 0021-1753. JSTOR 236738. S2CID 143956681.
• Mazzotti, Massimo (2002). "The Making of the Modern Engineer". In Mordechai Feingold (ed.). History of Universities: Volume XVII. Oxford University Press. pp. 121–161. ISBN 978-0-19-925636-5.
External links
• O'Connor, John J.; Robertson, Edmund F., "Vincenzo Flauti", MacTutor History of Mathematics Archive, University of St Andrews
• Menghini, Marta (1997). "FLAUTI, Vincenzo". Dizionario Biografico degli Italiani. Retrieved December 13, 2018.
• "VINCENZO FLAUTI". Matematica PRISTEM – Università Bocconi. Retrieved December 18, 2018.
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Vincenzo Mollame
Vincenzo Mollame (Naples, 4 July 1848 – Catania, 23 June 1912) was an Italian mathematician.
Mollame was privately tutored by Achille Sanni and then studied Mathematics at the University of Naples Federico II. After obtaining his degree, he became a high-school teacher, first at Benevento and after that at Naples, starting in 1878. He became a professor at the University of Catania in 1880 and remained there for the rest of his career, having retired in 1911, a few months before his death.[1]
His research area was the theory of equations and he proved in 1890 that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals,[2] probably unaware of the fact that Pierre Wantzel had already proved it in 1843. Molleme's research activity stopped in 1896, due to health problems.
Mollame was the author of a textbook on determinants.[3]
Notes
1. Marchisotto, Elena Anne; Smith, James (2007), "Life and works", The Legacy of Mario Pieri in Geometry and Arithmetic, Birkhäuser, ISBN 978-0-8176-3210-6
2. Mollame, Vincenzo (1890), "Sul casus irreductibilis dell'equazione cubica", Rendiconto dell'Accademia delle scienze fisiche e matematiche (Sezione della società Reale di Napoli) II (in Italian), 4: 167–171
3. Mollame, Vincenzo (1878), I determinanti e loro applicazioni all' algebra ed alla geometria analitica (in Italian), Tipografia dell'Accademia reale delle scienze
External links
• Short biography (in Italian)
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Vinculum (symbol)
A vinculum (from Latin vinculum 'fetter, chain, tie') is a horizontal line used in mathematical notation for various purposes. It may be placed as an overline (or underline) over (or under) a mathematical expression to indicate that the expression is to be considered grouped together. Historically, vincula were extensively used to group items together, especially in written mathematics, but in modern mathematics this function has almost entirely been replaced by the use of parentheses.[1] It was also used to mark Roman numerals whose values are multiplied by 1,000.[2] Today, however, the common usage of a vinculum to indicate the repetend of a repeating decimal[3][4] is a significant exception and reflects the original usage.
${\overline {\rm {AB}}}$
line segment from A to B
1⁄7 = 0.142857
repeated 0.1428571428571428571...
${\overline {a+bi}}$
complex conjugate
$Y={\overline {\rm {AB}}}$
boolean NOT (A AND B)
${\sqrt[{n}]{ab+2}}$
radical ab + 2
$a-{\overline {b+c}}$ = a − (b + c)
bracketing function
Vinculum usage
History
The vinculum, in its general use, was introduced by Frans van Schooten in 1646 as he edited the works of François Viète (who had himself not used this notation). However, earlier versions, such as using an underline as Chuquet did in 1484, or in limited form as Descartes did in 1637, using it only in relation to the radical sign, were common.[5]
Usage
Modern
A vinculum can indicate a line segment where A and B are the endpoints:
• ${\overline {\rm {AB}}}.$
A vinculum can indicate the repetend of a repeating decimal value:
• 1⁄7 = 0.142857 = 0.1428571428571428571...
A vinculum can indicate the complex conjugate of a complex number:
• ${\overline {2+3i}}=2-3i$
Logarithm of a number less than 1 can conveniently be represented using vinculum:
• $\log 2=0.301\Rightarrow \log 0.2={\overline {1}}.301=-0.699$
In Boolean algebra, a vinculum may be used to represent the operation of inversion (also known as the NOT function):
• $Y={\overline {\rm {AB}}},$
meaning that Y is false only when both A and B are both true - or by extension, Y is true when either A or B is false.
Similarly, it is used to show the repeating terms in a periodic continued fraction. Quadratic irrational numbers are the only numbers that have these.
Historical
Formerly its main use was as a notation to indicate a group (a bracketing device serving the same function as parentheses):
$a-{\overline {b+c}},$
meaning to add b and c first and then subtract the result from a, which would be written more commonly today as a − (b + c). Parentheses, used for grouping, are only rarely found in the mathematical literature before the eighteenth century. The vinculum was used extensively, usually as an overline, but Chuquet in 1484 used the underline version.[6]
In India, the use of this notation is still tested in primary school.[7]
As a part of a radical
The vinculum is used as part of the notation of a radical to indicate the radicand whose root is being indicated. In the following, the quantity $ab+2$ is the whole radicand, and thus has a vinculum over it:
${\sqrt[{n}]{ab+2}}.$
In 1637 Descartes was the first to unite the German radical sign √ with the vinculum to create the radical symbol in common use today.[8]
The symbol used to indicate a vinculum need not be a line segment (overline or underline); sometimes braces can be used (pointing either up or down).[9]
Encodings
Main article: Overline § Implementations
In Unicode
• U+0305 ◌̅ COMBINING OVERLINE
TeX
In LaTeX, a text <text> can be overlined with $\overline{\mbox{<text>}}$. The inner \mbox{} is necessary to override the math-mode (here invoked by the dollar signs) which the \overline{} demands.
See also
• Overline § Math and science similar-looking symbols
• Overline § Implementations in word processing and text editing software
• Underline
References
1. Cajori, Florian (2012) [1928]. A History of Mathematical Notations. Vol. I. Dover. p. 384. ISBN 978-0-486-67766-8.
2. Ifrah, Georges (2000). The Universal History of Numbers: From Prehistory to the Invention of the Computer. Translated by David Bellos, E. F. Harding, Sophie MENGNIU , Ian Monk. John Wiley & Sons.
3. Childs, Lindsay N. (2009). A Concrete Introduction to Higher Algebra (3rd ed.). Springer. pp. 183-188.
4. Conférence Intercantonale de l'Instruction Publique de la Suisse Romande et du Tessin (2011). Aide-mémoire. Mathématiques 9-10-11. LEP. pp. 20–21.
5. Cajori 2012, p. 386
6. Cajori 2012, pp. 390–391
7. https://www.khanacademy.org/math/middle-school-math-india/x888d92141b3e0e09:bridge-7th/x888d92141b3e0e09:untitled-302/e/b7-bodmas-1
8. Cajori 2012, p. 208
9. Abbott, Jacob (1847) [1847], Vulgar and decimal fractions (The Mount Vernon Arithmetic Part II), p. 27
External links
• Weisstein, Eric W. "Periodic Continued Fraction". MathWorld.
• Weisstein, Eric W. "Vinculum". MathWorld.
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Diffiety
In mathematics, a diffiety (/dəˈfaɪəˌtiː/) is a geometrical object which plays the same role in the modern theory of partial differential equations that algebraic varieties play for algebraic equations, that is, to encode the space of solutions in a more conceptual way. The term was coined in 1984 by Alexandre Mikhailovich Vinogradov as portmanteau from differential variety.[1]
Not to be confused with Diffeology.
Intuitive definition
In algebraic geometry the main objects of study (varieties) model the space of solutions of a system of algebraic equations (i.e. the zero locus of a set of polynomials), together with all their "algebraic consequences". This means that, applying algebraic operations to this set (e.g. adding those polynomials to each other or multiplying them with any other polynomials) will give rise to the same zero locus. In other words, one can actually consider the zero locus of the algebraic ideal generated by the initial set of polynomials.
When dealing with differential equations, apart from applying algebraic operations as above, one has also the option to differentiate the starting equations, obtaining new differential constraints. Therefore, the differential analogue of a variety should be the space of solutions of a system of differential equations, together with all their "differential consequences". Instead of considering the zero locus of an algebraic ideal, one needs therefore to work with a differential ideal.
An elementary diffiety will consist therefore of the infinite prolongation ${\mathcal {E}}^{\infty }$of a differential equation ${\mathcal {E}}\subset J^{k}(E,m)$, together with an extra structure provided by a special distribution. Elementary diffieties play the same role in the theory of differential equations as affine algebraic varieties do in the theory of algebraic equations. Accordingly, just like varieties or schemes are composed of irreducible affine varieties or affine schemes, one defines a (non-elementary) diffiety as an object that locally looks like an elementary diffiety.
Formal definition
The formal definition of a diffiety, which relies on the geometric approach to differential equations and their solutions, requires the notions of jets of submanifolds, prolongations, and Cartan distribution, which are recalled below.
Jet spaces of submanifolds
Let $E$ be an $(m+e)$-dimensional smooth manifold. Two $m$-dimensional submanifolds $M$, $M'$ of $E$ are tangent up to order $k$ at the point $p\in M\cap M'\subset E$ if one can locally describe both submanifolds as zeroes of functions defined in a neighbourhood of $p$, whose derivatives at $p$ agree up to order $k$.
One can show that being tangent up to order $k$ is a coordinate-invariant notion and an equivalence relation.[2] One says also that $M$ and $M'$ have same $k$-th order jet at $p$, and denotes their equivalence class by $[M]_{p}^{k}$ or $j_{p}^{k}M$.
The $k$-jet space of $k$-submanifolds of $E$, denoted by $J^{k}(E,m)$, is defined as the set of all $k$-jets of $m$-dimensional submanifolds of $E$ at all points of $E$:
$J^{k}(E,m):=\{[M]_{p}^{k}~|~p\in M,~{\text{dim}}(M)=m,M\subset E\ {\text{ submanifold}}\}$
As any given jet $[M]_{p}^{k}$ is locally determined by the derivatives up to order $k$ of the functions describing $M$ around $p$, one can use such functions to build local coordinates $(x^{i},u_{\sigma }^{j})$ and provide $J^{k}(E,m)$ with a natural structure of smooth manifold.[2]
For instance, for $k=1$ one recovers just points in $E$ and for $k=1$ one recovers the Grassmannian of $n$-dimensional subspaces of $TE$. More generally, all the projections $J^{k}(E)\to J^{k-1}E$ are fibre bundles.
As a particular case, when $E$ has a structure of fibred manifold over an $n$-dimensional manifold $X$, one can consider submanifolds of $E$ given by the graphs of local sections of $\pi :E\to X$. Then the notion of jet of submanifolds boils down to the standard notion of jet of sections, and the jet bundle $J^{k}(\pi )$ turns out to be an open and dense subset of $J^{k}(E,m)$.[3]
Prolongations of submanifolds
The $k$-jet prolongation of a submanifold $M\subseteq E$ is
$j^{k}(M):M\rightarrow J^{k}(E,m),\quad p\mapsto [M]_{p}^{k}$
The map $j^{k}(M)$ is a smooth embedding and its image $M^{k}:={\text{im}}(j^{k}(M))$, called the prolongation of the submanifold $M$, is a submanifold of $J^{k}(E,m)$ diffeomorphic to $M$.
Cartan distribution on jet spaces
A space of the form $T_{\theta }(M^{k})$, where $M$ is any submanifold of $E$ whose prolongation contains the point $\theta \in J^{k}(E,m)$, is called an $R$-plane (or jet plane, or Cartan plane) at $\theta $. The Cartan distribution on the jet space $J^{k}(E,m)$ is the distribution ${\mathcal {C}}\subseteq T(J^{k}(E,m))$ defined by
${\mathcal {C}}:J^{k}(E,m)\rightarrow TJ^{k}(E,m),\qquad \theta \mapsto {\mathcal {C}}_{\theta }\subset T_{\theta }(J^{k}(E,m))$
where ${\mathcal {C}}_{\theta }$ is the span of all $R$-planes at $\theta \in J^{k}(E,m)$.[4]
Differential equations
A differential equation of order $k$ on the manifold $E$ is a submanifold ${\mathcal {E}}\subset J^{k}(E,m)$; a solution is defined to be an $m$-dimensional submanifold $S\subset {\mathcal {E}}$ such that $S^{k}\subseteq {\mathcal {E}}$. When $E$ is a fibred manifold over $X$, one recovers the notion of partial differential equations on jet bundles and their solutions, which provide a coordinate-free way to describe the analogous notions of mathematical analysis. While jet bundles are enough to deal with many equations arising in geometry, jet spaces of submanifolds provide a greater generality, used to tackle several PDEs imposed on submanifolds of a given manifold, such as Lagrangian submanifolds and minimal surfaces.
As in the jet bundle case, the Cartan distribution is important in the algebro-geometric approach to differential equations because it allows to encode solutions in purely geometric terms. Indeed, a submanifold $S\subset {\mathcal {E}}$ is a solution if and only if it is an integral manifold for ${\mathcal {C}}$, i.e. $T_{\theta }S\subset {\mathcal {C}}_{\theta }$ for all $\theta \in S$.
One can also look at the Cartan distribution of a PDE ${\mathcal {E}}\subset J^{k}(E,m)$ more intrinsically, defining
${\mathcal {C}}({\mathcal {E}}):=\{{\mathcal {C}}_{\theta }\cap T_{\theta }({\mathcal {E}})~|~\theta \in {\mathcal {E}}\}$
In this sense, the pair $({\mathcal {E}},{\mathcal {C}}({\mathcal {E}}))$ encodes the information about the solutions of the differential equation ${\mathcal {E}}$.
Prolongations of PDEs
Given a differential equation ${\mathcal {E}}\subset J^{l}(E,m)$ of order $l$, its $k$-th prolongation is defined as
${\mathcal {E}}^{k}:=J^{k}({\mathcal {E}},m)\cap J^{k+l}(E,m)\subseteq J^{k+l}(E,m)$
where both $J^{k}({\mathcal {E}},m)$ and $J^{k+l}(E,m)$ are viewed as embedded submanifolds of $J^{k}(J^{l}(E,m),m)$, so that their intersection is well-defined. However, such an intersection is not necessarily a manifold again, hence ${\mathcal {E}}^{k}$ may not be an equation of order $k+l$. One therefore usually requires ${\mathcal {E}}$ to be "nice enough" such that at least its first prolongation is indeed a submanifold of $J^{k+1}(E,m)$.
Below we will assume that the PDE is formally integrable, i.e. all prolongations ${\mathcal {E}}^{k}$ are smooth manifolds and all projections ${\mathcal {E}}^{k}\to {\mathcal {E}}^{k-1}$ are smooth surjective submersions. Note that a suitable version of Cartan–Kuranishi prolongation theorem guarantees that, under minor regularity assumptions, checking the smoothness of a finite number of prolongations is enough. Then the inverse limit of the sequence $\{{\mathcal {E}}^{k}\}_{k\in \mathbb {N} }$ extends the definition of prolongation to the case when $k$ goes to infinity, and the space ${\mathcal {E}}^{\infty }$ has the structure of a profinite-dimensional manifold.[5]
Definition of a diffiety
An elementary diffiety is a pair $({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$ where ${\mathcal {E}}\subset J^{k}(E,m)$ is a $k$-th order differential equation on some manifold, ${\mathcal {E}}^{\infty }$ its infinite prolongation and ${\mathcal {C}}({\mathcal {E}}^{\infty })$ its Cartan distribution. Note that, unlike in the finite case, one can show that the Cartan distribution ${\mathcal {C}}({\mathcal {E}}^{\infty })$ is $m$-dimensional and involutive. However, due to the infinite-dimensionality of the ambient manifold, the Frobenius theorem does not hold, therefore ${\mathcal {C}}({\mathcal {E}}^{\infty })$ is not integrable
A diffiety is a triple $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$, consisting of
• a (generally infinite-dimensional) manifold ${\mathcal {O}}$
• the algebra of its smooth functions ${\mathcal {F}}({\mathcal {O}})$
• a finite-dimensional distribution ${\mathcal {C}}({\mathcal {O}})$,
such that $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$ is locally of the form $({\mathcal {E}}^{\infty },{\mathcal {F}}({\mathcal {E}}^{\infty }),{\mathcal {C}}({\mathcal {E}}^{\infty }))$, where $({\mathcal {E}}^{\infty },{\mathcal {C}}({\mathcal {E}}^{\infty }))$ is an elementary diffiety and ${\mathcal {F}}({\mathcal {E}}^{\infty })$ denotes the algebra of smooth functions on ${\mathcal {E}}^{\infty }$. Here locally means a suitable localisation with respect to the Zariski topology corresponding to the algebra ${\mathcal {F}}({\mathcal {O}})$.
The dimension of ${\mathcal {C}}({\mathcal {O}})$ is called dimension of the diffiety and its denoted by $\mathrm {Dim} ({\mathcal {O}})$, with a capital D (to distinguish it from the dimension of ${\mathcal {O}}$ as a manifold).
Morphisms of diffieties
A morphism between two diffieties $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$ and $({\mathcal {O}}',{\mathcal {F}}({\mathcal {O}}'),{\mathcal {C}}({\mathcal {O'}}))$ consists of a smooth map $\Phi :{\mathcal {O}}\rightarrow {\mathcal {O}}'$ :{\mathcal {O}}\rightarrow {\mathcal {O}}'} whose pushforward preserves the Cartan distribution, i.e. such that, for every point $\theta \in {\mathcal {O}}$, one has $d_{\theta }\Phi ({\mathcal {C}}_{\theta })\subseteq {\mathcal {C}}_{\Phi (\theta )}$.
Diffieties together with their morphisms define the category of differential equations.[3]
Applications
Vinogradov sequence
The Vinogradov ${\mathcal {C}}$-spectral sequence (or, for short, Vinogradov sequence) is a spectral sequence associated to a diffiety, which can be used to investigate certain properties of the formal solution space of a differential equation by exploiting its Cartan distribution ${\mathcal {C}}$.[6]
Given a diffiety $({\mathcal {O}},{\mathcal {F}}({\mathcal {O}}),{\mathcal {C}}({\mathcal {O}}))$, consider the algebra of differential forms over ${\mathcal {O}}$
$\Omega ({\mathcal {O}}):=\sum _{i\geq 0}\Omega ^{i}({\mathcal {O}})$
and the corresponding de Rham complex:
$C^{\infty }({\mathcal {O}})\longrightarrow \Omega ^{1}({\mathcal {O}})\longrightarrow \Omega ^{2}({\mathcal {O}})\longrightarrow \cdots $
Its cohomology groups $H^{i}({\mathcal {O}}):={\text{ker}}({\text{d}}_{i})/{\text{im}}({\text{d}}_{i-1})$ contain some structural information about the PDE; however, due to the Poincaré Lemma, they all vanish locally. In order to extract much more and even local information, one thus needs to take the Cartan distribution into account and introduce a more sophisticated sequence. To this end, let
${\mathcal {C}}\Omega ({\mathcal {O}})=\sum _{i\geq 0}{\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ({\mathcal {O}})$
be the submodule of differential forms over ${\mathcal {O}}$ whose restriction to the distribution ${\mathcal {C}}$ vanishes, i.e.
${\mathcal {C}}\Omega ^{p}({\mathcal {O}}):=\{w\in \Omega ^{p}({\mathcal {O}})\mid w(X_{1},\cdots ,X_{p})=0\quad \forall ~X_{1},\ldots ,X_{p}\in {\mathcal {C}}({\mathcal {O}})\}.$
Note that ${\mathcal {C}}\Omega ^{i}({\mathcal {O}})\subseteq \Omega ^{i}({\mathcal {O}})$ is actually a differential ideal since it is stable w.r.t. to the de Rham differential, i.e. ${\text{d}}({\mathcal {C}}\Omega ^{i}({\mathcal {O}}))\subset {\mathcal {C}}\Omega ^{i+1}({\mathcal {O}})$.
Now let ${\mathcal {C}}^{k}\Omega ({\mathcal {O}})$ be its $k$-th power, i.e. the linear subspace of ${\mathcal {C}}\Omega $ generated by $w_{1}\wedge \cdots \wedge w_{k},~w_{i}\in {\mathcal {C}}\Omega $. Then one obtains a filtration
$\Omega ({\mathcal {O}})\supset {\mathcal {C}}\Omega ({\mathcal {O}})\supset {\mathcal {C}}^{2}\Omega ({\mathcal {O}})\supset \cdots $
and since all ideals ${\mathcal {C}}^{k}\Omega $ are stable, this filtration completely determines the following spectral sequence:
${\mathcal {C}}E({\mathcal {O}})=\{E_{r}^{p,q},{\text{d}}_{r}^{p,q}\}\qquad {\text{where}}\qquad E_{0}^{p,q}:={\frac {{\mathcal {C}}^{p}\Omega ^{p+q}({\mathcal {O}})}{{\mathcal {C}}^{p+1}\Omega ^{p+q}({\mathcal {O}})}},\qquad {\text{and}}\qquad E_{r+1}^{p,q}:=H(E_{r}^{p,q},d_{r}^{p,q}).$
The filtration above is finite in each degree, i.e. for every $k\geq 0$
$\Omega ^{k}({\mathcal {O}})\supset {\mathcal {C}}^{1}\Omega ^{k}({\mathcal {O}})\supset \cdots \supset {\mathcal {C}}^{k+1}\Omega ^{k}({\mathcal {O}})=0,$
so that the spectral sequence converges to the de Rham cohomology $H({\mathcal {O}})$ of the diffiety. One can therefore analyse the terms of the spectral sequence order by order to recover information on the original PDE. For instance:[7]
• $E_{1}^{0,n}$ corresponds to action functionals constrained by the PDE ${\mathcal {E}}$. In particular, for ${\mathcal {L}}\in E_{1}^{0,n}$, the corresponding Euler-Lagrange equation is ${\text{d}}_{1}^{0,n}{\mathcal {L}}=0$.
• $E_{1}^{0,n-1}$ corresponds to conservation laws for solutions of ${\mathcal {E}}$.
• $E_{2}$ is interpreted as characteristic classes of bordisms of solutions of ${\mathcal {E}}$.
Many higher-order terms do not have an interpretation yet.
Variational bicomplex
As a particular case, starting with a fibred manifold $\pi :E\to X$ and its jet bundle $J^{k}(\pi )$ instead of the jet space $J^{k}(E,m)$, instead of the ${\mathcal {C}}$-spectral sequence one obtains the slightly less general variational bicomplex. More precisely, any bicomplex determines two spectral sequences: one of the two spectral sequences determined by the variational bicomplex is exactly the Vinogradov ${\mathcal {C}}$-spectral sequence. However, the variational bicomplex was developed independently from the Vinogradov sequence.[8][9]
Similarly to the terms of the spectral sequence, many terms of the variational bicomplex can be given a physical interpretation in classical field theory: for example, one obtains cohomology classes corresponding to action functionals, conserved currents, gauge charges, etc.[10]
Secondary calculus
Vinogradov developed a theory, known as secondary calculus, to formalise in cohomological terms the idea of a differential calculus on the space of solutions of a given system of PDEs (i.e. the space of integral manifolds of a given diffiety).[11][12][13][3]
In other words, secondary calculus provides substitutes for functions, vector fields, differential forms, differential operators, etc., on a (generically) very singular space where these objects cannot be defined in the usual (smooth) way on the space of solution. Furthermore, the space of these new objects are naturally endowed with the same algebraic structures of the space of the original objects.[14]
More precisely, consider the horizontal De Rham complex ${\overline {\Omega }}^{\bullet }({\mathcal {O}}):=\Gamma (\wedge ^{\bullet }{\mathcal {C(O)}}^{*})$ of a diffiety, which can be seen as the leafwise de Rham complex of the involutive distribution ${\mathcal {C(O)}}$or, equivalently, the Lie algebroid complex of the Lie algebroid ${\mathcal {C(O)}}$. Then the complex ${\overline {\Omega }}^{\bullet }({\mathcal {O}})$ becomes naturally a commutative DG algebra together with a suitable differential ${\overline {d}}$. Then, possibly tensoring with the normal bundle ${\mathcal {V}}:=T{\mathcal {O}}/{\mathcal {C(O)}}\to {\mathcal {O}}$, its cohomology is used to define the following "secondary objects":
• secondary functions are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}),{\overline {d}})$, which is naturally a commutative DG algebra (it is actually the first page of the ${\mathcal {C}}$-spectral sequence discussed above);
• secondary vector fields are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}},{\mathcal {V}})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes {\mathcal {V}}),{\overline {d}})$, which is naturally a Lie algebra; moreover, it forms a graded Lie-Rinehart algebra together with ${\overline {H}}^{\bullet }({\mathcal {O}})$;
• secondary differential $p$-forms are elements of the cohomology ${\overline {H}}^{\bullet }({\mathcal {O}},\wedge ^{p}{\mathcal {V}}^{*})=H^{\bullet }({\overline {\Omega }}^{\bullet }({\mathcal {O}}\otimes \wedge ^{p}{\mathcal {V}}^{*}),{\overline {d}})$, which is naturally a commutative DG algebra.
Secondary calculus can also be related to the covariant Phase Space, i.e. the solution space of the Euler-Lagrange equations associated to a Lagrangian field theory.[15]
See also
• Secondary calculus and cohomological physics
• Partial differential equations on Jet bundles
• Differential ideal
• Differential calculus over commutative algebras
Another way of generalizing ideas from algebraic geometry is differential algebraic geometry.
References
1. Vinogradov, A. M. (March 1984). "Local symmetries and conservation laws". Acta Applicandae Mathematicae. 2 (1): 21–78. doi:10.1007/BF01405491. ISSN 0167-8019. S2CID 121860845.
2. Saunders, D. J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Cambridge: Cambridge University Press. doi:10.1017/cbo9780511526411. ISBN 978-0-521-36948-0.
3. Vinogradov, A. M. (2001). Cohomological analysis of partial differential equations and secondary calculus. Providence, R.I.: American Mathematical Society. ISBN 0-8218-2922-X. OCLC 47296188.
4. Krasil'shchik, I. S.; Lychagin, V. V.; Vinogradov, A. M. (1986). Geometry of jet spaces and nonlinear partial differential equations. Adv. Stud. Contemp. Math., N. Y. Vol. 1. New York etc.: Gordon and Breach Science Publishers. ISBN 978-2-88124-051-5.
5. Güneysu, Batu; Pflaum, Markus J. (2017-01-10). "The Profinite Dimensional Manifold Structure of Formal Solution Spaces of Formally Integrable PDEs". SIGMA. Symmetry, Integrability and Geometry: Methods and Applications. 13: 003. arXiv:1308.1005. Bibcode:2017SIGMA..13..003G. doi:10.3842/SIGMA.2017.003. S2CID 15871902.
6. Vinogradov, A. M. (1978). "A spectral sequence associated with a nonlinear differential equation and algebro-geometric foundations of Lagrangian field theory with constraints". Soviet Math. Dokl. (in Russian). 19: 144–148 – via Math-Net.Ru.
7. Symmetries and conservation laws for differential equations of mathematical physics. A. V. Bocharov, I. S. Krasilʹshchik, A. M. Vinogradov. Providence, R.I.: American Mathematical Society. 1999. ISBN 978-1-4704-4596-6. OCLC 1031947580.{{cite book}}: CS1 maint: others (link)
8. Tulczyjew, W. M. (1980). García, P. L.; Pérez-Rendón, A.; Souriau, J. M. (eds.). "The Euler-Lagrange resolution". Differential Geometrical Methods in Mathematical Physics. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer. 836: 22–48. doi:10.1007/BFb0089725. ISBN 978-3-540-38405-2.
9. Tsujishita, Toru (1982). "On variation bicomplexes associated to differential equations". Osaka Journal of Mathematics. 19 (2): 311–363. ISSN 0030-6126.
10. "variational bicomplex in nLab". ncatlab.org. Retrieved 2021-12-11.
11. Vinogradov, A.M. (1984-04-30). "The b-spectral sequence, Lagrangian formalism, and conservation laws. I. The linear theory". Journal of Mathematical Analysis and Applications. 100 (1): 1–40. doi:10.1016/0022-247X(84)90071-4.
12. Vinogradov, A. M. (1984-04-30). "The b-spectral sequence, Lagrangian formalism, and conservation laws. II. The nonlinear theory". Journal of Mathematical Analysis and Applications. 100 (1): 41–129. doi:10.1016/0022-247X(84)90072-6. ISSN 0022-247X.
13. Henneaux, Marc; Krasil′shchik, Joseph; Vinogradov, Alexandre, eds. (1998). Secondary Calculus and Cohomological Physics. Contemporary Mathematics. Vol. 219. Providence, Rhode Island: American Mathematical Society. doi:10.1090/conm/219. ISBN 978-0-8218-0828-3.
14. Vitagliano, Luca (2014). "On the strong homotopy Lie–Rinehart algebra of a foliation". Communications in Contemporary Mathematics. 16 (6): 1450007. arXiv:1204.2467. doi:10.1142/S0219199714500072. ISSN 0219-1997. S2CID 119704524.
15. Vitagliano, Luca (2009-04-01). "Secondary calculus and the covariant phase space". Journal of Geometry and Physics. 59 (4): 426–447. arXiv:0809.4164. Bibcode:2009JGP....59..426V. doi:10.1016/j.geomphys.2008.12.001. ISSN 0393-0440. S2CID 21787052.
External links
• The Diffiety Institute (frozen since 2010)
• The Levi-Civita Institute (successor of above site)
• Geometry of Differential Equations
• Differential Geometry and PDEs
Manifolds (Glossary)
Basic concepts
• Topological manifold
• Atlas
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• Differential structure
• Smooth atlas
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• Pushforward
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• Differential form
• Vector field
Main results (list)
• Atiyah–Singer index
• Darboux's
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manifolds
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• Banach manifold
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Asymmetry
Asymmetry is the absence of, or a violation of, symmetry (the property of an object being invariant to a transformation, such as reflection).[1] Symmetry is an important property of both physical and abstract systems and it may be displayed in precise terms or in more aesthetic terms.[2] The absence of or violation of symmetry that are either expected or desired can have important consequences for a system.
This article is about the absence of symmetry. For a specific use in mathematics, see asymmetric relation. For other uses, see Asymmetry (disambiguation).
In organisms
Due to how cells divide in organisms, asymmetry in organisms is fairly usual in at least one dimension, with biological symmetry also being common in at least one dimension.
Louis Pasteur proposed that biological molecules are asymmetric because the cosmic [i.e. physical] forces that preside over their formation are themselves asymmetric. While at his time, and even now, the symmetry of physical processes are highlighted, it is known that there are fundamental physical asymmetries, starting with time.
Asymmetry in biology
Asymmetry is an important and widespread trait, having evolved numerous times in many organisms and at many levels of organisation (ranging from individual cells, through organs, to entire body-shapes). Benefits of asymmetry sometimes have to do with improved spatial arrangements, such as the left human lung being smaller, and having one fewer lobes than the right lung to make room for the asymmetrical heart. In other examples, division of function between the right and left half may have been beneficial and has driven the asymmetry to become stronger. Such an explanation is usually given for mammal hand or paw preference (handedness), an asymmetry in skill development in mammals. Training the neural pathways in a skill with one hand (or paw) may take less effort than doing the same with both hands.[3]
Nature also provides several examples of handedness in traits that are usually symmetric. The following are examples of animals with obvious left-right asymmetries:
• Most snails, because of torsion during development, show remarkable asymmetry in the shell and in the internal organs.[4]
• Male fiddler crabs have one big claw and one small claw.[5]
• The narwhal's tusk is a left incisor which can grow up to 10 feet in length and forms a left-handed helix.[6]
• Flatfish have evolved to swim with one side upward, and as a result have both eyes on one side of their heads.[7]
• Several species of owls exhibit asymmetries in the size and positioning of their ears, which is thought to help locate prey.[8]
• Many animals (ranging from insects to mammals) have asymmetric male genitalia. The evolutionary cause behind this is, in most cases, still a mystery.[9]
As an indicator of unfitness
• Certain disturbances during the development of the organism, resulting in birth defects.
• Injuries after cell division that cannot be biologically repaired, such as a lost limb from an accident.
Since birth defects and injuries are likely to indicate poor health of the organism, defects resulting in asymmetry often put an animal at a disadvantage when it comes to finding a mate. For example, a greater degree of facial symmetry is seen as more attractive in humans, especially in the context of mate selection. In general, there is a correlation between symmetry and fitness-related traits such as growth rate, fecundity and survivability for many species. This means that, through sexual selection, individuals with greater symmetry (and therefore fitness) tend to be preferred as mates, as they are more likely to produce healthy offspring.[10]
In structures
Pre-modern architectural styles tended to place an emphasis on symmetry, except where extreme site conditions or historical developments lead away from this classical ideal. To the contrary, modernist and postmodern architects became much more free to use asymmetry as a design element.
While most bridges employ a symmetrical form due to intrinsic simplicities of design, analysis and fabrication and economical use of materials, a number of modern bridges have deliberately departed from this, either in response to site-specific considerations or to create a dramatic design statement.
Some asymmetrical structures
• Eastern span replacement of the San Francisco – Oakland Bay Bridge
• Puente de la Mujer
• Auditorio de Tenerife
• Blohm & Voss BV 141 aircraft
• A proa, a form of outrigger canoe
In fire protection
In fire-resistance rated wall assemblies, used in passive fire protection, including, but not limited to, high-voltage transformer fire barriers, asymmetry is a crucial aspect of design. When designing a facility, it is not always certain, that in the event of fire, which side a fire may come from. Therefore, many building codes and fire test standards outline, that a symmetrical assembly, need only be tested from one side, because both sides are the same. However, as soon as an assembly is asymmetrical, both sides must be tested and the test report is required to state the results for each side. In practical use, the lowest result achieved is the one that turns up in certification listings. Neither the test sponsor, nor the laboratory can go by an opinion or deduction as to which side was in more peril as a result of contemplated testing and then test only one side. Both must be tested in order to be compliant with test standards and building codes.
In mathematics
See also: Symmetry in mathematics
In mathematics, asymmetry can arise in various ways. Examples include asymmetric relations, asymmetry of shapes in geometry, asymmetric graphs et cetera.
Asymmetric Relation
An asymmetric relation is a binary relation $R$ defined on a set of elements such that if $aRb$ holds for elements $a$ and $b$, then $bRa$ must be false. Stated differently, an asymmetric relation is characterized by a necessary absence of symmetry of the relation in the opposite direction.
