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https://en.wikipedia.org/wiki/Steffensen%27s%20inequality | Steffensen's inequality is an equation in mathematics named after Johan Frederik Steffensen.
It is an integral inequality in real analysis, stating:
If ƒ : [a, b] → R is a non-negative, monotonically decreasing, integrable function
and g : [a, b] → [0, 1] is another integrable function, then
where
References
External links
inequalities
real analysis |
https://en.wikipedia.org/wiki/Strassmann%27s%20theorem | In mathematics, Strassmann's theorem is a result in field theory. It states that, for suitable fields, suitable formal power series with coefficients in the valuation ring of the field have only finitely many zeroes.
History
It was introduced by .
Statement of the theorem
Let K be a field with a non-Archimedean absolute value | · | and let R be the valuation ring of K. Let f(x) be a formal power series with coefficients in R other than the zero series, with coefficients an converging to zero with respect to | · |. Then f(x) has only finitely many zeroes in R. More precisely, the number of zeros is at most N, where N is the largest index with |aN| = max |an|.
As a corollary, there is no analogue of Euler's identity, e2πi = 1, in Cp, the field of p-adic complex numbers.
See also
p-adic exponential function
References
External links
Field (mathematics)
Theorems in abstract algebra |
https://en.wikipedia.org/wiki/Spijker%27s%20lemma | In mathematics, Spijker's lemma is a result in the theory of rational mappings of the Riemann sphere. It states that the image of a circle under a complex rational map with numerator and denominator having degree at most n has length at most 2nπ.
Applications
Spijker's lemma can be used to derive a sharp bound version of Kreiss matrix theorem.
See also
Buffon's needle
External links
References
Theorems in complex analysis
Lemmas in analysis |
https://en.wikipedia.org/wiki/Quadrature%20of%20the%20Parabola | Quadrature of the Parabola () is a treatise on geometry, written by Archimedes in the 3rd century BC and addressed to his Alexandrian acquaintance Dositheus. It contains 24 propositions regarding parabolas, culminating in two proofs showing that the area of a parabolic segment (the region enclosed by a parabola and a line) is that of a certain inscribed triangle.
It is one of the best-known works of Archimedes, in particular for its ingenious use of the method of exhaustion and in the second part of a geometric series. Archimedes dissects the area into infinitely many triangles whose areas form a geometric progression. He then computes the sum of the resulting geometric series, and proves that this is the area of the parabolic segment. This represents the most sophisticated use of a reductio ad absurdum argument in ancient Greek mathematics, and Archimedes' solution remained unsurpassed until the development of integral calculus in the 17th century, being succeeded by Cavalieri's quadrature formula.
Main theorem
A parabolic segment is the region bounded by a parabola and line. To find the area of a parabolic segment, Archimedes considers a certain inscribed triangle. The base of this triangle is the given chord of the parabola, and the third vertex is the point on the parabola such that the tangent to the parabola at that point is parallel to the chord. Proposition 1 of the work states that a line from the third vertex drawn parallel to the axis divides the chord into equal segments. The main theorem claims that the area of the parabolic segment is that of the inscribed triangle.
Structure of the text
Conic sections such as the parabola were already well known in Archimedes' time thanks to Menaechmus a century earlier. However, before the advent of the differential and integral calculus, there were no easy means to find the area of a conic section. Archimedes provides the first attested solution to this problem by focusing specifically on the area bounded by a parabola and a chord.
Archimedes gives two proofs of the main theorem: one using abstract mechanics and the other one by pure geometry. In the first proof, Archimedes considers a lever in equilibrium under the action of gravity, with weighted segments of a parabola and a triangle suspended along the arms of a lever at specific distances from the fulcrum. When the center of gravity of the triangle is known, the equilibrium of the lever yields the area of the parabola in terms of the area of the triangle which has the same base and equal height. Archimedes here deviates from the procedure found in On the Equilibrium of Planes in that he has the centers of gravity at a level below that of the balance. The second and more famous proof uses pure geometry, particularly the sum of a geometric series.
Of the twenty-four propositions, the first three are quoted without proof from Euclid's Elements of Conics (a lost work by Euclid on conic sections). Propositions 4 and 5 establish element |
https://en.wikipedia.org/wiki/Permutohedron | In mathematics, the permutohedron (also spelled permutahedron) of order n is an (n − 1)-dimensional polytope embedded in an n-dimensional space. Its vertex coordinates (labels) are the permutations of the first n natural numbers. The edges identify the shortest possible paths (sets of transpositions) that connect two vertices (permutations). Two permutations connected by an edge differ in only two places (one transposition), and the numbers on these places are neighbors (differ in value by 1).
The image on the right shows the permutohedron of order 4, which is the truncated octahedron. Its vertices are the 24 permutations of (1, 2, 3, 4). Parallel edges have the same edge color. The 6 edge colors correspond to the 6 possible transpositions of 4 elements, i.e. they indicate in which two places the connected permutations differ. (E.g. red edges connect permutations that differ in the last two places.)
History
According to , permutohedra were first studied by . The name permutoèdre was coined by . They describe the word as barbaric, but easy to remember, and submit it to the criticism of their readers.
The alternative spelling permutahedron is sometimes also used. Permutohedra are sometimes called permutation polytopes, but this terminology is also used for the related Birkhoff polytope, defined as the convex hull of permutation matrices. More generally, uses that term for any polytope whose vertices have a bijection with the permutations of some set.
Vertices, edges, and facets
The permutohedron of order has vertices, each of which is adjacent to others.
The number of edges is , and their length is .
Two connected vertices differ by swapping two coordinates, whose values differ by 1. The pair of swapped places corresponds to the direction of the edge.
(In the example image the vertices and are connected by a blue edge and differ by swapping 2 and 3 on the first two places. The values 2 and 3 differ by 1. All blue edges correspond to swaps of coordinates on the first two places.)
The number of facets is , because they correspond to non-empty proper subsets of .
The vertices of a facet corresponding to subset have in common, that their coordinates on places in are smaller than the rest.
More generally, the faces of dimensions 0 (vertices) to (the permutohedron itself) correspond to the strict weak orderings of the set . So the number of all faces is the -th ordered Bell number.
A face of dimension corresponds to an ordering with equivalence classes.
|}
The number of faces of dimension in the permutohedron of order is given by the triangle :
with representing the Stirling numbers of the second kind
It is shown on the right together with its row sums, the ordered Bell numbers.
Other properties
The permutohedron is vertex-transitive: the symmetric group Sn acts on the permutohedron by permutation of coordinates.
The permutohedron is a zonotope; a translated copy of the permutohedron can be generated as the Minko |
https://en.wikipedia.org/wiki/Doubly%20linked%20face%20list | In applied mathematics, a doubly linked face list (DLFL) is an efficient data structure for storing 2-manifold mesh data. The structure stores linked lists for a 3D mesh's faces, edges, vertices, and corners. The structure guarantees the preservation of the manifold property.
References
3D imaging
Applied mathematics
Linked lists |
https://en.wikipedia.org/wiki/Shot%20quality | Shot Quality is a term used in the statistical analysis of ice hockey to indicate the probability that a given shot will result in a goal, based on factors such as the distance of the shot taken, the type of shot (wrist shot, slapshot, backhand, etc.) and other factors such as the number of players on the ice for each team. It is used to isolate the impact of goaltending performance. By comparing the number of goals allowed to the total of the Shot Quality figures for each shot against, a goaltender can be rated relative to average performance across the league.
As a theoretical example, if a wrist shot from 15 feet at even strength resulted in a goal 15% of the time, it would be assigned a Shot Quality of 0.15. If a goaltender faced ten such shots in a game, then they would be expected to yield 1.5 goals (10 * 0.15). Comparing actual results against the Expected Goals figure of 1.5 yields a measure of goaltending performance that is somewhat isolated from the effects of the other players on the ice.
The concept was first publicly presented in a paper by Alan Ryder of Hockey Analytics, and has since been utilized in slightly varying fashion by a number of blog-based hockey analysts such as the Forechecker at On The Forecheck, and JavaGeek at Hockey Numbers.
Ice hockey statistics |
https://en.wikipedia.org/wiki/Cocompact%20group%20action | In mathematics, an action of a group G on a topological space X is cocompact if the quotient space X/G is a compact space. If X is locally compact, then an equivalent condition is that there is a compact subset K of X such that the image of K under the action of G covers X. It is sometimes referred to as mpact, a tongue-in-cheek reference to dual notions where prefixing with "co-" twice would "cancel out".
References
Group actions (mathematics) |
https://en.wikipedia.org/wiki/Geometric%20group%20action | In mathematics, specifically geometric group theory, a geometric group action is a certain type of action of a discrete group on a metric space.
Definition
In geometric group theory, a geometry is any proper, geodesic metric space. An action of a finitely-generated group G on a geometry X is geometric if it satisfies the following conditions:
Each element of G acts as an isometry of X.
The action is cocompact, i.e. the quotient space X/G is a compact space.
The action is properly discontinuous, with each point having a finite stabilizer.
Uniqueness
If a group G acts geometrically upon two geometries X and Y, then X and Y are quasi-isometric. Since any group acts geometrically on its own Cayley graph, any space on which G acts geometrically is quasi-isometric to the Cayley graph of G.
Examples
Cannon's conjecture states that any hyperbolic group with a 2-sphere at infinity acts geometrically on hyperbolic 3-space.
References
Geometric group theory
Discrete groups |
https://en.wikipedia.org/wiki/Out%28Fn%29 | In mathematics, Out(Fn) is the outer automorphism group of a free group on n generators. These groups play an important role in geometric group theory.
Outer space
Out(Fn) acts geometrically on a cell complex known as Culler–Vogtmann Outer space, which can be thought of as the Teichmüller space for a bouquet of circles.
Definition
A point of the outer space is essentially an -graph X homotopy equivalent to a bouquet of n circles together with a certain choice of a free homotopy class of a homotopy equivalence from X to the bouquet of n circles. An -graph is just a weighted graph with weights in . The sum of all weights should be 1 and all weights should be positive. To avoid ambiguity (and to get a finite dimensional space) it is furthermore required that the valency of each vertex should be at least 3.
A more descriptive view avoiding the homotopy equivalence f is the following. We may fix an identification of the fundamental group of the bouquet of n circles with the free group in n variables. Furthermore, we may choose a maximal tree in X and choose for each remaining edge a direction. We will now assign to each remaining edge e a word in in the following way. Consider the closed path starting with e and then going back to the origin of e in the maximal tree. Composing this path with f we get a closed path in a bouquet of n circles and hence an element in its fundamental group . This element is not well defined; if we change f by a free homotopy we obtain another element. It turns out, that those two elements are conjugate to each other, and hence we can choose the unique cyclically reduced element in this conjugacy class. It is possible to reconstruct the free homotopy type of f from these data. This view has the advantage, that it avoids the extra choice of f and has the disadvantage that additional ambiguity arises, because one has to choose a maximal tree and an orientation of the remaining edges.
The operation of Out(Fn) on the outer space is defined as follows. Every automorphism g of induces a self homotopy equivalence g′ of the bouquet of n circles. Composing f with g′ gives the desired action. And in the other model it is just application of g and making the resulting word cyclically reduced.
Connection to length functions
Every point in the outer space determines a unique length function . A word in determines via the chosen homotopy equivalence a closed path in X. The length of the word is then the minimal length of a path in the free homotopy class of that closed path. Such a length function is constant on each conjugacy class. The assignment defines an embedding of the outer space to some infinite dimensional projective space.
Simplicial structure on the outer space
In the second model an open simplex is given by all those -graphs, which have combinatorically the same underlying graph and the same edges are labeled with the same words (only the length of the edges may differ). The boundary simplices of such a simple |
https://en.wikipedia.org/wiki/Outer%20space%20%28mathematics%29 | In the mathematical subject of geometric group theory, the Culler–Vogtmann Outer space or just Outer space of a free group Fn is a topological space consisting of the so-called "marked metric graph structures" of volume 1 on Fn. The Outer space, denoted Xn or CVn, comes equipped with a natural action of the group of outer automorphisms Out(Fn) of Fn. The Outer space was introduced in a 1986 paper of Marc Culler and Karen Vogtmann, and it serves as a free group analog of the Teichmüller space of a hyperbolic surface. Outer space is used to study homology and cohomology groups of Out(Fn) and to obtain information about algebraic, geometric and dynamical properties of Out(Fn), of its subgroups and individual outer automorphisms of Fn. The space Xn can also be thought of as the set of isometry types of minimal free discrete isometric actions of Fn on Fn on R-trees T such that the quotient metric graph T/Fn has volume 1.
History
The Outer space was introduced in a 1986 paper of Marc Culler and Karen Vogtmann, inspired by analogy with the Teichmüller space of a hyperbolic surface. They showed that the natural action of on is properly discontinuous, and that is contractible.
In the same paper Culler and Vogtmann constructed an embedding, via the translation length functions discussed below, of into the infinite-dimensional projective space , where is the set of nontrivial conjugacy classes of elements of . They also proved that the closure of in is compact.
Later a combination of the results of Cohen and Lustig and of Bestvina and Feighn identified (see Section 1.3 of ) the space with the space of projective classes of "very small" minimal isometric actions of on -trees.
Formal definition
Marked metric graphs
Let n ≥ 2. For the free group Fn fix a "rose" Rn, that is a wedge, of n circles wedged at a vertex v, and fix an isomorphism between Fn and the fundamental group 1(Rn, v) of Rn. From this point on we identify Fn and 1(Rn, v) via this isomorphism.
A marking on Fn consists of a homotopy equivalence f : Rn → Γ where Γ is a finite connected graph without degree-one and degree-two vertices. Up to a (free) homotopy, f is uniquely determined by the isomorphism f# : , that is by an isomorphism
A metric graph is a finite connected graph together with the assignment to every topological edge e of Γ of a positive real number L(e) called the length of e.
The volume of a metric graph is the sum of the lengths of its topological edges.
A marked metric graph structure on Fn consists of a marking f : Rn → Γ together with a metric graph structure L on Γ.
Two marked metric graph structures f1 : Rn → Γ1 and f2 : Rn → Γ2 are equivalent if there exists an isometry θ : Γ1 → Γ2 such that, up to free homotopy, we have θ o f1 = f2.
The Outer space Xn consists of equivalence classes of all the volume-one marked metric graph structures on Fn.
Weak topology on the Outer space
Open simplices
Let f : Rn → Γ where Γ is a marking and let k be the number |
https://en.wikipedia.org/wiki/Quasi-isometry | In mathematics, a quasi-isometry is a function between two metric spaces that respects large-scale geometry of these spaces and ignores their small-scale details. Two metric spaces are quasi-isometric if there exists a quasi-isometry between them. The property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
The concept of quasi-isometry is especially important in geometric group theory, following the work of Gromov.
Definition
Suppose that is a (not necessarily continuous) function from one metric space to a second metric space . Then is called a quasi-isometry from to if there exist constants , , and such that the following two properties both hold:
For every two points and in , the distance between their images is up to the additive constant within a factor of of their original distance. More formally:
Every point of is within the constant distance of an image point. More formally:
The two metric spaces and are called quasi-isometric if there exists a quasi-isometry from to .
A map is called a quasi-isometric embedding if it satisfies the first condition but not necessarily the second (i.e. it is coarsely Lipschitz but may fail to be coarsely surjective). In other words, if through the map, is quasi-isometric to a subspace of .
Two metric spaces M1 and M2 are said to be quasi-isometric, denoted , if there exists a quasi-isometry .
Examples
The map between the Euclidean plane and the plane with the Manhattan distance that sends every point to itself is a quasi-isometry: in it, distances are multiplied by a factor of at most . Note that there can be no isometry, since, for example, the points are of equal distance to each other in Manhattan distance, but in the Euclidean plane, there are no 4 points that are of equal distance to each other.
The map (both with the Euclidean metric) that sends every -tuple of integers to itself is a quasi-isometry: distances are preserved exactly, and every real tuple is within distance of an integer tuple. In the other direction, the discontinuous function that rounds every tuple of real numbers to the nearest integer tuple is also a quasi-isometry: each point is taken by this map to a point within distance of it, so rounding changes the distance between pairs of points by adding or subtracting at most .
Every pair of finite or bounded metric spaces is quasi-isometric. In this case, every function from one space to the other is a quasi-isometry.
Equivalence relation
If is a quasi-isometry, then there exists a quasi-isometry . Indeed, may be defined by letting be any point in the image of that is within distance of , and letting be any point in .
Since the identity map is a quasi-isometry, and the composition of two quasi-isometries is a quasi-isometry, it follows that the property of being quasi-isometric behaves like an equivalence relation on the class of metric spaces.
Use in geometric group theory
Given a finite generating set |
https://en.wikipedia.org/wiki/Quasivariety | In mathematics, a quasivariety is a class of algebraic structures generalizing the notion of variety by allowing equational conditions on the axioms defining the class.