Inequalities exemplify asymmetric relations. Consider elements $a$ and $b$. If $a$ is less than $b$ ($a<b$), then $a$ cannot be greater than $b$ ($a\ngtr b$).[11] This highlights how the relations "less than", and similarly "greater than", are not symmetric.
In contrast, if $a$ is equal to $b$ ($a=b$), then $b$ is also equal to $a$ ($b=a$). Thus the binary relation "equal to" is a symmetric one.
In chemistry
Certain molecules are chiral; that is, they cannot be superposed upon their mirror image. Chemically identical molecules with different chirality are called enantiomers; this difference in orientation can lead to different properties in the way they react with biological systems.
In physics
Asymmetry arises in physics in a number of different realms.
Thermodynamics
The original non-statistical formulation of thermodynamics was asymmetrical in time: it claimed that the entropy in a closed system can only increase with time. This was derived from the Second Law (any of the two, Clausius' or Lord Kelvin's statement can be used since they are equivalent) and using the Clausius' Theorem (see Kerson Huang ISBN 978-0471815181). The later theory of statistical mechanics, however, is symmetric in time. Although it states that a system significantly below maximum entropy is very likely to evolve towards higher entropy, it also states that such a system is very likely to have evolved from higher entropy.
Particle physics
Symmetry is one of the most powerful tools in particle physics, because it has become evident that practically all laws of nature originate in symmetries. Violations of symmetry therefore present theoretical and experimental puzzles that lead to a deeper understanding of nature. Asymmetries in experimental measurements also provide powerful handles that are often relatively free from background or systematic uncertainties.
Parity violation
Until the 1950s, it was believed that fundamental physics was left-right symmetric; i.e., that interactions were invariant under parity. Although parity is conserved in electromagnetism, strong interactions and gravity, it turns out to be violated in weak interactions. The Standard Model incorporates parity violation by expressing the weak interaction as a chiral gauge interaction. Only the left-handed components of particles and right-handed components of antiparticles participate in weak interactions in the Standard Model. A consequence of parity violation in particle physics is that neutrinos have only been observed as left-handed particles (and antineutrinos as right-handed particles).
In 1956–1957 Chien-Shiung Wu, E. Ambler, R. W. Hayward, D. D. Hoppes, and R. P. Hudson found a clear violation of parity conservation in the beta decay of cobalt-60. Simultaneously, R. L. Garwin, Leon Lederman, and R. Weinrich modified an existing cyclotron experiment and immediately verified parity violation.
CP violation
After the discovery of the violation of parity in 1956–57, it was believed that the combined symmetry of parity (P) and simultaneous charge conjugation (C), called CP, was preserved. For example, CP transforms a left-handed neutrino into a right-handed antineutrino. In 1964, however, James Cronin and Val Fitch provided clear evidence that CP symmetry was also violated in an experiment with neutral kaons.
CP violation is one of the necessary conditions for the generation of a baryon asymmetry in the early universe.
Combining the CP symmetry with simultaneous time reversal (T) produces a combined symmetry called CPT symmetry. CPT symmetry must be preserved in any Lorentz invariant local quantum field theory with a Hermitian Hamiltonian. As of 2006, no violations of CPT symmetry have been observed.
Baryon asymmetry of the universe
The baryons (i.e., the protons and neutrons and the atoms that they comprise) observed so far in the universe are overwhelmingly matter as opposed to anti-matter. This asymmetry is called the baryon asymmetry of the universe.
Isospin violation
Isospin is the symmetry transformation of the weak interactions. The concept was first introduced by Werner Heisenberg in nuclear physics based on the observations that the masses of the neutron and the proton are almost identical and that the strength of the strong interaction between any pair of nucleons is the same, independent of whether they are protons or neutrons. This symmetry arises at a more fundamental level as a symmetry between up-type and down-type quarks. Isospin symmetry in the strong interactions can be considered as a subset of a larger flavor symmetry group, in which the strong interactions are invariant under interchange of different types of quarks. Including the strange quark in this scheme gives rise to the Eightfold Way scheme for classifying mesons and baryons.
Isospin is violated by the fact that the masses of the up and down quarks are different, as well as by their different electric charges. Because this violation is only a small effect in most processes that involve the strong interactions, isospin symmetry remains a useful calculational tool, and its violation introduces corrections to the isospin-symmetric results.
In collider experiments
Because the weak interactions violate parity, collider processes that can involve the weak interactions typically exhibit asymmetries in the distributions of the final-state particles. These asymmetries are typically sensitive to the difference in the interaction between particles and antiparticles, or between left-handed and right-handed particles. They can thus be used as a sensitive measurement of differences in interaction strength and/or to distinguish a small asymmetric signal from a large but symmetric background.
• A forward-backward asymmetry is defined as AFB=(NF-NB)/(NF+NB), where NF is the number of events in which some particular final-state particle is moving "forward" with respect to some chosen direction (e.g., a final-state electron moving in the same direction as the initial-state electron beam in electron-positron collisions), while NB is the number of events with the final-state particle moving "backward". Forward-backward asymmetries were used by the LEP experiments to measure the difference in the interaction strength of the Z boson between left-handed and right-handed fermions, which provides a precision measurement of the weak mixing angle.
• A left-right asymmetry is defined as ALR=(NL-NR)/(NL+NR), where NL is the number of events in which some initial- or final-state particle is left-polarized, while NR is the corresponding number of right-polarized events. Left-right asymmetries in Z boson production and decay were measured at the Stanford Linear Collider using the event rates obtained with left-polarized versus right-polarized initial electron beams. Left-right asymmetries can also be defined as asymmetries in the polarization of final-state particles whose polarizations can be measured; e.g., tau leptons.
• A charge asymmetry or particle-antiparticle asymmetry is defined in a similar way. This type of asymmetry has been used to constrain the parton distribution functions of protons at the Tevatron from events in which a produced W boson decays to a charged lepton. The asymmetry between positively and negatively charged leptons as a function of the direction of the W boson relative to the proton beam provides information on the relative distributions of up and down quarks in the proton. Particle-antiparticle asymmetries are also used to extract measurements of CP violation from B meson and anti-B meson production at the BaBar and Belle experiments.
Lexical
Asymmetry is also relevant to grammar and linguistics, especially in the contexts of lexical analysis and transformational grammar.
Enumeration example: In English, there are grammatical rules for specifying coordinate items in an enumeration or series. Similar rules exist for programming languages and mathematical notation. These rules vary, and some require lexical asymmetry to be considered grammatically correct.
For example, in standard written English:
We sell domesticated cats, dogs, and goldfish. ### in-line asymmetric and grammatical
We sell domesticated animals (cats, dogs, goldfish). ### in-line symmetric and grammatical
We sell domesticated animals (cats, dogs, goldfish,). ### in-line symmetric and ungrammatical
We sell domesticated animals: ### outline symmetric and grammatical
- cats
- dogs
- goldfish
In fiction
• In Transformers: Rise of the Beasts, Apelinq's left shoulder seems to be asymmetrical to the right, being slightly more bulked up.
• Bagugan: New Vestroia featuresMira, the Token Girl, wearing a catsuit that goes halfway past the knee on the left side and barely past the waist on the right.
• Several of the title character's outfits in Cardcaptor Sakura have one short and one long stocking.
• Digimon features Angewoman wearing one glove, and also many Digimon including Zudomon owning asymmetric designs.
• In Dragon Ball, Cooler's troops all wear a single shoulder pad on their left side.
See also
• Information asymmetry
• Asymmetric multiprocessing
References
1. "Definition of ASYMMETRY". www.merriam-webster.com. 2023-07-19. Retrieved 2023-07-23.
2. "Definition of SYMMETRY". www.merriam-webster.com. 2023-07-22. Retrieved 2023-07-23.
3. Baofu, Peter (19 Mar 2009). The Future of Post-Human Geometry: A Preface to a New Theory of Infinity, Symmetry, and Dimensionality. p. 149. ISBN 978-1-4438-0524-7.
4. "Surprising Start for Snail Asymmetry". www.science.org. Retrieved 2023-06-04.
5. "Fiddler Crabs". biology-assets.anu.edu.au. Retrieved 2023-06-04.
6. Kingsley, Michael C.S.; Ramsay, Malcolm A. (September 1988). "The Spiral in the Tusk of the Narwhal". Arctic. 41 (3): 1. JSTOR 40510720 – via JSTOR.
7. Friedman, Matt (2008-07-10). "The evolutionary origin of flatfish asymmetry". Nature. 454 (7201): 209–212. doi:10.1038/nature07108. ISSN 1476-4687. PMID 18615083.
8. "Owl hearing | BTO - British Trust for Ornithology". www.bto.org. Retrieved 2023-06-04.
9. Schilthuizen, Menno (2013). "Something gone awry: unsolved mysteries in the evolution of asymmetric animal genitalia". Animal Biology. 63 (1): 1–20. doi:10.1163/15707563-00002398.
10. Little, Anthony C.; Jones, Benedict C.; DeBruine, Lisa M. (2011-06-12). "Facial attractiveness: evolutionary based research". Philosophical Transactions of the Royal Society B: Biological Sciences. 366 (1571): 1638–1659. doi:10.1098/rstb.2010.0404. ISSN 0962-8436. PMC 3130383. PMID 21536551.
11. Introduction to Set Theory, Third Edition, Revised and Expanded: Hrbacek, Jech.
Further reading
• Gardner, Martin (1990), The New Ambidextrous Universe: Symmetry and Asymmetry from Mirror Reflections to Superstrings, 3rd edition, W.H.Freeman & Co Ltd.
• Jan, Yuh-Nung; Yeh Jan, Lily (1999). "Asymmetry across species". Nature Cell Biology. 1 (2): E42–E44. doi:10.1038/10036. PMID 10559895. S2CID 9399564.
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Violet B. Haas
Violet Bushwick Haas (November 23, 1926 – January 21, 1986) was an American applied mathematician specializing in control theory and optimal estimation who became a professor of electrical engineering at Purdue University College of Engineering.
Violet B. Haas
BornNovember 23, 1926
Brooklyn, New York, U.S.
DiedJanuary 21, 1986(1986-01-21) (aged 59)
Lafayette, Indiana, U.S.
Alma materBrooklyn College
Massachusetts Institute of Technology
SpouseFelix Haas
Children3
Scientific career
FieldsControl theory, optimal estimation
InstitutionsUniversity of Detroit Mercy
Purdue University College of Engineering
Doctoral advisorNorman Levinson
Early life and education
Haas was born November 23, 1926 in Brooklyn.[1] She completed a A.B. in mathematics[2] at Brooklyn College in 1947.[1] Haas earned a M.S. (1949) and Ph.D. (1951) in mathematics from the MIT Department of Mathematics (MIT).[1] Her dissertation was titled Singular perturbations of an ordinary differential equation. Norman Levinson was her doctoral advisor.[3] She met her future husband, Felix Haas, a fellow mathematician at MIT.[4] Haas was selected as an American Association of University Women Vassie James Hill Fellow in 1951.[2][4] She was a member of Phi Beta Kappa, Sigma Xi, and Eta Kappa Nu.[2]
Career
Haas was a lecturer at Immaculata College from 1952 to 1955 and an instructor at the University of Connecticut from 1955 to 1956.[2] She was a faculty member at the University of Detroit from 1957 to 1962.[2] Haas also taught at Wayne State University.[4]
She joined the faculty at Purdue University in January 1962[1] as an assistant professor in the college of electrical engineering and computer engineering.[4] By 1978, Haas was a full professor of electrical engineering in the Purdue University College of Engineering.[4][2] Her areas of expertise included optimal control, nonlinear control, and optimal estimation.[2] Due to nepotism rules (her husband was a fellow mathematician), Haas took a position in electrical engineering rather than mathematics.[5]
Haas advocated for women in STEM fields. Some of her earlier academic environments were hostile to women.[5] In a few instances, she was the only department member excluded from grant proposals. This had largely improved by the early 1980s.[5] For 15 years, Haas was the counselor of the Purdue University student chapter of the Society of Women Engineers.[5]
Haas joined the Association for Women in Mathematics in 1975, serving as a coordinator for the speakers' bureau.[5] She was a member of the Institute of Electrical and Electronics Engineers (IEEE) committee on professional opportunities for women and the American Society for Engineering Education (ASEE) constituent committee on women in engineering.[5] For Haas' support and encouragement of women students in engineering, in 1977, she was elected as one of five "Very Important Women" on campus by the Association of Women Students.[6] In 1977, Haas received the D.D. Ewing Award as an outstanding teacher in the Purdue School of Electrical Engineering.[7] She received the 1978 Helen B. Schleman Medallion Award for her service and encouragement of women in academic and professional areas.[7] In the 1970s, Haas was a nominee for the distinguished science award of the Society of Women Engineers.[7]
From 1983 to 1984, Haas was a visiting professor at Massachusetts Institute of Technology through the National Science Foundation visiting professorships for women program. In this position, she was a full time researcher investigating control theory and infinite dimensional control problems.[5]
Haas was active in the control systems community and was on the program committee of the American Control Conference. She was also the Society for Industrial and Applied Mathematics (SIAM) representative to the IEEE conference on decision and control.[5] Mathematician and colleague Pamela G. Coxson stated that Haas' involvement "increased the participation of the mathematical community in these two annual conferences."[5] She is included in biographical listings of Who's Who in the Midwest, Who's Who of American Women, and American Men and Women of Science.[6] She was a former editor of the Women in Engineering Students Newsletter.[1] Haas was a member of the American Association of University Women,[6] League of Women Voters, YWCA, and served on the board of directors of the Lafayette Symphony.[1][2]
Haas and sociologist Carolyn C. Perrucci co-edited the book Women in Scientific and Engineering Professions. University of Michigan Press. 1984. ISBN 978-0-472-10049-1.[4]
Eight months after leaving MIT in 1984, Haas was diagnosed with a brain tumor and was soon unconscious.[5]
Personal life
Haas resided in West Lafayette, Indiana.[1] She was married to Felix Haas.[1] They had a daughter and two sons.[1]
Haas was unconscious from a brain tumor from 1984 until her death on the morning of January 21, 1986 at St. Elizabeth Hospital.[1][6]
In 1990, the Council on the Status of Women at Purdue University established Violet B. Haas award that recognizes people who promote the status of women at the university.[4]
References
Citations
1. Journal & Courier 1986, p. 12.
2. The Indianapolis Star 1986, p. 55.
3. Haas 1951.
4. Purdue University Archives and Special Collections.
5. Coxson 1986, p. 2.
6. Society for Industrial and Applied Mathematics 1986, p. 10.
7. Journal & Courier 1978, p. 6.
Bibliography
• "Obituary". Society for Industrial and Applied Mathematics News. March 1986.
• Coxson, Pamela G. (July 1986). "In Remembrance of Violet Bushwick Haas (1926-1986)". Association for Women in Mathematics Newsletter. Vol. 16, no. 4. pp. 2–3. Retrieved 2022-04-19.
• Haas, Violet B. (1951). Singular perturbations of an ordinary differential equation (Ph.D. thesis). MIT Department of Mathematics. OCLC 30350947.
• "Prof. Haas receives Schleman award for service". Journal & Courier. 1978-05-11. p. 6. Retrieved 2022-04-19.
• "Dr. Violet Haas, 59, dies; was on Purdue faculty". Journal & Courier. 1986-01-22. p. 12. Retrieved 2022-04-19.
• "Haas, Violet B." Purdue University Archives and Special Collections.
• "Memorial rites Friday for Professor Violet Haas". The Indianapolis Star. 1986-01-23. p. 55. Retrieved 2022-04-19.
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Viorel P. Barbu
Viorel P. Barbu (born 14 June 1941) is a Romanian mathematician, specializing in partial differential equations, control theory, and stochastic differential equations.
Biography
He was born in Deleni, Vaslui County, Romania.[1] He attended the Mihail Kogălniceanu High School in Vaslui and then the Costache Negruzzi National College in Iași. Barbu completed his undergraduate degree at the Alexandru Ioan Cuza University of Iași in 1964,[1] and his Ph.D. at the same university in 1969.[2] His doctoral advisor was Adolf Haimovici; his dissertation thesis was titled Regularity Theory of Pseudodifferential Operators.[2] He became a professor at the University of Iași in 1980.[1] His Ph.D. students there included Gheorghe Moroșanu and Daniel Tătaru.[2]
In 1993, he was elected a titular member of the Romanian Academy.[3] In 2011 he was awarded the Order of the Star of Romania, Knight rank by President Traian Băsescu.[4]
Bibliography
Some of his books and papers are:[5][6][7]
• Analysis And Control Of Nonlinear Infinite Dimensional Systems
• Optimization, Optimal Control and Partial Differential Equations
• Nonlinear semigroups and differential equations in Banach spaces
• Hamilton-Jacobi Equations on Hilbert Space
• Stochastic Porous Media Equations
• Nonlinear Differential Equations of Monotone Types in Banach Spaces
• Convexity and Optimization in Banach Spaces
• Optimal Control of Variational Inequalities
References
1. "Pagina personală Acad. Viorel Barbu". uaic.ro.
2. Viorel P. Barbu at the Mathematics Genealogy Project
3. (in Romanian) Membrii Academiei Române din 1866 până în prezent at the Romanian Academy site
4. "Decret nr. 639 din 7 iulie 2011 privind conferirea unor decorații". legislatie.just.ro (in Romanian). July 7, 2011. Retrieved May 1, 2021.
5. "Books by Viorel Barbu (Author of Differential Equations)". goodreads.com.
6. Viorel P. Barbu publications indexed by Google Scholar
7. "Viorel Barbu (Universitatea Alexandru Ioan Cuza, Iași) on ResearchGate - Expertise: Analysis, Civil Engineering, Structural Engineering". researchgate.net.
External links
• Official website
• Official website
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Virasoro conformal block
In two-dimensional conformal field theory, Virasoro conformal blocks (named after Miguel Ángel Virasoro) are special functions that serve as building blocks of correlation functions. On a given punctured Riemann surface, Virasoro conformal blocks form a particular basis of the space of solutions of the conformal Ward identities. Zero-point blocks on the torus are characters of representations of the Virasoro algebra; four-point blocks on the sphere reduce to hypergeometric functions in special cases, but are in general much more complicated. In two dimensions as in other dimensions, conformal blocks play an essential role in the conformal bootstrap approach to conformal field theory.
Definition
Definition from OPEs
Using operator product expansions (OPEs), an $N$-point function on the sphere can be written as a combination of three-point structure constants, and universal quantities called $N$-point conformal blocks.[1][2]
Given an $N$-point function, there are several types of conformal blocks, depending on which OPEs are used. In the case $N=4$, there are three types of conformal blocks, corresponding to three possible decompositions of the same four-point function. Schematically, these decompositions read
$\left\langle V_{1}V_{2}V_{3}V_{4}\right\rangle =\sum _{s}C_{12s}C_{s34}{\mathcal {F}}_{s}^{\text{(s-channel)}}=\sum _{t}C_{14t}C_{t23}{\mathcal {F}}_{t}^{\text{(t-channel)}}=\sum _{u}C_{13u}C_{24u}{\mathcal {F}}_{u}^{\text{(u-channel)}}\ ,$
where $C$ are structure constants and ${\mathcal {F}}$ are conformal blocks. The sums are over representations of the conformal algebra that appear in the CFT's spectrum. OPEs involve sums over the spectrum, i.e. over representations and over states in representations, but the sums over states are absorbed in the conformal blocks.
In two dimensions, the symmetry algebra factorizes into two copies of the Virasoro algebra, called left-moving and right-moving. If the fields are factorized too, then the conformal blocks factorize as well, and the factors are called Virasoro conformal blocks. Left-moving Virasoro conformal blocks are locally holomorphic functions of the fields' positions $z_{i}$; right-moving Virasoro conformal blocks are the same functions of ${\bar {z}}_{i}$. The factorization of a conformal block into Virasoro conformal blocks is of the type
${\mathcal {F}}_{s_{L}\otimes s_{R}}^{\text{(s-channel)}}(\{z_{i}\})={\mathcal {F}}_{s_{L}}^{\text{(s-channel, Virasoro)}}(\{z_{i}\}){\mathcal {F}}_{s_{R}}^{\text{(s-channel, Virasoro)}}(\{{\bar {z}}_{i}\})\ ,$
where $s_{L},s_{R}$ are representations of the left- and right-moving Virasoro algebras respectively.
Definition from Virasoro Ward identities
Conformal Ward identities are the linear equations that correlation functions obey, as a result of conformal symmetry.
In two dimensions, conformal Ward identities decompose into left-moving and right-moving Virasoro Ward identities. Virasoro conformal blocks are solutions of the Virasoro Ward identities.[3][4]
OPEs define specific bases of Virasoro conformal blocks, such as the s-channel basis in the case of four-point blocks. The blocks that are defined from OPEs are special cases of the blocks that are defined from Ward identities.
Properties
Any linear holomorphic equation that is obeyed by a correlation function, must also hold for the corresponding conformal blocks. In addition, specific bases of conformal blocks come with extra properties that are not inherited from the correlation function.
Conformal blocks that involve only primary fields have relatively simple properties. Conformal blocks that involve descendant fields can then be deduced using local Ward identities. An s-channel four-point block of primary fields depends on the four fields' conformal dimensions $\Delta _{i},$ on their positions $z_{i},$ and on the s-channel conformal dimension $\Delta _{s}$. It can be written as ${\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{i}|\{z_{i}\}),$ where the dependence on the Virasoro algebra's central charge is kept implicit.
Linear equations
From the corresponding correlation function, conformal blocks inherit linear equations: global and local Ward identities, and BPZ equations if at least one field is degenerate.[2]
In particular, in an $N$-point block on the sphere, global Ward identities reduce the dependence on the $N$ field positions to a dependence on $N-3$ cross-ratios. In the case $N=4,$
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|\{z_{i}\})=z_{23}^{\Delta _{1}-\Delta _{2}-\Delta _{3}+\Delta _{4}}z_{13}^{-2\Delta _{1}}z_{34}^{\Delta _{1}+\Delta _{2}-\Delta _{3}-\Delta _{4}}z_{24}^{-\Delta _{1}-\Delta _{2}+\Delta _{3}-\Delta _{4}}{\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z),$
where $z_{ij}=z_{i}-z_{j},$ and
$z={\frac {z_{12}z_{34}}{z_{13}z_{24}}}$
is the cross-ratio, and the reduced block ${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z)$ coincides with the original block where three positions are sent to $(0,\infty ,1),$
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z)={\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z,0,\infty ,1).$
Singularities
Like correlation functions, conformal blocks are singular when two fields coincide. Unlike correlation functions, conformal blocks have very simple behaviours at some of these singularities. As a consequence of their definition from OPEs, s-channel four-point blocks obey
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|z){\underset {z\to 0}{=}}z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}\left(1+\sum _{n=1}^{\infty }c_{n}z^{n}\right),$
for some coefficients $c_{n}.$ On the other hand, s-channel blocks have complicated singular behaviours at $z=1,\infty $: it is t-channel blocks that are simple at $z=1$, and u-channel blocks that are simple at $z=\infty .$
In a four-point block that obeys a BPZ differential equation, $z=0,1,\infty $ are regular singular points of the differential equation, and $\Delta _{s}-\Delta _{1}-\Delta _{2}$ is a characteristic exponent of the differential equation. For a differential equation of order $n$, the $n$ characteristic exponents correspond to the $n$ values of $\Delta _{s}$ that are allowed by the fusion rules.
Field permutations
Permutations of the fields $V_{i}(z_{i})$ leave the correlation function
$\left\langle \prod _{i=1}^{N}V_{i}(z_{i})\right\rangle $
invariant, and therefore relate different bases of conformal blocks with one another. In the case of four-point blocks, t-channel blocks are related to s-channel blocks by[2]
${\mathcal {F}}_{\Delta }^{(t)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z_{1},z_{2},z_{3},z_{4})={\mathcal {F}}_{\Delta }^{(s)}(\Delta _{1},\Delta _{4},\Delta _{3},\Delta _{2}|z_{1},z_{4},z_{3},z_{2}),$
or equivalently
${\mathcal {F}}_{\Delta }^{(t)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z)={\mathcal {F}}_{\Delta }^{(s)}(\Delta _{1},\Delta _{4},\Delta _{3},\Delta _{2}|1-z).$
Fusing matrix
The change of bases from s-channel to t-channel four-point blocks is characterized by the fusing matrix (or fusion kernel) $F$, such that
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\{\Delta _{i}\}|\{z_{i}\})=\int _{i\mathbb {R} }dP_{t}\ F_{\Delta _{s},\Delta _{t}}{\begin{bmatrix}\Delta _{2}&\Delta _{3}\\\Delta _{1}&\Delta _{4}\end{bmatrix}}{\mathcal {F}}_{\Delta _{t}}^{(t)}(\{\Delta _{i}\}|\{z_{i}\}).$
The fusing matrix is a function of the central charge and conformal dimensions, but it does not depend on the positions $z_{i}.$ The momentum $P_{t}$ is defined in terms of the dimension $\Delta _{t}$ by
$\Delta ={\frac {c-1}{24}}-P^{2}.$
The values $P\in i\mathbb {R} $ correspond to the spectrum of Liouville theory.
We also need to introduce two parameters $Q,b$ related to the central charge $c$,
$c=1+6Q^{2},\qquad Q=b+b^{-1}.$
Assuming $c\notin (-\infty ,1)$ and $P_{i}\in i\mathbb {R} $, the explicit expression of the fusing matrix is[5]
${\begin{aligned}F_{\Delta _{s},\Delta _{t}}&{\begin{bmatrix}\Delta _{2}&\Delta _{3}\\\Delta _{1}&\Delta _{4}\end{bmatrix}}=\\&=\left(\prod _{\pm }{\frac {\Gamma _{b}(Q\pm 2P_{s})}{\Gamma _{b}(\pm 2P_{t})}}\right){\frac {\Xi _{+}(P_{1},P_{4},P_{t})\Xi _{+}(P_{2},P_{3},P_{t})}{\Xi _{-}(P_{1},P_{2},P_{s})\Xi _{-}(P_{3},P_{4},P_{s})}}\times \\&\quad \times \int _{{\frac {Q}{4}}+i\mathbb {R} }du\ S_{b}\left(u-P_{12s}\right)S_{b}\left(u-P_{s34}\right)S_{b}\left(u-P_{23t}\right)S_{b}\left(u-P_{t14}\right)\\&\qquad \times S_{b}\left({\tfrac {Q}{2}}-u+P_{1234}\right)S_{b}\left({\tfrac {Q}{2}}-u+P_{st13}\right)S_{b}\left({\tfrac {Q}{2}}-u+P_{st24}\right)S_{b}\left({\tfrac {Q}{2}}-u\right)\end{aligned}}$
where $\Gamma _{b}$ is a double gamma function,
${\begin{aligned}S_{b}(x)&={\frac {\Gamma _{b}(x)}{\Gamma _{b}(Q-x)}}\\[6pt]\Xi _{\epsilon }(P_{1},P_{2},P_{3})&=\prod _{\underset {\epsilon _{1}\epsilon _{2}\epsilon _{3}=\epsilon }{\epsilon _{1},\epsilon _{2},\epsilon _{3}=\pm }}\Gamma _{b}\left({\tfrac {Q}{2}}+\sum _{i}\epsilon _{i}P_{i}\right)\\[6pt]P_{ijk}&=P_{i}+P_{j}+P_{k}\end{aligned}}$
Although its expression is simpler in terms of momentums $P_{i}$ than in terms of conformal dimensions $\Delta _{i}$, the fusing matrix is really a function of $\Delta _{i}$, i.e. a function of $P_{i}$ that is invariant under $P_{i}\to -P_{i}$. In the expression for the fusing matrix, the integral is a hyperbolic Barnes integral. Up to normalization, the fusing matrix coincides with Ruijsenaars' hypergeometric function, with the arguments $P_{s},P_{t}$ and parameters $b,b^{-1},P_{1},P_{2},P_{3},P_{4}$.[6]
In $N$-point blocks on the sphere, the change of bases between two sets of blocks that are defined from different sequences of OPEs can always be written in terms of the fusing matrix, and a simple matrix that describes the permutation of the first two fields in an s-channel block,[3]
${\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{1},\Delta _{2},\Delta _{3},\Delta _{4}|z_{1},z_{2},z_{3},z_{4})=e^{i\pi (\Delta _{s}-\Delta _{1}-\Delta _{2})}{\mathcal {F}}_{\Delta _{s}}^{(s)}(\Delta _{2},\Delta _{1},\Delta _{3},\Delta _{4}|z_{2},z_{1},z_{3},z_{4}).$
Computation of conformal blocks
From the definition
The definition from OPEs leads to an expression for an s-channel four-point conformal block as a sum over states in the s-channel representation, of the type [7]
${\mathcal {F}}_{\Delta _{s}}^{\text{(s)}}(\{\Delta _{i}\}|z)=z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}\sum _{L,L'}z^{|L|}f_{12s}^{L}Q_{L,L'}^{s}f_{43s}^{L'}\ .$
The sums are over creation modes $L,L'$ of the Virasoro algebra, i.e. combinations of the type $L=\prod _{i}L_{-n_{i}}$ of Virasoro generators with $1\leq n_{1}\leq n_{2}\leq \cdots $, whose level is $|L|=\sum n_{i}$. Such generators correspond to basis states in the Verma module with the conformal dimension $\Delta _{s}$. The coefficient $f_{12s}^{L}$ is a function of $\Delta _{1},\Delta _{2},\Delta _{s},L$, which is known explicitly. The matrix element $Q_{L,L'}^{s}$ is a function of $c,\Delta _{s},L,L'$ which vanishes if $|L|\neq |L'|$, and diverges for $|L|=N$ if there is a null vector at level $N$. Up to $|L|=1$, this reads
${\mathcal {F}}_{\Delta _{s}}^{\text{(s)}}(\{\Delta _{i}\}|z)=z^{\Delta _{s}-\Delta _{1}-\Delta _{2}}{\Bigg \{}1+{\frac {(\Delta _{s}+\Delta _{1}-\Delta _{2})(\Delta _{s}+\Delta _{4}-\Delta _{3})}{2\Delta _{s}}}z+O(z^{2}){\Bigg \}}\ .$
(In particular, $Q_{L_{-1},L_{-1}}^{s}={\frac {1}{2\Delta _{s}}}$ does not depend on the central charge $c$.)
Zamolodchikov's recursive representation
In Alexei Zamolodchikov's recursive representation of four-point blocks on the sphere, the cross-ratio $z$ appears via the nome
$q=\exp -\pi {\frac {F({\frac {1}{2}},{\frac {1}{2}},1,1-z)}{F({\frac {1}{2}},{\frac {1}{2}},1,z)}}{\underset {z\to 0}{=}}{\frac {z}{16}}+{\frac {z^{2}}{32}}+O(z^{3})\quad \iff \quad z={\frac {\theta _{2}(q)^{4}}{\theta _{3}(q)^{4}}}{\underset {q\to 0}{=}}16q-128q^{2}+O(q^{3})$
where $F$ is the hypergeometric function, and we used the Jacobi theta functions
$\theta _{2}(q)=2q^{\frac {1}{4}}\sum _{n=0}^{\infty }q^{n(n+1)}\quad ,\quad \theta _{3}(q)=\sum _{n\in {\mathbb {Z} }}q^{n^{2}}$
The representation is of the type
${\mathcal {F}}_{\Delta }^{(s)}(\{\Delta _{i}\}|z)=(16q)^{\Delta -{\frac {1}{4}}Q^{2}}z^{{\frac {1}{4}}Q^{2}-\Delta _{1}-\Delta _{2}}(1-z)^{{\frac {1}{4}}Q^{2}-\Delta _{1}-\Delta _{4}}\theta _{3}(q)^{3Q^{2}-4(\Delta _{1}+\Delta _{2}+\Delta _{3}+\Delta _{4})}H_{\Delta }(\{\Delta _{i}\}|q)\ .$
The function $H_{\Delta }(\{\Delta _{i}\}|q)$ is a power series in $q$, which is recursively defined by
$H_{\Delta }(\{\Delta _{i}\}|q)=1+\sum _{m,n=1}^{\infty }{\frac {(16q)^{mn}}{\Delta -\Delta _{(m,n)}}}R_{m,n}H_{\Delta _{(m,-n)}}(\{\Delta _{i}\}|q)\ .$
In this formula, the positions $\Delta _{(m,n)}$ of the poles are the dimensions of degenerate representations, which correspond to the momentums
$P_{(m,n)}={\frac {1}{2}}\left(mb+nb^{-1}\right)\ .$
The residues $R_{m,n}$ are given by
$R_{m,n}={\frac {2P_{(0,0)}P_{(m,n)}}{\prod _{r=1-m}^{m}\prod _{s=1-n}^{n}2P_{(r,s)}}}\prod _{r{\overset {2}{=}}1-m}^{m-1}\prod _{s{\overset {2}{=}}1-n}^{n-1}\prod _{\pm }(P_{2}\pm P_{1}+P_{(r,s)})(P_{3}\pm P_{4}+P_{(r,s)})\ ,$
where the superscript in ${\overset {2}{=}}$ indicates a product that runs by increments of $2$. The recursion relation for $H_{\Delta }(\{\Delta _{i}\}|q)$ can be solved, giving rise to an explicit (but impractical) formula.[2][8]
While the coefficients of the power series $H_{\Delta }(\{\Delta _{i}\}|q)$ need not be positive in unitary theories, the coefficients of $\prod _{k=1}^{\infty }(1-q^{2k})^{-{\frac {1}{2}}}H_{\Delta }(\{\Delta _{i}\}|q)$ are positive, due to this combination's interpretation in terms of sums of states in the pillow geometry.[9]
The recursive representation can be seen as an expansion around $\Delta =\infty $. It is sometimes called the $\Delta $-recursion, in order to distinguish it from the $c$-recursion: another recursive representation, also due to Alexei Zamolodchikov, which expands around $c=\infty $. Both representations can be generalized to $N$-point Virasoro conformal blocks on arbitrary Riemann surfaces.[10]
From the relation to instanton counting
The Alday–Gaiotto–Tachikawa relation between two-dimensional conformal field theory and supersymmetric gauge theory, more specifically, between the conformal blocks of Liouville theory and Nekrasov partition functions[11] of supersymmetric gauge theories in four dimensions, leads to combinatorial expressions for conformal blocks as sums over Young diagrams. Each diagram can be interpreted as a state in a representation of the Virasoro algebra, times an abelian affine Lie algebra.[12]
Special cases
Zero-point blocks on the torus
A zero-point block does not depend on field positions, but it depends on the moduli of the underlying Riemann surface. In the case of the torus
${\frac {\mathbb {C} }{\mathbb {Z} +\tau \mathbb {Z} }},$
that dependence is better written through $q=e^{2\pi i\tau }$ and the zero-point block associated to a representation ${\mathcal {R}}$ of the Virasoro algebra is
$\chi _{\mathcal {R}}(\tau )=\operatorname {Tr} _{\mathcal {R}}q^{L_{0}-{\frac {c}{24}}},$
where $L_{0}$ is a generator of the Virasoro algebra. This coincides with the character of ${\mathcal {R}}.$ The characters of some highest-weight representations are:[1]
• Verma module with conformal dimension $\Delta ={\tfrac {c-1}{24}}-P^{2}$:
$\chi _{P}(\tau )={\frac {q^{-P^{2}}}{\eta (\tau )}},$
where $\eta (\tau )$ is the Dedekind eta function.
• Degenerate representation with the momentum $P_{(r,s)}$:
$\chi _{(r,s)}(\tau )=\chi _{P_{(r,s)}}(\tau )-\chi _{P_{(r,-s)}}(\tau ).$
• Fully degenerate representation at rational $b^{2}=-{\tfrac {p}{q}}$:
$\chi _{(r,s)}(\tau )=\sum _{k\in \mathbb {Z} }\left(\chi _{P_{(r,s)}+ik{\sqrt {pq}}}(\tau )-\chi _{P_{(r,-s)}+ik{\sqrt {pq}}}(\tau )\right).$
The characters transform linearly under the modular transformations:
$\tau \to {\frac {a\tau +b}{c\tau +d}},\qquad {\begin{pmatrix}a&b\\c&d\end{pmatrix}}\in SL_{2}(\mathbb {Z} ).$
In particular their transformation under $\tau \to -{\tfrac {1}{\tau }}$ is described by the modular S-matrix. Using the S-matrix, constraints on a CFT's spectrum can be derived from the modular invariance of the torus partition function, leading in particular to the ADE classification of minimal models.[13]
One-point blocks on the torus
An arbitrary one-point block on the torus can be written in terms of a four-point block on the sphere at a different central charge. This relation maps the modulus of the torus to the cross-ratio of the four points' positions, and three of the four fields on the sphere have the fixed momentum $P_{(0,{\frac {1}{2}})}={\tfrac {1}{4b}}$:[14][15]
$H_{P'}^{\text{torus}}(P_{1}|q^{2})=H_{P}\left(\left.{\tfrac {1}{4b}},P_{2},{\tfrac {1}{4b}},{\tfrac {1}{4b}}\right|q\right)\quad {\text{with}}\quad \left\{{\begin{array}{l}b={\frac {b'}{\sqrt {2}}}\\P_{2}={\frac {P_{1}}{\sqrt {2}}}\\P={\sqrt {2}}P'\end{array}}\right.$
where
• $H_{P_{s}}\left(\left.P_{1},P_{2},P_{3},P_{4}\right|q\right)$ is the non-trivial factor of the sphere four-point block in Zamolodchikov's recursive representation, written in terms of momentums $P_{i}$ instead of dimensions $\Delta _{i}$.
• $H_{P}^{\text{torus}}(P_{1}|q)$ is the non-trivial factor of the torus one-point block ${\mathcal {F}}_{\Delta }^{\text{torus}}(\Delta _{1}|q)=q^{\Delta -{\frac {c-1}{24}}}\eta (q)^{-1}H_{\Delta }^{\text{torus}}(\Delta _{1}|q)$, where $\eta (q)$ is the Dedekind eta function, the modular parameter $\tau $ of the torus is such that $q=e^{2\pi i\tau }$, and the field on the torus has the dimension $\Delta _{1}$.
The recursive representation of one-point blocks on the torus is[16]
$H_{\Delta }^{\text{torus}}(\Delta _{1}|q)=1+\sum _{m,n=1}^{\infty }{\frac {q^{mn}}{\Delta -\Delta _{(m,n)}}}R_{m,n}^{\text{torus}}H_{\Delta _{(m,-n)}}^{\text{torus}}(\Delta _{1}|q)\ ,$
where the residues are
$R_{m,n}^{\text{torus}}={\frac {2P_{(0,0)}P_{(m,n)}}{\prod _{r=1-m}^{m}\prod _{s=1-n}^{n}2P_{(r,s)}}}\prod _{r{\overset {2}{=}}1-2m}^{2m-1}\prod _{s{\overset {2}{=}}1-2n}^{2n-1}\left(P_{1}+P_{(r,s)}\right)\ .$
Under modular transformations, one-point blocks on the torus behave as
${\mathcal {F}}_{P}^{\text{torus}}\left(P_{1}|-{\tfrac {1}{\tau }}\right)=\int _{i\mathbb {R} }dP'\ S_{P,P'}(P_{1}){\mathcal {F}}_{P'}^{\text{torus}}\left(P_{1}|\tau \right)\ ,$
where the modular kernel is[17][18]
$S_{P,P'}(P_{1})={\frac {2^{-{\frac {5}{2}}}}{S_{b}({\frac {Q}{2}}+P_{1})}}\prod _{\pm }{\frac {\Gamma _{b}(Q\pm 2P)}{\Gamma _{b}(\pm 2P')}}{\frac {\Gamma _{b}({\frac {Q}{2}}-P_{1}\pm 2P')}{\Gamma _{b}({\frac {Q}{2}}-P_{1}\pm 2P)}}\int _{i\mathbb {R} }du\ e^{4\pi iPu}\prod _{\pm ,\pm }S_{b}\left({\tfrac {Q}{4}}+{\tfrac {P_{1}}{2}}\pm u\pm P'\right)\ .$
Hypergeometric blocks
For a four-point function on the sphere
$\left\langle V_{\langle 2,1\rangle }(x)\prod _{i=1}^{3}V_{\Delta _{i}}(z_{i})\right\rangle $
where one field has a null vector at level two, the second-order BPZ equation reduces to the hypergeometric equation. A basis of solutions is made of the two s-channel conformal blocks that are allowed by the fusion rules, and these blocks can be written in terms of the hypergeometric function,
${\begin{aligned}{\mathcal {F}}_{P_{1}+\epsilon {\frac {b}{2}}}^{(s)}(z)&=z^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+b\epsilon P_{1}}(1-z)^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+bP_{3}}\\&\times F\left({\tfrac {1}{2}}+b(\epsilon P_{1}+P_{2}+P_{3}),{\tfrac {1}{2}}+b(\epsilon P_{1}-P_{2}+P_{3}),1+2b\epsilon P_{1},z\right),\end{aligned}}$
with $\epsilon \in \{+,-\}.$ Another basis is made of the two t-channel conformal blocks,
${\begin{aligned}{\mathcal {F}}_{P_{3}+\epsilon {\frac {b}{2}}}^{(t)}(z)&=z^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+bP_{1}}(1-z)^{{\frac {1}{2}}+{\frac {b^{2}}{2}}+b\epsilon P_{3}}\\&\times F\left({\tfrac {1}{2}}+b(P_{1}+P_{2}+\epsilon P_{3}),{\tfrac {1}{2}}+b(P_{1}-P_{2}+\epsilon P_{3}),1+2b\epsilon P_{3},1-z\right).\end{aligned}}$
The fusing matrix is the matrix of size two such that
${\mathcal {F}}_{P_{1}+\epsilon _{1}{\frac {b}{2}}}^{(s)}(x)=\sum _{\epsilon _{3}=\pm }F_{\epsilon _{1},\epsilon _{3}}{\mathcal {F}}_{P_{3}+\epsilon _{3}{\frac {b}{2}}}^{(t)}(x),$
whose explicit expression is
$F_{\epsilon _{1},\epsilon _{3}}={\frac {\Gamma (1-2b\epsilon _{1}P_{1})\Gamma (2b\epsilon _{3}P_{3})}{\prod _{\pm }\Gamma ({\frac {1}{2}}+b(-\epsilon _{1}P_{1}\pm P_{2}+\epsilon _{3}P_{3}))}}.$
Hypergeometric conformal blocks play an important role in the analytic bootstrap approach to two-dimensional CFT.[19][20]
Solutions of the Painlevé VI equation
If $c=1,$ then certain linear combinations of s-channel conformal blocks are solutions of the Painlevé VI nonlinear differential equation.[21] The relevant linear combinations involve sums over sets of momentums of the type $P_{s}+i\mathbb {Z} .$ This allows conformal blocks to be deduced from solutions of the Painlevé VI equation and vice versa. This also leads to a relatively simple formula for the fusing matrix at $c=1.$[22] Curiously, the $c=\infty $ limit of conformal blocks is also related to the Painlevé VI equation.[23] The relation between the $c=\infty $ and the $c=1$ limits, mysterious on the conformal field theory side, is explained naturally in the context of four dimensional gauge theories, using blowup equations,[24][25] and can be generalized to more general pairs $c,c'$of central charges.
Generalizations
Other representations of the Virasoro algebra
The Virasoro conformal blocks that are described in this article are associated to a certain type of representations of the Virasoro algebra: highest-weight representations, in other words Verma modules and their cosets.[2] Correlation functions that involve other types of representations give rise to other types of conformal blocks. For example:
• Logarithmic conformal field theory involves representations where the Virasoro generator $L_{0}$ is not diagonalizable, which give rise to blocks that depend logarithmically on field positions.
• Representations can be built from states on which some annihilation modes of the Virasoro algebra act diagonally, rather than vanishing. The corresponding conformal blocks have been called irregular conformal blocks.[26]
Larger symmetry algebras
In a theory whose symmetry algebra is larger than the Virasoro algebra, for example a WZW model or a theory with W-symmetry, correlation functions can in principle be decomposed into Virasoro conformal blocks, but that decomposition typically involves too many terms to be useful. Instead, it is possible to use conformal blocks based on the larger algebra: for example, in a WZW model, conformal blocks based on the corresponding affine Lie algebra, which obey Knizhnik–Zamolodchikov equations.
References
1. P. Di Francesco, P. Mathieu, and D. Sénéchal, Conformal Field Theory, 1997, ISBN 0-387-94785-X
2. Ribault, Sylvain (2014). "Conformal field theory on the plane". arXiv:1406.4290 [hep-th].
3. Moore, Gregory; Seiberg, Nathan (1989). "Classical and quantum conformal field theory". Communications in Mathematical Physics. 123 (2): 177–254. Bibcode:1989CMaPh.123..177M. doi:10.1007/BF01238857. S2CID 122836843.
4. Teschner, Joerg (2017). "A guide to two-dimensional conformal field theory". arXiv:1708.00680 [hep-th].
5. Teschner, J.; Vartanov, G. S. (2012). "6j symbols for the modular double, quantum hyperbolic geometry, and supersymmetric gauge theories". arXiv:1202.4698 [hep-th].
6. Roussillon, Julien (2021). "The Virasoro fusion kernel and Ruijsenaars' hypergeometric function". Letters in Mathematical Physics. 111 (1): 7. arXiv:2006.16101. Bibcode:2021LMaPh.111....7R. doi:10.1007/s11005-020-01351-4. PMC 7796901. PMID 33479555.
7. Marshakov, A.; Mironov, A.; Morozov, A. (2009). "On Combinatorial Expansions of Conformal Blocks". Theoretical and Mathematical Physics. 164: 831–852. arXiv:0907.3946. doi:10.1007/s11232-010-0067-6. S2CID 16017224.
8. Perlmutter, Eric (2015). "Virasoro conformal blocks in closed form". Journal of High Energy Physics. 2015 (8): 88. arXiv:1502.07742. Bibcode:2015JHEP...08..088P. doi:10.1007/JHEP08(2015)088. S2CID 54075672.
9. Maldacena, Juan; Simmons-Duffin, David; Zhiboedov, Alexander (2015-09-11). "Looking for a bulk point". arXiv:1509.03612 [hep-th].
10. Cho, Minjae; Collier, Scott; Yin, Xi (2017). "Recursive Representations of Arbitrary Virasoro Conformal Blocks". arXiv:1703.09805 [hep-th].
11. Nekrasov, Nikita (2004). "Seiberg-Witten Prepotential from Instanton Counting". Advances in Theoretical and Mathematical Physics. 7 (5): 831–864. arXiv:hep-th/0206161. doi:10.4310/ATMP.2003.v7.n5.a4. S2CID 2285041.
12. Alba, Vasyl A.; Fateev, Vladimir A.; Litvinov, Alexey V.; Tarnopolskiy, Grigory M. (2011). "On Combinatorial Expansion of the Conformal Blocks Arising from AGT Conjecture". Letters in Mathematical Physics. 98 (1): 33–64. arXiv:1012.1312. Bibcode:2011LMaPh..98...33A. doi:10.1007/s11005-011-0503-z. S2CID 119143670.
13. A. Cappelli, J-B. Zuber, "A-D-E Classification of Conformal Field Theories", Scholarpedia
14. Fateev, V. A.; Litvinov, A. V.; Neveu, A.; Onofri, E. (2009-02-08). "Differential equation for four-point correlation function in Liouville field theory and elliptic four-point conformal blocks". Journal of Physics A: Mathematical and Theoretical. 42 (30): 304011. arXiv:0902.1331. Bibcode:2009JPhA...42D4011F. doi:10.1088/1751-8113/42/30/304011. S2CID 16106733.
15. Hadasz, Leszek; Jaskolski, Zbigniew; Suchanek, Paulina (2010). "Modular bootstrap in Liouville field theory". Physics Letters B. 685 (1): 79–85. arXiv:0911.4296. Bibcode:2010PhLB..685...79H. doi:10.1016/j.physletb.2010.01.036. S2CID 118625083.
16. Fateev, V. A.; Litvinov, A. V. (2010). "On AGT conjecture". Journal of High Energy Physics. 2010 (2): 014. arXiv:0912.0504. Bibcode:2010JHEP...02..014F. doi:10.1007/JHEP02(2010)014. S2CID 118561574.
17. Teschner, J. (2003-08-05). "From Liouville Theory to the Quantum Geometry of Riemann Surfaces". arXiv:hep-th/0308031.
18. Nemkov, Nikita (2015-04-16). "On modular transformations of non-degenerate toric conformal blocks". Journal of High Energy Physics. 1510: 039. arXiv:1504.04360. doi:10.1007/JHEP10(2015)039. S2CID 73549642.
19. Teschner, Joerg. (1995). "On the Liouville three-point function". Physics Letters B. 363 (1–2): 65–70. arXiv:hep-th/9507109. Bibcode:1995PhLB..363...65T. doi:10.1016/0370-2693(95)01200-A. S2CID 15910029.
20. Migliaccio, Santiago; Ribault, Sylvain (2018). "The analytic bootstrap equations of non-diagonal two-dimensional CFT". Journal of High Energy Physics. 2018 (5): 169. arXiv:1711.08916. Bibcode:2018JHEP...05..169M. doi:10.1007/JHEP05(2018)169. S2CID 119385003.
21. Gamayun, O.; Iorgov, N.; Lisovyy, O. (2012). "Conformal field theory of Painlevé VI". Journal of High Energy Physics. 2012 (10): 038. arXiv:1207.0787. Bibcode:2012JHEP...10..038G. doi:10.1007/JHEP10(2012)038. S2CID 119610935.
22. Iorgov, N.; Lisovyy, O.; Tykhyy, Yu. (2013). "Painlevé VI connection problem and monodromy of c = 1 conformal blocks". Journal of High Energy Physics. 2013 (12): 029. arXiv:1308.4092. Bibcode:2013JHEP...12..029I. doi:10.1007/JHEP12(2013)029. S2CID 56401903.
23. Litvinov, Alexey; Lukyanov, Sergei; Nekrasov, Nikita; Zamolodchikov, Alexander (2014). "Classical conformal blocks and Painlevé VI". Journal of High Energy Physics. 2014 (7): 144. arXiv:1309.4700. Bibcode:2014JHEP...07..144L. doi:10.1007/JHEP07(2014)144. S2CID 119710593.
24. Nekrasov, Nikita (2020). "Blowups in BPS/CFT correspondence, and Painlevé VI". arXiv:2007.03646. {{cite journal}}: Cite journal requires |journal= (help)
25. Jeong, Saebyeok; Nekrasov, Nikita (2020). "Riemann-Hilbert correspondence and blown up surface defects". Journal of High Energy Physics. 2020 (12): 006. arXiv:2007.03660. Bibcode:2020JHEP...12..006J. doi:10.1007/JHEP12(2020)006. S2CID 220381427.
26. Gaiotto, D.; Teschner, J. (2012). "Irregular singularities in Liouville theory and Argyres-Douglas type gauge theories". Journal of High Energy Physics. 2012 (12): 50. arXiv:1203.1052. Bibcode:2012JHEP...12..050G. doi:10.1007/JHEP12(2012)050. S2CID 118380071.
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Virasoro conjecture
In algebraic geometry, the Virasoro conjecture states that a certain generating function encoding Gromov–Witten invariants of a smooth projective variety is fixed by an action of half of the Virasoro algebra. The Virasoro conjecture is named after theoretical physicist Miguel Ángel Virasoro. Tohru Eguchi, Kentaro Hori, and Chuan-Sheng Xiong (1997) proposed the Virasoro conjecture as a generalization of Witten's conjecture. Ezra Getzler (1999) gave a survey of the Virasoro conjecture.
References
• Getzler, Ezra (1999), "The Virasoro conjecture for Gromov-Witten invariants", in Wiśniewski, Jarosław; Szurek, Michał; Pragacz, Piotr (eds.), Algebraic geometry: Hirzebruch 70 (Warsaw, 1998), Contemporary Mathematics, vol. 241, Providence, R.I.: American Mathematical Society, pp. 147–176, arXiv:math/9812026, Bibcode:1998math.....12026G, doi:10.1090/conm/241/03634, ISBN 978-0-8218-1149-8, MR 1718143
• Eguchi, Tohru; Hori, Kentaro; Xiong, Chuan-Sheng (1997), "Quantum cohomology and Virasoro algebra", Physics Letters B, 402 (1): 71–80, arXiv:hep-th/9703086, Bibcode:1997PhLB..402...71E, doi:10.1016/S0370-2693(97)00401-2, ISSN 0370-2693, MR 1454328
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Virasoro group
In abstract algebra, the Virasoro group or Bott–Virasoro group (often denoted by Vir)[1] is an infinite-dimensional Lie group defined as the universal central extension of the group of diffeomorphisms of the circle. The corresponding Lie algebra is the Virasoro algebra, which has a key role in conformal field theory (CFT) and string theory.
The group is named after Miguel Ángel Virasoro and Raoul Bott.
Background
An orientation-preserving diffeomorphism of the circle $S^{1}$, whose points are labelled by a real coordinate $x$ subject to the identification $x\sim x+2\pi $, is a smooth map $f:\mathbb {R} \to \mathbb {R} :x\mapsto f(x)$ such that $f(x+2\pi )=f(x)+2\pi $ and $f'(x)>0$. The set of all such maps spans a group, with multiplication given by the composition of diffeomorphisms. This group is the universal cover of the group of orientation-preserving diffeomorphisms of the circle, denoted as ${\widetilde {\text{Diff}}}{}^{+}(S^{1})$.
Definition
The Virasoro group is the universal central extension of ${\widetilde {\text{Diff}}}{}^{+}(S^{1})$.[2]: sect. 4.4 The extension is defined by a specific two-cocycle, which is a real-valued function ${\mathsf {C}}(f,g)$ of pairs of diffeomorphisms. Specifically, the extension is defined by the Bott–Thurston cocycle:
${\mathsf {C}}(f,g)\equiv -{\frac {1}{48\pi }}\int _{0}^{2\pi }\log {\big [}f'{\big (}g(x){\big )}{\big ]}{\frac {g''(x)\,{\text{d}}x}{g'(x)}}.$
In these terms, the Virasoro group is the set ${\widetilde {\text{Diff}}}{}^{+}(S^{1})\times \mathbb {R} $ of all pairs $(f,\alpha )$, where $f$ is a diffeomorphism and $\alpha $ is a real number, endowed with the binary operation
$(f,\alpha )\cdot (g,\beta )={\big (}f\circ g,\alpha +\beta +{\mathsf {C}}(f,g){\big )}.$
This operation is an associative group operation. This extension is the only central extension of the universal cover of the group of circle diffeomorphisms, up to trivial extensions.[2] The Virasoro group can also be defined without the use explicit coordinates or an explicit choice of cocycle to represent the central extension, via a description the universal cover of the group.[2]
Virasoro algebra
Main article: Virasoro algebra
The Lie algebra of the Virasoro group is the Virasoro algebra. As a vector space, the Lie algebra of the Virasoro group consists of pairs $(\xi ,\alpha )$, where $\xi =\xi (x)\partial _{x}$ is a vector field on the circle and $\alpha $ is a real number as before. The vector field, in particular, can be seen as an infinitesimal diffeomorphism $f(x)=x+\epsilon \xi (x)$. The Lie bracket of pairs $(\xi ,\alpha )$ then follows from the multiplication defined above, and can be shown to satisfy[3]: sect. 6.4
${\big [}(\xi ,\alpha ),(\zeta ,\beta ){\big ]}={\bigg (}[\xi ,\zeta ],-{\frac {1}{24\pi }}\int _{0}^{2\pi }{\text{d}}x\,\xi (x)\zeta '''(x){\bigg )}$
where the bracket of vector fields on the right-hand side is the usual one: $[\xi ,\zeta ]=(\xi (x)\zeta '(x)-\zeta (x)\xi '(x))\partial _{x}$. Upon defining the complex generators
$L_{m}\equiv {\Big (}-ie^{imx}\partial _{x},-{\frac {i}{24}}\delta _{m,0}{\Big )},\qquad Z\equiv (0,-i),$
the Lie bracket takes the standard textbook form of the Virasoro algebra:[4]
$[L_{m},L_{n}]=(m-n)L_{m+n}+{\frac {Z}{12}}m(m^{2}-1)\delta _{m+n}.$
The generator $Z$ commutes with the whole algebra. Since its presence is due to a central extension, it is subject to a superselection rule which guarantees that, in any physical system having Virasoro symmetry, the operator representing $Z$ is a multiple of the identity. The coefficient in front of the identity is then known as a central charge.
Properties
Since each diffeomorphism $f$ must be specified by infinitely many parameters (for instance the Fourier modes of the periodic function $f(x)-x$), the Virasoro group is infinite-dimensional.
Coadjoint representation
The Lie bracket of the Virasoro algebra can be viewed as a differential of the adjoint representation of the Virasoro group. Its dual, the coadjoint representation of the Virasoro group, provides the transformation law of a CFT stress tensor under conformal transformations. From this perspective, the Schwarzian derivative in this transformation law emerges as a consequence of the Bott–Thurston cocycle; in fact, the Schwarzian is the so-called Souriau cocycle (referring to Jean-Marie Souriau) associated with the Bott–Thurston cocycle.[2]
References
1. Bahns, Dorothea; Bauer, Wolfram; Witt, Ingo (2016-02-11). Quantization, PDEs, and Geometry: The Interplay of Analysis and Mathematical Physics. Birkhäuser. ISBN 978-3-319-22407-7.
2. Guieu, Laurent; Roger, Claude (2007), L'algèbre et le groupe de Virasoro, Montréal: Centre de Recherches Mathématiques, ISBN 978-2921120449
3. Oblak, Blagoje (2016), BMS Particles in Three Dimensions, Springer Theses, Springer Theses, arXiv:1610.08526, doi:10.1007/978-3-319-61878-4, ISBN 978-3319618784, S2CID 119321869
4. Di Francesco, P.; Mathieu, P.; Sénéchal, D. (1997), Conformal Field Theory, New York: Springer Verlag, doi:10.1007/978-1-4612-2256-9, ISBN 9780387947853
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Virbhadra–Ellis lens equation
The Virbhadra-Ellis lens equation [1] in astronomy and mathematics relates to the angular positions of an unlensed source $\left(\beta \right)$, the image $\left(\theta \right)$, the Einstein bending angle of light $({\hat {\alpha }})$, and the angular diameter lens-source $\left(D_{ds}\right)$ and observer-source $\left(D_{s}\right)$ distances.
$\tan \beta =\tan \theta -{\frac {D_{ds}}{D_{s}}}\left[\tan \theta +\tan \left({\hat {\alpha }}-\theta \right)\right]$.
This lens equation is useful for studying gravitational lensing in a strong gravitational field.
References
1. Virbhadra, K. S.; Ellis, George F. R. (2000-09-08). "Schwarzschild black hole lensing". Physical Review D. American Physical Society (APS). 62 (8): 084003. arXiv:astro-ph/9904193. Bibcode:2000PhRvD..62h4003V. doi:10.1103/physrevd.62.084003. ISSN 0556-2821. S2CID 15956589.
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Virginia Newell
Virginia Kimbrough Newell (born October 7, 1917) is an American mathematics educator, author, politician, and centenarian.[1]
Early life and education
Virginia Kimbrough was born on October 7, 1917 in Advance, North Carolina,[1] one of nine children. Although her family was African American, she grew up playing with the white children in a white neighborhood; her father, a builder, had the right to vote because he had a white ancestor,[2] and both of her parents had studied at Shaw University, without finishing a degree.[3] Kimbrough learned arithmetic helping her father in his measurements, and won a mathematics competition in elementary school.[2]
Her family sent her away to live with a great aunt, so that she could obtain a better education at Atkins High School (North Carolina). There, she learned mathematics from teachers Togo West and Beatrice Armstead, earning straight A's and becoming a teacher's assistant.[3] After graduating in 1936,[4] she obtained scholarships from many colleges,[3] and chose to major in mathematics at Talladega College, a historically black college in Alabama.[1] Many of her teachers there had previously taught at Ivy League universities, and had come to Talladega to teach because of mandatory retirement at their former employers.[3]
She later earned a master's degree from New York University,[1] and took courses from the University of Wisconsin, Atlanta University, University of Chicago, and North Carolina State College.[5] She completed a doctorate in education at the University of Sarasota in 1976, with the dissertation Development of mathematics self-instructional learning packages with activities from the newspaper for prospective elementary school teachers enrolled at Winston-Salem State University.[6]
Mathematics
After college, Kimbrough returned to Atkins High School as a mathematics teacher.[7] There, in 1943,[8] she married George Newell, who had been her biology teacher at the same school, changing her name to Virginia Newell. They both taught at several institutions in Atlanta and Raleigh, North Carolina,[7] including Washington Graded and High School, John W. Ligon High School,[1] and Shaw University, where Virginia Newell was an associate professor of mathematics from 1960 to 1965.[5][2]
In 1965,[1] they both settled at Winston-Salem State University, where Virginia Newell became a mathematics professor.[7] At Winston-Salem State University, she chaired the mathematics department,[1] helped bring computers to the university and found the computer science program,[7] becoming founding chair of the computer science department in 1979.[8] She spearheaded several initiatives for middle school students, including the Math and Science Academy of Excellence, the New Directions for our Youth program aimed at preventing dropouts, and the Best Choice Center for after-school education. She was a co-founder and president of the North Carolina Council of Teachers on Mathematics[9]
In 1980, Newell became one of the coauthors of Black Mathematicians and Their Works (with Joella Gipson, L. Waldo Rich, and Beauregard Stubblefield, Dorrance & Company),[10] the first book to highlight the contributions of African American mathematicians. She was also editor of the newsletter of the National Association of Mathematicians, an organization for African American mathematicians, from 1974 into the 1980s.[8]
She retired after 20 years of service at Winston-Salem State, circa 1985,[1] as professor emerita.[7]
Politics and later life
As part of the 1972 US presidential campaign, Newell was co-chair of the Shirley Chisholm campaign in North Carolina.[1] In 1977, Newell was elected (with Vivian Burke) as one of the first two African American women to become aldermen of Winston-Salem, North Carolina; she represented its East Ward.[4] She served in that position for 16 years.[1]
Recognition
The computer science center at Winston-Salem State University is named for Newell, as is one of the streets in Winston-Salem, Virginia Newell Lane.[1]
In 2017, Newell was given the Order of the Long Leaf Pine, the highest honor of the governor of North Carolina. In 2018, the National Association of Mathematicians gave her their Centenarian Award.[8] In 2019, Newell was given the YWCA Women of Vision Lifetime Achievement Award.[9] She was listed in 2021 as a Black History Month Honoree by the Mathematically Gifted and Black website.[8]
References
1. "Happy birthday, Dr. Virginia Kimbrough", Congressional Record, 163 (159), 4 October 2017
2. Barr, Matthew, Oral history interview with Virginia Newell, University of North Carolina at Greensboro
3. Sua, Lou Sanders (2012), But Your Mother Was An Activist: Black Women's Activism in North Carolina (PDF) (Doctoral dissertation), University of North Carolina at Greensboro
4. Elam, Bridget (7 October 2020), "Virginia Newell turns 103", Winston-Salem Chronicle
5. "Associate professors", Shaw University Bulletin, XXX (1): 16, July 1961
6. WorldCat catalog entry for Development of mathematics self-instructional learning packages with activities from the newspaper for prospective elementary school teachers enrolled at Winston-Salem State University, retrieved 2021-09-28
7. Drabble, Jenny (8 October 2017), "Former Winston-Salem elected official turns 100", Winston-Salem Journal
8. "Dr. Virginia Newell", Black History Month 2021 Honoree, Mathematically Gifted and Black, 2021, retrieved 2021-09-28
9. Vickers, Talitha (24 April 2019), "YWCA Women of Vision: Lifetime Achievement Award recipient Virginia Newell", WXII 12 News, WXII
10. Reviews of Black Mathematicians and their Works:
• Goins, Edray (February 2021), "Mathematical comfort food", The American Mathematical Monthly, 128 (2): 188, doi:10.1080/00029890.2021.1853445
• Kenschaft, Patricia Clark (1997), "What next? A meta-history of black mathematicians", African Americans in mathematics: Proceedings of the second conference for African-American researchers in the mathematical sciences held at DIMACS, Piscataway, NJ, USA, June 26–28, 1996, Providence, RI: American Mathematical Society, pp. 183–186, ISBN 0-8218-0678-5, Zbl 1155.01347; review, p. 185
• Sims, Janet L. (Summer 1981), The Journal of Negro History, 66 (2): 160–161, doi:10.2307/2717293, JSTOR 2717293{{citation}}: CS1 maint: untitled periodical (link)
• Sonnabend, Tom (November 1980), The Mathematics Teacher, 73 (8): 629, JSTOR 27962208{{citation}}: CS1 maint: untitled periodical (link)
• Zaslavsky, Claudia (February 1983), Historia Mathematica, 10 (1): 105–115, doi:10.1016/0315-0860(83)90049-6{{citation}}: CS1 maint: untitled periodical (link)
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Virginia Kiryakova
Virginia S. Kiryakova (née Virdzhinia Stoinova Hristova) is a Bulgarian mathematician known for her work on the fractional calculus, on special functions in fractional calculus including the Mittag-Leffler functions, and on the history of calculus. She is a professor in the Institute of Mathematics and Informatics of the Bulgarian Academy of Sciences.
Education and career
As a high school student, Kiryakova competed for Bulgaria in the 1969 International Mathematical Olympiad, earning a bronze medal.[1][2] She graduated from Sofia University in 1975 with a combined bachelor's and master's degree in mathematics, and in the same year became a researcher in the Institute of Mathematics and Informatics. She earned a Ph.D. in 1987, with the thesis Generalized Operators of Integration and Differentiation of Fractional Order and Applications,[1] and completed a Dr.Sc. (habilitation) in 2010, with the thesis Generalized Fractional Calculus and Applications in Analysis, supervised by Ivan Dimovski.[1][3]
She is editor-in-chief of the journals Fractional Calculus and Applied Analysis and International Journal of Applied Mathematics.[1]
Selected publications
Kiryakova is the author of the research monograph Generalized Fractional Calculus and Applications (1993).[4] She has also coauthored highly cited work on the history of calculus.[5]
Recognition
Kiryakova won the 1996 Academic Prize for Mathematical Sciences of Bulgarian Academy of Sciences.[1] In 2012, at the 5th Symposium on Fractional Differentiation and its Applications, she was given the FDA Dissemination Award, for her "dissemination of fractional calculus among the scientific community, industry and society" over the previous five years.[1][6]
References
1. Curriculum vitae (PDF), 2017, retrieved 2022-02-24
2. Bulgaria in the 11th IMO, 1969, International Mathematical Olympiad, retrieved 2022-02-24
3. Virginia Kiryakova at the Mathematics Genealogy Project
4. Generalized Fractional Calculus and Applications (Pitman Research Notes in Mathematics Series 301, Longman Scientific and John Wiley & Sons, 1993). Reviews: Anatoly Kilbas (1995), MR1265940; S.L.Kalla, Zbl 0882.26003; A. C. McBride (1995), Proc. Edinburgh Math. Soc, doi:10.1017/S0013091500006325
5. Machado, J. Tenreiro; Kiryakova, Virginia; Mainardi, Francesco (2011), "Recent history of fractional calculus", Communications in Nonlinear Science and Numerical Simulation, 16 (3): 1140–1153, Bibcode:2011CNSNS..16.1140M, doi:10.1016/j.cnsns.2010.05.027, hdl:10400.22/4149, MR 2736622
6. "Awards", 5th Symposium on Fractional Differentiation and its Applications, Hohai University, archived from the original on 2017-10-03
External links
• Home page
• Virginia Kiryakova publications indexed by Google Scholar
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Virginia Ragsdale
Virginia Ragsdale (December 13, 1870 – June 4, 1945) was a teacher and mathematician specializing in algebraic curves. She is most known as the creator of the Ragsdale conjecture.