Definition
A trivial algebra contains just one element. A quasivariety is a class K of algebras with a specified signature satisfying any of the following equivalent conditions.
1. K is a pseudoelementary class closed under subalgebras and direct products.
2. K is the class of all models of a set of quasi-identities, that is, implications of the form , where are terms built up from variables using the operation symbols of the specified signature.
3. K contains a trivial algebra and is closed under isomorphisms, subalgebras, and reduced products.
4. K contains a trivial algebra and is closed under isomorphisms, subalgebras, direct products, and ultraproducts.
Examples
Every variety is a quasivariety by virtue of an equation being a quasi-identity for which n = 0.
The cancellative semigroups form a quasivariety.
Let K be a quasivariety. Then the class of orderable algebras from K forms a quasivariety, since the preservation-of-order axioms are Horn clauses.
References
Universal algebra |
https://en.wikipedia.org/wiki/Reduct | In universal algebra and in model theory, a reduct of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The opposite of "reduct" is "expansion".
Definition
Let A be an algebraic structure (in the sense of universal algebra) or a structure in the sense of model theory, organized as a set X together with an indexed family of operations and relations φi on that set, with index set I. Then the reduct of A defined by a subset J of I is the structure consisting of the set X and J-indexed family of operations and relations whose j-th operation or relation for j ∈ J is the j-th operation or relation of A. That is, this reduct is the structure A with the omission of those operations and relations φi for which i is not in J.
A structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.
Examples
The monoid (Z, +, 0) of integers under addition is a reduct of the group (Z, +, −, 0) of integers under addition and negation, obtained by omitting negation. By contrast, the monoid (N, +, 0) of natural numbers under addition is not the reduct of any group.
Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.
References
Algebra
Mathematical relations
Model theory
Universal algebra |
https://en.wikipedia.org/wiki/Quasi-identity | In universal algebra, a quasi-identity is an implication of the form
s1 = t1 ∧ … ∧ sn = tn → s = t
where s1, ..., sn, t1, ..., tn, s, and t are terms built up from variables using the operation symbols of the specified signature.
A quasi-identity amounts to a conditional equation for which the conditions themselves are equations. Alternatively, it can be seen as a disjunction of inequations and one equation s1 ≠ t1 ∨ ... ∨ sn ≠ tn ∨ s = t—that is, as a definite Horn clause. A quasi-identity with n = 0 is an ordinary identity or equation, so quasi-identities are a generalization of identities.
See also
Quasivariety
References
Free online edition.
Universal algebra |
https://en.wikipedia.org/wiki/Maurice%20Karnaugh | Maurice Karnaugh (; October 4, 1924 – November 8, 2022) was an American physicist, mathematician, computer scientist, and inventor known for the Karnaugh map used in Boolean algebra.
Career
Karnaugh studied mathematics and physics at City College of New York (1944 to 1948) and transferred to Yale University to complete his B.Sc. (1949), M.Sc. (1950) and Ph.D. in physics with a thesis on The Theory of Magnetic Resonance and Lambda-Type Doubling in Nitric-Oxide (1952).
Karnaugh worked at Bell Labs (1952 to 1966), developing the Karnaugh map (1954) as well as patents for PCM encoding and magnetic logic circuits and coding. He later worked at IBM's Federal Systems Division in Gaithersburg (1966 to 1970) and at the IBM Thomas J. Watson Research Center (1970 to 1994), studying multistage interconnection networks.
Karnaugh was elected an IEEE Fellow in 1976, and held an adjunct position at Polytechnic University of New York (now New York University Tandon School of Engineering) at the Westchester campus from 1980 to 1999.
Personal life and death
Karnaugh was married to the former Linn Blank Weil from 1970 until his death in 2022. He had two sons, Robert Victor Karnaugh and Paul Joseph Karnaugh, from his first marriage.
Karnaugh died in The Bronx on November 8, 2022, at the age of 98.
Publications
(61 pages)
See also
List of pioneers in computer science
References
External links
Publications at DBLP
1924 births
2022 deaths
21st-century American physicists
American telecommunications engineers
Scientists at Bell Labs
Fellow Members of the IEEE
Yale University alumni
IBM employees
Polytechnic Institute of New York University faculty
Scientists from New York City |
https://en.wikipedia.org/wiki/UKSA | UKSA may refer to:
United Kingdom
UK Statistics Authority, the board which assures the quality of official statistics
UK Space Agency, an agency of the government
UKSA (maritime charity) (previously called UK Sailing Academy), in Cowes, England
UK Shareholders Association, which represents private shareholders
Elsewhere
Unió Korfbalera Sant Adrià de Besòs, the korfbal team of Sant Adrià de Besòs city, Catalonia, Spain |
https://en.wikipedia.org/wiki/Euclidean%20field | In mathematics, a Euclidean field is an ordered field for which every non-negative element is a square: that is, in implies that for some in .
The constructible numbers form a Euclidean field. It is the smallest Euclidean field, as every Euclidean field contains it as an ordered subfield. In other words, the constructible numbers form the Euclidean closure of the rational numbers.
Properties
Every Euclidean field is an ordered Pythagorean field, but the converse is not true.
If E/F is a finite extension, and E is Euclidean, then so is F. This "going-down theorem" is a consequence of the Diller–Dress theorem.
Examples
The real constructible numbers, those (signed) lengths which can be constructed from a rational segment by ruler and compass constructions, form a Euclidean field.
Every real closed field is a Euclidean field. The following examples are also real closed fields.
The real numbers with the usual operations and ordering form a Euclidean field.
The field of real algebraic numbers is a Euclidean field.
The field of hyperreal numbers is a Euclidean field.
Counterexamples
The rational numbers with the usual operations and ordering do not form a Euclidean field. For example, 2 is not a square in since the square root of 2 is irrational. By the going-down result above, no algebraic number field can be Euclidean.
The complex numbers do not form a Euclidean field since they cannot be given the structure of an ordered field.
Euclidean closure
The Euclidean closure of an ordered field is an extension of in the quadratic closure of which is maximal with respect to being an ordered field with an order extending that of . It is also the smallest subfield of the algebraic closure of that is a Euclidean field and is an ordered extension of .
References
External links
Field (mathematics) |
https://en.wikipedia.org/wiki/Feit%E2%80%93Thompson%20conjecture | In mathematics, the Feit–Thompson conjecture is a conjecture in number theory, suggested by . The conjecture states that there are no distinct prime numbers p and q such that
divides .
If the conjecture were true, it would greatly simplify the final chapter of the proof of the Feit–Thompson theorem that every finite group of odd order is solvable. A stronger conjecture that the two numbers are always coprime was disproved by with the counterexample p = 17 and q = 3313 with common factor 2pq + 1 = 112643.
It is known that the conjecture is true for q = 3 .
Informal probability arguments suggest that the "expected" number of counterexamples to the Feit–Thompson conjecture is very close to 0, suggesting that the Feit–Thompson conjecture is likely to be true.
See also
Cyclotomic polynomials
Goormaghtigh conjecture
References
External links
(This article confuses the Feit–Thompson conjecture with the stronger disproved conjecture mentioned above.)
Conjectures
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Flat%20manifold | In mathematics, a Riemannian manifold is said to be flat if its Riemann curvature tensor is everywhere zero. Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.
The universal cover of a complete flat manifold is Euclidean space. This can be used to prove the theorem of
that all compact flat manifolds are finitely covered by tori; the 3-dimensional case was proved earlier by .
Examples
The following manifolds can be endowed with a flat metric. Note that this may not be their 'standard' metric (for example, the flat metric on the 2-dimensional torus is not the metric induced by its usual embedding into ).
Dimension 1
Every one-dimensional Riemannian manifold is flat. Conversely, given that every connected one-dimensional smooth manifold is diffeomorphic to either or it is straightforward to see that every connected one-dimensional Riemannian manifold is isometric to one of the following (each with their standard Riemannian structure):
the real line
the open interval for some number
the open interval
the circle of radius for some number
Only the first and last are complete. If one includes Riemannian manifolds-with-boundary, then the half-open and closed intervals must also be included.
The simplicity of a complete description in this case could be ascribed to the fact that every one-dimensional Riemannian manifold has a smooth unit-length vector field, and that an isometry from one of the above model examples is provided by considering an integral curve.
Dimension 2
The five possibilities, up to diffeomorphism
If is a smooth two-dimensional connected complete flat Riemannian manifold, then must be diffeomorphic to the Möbius strip, or the Klein bottle. Note that the only compact possibilities are and the Klein bottle, while the only orientable possibilities are and
It takes more effort to describe the distinct complete flat Riemannian metrics on these spaces. For instance, the two factors of can have any two real numbers as their radii. These metrics are distinguished from each other by the ratio of their two radii, so this space has infinitely many different flat product metrics which are not isometric up to a scale factor. In order to talk uniformly about the five possibilities, and in particular to work concretely with the Möbius strip and the Klein bottle as abstract manifolds, it is useful to use the language of group actions.
The five possibilities, up to isometry
Given let denote the translation given by Let denote the reflection given by Given two positive numbers consider the following subgroups of the group of isometries of with its standard metric.
provided
These are all groups acting freely and properly discontinuously on and so the various coset spaces all naturally have the structure of two-dimensional complete flat Riemannian manifolds. None of them are isometric |
https://en.wikipedia.org/wiki/Freiheitssatz | In mathematics, the Freiheitssatz (German: "freedom/independence theorem": Freiheit + Satz) is a result in the presentation theory of groups, stating that certain subgroups of a one-relator group are free groups.
Statement
Consider a group presentation
given by generators and a single cyclically reduced relator . If appears in , then (according to the freiheitssatz) the subgroup of generated by is a free group, freely generated by . In other words, the only relations involving are the trivial ones.
History
The result was proposed by the German mathematician Max Dehn and proved by his student, Wilhelm Magnus, in his doctoral thesis. Although Dehn expected Magnus to find a topological proof, Magnus instead found a proof based on mathematical induction and amalgamated products of groups. Different induction-based proofs were given later by and .
Significance
The freiheitssatz has become "the cornerstone of one-relator group theory", and motivated the development of the theory of amalgamated products. It also provides an analogue, in non-commutative group theory, of certain results on vector spaces and other commutative groups.
References
Combinatorial group theory |
https://en.wikipedia.org/wiki/Sandro%20Ga%C3%BAcho | Sandro Araújo da Silva (born May 19, 1974, in Restinga Seca, Rio Grande do Sul), known as Sandro Gaúcho, is a former Brazilian football midfielder.
Club career
Club Career Statistics
Last Update 2 May 2010
External links
Soccer Terminal Profile
Living people
1974 births
Brazilian men's footballers
Grêmio Foot-Ball Porto Alegrense players
C.F. Os Belenenses players
Foolad F.C. players
Sanat Naft Abadan F.C. players
Expatriate men's footballers in Iran
Men's association football midfielders |
https://en.wikipedia.org/wiki/Apeirotope | In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many facets.
Definition
Abstract apeirotope
An abstract n-polytope is a partially ordered set P (whose elements are called faces) such that P contains a least face and a greatest face, each maximal totally ordered subset (called a flag) contains exactly n + 2 faces, P is strongly connected, and there are exactly two faces that lie strictly between a and b are two faces whose ranks differ by two. An abstract polytope is called an abstract apeirotope if it has infinitely many faces.
An abstract polytope is called regular if its automorphism group Γ(P) acts transitively on all of the flags of P.
Classification
There are two main geometric classes of apeirotope:
honeycombs in n dimensions, which completely fill an n-dimensional space.
skew apeirotopes, comprising an n-dimensional manifold in a higher space
Honeycombs
In general, a honeycomb in n dimensions is an infinite example of a polytope in n + 1 dimensions.
Tilings of the plane and close-packed space-fillings of polyhedra are examples of honeycombs in two and three dimensions respectively.
A line divided into infinitely many finite segments is an example of an apeirogon.
Skew apeirotopes
Skew apeirogons
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.
Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular skew apeirogon traces out a helical spiral and may be either left- or right-handed.
Infinite skew polyhedra
There are three regular skew apeirohedra, which look rather like polyhedral sponges:
6 squares around each vertex, Coxeter symbol {4,6|4}
4 hexagons around each vertex, Coxeter symbol {6,4|4}
6 hexagons around each vertex, Coxeter symbol {6,6|3}
There are thirty regular apeirohedra in Euclidean space. These include those listed above, as well as (in the plane) polytopes of type: {∞,3}, {∞,4}, {∞,6} and in 3-dimensional space, blends of these with either an apeirogon or a line segment, and the "pure" 3-dimensional apeirohedra (12 in number)
References
Bibliography
Multi-dimensional geometry |
https://en.wikipedia.org/wiki/%C5%81ukasz%20Bro%C5%BA | Łukasz Broź (born 17 December 1985 in Giżycko) is a Polish professional footballer who plays as a defender for Polish IV liga side Mazovia Mińsk Mazowiecki.
Career statistics
Club
1 All appearances in Ekstraklasa Cup.
2 All appearances in Polish Super Cup.
References
External links
1985 births
Living people
Polish men's footballers
Poland men's international footballers
Widzew Łódź players
Legia Warsaw players
Ekstraklasa players
I liga players
III liga players
IV liga players
People from Giżycko
Footballers from Warmian-Masurian Voivodeship
Men's association football defenders |
https://en.wikipedia.org/wiki/Banach%E2%80%93Stone%20theorem | In mathematics, the Banach–Stone theorem is a classical result in the theory of continuous functions on topological spaces, named after the mathematicians Stefan Banach and Marshall Stone.
In brief, the Banach–Stone theorem allows one to recover a compact Hausdorff space X from the Banach space structure of the space C(X) of continuous real- or complex-valued functions on X. If one is allowed to invoke the algebra structure of C(X) this is easy – we can identify X with the spectrum of C(X), the set of algebra homomorphisms into the scalar field, equipped with the weak*-topology inherited from the dual space C(X)*. The Banach-Stone theorem avoids reference to multiplicative structure by recovering X from the extreme points of the unit ball of C(X)*.
Statement
For a compact Hausdorff space X, let C(X) denote the Banach space of continuous real- or complex-valued functions on X, equipped with the supremum norm ‖·‖∞.
Given compact Hausdorff spaces X and Y, suppose T : C(X) → C(Y) is a surjective linear isometry. Then there exists a homeomorphism φ : Y → X and a function g ∈ C(Y) with
such that
The case where X and Y are compact metric spaces is due to Banach, while the extension to compact Hausdorff spaces is due to Stone. In fact, they both prove a slight generalization—they do not assume that T is linear, only that it is an isometry in the sense of metric spaces, and use the Mazur–Ulam theorem to show that T is affine, and so is a linear isometry.
Generalizations
The Banach–Stone theorem has some generalizations for vector-valued continuous functions on compact, Hausdorff topological spaces. For example, if E is a Banach space with trivial centralizer and X and Y are compact, then every linear isometry of C(X; E) onto C(Y; E) is a strong Banach–Stone map.
A similar technique has also been used to recover a space X from the extreme points of the duals of some other spaces of functions on X.
The noncommutative analog of the Banach-Stone theorem is the folklore theorem that two unital C*-algebras are isomorphic if and only if they are completely isometric (i.e., isometric at all matrix levels). Mere isometry is not enough, as shown by the existence of a C*-algebra that is not isomorphic to its opposite algebra (which trivially has the same Banach space structure).
See also
References
Theory of continuous functions
Operator theory
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Multipliers%20and%20centralizers%20%28Banach%20spaces%29 | In mathematics, multipliers and centralizers are algebraic objects in the study of Banach spaces. They are used, for example, in generalizations of the Banach–Stone theorem.
Definitions
Let (X, ‖·‖) be a Banach space over a field K (either the real or complex numbers), and let Ext(X) be the set of extreme points of the closed unit ball of the continuous dual space X∗.
A continuous linear operator T : X → X is said to be a multiplier if every point p in Ext(X) is an eigenvector for the adjoint operator T∗ : X∗ → X∗. That is, there exists a function aT : Ext(X) → K such that
making the eigenvalue corresponding to p. Given two multipliers S and T on X, S is said to be an adjoint for T if
i.e. aS agrees with aT in the real case, and with the complex conjugate of aT in the complex case.
The centralizer (or commutant) of X, denoted Z(X), is the set of all multipliers on X for which an adjoint exists.
Properties
The multiplier adjoint of a multiplier T, if it exists, is unique; the unique adjoint of T is denoted T∗.
If the field K is the real numbers, then every multiplier on X lies in the centralizer of X.
See also
Centralizer and normalizer
References
Banach spaces
Operator theory |
https://en.wikipedia.org/wiki/Brule%2C%20Alberta | Brule is a hamlet in west-central Alberta, Canada within Yellowhead County. It is located on the northwest shore of Brûlé Lake, approximately west of Hinton. It has an elevation of .