Virginia Ragsdale
Virginia Ragsdale
Born(1870-12-13)December 13, 1870
Jamestown, North Carolina, US
DiedJune 4, 1945(1945-06-04) (aged 74)
Greensboro, North Carolina, US
Alma materGuilford College
Bryn Mawr College
Known forRagsdale conjecture
Scientific career
FieldsMathematics
InstitutionsWoman's College in Greensboro
ThesisOn the Arrangement of the Real Branches of Plane Algebraic Curves (1904)
Doctoral advisorCharlotte Scott
Early life
Ragsdale was born on a farm in Jamestown, North Carolina the third child of John Sinclair Ragsdale and Emily Jane Idol.[1] John was an officer in the Civil War, a teacher in the Flint Hill School, and later a state legislator.[1]
Virginia Ragsdale descended from Godfrey Ragsdale, a settler of the new Jamestown colony. Jamestown was raided by a native-American tribe in 1644 led by the uncle of Pocahontas, during which Godfrey and his wife were killed, but their infant son, Godfrey, Jr., survived. Ragsdale was then descended from the infant.[2]
Virginia documented her early years in a paper titled "Our Early Home and Childhood", writing:
One of my earliest recollections was a little trundle bed where Ida [her sister] and I slept together. … The house had no conveniences. Water had to be carried from a spring at the foot of the hill, milk and butter were kept there, washing was done there.
In the first years or two, there were three or four boarders, boys or young men, who came to attend Father's school.
Grandma (Idol), mother and Aunt Julia had all done their bit before and during the war, weaving blankets (and) jeans for men's suits, which were sold to Greensboro merchants in exchange for silk and other goods.
— Virginia Ragsdale, Our Early Home and Childhood[1]
Study
As a junior, Ragsdale entered Salem Academy, and graduated in 1887 as valedictorian with an extra diploma in piano.[2] Ragsdale attended Guilford College in Greensboro, North Carolina, where she earned her B.S. in 1892.[2] She was active in student life, establishing a Y.M.C.A. on campus, expanding collegiate athletics, and contributing to the formation the Guilford's Alumni Association.[2]
Ragsdale was awarded the first scholarship from Bryn Mawr College for the top scholar Guilford College.[1] She studied physics at Bryn Mawr College, obtaining an A.B. degree in 1896.[3] She was elected European fellow for the class of 1896, but waited a year before traveling, working as an assistant demonstrator in physics and mathematics graduate student at Bryn Mawr.[3]
Together with two of her colleagues (including Emilie Martin),[4] she spent 1897-98 abroad at the University of Göttingen, attending lectures of Felix Klein and David Hilbert.[3] After her return to the United States, she taught in Baltimore for three years until a second scholarship, by the Baltimore Association for the Promotion of University Education of Women,[3] permitted her to return to Bryn Mawr college to complete her Ph.D. under the direction of Charlotte Scott.[2]
Her dissertation, "On the Arrangement of the Real Branches of Plane Algebraic Curves," was published in 1906 by the American Journal of Mathematics.[2] Her dissertation addressed the 16th of Hilbert's problems, for which Ragsdale formulated a conjecture that provided an upper bound on the number of topological circles of a certain type.[2] Her result is called the Ragsdale conjecture; it was an open problem for 90 years until counterexamples were derived by Oleg Viro (1979) and Ilya Itenberg (1994).[2]
Career
After completing her degree, Ragsdale taught in New York City and Dr. Sach's School for Girls until 1905.[3] She was head of the Baldwin School in Bryn Mawr from 1906 to 1911, and a reader for Charlotte Scott from 1908 to 1910.[3] Ragsdale returned to North Carolina in 1911 to accept a mathematics position at Woman's College in Greensboro (now known as the University of North Carolina at Greensboro).[2] She remained there for almost two decades and served as department head from 1926 to 1928.[2] She encouraged the school to buy a telescope, and the math department to add statistics to the curriculum.[2]
In 1928, she retired from teaching in order to care for her mother's health and help manage the family farm.[3] After the death of her mother in 1934, she built a house at Guilford College, where she spent her last years gardening, working with furniture,[2] working on family genealogy, holding book clubs, and visiting with students.[1] Upon her death, she donated her house to Guilford College, where it housed the faculty, alumni, and visitors.[2] In 1965 President of Guilford Grimsley Hobbs moved into Ragsdale's house, and it has been the home of the college's president ever since.[1]
See also
• Ragsdale conjecture
• Algebraic curves
• Emilie Martin
References
1. Brooks, Carol (March 21, 2012). "Virginia Ragsdale: From farm girl to Ph.D." Jamestown News. Retrieved 3 January 2014.
2. De Loera, Jesús; Wicklin, Frederick J. "Biographies of Women in Mathematics: Virginia Ragsdale". Anges Scott College. Retrieved 3 January 2014.
3. Green, Judy; LaDuke, Jeanne (2009). Pioneering Women in American Mathematics: The Pre-1940 PhD's. Providence, Rhode Island: American Mathematical Society. https://books.google.com/books?id=jUrq3bUvQlYC&pg=PA271 pp. 271–272]. ISBN 978-0-8218-4376-5. Biography on p.503-505 of the Supplementary Material at AMS
4. Green & LaDuke (2009), p. 235.
External links
• Works by or about Virginia Ragsdale at Internet Archive
• Virginia Ragsdale at the Mathematics Genealogy Project
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Virginia Torczon
Virginia Joanne Torczon is an American applied mathematician and computer scientist known for her research on nonlinear optimization methods including pattern search. She is dean of graduate studies and research, and chancellor professor of computer science, at the College of William & Mary.[1]
Education and career
Torczon majored in history as an undergraduate at Wesleyan University.[2] She earned her Ph.D. in mathematical sciences in 1989 from Rice University.[2][3] Her dissertation, Multi-Directional Search: a Direct Search Algorithm for Parallel Machines, was supervised by John E. Dennis.[3]
Before becoming dean of graduate studies and research at William & Mary, she was the first female chair of the computer science department there.[4]
Recognition
Torczon's paper "On the Convergence of Pattern Search Algorithms" won the inaugural Society for Industrial and Applied Mathematics (SIAM) Outstanding Paper Prize for the best paper published in a SIAM journal in 1999.[5]
References
1. "Virginia Torczon", Computer Science Faculty, College of William & Mary, retrieved 2021-02-18
2. "Virginia Torczon", Parallel Profile, Parallel Computing Research, 5 (1), Winter 1997
3. Virginia Torczon at the Mathematics Genealogy Project
4. Berard, Adrienne (November 11, 2018), "Women in computer science: Taking the 'brogrammer' out of the algorithm", Williamsburg Yorktown Daily
5. The SIAM Outstanding Paper Prizes, Society for Industrial and Applied Mathematics, retrieved 2021-02-18
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Virginia Vassilevska Williams
Virginia Vassilevska Williams (née Virginia Panayotova Vassilevska)[1] is a theoretical computer scientist and mathematician known for her research in computational complexity theory and algorithms. She is currently the Steven and Renee Finn Career Development Associate Professor of Electrical Engineering and Computer Science at the Massachusetts Institute of Technology.[2] She is notable for her breakthrough results in fast matrix multiplication,[3] for her work on dynamic algorithms,[4] and for helping to develop the field of fine-grained complexity.[5]
Virginia Vassilevska Williams
Vassilevska Williams at Oberwolfach, 2012
NationalityBulgarian American
Alma mater
• Carnegie Mellon University (PhD, 2008)
• Caltech (BS, 2003)
Known for
• Matrix multiplication
• Graph algorithms
• Dynamic algorithms
• Fine-grained complexity theory
Scientific career
Fields
• Complexity theory
• Algorithms
Institutions
• MIT
• Stanford
• UC Berkeley
• Institute for Advanced Study
Doctoral advisorGuy Blelloch
Education and career
Williams is originally from Bulgaria, and attended a German-language high school in Sofia.[6] She graduated from the California Institute of Technology in 2003, and completed her Ph.D. at Carnegie Mellon University in 2008.[1] Her dissertation, Efficient Algorithms for Path Problems in Weighted Graphs, was supervised by Guy Blelloch.[7]
After postdoctoral research at the Institute for Advanced Study and University of California, Berkeley, Williams became an assistant professor of computer science at Stanford University in 2013.[1] She moved to MIT as an associate professor in 2017.[2]
Research
In 2011, Williams found an algorithm for multiplying two $n\times n$ matrices in time $O(n^{2.373})$. This improved a previous time bound for matrix multiplication algorithms, the Coppersmith–Winograd algorithm, that had stood as the best known for 24 years. Her initial improvement was independent of Andrew Stothers, who also improved the same bound a year earlier; after learning of Stothers' work, she combined ideas from both methods to improve his bound as well.[8][3] As of 2020, her work also establishes the current best-known algorithm for matrix multiplication with Josh Alman, in time $O(n^{2.3728596})$.[9]
Recognition
Williams was an NSF Computing Innovation Fellow for 2009–2011,[1] and won a Sloan Research Fellowship in 2017.[2] She was an invited speaker at the 2018 International Congress of Mathematicians, speaking in the section on Mathematical Aspects of Computer Science.[10]
Personal life
Williams is the daughter of applied mathematicians Panayot Vassilevski and Tanya Kostova-Vassilevska.[11] She is married to Ryan Williams, also a computer science professor at MIT; they have worked together in the field of fine-grained complexity.[6]
References
1. Curriculum vitae (PDF), retrieved 2018-02-24
2. Three EECS professors receive 2017 Sloan Research Fellowships, Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science, February 22, 2017
3. Virginia Vassilevska Williams (2012), "Multiplying Matrices Faster than Coppersmith-Winograd", in Howard J. Karloff and Toniann Pitassi (ed.), Proceedings of the 44th Symposium on Theory of Computing (STOC), ACM, pp. 887–898, doi:10.1145/2213977.2214056, S2CID 14350287
4. Abboud, Amir; Williams, Virginia Vassilevska (2014), "Popular Conjectures Imply Strong Lower Bounds for Dynamic Problems", 2014 IEEE 55th Annual Symposium on Foundations of Computer Science, pp. 434–443, arXiv:1402.0054, doi:10.1109/FOCS.2014.53, ISBN 978-1-4799-6517-5, S2CID 2267837
5. Williams, V. V. (2019), "On Some Fine-Grained Questions in Algorithms and Complexity", Proceedings of the International Congress of Mathematicians (ICM 2018): 3447–3487, doi:10.1142/9789813272880_0188, ISBN 978-981-327-287-3, S2CID 19282287
6. Matheson, Rob (January 7, 2020), "Finding the true potential of algorithms: Using mathematical theory, Virginia Williams coaxes algorithms to run faster or proves they've hit their maximum speed", MIT News, retrieved 2021-12-18
7. Virginia Vassilevska Williams at the Mathematics Genealogy Project
8. Aron, Jacob (December 9, 2011), "Key mathematical tool sees first advance in 24 years", New Scientist
9. Hartnett, Kevin (March 23, 2021), "Matrix Multiplication Inches Closer to Mythic Goal", Quanta Magazine, retrieved 2021-04-01
10. "Speakers", ICM 2018, archived from the original on 2017-12-15, retrieved 2018-02-24
11. "Vassilevska, Williams to wed", Engagements, Hartselle Enquirer, August 28, 2008, retrieved 2022-07-10
External links
• Home page
• Virginia Vassilevska Williams publications indexed by Google Scholar
• Virginia Vassilevska Williams at DBLP Bibliography Server
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Virginia Warfield
Virginia "Ginger" Patricia McShane Warfield is an American mathematician and mathematical educator. She received the Louise Hay Award from the Association for Women in Mathematics in 2007.[1]
Virginia Warfield
Born
Virginia Patricia McShane
NationalityAmerican
Alma materBrown University,
AwardsLouise Hay Award
Scientific career
InstitutionsUniversity of Washington
Doctoral advisorWendell Fleming
Education
Warfield's father was mathematician Edward J. McShane.[2] She received her Ph.D. in mathematics from Brown University in 1971. Her doctoral advisor was Wendell Fleming and the title of her dissertation was A Stochastic Maximum Principle.[3]
Career
While making contributions to the field of stochastic analysis after her Ph.D., Warfield became more and more engrossed by the problems of mathematics education. She worked with Project SEED, a highly regarded mathematics program whose goal was to promote sense-making mathematical activities for fourth through sixth graders. She addressed issues of teacher preparation and enhancement. She collaborated with the French mathematician Guy Brousseau, a pioneer in the “didactics of mathematics,” the scientific study of issues in mathematics teaching and learning.[4] She has been an active member of the Association for Women in Mathematics (AWM). She has chaired the Education Committee, has served as Education Column Editor for the AWM Newsletter, and was elected as a Member-at-large to the Executive Committee. She has been a member of the Mathematical Association of America’s committees on Professional Development and Mathematical Education of Teachers.[4]
Books
Warfield is the author of the book Invitation to Didactique (self-published, 2007, and Springer Briefs in Education, 2014)[5] and the co-author of Teaching Fractions through Situations: A Fundamental Experiment (with Guy Brousseau and Nadine Brousseau, Springer 2013).[6]
References
1. Kelley, Peter. "Teacher's teacher: 'Ginger' Warfield wins national math education award". UW News. Retrieved 7 April 2019.
2. New York Times:Edward McShane, 85, Mathematician, Dies; June 06, 1989
3. "Virginia P. Warfield". Mathematics Genealogy Project. Retrieved 7 April 2019.
4. "Virginia McShane Warfield Honored With Hay Award". Association for Women in Mathematics. Retrieved 7 April 2019.
5. Lancaster, Stephen (January 16, 2008). "Review of Invitation to Didactique". MAA Reviews. Mathematical Association of America.
6. Selden, Annie (February 22, 2014). "Review of Teaching Fractions through Situations". MAA Reviews. Mathematical Association of America.
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Virginie Bonnaillie-Noël
Virginie Bonnaillie-Noël (born on 3 October 1976 in Calais) is a French mathematician and research director specializing in numerical analysis. Her research topics concern partial differential equations, asymptotic, spectral and numerical analysis of problems arising from physics or mechanics.
Virginie Bonnaillie-Noël
Born
Virginie Bonnaillie
3 October 1976
Calais, France
NationalityFrench
Alma materParis-Sud University
Ecole Normale Supérieure Paris-Saclay
Occupation(s)Mathematician and research director
Known forIrène-Joliot-Curie Prize
Life and work
After completing her preparatory classes at Faidherbe high school in Lille between 1994 and 1997, Bonnaillie-Noël entered Paris-Sud University where she obtained a bachelor's degree in 1998 and a master's degree in 1999. The same year, she was admitted to the Ecole Normale Supérieure Paris-Saclay (ENS) where she obtained the aggregation in 2000 with the option of numerical analysis. In 2001, she obtained a Diploma of Advanced Studies (DEA) in Numerical Analysis and Partial Differential Equations.[1][2]
Between 2001 and 2003 she completed a thesis (as Virginie Bonnaillie) under the supervision of François Alouges and Bernard Helffer titled Mathematical analysis of superconductivity in a corner domain: semi-classical and numerical methods, which explored interdisciplinarity between physics and mathematics, at the border of numerical analysis, partial differential equations and spectral theory.[1][3]
In 2004, she joined the Mathematical Research Institute of Rennes (IRMAR) as a research fellow. In 2011, she obtained authorization to direct research at the University of Rennes I. In 2014, she left IRMAR to direct research in the Mathematics and Applications Department of ENS.[1]
In addition to her official work, Bonnaillie-Noël has frequently spoken about gender parity in science and research to encourage more young people to participate in the sciences.[1]
Distinctions
• 2008: CNRS bronze medal[1][4]
• 2009: Irène-Joliot-Curie Prize in the young female scientist category[1][2][5]
• 2011: Chevalier of the National Order of Merit by Cédric Villani[2]
• 2021: Officer of the National Order of Merit[6]
Selected publications
• Bonnaillie-Noël, V., & Dauge, M. (2006, August). Asymptotics for the low-lying eigenstates of the Schrödinger operator with magnetic field near corners. In Annales Henri Poincaré (Vol. 7, No. 5, pp. 899-931). Birkhäuser-Verlag.
• Bonnaillie-Noël, V., & Fournais, S. (2007). Superconductivity in domains with corners. Reviews in Mathematical Physics, 19(06), 607-637.
• Bonnaillie-Noël, V., Dambrine, M., Tordeux, S., & Vial, G. (2009). Interactions between moderately close inclusions for the Laplace equation. Mathematical Models and Methods in Applied Sciences, 19(10), 1853-1882.
• Bonnaillie-Noël, V., Dambrine, M., Hérau, F., & Vial, G. (2010). On generalized Ventcel's type boundary conditions for Laplace operator in a bounded domain. SIAM journal on mathematical analysis, 42(2), 931-945.
• Bonnaillie-Noël, V., Helffer, B., & Vial, G. (2010). Numerical simulations for nodal domains and spectral minimal partitions. ESAIM: Control, Optimisation and Calculus of Variations, 16(1), 221-246.
References
1. "PRIX DE LA JEUNE FEMME SCIENTIFIQUE, Virginie BONNAILLIE-NOËL" (PDF). 2009. Retrieved 2022-07-22.
2. Jeune, Cecile Le (September 11, 2012). "ENS Rennes - Remise de l'insigne de Chevalier de l'Ordre National du mérite à Virginie Bonnaillie-Noël par Cédric Villani". ENS Rennes (in French). Retrieved 2022-07-22.
3. Bonnaillie, Virginie. "Thesis".
4. "INSMI - Institut national des sciences mathématiques et de leurs interactions - Remise de la médaille de bronze du CNRS à Virginie Bonnaillie-Noël". archive.wikiwix.com. Retrieved 2022-07-22.
5. "Les lauréates du Prix Irène Joliot-Curie 2009 - ESR : enseignementsup-recherche.gouv.fr". www.enseignementsup-recherche.gouv.fr (in French). Retrieved 2022-07-22.
6. "Promotions". www.legifrance.gouv.fr. Retrieved 2022-07-22.
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Virtual knot
In knot theory, a virtual knot is a generalization of knots in 3-dimensional Euclidean space, R3, to knots in thickened surfaces $\Sigma \times [0,1]$ modulo an equivalence relation called stabilization/destabilization. Here $\Sigma $ is required to be closed and oriented. Virtual knots were first introduced by Kauffman (1999).
Unsolved problem in mathematics:
[Extension of Jones polynomial to general 3-manifolds.] Can the original Jones polynomial, which is defined for 1-links in the 3-sphere (the 3-ball, the 3-space R3), be extended for 1-links in any 3-manifold?
(more unsolved problems in mathematics)
Overview
In the theory of classical knots, knots can be considered equivalence classes of knot diagrams under the Reidemeister moves. Likewise a virtual knot can be considered an equivalence of virtual knot diagrams that are equivalent under generalized Reidemeister moves. Virtual knots allow for the existence of, for example, knots whose Gauss codes which could not exist in 3-dimensional Euclidean space. A virtual knot diagram is a 4-valent planar graph, but each vertex is now allowed to be a classical crossing or a new type called virtual. The generalized moves show how to manipulate such diagrams to obtain an equivalent diagram; one move called the semi-virtual move involves both classical and virtual crossings, but all the other moves involve only one variety of crossing.
Virtual knots are important, and there is a strong relation between Quantum Field Theory and virtual knots.
Virtual knots themselves are fascinating objects, and having many connections to other areas of mathematics. Virtual knots have many exciting connections with other fields of knots theory. The unsolved problem shown is an important motivation to the study of virtual knots.
See section 1.1 of this paper [KOS] [1] for the background and the history of this problem. Kauffman submitted a solution in the case of the product manifold of closed oriented surface and the closed interval, by introducing virtual 1-knots .[2] It is open in the other cases. Witten’s path integral for Jones polynomial is written for links in any compact 3-manifold formally, but the calculus is not done even in physics level in any case other than the 3-sphere (the 3-ball, the 3-space R3). This problem is also open in physics level. In the case of Alexander polynomial, this problem is solved.
A classical knot can also be considered an equivalence class of Gauss diagrams under certain moves coming from the Reidemeister moves. Not all Gauss diagrams are realizable as knot diagrams, but by considering all equivalence classes of Gauss diagrams we obtain virtual knots.
A classical knot can be considered an ambient isotopy class of embeddings of the circle into a thickened 2-sphere. This can be generalized by considering such classes of embeddings into thickened higher-genus surfaces. This is not quite what we want since adding a handle to a (thick) surface will create a higher-genus embedding of the original knot. The adding of a handle is called stabilization and the reverse process destabilization. Thus a virtual knot can be considered an ambient isotopy class of embeddings of the circle into thickened surfaces with the equivalence given by (de)stabilization.
Some basic theorems relating classical and virtual knots:
• If two classical knots are equivalent as virtual knots, they are equivalent as classical knots.
• There is an algorithm to determine if a virtual knot is classical.
• There is an algorithm to determine if two virtual knots are equivalent.
It is important that there is a relation among the following. See the paper [KOS] cited above and below.
• Virtual equivalence of virtual 1-knot diagrams, which is a set of virtual 1-knots.
• Welded equivalence of virtual 1-knot diagrams
• Rotational welded equivalence of virtual 1-knot diagrams
• Fiberwise equivalence of virtual 1-knot diagrams
Virtual 2-knots are also defined. See the paper cited above.
See also
• Knots and graphs
References
1. Kauffman, L.H; Ogasa, E; Schneider, J (2018), A spinning construction for virtual 1-knots and 2-knots, and the fiberwise and welded equivalence of virtual 1-knots, arXiv:1808.03023
2. Kauffman, L.E. (1998), Talks at MSRI Meeting in January 1997, AMS Meeting at University of Maryland, College Park in March 1997, Isaac Newton Institute Lecture in November 1997, Knots in Hellas Meeting in Delphi, Greece in July 1998, APCTP-NANKAI Symposium on Yang-Baxter Systems, Non-Linear Models and Applications at Seoul, Korea in October 1998, and Kauffman's paper1999 cited below., arXiv:math/9811028
• Boden, Hans; Nagel, Matthias (2017). "Concordance group of virtual knots". Proceedings of the American Mathematical Society. 145 (12): 5451–5461. doi:10.1090/proc/13667. S2CID 119139769.
• Carter, J. Scott; Kamada, Seiichi; Saito, Masahico (2002). "Stable equivalence of knots on surfaces and virtual knot cobordisms. Knots 2000 Korea, Vol. 1 (Yongpyong)". J. Knot Theory Ramifications. 11 (3): 311–322.
• Carter, J. Scott; Silver, Daniel; Williams, Susan (2014). "Invariants of links in thickened surfaces". Algebraic & Geometric Topology. 14 (3): 1377–1394. doi:10.2140/agt.2014.14.1377. S2CID 53137201.
• Dye, Heather A (2016). An Invitation to Knot Theory : Virtual and Classical (First ed.). Chapman and Hall/CRC. ISBN 9781315370750.
• Goussarov, Mikhail; Polyak, Michael; Viro, Oleg (2000). "Finite-type invariants of classical and virtual knots". Topology. 39 (5): 1045–1068. arXiv:math/9810073. doi:10.1016/S0040-9383(99)00054-3. S2CID 8871411.
• Kamada, Naoko; Kamda, Seiichi (2000). "Abstract link diagrams and virtual knots". Journal of Knot Theory and Its Ramifications. 9 (1): 93–106. doi:10.1142/S0218216500000049.
• Kauffman, Louis H. (1999). "Virtual knot theory" (PDF). European Journal of Combinatorics. 20 (7): 663–690. doi:10.1006/eujc.1999.0314. ISSN 0195-6698. MR 1721925. S2CID 5993431.
• Kauffman, Louis H.; Manturov, Vassily Olegovich (2005). "Virtual Knots and Links". arXiv:math.GT/0502014.
• Kuperberg, Greg (2003). "What is a virtual link?". Algebraic & Geometric Topology. 3: 587–591. doi:10.2140/agt.2003.3.587. S2CID 16803280.
• Manturov, Vassily (2004). Knot Theory. CRC Press. ISBN 978-0-415-31001-7.
• Manturov, Vassily Olegovich (2004). "Virtual knots and infinite dimensional Lie algebras". Acta Applicandae Mathematicae. 83 (3): 221–233. doi:10.1023/B:ACAP.0000038944.29820.5e. S2CID 124019548.
• Turaev, Vladimir (2008). "Cobordism of knots on surfaces". Journal of Topology. 1 (2): 285–305. arXiv:math/0703055. doi:10.1112/jtopol/jtn002. S2CID 17888102.
External links
• A Table of Virtual Knots
• Elementary explanation with diagrams
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Virtually Haken conjecture
In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.
After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.
The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968,[1] although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list.
A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica.[2] The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes)[3] It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.[4][5] Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise[6] and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.[7]
In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Hanken[8][9]
See also
• Virtually fibered conjecture
• Surface subgroup conjecture
• Ehrenpreis conjecture
Notes
1. Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. JSTOR 1970594. MR 0224099.
2. Agol, Ian (2013). With an appendix by Ian Agol, Daniel Groves, and Jason Manning. "The virtual Haken Conjecture". Doc. Math. 18: 1045–1087. MR 3104553.
3. Haglund, Frédéric; Wise, Daniel (2012). "A combination theorem for special cube complexes". Annals of Mathematics. 176 (3): 1427–1482. doi:10.4007/annals.2012.176.3.2. MR 2979855.
4. Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4. MR 2912704. S2CID 32593851.
5. Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology. 16 (1): 601–624. arXiv:1012.2828. doi:10.2140/gt.2012.16.601. MR 2916295.
6. Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
7. Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics. 134 (3): 843–859. arXiv:0908.3609. doi:10.1353/ajm.2012.0020. MR 2931226. S2CID 14128842.
8. Przytycki, Piotr; Wise, Daniel (2017-10-19). "Mixed 3-manifolds are virtually special". Journal of the American Mathematical Society. 31 (2): 319–347. doi:10.1090/jams/886. ISSN 0894-0347. S2CID 39611341.
9. "Piotr Przytycki and Daniel Wise receive 2022 Moore Prize". American Mathematical Society.
References
• Dunfield, Nathan; Thurston, William (2003), "The virtual Haken conjecture: experiments and examples", Geometry and Topology, 7: 399–441, arXiv:math/0209214, doi:10.2140/gt.2003.7.399, MR 1988291, S2CID 6265421.
• Kirby, Robion (1978), "Problems in low dimensional manifold theory.", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), vol. 7, pp. 273–312, ISBN 9780821867891, MR 0520548.
External links
• Klarreich, Erica (2012-10-02). "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back". Quanta Magazine.
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Virtually
In mathematics, especially in the area of abstract algebra that studies infinite groups, the adverb virtually is used to modify a property so that it need only hold for a subgroup of finite index. Given a property P, the group G is said to be virtually P if there is a finite index subgroup $H\leq G$ such that H has property P.
For a definition of the term "virtually", see the Wiktionary entry virtually.
Common uses for this would be when P is abelian, nilpotent, solvable or free. For example, virtually solvable groups are one of the two alternatives in the Tits alternative, while Gromov's theorem states that the finitely generated groups with polynomial growth are precisely the finitely generated virtually nilpotent groups.
This terminology is also used when P is just another group. That is, if G and H are groups then G is virtually H if G has a subgroup K of finite index in G such that K is isomorphic to H.
In particular, a group is virtually trivial if and only if it is finite. Two groups are virtually equal if and only if they are commensurable.
Examples
Virtually abelian
The following groups are virtually abelian.
• Any abelian group.
• Any semidirect product $N\rtimes H$ where N is abelian and H is finite. (For example, any generalized dihedral group.)
• Any semidirect product $N\rtimes H$ where N is finite and H is abelian.
• Any finite group (since the trivial subgroup is abelian).
Virtually nilpotent
• Any group that is virtually abelian.
• Any nilpotent group.
• Any semidirect product $N\rtimes H$ where N is nilpotent and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is nilpotent.
Gromov's theorem says that a finitely generated group is virtually nilpotent if and only if it has polynomial growth.
Virtually polycyclic
Main article: virtually polycyclic group
Virtually free
• Any free group.
• Any virtually cyclic group.
• Any semidirect product $N\rtimes H$ where N is free and H is finite.
• Any semidirect product $N\rtimes H$ where N is finite and H is free.
• Any free product $H*K$, where H and K are both finite. (For example, the modular group $\operatorname {PSL} (2,\mathbb {Z} )$.)
It follows from Stalling's theorem that any torsion-free virtually free group is free.
Others
The free group $F_{2}$ on 2 generators is virtually $F_{n}$ for any $n\geq 2$ as a consequence of the Nielsen–Schreier theorem and the Schreier index formula.
The group $\operatorname {O} (n)$ is virtually connected as $\operatorname {SO} (n)$ has index 2 in it.
References
Look up virtually in Wiktionary, the free dictionary.
• Schneebeli, Hans Rudolf (1978). "On virtual properties and group extensions". Mathematische Zeitschrift. 159: 159–167. doi:10.1007/bf01214488. Zbl 0358.20048.
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Virtually fibered conjecture
In the mathematical subfield of 3-manifolds, the virtually fibered conjecture, formulated by American mathematician William Thurston, states that every closed, irreducible, atoroidal 3-manifold with infinite fundamental group has a finite cover which is a surface bundle over the circle.
A 3-manifold which has such a finite cover is said to virtually fiber. If M is a Seifert fiber space, then M virtually fibers if and only if the rational Euler number of the Seifert fibration or the (orbifold) Euler characteristic of the base space is zero.
The hypotheses of the conjecture are satisfied by hyperbolic 3-manifolds. In fact, given that the geometrization conjecture is now settled, the only case needed to be proven for the virtually fibered conjecture is that of hyperbolic 3-manifolds.
The original interest in the virtually fibered conjecture (as well as its weaker cousins, such as the virtually Haken conjecture) stemmed from the fact that any of these conjectures, combined with Thurston's hyperbolization theorem, would imply the geometrization conjecture. However, in practice all known attacks on the "virtual" conjecture take geometrization as a hypothesis, and rely on the geometric and group-theoretic properties of hyperbolic 3-manifolds.
The virtually fibered conjecture was not actually conjectured by Thurston. Rather, he posed it as a question and has stated that it was intended as a challenge and not meant to indicate he believed it, although he wrote that "[t]his dubious-sounding question seems to have a definite chance for a positive answer".[1]
The conjecture was finally settled in the affirmative in a series of papers from 2009 to 2012. In a posting on the ArXiv on 25 Aug 2009,[2] Daniel Wise implicitly implied (by referring to a then-unpublished longer manuscript) that he had proven the conjecture for the case where the 3-manifold is closed, hyperbolic, and Haken. This was followed by a survey article in Electronic Research Announcements in Mathematical Sciences.[3][4][5][6] have followed, including the aforementioned longer manuscript by Wise.[7] In March 2012, during a conference at Institut Henri Poincaré in Paris, Ian Agol announced he could prove the virtually Haken conjecture for closed hyperbolic 3-manifolds .[8] Taken together with Daniel Wise's results, this implies the virtually fibered conjecture for all closed hyperbolic 3-manifolds.
See also
• Virtually Haken conjecture
• Surface subgroup conjecture
• Ehrenpreis conjecture
• positive virtual Betti number conjecture
Notes
1. Thurston 1982, p. 380.
2. Bergeron, Nicolas; Wise, Daniel T. (2009). "A boundary criterion for cubulation". arXiv:0908.3609. {{cite journal}}: Cite journal requires |journal= (help)
3. Wise, Daniel (2009). "Research announcement: The structure of groups with a quasiconvex hierarchy". Electronic Research Announcements in Mathematical Sciences. 16: 44–55. doi:10.3934/era.2009.16.44.
4. Haglund, Frédéric; Wise, Daniel (2012). "A combination theorem for special cube complexes". Annals of Mathematics. 176 (3): 1427–1482. doi:10.4007/annals.2012.176.3.2.
5. Christopher Hruska, G. C.; Wise, Daniel T. (2014). "Finiteness properties of cubulated groups". Compositio Mathematica. 150 (3): 453–506. arXiv:1209.1074. doi:10.1112/S0010437X13007112. S2CID 119341019.
6. Hsu, Tim; Wise, Daniel T. (2015). "Cubulating malnormal amalgams". Inventiones Mathematicae. 199 (2): 293–331. Bibcode:2015InMat.199..293H. doi:10.1007/s00222-014-0513-4. S2CID 122292998.
7. Wise, Daniel T. The structure of groups with a quasiconvex hierarchy (PDF).
8. Agol, Ian; Groves, Daniel; Manning, Jason (2012). "The virtual Haken conjecture". arXiv:1204.2810. {{cite journal}}: Cite journal requires |journal= (help)
References
• Thurston, William P. (1982). "Three dimensional manifolds, Kleinian groups and hyperbolic geometry". Bulletin of the American Mathematical Society. 6 (3): 357–382. doi:10.1090/S0273-0979-1982-15003-0.
• D. Gabai, On 3-manifold finitely covered by surface bundles, Low Dimensional Topology and Kleinian Groups (ed: D.B.A. Epstein), London Mathematical Society Lecture Note Series vol 112 (1986), p. 145-155.
• Agol, Ian (2008). "Criteria for virtual fibering". Journal of Topology. 1 (2): 269–284. arXiv:0707.4522. doi:10.1112/jtopol/jtn003. S2CID 3028314.
External links
• Klarreich, Erica (2012-10-02). "Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back". Quanta Magazine.
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Michael Viscardi
Michael Anthony Viscardi (born February 22, 1989 in Plano, Texas) of San Diego, California is an American mathematician who, as a highschooler, won the 2005 Siemens Competition and Davidson Fellowship with a mathematical project on the Dirichlet problem, whose applications include describing the flow of heat across a metal surface, winning $100,000 and $50,000 in scholarships, respectively.[1][2] Viscardi's theorem is an expansion of the 19th-century work of Peter Gustav Lejeune Dirichlet.[3] He was also named a finalist with the same project in the Intel Science Talent Search. Viscardi placed Best of Category in Mathematics at the International Science and Engineering Fair (ISEF) in May 2006. Viscardi also qualified for the United States of America Mathematical Olympiad and the Junior Science and Humanities Symposium.
Michael Viscardi
Born (1989-02-22) February 22, 1989
Plano, Texas, United States
NationalityAmerican
Alma materHarvard University
Massachusetts Institute of Technology
Known forSiemens Competition winner
Awards2010 Hoopes Prize
Scientific career
FieldsMathematics
Doctoral advisorRoman Bezrukavnikov
Other academic advisorsShing-Tung Yau
Joe Harris
Life
Viscardi was homeschooled for high school, supplemented with mathematics classes at the University of California, San Diego.[4][5] He is also a pianist and violinist, and onetime concertmaster of the San Diego Youth Symphony.[5]
Viscardi is a member of the Harvard College class of 2010.[6] He graduated summa cum laude from Harvard, receiving a 2010 Thomas T. Hoopes, Class of 1919, Prize, and earning the 2011 Morgan Prize honorable mention for his senior thesis "Alternate Compactifications of the Moduli Space of Genus One Maps".[7] He worked as a postdoc at UC Berkeley from 2016 to 2018.[8]
Selected publication
• ———; Ebenfelt, Peter (2007), "An Explicit Solution to the Dirichlet Problem with Rational Holomorphic Data in Terms of a Riemann Mapping", Computational Methods and Function Theory, 7 (1): 127–140, doi:10.1007/BF03321636, S2CID 120812150.
References
1. Briggs, Tracey Wong (December 5, 2005), "Problems no problem for Siemens winners", USA Today.
2. Rodriguez, Juan-Carlos (December 6, 2005), "California teen wins science competition", Seattle Times.
3. "Teen Updates 19th-Century Mathematical Law", ABC News, December 9, 2005.
4. "Homeschooled teen wins top science honor", Associated Press, 2005
5. "Mathematics Student Wins the Siemens-Westinghouse Competition", FOCUS, Mathematical Association of America, vol. 26, no. 1, p. 3, January 2006
6. Herchel Smith Research Fellows to begin this summer
7. Viscardi, Michael (2010). "Alternate compactifications of the moduli space of genus one maps". arXiv:1005.1431 [math.AG].
8. Viscardi's webpage at Berkeley
External links
• Viscardi's website at MIT
• Michael Viscardi: Person of the Week
• Michael's Presentation
• Biography at Davidson Institute site
Authority control: Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
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Viscosity solution
In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in dynamic programming (the Hamilton–Jacobi–Bellman equation), differential games (the Hamilton–Jacobi–Isaacs equation) or front evolution problems,[1][2] as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games.
The classical concept was that a PDE
$F(x,u,Du,D^{2}u)=0$
over a domain $x\in \Omega $ has a solution if we can find a function u(x) continuous and differentiable over the entire domain such that $x$, $u$, $Du$, $D^{2}u$ satisfy the above equation at every point.
If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called viscosity solution. Under the viscosity solution concept, u does not need to be everywhere differentiable. There may be points where either $Du$ or $D^{2}u$ does not exist and yet u satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations.
Definition
There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book[3] or the definition using semi-jets in the Users Guide.[4]
Degenerate elliptic
An equation $F(x,u,Du,D^{2}u)=0$ in a domain $\Omega $ is defined to be degenerate elliptic if for any two symmetric matrices $X$ and $Y$ such that $Y-X$ is positive definite, and any values of $x\in \Omega $, $u\in \mathbb {R} $ and $p\in \mathbb {R} ^{n}$, we have the inequality $F(x,u,p,X)\geq F(x,u,p,Y)$. For example, $-\Delta u=0$ (where $\Delta $ denotes the Laplacian) is degenerate elliptic since in this case, $F(x,u,p,X)=-{\text{trace}}(X)$, and the trace of $X$ is the sum of its eigenvalues. Any real first- order equation is degenerate elliptic.
Viscosity subsolution
An upper semicontinuous function $u$ in $\Omega $ is defined to be a subsolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \geq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\leq 0$.
Viscosity supersolution
A lower semicontinuous function $u$ in $\Omega $ is defined to be a supersolution of the above degenerate elliptic equation in the viscosity sense if for any point $x_{0}\in \Omega $ and any $C^{2}$ function $\phi $ such that $\phi (x_{0})=u(x_{0})$ and $\phi \leq u$ in a neighborhood of $x_{0}$, we have $F(x_{0},\phi (x_{0}),D\phi (x_{0}),D^{2}\phi (x_{0}))\geq 0$.
Viscosity solution
A continuous function u is a viscosity solution of the PDE $F(x,u,Du,D^{2}u)=0$ in $\Omega $ if it is both a supersolution and a subsolution. Note that the boundary condition in the viscosity sense has not been discussed here.
Example
Consider the boundary value problem $|u'(x)|=1$, or $F(u')=|u'|-1=0$, on $(-1,1)$ with boundary conditions $u(-1)=u(1)=0$. Then, the function $u(x)=1-|x|$ is a viscosity solution.
Indeed, note that the boundary conditions are satisfied classically, and $|u'(x)|=1$ is well-defined in the interior except at $x=0$. Thus, it remains to show that the conditions for viscosity subsolution and viscosity supersolution hold at $x=0$. Suppose that $\phi (x)$ is any function differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\geq u(x)$ near $x=0$. From these assumptions, it follows that $\phi (x)-\phi (0)\geq -|x|$. For positive $x$, this inequality implies $\lim _{x\to 0^{+}}{\frac {\phi (x)-\phi (0)}{x}}\geq -1$, using that $|x|/x=sgn(x)=1$ for $x>0$. On the other hand, for $x<0$, we have that $\lim _{x\to 0^{-}}{\frac {\phi (x)-\phi (0)}{x}}\leq 1$. Because $\phi $ is differentiable, the left and right limits agree and are equal to $\phi '(0)$, and we therefore conclude that $|\phi '(0)|\leq 1$, i.e., $F(\phi '(0))\leq 0$. Thus, $u$ is a viscosity subsolution. Moreover, the fact that $u$ is a supersolution holds vacuously, since there is no function $\phi (x)$ differentiable at $x=0$ with $\phi (0)=u(0)=1$ and $\phi (x)\leq u(x)$ near $x=0$. This implies that $u$ is a viscosity solution.
In fact, one may prove that $u$ is the unique viscosity solution for such problem. The uniqueness part involves a more refined argument.
Discussion
The previous boundary value problem is an eikonal equation in a single spatial dimension with $f=1$, where the solution is known to be the signed distance function to the boundary of the domain. Note also in the previous example, the importance of the sign of $F$. In particular, the viscosity solution to the PDE $-F=0$ with the same boundary conditions is $u(x)=|x|-1$. This can be explained by observing that the solution $u(x)=1-|x|$ is the limiting solution of the vanishing viscosity problem $F(u')=[u']^{2}-1=\epsilon u''$ as $\epsilon $ goes to zero, while $u(x)=|x|-1$ is the limit solution of the vanishing viscosity problem $-F(u')=1-[u']^{2}=\epsilon u''$.[5] One can readily confirm that $u_{\epsilon }(x)=\epsilon [\ln(\cosh(1/\epsilon ))-\ln(\cosh(x/\epsilon ))]$ solves the PDE $F(u')=[u']^{2}-1=\epsilon u''$ for each $\epsilon >0$. Further, the family of solutions $u_{\epsilon }$ converges toward the solution $u=1-|x|$ as $\epsilon $ vanishes (see Figure).
Basic properties
The three basic properties of viscosity solutions are existence, uniqueness and stability.
• The uniqueness of solutions requires some extra structural assumptions on the equation. Yet it can be shown for a very large class of degenerate elliptic equations.[4] It is a direct consequence of the comparison principle. Some simple examples where comparison principle holds are
1. $u+H(x,\nabla u)=0$ with H uniformly continuous in both variables.
2. (Uniformly elliptic case) $F(D^{2}u,Du,u)=0$ so that $F$ is Lipschitz with respect to all variables and for every $r\leq s$ and $X\geq Y$, $F(Y,p,s)\geq F(X,p,r)+\lambda ||X-Y||$ for some $\lambda >0$.
• The existence of solutions holds in all cases where the comparison principle holds and the boundary conditions can be enforced in some way (through barrier functions in the case of a Dirichlet boundary condition). For first order equations, it can be obtained using the vanishing viscosity method[6][2] or for most equations using Perron's method.[7][8][2] There is a generalized notion of boundary condition, in the viscosity sense. The solution to a boundary problem with generalized boundary conditions is solvable whenever the comparison principle holds.[4]
• The stability of solutions in $L^{\infty }$ holds as follows: a locally uniform limit of a sequence of solutions (or subsolutions, or supersolutions) is a solution (or subsolution, or supersolution). More generally, the notions of viscosity sub- and supersolution are also conserved by half-relaxed limits.[4]
History
The term viscosity solutions first appear in the work of Michael G. Crandall and Pierre-Louis Lions in 1983 regarding the Hamilton–Jacobi equation.[6] The name is justified by the fact that the existence of solutions was obtained by the vanishing viscosity method. The definition of solution had actually been given earlier by Lawrence C. Evans in 1980.[9] Subsequently the definition and properties of viscosity solutions for the Hamilton–Jacobi equation were refined in a joint work by Crandall, Evans and Lions in 1984.[10]
For a few years the work on viscosity solutions concentrated on first order equations because it was not known whether second order elliptic equations would have a unique viscosity solution except in very particular cases. The breakthrough result came with the method introduced by Robert Jensen in 1988 to prove the comparison principle using a regularized approximation of the solution which has a second derivative almost everywhere (in modern versions of the proof this is achieved with sup-convolutions and Alexandrov theorem).[11]
In subsequent years the concept of viscosity solution has become increasingly prevalent in analysis of degenerate elliptic PDE. Based on their stability properties, Barles and Souganidis obtained a very simple and general proof of convergence of finite difference schemes.[12] Further regularity properties of viscosity solutions were obtained, especially in the uniformly elliptic case with the work of Luis Caffarelli.[13] Viscosity solutions have become a central concept in the study of elliptic PDE. In particular, Viscosity solutions are essential in the study of the infinity Laplacian.[14]
In the modern approach, the existence of solutions is obtained most often through the Perron method.[4] The vanishing viscosity method is not practical for second order equations in general since the addition of artificial viscosity does not guarantee the existence of a classical solution. Moreover, the definition of viscosity solutions does not generally involve physical viscosity. Nevertheless, while the theory of viscosity solutions is sometimes considered unrelated to viscous fluids, irrotational fluids can indeed be described by a Hamilton-Jacobi equation.[15] In this case, viscosity corresponds to the bulk viscosity of an irrotational, incompressible fluid. Other names that were suggested were Crandall–Lions solutions, in honor to their pioneers, $L^{\infty }$-weak solutions, referring to their stability properties, or comparison solutions, referring to their most characteristic property.
References
1. Dolcetta, I.; Lions, P., eds. (1995). Viscosity Solutions and Applications. Berlin: Springer. ISBN 3-540-62910-6.
2. Tran, Hung V. (2021). Hamilton-Jacobi Equations : Theory and Applications. Providence, Rhode Island. ISBN 978-1-4704-6511-7. OCLC 1240263322.{{cite book}}: CS1 maint: location missing publisher (link)
3. Wendell H. Fleming, H. M . Soner, (2006), Controlled Markov Processes and Viscosity Solutions. Springer, ISBN 978-0-387-26045-7.
4. Crandall, Michael G.; Ishii, Hitoshi; Lions, Pierre-Louis (1992), "User's guide to viscosity solutions of second order partial differential equations", Bulletin of the American Mathematical Society, New Series, 27 (1): 1–67, arXiv:math/9207212, Bibcode:1992math......7212C, doi:10.1090/S0273-0979-1992-00266-5, ISSN 0002-9904, S2CID 119623818
5. Barles, Guy (2013). "An Introduction to the Theory of Viscosity Solutions for First-Order Hamilton–Jacobi Equations and Applications". Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications. Lecture Notes in Mathematics. Vol. 2074. Berlin: Springer. pp. 49–109. doi:10.1007/978-3-642-36433-4_2. ISBN 978-3-642-36432-7. S2CID 55804130.
6. Crandall, Michael G.; Lions, Pierre-Louis (1983), "Viscosity solutions of Hamilton-Jacobi equations", Transactions of the American Mathematical Society, 277 (1): 1–42, doi:10.2307/1999343, ISSN 0002-9947, JSTOR 1999343
7. Ishii, Hitoshi (1987), "Perron's method for Hamilton-Jacobi equations", Duke Mathematical Journal, 55 (2): 369–384, doi:10.1215/S0012-7094-87-05521-9, ISSN 0012-7094
8. Ishii, Hitoshi (1989), "On uniqueness and existence of viscosity solutions of fully nonlinear second-order elliptic PDEs", Communications on Pure and Applied Mathematics, 42 (1): 15–45, doi:10.1002/cpa.3160420103, ISSN 0010-3640
9. Evans, Lawrence C. (1980), "On solving certain nonlinear partial differential equations by accretive operator methods", Israel Journal of Mathematics, 36 (3): 225–247, doi:10.1007/BF02762047, ISSN 0021-2172
10. Crandall, Michael G.; Evans, Lawrence C.; Lions, Pierre-Louis (1984), "Some properties of viscosity solutions of Hamilton–Jacobi equations", Transactions of the American Mathematical Society, 282 (2): 487–502, doi:10.2307/1999247, ISSN 0002-9947, JSTOR 1999247
11. Jensen, Robert (1988), "The maximum principle for viscosity solutions of fully nonlinear second order partial differential equations", Archive for Rational Mechanics and Analysis, 101 (1): 1–27, Bibcode:1988ArRMA.101....1J, doi:10.1007/BF00281780, ISSN 0003-9527, S2CID 5776251
12. Barles, G.; Souganidis, P. E. (1991), "Convergence of approximation schemes for fully nonlinear second order equations", Asymptotic Analysis, 4 (3): 271–283, doi:10.3233/ASY-1991-4305, ISSN 0921-7134
13. Caffarelli, Luis A.; Cabré, Xavier (1995), Fully nonlinear elliptic equations, American Mathematical Society Colloquium Publications, vol. 43, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-0437-7
14. Crandall, Michael G.; Evans, Lawrence C.; Gariepy, Ronald F. (2001), "Optimal Lipschitz extensions and the infinity Laplacian", Calculus of Variations and Partial Differential Equations, 13 (2): 123–129, doi:10.1007/s005260000065, S2CID 1529607
15. Westernacher-Schneider, John Ryan; Markakis, Charalampos; Tsao, Bing Jyun (2020). "Hamilton-Jacobi hydrodynamics of pulsating relativistic stars". Classical and Quantum Gravity. 37 (15): 155005. arXiv:1912.03701. Bibcode:2020CQGra..37o5005W. doi:10.1088/1361-6382/ab93e9. S2CID 208909879.
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Visibility graph analysis
In architecture, visibility graph analysis (VGA) is a method of analysing the inter-visibility connections within buildings or urban networks. Visibility graph analysis was developed from the architectural theory of space syntax by Turner et al. (2001), and is applied through the construction of a visibility graph within the open space of a plan.
"Visibility analysis" redirects here. Not to be confused with Visibility (geometry).
Visibility graph analysis uses various measures from the theory of small-world networks and centrality in network theory in order to assess perceptual qualities of space and the possible usage of it.
Visibility graph analysis was firstly implemented in Turner's Depthmap software and is now widely used by both academics and practitioners through the open source and multi-platform depthmapX developed by Tasos Varoudis.
Another opensource and multi-platform software that implements visibility graphs is topologicpy developed by Wassim Jabi.
See also
• Fuzzy architectural spatial analysis
• Isovist
• Spatial network analysis software
• Viewshed analysis
References
• A. Turner; Doxa, M.; O'Sullivan, D.; Penn, A. (2001). "From isovists to visibility graphs: a methodology for the analysis of architectural space" (PDF). Environment and Planning B. 28 (1): 103–121. doi:10.1068/b2684.
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Visibility graph
In computational geometry and robot motion planning,[1] a visibility graph is a graph of intervisible locations, typically for a set of points and obstacles in the Euclidean plane. Each node in the graph represents a point location, and each edge represents a visible connection between them. That is, if the line segment connecting two locations does not pass through any obstacle, an edge is drawn between them in the graph. When the set of locations lies in a line, this can be understood as an ordered series. Visibility graphs have therefore been extended to the realm of time series analysis.
Applications
Visibility graphs may be used to find Euclidean shortest paths among a set of polygonal obstacles in the plane: the shortest path between two obstacles follows straight line segments except at the vertices of the obstacles, where it may turn, so the Euclidean shortest path is the shortest path in a visibility graph that has as its nodes the start and destination points and the vertices of the obstacles.[2] Therefore, the Euclidean shortest path problem may be decomposed into two simpler subproblems: constructing the visibility graph, and applying a shortest path algorithm such as Dijkstra's algorithm to the graph. For planning the motion of a robot that has non-negligible size compared to the obstacles, a similar approach may be used after expanding the obstacles to compensate for the size of the robot.[2] Lozano-Pérez & Wesley (1979) attribute the visibility graph method for Euclidean shortest paths to research in 1969 by Nils Nilsson on motion planning for Shakey the robot, and also cite a 1973 description of this method by Russian mathematicians M. B. Ignat'yev, F. M. Kulakov, and A. M. Pokrovskiy.
Visibility graphs may also be used to calculate the placement of radio antennas, or as a tool used within architecture and urban planning through visibility graph analysis.
The visibility graph of a set of locations that lie in a line can be interpreted as a graph-theoretical representation of a time series.[3] This particular case builds a bridge between time series, dynamical systems and graph theory.
Characterization
The visibility graph of a simple polygon has the polygon's vertices as its point locations, and the exterior of the polygon as the only obstacle. Visibility graphs of simple polygons must be Hamiltonian graphs: the boundary of the polygon forms a Hamiltonian cycle in the visibility graph. It is known that not all visibility graphs induce a simple polygon. However, an efficient algorithmic characterization of the visibility graphs of simple polygons remains unknown. These graphs do not fall into many known families of well-structured graphs: they might not be perfect graphs, circle graphs, or chordal graphs.[4] An exception to this phenomenon is that the visibility graphs of simple polygons are cop-win graphs.[5]
Related problems
The art gallery problem is the problem of finding a small set of points such that all other non-obstacle points are visible from this set. Certain forms of the art gallery problem may be interpreted as finding a dominating set in a visibility graph.
The bitangents of a system of polygons or curves are lines that touch two of them without penetrating them at their points of contact. The bitangents of a set of polygons form a subset of the visibility graph that has the polygon's vertices as its nodes and the polygons themselves as the obstacles. The visibility graph approach to the Euclidean shortest path problem may be sped up by forming a graph from the bitangents instead of using all visibility edges, since a Euclidean shortest path may only enter or leave the boundary of an obstacle along a bitangent.[6]
See also
• Visibility graph analysis
• Fuzzy architectural spatial analysis
• Space syntax
Notes
1. Niu, Hanlin; Savvaris, Al; Tsourdos, Antonios; Ji, Ze (2019). "Voronoi-Visibility Roadmap-based Path Planning Algorithm for Unmanned Surface Vehicles". Journal of Navigation. 72 (04): 850–874. doi:10.1017/S0373463318001005. ISSN 0373-4633.
2. de Berg et al. (2000), sections 5.1 and 5.3; Lozano-Pérez & Wesley (1979).
3. Lacasa, Lucas; Luque, Bartolo; Ballesteros, Fernando; Luque, Jordi; Nuño, Juan Carlos (2008). "From time series to complex networks: The visibility graph". Proceedings of the National Academy of Sciences. 105 (13): 4972–4975. arXiv:0810.0920. doi:10.1073/pnas.0709247105. PMC 2278201. PMID 18362361.
4. Ghosh, S. K. (1997-03-01). "On recognizing and characterizing visibility graphs of simple polygons". Discrete & Computational Geometry. 17 (2): 143–162. doi:10.1007/BF02770871. ISSN 0179-5376.
5. Lubiw, Anna; Snoeyink, Jack; Vosoughpour, Hamideh (2017). "Visibility graphs, dismantlability, and the cops and robbers game". Computational Geometry. 66: 14–27. arXiv:1601.01298. doi:10.1016/j.comgeo.2017.07.001. MR 3693353.
6. de Berg et al. (2000), p. 316.
References
• de Berg, Mark; van Kreveld, Marc; Overmars, Mark; Schwarzkopf, Otfried (2000), "Chapter 15: Visibility Graphs", Computational Geometry (2nd ed.), Springer-Verlag, pp. 307–317, ISBN 978-3-540-65620-3.
• Lozano-Pérez, Tomás; Wesley, Michael A. (1979), "An algorithm for planning collision-free paths among polyhedral obstacles", Communications of the ACM, 22 (10): 560–570, doi:10.1145/359156.359164, S2CID 17397594.
External links
• VisiLibity: A free open source C++ library of floating-point visibility algorithms and supporting data types. This software can be used for calculating visibility graphs of polygonal environments with polygonal holes. A Matlab interface is also included.
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Visibility (geometry)
In geometry, visibility is a mathematical abstraction of the real-life notion of visibility.
Given a set of obstacles in the Euclidean space, two points in the space are said to be visible to each other, if the line segment that joins them does not intersect any obstacles. (In the Earth's atmosphere light follows a slightly curved path that is not perfectly predictable, complicating the calculation of actual visibility.)
Computation of visibility is among the basic problems in computational geometry and has applications in computer graphics, motion planning, and other areas.
Concepts and problems
• Point visibility
• Edge visibility[1][2]
• Visibility polygon
• Weak visibility
• Art gallery problem or museum problem
• Visibility graph
• Visibility graph of vertical line segments
• Watchman route problem
• Computer graphics applications:
• Hidden surface determination
• Hidden line removal
• z-buffering
• portal engine
• Star-shaped polygon
• Kernel of a polygon
• Isovist
• Viewshed
• Zone of Visual Influence
• Painter's algorithm
References
• O'Rourke, Joseph (1987). Art Gallery Theorems and Algorithms. Oxford University Press. ISBN 0-19-503965-3.
• Ghosh, Subir Kumar (2007). Visibility Algorithms in the Plane. Cambridge University Press. ISBN 978-0-521-87574-5.
• Mark de Berg, Marc van Kreveld, Mark Overmars, and Otfried Schwarzkopf (2000). Computational Geometry (2nd revised ed.). Springer-Verlag. ISBN 3-540-65620-0. 1st edition (1987).{{cite book}}: CS1 maint: multiple names: authors list (link) Chapter 15: "Visibility graphs"
1. D. Avis and G. T. Toussaint, "An optimal algorithm for determining the visibility of a polygon from an edge," IEEE Transactions on Computers, vol. C-30, No. 12, December 1981, pp. 910-914.
2. E. Roth, G. Panin and A. Knoll, "Sampling feature points for contour tracking with graphics hardware", "In International Workshop on Vision, Modeling and Visualization (VMV)", Konstanz, Germany, October 2008.
External links
Software
• VisiLibity: A free open source C++ library of floating-point visibility algorithms and supporting data types
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Visual calculus
Visual calculus, invented by Mamikon Mnatsakanian (known as Mamikon), is an approach to solving a variety of integral calculus problems.[1] Many problems that would otherwise seem quite difficult yield to the method with hardly a line of calculation, often reminiscent of what Martin Gardner called "aha! solutions" or Roger Nelsen a proof without words.[2][3]
Description
Mamikon devised his method in 1959 while an undergraduate, first applying it to a well-known geometry problem: find the area of a ring (annulus), given the length of a chord tangent to the inner circumference. Perhaps surprisingly, no additional information is needed; the solution does not depend on the ring's inner and outer dimensions.
The traditional approach involves algebra and application of the Pythagorean theorem. Mamikon's method, however, envisions an alternate construction of the ring: first the inner circle alone is drawn, then a constant-length tangent is made to travel along its circumference, "sweeping out" the ring as it goes.
Now if all the (constant-length) tangents used in constructing the ring are translated so that their points of tangency coincide, the result is a circular disk of known radius (and easily computed area). Indeed, since the inner circle's radius is irrelevant, one could just as well have started with a circle of radius zero (a point)—and sweeping out a ring around a circle of zero radius is indistinguishable from simply rotating a line segment about one of its endpoints and sweeping out a disk.
Mamikon's insight was to recognize the equivalence of the two constructions; and because they are equivalent, they yield equal areas. Moreover, so long as it is given that the tangent length is constant, the two starting curves need not be circular—a finding not easily proven by more traditional geometric methods. This yields Mamikon's theorem:
The area of a tangent sweep is equal to the area of its tangent cluster, regardless of the shape of the original curve.
Applications
Area of a cycloid
The area of a cycloid can be calculated by considering the area between it and an enclosing rectangle. These tangents can all be clustered to form a circle. If the circle generating the cycloid has radius r then this circle also has radius r and area πr2. The area of the rectangle is 2r × 2πr = 4πr2. Therefore the area of the cycloid is 3πr2: it is 3 times the area of the generating circle.
The tangent cluster can be seen to be a circle because the cycloid is generated by a circle and the tangent to the cycloid will be at right angle to the line from the generating point to the rolling point. Thus the tangent and the line to the contact point form a right-angled triangle in the generating circle. This means that clustered together the tangents will describe the shape of the generating circle.[5]
See also
• Cavalieri's principle
• Hodograph – This is a related construct that maps the velocity of a point using a polar diagram.
• The Method of Mechanical Theorems
• Pappus's centroid theorem
• Planimeter
References
1. Visual Calculus Mamikon Mnatsakanian
2. Nelsen, Roger B. (1993). Proofs without Words, Cambridge University Press. ISBN 978-0-88385-700-7.
3. Martin Gardner (1978) Aha! Insight, W.H. Freeman & Company; ISBN 0-7167-1017-X
4. Haunsperger, Deanna; Kennedy, Stephen (2006). The Edge of the Universe: Celebrating Ten Years of Math Horizons. ISBN 9780883855553. Retrieved May 9, 2017.
5. Apostol, Mnatsakanian (2012). New Horizons in Geometry. Mathematical Association of America. ISBN 9781614442103.
External links
• ProjMath Mamikon
• Proof without Words from MathWorld
• Wolfram Interactive Demonstration of Mamikon's theorem
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Tristan Needham
Tristan Needham is a British mathematician and professor of mathematics at the University of San Francisco.
Education, career and publications
Tristan is the son of social anthropologist Rodney Needham of Oxford, England. He attended the Dragon School. Later Needham attended the University of Oxford and studied physics at Merton College, and then transferred to the Mathematical Institute where he studied under Roger Penrose. He obtained his D.Phil. in 1987 and in 1989 took up his post at University of San Francisco.[1][2]
In 1993 he published A Visual Explanation of Jensen's inequality.[3] The following year he published The Geometry of Harmonic Functions, which won the Carl B. Allendoerfer Award for 1995.[4][5]
Needham wrote the book Visual Complex Analysis, which has received positive reviews.[6] Though it is described as a "radical first course in complex analysis aimed at undergraduates", writing in Mathematical Reviews D.H. Armitage said that "the book will be appreciated most by those who already know some complex analysis."[7] In fact Douglas Hofstadter wrote "Needham's work of art with its hundreds and hundreds of beautiful figures á la Latta, brings complex analysis alive in an unprecedented manner".[8] Hofstadter had studied complex analysis at Stanford with Gordon Latta, and he recalled "Latta's amazingly precise and elegant blackboard diagrams". In 2001 a German language version, translated by Norbert Herrmann and Ina Paschen, was published by R. Oldenbourg Verlag, Munich.
In 2021, Needham published Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts (Princeton University Press)[9]. (The original title was Visual Differential Geometry.) Much of this material was already developed in the writing of Visual Complex Analysis.
See also
• Amplitwist
Bibliography
• Needham, Tristan. Visual Complex Analysis. The Clarendon Press, Oxford University Press, New York, 1997 ISBN 0-19-853447-7.[10][11]
• Needham, Tristan. Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton University Press, Princeton, 2021 ISBN 9780691203706.[12]
Notes
1. Faculty profile Archived 2012-06-07 at the Wayback Machine from University of San Francisco
2. University of San Francisco website – History of the Sciences: Changing Course.
3. Needham, Tristan (1993). "A Visual Explanation of Jensen's Inequality". The American Mathematical Monthly. 100 (8): 768–771. doi:10.2307/2324783. JSTOR 2324783.
4. Needham, Tristan (1994). "The Geometry of Harmonic Functions". Mathematics Magazine. 67 (2): 92–108. doi:10.1080/0025570X.1994.11996195. ISSN 0025-570X.
5. Allendoerfer Award from Mathematics Association of America
6. Frank A. Farris (1998) American Mathematical Monthly, 105(6):570: "Visual Complex Analysis will show you the field of complex analysis in a way you almost certainly have not seen it before".
7. Review of Visual Complex Analysis from Mathematical Reviews
8. Preface page xvi of Chris Pritchard (2003) Changing Shape of Geometry, Cambridge University Press ISBN 0521531624
9. Needham, Tristan (2021). Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts. Princeton University Press. ISBN 9780691203690.
10. Farris, Frank A. (1998-01-01). "Review of Visual Complex Analysis". The American Mathematical Monthly. 105 (6): 570–576. doi:10.2307/2589427. JSTOR 2589427.
11. Shiu, P. (1999-01-01). "Review of Visual Complex Analysis". The Mathematical Gazette. 83 (496): 182–183. doi:10.2307/3618747. JSTOR 3618747.
12. Bultheel, Adhemar (2021-01-10). "Book Review: Visual Differential Geometry and Forms (T. Needham)". MAA Publications. Mathematical Association of America (MAA). Retrieved 2022-09-03.
External links
• Tristan Needham at the Mathematics Genealogy Project
• Author website for the book Visual Complex Analysis
• Princeton University Press website for the book Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
• Author website (including Errata) for the book Visual Differential Geometry and Forms: A Mathematical Drama in Five Acts
Authority control
International
• ISNI
• VIAF
National
• France
• BnF data
• Italy
• Israel
• United States
• Japan
• Czech Republic
Academics
• MathSciNet
• Mathematics Genealogy Project
• zbMATH
Other
• IdRef
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Visual Turing Test
The Visual Turing Test,[1] it is “an operator-assisted device that produces a stochastic sequence of binary questions from a given test image”.[1] The query engine produces a sequence of questions that have unpredictable answers given the history of questions. The test is only about vision and does not require any natural language processing. The job of the human operator is to provide the correct answer to the question or reject it as ambiguous. The query generator produces questions such that they follow a “natural story line”, similar to what humans do when they look at a picture.
History
Research in computer vision dates back to the 1960s when Seymour Papert first attempted to solve the problem. This unsuccessful attempt was referred to as the Summer Vision Project. The reason why it was not successful was because computer vision is more complicated than what people think. The complexity is in alignment with the human visual system. Roughly 50% of the human brain is devoted in processing vision, which indicates that it is a difficult problem.
Later there were attempts to solve the problems with models inspired by the human brain. Perceptrons by Frank Rosenblatt, which is a form of the neural networks, was one of the first such approaches. These simple neural networks could not live up to their expectations and had certain limitations due to which they were not considered in future research.
Later with the availability of the hardware and some processing power the research shifted to image processing which involves pixel-level operations, like finding edges, de-noising images or applying filters to name a few. There was some great progress in this field but the problem of vision which was to make the machines understand the images was still not being addressed. During this time the neural networks also resurfaced as it was shown that the limitations of the perceptrons can be overcome by Multi-layer perceptrons. Also in the early 1990s convolutional neural networks were born which showed great results on digit recognition but did not scale up well on harder problems.
The late 1990s and early 2000s saw the birth of modern computer vision. One of the reasons this happened was due to the availability of key, feature extraction and representation algorithms. Features along with the already present machine learning algorithms were used to detect, localise and segment objects in Images.
While all these advancements were being made, the community felt the need to have standardised datasets and evaluation metrics so the performances can be compared. This led to the emergence of challenges like the Pascal VOC challenge and the ImageNet challenge. The availability of standard evaluation metrics and the open challenges gave directions to the research. Better algorithms were introduced for specific tasks like object detection and classification.
Visual Turing Test aims to give a new direction to the computer vision research which would lead to the introduction of systems that will be one step closer to understanding images the way humans do.
Current evaluation practices
A large number of datasets have been annotated and generalised to benchmark performances of difference classes of algorithms to assess different vision tasks (e.g., object detection/recognition) on some image domain (e.g., scene images).
One of the most famous datasets in computer vision is ImageNet which is used to assess the problem of object level Image classification. ImageNet is one of the largest annotated datasets available and has over one million images. The other important vision task is object detection and localisation which refers to detecting the object instance in the image and providing the bounding box coordinates around the object instance or segmenting the object. The most popular dataset for this task is the Pascal dataset. Similarly there are other datasets for specific tasks like the H3D[2] dataset for human pose detection, Core dataset to evaluate the quality of detected object attributes such as colour, orientation, and activity.
Having these standard datasets has helped the vision community to come up with extremely well performing algorithms for all these tasks. The next logical step is to create a larger task encompassing of these smaller subtasks. Having such a task would lead to building systems that would understand images, as understanding images would inherently involve detecting objects, localising them and segmenting them.
Details
The Visual Turing Test (VTT) unlike the Turing test has a query engine system which interrogates a computer vision system in the presence of a human co-ordinator.
It is a system that generates a random sequence of binary questions specific to the test image, such that the answer to any question k is unpredictable given the true answers to the previous k − 1 questions (also known as history of questions).
The test happens in the presence of a human operator who serves two main purposes: removing the ambiguous questions and providing the correct answers to the unambiguous questions. Given an Image infinite possible binary questions can be asked and a lot of them are bound to be ambiguous. These questions if generated by the query engine are removed by the human moderator and instead the query engine generates another question such that the answer to it is unpredictable given the history of the questions.
The aim of the Visual Turing Test is to evaluate the Image understanding of a computer system, and an important part of image understanding is the story line of the image. When humans look at an image, they do not think that there is a car at ‘x’ pixels from the left and ‘y’ pixels from the top, but instead they look at it as a story, for e.g. they might think that there is a car parked on the road, a person is exiting the car and heading towards a building. The most important elements of the story line are the objects and so to extract any story line from an image the first and the most important task is to instantiate the objects in it, and that is what the query engine does.
Query engine
The query engine is the core of the Visual Turing Test and it comprises two main parts : Vocabulary and Questions
Vocabulary
Vocabulary is a set of words that represent the elements of the images. This vocabulary when used with appropriate grammar leads to a set of questions. The grammar is defined in the next section in a way that it leads to a space of binary questions.
The vocabulary ${\mathcal {V}}$ consist of three components:
1. Types of Objects ${\mathcal {T}}$
2. Type-dependent attributes of objects ${\mathcal {A}}(t)$
3. Type-dependent relationships between two objects ${\mathcal {R}}(t,t')$
For Images of urban street scenes the types of objects include people, vehicle and buildings. Attributes refer to the properties of these objects, for e.g. female, child, wearing a hat or carrying something, for people and moving, parked, stopped, one tire visible or two tires visible for vehicles. Relationships between each pair of object classes can be either “ordered” or “unordered”. The unordered relationships may include talking, walking together and the ordered relationships include taller, closer to the camera, occluding, being occluded etc.