Statistics Canada recognizes Brule as a designated place.
The hamlet is located in Census Division No. 14 and in the federal riding of Yellowhead.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Brule had a population of 127 living in 53 of its 57 total private dwellings, a change of from its 2016 population of 74. With a land area of , it had a population density of in 2021.
As a designated place in the 2016 Census of Population conducted by Statistics Canada, Brule had a population of 31 living in 14 of its 19 total private dwellings, a change of from its 2011 population of 76. With a land area of , it had a population density of in 2016.
Climate
Brule has a subarctic climate (Köppen Dfc).
See also
List of communities in Alberta
List of designated places in Alberta
List of hamlets in Alberta
References
Designated places in Alberta
Hamlets in Alberta
Yellowhead County |
https://en.wikipedia.org/wiki/Aronszajn%20tree | In set theory, an Aronszajn tree is a tree of uncountable height with no uncountable branches and no uncountable levels. For example, every Suslin tree is an Aronszajn tree. More generally, for a cardinal κ, a κ-Aronszajn tree is a tree of height κ in which all levels have size less than κ and all branches have height less than κ (so Aronszajn trees are the same as -Aronszajn trees). They are named for Nachman Aronszajn, who constructed an Aronszajn tree in 1934; his construction was described by .
A cardinal κ for which no κ-Aronszajn trees exist is said to have the tree property
(sometimes the condition that κ is regular and uncountable is included).
Existence of κ-Aronszajn trees
Kőnig's lemma states that -Aronszajn trees do not exist.
The existence of Aronszajn trees (-Aronszajn trees) was proven by Nachman Aronszajn, and implies that the analogue of Kőnig's lemma does not hold for uncountable trees.
The existence of -Aronszajn trees is undecidable in ZFC: more precisely, the continuum hypothesis implies the existence of an -Aronszajn tree, and Mitchell and Silver showed that it is consistent (relative to the existence of a weakly compact cardinal) that no -Aronszajn trees exist.
Jensen proved that V = L implies that there is a κ-Aronszajn tree (in fact a κ-Suslin tree) for every infinite successor cardinal κ.
showed (using a large cardinal axiom) that it is consistent that no -Aronszajn trees exist for any finite n other than 1.
If κ is weakly compact then no κ-Aronszajn trees exist. Conversely, if κ is inaccessible and no κ-Aronszajn trees exist, then κ is weakly compact.
Special Aronszajn trees
An Aronszajn tree is called special if there is a function f from the tree to the rationals so that
f(x) < f(y) whenever x < y. Martin's axiom MA() implies that all Aronszajn trees are special, a proposition sometimes abbreviated by EATS. The stronger proper forcing axiom implies the stronger statement that for any two Aronszajn trees there is a club set of levels such that the restrictions of the trees to this set of levels are isomorphic, which says that in some sense any two Aronszajn trees are essentially isomorphic . On the other hand, it is consistent that non-special Aronszajn trees exist, and this is also consistent with the generalized continuum hypothesis plus Suslin's hypothesis .
Construction of a special Aronszajn tree
A special Aronszajn tree can be constructed as follows.
The elements of the tree are certain well-ordered sets of rational numbers with supremum that is rational or −∞. If x and y are two of these sets then we define x ≤ y (in the tree order) to mean that x is an initial segment of the ordered set y. For each countable ordinal α we write Uα for the elements of the tree of level α, so that the elements of Uα are certain sets of rationals with order type α. The special Aronszajn tree T is the union of the sets Uα for all countable α.
We construct the countable levels Uα by transfinite induction on α as f |
https://en.wikipedia.org/wiki/Ath%C3%A9n%C3%A9e%20de%20Luxembourg | {
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}The Athénée de Luxembourg (), is a high school situated in Luxembourg City, in southern Luxembourg. Throughout the school's history of more than 400 years, its name was changed repeatedly. It is nowadays popularly known as the Stater Kolléisch or De Kolléisch, and is the nation's oldest school still in existence.
History
Jesuit Origins
On 15 May 1585, Pope Sixtus V signed a Papal bull granting the Jesuit Order the right to establish a school in Luxembourg. The school was eventually founded in 1603 by the Jesuit Order, and was located next to the Notre Dame Cathedral, in the Ville Haute quarter. It was modelled on the Jesuit school in Trier. The school flourished and in 1684 it was expanded. After the Suppression of the Society of Jesus by Pope Clement XIV in 1773, the school was renamed the Collège royal, and was put under auspices of the clergy. Furthermore, the school's curriculum was reformed and expanded.
Secularization
In the course of the French Revolution and the political changes that followed (notably the Napoleonic regime), the school was reorganized along French educational lines and renamed several times: École centrale (1795-1802), École Secondaire (1802-1808), and Collège municipal (1808-1817).
In 1817, the school was renamed "Athénée royal grand-ducal". To commemorate this event, a chronogram ATHENAEVM SIT LVCELBVRGI DECOR (=1817) was placed on the backside of a portal at the school's old premises.
In the course of the 19th century, the curriculum was expanded and modernized.
Second World War
When Luxembourg was occupied by Nazi forces in World War II in 1940, the school was forcibly Germanized, renamed the Gymnasium mit Oberschule für Jungen, and the French language was forbidden. These policies were met with considerable resistance. Infamously, when the Germans dismantled the Gëlle Fra memorial, several hundred of the school's students protested. Two professors and 76 students of the Athénéé lost their lives during the war.
Post-World War II: A New Building
After the Second World War, the school's premises became too small, and the school was relocated in 1964 to the Hollerich quarter, in the south-west of the city. Since the Athénée moved to this location, other schools have moved to the site as well, creating the school’s Campus Geesseknäppchen, just north of the terminus of the A4 motorway. The old site of the city Athenaeum was host to the National Library until October 2019, date at which the library moved to Kirchberg (in Luxembourg City).
21st century
In 2003, the school celebrated its 400th anniversary with a series of events and the publication of a four-volume study of the school's rich history. The school's official motto is 'Tradition & Innovation', a phrase that the |
https://en.wikipedia.org/wiki/Multiplicative%20cascade | In mathematics, a multiplicative cascade is a fractal/multifractal distribution of points produced via an iterative and multiplicative random process.
Definition
The plots above are examples of multiplicative cascade multifractals.
To create these distributions there are a few steps to take. Firstly, we must create a lattice of cells which will be our underlying probability density field.
Secondly, an iterative process is followed to create multiple levels of the lattice: at each iteration the cells are split into four equal parts (cells). Each new cell is then assigned a probability randomly from the set without replacement, where . This process is continued to the Nth level. For example, in constructing such a model down to level 8 we produce a 48 array of cells.
Thirdly, the cells are filled as follows: We take the probability of a cell being occupied as the product of the cell's own pi and those of all its parents (up to level 1). A Monte Carlo rejection scheme is used repeatedly until the desired cell population is obtained, as follows: x and y cell coordinates are chosen randomly, and a random number between 0 and 1 is assigned; the (x, y) cell is then populated depending on whether the assigned number is lesser than (outcome: not populated) or greater or equal to (outcome: populated) the cell's occupation probability.
Examples
To produce the plots above we filled the probability density field with 5,000 points in a space of 256 × 256.
An example of the probability density field:
The fractals are generally not scale-invariant and therefore cannot be considered standard fractals. They can however be considered multifractals. The Rényi (generalized) dimensions can be theoretically predicted. It can be shown that as ,
where N is the level of the grid refinement and,
See also
Fractal dimension
Hausdorff dimension
Scale invariance
References
Fractals |
https://en.wikipedia.org/wiki/Yamabe%20invariant | In mathematics, in the field of differential geometry, the Yamabe invariant, also referred to as the sigma constant, is a real number invariant associated to a smooth manifold that is preserved under diffeomorphisms. It was first written down independently by O. Kobayashi and R. Schoen and takes its name from H. Yamabe. Used by Vincent Moncrief and Arthur Fischer to study reduced Hamiltonian for Einstein's equations.
Definition
Let be a compact smooth manifold (without boundary) of dimension . The normalized Einstein–Hilbert functional assigns to each Riemannian metric on a real number as follows:
where is the scalar curvature of and is the volume density associated to the metric . The exponent in the denominator is chosen so that the functional is scale-invariant: for every positive real constant , it satisfies . We may think of as measuring the average scalar curvature of over . It was conjectured by Yamabe that every conformal class of metrics contains a metric of constant scalar curvature (the so-called Yamabe problem); it was proven by Yamabe, Trudinger, Aubin, and Schoen that a minimum value of is attained in each conformal class of metrics, and in particular this minimum is achieved by a metric of constant scalar curvature.
We define
where the infimum is taken over the smooth real-valued functions on . This infimum is finite (not ): Hölder's inequality implies . The number is sometimes called the conformal Yamabe energy of (and is constant on conformal classes).
A comparison argument due to Aubin shows that for any metric , is bounded above by , where
is the standard metric on the -sphere . It follows that if we define
where the supremum is taken over all metrics on , then (and is in particular finite). The
real number is called the Yamabe invariant of .
The Yamabe invariant in two dimensions
In the case that , (so that M is a closed surface) the Einstein–Hilbert functional is given by
where is the Gauss curvature of g. However, by the Gauss–Bonnet theorem, the integral of the Gauss curvature is given by , where is the Euler characteristic of M. In particular, this number does not depend on the choice of metric. Therefore, for surfaces, we conclude that
For example, the 2-sphere has Yamabe invariant equal to , and the 2-torus has Yamabe invariant equal to zero.
Examples
In the late 1990s, the Yamabe invariant was computed for large classes of 4-manifolds by Claude LeBrun and his collaborators. In particular, it was shown that most compact complex surfaces have negative, exactly computable Yamabe invariant, and that any Kähler–Einstein metric of negative scalar curvature realizes the Yamabe invariant in dimension 4. It was also shown that the Yamabe invariant of is realized by the Fubini–Study metric, and so is less than that of the 4-sphere. Most of these arguments involve Seiberg–Witten theory, and so are specific to dimension 4.
An important result due to Petean states that |
https://en.wikipedia.org/wiki/List%20of%20Manchester%20United%20F.C.%20records%20and%20statistics | Manchester United Football Club is an English professional football club based in Old Trafford, Greater Manchester. The club was founded as Newton Heath LYR F.C. in 1878 and turned professional in 1885, before joining the Football League in 1892. After a brush with bankruptcy in 1901, the club reformed as Manchester United in 1902. Manchester United currently play in the Premier League, the top tier of English football. They have not been out of the top tier since 1975, and they have never been lower than the second tier. They have also been involved in European football ever since they became the first English club to enter the European Cup in 1956.
This list encompasses the major honours won by Manchester United and records set by the club, their managers and their players. The player records section includes details of the club's leading goalscorers and those who have made most appearances in first-team competitions. It also records notable achievements by Manchester United players on the international stage, and the highest transfer fees paid and received by the club. The club's attendance records, both at Old Trafford, their home since 1910, and Maine Road, their temporary home from 1946 to 1949, are also included in the list.
The club currently holds the record for the most Premier League titles with 13, and the highest number of English top-flight titles with 20. The club's record appearance maker is Ryan Giggs, who made 963 appearances between 1991 and 2014, and the club's record goalscorer is Wayne Rooney, who scored 253 goals in 559 appearances between 2004 and 2017.
Honours
Manchester United's first trophy was the Manchester Cup, which they won as Newton Heath LYR in 1886. Their first national senior honour came in 1908, when they won the 1907–08 Football League First Division title. The club also won the FA Cup for the first time the following year. In terms of the number of trophies won, the 1990s were Manchester United's most successful decade, during which they won five league titles, four FA Cups, one League Cup, five Charity Shields (one shared), one Champions League, one Cup Winners' Cup, one Super Cup and one Intercontinental Cup.
The club currently holds the record for most top-division titles, with 20. They were also the first team to win the Premier League, as well as holding the record for the most Premier League titles (13), and became the first English team to win the European Cup when they won it in 1968. Their most recent trophy came in February 2023, when they won the EFL Cup.
Domestic
League
First Division / Premier League (Level 1): 20 – record
1907–08, 1910–11, 1951–52, 1955–56, 1956–57, 1964–65, 1966–67, 1992–93, 1993–94, 1995–96, 1996–97, 1998–99, 1999–2000, 2000–01, 2002–03, 2006–07, 2007–08, 2008–09, 2010–11, 2012–13
Second Division (Level 2): 2
1935–36, 1974–75
Cups
FA Cup: 12
1908–09, 1947–48, 1962–63, 1976–77, 1982–83, 1984–85, 1989–90, 1993–94, 1995–96, 1998–99, 2003–04, 2015–16
League/EFL Cup: 6 |
https://en.wikipedia.org/wiki/Buffon%27s%20noodle | In geometric probability, the problem of Buffon's noodle is a variation on the well-known problem of Buffon's needle, named after Georges-Louis Leclerc, Comte de Buffon who lived in the 18th century. This approach to the problem was published by Joseph-Émile Barbier in 1860.
Buffon's needle
Suppose there exist infinitely many equally spaced parallel lines, and we were to randomly toss a needle whose length is less than or equal to the distance between adjacent lines. What is the probability that the needle will lie across a line upon landing?
To solve this problem, let be the length of the needle and be the distance between two adjacent lines. Then, let be the acute angle the needle makes with the horizontal, and let be the distance from the center of the needle to the nearest line.
The needle lies across the nearest line if and only if . We see this condition from the right triangle formed by the needle, the nearest line, and the line of length when the needle lies across the nearest line.
Now, we assume that the values of are randomly determined when they land, where , since , and . The sample space for is thus a rectangle of side lengths and .
The probability of the event that the needle lies across the nearest line is the fraction of the sample space that intersects with . Since , the area of this intersection is given by
Now, the area of the sample space is
Hence, the probability of the event is
Bending the needle
The formula stays the same even when the needle is bent in any way (subject to the constraint that it must lie in a plane), making it a "noodle"—a rigid plane curve. We drop the assumption that the length of the noodle is no more than the distance between the parallel lines.
The probability distribution of the number of crossings depends on the shape of the noodle, but the expected number of crossings does not; it depends only on the length L of the noodle and the distance D between the parallel lines (observe that a curved noodle may cross a single line multiple times).
This fact may be proved as follows (see Klain and Rota). First suppose the noodle is piecewise linear, i.e. consists of n straight pieces. Let Xi be the number of times the ith piece crosses one of the parallel lines. These random variables are not independent, but the expectations are still additive due to the linearity of expectation:
Regarding a curved noodle as the limit of a sequence of piecewise linear noodles, we conclude that the expected number of crossings per toss is proportional to the length; it is some constant times the length L. Then the problem is to find the constant. In case the noodle is a circle of diameter equal to the distance D between the parallel lines, then L = D and the number of crossings is exactly 2, with probability 1. So when L = D then the expected number of crossings is 2. Therefore, the expected number of crossings must be 2L/(D).
Barbier's theorem
Extending this argument slightly, if is a convex co |
https://en.wikipedia.org/wiki/Sigma%20constant | Sigma constant (σ constant) may refer to:
Yamabe invariant, in mathematics
a parameter of the Hammett equation in organic chemistry |
https://en.wikipedia.org/wiki/Iraqis%20in%20Lebanon | Iraqis in Lebanon are people of Iraqi origin residing in Lebanon and Lebanese citizens of Iraqi ancestry. Statistics for Iraqi refugees in Lebanon vary, but typically put the number at around 50,000.
History
Iraqis have been present in Lebanon for decades. However, the first real influx of a large number of Iraqis to Lebanon started in earnest in the 1990s, with Iraqis fleeing Saddam Hussein's regime as well as the hardships of international sanctions. Most of the Iraqis during this period were Shia, fleeing Saddam's regime, or Christians, seeking exile in an Arab country with a significant local Christian population. Human Rights Watch puts the pre-2003 number of Iraqis in Lebanon at about 10,000.
Recent migration
After the 2003 invasion of Iraq, the first wave of Iraqi refugees fleeing the war began. By the middle of 2005, the number of Iraqis in Lebanon had doubled from the pre-Iraq War figure to 20,000. This number more than doubled with the second wave of Iraqi refugees fleeing the country after the February 2006 bombing of al-Askari Mosque in Samarra. By 2007 the numbers of Iraqis in Lebanon increases to between 26,000 and 100,000, but usually set at 50,000 by international agencies. Reliable and irrefutable statistics are difficult to come by with the majority of refugees in legal limbo. Variations in statistics as well as many of the issues that Iraqi refugees in Lebanon face are also linked to the 'invisible' nature of urban refugees.
Of the 8,090 Iraqi refugees actively registered with the UNHCR in Lebanon, over half have arrived since 2009. Of the same group of refugees, most are either Christians (42.0%) or Shia Muslims (39.2%) with a minority of Sunni Muslims (15.6%) and other sects or religions, including Mandeans and Yezidis (less than 1%, each). Most Iraqis in Lebanon are from Baghdad, having entered the country via Syria.
As a direct result of the instability and violence that followed the 2003 invasion of Iraq, the number of Iraqis in the country changed. Statistics for Iraqi refugees in Lebanon vary, as 2007, the number was around 50,000. As of 2007, statistics from the United Nations High Commissioner for Refugees put the current number of Iraqi refugees at just under 30,000.
Legal status
Lebanon is not a signatory of the 1951 Convention Relating to the Status of Refugees, nor the 1967 Protocol, leaving the 1962 law regarding the entry and stay of foreigners as the legal status determinant. 71 per cent of Iraqis surveyed in 2007 by the Danish Refugee Council had illegal status, and 95 per cent of respondents reached Lebanon by being smuggled across the Syrian-Lebanese border.
Education and health care
Since Iraqis rely on children as a source of income, impacting the enrollment levels for boys and girls. Other factors, including cost, lack of documentation as well as language difficulties from dialectal differences impact education for this population in general. Attendance rates in school amongst youths of the ages |
https://en.wikipedia.org/wiki/Vernon%20McCain | Vernon E. "Skip" McCain (June 4, 1908 – April 5, 1993) was an American football and basketball coach and mathematics professor. He served as the head football coach at Maryland State College—now known as the University of Maryland Eastern Shore—from 1948 to 1963, compiling a record of 100–21–5. McCain was inducted into the College Football Hall of Fame in 2006.
McCain was born in Marietta, Oklahoma. He played college football as a quarterback at Langston University in 1930. Prior to being hired at Maryland State in 1948, McCain was an assistant coach at Tennessee Agricultural & Industrial State College—now known as Tennessee State University. He died on April 5, 1993, at his home in Oxon Hill, Maryland.
Head coaching record
Football
References
External links
1908 births
1993 deaths
American football quarterbacks
Langston Lions football players
Maryland Eastern Shore Hawks athletic directors
Maryland Eastern Shore Hawks football coaches
Maryland Eastern Shore Hawks men's basketball coaches
Tennessee State Tigers football coaches
College Football Hall of Fame inductees
University of Maryland Eastern Shore faculty
People from Marietta, Oklahoma
Coaches of American football from Oklahoma
Players of American football from Oklahoma
Basketball coaches from Oklahoma
African-American coaches of American football
African-American players of American football
African-American basketball coaches
20th-century African-American sportspeople |
https://en.wikipedia.org/wiki/Bahador%20Abdi | Bahador Abdi (, born on 1 May 1984 in Tehran) is an Iranian football midfielder who currently plays for Aluminium Arak in the Azadegan League.
Club career
Club Career Statistics
Last Update 7 August 2014
Assist Goals
International career
He started his international career under head coach Afshin Ghotbi in November 2010 against Nigeria.
Honours
Iran's Premier Football League Winner: 1
2007–08 with Persepolis
References
Iranian men's footballers
Persian Gulf Pro League players
Azadegan League players
Persepolis F.C. players
Shahin Bushehr F.C. players
Footballers from Tehran
1984 births
Living people
Rah Ahan Tehran F.C. players
Men's association football midfielders
Persepolis F.C. non-playing staff |
https://en.wikipedia.org/wiki/Singular%20integral | In mathematics, singular integrals are central to harmonic analysis and are intimately connected with the study of partial differential equations. Broadly speaking a singular integral is an integral operator
whose kernel function K : Rn×Rn → R is singular along the diagonal x = y. Specifically, the singularity is such that |K(x, y)| is of size |x − y|−n asymptotically as |x − y| → 0. Since such integrals may not in general be absolutely integrable, a rigorous definition must define them as the limit of the integral over |y − x| > ε as ε → 0, but in practice this is a technicality. Usually further assumptions are required to obtain results such as their boundedness on Lp(Rn).
The Hilbert transform
The archetypal singular integral operator is the Hilbert transform H. It is given by convolution against the kernel K(x) = 1/(πx) for x in R. More precisely,
The most straightforward higher dimension analogues of these are the Riesz transforms, which replace K(x) = 1/x with
where i = 1, ..., n and is the i-th component of x in Rn. All of these operators are bounded on Lp and satisfy weak-type (1, 1) estimates.
Singular integrals of convolution type
A singular integral of convolution type is an operator T defined by convolution with a kernel K that is locally integrable on Rn\{0}, in the sense that
Suppose that the kernel satisfies:
The size condition on the Fourier transform of K
The smoothness condition: for some C > 0,
Then it can be shown that T is bounded on Lp(Rn) and satisfies a weak-type (1, 1) estimate.
Property 1. is needed to ensure that convolution () with the tempered distribution p.v. K given by the principal value integral
is a well-defined Fourier multiplier on L2. Neither of the properties 1. or 2. is necessarily easy to verify, and a variety of sufficient conditions exist. Typically in applications, one also has a cancellation condition
which is quite easy to check. It is automatic, for instance, if K is an odd function. If, in addition, one assumes 2. and the following size condition
then it can be shown that 1. follows.
The smoothness condition 2. is also often difficult to check in principle, the following sufficient condition of a kernel K can be used:
Observe that these conditions are satisfied for the Hilbert and Riesz transforms, so this result is an extension of those result.
Singular integrals of non-convolution type
These are even more general operators. However, since our assumptions are so weak, it is not necessarily the case that these operators are bounded on Lp.
Calderón–Zygmund kernels
A function is said to be a Calderón–Zygmund kernel if it satisfies the following conditions for some constants C > 0 and δ > 0.
Singular integrals of non-convolution type
T is said to be a singular integral operator of non-convolution type associated to the Calderón–Zygmund kernel K if
whenever f and g are smooth and have disjoint support. Such operators need not be bounded on Lp
Calder |
https://en.wikipedia.org/wiki/Word%20%28group%20theory%29 | In group theory, a word is any written product of group elements and their inverses. For example, if x, y and z are elements of a group G, then xy, z−1xzz and y−1zxx−1yz−1 are words in the set {x, y, z}. Two different words may evaluate to the same value in G, or even in every group. Words play an important role in the theory of free groups and presentations, and are central objects of study in combinatorial group theory.
Definitions
Let G be a group, and let S be a subset of G. A word in S is any expression of the form
where s1,...,sn are elements of S, called generators, and each εi is ±1. The number n is known as the length of the word.
Each word in S represents an element of G, namely the product of the expression. By convention, the unique identity element can be represented by the empty word, which is the unique word of length zero.
Notation
When writing words, it is common to use exponential notation as an abbreviation. For example, the word
could be written as
This latter expression is not a word itself—it is simply a shorter notation for the original.
When dealing with long words, it can be helpful to use an overline to denote inverses of elements of S. Using overline notation, the above word would be written as follows:
Reduced words
Any word in which a generator appears next to its own inverse (xx−1 or x−1x) can be simplified by omitting the redundant pair:
This operation is known as reduction, and it does not change the group element represented by the word. Reductions can be thought of as relations (defined below) that follow from the group axioms.
A reduced word is a word that contains no redundant pairs. Any word can be simplified to a reduced word by performing a sequence of reductions:
The result does not depend on the order in which the reductions are performed.
A word is cyclically reduced if and only if every cyclic permutation of the word is reduced.
Operations on words
The product of two words is obtained by concatenation:
Even if the two words are reduced, the product may not be.
The inverse of a word is obtained by inverting each generator, and reversing the order of the elements:
The product of a word with its inverse can be reduced to the empty word:
You can move a generator from the beginning to the end of a word by conjugation:
Generating set of a group
A subset S of a group G is called a generating set if every element of G can be represented by a word in S.
When S is not a generating set for G, the set of elements represented by words in S is a subgroup of G, known as the subgroup of G generated by S and usually denoted . It is the smallest subgroup of G that contains the elements of S.
Normal forms
A normal form for a group G with generating set S is a choice of one reduced word in S for each element of G. For example:
The words 1, i, j, ij are a normal form for the Klein four-group with } and 1 representing the empty word (the identity element for the group).
The words 1, r, r2, |
https://en.wikipedia.org/wiki/Alexis%20Allart | Alexis Allart (born 7 August 1986) is a French footballer who plays as a forward for USL Dunkerque of the Championnat National.
Career statistics
References
External links
France Football
Transfers Thai Premier League 2015-16, thai-fussball.com
1986 births
Living people
People from Charleville-Mézières
Footballers from Ardennes (department)
French men's footballers
Men's association football forwards
AS Monaco FC players
Louhans-Cuiseaux FC players
R.E. Mouscron players
CS Sedan Ardennes players
US Boulogne players
LB Châteauroux players
FC Istres players
CA Bastia players
Ligue 1 players
Ligue 2 players
Championnat National players
Belgian Pro League players
Expatriate men's footballers in Thailand
French expatriate sportspeople in Thailand
French expatriate men's footballers
French expatriate sportspeople in Belgium
Expatriate men's footballers in Belgium |
https://en.wikipedia.org/wiki/Philosophy%20of%20Mathematics%20Education%20Journal | The Philosophy of Mathematics Education Journal is a peer-reviewed open-access academic journal published and edited by Paul Ernest (University of Exeter). It publishes articles relevant to the philosophy of mathematics education, a subfield of mathematics education that often draws in issues from the philosophy of mathematics. The journal includes articles, graduate student assignments, theses, and other pertinent resources.
Special issues of the journal have focussed on
social justice issues in mathematics education, part 1 (issue no. 20, 2007)
semiotics of mathematics education (issue no. 10, 1997)
See also
List of scientific journals in mathematics education
External links
Philosophy journals
Open access journals
Academic journals established in 1990
English-language journals
Mathematics education journals
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/Laplace%20principle%20%28large%20deviations%20theory%29 | In mathematics, Laplace's principle is a basic theorem in large deviations theory which is similar to Varadhan's lemma. It gives an asymptotic expression for the Lebesgue integral of exp(−θφ(x)) over a fixed set A as θ becomes large. Such expressions can be used, for example, in statistical mechanics to determining the limiting behaviour of a system as the temperature tends to absolute zero.
Statement of the result
Let A be a Lebesgue-measurable subset of d-dimensional Euclidean space Rd and let φ : Rd → R be a measurable function with
Then
where ess inf denotes the essential infimum. Heuristically, this may be read as saying that for large θ,
Application
The Laplace principle can be applied to the family of probability measures Pθ given by
to give an asymptotic expression for the probability of some event A as θ becomes large. For example, if X is a standard normally distributed random variable on R, then
for every measurable set A.
See also
Laplace's method
References
Asymptotic analysis
Large deviations theory
Probability theorems
Statistical mechanics
Mathematical principles
Theorems in analysis |
https://en.wikipedia.org/wiki/Hadwiger%E2%80%93Finsler%20inequality | In mathematics, the Hadwiger–Finsler inequality is a result on the geometry of triangles in the Euclidean plane. It states that if a triangle in the plane has side lengths a, b and c and area T, then
Related inequalities
Weitzenböck's inequality is a straightforward corollary of the Hadwiger–Finsler inequality: if a triangle in the plane has side lengths a, b and c and area T, then
Hadwiger–Finsler inequality is actually equivalent to Weitzenböck's inequality. Applying (W) to the circummidarc triangle gives (HF)
Weitzenböck's inequality can also be proved using Heron's formula, by which route it can be seen that equality holds in (W) if and only if the triangle is an equilateral triangle, i.e. a = b = c.
A version for quadrilateral: Let ABCD be a convex quadrilateral with the lengths a, b, c, d and the area T then:
with equality only for a square.
Where
Proof
From the cosines law we have:
α being the angle between b and c. This can be transformed into:
Since A=1/2bcsinα we have:
Now remember that
and
Using this we get:
Doing this for all sides of the triangle and adding up we get:
β and γ being the other angles of the triangle. Now since the halves of the triangle’s angles are less than π/2 the function tan is convex we have:
Using this we get:
This is the Hadwiger-Finsler inequality.
History
The Hadwiger–Finsler inequality is named after , who also published in the same paper the Finsler–Hadwiger theorem on a square derived from two other squares that share a vertex.
See also
List of triangle inequalities
Isoperimetric inequality
References
Claudi Alsina, Roger B. Nelsen: When Less is More: Visualizing Basic Inequalities. MAA, 2009, , pp. 84-86
External links
Triangle inequalities |
https://en.wikipedia.org/wiki/Jaffard%20ring | In mathematics, a Jaffard ring is a type of ring, more general than a Noetherian ring, for which Krull dimension behaves as expected in polynomial extensions. They are named for Paul Jaffard who first studied them in 1960.
Formally, a Jaffard ring is a ring R such that the polynomial ring
where "dim" denotes Krull dimension. A Jaffard ring that is also an integral domain is called a Jaffard domain.
The Jaffard property is satisfied by any Noetherian ring R, and examples of non-Noetherian rings might appear to be quite difficult to find, however they do arise naturally. For example, the ring of (all) algebraic integers, or more generally, any Prüfer domain. Another example is obtained by "pinching" formal power series at the origin along a subfield of infinite extension degree, such as the example given in 1953 by Abraham Seidenberg: the subring of
consisting of those formal power series whose constant term is rational.
References
External links
Ring theory |
https://en.wikipedia.org/wiki/Hurwitz%20surface | In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(g − 1) automorphisms, where g is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms . They are also referred to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2).
The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The finite quotient group is precisely the automorphism group.
Automorphisms of complex algebraic curves are orientation-preserving automorphisms of the underlying real surface; if one allows orientation-reversing isometries, this yields a group twice as large, of order 168(g − 1), which is sometimes of interest.
A note on terminology – in this and other contexts, the "(2,3,7) triangle group" most often refers, not to the full triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ordinary triangle group (the von Dyck group) D(2,3,7) of orientation-preserving maps (the rotation group), which is index 2. The group of complex automorphisms is a quotient of the ordinary (orientation-preserving) triangle group, while the group of (possibly orientation-reversing) isometries is a quotient of the full triangle group.
Classification by genus
Only finitely many Hurwitz surfaces occur with each genus. The function mapping the genus to the number of Hurwitz surfaces with that genus is unbounded, even though most of its values are zero. The sum
converges for , implying in an approximate sense that the genus of the th Hurwitz surface grows at least as a cubic function of .
The Hurwitz surface of least genus is the Klein quartic of genus 3, with automorphism group the projective special linear group PSL(2,7), of order 84(3 − 1) = 168 = 23·3·7, which is a simple group; (or order 336 if one allows orientation-reversing isometries). The next possible genus is 7, possessed by the Macbeath surface, with automorphism group PSL(2,8), which is the simple group of order 84(7 − 1) = 504 = 23·32·7; if one includes orientation-reversing isometries, the group is of order 1,008.
An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14 − 1) = 1092 = 22·3·7·13). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet.
The sequence of allowable values for the genus of a Hurwitz surface begins
3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, |
https://en.wikipedia.org/wiki/Ono%27s%20inequality | In mathematics, Ono's inequality is a theorem about triangles in the Euclidean plane. In its original form, as conjectured by T. Ono in 1914, the inequality is actually false; however, the statement is true for acute triangles and right triangles, as shown by F. Balitrand in 1916.
Statement of the inequality
Consider an acute or right triangle in the Euclidean plane with side lengths a, b and c and area A. Then
This inequality fails for general triangles (to which Ono's original conjecture applied), as shown by the counterexample
The inequality holds with equality in the case of an equilateral triangle, in which up to similarity we have sides and area
See also
List of triangle inequalities
References
External links
Disproved conjectures
Triangle inequalities |
https://en.wikipedia.org/wiki/Ali%20Akansu | Ali Naci Akansu (born May 6, 1958) is a Turkish-American Professor of electrical & computer engineering and scientist in applied mathematics.
He is best known for his seminal contributions to the theory and applications of linear subspace methods including sub-band and wavelet transforms, particularly the binomial QMF (also known as Daubechies wavelet) and the multivariate framework to design statistically optimized filter bank (eigen filter bank).
Biography
Akansu received his B.S. degree from the Istanbul Technical University, Turkey, in 1980, his M.S. and PhD degrees from the Polytechnic University (now New York University), Brooklyn, New York, in 1983 and 1987, respectively, all in Electrical Engineering. Since 1987, he has been with the New Jersey Institute of Technology where he is a Professor of Electrical and Computer Engineering. He was a Visiting Professor at Courant Institute of Mathematical Sciences of the New York University, 2009–2010.