Additionally all of this vocabulary is used in context of rectangular image regions w \in W which allow for the localisation of objects in the image. An extremely large number of such regions are possible and this complicates the problem, so for this test, regions at specific scales are only used which include 1/16 the size of image, 1/4 the size of image, 1/2 the size of image or larger.
Questions
The question space is composed of four types of questions:
• Existence questions: The aim of the existence questions is to find new objects in the image that have not been uniquely identified previously.
They are of the form :
Qexist = 'Is there an instance of an object of type t with attributes A partially visible in region w that was not previously instantiated?'
• Uniqueness questions: A uniqueness question tries to uniquely identify an object to instantiate it.
Quniq = 'Is there a unique instance of an object of type t with attributes A partially visible in region w that was not previously instantiated?'
The uniqueness questions along with the existence questions form the instantiation questions. As mentioned earlier instantiating objects leads to other interesting questions and eventually a story line. Uniqueness questions follow the existence questions and a positive answer to it leads to instantiation of an object.
• Attribute questions: An attribute question tries to find more about the object once it has been instantiated. Such questions can query about a single attribute, conjunction of two attributes or disjunction of two attributes.
Qatt(ot) = {'Does object ot have attribute a?' , 'Does object ot have attribute a1 or attribute a2?' , 'Does object ot have attribute a1 and attribute a2?'}
• Relationship questions: Once multiple objects have been instantiated, a relationship question explores the relationship between pairs of objects.
Qrel(ot,ot') = 'Does object ot have relationship r with object ot'?'
Implementation details
As mentioned before the core of the Visual Turing Test is the query generator which generates a sequence of binary questions such that the answer to any question k is unpredictable given the correct answers to the previous k − 1 questions. This is a recursive process, given a history of questions and their correct answers, the query generator either stops because there are no more unpredictable questions, or randomly selects an unpredictable question and adds it to the history.
The question space defined earlier implicitly imposes a constraint on the flow of the questions. To make it more clear this means that the attribute and relationship questions can not precede the instantiation questions. Only when the objects have been instantiated, can they be queried about their attributes and relations to other previously instantiated objects. Thus given a history we can restrict the possible questions that can follow it, and this set of questions are referred to as the candidate questions $Q_{\text{can}}$.
The task is to choose an unpredictable question from these candidate questions such that it conforms with the question flow that we will describe in the next section. For this, find the unpredictability of every question among the candidate questions.
Let $H$ be a binary random variable, where $H(I)=1$, if the history $H$ is valid for the Image $I$ and $0$ otherwise. Let $q\in Q$ can be the proposed question, and $X_{q}$ be the answer to the question $q$.
Then, find the conditional probability of getting the answer Xq to the question q given the history H.
$P_{H}(X_{q}=x)={\frac {P\{I:H(I)=1,X_{q}(I)=x\}}{P\{I:H(I)=1\}}}$
Given this probability the measure of the unpredictability is given by:
$\rho _{H}(q)=|P_{H}(X_{Q}=1)-0.5|$
The closer $\rho _{H}(q)$ is to 0, the more unpredictable the question is. $\rho _{H}(q)$ for every question is calculated. The questions for which $\rho _{H}(q)<\epsilon $, are the set of almost unpredictable questions and the next question is randomly picked from these.
Question flow
As discussed in the previous section there is an implicit ordering in the question space, according to which the attribute questions come after the instantiation questions and the relationship questions come after the attribute questions, once multiple objects have been instantiated.
Therefore, the query engine follows a loop structure where it first instantiates an object with the existence and uniqueness questions, then queries about its attributes, and then the relationship questions are asked for that object with all the previously instantiated objects.
Look-ahead search
It is clear that the interesting questions about the attributes and the relations come after the instantiation questions, and so the query generator aims at instantiating as many objects as possible.
Instantiation questions are composed of both the existence and the uniqueness questions, but it is the uniqueness questions that actually instantiate an object if they get a positive response. So if the query generator has to randomly pick an instantiation question, it prefers to pick an unpredictable uniqueness question if present. If such a question is not present, the query generator picks an existence question such that it will lead to a uniqueness question with a high probability in the future. Thus the query generator performs a look-ahead search in this case.
Story line
An integral part of the ultimate aim of building systems that can understand images the way humans do, is the story line. Humans try to figure out a story line in the Image they see. The query generator achieves this by a continuity in the question sequences.
This means that once the object has been instantiated it tries to explore it in more details. Apart from finding its attributes and relation to the other objects, localisation is also an important step. Thus, as a next step the query generator tries to localise the object in the region it was first identified, so it restricts the set of instantiation questions to the regions within the original region.
Simplicity preference
Simplicity preference states that the query generator should pick simpler questions over the more complicated ones. Simpler questions are the ones that have fewer attributes in them. So this gives an ordering to the questions based on the number of attributes, and the query generator prefers the simpler ones.
Estimating predictability
To select the next question in the sequence, VTT has to estimate the predictability of every proposed question. This is done using the annotated training set of Images. Each Image is annotated with bounding box around the objects and labelled with the attributes, and pairs of objects are labelled with the relations.
Consider each question type separately:
1. Instantiation questions: The conditional probability estimator for instantiation questions can be represented as:
$\quad {\widehat {P}}(X_{q}=1)={\frac {\#\{I\in T,H(I)=1,X_{q}(I)=1\}}{\#\{I\in T,H(I)=1\}}}$
The question is only considered if the denominator is at least 80 images. The condition of $H(I)=1$ is very strict and may not be true for a large number of Images, as every question in the history eliminates approximately half of the candidates (Images in this case). As a result, the history is pruned and the questions which may not alter the conditional probability are eliminated. Having a shorter history lets us consider a larger number of Images for the probability estimation.
The history pruning is done in two stages:
• In the first stage all the attribute and relationship questions are removed, under the assumption that the presence and instantiation of objects only depends on other objects and not their attributes or relations. Also, all the existence questions referring to regions disjoint from the region being referred to in the proposed question, are dropped with the assumption being that the probability of the presence of an object at a location $w$ does not change with the presence or absence of objects at locations other than $w$. And finally all the uniqueness questions with a negative response referring to regions disjointed from the region being referred to in the proposed question, are dropped with the assumption that the uniqueness questions with a positive response if dropped can alter the response of the future instantiation questions. The history of questions obtained after this first stage of pruning can be referred to as $H_{q}'$.
• In the second stage an image-by-image pruning is performed. Let $q_{i}$ be a uniqueness question in $H$ that has not been pruned and is preserved in $H_{q}'$. If this question is in context of a region which is disjoint from the region being referenced in the proposed question, then the expected answer to this question will be $1$, because of the constraints in the first stage. But if the actual answer to this question for the training image is $0$, then that training image is not considered for the probability estimation, and the question $q_{i}$ is also dropped. The final history of questions after this is ${\tilde {H}}(q,I)$, and the probability is given by:
$\quad {\widehat {P}}(X_{q}=1)={\frac {\#\{I\in T,{\tilde {H}}(q,I)=1,X_{q}(I)=1\}}{\#\{I\in T,{\tilde {H}}(q,I)=1\}}}$
2. Attribute questions: The probability estimator for attribute questions is dependent on the number of labeled objects rather than the images unlike the instantiation questions.
Consider an attribute question of the form : ‘Does object ot have attribute a?’, where $o_{t}$ is an object of type $t$ and $a\in A_{t}$. Let $A$ be the set of attributes already known to belong to $o_{t}$ because of the history. Let ${\mathcal {O}}_{\mathbb {T} }$ be the set of all the annotated objects (ground truth) in the training set, and for each $o\in {\mathcal {O}}_{\mathbb {T} }$, let ${\mathcal {T}}_{\mathbb {T} }(o)$ be the type of object, and ${\mathcal {A}}_{\mathbb {T} }(o)$ be the set of attributes belonging to $o$. Then the estimator is given by:
$\quad P(X_{q}=1)={\frac {\#\{o\in {\mathcal {O}}_{\mathbb {T} }:{\mathcal {T}}_{\mathbb {T} }(o)=t,A\cup \{a\}\subseteq {\mathcal {A}}_{\mathbb {T} }(o)\}}{\#\{o\in {\mathcal {O}}_{\mathbb {T} }:{\mathcal {T}}_{\mathbb {T} }(o)=t,A\subseteq {\mathcal {A}}_{\mathbb {T} }(o)\}}}$
This is basically the ratio of the number of times the object $o$ of type $t$ with attributes $A\cup \{a\}$ occurs in the training data, to the number of times the object $o$ of type $t$ with attributes $A$ occurs in the training data. A high number of attributes in $A$ leads to a sparsity problem similar to the instantiation questions. To deal with it we partition the attributes into subsets that are approximately independent conditioned on belonging to the object $o_{t}$. For e.g. for $t={}$person, attributes like crossing a street and standing still are not independent, but both are fairly independent of the sex of the person, whether the person is child or adult, and whether they are carrying something or not. These conditional independencies reduce the size of the set $A$, and thereby overcome the problem of sparsity.
3. Relationship questions: The approach for relationship questions is the same as the attribute questions, where instead of the number of objects, number of pair of objects is considered and for the independence assumption, the relationships that are independent of the attributes of the related objects and the relationships that are independent of each other are included.
Example
Detailed example sequences can be found here.[3]
Dataset
The Images considered for the Geman et al.[1] work are that of ‘Urban street scenes’ dataset,[1] which has scenes of streets from different cities across the world. This why the types of objects are constrained to people and vehicles for this experiment.
Another dataset introduced by the Max Planck Institute for Informatics is known as DAQUAR[4][5] dataset which has real world images of indoor scenes. But they[4] propose a different version of the visual Turing test which takes on a holistic approach and expects the participating system to exhibit human like common sense.
Conclusion
This is a very recent work published on March 9, 2015, in the journal Proceedings of the National Academy of Sciences, by researchers from Brown University and Johns Hopkins University. It evaluates how the computer vision systems understand the Images as compared to humans. Currently the test is written and the interrogator is a machine because having an oral evaluation by a human interrogator gives the humans an undue advantage of being subjective, and also expects real time answers.
The Visual Turing Test is expected to give a new direction to the computer vision research. Companies like Google and Facebook are investing millions of dollars into computer vision research, and are trying to build systems that closely resemble the human visual system. Recently Facebook announced its new platform M, which looks at an image and provides a description of it to help the visually impaired.[6] Such systems might be able to perform well on the VTT.
References
1. Geman, Donald; Geman, Stuart; Hallonquist, Neil; Younes, Laurent (2015-03-24). "Visual Turing test for computer vision systems". Proceedings of the National Academy of Sciences. 112 (12): 3618–3623. Bibcode:2015PNAS..112.3618G. doi:10.1073/pnas.1422953112. ISSN 0027-8424. PMC 4378453. PMID 25755262.
2. "H3D". www.eecs.berkeley.edu. Retrieved 2015-11-19.
3. "Visual Turing Test | Division of Applied Mathematics". www.brown.edu. Retrieved 2015-11-19.
4. "Max-Planck-Institut für Informatik: Visual Turing Challenge". www.mpi-inf.mpg.de. Retrieved 2015-11-19.
5. Malinowski, Mateusz; Fritz, Mario (2014-10-29). "Towards a Visual Turing Challenge". arXiv:1410.8027 [cs.AI].
6. Metz, Cade (27 October 2015). "Facebook's AI Can Caption Photos for the Blind on Its Own". WIRED. Retrieved 2015-11-19.
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Implicit curve
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation $x^{2}+y^{2}=1$. In general, every implicit curve is defined by an equation of the form
$F(x,y)=0$
for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
If $F(x,y)$ is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation $y=f(x)$ in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by two functions x(t), y(t) both of whose functional forms are explicitly stated, and which are dependent on a common parameter $t.$
Examples of implicit curves include:
1. a line: $x+2y-3=0,$
2. a circle: $x^{2}+y^{2}-4=0,$
3. the semicubical parabola: $x^{3}-y^{2}=0,$
4. Cassini ovals $(x^{2}+y^{2})^{2}-2c^{2}(x^{2}-y^{2})-(a^{4}-c^{4})=0$ (see diagram),
5. $\sin(x+y)-\cos(xy)+1=0$ (see diagram).
The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.
The implicit function theorem describes conditions under which an equation $F(x,y)=0$ can be solved implicitly for x and/or y – that is, under which one can validly write $x=g(y)$ or $y=f(x)$. This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.
An implicit curve with an equation $F(x,y)=0$ can be considered as the level curve of level 0 of the surface $z=F(x,y)$ (see third diagram).
Slope and curvature
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
There are several possible ways to compute these quantities for a given implicit curve. One method is to use implicit differentiation to compute the derivatives of y with respect to x. Alternatively, for a curve defined by the implicit equation $F(x,y)=0$, one can express these formulas directly in terms of the partial derivatives of $F$. In what follows, the partial derivatives are denoted $F_{x}$ (for the derivative with respect to x), $F_{y}$, $F_{xx}$ (for the second partial with respect to x), $F_{xy}$ (for the mixed second partial), $F_{yy}.$
Tangent and normal vector
A curve point $(x_{0},y_{0})$ is regular if the first partial derivatives $F_{x}(x_{0},y_{0})$ and $F_{y}(x_{0},y_{0})$ are not both equal to 0.
The equation of the tangent line at a regular point $(x_{0},y_{0})$ is
$F_{x}(x_{0},y_{0})(x-x_{0})+F_{y}(x_{0},y_{0})(y-y_{0})=0,$
so the slope of the tangent line, and hence the slope of the curve at that point, is
${\text{slope}}=-{\frac {F_{x}(x_{0},y_{0})}{F_{y}(x_{0},y_{0})}}.$
If $F_{y}(x,y)=0\neq F_{x}(x,y)$ at $(x_{0},y_{0}),$ the curve is vertical at that point, while if both $F_{y}(x,y)=0$ and $F_{x}(x,y)=0$ at that point then the curve is not differentiable there, but instead is a singular point – either a cusp or a point where the curve intersects itself.
A normal vector to the curve at the point is given by
$\mathbf {n} (x_{0},y_{0})=(F_{x}(x_{0},y_{0}),F_{y}(x_{0},y_{0}))$
(here written as a row vector).
Curvature
For readability of the formulas, the arguments $(x_{0},y_{0})$ are omitted. The curvature $\kappa $ at a regular point is given by the formula
$\kappa ={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{(F_{x}^{2}+F_{y}^{2})^{3/2}}}$.[1]
Derivation of the formulas
The implicit function theorem guarantees within a neighborhood of a point $(x_{0},y_{0})$ the existence of a function $f$ such that $F(x,f(x))=0$. By the chain rule, the derivatives of function $f$ are
$f'(x)=-{\frac {F_{x}(x,f(x))}{F_{y}(x,f(x))}}$ and $f''(x)={\frac {-F_{y}^{2}F_{xx}+2F_{x}F_{y}F_{xy}-F_{x}^{2}F_{yy}}{F_{y}^{3}}}$
(where the arguments $(x,f(x))$ on the right side of the second formula are omitted for ease of reading).
Inserting the derivatives of function $f$ into the formulas for a tangent and curvature of the graph of the explicit equation $y=f(x)$ yields
$y=f(x_{0})+f'(x_{0})(x-x_{0})$ (tangent)
$\kappa (x_{0})={\frac {f''(x_{0})}{(1+f'(x_{0})^{2})^{3/2}}}$ (curvature).
Advantage and disadvantage of implicit curves
Disadvantage
The essential disadvantage of an implicit curve is the lack of an easy possibility to calculate single points which is necessary for visualization of an implicit curve (see next section).
Advantages
1. Implicit representations facilitate the computation of intersection points: If one curve is represented implicitly and the other parametrically the computation of intersection points needs only a simple (1-dimensional) Newton iteration, which is contrary to the cases implicit-implicit and parametric-parametric (see Intersection).
2. An implicit representation $F(x,y)=0$ gives the possibility of separating points not on the curve by the sign of $F(x,y)$. This may be helpful for example applying the false position method instead of a Newton iteration.
3. It is easy to generate curves which are almost geometrically similar to the given implicit curve $F(x,y)=0,$ by just adding a small number: $F(x,y)-c=0$ (see section #Smooth approximations).
Applications of implicit curves
Within mathematics implicit curves play a prominent role as algebraic curves. In addition, implicit curves are used for designing curves of desired geometrical shapes. Here are two examples.
Convex polygons
A smooth approximation of a convex polygon can be achieved in the following way: Let $g_{i}(x,y)=a_{i}x+b_{i}y+c_{i}=0,\ i=1,\dotsc ,n$ be the equations of the lines containing the edges of the polygon such that for an inner point of the polygon $g_{i}$ is positive. Then a subset of the implicit curve
$F(x,y)=g_{1}(x,y)\cdots g_{n}(x,y)-c=0$
with suitable small parameter $c$ is a smooth (differentiable) approximation of the polygon. For example, the curves
$F(x,y)=(x+1)(-x+1)y(-x-y+2)(x-y+2)-c=0$ for $c=0.03,\dotsc ,0.6$
contain smooth approximations of a polygon with 5 edges (see diagram).
Pairs of lines
In case of two lines
$F(x,y)=g_{1}(x,y)g_{2}(x,y)-c=0$
one gets
a pencil of parallel lines, if the given lines are parallel or
the pencil of hyperbolas, which have the given lines as asymptotes.
For example, the product of the coordinate axes variables yields the pencil of hyperbolas $xy-c=0,\ c\neq 0$, which have the coordinate axes as asymptotes.
Others
If one starts with simple implicit curves other than lines (circles, parabolas,...) one gets a wide range of interesting new curves. For example,
$F(x,y)=y(-x^{2}-y^{2}+1)-c=0$
(product of a circle and the x-axis) yields smooth approximations of one half of a circle (see picture), and
$F(x,y)=(-x^{2}-(y+1)^{2}+4)(-x^{2}-(y-1)^{2}+4)-c=0$
(product of two circles) yields smooth approximations of the intersection of two circles (see diagram).
Blending curves
In CAD one uses implicit curves for the generation of blending curves,[2][3] which are special curves establishing a smooth transition between two given curves. For example,
$F(x,y)=(1-\mu )f_{1}f_{2}-\mu (g_{1}g_{2})^{3}=0$
generates blending curves between the two circles
$f_{1}(x,y)=(x-x_{1})^{2}+y^{2}-r_{1}^{2}=0,$
$f_{2}(x,y)=(x-x_{2})^{2}+y^{2}-r_{2}^{2}=0.$
The method guarantees the continuity of the tangents and curvatures at the points of contact (see diagram). The two lines
$g_{1}(x,y)=x-x_{1}=0,\ g_{2}(x,y)=x-x_{2}=0$
determine the points of contact at the circles. Parameter $\mu $ is a design parameter. In the diagram, $\mu =0.05,\dotsc ,0.2$.
Equipotential curves of two point charges
Equipotential curves of two equal point charges at the points $P_{1}=(1,0),\;P_{2}=(-1,0)$ can be represented by the equation
$f(x,y)={\frac {1}{|PP_{1}|}}+{\frac {1}{|PP_{2}|}}-c$
$={\frac {1}{\sqrt {(x-1)^{2}+y^{2}}}}+{\frac {1}{\sqrt {(x+1)^{2}+y^{2}}}}-c=0.$
The curves are similar to Cassini ovals, but they are not such curves.
Visualization of an implicit curve
To visualize an implicit curve one usually determines a polygon on the curve and displays the polygon. For a parametric curve this is an easy task: One just computes the points of a sequence of parametric values. For an implicit curve one has to solve two subproblems:
1. determination of a first curve point to a given starting point in the vicinity of the curve,
2. determination of a curve point starting from a known curve point.
In both cases it is reasonable to assume $\operatorname {grad} F\neq (0,0)$. In practice this assumption is violated at single isolated points only.
Point algorithm
For the solution of both tasks mentioned above it is essential to have a computer program (which we will call ${\mathsf {CPoint}}$), which, when given a point $Q_{0}=(x_{0},y_{0})$ near an implicit curve, finds a point $P$ that is exactly on the curve:
(P1) for the start point is $j=0$
(P2) repeat
$(x_{j+1},y_{j+1})=(x_{j},y_{j})-{\frac {F(x_{j},y_{j})}{F_{x}(x_{j},y_{j})^{2}+F_{y}(x_{j},y_{j})^{2}}}\,\left(F_{x}(x_{j},y_{j}),F_{y}(x_{j},y_{j})\right)$
( Newton step for function $g(t)=F\left(x_{j}+tF_{x}(x_{j},y_{j}),y_{j}+tF_{y}(x_{j},y_{j})\right)\ .$)
(P3) until the distance between the points $(x_{j+1},y_{j+1}),\,(x_{j},y_{j})$ is small enough.
(P4) $P=(x_{j+1},y_{j+1})$ is the curve point near the start point $Q_{0}$.
Tracing algorithm
In order to generate a nearly equally spaced polygon on the implicit curve one chooses a step length $s$ and
(T1) chooses a suitable starting point in the vicinity of the curve
(T2) determines a first curve point $P_{1}$ using program ${\mathsf {CPoint}}$
(T3) determines the tangent (see above), chooses a starting point on the tangent using step length $s$ (see diagram) and determines a second curve point $P_{2}$ using program ${\mathsf {CPoint}}$ .
$\cdots $
Because the algorithm traces the implicit curve it is called a tracing algorithm. The algorithm traces only connected parts of the curve. If the implicit curve consists of several parts it has to be started several times with suitable starting points.
Raster algorithm
If the implicit curve consists of several or even unknown parts, it may be better to use a rasterisation algorithm. Instead of exactly following the curve, a raster algorithm covers the entire curve in so many points that they blend together and look like the curve.
(R1) Generate a net of points (raster) on the area of interest of the x-y-plane.
(R2) For every point $P$ in the raster, run the point algorithm ${\mathsf {CPoint}}$ starting from P, then mark its output.
If the net is dense enough, the result approximates the connected parts of the implicit curve. If for further applications polygons on the curves are needed one can trace parts of interest by the tracing algorithm.
Implicit space curves
Any space curve which is defined by two equations
${\begin{matrix}F(x,y,z)=0,\\G(x,y,z)=0\end{matrix}}$
is called an implicit space curve.
A curve point $(x_{0},y_{0},z_{0})$ is called regular if the cross product of the gradients $F$ and $G$ is not $(0,0,0)$ at this point:
$\mathbf {t} (x_{0},y_{0},z_{0})=\operatorname {grad} F(x_{0},y_{0},z_{0})\times \operatorname {grad} G(x_{0},y_{0},z_{0})\neq (0,0,0);$
otherwise it is called singular. Vector $\mathbf {t} (x_{0},y_{0},z_{0})$ is a tangent vector of the curve at point $(x_{0},y_{0},z_{0}).$
Examples:
$(1)\quad x+y+z-1=0\ ,\ x-y+z-2=0$
is a line.
$(2)\quad x^{2}+y^{2}+z^{2}-4=0\ ,\ x+y+z-1=0$
is a plane section of a sphere, hence a circle.
$(3)\quad x^{2}+y^{2}-1=0\ ,\ x+y+z-1=0$
is an ellipse (plane section of a cylinder).
$(4)\quad x^{2}+y^{2}+z^{2}-16=0\ ,\ (y-y_{0})^{2}+z^{2}-9=0$
is the intersection curve between a sphere and a cylinder.
For the computation of curve points and the visualization of an implicit space curve see Intersection.
See also
• Implicit surface
References
1. Goldman, R. (2005). "Curvature formulas for implicit curves and surfaces". Computer Aided Geometric Design. 22 (7): 632. CiteSeerX 10.1.1.413.3008. doi:10.1016/j.cagd.2005.06.005.
2. C. Hoffmann & J. Hopcroft: The potential method for blending surfaces and corners in G. Farin (Ed) Geometric-Modeling, SIAM, Philadelphia, pp. 347-365
3. E. Hartmann: Blending of implicit surfaces with functional splines, CAD,Butterworth-Heinemann, Volume 22 (8), 1990, p. 500-507
4. G. Taubin: Distance Approximations for Rastering Implicit Curves. ACM Transactions on Graphics, Vol. 13, No. 1, 1994.
• Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
• C:L: Bajaj, C.M. Hoffmann, R.E. Lynch: Tracing surface intersections, Comp. Aided Geom. Design 5 (1988), 285-307.
• Geometry and Algorithms for COMPUTER AIDED DESIGN
External links
Wikimedia Commons has media related to Implicit curves.
• Famous Curves
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Implicit surface
In mathematics, an implicit surface is a surface in Euclidean space defined by an equation
$F(x,y,z)=0.$
An implicit surface is the set of zeros of a function of three variables. Implicit means that the equation is not solved for x or y or z.
The graph of a function is usually described by an equation $z=f(x,y)$ and is called an explicit representation. The third essential description of a surface is the parametric one: $(x(s,t),y(s,t),z(s,t))$, where the x-, y- and z-coordinates of surface points are represented by three functions $x(s,t)\,,y(s,t)\,,z(s,t)$ depending on common parameters $s,t$. Generally the change of representations is simple only when the explicit representation $z=f(x,y)$ is given: $z-f(x,y)=0$ (implicit), $(s,t,f(s,t))$ (parametric).
Examples:
1. The plane $x+2y-3z+1=0.$
2. The sphere $x^{2}+y^{2}+z^{2}-4=0.$
3. The torus $(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0.$
4. A surface of genus 2: $2y(y^{2}-3x^{2})(1-z^{2})+(x^{2}+y^{2})^{2}-(9z^{2}-1)(1-z^{2})=0$ (see diagram).
5. The surface of revolution $x^{2}+y^{2}-(\ln(z+3.2))^{2}-0.02=0$ (see diagram wineglass).
For a plane, a sphere, and a torus there exist simple parametric representations. This is not true for the fourth example.
The implicit function theorem describes conditions under which an equation $F(x,y,z)=0$ can be solved (at least implicitly) for x, y or z. But in general the solution may not be made explicit. This theorem is the key to the computation of essential geometric features of a surface: tangent planes, surface normals, curvatures (see below). But they have an essential drawback: their visualization is difficult.
If $F(x,y,z)$ is polynomial in x, y and z, the surface is called algebraic. Example 5 is non-algebraic.
Despite difficulty of visualization, implicit surfaces provide relatively simple techniques to generate theoretically (e.g. Steiner surface) and practically (see below) interesting surfaces.
Formulas
Throughout the following considerations the implicit surface is represented by an equation $F(x,y,z)=0$ where function $F$ meets the necessary conditions of differentiability. The partial derivatives of $F$ are $F_{x},F_{y},F_{z},F_{xx},\ldots $.
Tangent plane and normal vector
A surface point $(x_{0},y_{0},z_{0})$ is called regular if and only if the gradient of $F$ at $(x_{0},y_{0},z_{0})$ is not the zero vector $(0,0,0)$, meaning
$(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))\neq (0,0,0)$.
If the surface point $(x_{0},y_{0},z_{0})$ is not regular, it is called singular.
The equation of the tangent plane at a regular point $(x_{0},y_{0},z_{0})$ is
$F_{x}(x_{0},y_{0},z_{0})(x-x_{0})+F_{y}(x_{0},y_{0},z_{0})(y-y_{0})+F_{z}(x_{0},y_{0},z_{0})(z-z_{0})=0,$
and a normal vector is
$\mathbf {n} (x_{0},y_{0},z_{0})=(F_{x}(x_{0},y_{0},z_{0}),F_{y}(x_{0},y_{0},z_{0}),F_{z}(x_{0},y_{0},z_{0}))^{T}.$
Normal curvature
In order to keep the formula simple the arguments $(x_{0},y_{0},z_{0})$ are omitted:
$\kappa _{n}={\frac {\mathbf {v} ^{\top }H_{F}\mathbf {v} }{\|\operatorname {grad} F\|}}$
is the normal curvature of the surface at a regular point for the unit tangent direction $\mathbf {v} $. $H_{F}$ is the Hessian matrix of $F$ (matrix of the second derivatives).
The proof of this formula relies (as in the case of an implicit curve) on the implicit function theorem and the formula for the normal curvature of a parametric surface.
Applications of implicit surfaces
As in the case of implicit curves it is an easy task to generate implicit surfaces with desired shapes by applying algebraic operations (addition, multiplication) on simple primitives.
Equipotential surface of point charges
The electrical potential of a point charge $q_{i}$ at point $\mathbf {p} _{i}=(x_{i},y_{i},z_{i})$ generates at point $\mathbf {p} =(x,y,z)$ the potential (omitting physical constants)
$F_{i}(x,y,z)={\frac {q_{i}}{\|\mathbf {p} -\mathbf {p} _{i}\|}}.$
The equipotential surface for the potential value $c$ is the implicit surface $F_{i}(x,y,z)-c=0$ which is a sphere with center at point $\mathbf {p} _{i}$.
The potential of $4$ point charges is represented by
$F(x,y,z)={\frac {q_{1}}{\|\mathbf {p} -\mathbf {p} _{1}\|}}+{\frac {q_{2}}{\|\mathbf {p} -\mathbf {p} _{2}\|}}+{\frac {q_{3}}{\|\mathbf {p} -\mathbf {p} _{3}\|}}+{\frac {q_{4}}{\|\mathbf {p} -\mathbf {p} _{4}\|}}.$
For the picture the four charges equal 1 and are located at the points $(\pm 1,\pm 1,0)$. The displayed surface is the equipotential surface (implicit surface) $F(x,y,z)-2.8=0$.
Constant distance product surface
A Cassini oval can be defined as the point set for which the product of the distances to two given points is constant (in contrast, for an ellipse the sum is constant). In a similar way implicit surfaces can be defined by a constant distance product to several fixed points.
In the diagram metamorphoses the upper left surface is generated by this rule: With
${\begin{aligned}F(x,y,z)={}&{\Big (}{\sqrt {(x-1)^{2}+y^{2}+z^{2}}}\cdot {\sqrt {(x+1)^{2}+y^{2}+z^{2}}}\\&\qquad \cdot {\sqrt {x^{2}+(y-1)^{2}+z^{2}}}\cdot {\sqrt {x^{2}+(y+1)^{2}+z^{2}}}{\Big )}\end{aligned}}$
the constant distance product surface $F(x,y,z)-1.1=0$ is displayed.
Metamorphoses of implicit surfaces
A further simple method to generate new implicit surfaces is called metamorphosis of implicit surfaces:
For two implicit surfaces $F_{1}(x,y,z)=0,F_{2}(x,y,z)=0$ (in the diagram: a constant distance product surface and a torus) one defines new surfaces using the design parameter $\mu \in [0,1]$:
$F(x,y,z)=\mu F_{1}(x,y,z)+(1-\mu )F_{2}(x,y,z)=0$
In the diagram the design parameter is successively $\mu =0,\,0.33,\,0.66,\,1$ .
Smooth approximations of several implicit surfaces
$\Pi $-surfaces [1] can be used to approximate any given smooth and bounded object in $R^{3}$ whose surface is defined by a single polynomial as a product of subsidiary polynomials. In other words, we can design any smooth object with a single algebraic surface. Let us denote the defining polynomials as $f_{i}\in \mathbb {R} [x_{1},\ldots ,x_{n}](i=1,\ldots ,k)$. Then, the approximating object is defined by the polynomial
$F(x,y,z)=\prod _{i}f_{i}(x,y,z)-r$[1]
where $r\in \mathbb {R} $ stands for the blending parameter that controls the approximating error.
Analogously to the smooth approximation with implicit curves, the equation
$F(x,y,z)=F_{1}(x,y,z)\cdot F_{2}(x,y,z)\cdot F_{3}(x,y,z)-r=0$
represents for suitable parameters $c$ smooth approximations of three intersecting tori with equations
${\begin{aligned}F_{1}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+y^{2})=0,\\[3pt]F_{2}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(x^{2}+z^{2})=0,\\[3pt]F_{3}=(x^{2}+y^{2}+z^{2}+R^{2}-a^{2})^{2}-4R^{2}(y^{2}+z^{2})=0.\end{aligned}}$
(In the diagram the parameters are $R=1,\,a=0.2,\,r=0.01.$)
Visualization of implicit surfaces
There are various algorithms for rendering implicit surfaces,[2] including the marching cubes algorithm.[3] Essentially there are two ideas for visualizing an implicit surface: One generates a net of polygons which is visualized (see surface triangulation) and the second relies on ray tracing which determines intersection points of rays with the surface.[4] The intersection points can be approximated by sphere tracing, using a signed distance function to find the distance to the surface.[5]
See also
• Implicit curve
References
1. Adriano N. Raposo; Abel J.P. Gomes (2019). "Pi-surfaces: products of implicit surfaces towards constructive composition of 3D objects". WSCG 2019 27. International Conference in Central Europe on Computer Graphics, Visualization and Computer Vision. arXiv:1906.06751.
2. Jules Bloomenthal; Chandrajit Bajaj; Brian Wyvill (15 August 1997). Introduction to Implicit Surfaces. Morgan Kaufmann. ISBN 978-1-55860-233-5.
3. Ian Stephenson (1 December 2004). Production Rendering: Design and Implementation. Springer Science & Business Media. ISBN 978-1-85233-821-3.
4. Eric Haines, Tomas Akenine-Moller: Ray Tracing Gems, Springer, 2019, ISBN 978-1-4842-4427-2
5. Hardy, Alexandre; Steeb, Willi-Hans (2008). Mathematical Tools in Computer Graphics with C# Implementations. World Scientific. ISBN 978-981-279-102-3.
Further reading
• Gomes, A., Voiculescu, I., Jorge, J., Wyvill, B., Galbraith, C.: Implicit Curves and Surfaces: Mathematics, Data Structures and Algorithms, 2009, Springer-Verlag London, ISBN 978-1-84882-405-8
• Thorpe: Elementary Topics in Differential Geometry, Springer-Verlag, New York, 1979, ISBN 0-387-90357-7
External links
• Sultanow: Implizite Flächen
• Hartmann: Geometry and Algorithms for COMPUTER AIDED DESIGN
• GEOMVIEW
• K3Dsurf: 3d surface generator
• SURF: Visualisierung algebraischer Flächen
Dimension
Dimensional spaces
• Vector space
• Euclidean space
• Affine space
• Projective space
• Free module
• Manifold
• Algebraic variety
• Spacetime
Other dimensions
• Krull
• Lebesgue covering
• Inductive
• Hausdorff
• Minkowski
• Fractal
• Degrees of freedom
Polytopes and shapes
• Hyperplane
• Hypersurface
• Hypercube
• Hyperrectangle
• Demihypercube
• Hypersphere
• Cross-polytope
• Simplex
• Hyperpyramid
Dimensions by number
• Zero
• One
• Two
• Three
• Four
• Five
• Six
• Seven
• Eight
• n-dimensions
See also
• Hyperspace
• Codimension
Category
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Vitale's random Brunn–Minkowski inequality
In mathematics, Vitale's random Brunn–Minkowski inequality is a theorem due to Richard Vitale that generalizes the classical Brunn–Minkowski inequality for compact subsets of n-dimensional Euclidean space Rn to random compact sets.