In 1990, he showed that the binomial quadrature mirror filter bank (binomial QMF) is identical to the Daubechies wavelet filter, and interpreted and evaluated its performance from a discrete-time signal processing perspective. It was an extension of his prior work on Binomial coefficient and Hermite polynomials that he developed the Modified Hermite Transformation (MHT) in 1987. The magnitude square functions of Binomial-QMF filters are the unique maximally flat functions in a two-band PR-QMF design formulation. He organized the first wavelet conference in the United States at NJIT in April 1990, and, then in 1992 and 1994. He published the first wavelet-related engineering book in the literature entitled Multiresolution Signal Decomposition: Transforms, Subbands and Wavelets in 1992.
He made contributions in the areas of optimal filter banks, nonlinear phase extensions of discrete Walsh-Hadamard transform and discrete Fourier transform, principal component analysis of first-order autoregressive process, sparse approximation, digital watermarking,
financial signal processing and quantitative finance. His publications include the books A Primer for Financial Engineering: Financial Signal Processing and Electronic Trading and Financial Signal Processing and Machine Learning.
He was a founding director of the New Jersey Center for Multimedia Research (NJCMR), 1996–2000, and NSF Industry-University Cooperative Research Center (IUCRC) for Digital Video, 1998–2000. He was the vice president for research and development of the IDT Corporation, 2000–2001, the founding president and CEO of PixWave, Inc. (an IDT Entertainment subsidiary) that has built the technology for secure peer-to-peer video distribution over the Internet. He was an academic visitor at David Sarnoff Research Center (Sarnoff Corporation), at IBM's Thomas J. Watson Research Center, and at Marconi Electronic Systems.
He is an IEEE Fellow (since 2008) with the citation for contributions to optimal design of transforms and filter b |
https://en.wikipedia.org/wiki/Presentation%20complex | In geometric group theory, a presentation complex is a 2-dimensional cell complex associated to any presentation of a group G. The complex has a single vertex, and one loop at the vertex for each generator of G. There is one 2-cell for each relation in the presentation, with the boundary of the 2-cell attached along the appropriate word.
Properties
The fundamental group of the presentation complex is the group G itself.
The universal cover of the presentation complex is a Cayley complex for G, whose 1-skeleton is the Cayley graph of G.
Any presentation complex for G is the 2-skeleton of an Eilenberg–MacLane space .
Examples
Let be the two-dimensional integer lattice, with presentation
Then the presentation complex for G is a torus, obtained by gluing the opposite sides of a square, the 2-cell, which are labelled x and y. All four corners of the square are glued into a single vertex, the 0-cell of the presentation complex, while a pair consisting of a longtitudal and meridian circles on the torus, intersecting at the vertex, constitutes its 1-skeleton.
The associated Cayley complex is a regular tiling of the plane by unit squares. The 1-skeleton of this complex is a Cayley graph for .
Let be the Infinite dihedral group, with presentation . The presentation complex for is , the wedge sum of projective planes. For each path, there is one 2-cell glued to each loop, which provides the standard cell structure for each projective plane. The Cayley complex is an infinite string of spheres.
References
Roger C. Lyndon and Paul E. Schupp, Combinatorial group theory. Reprint of the 1977 edition (Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89). Classics in Mathematics. Springer-Verlag, Berlin, 2001
Ronald Brown and Johannes Huebschmann, Identities among relations, in Low dimensional topology, London Math. Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge University Press, 1982), pp. 153–202.
Hog-Angeloni, Cynthia, Metzler, Wolfgang and Sieradski, Allan J. (eds.). Two-dimensional homotopy and combinatorial group theory, London Mathematical Society Lecture Note Series, Volume 197. Cambridge University Press, Cambridge (1993).
Algebraic topology
Geometric group theory |
https://en.wikipedia.org/wiki/List%20of%20A1%20Grand%20Prix%20teams | The following is a list of teams which competed in the A1 Grand Prix series. 29 teams participated in at least one A1 Grand Prix race.
A1 team list and statistics
Notes
References
Teams |
https://en.wikipedia.org/wiki/Macbeath%20surface | In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface.
The automorphism group of the Macbeath surface is the simple group PSL(2,8), consisting of 504 symmetries.
Triangle group construction
The surface's Fuchsian group can be constructed as the principal congruence subgroup of the (2,3,7) triangle group in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and Hurwitz quaternion order are described at the triangle group page. Choosing the ideal in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its systole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.
It is possible to realize the resulting triangulated surface as a non-convex polyhedron without self-intersections.
Historical note
This surface was originally discovered by , but named after Alexander Murray Macbeath due to his later independent rediscovery of the same curve. Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by Serre in a 24.vii.1990 letter to Abhyankar".
See also
Klein quartic
First Hurwitz triplet
Notes
References
.
.
.
.
.
. Translation in Moscow Univ. Math. Bull. 44 (1989), no. 5, 37–40.
.
.
. Corrigendum, vol. 28, no. 2, 1986, p. 241, .
Hyperbolic geometry
Riemann surfaces
Riemannian geometry
Differential geometry of surfaces
Systolic geometry |
https://en.wikipedia.org/wiki/Group%20Hopf%20algebra | In mathematics, the group Hopf algebra of a given group is a certain construct related to the symmetries of group actions. Deformations of group Hopf algebras are foundational in the theory of quantum groups.
Definition
Let G be a group and k a field. The group Hopf algebra of G over k, denoted kG (or k[G]), is as a set (and a vector space) the free vector space on G over k. As an algebra, its product is defined by linear extension of the group composition in G, with multiplicative unit the identity in G; this product is also known as convolution.
Note that while the group algebra of a finite group can be identified with the space of functions on the group, for an infinite group these are different. The group algebra, consisting of finite sums, corresponds to functions on the group that vanish for cofinitely many points; topologically (using the discrete topology), these correspond to functions which are non-zero only on a finite set.
However, the group algebra and – the commutative algebra of functions of G into k – are dual: given an element of the group algebra and a function on the group these pair to give an element of k via which is a well-defined sum because it is finite.
Hopf algebra structure
We give kG the structure of a cocommutative Hopf algebra by defining the coproduct, counit, and antipode to be the linear extensions of the following maps defined on G:
The required Hopf algebra compatibility axioms are easily checked. Notice that , the set of group-like elements of kG (i.e. elements such that and ), is precisely G.
Symmetries of group actions
Let G be a group and X a topological space. Any action of G on X gives a homomorphism , where F(X) is an appropriate algebra of k-valued functions, such as the Gelfand–Naimark algebra of continuous functions vanishing at infinity. The homomorphism is defined by , with the adjoint defined by
for , and .
This may be described by a linear mapping
where , are the elements of G, and , which has the property that group-like elements in give rise to automorphisms of F(X).
endows F(X) with an important extra structure, described below.
Hopf module algebras and the Hopf smash product
Let H be a Hopf algebra. A (left) Hopf H-module algebra A is an algebra which is a (left) module over the algebra H such that and
whenever , and in sumless Sweedler notation. When has been defined as in the previous section, this turns F(X) into a left Hopf kG-module algebra, which allows the following construction.
Let H be a Hopf algebra and A a left Hopf H-module algebra. The smash product algebra is the vector space with the product
,
and we write for in this context.
In our case, and , and we have
.
In this case the smash product algebra is also denoted by .
The cyclic homology of Hopf smash products has been computed. However, there the smash product is called a crossed product and denoted - not to be confused with the crossed product derived from -dynamical systems.
Re |
https://en.wikipedia.org/wiki/Wiedersehen%20pair | In mathematics—specifically, in Riemannian geometry—a Wiedersehen pair is a pair of distinct points x and y on a (usually, but not necessarily, two-dimensional) compact Riemannian manifold (M, g) such that every geodesic through x also passes through y, and the same with x and y interchanged.
For example, on an ordinary sphere where the geodesics are great circles, the Wiedersehen pairs are exactly the pairs of antipodal points.
If every point of an oriented manifold (M, g) belongs to a Wiedersehen pair, then (M, g) is said to be a Wiedersehen manifold. The concept was introduced by the Austro-Hungarian mathematician Wilhelm Blaschke and comes from the German term meaning "seeing again". As it turns out, in each dimension n the only Wiedersehen manifold (up to isometry) is the standard Euclidean n-sphere. Initially known as the Blaschke conjecture, this result was established by combined works of Berger, Kazdan, Weinstein (for even n), and Yang (odd n).
See also
Cut locus (Riemannian manifold)
References
External links
Riemannian geometry
Equations
Manifolds |
https://en.wikipedia.org/wiki/Borel%20fixed-point%20theorem | In mathematics, the Borel fixed-point theorem is a fixed-point theorem in algebraic geometry generalizing the Lie–Kolchin theorem. The result was proved by .
Statement
If G is a connected, solvable, linear algebraic group acting regularly on a non-empty, complete algebraic variety V over an algebraically closed field k, then there is a G fixed-point of V.
A more general version of the theorem holds over a field k that is not necessarily algebraically closed. A solvable algebraic group G is split over k or k-split if G admits a composition series whose composition factors are isomorphic (over k) to the additive group or the multiplicative group . If G is a connected, k-split solvable algebraic group acting regularly on a complete variety V having a k-rational point, then there is a G fixed-point of V.
References
External links
Fixed-point theorems
Group actions (mathematics)
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Berger%27s%20isoembolic%20inequality | In mathematics, Berger's isoembolic inequality is a result in Riemannian geometry that gives a lower bound on the volume of a Riemannian manifold and also gives a necessary and sufficient condition for the manifold to be isometric to the -dimensional sphere with its usual "round" metric. The theorem is named after the mathematician Marcel Berger, who derived it from an inequality proved by Jerry Kazdan.
Statement of the theorem
Let be a closed -dimensional Riemannian manifold with injectivity radius . Let denote the Riemannian volume of and let denote the volume of the standard -dimensional sphere of radius one. Then
with equality if and only if is isometric to the -sphere with its usual round metric. This result is known as Berger's isoembolic inequality. The proof relies upon an analytic inequality proved by Kazdan. The original work of Berger and Kazdan appears in the appendices of Arthur Besse's book "Manifolds all of whose geodesics are closed." At this stage, the isoembolic inequality appeared with a non-optimal constant. Sometimes Kazdan's inequality is called Berger–Kazdan inequality.
References
Books.
External links
Geometric inequalities
Theorems in Riemannian geometry |
https://en.wikipedia.org/wiki/Berezin%20transform | In mathematics — specifically, in complex analysis — the Berezin transform is an integral operator acting on functions defined on the open unit disk D of the complex plane C. Formally, for a function ƒ : D → C, the Berezin transform of ƒ is a new function Bƒ : D → C defined at a point z ∈ D by
where denotes the complex conjugate of w and is the area measure. It is named after Felix Alexandrovich Berezin.
References
External links
Complex analysis
Operator theory |
https://en.wikipedia.org/wiki/Constant | Constant or The Constant may refer to:
Mathematics
Constant (mathematics), a non-varying value
Mathematical constant, a special number that arises naturally in mathematics, such as or
Other concepts
Control variable or scientific constant, in experimentation the unchanging or constant variable
Physical constant, a physical quantity generally believed to be universal and unchanging
Constant (computer programming), a value that, unlike a variable, cannot be reassociated with a different value
Logical constant, a symbol in symbolic logic that has the same meaning in all models, such as the symbol "=" for "equals"
People
Constant (given name)
Constant (surname)
John, Elector of Saxony (1468–1532), known as John the Constant
Constant Nieuwenhuys (1920-2005), better known as Constant
Places
Constant, Barbados, a populated place
Arts and entertainment
"The Constant", a 2008 episode of the television show Lost
The Constant (Story of the Year album)
The Constant (I Blame Coco album)
Constants (band), an American rock band |
https://en.wikipedia.org/wiki/Earl%20D.%20Rainville | Professor Earl David Rainville (5 November 1907 – 29 April 1966) taught in the Department of Engineering Mathematics at the University of Michigan, where he began as an assistant professor in 1941. He studied at the University of Colorado, receiving his B.A. there in 1930 before going on to graduate studies at Michigan, where he received his Ph.D. in 1939 under the supervision of Ruel Churchill.
He was the author of several textbooks.
Books
Linear Differential Invariance Under an Operator Related to the Laplace Transformation, Univ. of Michigan, 1940, reprinted from American Journal of Mathematics, vol. 62. (Rainville's Ph.D. thesis.)
Intermediate Course in Differential Equations, Chapman & Hall, 1943.
Analytic Geometry, with Clyde E. Love, Macmillan, 1955.
Special Functions, Macmillan, 1960.
Unified Calculus and Analytic Geometry, Macmillan, 1961.
Differential and Integral Calculus, with Clyde E. Love, Macmillan, 1962.
Laplace Transform: An Introduction, 1963.
Intermediate Differential Equations, Macmillan, 1964.
Infinite Series, Macmillan, 1967.
Elementary Differential Equations, with Phillip E. Bedient, Macmillan, 1969. Eighth edition published by Prentice Hall, 1997, .
A Short Course in Differential Equations, with Phillip E. Bedient, Macmillan, 1969.
See also
Rainville polynomials
References
1907 births
1966 deaths
20th-century American mathematicians
University of Colorado alumni
University of Michigan alumni
University of Michigan faculty
American textbook writers |
https://en.wikipedia.org/wiki/Heteroskedasticity-consistent%20standard%20errors | The topic of heteroskedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as heteroskedasticity-robust standard errors (or simply robust standard errors), Eicker–Huber–White standard errors (also Huber–White standard errors or White standard errors), to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White.
In regression and time-series modelling, basic forms of models make use of the assumption that the errors or disturbances ui have the same variance across all observation points. When this is not the case, the errors are said to be heteroskedastic, or to have heteroskedasticity, and this behaviour will be reflected in the residuals estimated from a fitted model. Heteroskedasticity-consistent standard errors are used to allow the fitting of a model that does contain heteroskedastic residuals. The first such approach was proposed by Huber (1967), and further improved procedures have been produced since for cross-sectional data, time-series data and GARCH estimation.
Heteroskedasticity-consistent standard errors that differ from classical standard errors may indicate model misspecification. Substituting heteroskedasticity-consistent standard errors does not resolve this misspecification, which may lead to bias in the coefficients. In most situations, the problem should be found and fixed. Other types of standard error adjustments, such as clustered standard errors or HAC standard errors, may be considered as extensions to HC standard errors.
History
Heteroskedasticity-consistent standard errors are introduced by Friedhelm Eicker, and popularized in econometrics by Halbert White.
Problem
Consider the linear regression model for the scalar .
where is a k x 1 column vector of explanatory variables (features), is a k × 1 column vector of parameters to be estimated, and is the residual error.
The ordinary least squares (OLS) estimator is
where is a vector of observations , and denotes the matrix of stacked values observed in the data.
If the sample errors have equal variance and are uncorrelated, then the least-squares estimate of is BLUE (best linear unbiased estimator), and its variance is estimated with
where are the regression residuals.
When the error terms do not have constant variance (i.e., the assumption of is untrue), the OLS estimator loses its desirable properties. The formula for variance now cannot be simplified:
where
While the OLS point estimator remains unbiased, it is not "best" in the sense of having minimum mean square error, and the OLS variance estimator does not provide a consistent estimate of the variance of the OLS estimates.
For any non-linear model (for instance logit and probit models), however, heteroskedasticity has more severe consequences: the maximum likelihood estimates of the parameters will be biased (in an unknown direction), as well a |
https://en.wikipedia.org/wiki/Filling%20radius | In Riemannian geometry, the filling radius of a Riemannian manifold X is a metric invariant of X. It was originally introduced in 1983 by Mikhail Gromov, who used it to prove his systolic inequality for essential manifolds, vastly generalizing Loewner's torus inequality and Pu's inequality for the real projective plane, and creating systolic geometry in its modern form.
The filling radius of a simple loop C in the plane is defined as the largest radius, R > 0, of a circle that fits inside C:
Dual definition via neighborhoods
There is a kind of a dual point of view that allows one to generalize this notion in an extremely fruitful way, as shown by Gromov. Namely, we consider the -neighborhoods of the loop C, denoted
As increases, the -neighborhood swallows up more and more of the interior of the loop. The last point to be swallowed up is precisely the center of a largest inscribed circle. Therefore, we can reformulate the above definition by defining
to be the infimum of such that the loop C contracts to a point in .
Given a compact manifold X imbedded in, say, Euclidean space E, we could define the filling radius relative to the imbedding, by minimizing the size of the neighborhood in which X could be homotoped to something smaller dimensional, e.g., to a lower-dimensional polyhedron. Technically it is more convenient to work with a homological definition.
Homological definition
Denote by A the coefficient ring or , depending on whether or not X is orientable. Then the fundamental class, denoted [X], of a compact n-dimensional manifold X, is a generator of the homology group , and we set
where is the inclusion homomorphism.
To define an absolute filling radius in a situation where X is equipped with a Riemannian metric g, Gromov proceeds as follows.
One exploits Kuratowski embedding.
One imbeds X in the Banach space of bounded Borel functions on X, equipped with the sup norm . Namely, we map a point to the function defined by the formula
for all , where d is the distance function defined by the metric. By the triangle inequality we have and therefore the imbedding is strongly isometric, in the precise sense that internal distance and ambient distance coincide. Such a strongly isometric imbedding is impossible if the ambient space is a Hilbert space, even when X is the Riemannian circle (the distance between opposite points must be
, not 2!). We then set in the formula above, and define
Properties
The filling radius is at most a third of the diameter (Katz, 1983).
The filling radius of real projective space with a metric of constant curvature is a third of its Riemannian diameter, see (Katz, 1983). Equivalently, the filling radius is a sixth of the systole in these cases.
The filling radius of the Riemannian circle of length 2π, i.e. the unit circle with the induced Riemannian distance function, equals π/3, i.e. a sixth of its length. This follows by combining the diameter upper bound mentioned above with |
https://en.wikipedia.org/wiki/Jerry%20Kazdan | Jerry Lawrence Kazdan (born 31 October 1937 in Detroit, Michigan) is an American mathematician noted for his work in differential geometry and the study of partial differential equations. His contributions include the Berger–Kazdan comparison theorem, which was a key step in the proof of the Blaschke conjecture and the classification of Wiedersehen manifolds. His best-known work, done in collaboration with Frank Warner, dealt with the problem of prescribing the scalar curvature of a Riemannian metric.
Biography
Kazdan received his bachelor's degree in 1959 from Rensselaer Polytechnic Institute and his master's degree in 1961 from NYU. He obtained his PhD in 1963 from the Courant Institute of Mathematical Sciences at New York University; his thesis was entitled A Boundary Value Problem Arising in the Theory of Univalent Functions and was supervised by Paul Garabedian. He then took a position as a Benjamin Peirce Instructor at Harvard University. Since 1966, he has been a Professor of Mathematics at the University of Pennsylvania.
Dennis DeTurck was a student of his.
Honours
In 1999 he received the Lester Randolph Ford Award for his expository article Solving equations, an elegant legacy. In 2012 he became a fellow of the American Mathematical Society.
Major publications
DeTurck, Dennis M.; Kazdan, Jerry L. Some regularity theorems in Riemannian geometry. Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 3, 249–260.
Kazdan, Jerry L.; Warner, F.W. Curvature functions for compact 2-manifolds. Ann. of Math. (2) 99 (1974), 14–47.
Kazdan, Jerry L.; Warner, F.W. Remarks on some quasilinear elliptic equations. Comm. Pure Appl. Math. 28 (1975), no. 5, 567–597.
Kazdan, Jerry L.; Warner, F.W. Scalar curvature and conformal deformation of Riemannian structure. Journal of Differential Geometry 10 (1975), 113–134.
Kazdan, Jerry L.; Warner, F.W. Existence and conformal deformation of metrics with prescribed Gaussian and scalar curvatures. Ann. of Math. (2) 101 (1975), 317–331.
Books
Lectures on Complex Numbers and Infinite Series (1966)
Calculus Two: Linear and Nonlinear Functions (1971, with Francis J. Flanigan)
Intermediate Calculus And Linear Algebra (1975)
Prescribing the Curvature of a Riemannian Manifold (1985)
See also
Prescribed scalar curvature problem
References
External links
Jerry Kazdan's homepage
Brief biography on the occasion of receiving the Lester R. Ford award
20th-century American mathematicians
21st-century American mathematicians
Harvard University Department of Mathematics faculty
Harvard University faculty
Courant Institute of Mathematical Sciences alumni
Rensselaer Polytechnic Institute alumni
University of Pennsylvania faculty
Mathematicians at the University of Pennsylvania
Fellows of the American Mathematical Society
Living people
1937 births
American textbook writers
Differential geometers |
https://en.wikipedia.org/wiki/EAS3 | EAS3 (EAS = Ein-Ausgabe-System) is a software toolkit for reading and writing structured binary data with geometry information and for postprocessing of these data. It is meant to exchange floating-point data according to IEEE standard between different computers, to modify them or to convert them into other file formats. It can be used for all kinds of structured data sets. It is mainly used in the field of direct numerical simulations.
EAS3 package
The complete package consists of libraries intended for usage in own codes and a separate command-line tool. It is written in Fortran and C and runs on all POSIX operating systems. The libraries include different numerical algorithms and subroutines for reading and writing files in the binary EAS3 file format. The read/write routines are provided in Fortran and C. Implemented numerical methods include, for example, Fast Fourier transform, Thomas algorithm and interpolation routines. The libraries are also suitable for vector computers.
History
EAS3 has been developed at the Institut für Aerodynamik und Gasdynamik (IAG) of the University of Stuttgart. The previous versions (EAS, EAS2) range back to the end of the 1980s, when computer power allowed the first spatial DNS computations. The upcoming amount of data required efficient handling and postprocessing. Typically, simulations were, and are still today, performed on a high-performance computer and afterwards postprocessed on other machines of opposite endianness. This required an endianness-independent file format for data handling.
Since the publication of EAS3 in the 1999, the software has been developed continuously by members of the involved institutes. Since 2007, EAS3 is also available via the heise software directory. EAS3 is used by applications within the European PRACE project. The current version number is 1.6.7 from April, 2009.
File Format
The EAS3 file format is used to store floating point data in IEEE format and to exchange the files between different computer architectures (little/big endian). The data is organized as parameters with one parameter being a one-, two- or three-dimensional floating point array. Several of these parameters may be combined to one time step. This allows to store five-dimensional arrays. Data can be written in single-precision (32 Bit), double-precision (64 Bit) or quadruple-precision (128 Bit). Geometry information for the different directions are saved in the header of the file. It is also possible to store additional information in user defined arrays there. With the file size being limited only by the computer itself (e.g. file system), EAS3 files are suitable for large simulations and thus for high-performance computing.
Functionality
The actual EAS3 executable is a command-line interface for alteration of EAS3 files. The implemented commands range from basic operations, e.g. simple computations, file operations, to rather complex operations like Fourier transformation or the computation of der |
https://en.wikipedia.org/wiki/Porous%20set | In mathematics, a porous set is a concept in the study of metric spaces. Like the concepts of meagre and measure zero sets, a porous set can be considered "sparse" or "lacking bulk"; however, porous sets are not equivalent to either meagre sets or measure zero sets, as shown below.
Definition
Let (X, d) be a complete metric space and let E be a subset of X. Let B(x, r) denote the closed ball in (X, d) with centre x ∈ X and radius r > 0. E is said to be porous if there exist constants 0 < α < 1 and r0 > 0 such that, for every 0 < r ≤ r0 and every x ∈ X, there is some point y ∈ X with
A subset of X is called σ-porous if it is a countable union of porous subsets of X.
Properties
Any porous set is nowhere dense. Hence, all σ-porous sets are meagre sets (or of the first category).
If X is a finite-dimensional Euclidean space Rn, then porous subsets are sets of Lebesgue measure zero.
However, there does exist a non-σ-porous subset P of Rn which is of the first category and of Lebesgue measure zero. This is known as Zajíček's theorem.
The relationship between porosity and being nowhere dense can be illustrated as follows: if E is nowhere dense, then for x ∈ X and r > 0, there is a point y ∈ X and s > 0 such that
However, if E is also porous, then it is possible to take s = αr (at least for small enough r), where 0 < α < 1 is a constant that depends only on E.
References
Metric geometry |
https://en.wikipedia.org/wiki/Toric%20manifold | In mathematics, a toric manifold is a topological analogue of toric variety in algebraic geometry. It is an even-dimensional manifold with an effective smooth action of an -dimensional compact torus which is locally standard with the orbit space a simple convex polytope.
The aim is to do combinatorics on the quotient polytope and obtain information on the manifold above. For example, the Euler characteristic and the cohomology ring of the manifold can be described in terms of the polytope.
The Atiyah and Guillemin-Sternberg theorem
This theorem states that the image of the moment map of a Hamiltonian toric action is the convex hull of the set of moments of the points fixed by the action. In particular, this image is a convex polytope.
References
Structures on manifolds
Manifolds
Topology |
https://en.wikipedia.org/wiki/Discontinuous%20Galerkin%20method | In applied mathematics, discontinuous Galerkin methods (DG methods) form a class of numerical methods for solving differential equations. They combine features of the finite element and the finite volume framework and have been successfully applied to hyperbolic, elliptic, parabolic and mixed form problems arising from a wide range of applications. DG methods have in particular received considerable interest for problems with a dominant first-order part, e.g. in electrodynamics, fluid mechanics and plasma physics.
Discontinuous Galerkin methods were first proposed and analyzed in the early 1970s as a technique to numerically solve partial differential equations. In 1973 Reed and Hill introduced a DG method to solve the hyperbolic neutron transport equation.
The origin of the DG method for elliptic problems cannot be traced back to a single publication as features such as jump penalization in the modern sense were developed gradually. However, among the early influential contributors were Babuška, J.-L. Lions, Joachim Nitsche and Miloš Zlámal. DG methods for elliptic problems were already developed in a paper by Garth Baker in the setting of 4th order equations in 1977. A more complete account of the historical development and an introduction to DG methods for elliptic problems is given in a publication by Arnold, Brezzi, Cockburn and Marini. A number of research directions and challenges on DG methods are collected in the proceedings volume edited by Cockburn, Karniadakis and Shu.
Overview
Much like the continuous Galerkin (CG) method, the discontinuous Galerkin (DG) method is a finite element method formulated relative to a weak formulation of a particular model system. Unlike traditional CG methods that are conforming, the DG method works over a trial space of functions that are only piecewise continuous, and thus often comprise more inclusive function spaces than the finite-dimensional inner product subspaces utilized in conforming methods.
As an example, consider the continuity equation for a scalar unknown in a spatial domain without "sources" or "sinks" :
where is the flux of .
Now consider the finite-dimensional space of discontinuous piecewise polynomial functions over the spatial domain restricted to a discrete triangulation , written as
for the space of polynomials with degrees less than or equal to over element indexed by . Then for finite element shape functions the solution is represented by
Then similarly choosing a test function
multiplying the continuity equation by and integrating by parts in space, the semidiscrete DG formulation becomes:
Scalar hyperbolic conservation law
A scalar hyperbolic conservation law is of the form
where one tries to solve for the unknown scalar function , and the functions are typically given.
Space discretization
The -space will be discretized as
Furthermore, we need the following definitions
Basis for function space
We derive the basis representation for the function sp |
https://en.wikipedia.org/wiki/Generic%20property | In mathematics, properties that hold for "typical" examples are called generic properties. For instance, a generic property of a class of functions is one that is true of "almost all" of those functions, as in the statements, "A generic polynomial does not have a root at zero," or "A generic square matrix is invertible." As another example, a generic property of a space is a property that holds at "almost all" points of the space, as in the statement, "If is a smooth function between smooth manifolds, then a generic point of is not a critical value of ." (This is by Sard's theorem.)
There are many different notions of "generic" (what is meant by "almost all") in mathematics, with corresponding dual notions of "almost none" (negligible set); the two main classes are:
In measure theory, a generic property is one that holds almost everywhere, with the dual concept being null set, meaning "with probability 0".
In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set, with the dual concept being a nowhere dense set, or more generally a meagre set.
There are several natural examples where those notions are not equal. For instance, the set of Liouville numbers is generic in the topological sense, but has Lebesgue measure zero.
In measure theory
In measure theory, a generic property is one that holds almost everywhere. The dual concept is a null set, that is, a set of measure zero.
In probability
In probability, a generic property is an event that occurs almost surely, meaning that it occurs with probability 1. For example, the law of large numbers states that the sample mean converges almost surely to the population mean. This is the definition in the measure theory case specialized to a probability space.
In discrete mathematics
In discrete mathematics, one uses the term almost all to mean cofinite (all but finitely many), cocountable (all but countably many), for sufficiently large numbers, or, sometimes, asymptotically almost surely. The concept is particularly important in the study of random graphs.
In topology
In topology and algebraic geometry, a generic property is one that holds on a dense open set, or more generally on a residual set (a countable intersection of dense open sets), with the dual concept being a closed nowhere dense set, or more generally a meagre set (a countable union of nowhere dense closed sets).
However, density alone is not sufficient to characterize a generic property. This can be seen even in the real numbers, where both the rational numbers and their complement, the irrational numbers, are dense. Since it does not make sense to say that both a set and its complement exhibit typical behavior, both the rationals and irrationals cannot be examples of sets large enough to be typical. Consequently, we rely on the stronger definition above which implies that the irrationals are typical and the rationals are not.
For applications, |
https://en.wikipedia.org/wiki/Graph%20cut | Graph cut may refer to:
Cut (graph theory), in mathematics
Graph cut optimization
Graph cuts in computer vision |
https://en.wikipedia.org/wiki/Tim%20Hrynewich | Tim Hrynewich (born October 2, 1963) is a Canadian former professional ice hockey left winger. He played 55 NHL games for the Pittsburgh Penguins between 1982 and 1984.
Career statistics
Regular season and playoffs
References
External links
1963 births
Living people
Baltimore Skipjacks players
Canadian ice hockey left wingers
Flint Spirits players
Fort Wayne Komets players
Ice hockey people from Ontario
Milwaukee Admirals (IHL) players
Muskegon Lumberjacks players
People from Leamington, Ontario
Sportspeople from Essex County, Ontario
Pittsburgh Penguins players
Pittsburgh Penguins draft picks
SaiPa players
Sudbury Wolves players
Toledo Goaldiggers players |
https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20on-base%20percentage%20leaders | In baseball statistics, on-base percentage (OBP) is a measure of how often a batter reaches base for any reason other than a fielding error, fielder's choice, dropped or uncaught third strike, fielder's obstruction, or catcher's interference. OBP is calculated in Major League Baseball (MLB) by dividing the sum of hits, walks, and times hit by a pitch by the sum of at-bats, walks, times hit by pitch and sacrifice flies. A hitter with a .400 on-base percentage is considered to be great and rare; only 61 players in MLB history with at least 3,000 career plate appearances (PA) have maintained such an OBP. Left fielder Ted Williams, who played 19 seasons for the Boston Red Sox, has the highest career on-base percentage, .4817, in MLB history. Williams led the American League (AL) in on-base percentage in twelve seasons, the most such seasons for any player in the major leagues. Barry Bonds led the National League (NL) in ten seasons, a NL record. Williams also posted the then-highest single-season on-base percentage of .5528 in 1941, a record that stood for 61 years until Bonds broke it with a .5817 OBP in 2002. Bonds broke his own record in 2004, setting the current single-season mark of .6094.
Players are eligible for the Hall of Fame if they have played at least 10 major league seasons, have been either retired for five seasons or deceased for six months, and have not been banned from MLB. These requirements leave 6 living players ineligible who have played in the past 5 seasons; 5 players (Bill Joyce, Ferris Fain, Jake Stenzel, Bill Lange, and George Selkirk) who did not play 10 seasons in MLB; and Shoeless Joe Jackson, who was banned for his role in the Black Sox Scandal.
Key
List
Stats updated as October 1, 2023.
See also
List of Major League Baseball career batting average leaders
List of Major League Baseball career slugging percentage leaders
List of Major League Baseball career OPS leaders
Notes
References
External links
On-base
Major League Baseball statistics |
https://en.wikipedia.org/wiki/List%20of%20Major%20League%20Baseball%20career%20slugging%20percentage%20leaders | In baseball statistics, slugging percentage (SLG) is a measure of the batting productivity of a hitter. It is calculated as total bases divided by at bats. Unlike batting average, slugging percentage gives more weight to extra-base hits with doubles, triples, and home runs, relative to singles. Plate appearances ending in walks are specifically excluded from this calculation, as an appearance that ends in a walk is not counted as an at bat.
Babe Ruth is the all-time leader with a career slugging percentage of .6897. Ted Williams (.6338), Lou Gehrig (.6324), Mule Suttles (.6179), Turkey Stearnes (.6165), Oscar Charleston (.6145), Jimmie Foxx (.6093), Barry Bonds (.6069), and Hank Greenberg (.6050) are the only other players with a career slugging percentage over .600.
Key
List
Stats updated as of October 1, 2023.
Notes
Sources
Slugging percentage 5
Major League Baseball statistics |
https://en.wikipedia.org/wiki/Statistics%20and%20Registration%20Service%20Act%202007 | The Statistics and Registration Service Act 2007 (c 18) is an Act of the Parliament of the United Kingdom which established the UK Statistics Authority (UKSA). It came into force in April 2008. Sir Michael Scholar was appointed as the first Chair of the UKSA.
The Act established the UK Statistics Authority as a non-ministerial department that employs the National Statistician. The National Statistician has an office to support them, the Office for National Statistics.
References
External links
Guide to the Act from the Office for National Statistics
United Kingdom Acts of Parliament 2007
Office for National Statistics |
https://en.wikipedia.org/wiki/Northicote%20School | Northicote School was a co-educational secondary school located in the city Wolverhampton, West Midlands, England. The age range of the school was 11-18. It had specialist status in mathematics and computing.