Statement of the inequality
Let X be a random compact set in Rn; that is, a Borel–measurable function from some probability space (Ω, Σ, Pr) to the space of non-empty, compact subsets of Rn equipped with the Hausdorff metric. A random vector V : Ω → Rn is called a selection of X if Pr(V ∈ X) = 1. If K is a non-empty, compact subset of Rn, let
$\|K\|=\max \left\{\left.\|v\|_{\mathbb {R} ^{n}}\right|v\in K\right\}$
and define the set-valued expectation E[X] of X to be
$\mathrm {E} [X]=\{\mathrm {E} [V]|V{\mbox{ is a selection of }}X{\mbox{ and }}\mathrm {E} \|V\|<+\infty \}.$
Note that E[X] is a subset of Rn. In this notation, Vitale's random Brunn–Minkowski inequality is that, for any random compact set X with $E[\|X\|]<+\infty $,
$\left(\mathrm {vol} _{n}\left(\mathrm {E} [X]\right)\right)^{1/n}\geq \mathrm {E} \left[\mathrm {vol} _{n}(X)^{1/n}\right],$
where "$vol_{n}$" denotes n-dimensional Lebesgue measure.
Relationship to the Brunn–Minkowski inequality
If X takes the values (non-empty, compact sets) K and L with probabilities 1 − λ and λ respectively, then Vitale's random Brunn–Minkowski inequality is simply the original Brunn–Minkowski inequality for compact sets.
References
• Gardner, Richard J. (2002). "The Brunn-Minkowski inequality" (PDF). Bull. Amer. Math. Soc. (N.S.). 39 (3): 355–405 (electronic). doi:10.1090/S0273-0979-02-00941-2.
• Vitale, Richard A. (1990). "The Brunn-Minkowski inequality for random sets". J. Multivariate Anal. 33 (2): 286–293. doi:10.1016/0047-259X(90)90052-J.
Lp spaces
Basic concepts
• Banach & Hilbert spaces
• Lp spaces
• Measure
• Lebesgue
• Measure space
• Measurable space/function
• Minkowski distance
• Sequence spaces
L1 spaces
• Integrable function
• Lebesgue integration
• Taxicab geometry
L2 spaces
• Bessel's
• Cauchy–Schwarz
• Euclidean distance
• Hilbert space
• Parseval's identity
• Polarization identity
• Pythagorean theorem
• Square-integrable function
$L^{\infty }$ spaces
• Bounded function
• Chebyshev distance
• Infimum and supremum
• Essential
• Uniform norm
Maps
• Almost everywhere
• Convergence almost everywhere
• Convergence in measure
• Function space
• Integral transform
• Locally integrable function
• Measurable function
• Symmetric decreasing rearrangement
Inequalities
• Babenko–Beckner
• Chebyshev's
• Clarkson's
• Hanner's
• Hausdorff–Young
• Hölder's
• Markov's
• Minkowski
• Young's convolution
Results
• Marcinkiewicz interpolation theorem
• Plancherel theorem
• Riemann–Lebesgue
• Riesz–Fischer theorem
• Riesz–Thorin theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Bochner space
• Fourier analysis
• Lorentz space
• Probability theory
• Quasinorm
• Real analysis
• Sobolev space
• *-algebra
• C*-algebra
• Von Neumann
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
|
Vitali convergence theorem
In real analysis and measure theory, the Vitali convergence theorem, named after the Italian mathematician Giuseppe Vitali, is a generalization of the better-known dominated convergence theorem of Henri Lebesgue. It is a characterization of the convergence in Lp in terms of convergence in measure and a condition related to uniform integrability.
Preliminary definitions
Let $(X,{\mathcal {A}},\mu )$ be a measure space, i.e. $\mu :{\mathcal {A}}\to [0,\infty ]$ :{\mathcal {A}}\to [0,\infty ]} is a set function such that $\mu (\emptyset )=0$ and $\mu $ is countably-additive. All functions considered in the sequel will be functions $f:X\to \mathbb {K} $, where $\mathbb {K} =\mathbb {R} $ or $\mathbb {C} $. We adopt the following definitions according to Bogachev's terminology.[1]
• A set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is called uniformly integrable if $\lim _{M\to +\infty }\sup _{f\in {\mathcal {F}}}\int _{\{|f|>M\}}|f|\,d\mu =0$, i.e $\forall \ \varepsilon >0,\ \exists \ M_{\varepsilon }>0:\sup _{f\in {\mathcal {F}}}\int _{\{|f|\geq M_{\varepsilon }\}}|f|\,d\mu <\varepsilon $.
• A set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is said to have uniformly absolutely continuous integrals if $\lim _{\mu (A)\to 0}\sup _{f\in {\mathcal {F}}}\int _{A}|f|\,d\mu =0$, i.e. $\forall \ \varepsilon >0,\ \exists \ \delta _{\varepsilon }>0,\ \forall \ A\in {\mathcal {A}}:\mu (A)<\delta _{\varepsilon }\Rightarrow \sup _{f\in {\mathcal {F}}}\int _{A}|f|\,d\mu <\varepsilon $. This definition is sometimes used as a definition of uniform integrability. However, it differs from the definition of uniform integrability given above.
When $\mu (X)<\infty $, a set of functions ${\mathcal {F}}\subset L^{1}(X,{\mathcal {A}},\mu )$ is uniformly integrable if and only if it is bounded in $L^{1}(X,{\mathcal {A}},\mu )$ and has uniformly absolutely continuous integrals. If, in addition, $\mu $ is atomless, then the uniform integrability is equivalent to the uniform absolute continuity of integrals.
Finite measure case
Let $(X,{\mathcal {A}},\mu )$ be a measure space with $\mu (X)<\infty $. Let $(f_{n})\subset L^{p}(X,{\mathcal {A}},\mu )$ and $f$ be an ${\mathcal {A}}$-measurable function. Then, the following are equivalent :
1. $f\in L^{p}(X,{\mathcal {A}},\mu )$ and $(f_{n})$ converges to $f$ in $L^{p}(X,{\mathcal {A}},\mu )$ ;
2. The sequence of functions $(f_{n})$ converges in $\mu $-measure to $f$ and $(|f_{n}|^{p})_{n\geq 1}$ is uniformly integrable ;
For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]
Infinite measure case
Let $(X,{\mathcal {A}},\mu )$ be a measure space and $1\leq p<\infty $. Let $(f_{n})_{n\geq 1}\subseteq L^{p}(X,{\mathcal {A}},\mu )$ and $f\in L^{p}(X,{\mathcal {A}},\mu )$. Then, $(f_{n})$ converges to $f$ in $L^{p}(X,{\mathcal {A}},\mu )$ if and only if the following holds :
1. The sequence of functions $(f_{n})$ converges in $\mu $-measure to $f$ ;
2. $(f_{n})$ has uniformly absolutely continuous integrals;
3. For every $\varepsilon >0$, there exists $X_{\varepsilon }\in {\mathcal {A}}$ such that $\mu (X_{\varepsilon })<\infty $ and $\sup _{n\geq 1}\int _{X\setminus X_{\varepsilon }}|f_{n}|^{p}\,d\mu <\varepsilon .$
When $\mu (X)<\infty $, the third condition becomes superfluous (one can simply take $X_{\varepsilon }=X$) and the first two conditions give the usual form of Lebesgue-Vitali's convergence theorem originally stated for measure spaces with finite measure. In this case, one can show that conditions 1 and 2 imply that the sequence $(|f_{n}|^{p})_{n\geq 1}$ is uniformly integrable.
Converse of the theorem
Let $(X,{\mathcal {A}},\mu )$ be measure space. Let $(f_{n})_{n\geq 1}\subseteq L^{1}(X,{\mathcal {A}},\mu )$ and assume that $\lim _{n\to \infty }\int _{A}f_{n}\,d\mu $ exists for every $A\in {\mathcal {A}}$. Then, the sequence $(f_{n})$ is bounded in $L^{1}(X,{\mathcal {A}},\mu )$ and has uniformly absolutely continuous integrals. In addition, there exists $f\in L^{1}(X,{\mathcal {A}},\mu )$ such that $\lim _{n\to \infty }\int _{A}f_{n}\,d\mu =\int _{A}f\,d\mu $ for every $A\in {\mathcal {A}}$.
When $\mu (X)<\infty $, this implies that $(f_{n})$ is uniformly integrable.
For a proof, see Bogachev's monograph "Measure Theory, Volume I".[1]
Citations
1. Bogachev, Vladimir I. (2007). Measure Theory Volume I. New York: Springer. pp. 267–271. ISBN 978-3-540-34513-8.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
|
Vitali–Carathéodory theorem
In mathematics, the Vitali–Carathéodory theorem is a result in real analysis that shows that, under the conditions stated below, integrable functions can be approximated in L1 from above and below by lower- and upper-semicontinuous functions, respectively. It is named after Giuseppe Vitali and Constantin Carathéodory.
Statement of the theorem
Let X be a locally compact Hausdorff space equipped with a Borel measure, µ, that is finite on every compact set, outer regular, and tight when restricted to any Borel set that is open or of finite mass. If f is an element of L1(µ) then, for every ε > 0, there are functions u and v on X such that u ≤ f ≤ v, u is upper-semicontinuous and bounded above, v is lower-semicontinuous and bounded below, and
$\int _{X}(v-u)\,\mathrm {d} \mu <\varepsilon .$
References
• Rudin, Walter (1986). Real and Complex Analysis (third ed.). McGraw-Hill. pp. 56–57. ISBN 978-0-07-054234-1.
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Vitali covering lemma
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.[1] The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
Vitali covering lemma
There are two basic version of the lemma, a finite version and an infinite version. Both lemmas can be proved in the general setting of a metric space, typically these results are applied to the special case of the Euclidean space $\mathbb {R} ^{d}$. In both theorems we will use the following notation: if $ B=B(x,r)$ is a ball and $c\in \mathbb {R} $, we will write $cB$ for the ball $ B(x,cr)$.
Finite version
Theorem (Finite Covering Lemma). Let $B_{1},\dots ,B_{n}$ be any finite collection of balls contained in an arbitrary metric space. Then there exists a subcollection $B_{j_{1}},B_{j_{2}},\dots ,B_{j_{m}}$ of these balls which are disjoint and satisfy
$B_{1}\cup B_{2}\cup \dots \cup B_{n}\subseteq 3B_{j_{1}}\cup 3B_{j_{2}}\cup \dots \cup 3B_{j_{m}}.$
Proof: Without loss of generality, we assume that the collection of balls is not empty; that is, n > 0. Let $B_{j_{1}}$ be the ball of largest radius. Inductively, assume that $B_{j_{1}},\dots ,B_{j_{k}}$ have been chosen. If there is some ball in $B_{1},\dots ,B_{n}$ that is disjoint from $B_{j_{1}}\cup B_{j_{2}}\cup \dots \cup B_{j_{k}}$, let $B_{j_{k+1}}$ be such ball with maximal radius (breaking ties arbitrarily), otherwise, we set m := k and terminate the inductive definition.
Now set $ X:=\bigcup _{k=1}^{m}3\,B_{j_{k}}$. It remains to show that $B_{i}\subset X$ for every $i=1,2,\dots ,n$. This is clear if $i\in \{j_{1},\dots ,j_{m}\}$. Otherwise, there necessarily is some $k\in \{1,\dots ,m\}$ such that $B_{i}$ intersects $B_{j_{k}}$ and the radius of $B_{j_{k}}$ is at least as large as that of $B_{i}$. The triangle inequality then easily implies that $B_{i}\subset 3\,B_{j_{k}}\subset X$, as needed. This completes the proof of the finite version.
Infinite version
Theorem (Infinite Covering Lemma). Let $\mathbf {F} $ be an arbitrary collection of balls in a separable metric space such that
$R:=\sup \,\{\mathrm {rad} (B):B\in \mathbf {F} \}<\infty $
where $\mathrm {rad} (B)$ denotes the radius of the ball B. Then there exists a countable sub-collection $\mathbf {G} \subset \mathbf {F} $ such that the balls of $\mathbf {G} $ are pairwise disjoint, and satisfy
$\bigcup _{B\in \mathbf {F} }B\subseteq \bigcup _{C\in \mathbf {G} }5\,C.$
And moreover, each $B\in \mathbf {F} $ intersects some $C\in \mathbf {G} $ with $B\subset 5C$.
Proof: Consider the partition of F into subcollections Fn, n ≥ 0, defined by
$\mathbf {F} _{n}=\{B\in \mathbf {F} :2^{-n-1}R<{\text{rad}}(B)\leq 2^{-n}R\}.$
That is, $ \mathbf {F} _{n}$ consists of the balls B whose radius is in (2−n−1R, 2−nR]. A sequence Gn, with Gn ⊂ Fn, is defined inductively as follows. First, set H0 = F0 and let G0 be a maximal disjoint subcollection of H0 (such a subcollection exists by Zorn's lemma). Assuming that G0,...,Gn have been selected, let
$\mathbf {H} _{n+1}=\{B\in \mathbf {F} _{n+1}:\ B\cap C=\emptyset ,\ \ \forall C\in \mathbf {G} _{0}\cup \mathbf {G} _{1}\cup \dots \cup \mathbf {G} _{n}\},$
and let Gn+1 be a maximal disjoint subcollection of Hn+1. The subcollection
$\mathbf {G} :=\bigcup _{n=0}^{\infty }\mathbf {G} _{n}$ :=\bigcup _{n=0}^{\infty }\mathbf {G} _{n}}
of F satisfies the requirements of the theorem: G is a disjoint collection, and is thus countable since the given metric space is separable. Moreover, every ball B ∈ F intersects a ball C ∈ G such that B ⊂ 5 C.
Indeed, if we are given some $B\in \mathbf {F} $, there must be some n be such that B belongs to Fn. Either B does not belong to Hn, which implies n > 0 and means that B intersects a ball from the union of G0, ..., Gn−1, or B ∈ Hn and by maximality of Gn, B intersects a ball in Gn. In any case, B intersects a ball C that belongs to the union of G0, ..., Gn. Such a ball C must have a radius larger than 2−n−1R. Since the radius of B is less than or equal to 2−nR, we can conclude by the triangle inequality that B ⊂ 5 C, as claimed. From this $\bigcup _{B\in \mathbf {F} }B\subseteq \bigcup _{C\in \mathbf {G} }5\,C$ immediately follows, completing the proof.[2]
Remarks
• In the infinite version, the initial collection of balls can be countable or uncountable. In a separable metric space, any pairwise disjoint collection of balls must be countable. In a non-separable space, the same argument shows a pairwise disjoint subfamily exists, but that family need not be countable.
• The result may fail if the radii are not bounded: consider the family of all balls centered at 0 in Rd; any disjoint subfamily consists of only one ball B, and 5 B does not contain all the balls in this family.
• The constant 5 is not optimal. If the scale c−n, c > 1, is used instead of 2−n for defining Fn, the final value is 1 + 2c instead of 5. Any constant larger than 3 gives a correct statement of the lemma, but not 3.
• Using a finer analysis, when the original collection F is a Vitali covering of a subset E of Rd, one shows that the subcollection G, defined in the above proof, covers E up to a Lebesgue-negligible set.[3]
Applications and method of use
An application of the Vitali lemma is in proving the Hardy–Littlewood maximal inequality. As in this proof, the Vitali lemma is frequently used when we are, for instance, considering the d-dimensional Lebesgue measure, $\lambda _{d}$, of a set E ⊂ Rd, which we know is contained in the union of a certain collection of balls $\{B_{j}:j\in J\}$, each of which has a measure we can more easily compute, or has a special property one would like to exploit. Hence, if we compute the measure of this union, we will have an upper bound on the measure of E. However, it is difficult to compute the measure of the union of all these balls if they overlap. By the Vitali lemma, we may choose a subcollection $\left\{B_{j}:j\in J'\right\}$ which is disjoint and such that $ \bigcup _{j\in J'}5B_{j}\supset \bigcup _{j\in J}B_{j}\supset E$. Therefore,
$\lambda _{d}(E)\leq \lambda _{d}{\biggl (}\bigcup _{j\in J}B_{j}{\biggr )}\leq \lambda _{d}{\biggl (}\bigcup _{j\in J'}5B_{j}{\biggr )}\leq \sum _{j\in J'}\lambda _{d}(5B_{j}).$
Now, since increasing the radius of a d-dimensional ball by a factor of five increases its volume by a factor of 5d, we know that
$\sum _{j\in J'}\lambda _{d}(5B_{j})=5^{d}\sum _{j\in J'}\lambda _{d}(B_{j})$
and thus
$\lambda _{d}(E)\leq 5^{d}\sum _{j\in J'}\lambda _{d}(B_{j}).$
Vitali covering theorem
In the covering theorem, the aim is to cover, up to a "negligible set", a given set E ⊆ Rd by a disjoint subcollection extracted from a Vitali covering for E : a Vitali class or Vitali covering ${\mathcal {V}}$ for E is a collection of sets such that, for every x ∈ E and δ > 0, there is a set U in the collection ${\mathcal {V}}$ such that x ∈ U and the diameter of U is non-zero and less than δ.
In the classical setting of Vitali,[1] the negligible set is a Lebesgue negligible set, but measures other than the Lebesgue measure, and spaces other than Rd have also been considered, as is shown in the relevant section below.
The following observation is useful: if ${\mathcal {V}}$ is a Vitali covering for E and if E is contained in an open set Ω ⊆ Rd, then the subcollection of sets U in ${\mathcal {V}}$ that are contained in Ω is also a Vitali covering for E.
Vitali's covering theorem for the Lebesgue measure
The next covering theorem for the Lebesgue measure λd is due to Lebesgue (1910). A collection ${\mathcal {V}}$ of measurable subsets of Rd is a regular family (in the sense of Lebesgue) if there exists a constant C such that
$\operatorname {diam} (V)^{d}\leq C\,\lambda _{d}(V)$
for every set V in the collection ${\mathcal {V}}$.
The family of cubes is an example of regular family ${\mathcal {V}}$, as is the family ${\mathcal {V}}(m)$ of rectangles in R2 such that the ratio of sides stays between m−1 and m, for some fixed m ≥ 1. If an arbitrary norm is given on Rd, the family of balls for the metric associated to the norm is another example. To the contrary, the family of all rectangles in R2 is not regular.
Theorem — Let E ⊆ Rd be a measurable set with finite Lebesgue measure, and let ${\mathcal {V}}$ be a regular family of closed subsets of Rd that is a Vitali covering for E. Then there exists a finite or countably infinite disjoint subcollection $\{U_{j}\}\subseteq {\mathcal {V}}$ such that
$\lambda _{d}{\biggl (}E\setminus \bigcup _{j}U_{j}{\biggr )}=0.$
The original result of Vitali (1908) is a special case of this theorem, in which d = 1 and ${\mathcal {V}}$ is a collection of intervals that is a Vitali covering for a measurable subset E of the real line having finite measure.
The theorem above remains true without assuming that E has finite measure. This is obtained by applying the covering result in the finite measure case, for every integer n ≥ 0, to the portion of E contained in the open annulus Ωn of points x such that n < |x| < n+1.[4]
A somewhat related covering theorem is the Besicovitch covering theorem. To each point a of a subset A ⊆ Rd, a Euclidean ball B(a, ra) with center a and positive radius ra is assigned. Then, as in the Vitali theorem, a subcollection of these balls is selected in order to cover A in a specific way. The main differences with the Vitali covering theorem are that on one hand, the disjointness requirement of Vitali is relaxed to the fact that the number Nx of the selected balls containing an arbitrary point x ∈ Rd is bounded by a constant Bd depending only upon the dimension d; on the other hand, the selected balls do cover the set A of all the given centers.[5]
Vitali's covering theorem for the Hausdorff measure
One may have a similar objective when considering Hausdorff measure instead of Lebesgue measure. The following theorem applies in that case.[6]
Theorem — Let Hs denote s-dimensional Hausdorff measure, let E ⊆ Rd be an Hs-measurable set and ${\mathcal {V}}$ a Vitali class of closed sets for E. Then there exists a (finite or countably infinite) disjoint subcollection $\{U_{j}\}\subseteq {\mathcal {V}}$ such that either
$H^{s}\left(E\setminus \bigcup _{j}U_{j}\right)=0$
or
$\sum _{j}\operatorname {diam} (U_{j})^{s}=\infty .$
Furthermore, if E has finite s-dimensional Hausdorff measure, then for any ε > 0, we may choose this subcollection {Uj} such that
$H^{s}(E)\leq \sum _{j}\mathrm {diam} (U_{j})^{s}+\varepsilon .$
This theorem implies the result of Lebesgue given above. Indeed, when s = d, the Hausdorff measure Hs on Rd coincides with a multiple of the d-dimensional Lebesgue measure. If a disjoint collection $\{U_{j}\}$ is regular and contained in a measurable region B with finite Lebesgue measure, then
$\sum _{j}\operatorname {diam} (U_{j})^{d}\leq C\sum _{j}\lambda _{d}(U_{j})\leq C\,\lambda _{d}(B)<+\infty $
which excludes the second possibility in the first assertion of the previous theorem. It follows that E is covered, up to a Lebesgue-negligible set, by the selected disjoint subcollection.
From the covering lemma to the covering theorem
The covering lemma can be used as intermediate step in the proof of the following basic form of the Vitali covering theorem.
Theorem — For every subset E of Rd and every Vitali cover of E by a collection F of closed balls, there exists a disjoint subcollection G which covers E up to a Lebesgue-negligible set.
Proof: Without loss of generality, one can assume that all balls in F are nondegenerate and have radius less than or equal to 1. By the infinite form of the covering lemma, there exists a countable disjoint subcollection $\mathbf {G} $ of F such that every ball B ∈ F intersects a ball C ∈ G for which B ⊂ 5 C. Let r > 0 be given, and let Z denote the set of points z ∈ E that are not contained in any ball from G and belong to the open ball B(r) of radius r, centered at 0. It is enough to show that Z is Lebesgue-negligible, for every given r.
Let $\mathbf {G} _{r}=\{C_{n}\}_{n}$ denote the subcollection of those balls in G that meet B(r). Note that $\mathbf {G} _{r}$ may be finite or countably infinite. Let z ∈ Z be fixed. For each N, z does not belong to the closed set $K=\bigcup _{n\leq N}C_{n}$ by the definition of Z. But by the Vitali cover property, one can find a ball B ∈ F containing z, contained in B(r), and disjoint from K. By the property of G, the ball B intersects some ball $C_{i}\in \mathbf {G} $ and is contained in $5C_{i}$. But because K and B are disjoint, we must have i > N. So $z\in 5C_{i}$ for some i > N, and therefore
$Z\subset \bigcup _{n>N}5C_{n}.$
This gives for every N the inequality
$\lambda _{d}(Z)\leq \sum _{n>N}\lambda _{d}(5C_{n})=5^{d}\sum _{n>N}\lambda _{d}(C_{n}).$
But since the balls of $\mathbf {G} _{r}$ are contained in B(r+2), and these balls are disjoint we see
$\sum _{n}\lambda _{d}(C_{n})<\infty .$
Therefore, the term on the right side of the above inequality converges to 0 as N goes to infinity, which shows that Z is negligible as needed.[7]
Infinite-dimensional spaces
The Vitali covering theorem is not valid in infinite-dimensional settings. The first result in this direction was given by David Preiss in 1979:[8] there exists a Gaussian measure γ on an (infinite-dimensional) separable Hilbert space H so that the Vitali covering theorem fails for (H, Borel(H), γ). This result was strengthened in 2003 by Jaroslav Tišer: the Vitali covering theorem in fact fails for every infinite-dimensional Gaussian measure on any (infinite-dimensional) separable Hilbert space.[9]
See also
• Besicovitch covering theorem
Notes
1. (Vitali 1908).
2. The proof given is based on (Evans & Gariepy 1992, section 1.5.1)
3. See the "From the covering lemma to the covering theorem" section of this entry.
4. See (Evans & Gariepy 1992).
5. Vitali (1908) allowed a negligible error.
6. (Falconer 1986).
7. The proof given is based on (Natanson 1955), with some notation taken from (Evans & Gariepy 1992).
8. (Preiss 1979).
9. (Tišer 2003).
References
• Evans, Lawrence C.; Gariepy, Ronald F. (1992), Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, Boca Raton, FL: CRC Press, pp. viii+268, ISBN 0-8493-7157-0, MR 1158660, Zbl 0804.28001
• Falconer, Kenneth J. (1986), The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge: Cambridge University Press, pp. xiv+162, ISBN 0-521-25694-1, MR 0867284, Zbl 0587.28004
• "Vitali theorem", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Lebesgue, Henri (1910), "Sur l'intégration des fonctions discontinues", Annales Scientifiques de l'École Normale Supérieure, 27: 361–450, doi:10.24033/asens.624, JFM 41.0457.01
• Natanson, I. P (1955), Theory of functions of a real variable, New York: Frederick Ungar Publishing Co., p. 277, MR 0067952, Zbl 0064.29102
• Preiss, David (1979), "Gaussian measures and covering theorems", Commentatione Mathematicae Universitatis Carolinae, 20 (1): 95–99, ISSN 0010-2628, MR 0526149, Zbl 0386.28015
• Stein, Elias M.; Shakarchi, Rami (2005), Real analysis. Measure theory, integration, and Hilbert spaces, Princeton Lectures in Analysis, III, Princeton, NJ: Princeton University Press, pp. xx+402, ISBN 0-691-11386-6, MR 2129625, Zbl 1081.28001
• Tišer, Jaroslav (2003), "Vitali covering theorem in Hilbert space", Transactions of the American Mathematical Society, 355 (8): 3277–3289 (electronic), doi:10.1090/S0002-9947-03-03296-3, MR 1974687, Zbl 1042.28014
• Vitali, Giuseppe (1908) [17 December 1907], "Sui gruppi di punti e sulle funzioni di variabili reali", Atti dell'Accademia delle Scienze di Torino (in Italian), 43: 75–92, JFM 39.0101.05 (Title translation) "On groups of points and functions of real variables" is the paper containing the first proof of Vitali covering theorem.
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Vitali set
In mathematics, a Vitali set is an elementary example of a set of real numbers that is not Lebesgue measurable, found by Giuseppe Vitali in 1905.[1] The Vitali theorem is the existence theorem that there are such sets. There are uncountably many Vitali sets, and their existence depends on the axiom of choice. In 1970, Robert Solovay constructed a model of Zermelo–Fraenkel set theory without the axiom of choice where all sets of real numbers are Lebesgue measurable, assuming the existence of an inaccessible cardinal (see Solovay model).[2]
Measurable sets
Certain sets have a definite 'length' or 'mass'. For instance, the interval [0, 1] is deemed to have length 1; more generally, an interval [a, b], a ≤ b, is deemed to have length b − a. If we think of such intervals as metal rods with uniform density, they likewise have well-defined masses. The set [0, 1] ∪ [2, 3] is composed of two intervals of length one, so we take its total length to be 2. In terms of mass, we have two rods of mass 1, so the total mass is 2.
There is a natural question here: if E is an arbitrary subset of the real line, does it have a 'mass' or 'total length'? As an example, we might ask what is the mass of the set of rational numbers between 0 and 1, given that the mass of the interval [0, 1] is 1. The rationals are dense in the reals, so any value between and including 0 and 1 may appear reasonable.
However the closest generalization to mass is sigma additivity, which gives rise to the Lebesgue measure. It assigns a measure of b − a to the interval [a, b], but will assign a measure of 0 to the set of rational numbers because it is countable. Any set which has a well-defined Lebesgue measure is said to be "measurable", but the construction of the Lebesgue measure (for instance using Carathéodory's extension theorem) does not make it obvious whether non-measurable sets exist. The answer to that question involves the axiom of choice.
Construction and proof
A Vitali set is a subset $V$ of the interval $[0,1]$ of real numbers such that, for each real number $r$, there is exactly one number $v\in V$ such that $v-r$ is a rational number. Vitali sets exist because the rational numbers $\mathbb {Q} $ form a normal subgroup of the real numbers $\mathbb {R} $ under addition, and this allows the construction of the additive quotient group $\mathbb {R} /\mathbb {Q} $ of these two groups which is the group formed by the cosets $r+\mathbb {Q} $ of the rational numbers as a subgroup of the real numbers under addition. This group $\mathbb {R} /\mathbb {Q} $ consists of disjoint "shifted copies" of $\mathbb {Q} $ in the sense that each element of this quotient group is a set of the form $r+\mathbb {Q} $ for some $r$ in $\mathbb {R} $. The uncountably many elements of $\mathbb {R} /\mathbb {Q} $ partition $\mathbb {R} $ into disjoint sets, and each element is dense in $\mathbb {R} $. Each element of $\mathbb {R} /\mathbb {Q} $ intersects $[0,1]$, and the axiom of choice guarantees the existence of a subset of $[0,1]$ containing exactly one representative out of each element of $\mathbb {R} /\mathbb {Q} $. A set formed this way is called a Vitali set.
Every Vitali set $V$ is uncountable, and $v-u$ is irrational for any $u,v\in V,u\neq v$.
Non-measurability
A Vitali set is non-measurable. To show this, we assume that $V$ is measurable and we derive a contradiction. Let $q_{1},q_{2},\dots $ be an enumeration of the rational numbers in $[-1,1]$ (recall that the rational numbers are countable). From the construction of $V$, note that the translated sets $V_{k}=V+q_{k}=\{v+q_{k}:v\in V\}$, $k=1,2,\dots $ are pairwise disjoint, and further note that
$[0,1]\subseteq \bigcup _{k}V_{k}\subseteq [-1,2].$
To see the first inclusion, consider any real number $r$ in $[0,1]$ and let $v$ be the representative in $V$ for the equivalence class $[r]$; then $r-v=q_{i}$ for some rational number $q_{i}$ in $[-1,1]$ which implies that $r$ is in $V_{i}$.
Apply the Lebesgue measure to these inclusions using sigma additivity:
$1\leq \sum _{k=1}^{\infty }\lambda (V_{k})\leq 3.$
Because the Lebesgue measure is translation invariant, $\lambda (V_{k})=\lambda (V)$ and therefore
$1\leq \sum _{k=1}^{\infty }\lambda (V)\leq 3.$
But this is impossible. Summing infinitely many copies of the constant $\lambda (V)$ yields either zero or infinity, according to whether the constant is zero or positive. In neither case is the sum in $[1,3]$. So $V$ cannot have been measurable after all, i.e., the Lebesgue measure $\lambda $ must not define any value for $\lambda (V)$.
See also
• Banach–Tarski paradox – Geometric theorem
• Carathéodory's criterion – necessary and sufficient condition for a measurable setPages displaying wikidata descriptions as a fallback
• Non-measurable set – Set which cannot be assigned a meaningful "volume"
• Outer measure – Mathematical function
References
1. Vitali, Giuseppe (1905). "Sul problema della misura dei gruppi di punti di una retta". Bologna, Tip. Gamberini e Parmeggiani.
2. Solovay, Robert M. (1970), "A model of set-theory in which every set of reals is Lebesgue measurable", Annals of Mathematics, Second Series, 92 (1): 1–56, doi:10.2307/1970696, ISSN 0003-486X, JSTOR 1970696, MR 0265151
Bibliography
• Herrlich, Horst (2006). Axiom of Choice. Springer. p. 120. ISBN 9783540309895.
• Vitali, Giuseppe (1905). "Sul problema della misura dei gruppi di punti di una retta". Bologna, Tip. Gamberini e Parmeggiani.
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
Real numbers
• 0.999...
• Absolute difference
• Cantor set
• Cantor–Dedekind axiom
• Completeness
• Construction
• Decidability of first-order theories
• Extended real number line
• Gregory number
• Irrational number
• Normal number
• Rational number
• Rational zeta series
• Real coordinate space
• Real line
• Tarski axiomatization
• Vitali set
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Vitali–Hahn–Saks theorem
In mathematics, the Vitali–Hahn–Saks theorem, introduced by Vitali (1907), Hahn (1922), and Saks (1933), proves that under some conditions a sequence of measures converging point-wise does so uniformly and the limit is also a measure.
Statement of the theorem
If $(S,{\mathcal {B}},m)$ is a measure space with $m(S)<\infty ,$ and a sequence $\lambda _{n}$ of complex measures. Assuming that each $\lambda _{n}$ is absolutely continuous with respect to $m,$ and that a for all $B\in {\mathcal {B}}$ the finite limits exist $\lim _{n\to \infty }\lambda _{n}(B)=\lambda (B).$ Then the absolute continuity of the $\lambda _{n}$ with respect to $m$ is uniform in $n,$ that is, $\lim _{B}m(B)=0$ implies that $\lim _{B}\lambda _{n}(B)=0$ uniformly in $n.$ Also $\lambda $ is countably additive on ${\mathcal {B}}.$
Preliminaries
Given a measure space $(S,{\mathcal {B}},m),$ a distance can be constructed on ${\mathcal {B}}_{0},$ the set of measurable sets $B\in {\mathcal {B}}$ with $m(B)<\infty .$ This is done by defining
$d(B_{1},B_{2})=m(B_{1}\Delta B_{2}),$ where $B_{1}\Delta B_{2}=(B_{1}\setminus B_{2})\cup (B_{2}\setminus B_{1})$ is the symmetric difference of the sets $B_{1},B_{2}\in {\mathcal {B}}_{0}.$
This gives rise to a metric space ${\tilde {{\mathcal {B}}_{0}}}$ by identifying two sets $B_{1},B_{2}\in {\mathcal {B}}_{0}$ when $m(B_{1}\Delta B_{2})=0.$ Thus a point ${\overline {B}}\in {\tilde {{\mathcal {B}}_{0}}}$ with representative $B\in {\mathcal {B}}_{0}$ is the set of all $B_{1}\in {\mathcal {B}}_{0}$ such that $m(B\Delta B_{1})=0.$
Proposition: ${\tilde {{\mathcal {B}}_{0}}}$ with the metric defined above is a complete metric space.
Proof: Let
$\chi _{B}(x)={\begin{cases}1,&x\in B\\0,&x\notin B\end{cases}}$
Then
$d(B_{1},B_{2})=\int _{S}|\chi _{B_{1}}(s)-\chi _{B_{2}}(x)|dm$
This means that the metric space ${\tilde {{\mathcal {B}}_{0}}}$ can be identified with a subset of the Banach space $L^{1}(S,{\mathcal {B}},m)$.
Let $B_{n}\in {\mathcal {B}}_{0}$, with
$\lim _{n,k\to \infty }d(B_{n},B_{k})=\lim _{n,k\to \infty }\int _{S}|\chi _{B_{n}}(x)-\chi _{B_{k}}(x)|dm=0$
Then we can choose a sub-sequence $\chi _{B_{n'}}$ such that $\lim _{n'\to \infty }\chi _{B_{n'}}(x)=\chi (x)$ exists almost everywhere and $\lim _{n'\to \infty }\int _{S}|\chi (x)-\chi _{B_{n'}(x)}|dm=0$. It follows that $\chi =\chi _{B_{\infty }}$ for some $B_{\infty }\in {\mathcal {B}}_{0}$ (furthermore $\chi (x)=1$ if and only if $\chi _{B_{n'}}(x)=1$ for $n'$ large enough, then we have that $B_{\infty }=\liminf _{n'\to \infty }B_{n'}={\bigcup _{n'=1}^{\infty }}\left({\bigcap _{m=n'}^{\infty }}B_{m}\right)$ the limit inferior of the sequence) and hence $\lim _{n\to \infty }d(B_{\infty },B_{n})=0.$ Therefore, ${\tilde {{\mathcal {B}}_{0}}}$ is complete.
Proof of Vitali-Hahn-Saks theorem
Each $\lambda _{n}$ defines a function ${\overline {\lambda }}_{n}({\overline {B}})$ on ${\tilde {\mathcal {B}}}$ by taking ${\overline {\lambda }}_{n}({\overline {B}})=\lambda _{n}(B)$. This function is well defined, this is it is independent on the representative $B$ of the class ${\overline {B}}$ due to the absolute continuity of $\lambda _{n}$ with respect to $m$. Moreover ${\overline {\lambda }}_{n}$ is continuous.
For every $\epsilon >0$ the set
$F_{k,\epsilon }=\{{\overline {B}}\in {\tilde {\mathcal {B}}}:\ \sup _{n\geq 1}|{\overline {\lambda }}_{k}({\overline {B}})-{\overline {\lambda }}_{k+n}({\overline {B}})|\leq \epsilon \}$
is closed in ${\tilde {\mathcal {B}}}$, and by the hypothesis $\lim _{n\to \infty }\lambda _{n}(B)=\lambda (B)$ we have that
${\tilde {\mathcal {B}}}=\bigcup _{k=1}^{\infty }F_{k,\epsilon }$
By Baire category theorem at least one $F_{k_{0},\epsilon }$ must contain a non-empty open set of ${\tilde {\mathcal {B}}}$. This means that there is ${\overline {B_{0}}}\in {\tilde {\mathcal {B}}}$ and a $\delta >0$ such that
$d(B,B_{0})<\delta $
implies $\sup _{n\geq 1}|{\overline {\lambda }}_{k_{0}}({\overline {B}})-{\overline {\lambda }}_{k_{0}+n}({\overline {B}})|\leq \epsilon $
On the other hand, any $B\in {\mathcal {B}}$ with $m(B)\leq \delta $ can be represented as $B=B_{1}\setminus B_{2}$ with $d(B_{1},B_{0})\leq \delta $ and $d(B_{2},B_{0})\leq \delta $. This can be done, for example by taking $B_{1}=B\cup B_{0}$ and $B_{2}=B_{0}\setminus (B\cap B_{0})$. Thus, if $m(B)\leq \delta $ and $k\geq k_{0}$ then
${\begin{aligned}|\lambda _{k}(B)|&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B)-\lambda _{k}(B)|\\&\leq |\lambda _{k_{0}}(B)|+|\lambda _{k_{0}}(B_{1})-\lambda _{k}(B_{1})|+|\lambda _{k_{0}}(B_{2})-\lambda _{k}(B_{2})|\\&\leq |\lambda _{k_{0}}(B)|+2\epsilon \end{aligned}}$
Therefore, by the absolute continuity of $\lambda _{k_{0}}$ with respect to $m$, and since $\epsilon $ is arbitrary, we get that $m(B)\to 0$ implies $\lambda _{n}(B)\to 0$ uniformly in $n.$ In particular, $m(B)\to 0$ implies $\lambda (B)\to 0.$
By the additivity of the limit it follows that $\lambda $ is finitely-additive. Then, since $\lim _{m(B)\to 0}\lambda (B)=0$ it follows that $\lambda $ is actually countably additive.
References
• Hahn, H. (1922), "Über Folgen linearer Operationen", Monatsh. Math. (in German), 32: 3–88, doi:10.1007/bf01696876
• Saks, Stanislaw (1933), "Addition to the Note on Some Functionals", Transactions of the American Mathematical Society, 35 (4): 965–970, doi:10.2307/1989603, JSTOR 1989603
• Vitali, G. (1907), "Sull' integrazione per serie", Rendiconti del Circolo Matematico di Palermo (in Italian), 23: 137–155, doi:10.1007/BF03013514
• Yosida, K. (1971), Functional Analysis, Springer, pp. 70–71, ISBN 0-387-05506-1
Measure theory
Basic concepts
• Absolute continuity of measures
• Lebesgue integration
• Lp spaces
• Measure
• Measure space
• Probability space
• Measurable space/function
Sets
• Almost everywhere
• Atom
• Baire set
• Borel set
• equivalence relation
• Borel space
• Carathéodory's criterion
• Cylindrical σ-algebra
• Cylinder set
• 𝜆-system
• Essential range
• infimum/supremum
• Locally measurable
• π-system
• σ-algebra
• Non-measurable set
• Vitali set
• Null set
• Support
• Transverse measure
• Universally measurable
Types of Measures
• Atomic
• Baire
• Banach
• Besov
• Borel
• Brown
• Complex
• Complete
• Content
• (Logarithmically) Convex
• Decomposable
• Discrete
• Equivalent
• Finite
• Inner
• (Quasi-) Invariant
• Locally finite
• Maximising
• Metric outer
• Outer
• Perfect
• Pre-measure
• (Sub-) Probability
• Projection-valued
• Radon
• Random
• Regular
• Borel regular
• Inner regular
• Outer regular
• Saturated
• Set function
• σ-finite
• s-finite
• Signed
• Singular
• Spectral
• Strictly positive
• Tight
• Vector
Particular measures
• Counting
• Dirac
• Euler
• Gaussian
• Haar
• Harmonic
• Hausdorff
• Intensity
• Lebesgue
• Infinite-dimensional
• Logarithmic
• Product
• Projections
• Pushforward
• Spherical measure
• Tangent
• Trivial
• Young
Maps
• Measurable function
• Bochner
• Strongly
• Weakly
• Convergence: almost everywhere
• of measures
• in measure
• of random variables
• in distribution
• in probability
• Cylinder set measure
• Random: compact set
• element
• measure
• process
• variable
• vector
• Projection-valued measure
Main results
• Carathéodory's extension theorem
• Convergence theorems
• Dominated
• Monotone
• Vitali
• Decomposition theorems
• Hahn
• Jordan
• Maharam's
• Egorov's
• Fatou's lemma
• Fubini's
• Fubini–Tonelli
• Hölder's inequality
• Minkowski inequality
• Radon–Nikodym
• Riesz–Markov–Kakutani representation theorem
Other results
• Disintegration theorem
• Lifting theory
• Lebesgue's density theorem
• Lebesgue differentiation theorem
• Sard's theorem
For Lebesgue measure
• Isoperimetric inequality
• Brunn–Minkowski theorem
• Milman's reverse
• Minkowski–Steiner formula
• Prékopa–Leindler inequality
• Vitale's random Brunn–Minkowski inequality
Applications & related
• Convex analysis
• Descriptive set theory
• Probability theory
• Real analysis
• Spectral theory
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Vitaly Khonik
Khonik Vitaly Alexandrovich (Russian: Хоник Виталий Александрович; born 17 December 1955) is a Russian physicist, doctor of physics and mathematics, professor, head of a laboratory researching the physics of non-crystalline materials, and head of the Department of General Physics at Voronezh State Pedagogical University (VSPU). He was born in Kemerovo, USSR.[1]
Vitaly Khonik
Born17 December 1955 (1955-12-17) (age 67)
CitizenshipRussian Federation
EducationDoctor of Science (physics and mathematics)
Alma materVoronezh State Technical University (VSTU)
AwardsHonored Worker in Higher Professional Education
Scientific career
InstitutionsVoronezh State Pedagogical University
Websitehosting.vspu.ac.ru/~khonik
His laboratory collaborates with the Institute of Solid State Physics of the Russian Academy of Sciences, the Institute of Physics of the Slovak Academy of Sciences, the Institut für Materialphysik in Germany and the School of Mechanics and Civil Architecture of Northwestern Polytechical University in China.
Education, academic degrees and titles
• 1994 - Professor
• 1992 - Doctor of Science (physics & mathematics), focusing on solid state physics
• 1991 - Senior researcher in solid state physics
• 1983 - Candidate for a doctoral degree in solid state physics
• 1978 - Graduated from Voronezh Polytechnic Institute (VPI), majoring in solid state physics
Employment history
• 2010 to present - Head of the Department of General Physics at VSPU
• 1992 to 2010 - Professor at VSPU
• 1992 - Associate professor at VSPU
• 1991-1992 - Associate professor at VPI
• 1985-1991 - Senior researcher at VPI
• 1984-1985 - Junior researcher at VPI
• 1981-1983 - Doctoral student at VPI
• 1978-1981 - Engineer and physicist at VPI
Academic awards
• Awarded the title "Soros Professor" in 1997, 1998 and 1999.
• Honored Worker in Higher Professional Education (2011).
International experience
• July 2019 - Visiting professor at Northwestern Polytechical University, Xi'an, China
• July 2018 - Visiting professor at Northwestern Polytechical University, Xi'an, China
• October 2016 - Visiting professor at the Institute of Physics, Chinese Academy of Sciences, Beijing, China
• August 2012 - Visiting professor at the department of physics, University of Illinois at Urbana-Champaign, USA
• May 2009 - Guest professor at the school of materials science, Harbin Institute of Technology, China
• April 2007 – Guest professor at Roskilde University, Denmark
• January 2007 to February 2007 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• January 2006 to March 2006 – Scholar of the Japanese Society for the Promotion of Science (JSPS) at the graduate school of natural science and technology of Kanazawa University, Japan
• January 2005 to February 2005 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• April 2003 to August 2003 – Visiting scholar at the physics department, University of Illinois at Urbana-Champaign, USA
• October 2002 to December 2002 – Scholar of the German Service for Academic Exchanges (DAAD), Technical University Carolo-Wilhelmina, Braunschweig, Germany
• May 1999 to April 2000 – Associate professor of the mechanical system engineering department, Kanazawa University, Kanazawa, Japan
• Visiting professor at the Institute of Experimental Physics, Slovak Academy of Sciences, Kosice, Slovakia (two to four week visits in 1996, 1998 and 2001)
International conferences and workshops
• Internal Friction and Ultrasonic Attenuation (ICIFUAS, Italy 1993, France 1996, Spain 2002)
• Mechanical Spectroscopy (Poland 2000)
• Structure of Non-Crystalline Solids (Czech Republic 1996)
• 18th International Congress on Glass (USA 1998)
• Physics of Amorphous Solids: Mechanical Properties and Plasticity (France, Les Houches, March 2010 )
• ACAM Workshop on Multiscale Modelling of Amorphous Materials: from Structure to Mechnical Properties (Dublin, Ireland, July 2011)
• 8th International Discussion Meeting on Relaxations in Complex Systems (Wisla, Poland, July 2017).
Major scientific projects
• Ministry of Education and Science of the Russian Federation, No 3.114.2014/К, "Nature of relaxation phenomena in non-crystalline metallic materials - new theoretical concepts and experiments", 2014–2016.
• Ministry of Education and Science of the Russian Federation, No 3.1310.2017/4.6, "Shear elasticity relaxation as a fundamental basis for the description and prediction of the physical properties of amorphous alloys", 2017–2019.
• Russian Science Foundation, No 20-62-46003, "Amorphous alloys: a new approach to the understanding of the defect structure and its influence on physical properties", 2020 – present.
References
1. "Professor V.A. Khonik". hosting.vspu.ac.ru. Retrieved 2020-11-22.
External links
• "List of papers". hosting.vspu.ac.ru. Retrieved 2020-11-22.
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Vittorio Francesco Stancari
Vittorio Francesco Stancari (1678 – 1709) was a professor of mathematics at the University of Bologna who undertook research into the measurement of sounds, and into optics and hydrostatics.
Vittorio Francesco Stancari
Born1678
Bologna, Papal States
Died1709
Bologna, Papal States
NationalityBolognese
OccupationMathematician
Known forMeasurement of the pitch of sounds
Career
Vittorio Francesco Stancari was born in Bologna in 1678. In 1698 he became a professor of mathematics at the University of Bologna.[1] Stancari was one of a group of young men at the University who became interested in the techniques of Cartesian geometry and differential calculus, and who engaged in experiments and astronomical observation. Others were Eustachio Manfredi, his brother Gabriele Manfredi and Giuseppe Sentenziola Verzaglia. Of these, Gabriele Manfredi developed the most advanced understanding of mathematics.[2] Stancari was awarded the chair of infinitesimal calculus in Bologna in 1708.[3] He died in Bologna in 1709, aged about 31.[1]
Work
Stancari's dissertations and manuscripts show that he applied Leibnizian calculus to problems of physics, hydrodynamics, meteorology and mechanics.[3] He was also aware of Sir Isaac Newton's Principia Mathematica, and discussed Newton in lectures before the Accademia degli Inquieti in Bologna.[4]
Stancari developed a method of measuring the pitch of sound in 1706, using foil that was excited into vibration by rotating toothed wheels.[1] Working in the observatory founded by Count Marsigli,[5] Stancari and Eustachio Manfredi discovered the comet C/1707 W1 in the evening of 25 November 1707. They described it as visible to the naked eye, white, irregular and with a short, faint tail. It had the same apparent size as Jupiter.[6]
Stancari experimented with Guillaume Amontons' air thermometer, where air in the bulb pushes up a column of mercury as it expands due to rising temperature. He discovered that the humidity of the air in the bulb had a significant effect on the readings.[7]
Bibliography
• Stancari, Vittorio Francesco; Manfredi, Eustachio (1713). Victorij Francisci Stancarij philosophiae doctoris Bononiensis et patrio archigymnasio analyticae lectoris Schedae mathematicae: post ejus obitum collectae ejusdem observationes astronomicae. Typis Jo: Petri Barbiroli sub signo Rose propè Archigymnasium. Retrieved 21 January 2013.
• Stancari, Vittorio Francesco (1733). Lettera del signor Vittorio Francesco Stancari ... in cui parla della figura del seme del GebelIndi guardata e disaminata col microscopio, dell'ovaja delle anguille, del camaleonte e suoi occhi, come le uova empiatrate poco o nulla traspirino de'fonti o pozzi osservati sulle cime de'monti. Retrieved 21 January 2013.
References
Citations
1. Stancari, Vittorio Francesco – Treccani.
2. Feingold & Brotons 2006, p. 133.
3. Olschki 1996, p. 307.
4. Feingold 2004, p. 81.
5. Dizionario biografico universale, Volume 5, by Felice Scifoni, Publisher Davide Passagli, Florence (1849); page 172.
6. Kronk 1999, p. 389.
7. Camuffo & Jones 2002, p. 299.
Sources
• Camuffo, Dario; Jones, Phil D. (31 May 2002). Improved Understanding of Past Climatic Variability from Early Daily European Instrumental Sources. Springer. ISBN 978-1-4020-0556-5. Retrieved 21 January 2013.
• Feingold, Mordechai (2004). The Newtonian moment: Isaac Newton and the making of modern culture. New York Public Library. ISBN 978-0-19-517735-0. Retrieved 21 January 2013.
• Feingold, Mordechai; Brotons, Víctor Navarro (1 January 2006). Universities And Science in the Early Modern Period. Springer. ISBN 978-1-4020-3975-1. Retrieved 21 January 2013.
• Kronk, Gary W. (28 September 1999). Cometography: Volume 1, Ancient–1799: A Catalog of Comets. Cambridge University Press. ISBN 978-0-521-58504-0. Retrieved 21 January 2013.
• Olschki, L. S. (1996). Physis; rivista internazionale di storia della scienza. Consiglio Nazionale delle Ricerche. Retrieved 21 January 2013.
• Stancari, Vittorio Francesco. Treccani. Retrieved 2013-01-21.
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Vittorio Grünwald
Vittorio Grünwald (Verona, Italy, 13 June 1855 – Florence, Italy, March 1943) was an Italian professor of mathematics and German language. His father Guglielmo (Willhelm) Grünwald (son of Aronne and Regina) was Hungarian, his mother Fortuna Marini (daughter of Mandolino Marini and Ricca Bassani) was Italian. In 1861 he moved to Hungary with his family, then came back in 1877 to Verona, later in November 1885 they moved to Brescia, and then to Venice. He studied at the Technische Universität Wien, where he graduated in mathematics. After coming back to Italy, he taught mathematics and German language in several schools (such as in Livorno and Venice), and then he settled in Florence.
Vittorio Grünwald
Vittorio Grünwald
Born(1855-06-13)13 June 1855
Verona, Italy
DiedMarch 1943 (1943-04) (aged 87)
Firenze, Italy
He married Dora Olschky, born in Berlin, and had three kids: Marta Grünwald, Beniamino (Benno) Grünwald, and Emanuele Grünwald.
He was a librarian and a teacher at the Rabbinical College of Florence. He died at 88 in Florence, a few months before Nazi's persecutions hit Jewish families in Central Italy. He published several contributions in mathematics, including a seminal work on negative numerical bases. He also published an Italian-German vocabulary.
References
• Vittorio Grünwald. Saggio di aritmetica non decimale con applicazioni del calcolo duodecima/e e trigesimale a problemi sui numeri complessi (Verona, 1884)
• Vittorio Grünwald. Intorno all'aritmetica dei sistemi numerici a base negativa con particolare riguardo al sistema numerico a base negativo-decimale per lo studio delle sue analogie coll'aritmetica ordinaria (decimale), Giornale di Matematiche di Battaglini (1885), 203-221, 367
• Vittorio Grünwald and Garibaldi Menotti Gatti, Vocabolario delle lingue Italiana e Tedesca. Ed. Belforte.
• Gianfranco Di Segni, In ricordo del prof. Vittorio Grünwald, Firenze Ebraica, Anno 25 n. 5, Settembre-Ottobre 2012.
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Vittorio Siri
Vittorio Siri or Francesco Siri (1608–1685) was an Italian mathematician, monk and historian.[1]
Vittorio Siri
Title page of the third volume of Vittorio Siri's Il Mercurio, 1652, etched by Stefano della Bella
Born
Francesco Siri
(1608-11-02)2 November 1608
Parma, Duchy of Parma
Died6 October 1685(1685-10-06) (aged 76)
Paris, Kingdom of France
Occupations
• Christian monk
• Historian
• Diplomat
Parent(s)Ottavio Siri
Maria Caterina Siri
Writing career
LanguageItalian, Latin
Notable worksIl Mercurio overo historia de' correnti tempi
Life
Siri was born in Parma, and studied at the Benedictine convent of San Giovanni Evangelista, Parma, where he pronounced his vows on December 25, 1625. At first, he specialized in geometry, and taught mathematics in Venice.[2] There he befriended the French ambassador and took a liking to political matters.[2]
In 1640, Siri published a book about the occupation of Casale Monferrato (Il politico soldato Monferrino) defending the French position. This earned him the patronage of Cardinal Richelieu, who granted him access to the French archives. Based on what he found in the archives, Siri set up to publish Il Mercurio overo historia de' correnti tempi ('Mercury, or the History of Current Times'), a monumental work in 15 volumes, published in Venice between 1644 and 1682 and translated into French by Jean Baptiste Requier.[3] Besides the Mercurio Politico Siri wrote another historical work, entitled Memorie Recondite, which fills eight volumes. Both these works contain a vast number of valuable authentic documents.[4]
In 1648 Genoese historian and polygraph Giovanni Battista Birago Avogadro offended Siri by publishing a survey of Europe in the year 1642 which he called Mercurio veridico, an undisguised slight of the latter's Mercurio, whose second volume appeared that same year. The affront was answered by Siri in 1653 with a whole book that enumerated Birago's mistakes and charged him with dishonesty (Bollo di D. Vittorio Siri).[5]
Cardinal Mazarin honored Siri with a pension and the title of Counsellor of State, chaplain and historian of the king of France. Siri therefore moved to France in 1649 and from 1655 he lived at the court. In the meanwhile he served as the representative in France of the duke of Parma and wrote newsletters for that duke as well as for the rival duke of Modena.[6] He died in Paris on 6 October 1685.[2]
Works
• Problemata et theoremata geometrica et mecanica, Bologna, 1633.
• Siri, Vittorio (1634). Propositiones mathematicae (in Latin). Bologna: Nicolò Tebaldini.
• Il politico Soldato Monferrino, ovvero discorso politico sopra gli affari di Casale published under the pseudonym Capitano Latino Verità, Casale (Venice), 1640.
• Il Mercurio overo historia de' correnti tempi in 15 volumes in-4°, 1644–1682.
• Memorie recondite in 8 volumes in-4°, 1676-79.
Bibliography
• Affò, Ireneo (1797). Memorie degli scrittori e letterati parmigiani. Vol. V. Parma. pp. 205–336.
References
1. "Siri, Vittorio (1608–1685) in Cerl Thesaurus".
2. "Vittorio Siri in Treccani.it".
3. Mercure de Vittorio Siri, conseiller d'État et historiographe de sa majesté très chretienne, contenant l'histoire generale de l'Europe, depuis 1640 jusqu'en 1655, Didot, Paris (voll. 1-2), Durand, Paris (voll. 3-18) 1756-1759.
4. Jean Le Clerc. Bibliothèque Choisie. Vol. IV. p. 158.
5. Ilan Rachum (1995). "Italian Historians and the Emergence of the Term 'Revolution', 1644–1659". History. 80 (259): 197–198. doi:10.1111/j.1468-229X.1995.tb01666.x.
6. Brendan Maurice Dooley (1999). The Social History of Skepticism. Experience and Doubt in Early Modern Culture. Johns Hopkins University Press. p. 98. ISBN 978-0801861420.
External links
• Ceccarelli, Alessia (2018). "SIRI, Vittorio". Dizionario Biografico degli Italiani, Volume 92: Semino–Sisto IV (in Italian). Rome: Istituto dell'Enciclopedia Italiana. ISBN 978-8-81200032-6.
• Villani, Stefano (2001). "La prima rivoluzione inglese nelle pagine del 'Mercurio' di Vittorio Siri". L'Informazione politica in Italia (Secoli XVI-XVIII). Atti del seminario organizzato dalla Scuola Normale Superiore di Pisa e dal Dipartimento di Storia moderna e contemporanea dell'Università di Pisa. Pisa, 23 e 24 giugno 1997. Pisa: Scuola Normale Superiore: 137–172.
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Vivanti–Pringsheim theorem
The Vivanti–Pringsheim theorem is a mathematical statement in complex analysis, that determines a specific singularity for a function described by certain type of power series. The theorem was originally formulated by Giulio Vivanti in 1893 and proved in the following year by Alfred Pringsheim.
More precisely the theorem states the following:
A complex function defined by a power series
$f(z)=\sum _{n=0}^{\infty }a_{n}z^{n}$
with non-negative real coefficients $a_{n}$ and a radius of convergence $R$ has a singularity at $z=R$.
A simple example is the (complex) geometric series
$f(z)=\sum _{n=0}^{\infty }z^{n}={\frac {1}{1-z}}$
with a singularity at $z=1$.
References
• Reinhold Remmert: The Theory of Complex Functions. Springer Science & Business Media, 1991, ISBN 9780387971957, p. 235
• I-hsiung Lin: Classical Complex Analysis: A Geometric Approach (Volume 2). World Scientific Publishing Company, 2010, ISBN 9789813101074, p. 45
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Viveka Erlandsson
Viveka Erlandsson is a Swedish[1] mathematician specialising in low-dimensional topology and geometry, and known in particular for extending the work of Maryam Mirzakhani on counting geodesics on hyperbolic manifolds.[2][3] She is a lecturer at the University of Bristol.
Education and career
Erlandsson earned a bachelor's degree in applied mathematics from San Francisco State University in 2004, and continued at the same university for a master's degree in 2006. She became a lecturer at Baruch College and Hunter College in the City University of New York system, while pursuing a doctorate in mathematics through the Graduate Center of the City University of New York, which she completed in 2013.[4] Her dissertation, The Margulis region in hyperbolic 4-space, was supervised by Ara Basmajian.[5]
After postdoctoral research at Aalto University and the University of Helsinki in Finland, she became a lecturer in mathematics at the University of Bristol in 2017.[4]
Book
Erlandsson is the coauthor of the book Geodesic Currents and Mirzakhani’s Curve Counting, with Juan Souto, to be published by Springer in 2022.[3][4]
Recognition
Erlandsson is the 2021 winner of the Anne Bennett Prize of the London Mathematical Society, given to her "for her outstanding achievements in geometry and topology and her inspirational active role in promoting women mathematicians".[2][3]
References
1. Curriculum vitae (PDF), Aalto University, 2016, retrieved 2022-02-04
2. Anne Bennett Prize: citation for Viveka Erlandsson (PDF), London Mathematical Society, 2021, retrieved 2022-02-04
3. Alumna Viveka Erlandsson wins the Anne Bennett Prize from the London Mathematical Society, CUNY Graduate Center, retrieved 2022-02-04
4. Curriculum vitae, retrieved 2022-02-04
5. Viveka Erlandsson at the Mathematics Genealogy Project
External links
• Home page
• Viveka Erlandsson publications indexed by Google Scholar
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Viviane Baladi
Viviane Baladi (born 23 May 1963) is a mathematician who works as a director of research at the Centre national de la recherche scientifique (CNRS) in France. Originally Swiss, she has become a naturalized citizen of France.[1] Her research concerns dynamical systems.
Viviane Baladi
Baladi at Oberwolfach, 2009
Born (1963-05-23) 23 May 1963
Switzerland
NationalitySwiss
Alma materUniversity of Geneva
Scientific career
FieldsMathematics
Doctoral advisorJean-Pierre Eckmann
Education and career
Baladi earned master's degrees in mathematics and computer science in 1986 from the University of Geneva.[1] She stayed in Geneva for her doctoral studies, finishing a Ph.D. in 1989 under the supervision of Jean-Pierre Eckmann, with a dissertation concerning the zeta functions of dynamical systems.[2]
She worked at CNRS beginning in 1990, with a leave of absence from 1993 to 1999 when she taught at ETH Zurich and the University of Geneva. She also spent a year as a professor at the University of Copenhagen in 2012–2013.[1]
Books
She is the author of the book Positive Transfer Operators and Decay of Correlation (Advanced Series in Nonlinear Dynamics 16, World Scientific, 2000)[3] and of Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps: A Functional Approach (Ergebnisse der Mathematik und ihrer Grenzgebiete 68, Springer, 2018).[4]
Recognition
She was an invited speaker at the International Congress of Mathematicians in 2014, speaking in the section on "Dynamical Systems and Ordinary Differential Equations".[5] She became a member of the Academia Europaea in 2018.[6] Baladi was awarded the CNRS Silver Medal in 2019.[7]
References
1. Curriculum vitae: Viviane Baladi, Centre national de la recherche scientifique, retrieved 2015-10-14.
2. Viviane Baladi at the Mathematics Genealogy Project.
3. Review of Positive Transfer Operators and Decay of Correlation by Jérôme Buzzi (2001), MR1793194.
4. Reviews of Dynamical Zeta Functions and Dynamical Determinants for Hyperbolic Maps: Claudio Bonanno, MR3837132; Kazuhiro Sakai, Zbl 1405.37001
5. ICM Plenary and Invited Speakers since 1897, International Mathematical Union, retrieved 2015-10-01.
6. List of members, Academia Europaea, retrieved 2020-10-02
7. Médaille d'argent du CNRS, 26 January 2023
External links
• Home page
• Viviane Baladi publications indexed by Google Scholar
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Viviani's theorem
Viviani's theorem, named after Vincenzo Viviani, states that the sum of the distances from any interior point to the sides of an equilateral triangle equals the length of the triangle's altitude.[1] It is a theorem commonly employed in various math competitions, secondary school mathematics examinations, and has wide applicability to many problems in the real world.
Proof
This proof depends on the readily-proved proposition that the area of a triangle is half its base times its height—that is, half the product of one side with the altitude from that side.[2]
Let ABC be an equilateral triangle whose height is h and whose side is a.
Let P be any point inside the triangle, and u, s, t the distances of P from the sides. Draw a line from P to each of A, B, and C, forming three triangles PAB, PBC, and PCA.
Now, the areas of these triangles are ${\frac {u\cdot a}{2}}$, ${\frac {s\cdot a}{2}}$, and ${\frac {t\cdot a}{2}}$. They exactly fill the enclosing triangle, so the sum of these areas is equal to the area of the enclosing triangle. So we can write:
${\frac {u\cdot a}{2}}+{\frac {s\cdot a}{2}}+{\frac {t\cdot a}{2}}={\frac {h\cdot a}{2}}$
and thus
$u+s+t=h$
Q.E.D.
Converse
The converse also holds: If the sum of the distances from an interior point of a triangle to the sides is independent of the location of the point, the triangle is equilateral.[3]
Applications
Further information: Ternary plot
Viviani's theorem means that lines parallel to the sides of an equilateral triangle give coordinates for making ternary plots, such as flammability diagrams.
More generally, they allow one to give coordinates on a regular simplex in the same way.
Extensions
Parallelogram
The sum of the distances from any interior point of a parallelogram to the sides is independent of the location of the point. The converse also holds: If the sum of the distances from a point in the interior of a quadrilateral to the sides is independent of the location of the point, then the quadrilateral is a parallelogram.[3]
The result generalizes to any 2n-gon with opposite sides parallel. Since the sum of distances between any pair of opposite parallel sides is constant, it follows that the sum of all pairwise sums between the pairs of parallel sides, is also constant. The converse in general is not true, as the result holds for an equilateral hexagon, which does not necessarily have opposite sides parallel.
Regular polygon
If a polygon is regular (both equiangular and equilateral), the sum of the distances to the sides from an interior point is independent of the location of the point. Specifically, it equals n times the apothem, where n is the number of sides and the apothem is the distance from the center to a side.[3][4] However, the converse does not hold; the non-square parallelogram is a counterexample.[3]
Equiangular polygon
The sum of the distances from an interior point to the sides of an equiangular polygon does not depend on the location of the point.[1]
Convex polygon
A necessary and sufficient condition for a convex polygon to have a constant sum of distances from any interior point to the sides is that there exist three non-collinear interior points with equal sums of distances.[1]
Regular polyhedron
The sum of the distances from any point in the interior of a regular polyhedron to the sides is independent of the location of the point. However, the converse does not hold, not even for tetrahedra.[3]
References
1. Abboud, Elias (2010). "On Viviani's Theorem and its Extensions". College Mathematics Journal. 43 (3): 203–211. arXiv:0903.0753. doi:10.4169/074683410X488683. S2CID 118912287.
2. Claudi Alsina, Roger B. Nelsen: Charming Proofs: A Journey Into Elegant Mathematics. MAA 2010, ISBN 9780883853481, p. 96 (excerpt (Google), p. 96, at Google Books)
3. Chen, Zhibo; Liang, Tian (2006). "The converse of Viviani's theorem". The College Mathematics Journal. 37 (5): 390–391. doi:10.2307/27646392. JSTOR 27646392.
4. Pickover, Clifford A. (2009). The Math Book. Stirling. p. 150. ISBN 978-1402788291.
Further reading
• Gueron, Shay; Tessler, Ran (2002). "The Fermat-Steiner problem". Amer. Math. Monthly. 109 (5): 443–451. doi:10.2307/2695644. JSTOR 2695644.
• Samelson, Hans (2003). "Proof without words: Viviani's theorem with vectors". Math. Mag. 76 (3): 225. doi:10.2307/3219327. JSTOR 3219327.
• Chen, Zhibo; Liang, Tian (2006). "The converse of Viviani's theorem". The College Mathematics Journal. 37 (5): 390–391. doi:10.2307/27646392. JSTOR 27646392.
• Kawasaki, Ken-Ichiroh; Yagi, Yoshihiro; Yanagawa, Katsuya (2005). "On Viviani's theorem in three dimensions". Math. Gaz. 89 (515): 283–287. doi:10.1017/S002555720017785X. JSTOR 3621243. S2CID 126113074.
• Zhou, Li (2012). "Viviani polytopes and Fermat Points". Coll. Math. J. 43 (4): 309–312. arXiv:1008.1236. CiteSeerX 10.1.1.740.7670. doi:10.4169/college.math.j.43.4.309. S2CID 117039483.
External links
• Weisstein, Eric W. "Viviani's Theorem". MathWorld.
• Li Zhou, Viviani Polytopes and Fermat Points
• "Viviani's Theorem: What is it?". at Cut the knot.
• Warendorff, Jay. "Viviani's Theorem". the Wolfram Demonstrations Project.
• "A variation of Viviani's theorem & some generalizations". at Dynamic Geometry Sketches, an interactive dynamic geometry sketch.
• Abboud, Elias (2017). "Loci of points inspired by Viviani's theorem". arXiv:1701.07339 [math.HO].
• Armstrong, Addie; McQuillan, Dan (2017). "Specialization, generalization, and a new proof of Viviani's theorem". arXiv:1701.01344 [math.HO].
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