It was the first school in Britain to be condemned as "failing" by OFSTED shortly after the organisation's creation in 1992, but within five years had been transformed to a "successful and over-subscribed school" — a remarkable turnaround that saw head teacher Geoff Hampton knighted for his services to education. Sir Geoff has since departed for a Professor's role at University of Wolverhampton. The last headteacher of the school was Mr R. Davis.
The Northicote School was built as a bilateral school, having both secondary modern and grammar streams during the 1950s to serve the expanding Bushbury area of Wolverhampton, though during the 1970s it converted to a comprehensive school. The school was informed in 2007 that it was being merged with Pendeford Business and Enterprise College to form an academy under controversial plans.
In the academic year 2010–11 the school merged with Pendeford Business and Enterprise College to become the North East Wolverhampton Academy. The combined school was originally located over both former school sites before relocating to a newly constructed and refurbished campus in September 2014 at the former Pendeford Business and Enterprise College site. Northicote Campus was subsequently demolished although a few months later contractors working for the council repainted the "School Keep Clear" markings outside the former site.
External links
North East Wolverhampton Academy website
Defunct schools in Wolverhampton
Educational institutions disestablished in 2011
2011 disestablishments in England |
https://en.wikipedia.org/wiki/Mihajlo%20D.%20Mesarovic | Mihajlo D. Mesarovic (Serbian Latin: Mihajlo D. Mesarović, Serbian Cyrillic: Михајло Д. Месаровић; born 2 July 1928) is a Serbian scientist, who is a professor of Systems Engineering and Mathematics at Case Western Reserve University. Mesarovic has been a pioneer in the field of systems theory, he was UNESCO Scientific Advisor on Global change and also a member of the Club of Rome.
Biography
Mihajlo D. Mesarović was born on 2 July 1928, in Zrenjanin, Yugoslavia. He was awarded the B.S. from the University of Belgrade Faculty of Electrical Engineering in 1951. In 1955 he received a Ph.D. in Technical sciences from the Serbian Academy of Sciences and Arts.
From 1951 to 1955, Mesarović was a research assistant at the Nikola Tesla Institute in Belgrade. From 1955 to 1958 he was head of the inspection department of the Institute. At the same time, Mesarović held academic positions at University of Belgrade, Yugoslavia from 1954 to 1958. In 1958 he became professor at the Massachusetts Institute of Technology / USA (MIT), where he served until 1959. He was associate professor at Case Western Reserve University from 1959 to 1964 and professor from 1964 to 1978. In that time he was head of the Systems Engineering Group 1965–68, head of the Systems Engineering Department 1968-72 and director of the Systems Research Center 1968–78. Starting 1978, Prof. Mesarović has been the Cady Staley Professor of Systems Engineering and Mathematics. One of his students was Roger W. Brockett.
He has lectured in more than 60 countries, advised government officials on a variety of issues, consulted for international organizations, and published widely. He was also the founder of the 'Mathematical Theory of General Systems' Journal, Springer Verlag.
In 1999, he was appointed a Scientific Advisor on Global Change by Federico Mayor, Director-General of the UNESCO. In that role, Mesarović traveled to UNESCO's headquarters in Paris and advised the director general's office on issues such as climate change, economics, population, technology transfer, and the education of women in developing countries.
In 2005 he was awarded the Hovorka Prize from Case Western Reserve University for exceptional achievements.
In 2005 he was awarded the USA Club of Rome Lifetime Achievement Award at the United Nations.
Work
His research interests include the areas and topics like complexity, complex systems theory, global change and sustainable human development, hierarchical systems, large-scale systems theory, mathematical theory of general systems, multi-level systems, systems biology, and world and regional modeling. In the field of mathematics he is considered to be founder of:
Mathematical theory of coordination
Multi-level Hierarchical Systems Developer,
Negotiation Support Software System
Publications
Mesarovic published several books and numerous articles. A selection:
1960. Multi-variable Control Systems. MIT Press.
1962. General Systems Theory and Systems Research Contrast |
https://en.wikipedia.org/wiki/W%C5%82odzimierz%20Kuperberg | Włodzimierz Kuperberg (born January 19, 1941) is a professor of mathematics at Auburn University, with research interests in geometry and topology.
Biography
Although Kuperberg is Polish-American, he was born in what is now Belarus, where his parents and older siblings had traveled east to escape World War II. In 1946, the family returned to Poland, resettling in Szczecin, where Kuperberg grew up. He began his studies at the University of Warsaw in 1959, and received his Ph.D. from the same institution in 1969, under the supervision of Karol Borsuk. During his time at Warsaw, he published three high school textbooks in Polish. Kuperberg left Poland due to the anti-semitic aspects of the 1967-1968 Polish political crisis, and worked at Stockholm University until 1972, when he assumed a visiting position at the University of Houston. In 1974, Kuperberg took a position at Auburn where he remains.
Kuperberg married mathematician Krystyna Kuperberg in 1964, and their son Greg Kuperberg is also a professional mathematician, while their daughter Anna Kuperberg is a photographer.
Research highlights
Although much of Kuperberg's early mathematical work is in topology, he is best known today for his work in geometry, and in particular on packing and covering problems. His first paper in this area (1982) showed that the ratio of packing density to covering density of any convex body in the plane is at least 3/4. His 1990 paper on double lattices with his son Greg provides the best lower bound known at that time for packing densities of arbitrary two-dimensional convex bodies; with Bezdek (1990) he calculated the exact packing density of the infinite cylinder, which prior to Hales' 1998 solution of the Kepler conjecture was the first nontrivial calculation of the packing density of any three-dimensional convex body.
Awards and honors
As a high school student, Kuperberg won first prize in the 10th Polish Mathematical Olympiad, leading him to enroll in mathematics when he began his college studies. While at the University of Warsaw he received both the university's Excellence in Teaching and Research Award and the Polish Mathematical Society Award for Young Mathematicians. He was honored again at Auburn by a five-year Alumni Professor chairship in 1996, and again by an Erdős Professorship in 1999, which he used to visit the Hungarian Academy of Sciences in Budapest. In 2003, his colleagues presented him with a festschrift edited by András Bezdek, consisting of 34 papers in discrete geometry.
References
.
Selected publications
.
.
.
External links
Kuperberg's home page at Auburn U.
Geometers
Topologists
20th-century Polish mathematicians
21st-century Polish mathematicians
20th-century American mathematicians
21st-century American mathematicians
University of Warsaw alumni
Polish emigrants to the United States
University of Houston staff
Auburn University faculty
Living people
1941 births |
https://en.wikipedia.org/wiki/Rick%20Hodgson | Richard S. Hodgson (born May 23, 1956) is a Canadian former professional ice hockey defenceman. He played 6 games for the Hartford Whalers during the 1979–80 season.
Career statistics
Regular season and playoffs
International
External links
1956 births
Living people
Atlanta Flames draft picks
Calgary Centennials players
Canadian ice hockey defencemen
Hartford Whalers players
Kamloops Rockets players
San Diego Mariners draft picks
Ice hockey people from Medicine Hat
Springfield Indians players
Tulsa Oilers (1964–1984) players |
https://en.wikipedia.org/wiki/Applied%20Probability%20Trust | The Applied Probability Trust is a UK-based non-profit foundation for study and research in the mathematical sciences, founded in 1964 and based in the School of Mathematics and Statistics at the University of Sheffield, which it has been affiliated with since 1964.
Publications
The Applied Probability Trust (APT) published two world leading research journals, the Journal of Applied Probability and Advances in Applied Probability, until 2016. Joe Gani, founding editor for the two journals, intended to create outlets for researchers in applied probability, as they increasingly had difficulty in getting published in the few journals in probability and statistics that existed at that time. The Journal of Applied Probability appeared first, in 1964, and with a prominent editorial board from the beginning, it secured contributions from renowned probabilists. The Advances in Applied Probability started in 1969. In 2016, Cambridge University Press took over the publication of the two journals.
In addition to these two journals, two further magazine style publications have been published, The Mathematical Scientist and Mathematical Spectrum.
Journal of Applied Probability (1964 – present)
Advances in Applied Probability (1972 – present)
The Mathematical Scientist (1976 – 2018)
Mathematical Spectrum (1968 – 2016)
To mark special occasions, the Applied Probability Trust commissions special issues of the journal. These include:
Perspectives in Probability and Statistics (1975)
Essays in Statistical Science (1982)
Essays in Time Series and Allied Processes (1986)
A Celebration of Applied Probability (1988)
Studies in Applied Probability (1994)
Probability, Statistics and Seismology (2001)
Stochastic Methods and Their Applications (2004)
New Frontiers in Applied Probability (2011)
Celebrating 50 Years of the Applied Probability Trust (2014)
Probability, Analysis and Number Theory (2016)
Branching and Applied Probability (2018)
Board of Trustees, Past and Present
The Applied Probability Trust was set up by Joe Gani in 1964 along with Norma McArthur, Edward Hannan and support from the London Mathematical Society. Over the history of the APT, many world renowned probabilists have accepted the invitation to join the APT board of trustees. A complete list of trustees, past and present can be found below:
Joe Gani (1964 – 2016)
Norma McArthur (1964 – 1984)
Edward J. Hannan (1964 – 1994)
London Mathematical Society (1964 – 2008)
Chris Heyde (1984 – 2008)
Daryl Daley, Australian National University (1997 – present)
Søren Asmussen (2008 – 2020)
Peter Taylor, University of Melbourne (2008 – present)
Frank Kelly (2008 – 2021)
Peter Glynn, Stanford University (2014 – present)
Ilya Molchanov, University of Bern (2019 – present)
Jiangang (Jim) Dai, Cornell University (2019 – present)
Remco van der Hofstad, Eindhoven University of Technology (2019 – present)
Christina Goldschmidt, Oxford University (2021 – present)
Nigel Bean, University |
https://en.wikipedia.org/wiki/Pollard%27s%20kangaroo%20algorithm | In computational number theory and computational algebra, Pollard's kangaroo algorithm (also Pollard's lambda algorithm, see Naming below) is an algorithm for solving the discrete logarithm problem. The algorithm was introduced in 1978 by the number theorist John M. Pollard, in the same paper as his better-known Pollard's rho algorithm for solving the same problem. Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a generic discrete logarithm algorithm—it will work in any finite cyclic group.
Algorithm
Suppose is a finite cyclic group of order which is generated by the element , and we seek to find the discrete logarithm of the element to the base . In other words, one seeks such that . The lambda algorithm allows one to search for in some interval . One may search the entire range of possible logarithms by setting and .
1. Choose a set of positive integers of mean roughly and define a pseudorandom map .
2. Choose an integer and compute a sequence of group elements according to:
3. Compute
Observe that:
4. Begin computing a second sequence of group elements according to:
and a corresponding sequence of integers according to:
.
Observe that:
5. Stop computing terms of and when either of the following conditions are met:
A) for some . If the sequences and "collide" in this manner, then we have:
and so we are done.
B) . If this occurs, then the algorithm has failed to find . Subsequent attempts can be made by changing the choice of and/or .
Complexity
Pollard gives the time complexity of the algorithm as , using a probabilistic argument based on the assumption that acts pseudorandomly. Since can be represented using bits, this is exponential in the problem size (though still a significant improvement over the trivial brute-force algorithm that takes time ). For an example of a subexponential time discrete logarithm algorithm, see the index calculus algorithm.
Naming
The algorithm is well known by two names.
The first is "Pollard's kangaroo algorithm". This name is a reference to an analogy used in the paper presenting the algorithm, where the algorithm is explained in terms of using a tame kangaroo to trap a wild kangaroo. Pollard has explained that this analogy was inspired by a "fascinating" article published in the same issue of Scientific American as an exposition of the RSA public key cryptosystem. The article described an experiment in which a kangaroo's "energetic cost of locomotion, measured in terms of oxygen consumption at various speeds, was determined by placing kangaroos on a treadmill".
The second is "Pollard's lambda algorithm". Much like the name of another of Pollard's discrete logarithm algorithms, Pollard's rho algorithm, this name refers to the similarity between a visualisation of the algorithm and the Greek letter lambda (). The shorter stroke of the letter lambda co |
https://en.wikipedia.org/wiki/Dixmier%20conjecture | In algebra the Dixmier conjecture, asked by Jacques Dixmier in 1968, is the conjecture that any endomorphism of a Weyl algebra is an automorphism.
Tsuchimoto in 2005, and independently Belov-Kanel and Kontsevich in 2007, showed that the Dixmier conjecture is stably equivalent to the Jacobian conjecture.
References
Abstract algebra
Conjectures
Unsolved problems in mathematics |
https://en.wikipedia.org/wiki/Functional-theoretic%20algebra | Any vector space can be made into a unital associative algebra, called functional-theoretic algebra, by defining products in terms of two linear functionals. In general, it is a non-commutative algebra. It becomes commutative when the two functionals are the same.
Definition
Let AF be a vector space over a field F, and let L1 and L2 be two linear functionals on AF with the property L1(e) = L2(e) = 1F for some e in AF. We define multiplication of two elements x, y in AF by
It can be verified that the above multiplication is associative and that e is the identity of this multiplication.
So, AF forms an associative algebra with unit e and is called a functional theoretic algebra(FTA).
Suppose the two linear functionals L1 and L2 are the same, say L. Then AF becomes a commutative algebra with multiplication defined by
Example
X is a nonempty set and F a field. FX is the set of functions from X to F.
If f, g are in FX, x in X and α in F, then define
and
With addition and scalar multiplication defined as this, FX is a vector space over F.
Now, fix two elements a, b in X and define a function e from X to F by e(x) = 1F for all x in X.
Define L1 and L2 from FX to F by L1(f) = f(a) and L2(f) = f(b).
Then L1 and L2 are two linear functionals on FX such that L1(e)= L2(e)= 1F
For f, g in FX define
Then FX becomes a non-commutative function algebra with the function e as the identity of multiplication.
Note that
FTA of Curves in the Complex Plane
Let C denote the field of
Complex numbers.
A continuous function γ from the closed
interval [0, 1] of real numbers to the field C is called a
curve. The complex numbers γ(0) and γ(1) are, respectively,
the initial and terminal points of the curve.
If they coincide, the
curve is called a loop.
The set V[0, 1] of all the curves is a
vector space over C.
We can make this vector space of curves into an
algebra by defining multiplication as above.
Choosing we have for α,β in C[0, 1],
Then, V[0, 1] is a non-commutative algebra with e as the unity.
We illustrate
this with an example.
Example of f-Product of Curves
Let us take (1) the line segment joining the points (1, 0) and (0, 1) and (2) the unit circle with center at the
origin.
As curves in V[0, 1], their equations can be obtained as
Since the circle g
is a loop.
The line segment f starts from :
and ends at
Now, we get two f-products
given by
and
See the Figure.
Observe that showing that
multiplication is non-commutative. Also both the products starts from
See also
N-curve
References
Sebastian Vattamattam and R. Sivaramakrishnan, A Note on Convolution Algebras, in Recent Trends in Mathematical Analysis, Allied Publishers, 2003.
Sebastian Vattamattam and R. Sivaramakrishnan, Associative Algebras via Linear Functionals, Proceedings of the Annual Conference of K.M.A., Jan. 17 - 19, 2000, pp. 81-89
Sebastian Vattamattam, Non-Commutative Function Algebras, in Bulletin of Kerala Mathematical Association, Vol. 4, |
https://en.wikipedia.org/wiki/Kopalasingham%20Sritharan | Kopalasingham Sritharan is a Tamil Human Rights activist who along with Rajan Hoole ran the University Teachers for Human Rights while affiliated to the Department of Mathematics, University of Jaffna. He was awarded the prestigious Martin Ennals Award for Human Rights Defenders in 2007 along with Rajan Hoole for his work in documenting Human Rights violations and abuses in the civil conflict in Sri Lanka. He also worked in Afghanistan and Nepal in UN missions as a Human Rights Officer and a Civil Affairs Officer respectively. He is a co-author of Broken Palmyra, which was the first book published in English analyzing the violent nature of politics in the North and East of Sri Lanka and its effects on civilians. Since the murder of co-author Rajani Thiranagama, he has rarely appeared in public. His place of residence is not known to many and he does not divulge this in fear of reprisals.
References
Sri Lankan Tamil activists
Sri Lankan human rights activists
Sri Lankan Hindus
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Stein%27s%20method | Stein's method is a general method in probability theory to obtain bounds on the distance between two probability distributions with respect to a probability metric. It was introduced by Charles Stein, who first published it in 1972, to obtain a bound between the distribution of a sum of -dependent sequence of random variables and a standard normal distribution in the Kolmogorov (uniform) metric and hence to prove not only a central limit theorem, but also bounds on the rates of convergence for the given metric.
History
At the end of the 1960s, unsatisfied with the by-then known proofs of a specific central limit theorem, Charles Stein developed a new way of proving the theorem for his statistics lecture. His seminal paper was presented in 1970 at the sixth Berkeley Symposium and published in the corresponding proceedings.
Later, his Ph.D. student Louis Chen Hsiao Yun modified the method so as to obtain approximation results for the Poisson distribution; therefore the Stein method applied to the problem of Poisson approximation is often referred to as the Stein–Chen method.
Probably the most important contributions are the monograph by Stein (1986), where he presents his view of the method and the concept of auxiliary randomisation, in particular using exchangeable pairs, and the articles by Barbour (1988) and Götze (1991), who introduced the so-called generator interpretation, which made it possible to easily adapt the method to many other probability distributions. An important contribution was also an article by Bolthausen (1984) on the so-called combinatorial central limit theorem.
In the 1990s the method was adapted to a variety of distributions, such as Gaussian processes by Barbour (1990), the binomial distribution by Ehm (1991), Poisson processes by Barbour and Brown (1992), the Gamma distribution by Luk (1994), and many others.
The method gained further popularity in the machine learning community in the mid 2010s, following the development of computable Stein discrepancies and the diverse applications and algorithms based on them.
The basic approach
Probability metrics
Stein's method is a way to bound the distance between two probability distributions using a specific probability metric.
Let the metric be given in the form
Here, and are probability measures on a measurable space , and are random variables with distribution and respectively, is the usual expectation operator and is a set of functions from to the set of real numbers. Set has to be large enough, so that the above definition indeed yields a metric.
Important examples are the total variation metric, where we let consist of all the indicator functions of measurable sets, the Kolmogorov (uniform) metric for probability measures on the real numbers, where we consider all the half-line indicator functions, and the Lipschitz (first order Wasserstein; Kantorovich) metric, where the underlying space is itself a metric space and we take the set to be all Lip |
https://en.wikipedia.org/wiki/Lune%20%28geometry%29 | In plane geometry, a lune () is the concave-convex region bounded by two circular arcs. It has one boundary portion for which the connecting segment of any two nearby points moves outside the region and another boundary portion for which the connecting segment of any two nearby points lies entirely inside the region. A convex-convex region is termed a lens.
Formally, a lune is the relative complement of one disk in another (where they intersect but neither is a subset of the other). Alternatively, if and are disks, then is a lune.
Squaring the lune
In the 5th century BC, Hippocrates of Chios showed that the Lune of Hippocrates and two other lunes could be exactly squared (converted into a square having the same area) by straightedge and compass. In 1766 the Finnish mathematician Daniel Wijnquist, quoting Daniel Bernoulli, listed all five geometrical squareable lunes, adding to those known by Hippocrates. In 1771 Leonard Euler gave a general approach and obtained certain equation to the problem. In 1933 and 1947 it was proven by Nikolai Chebotaryov and his student Anatoly Dorodnov that these five are the only squarable lunes.
Area
The area of a lune formed by circles of radii a and b (b>a) with distance c between their centers is
where is the inverse function of the secant function, and where
is the area of a triangle with sides a, b and c.
See also
Arbelos
Crescent
Gauss–Bonnet theorem
Lens
References
External links
The Five Squarable Lunes at MathPages
Piecewise-circular curves |
https://en.wikipedia.org/wiki/Quasithin%20group | In mathematics, a quasithin group is a finite simple group that resembles a group of Lie type of rank at most 2 over a field of characteristic 2. More precisely it is a finite simple group of characteristic 2 type and width 2. Here characteristic 2 type means that its centralizers of involutions resemble those of groups of Lie type over fields of characteristic 2, and the width is roughly the maximal rank of an abelian group of odd order normalizing a non-trivial 2-subgroup of G. When G is a group of Lie type of characteristic 2 type, the width is usually the rank (the dimension of a maximal torus of the algebraic group).
Classification
The classification of quasithin groups is a crucial part of the classification of finite simple groups. The quasithin groups were classified in a 1221-page paper by . An earlier announcement by of the classification, on the basis of which the classification of finite simple groups was announced as finished in 1983, was premature as the unpublished manuscript of his work was incomplete and contained serious gaps.
According to , the finite simple quasithin groups of even characteristic are given by
Groups of Lie type of characteristic 2 and rank 1 or 2, except that U5(q) only occurs for q = 4
PSL4(2), PSL5(2), Sp6(2)
The alternating groups on 5, 6, 8, 9, points
PSL2(p) for p a Fermat or Mersenne prime, L(3), L(3), G2(3)
The Mathieu groups M11, M12, M22, M23, M24, The Janko groups J2, J3, J4, the Higman-Sims group, the Held group, and the Rudvalis group.
If the condition "even characteristic" is relaxed to "even type" in the sense of the revision of the classification by Daniel Gorenstein, Richard Lyons, and Ronald Solomon, then the only extra group that appears is the Janko group J1.
References
(unpublished typescript)
Finite groups |
https://en.wikipedia.org/wiki/Fumiya%20Nishiguchi | , (born September 26, 1972) is a Japanese baseball player. He is a right-handed pitcher in Japan's Nippon Professional Baseball for the Saitama Seibu Lions.
Career statistics
Bold indicates league leader; statistics current as of December 25, 2013
External links
1972 births
Living people
Sportspeople from Wakayama (city)
Japanese baseball players
Nippon Professional Baseball pitchers
Seibu Lions players
Saitama Seibu Lions players
Nippon Professional Baseball MVP Award winners
Japanese baseball coaches
Nippon Professional Baseball coaches |
https://en.wikipedia.org/wiki/Local%20flatness | In topology, a branch of mathematics, local flatness is a smoothness condition that can be imposed on topological submanifolds. In the category of topological manifolds, locally flat submanifolds play a role similar to that of embedded submanifolds in the category of smooth manifolds. Violations of local flatness describe ridge networks and crumpled structures, with applications to materials processing and mechanical engineering.
Definition
Suppose a d dimensional manifold N is embedded into an n dimensional manifold M (where d < n). If we say N is locally flat at x if there is a neighborhood of x such that the topological pair is homeomorphic to the pair , with the standard inclusion of That is, there exists a homeomorphism such that the image of coincides with . In diagrammatic terms, the following square must commute:
We call N locally flat in M if N is locally flat at every point. Similarly, a map is called locally flat, even if it is not an embedding, if every x in N has a neighborhood U whose image is locally flat in M.
In manifolds with boundary
The above definition assumes that, if M has a boundary, x is not a boundary point of M. If x is a point on the boundary of M then the definition is modified as follows. We say that N is locally flat at a boundary point x of M if there is a neighborhood of x such that the topological pair is homeomorphic to the pair , where is a standard half-space and is included as a standard subspace of its boundary.
Consequences
Local flatness of an embedding implies strong properties not shared by all embeddings. Brown (1962) proved that if d = n − 1, then N is collared; that is, it has a neighborhood which is homeomorphic to N × [0,1] with N itself corresponding to N × 1/2 (if N is in the interior of M) or N × 0 (if N is in the boundary of M).
See also
Euclidean space
Neat submanifold
References
Brown, Morton (1962), Locally flat imbeddings of topological manifolds. Annals of Mathematics, Second series, Vol. 75 (1962), pp. 331–341.
Mazur, Barry. On embeddings of spheres. Bulletin of the American Mathematical Society, Vol. 65 (1959), no. 2, pp. 59–65. http://projecteuclid.org/euclid.bams/1183523034.
Topology
Geometric topology |
https://en.wikipedia.org/wiki/CAT%28k%29%20space | In mathematics, a space, where is a real number, is a specific type of metric space. Intuitively, triangles in a space are "slimmer" than corresponding "model triangles" in a standard space of constant curvature . In a space, the curvature is bounded from above by . A notable special case is ; complete spaces are known as "Hadamard spaces" after the French mathematician Jacques Hadamard.
Originally, Aleksandrov called these spaces “ domain”.
The terminology was coined by Mikhail Gromov in 1987 and is an acronym for Élie Cartan, Aleksandr Danilovich Aleksandrov and Victor Andreevich Toponogov (although Toponogov never explored curvature bounded above in publications).
Definitions
For a real number , let denote the unique complete simply connected surface (real 2-dimensional Riemannian manifold) with constant curvature . Denote by the diameter of , which is if and is if .
Let be a geodesic metric space, i.e. a metric space for which every two points can be joined by a geodesic segment, an arc length parametrized continuous curve , whose length
is precisely . Let be a triangle in with geodesic segments as its sides. is said to satisfy the inequality if there is a comparison triangle in the model space , with sides of the same length as the sides of , such that distances between points on are less than or equal to the distances between corresponding points on .
The geodesic metric space is said to be a space if every geodesic triangle in with perimeter less than satisfies the inequality. A (not-necessarily-geodesic) metric space is said to be a space with curvature if every point of has a geodesically convex neighbourhood. A space with curvature may be said to have non-positive curvature.
Examples
Any space is also a space for all . In fact, the converse holds: if is a space for all , then it is a space.
The -dimensional Euclidean space with its usual metric is a space. More generally, any real inner product space (not necessarily complete) is a space; conversely, if a real normed vector space is a space for some real , then it is an inner product space.
The -dimensional hyperbolic space with its usual metric is a space, and hence a space as well.
The -dimensional unit sphere is a space.
More generally, the standard space is a space. So, for example, regardless of dimension, the sphere of radius (and constant curvature ) is a space. Note that the diameter of the sphere is (as measured on the surface of the sphere) not (as measured by going through the centre of the sphere).
The punctured plane is not a space since it is not geodesically convex (for example, the points and cannot be joined by a geodesic in with arc length 2), but every point of does have a geodesically convex neighbourhood, so is a space of curvature .
The closed subspace of given by equipped with the induced length metric is not a space for any .
Any product of spaces is . (This does not hold for negative |
https://en.wikipedia.org/wiki/CAT%28k%29%20group | In mathematics, a CAT(k) group is a group that acts discretely, cocompactly and isometrically on a CAT(k) space.
References
Group actions (mathematics) |
https://en.wikipedia.org/wiki/Gromov%20product | In mathematics, the Gromov product is a concept in the theory of metric spaces named after the mathematician Mikhail Gromov. The Gromov product can also be used to define δ-hyperbolic metric spaces in the sense of Gromov.
Definition
Let (X, d) be a metric space and let x, y, z ∈ X. Then the Gromov product of y and z at x, denoted (y, z)x, is defined by
Motivation
Given three points x, y, z in the metric space X, by the triangle inequality there exist non-negative numbers a, b, c such that . Then the Gromov products are . In the case that the points x, y, z are the outer nodes of a tripod then these Gromov products are the lengths of the edges.
In the hyperbolic, spherical or euclidean plane, the Gromov product (A, B)C equals the distance p between C and the point where the incircle of the geodesic triangle ABC touches the edge CB or CA. Indeed from the diagram , so that . Thus for any metric space, a geometric interpretation of (A, B)C is obtained by isometrically embedding (A, B, C) into the euclidean plane.
Properties
The Gromov product is symmetric: (y, z)x = (z, y)x.
The Gromov product degenerates at the endpoints: (y, z)y = (y, z)z = 0.
For any points p, q, x, y and z,
Points at infinity
Consider hyperbolic space Hn. Fix a base point p and let and be two distinct points at infinity. Then the limit
exists and is finite, and therefore can be considered as a generalized Gromov product. It is actually given by the formula
where is the angle between the geodesic rays and .
δ-hyperbolic spaces and divergence of geodesics
The Gromov product can be used to define δ-hyperbolic spaces in the sense of Gromov.: (X, d) is said to be δ-hyperbolic if, for all p, x, y and z in X,
In this case. Gromov product measures how long geodesics remain close together. Namely, if x, y and z are three points of a δ-hyperbolic metric space then the initial segments of length (y, z)x of geodesics from x to y and x to z are no further than 2δ apart (in the sense of the Hausdorff distance between closed sets).
Notes
References
Metric geometry
Hyperbolic metric space |
https://en.wikipedia.org/wiki/Elias%20Charalambous | Elias Charalambous (; born 25 September 1980) is a Cypriot football manager and former player, who is in charge of Liga I club FCSB.
Career statistics
International
Managerial
Honours
Omonia
Cypriot First Division: 2000–01, 2002–03, 2009–10
Cypriot Cup: 1999–2000, 2004–05, 2010–11
Cypriot Super Cup: 2001, 2003, 2010
PAOK
Greek Cup runner-up: 2005–06
References
External links
1980 births
Living people
Sportspeople from East London, South Africa
Soccer players from the Eastern Cape
South African people of Greek Cypriot descent
Greek Cypriot people
Cypriot men's footballers
Cyprus men's international footballers
Men's association football defenders
AC Omonia players
PAOK FC players
Alki Larnaca FC players
CS Sporting Vaslui players
Karlsruher SC players
Doxa Katokopias FC players
Levadiakos F.C. players
AEK Larnaca FC players
Cypriot First Division players
Liga I players
Super League Greece players
2. Bundesliga players
Cypriot expatriate men's footballers
Cypriot expatriate sportspeople in Greece
Cypriot expatriate sportspeople in Romania
Cypriot expatriate sportspeople in Germany
Expatriate men's footballers in Greece
Expatriate men's footballers in Germany
Expatriate men's footballers in Romania
White South African people
Cypriot football managers
AEK Larnaca FC managers
Ethnikos Achna FC managers
Doxa Katokopias FC managers
FC Steaua București managers
Cypriot First Division managers
Liga I managers
Cypriot expatriate football managers
Expatriate football managers in Romania |
https://en.wikipedia.org/wiki/Algebraic%20topology%20%28object%29 | In mathematics, the algebraic topology on the set of group representations from G to a topological group H is the topology of pointwise convergence, i.e. pi converges to p if the limit of pi(g) = p(g) for every g in G.
This terminology is often used in the case of the algebraic topology on the set of discrete, faithful representations of a Kleinian group into PSL(2,C). Another topology, the geometric topology (also called the Chabauty topology), can be put on the set of images of the representations, and its closure can include extra Kleinian groups that are not images of points in the closure in the algebraic topology. This fundamental distinction is behind the phenomenon of hyperbolic Dehn surgery and plays an important role in the general theory of hyperbolic 3-manifolds.
References
William Thurston, The geometry and topology of 3-manifolds, Princeton lecture notes (1978–1981).
3-manifolds
Algebraic topology |
https://en.wikipedia.org/wiki/Per%20Wimmer | Per Wimmer (born 1968) is a Danish space advocate, entrepreneur, financier and author.
Education
Per Wimmer graduated in 1987 with concentrations in mathematics and physics section from Slagelse Gymnasium. In 1988, Per Wimmer has a French Baccalaureate with concentrations in philosophy and French literature.
Per Wimmer graduated with a Master of Laws degree from University of London in 1991-1992 and in 1988-1993 he received Bachelor of Arts and Master of Arts degrees in law from University of Copenhagen whilst serving as student member of the Governing Board of the University of Copenhagen Faculty of Law.
Wimmer graduated with a Master of Public Administration degree from Harvard University with concentrations in business, finance and international relations.
Career
Until the end of the Jacques Delors administration in early 1996, Per Wimmer worked in the cabinet of the Vice-President of the European Commission, Mr. Henning Christophersen, in a junior capacity. The VP responsibilities included the Monetary Union and the EU budget.
During 1996-1997 Per Wimmer was an Associate management consultant at McKinsey & Co. with a particular focus in the media sector.
In 1998-2002, prior to founding his London-based corporate advisory firm, he worked in New York City and London for Goldman Sachs leaving the company as Executive Director for Institutional Sales of European Equity products advising on investments to Scandinavian-based financial institutions. In 2002, Wimmer left Goldman Sachs in favor of similar positions at i.a. Collins Stewart and MF Global/Man Securities, the latter of which is part of the world's largest hedge fund Man Group.
Wimmer owns and administers his own international corporate financial advisory firm, Wimmer Financial, which he founded on the 50th anniversary of Sputnik day, October 4, 2007. Wimmer Financial specializes in global corporate finance, shares trading, real estate and natural resources.
Other occupations
On Oct 6, 2008, Per Wimmer participated in the World's first tandem skydive above Mount Everest, the highest point on Earth, with Ralph Mitchell as tandem master.
Wimmer is a founding astronaut with Sir Richard Branson's Virgin Galactic. In 2001 holds a trip to space reservation with Space Adventures. In December 4, 2008 held a ticket to be the first astronaut on board the XCOR Lynx rocketplane (now canceled). He was expecting to be the first Danish citizen to enter space on board SpaceShipTwo, until Andreas Mogensen did it in 2015.
Publications
In September 2011, Per Wimmer published the book "Wall Street", about the bubbles in the financial markets, with anecdotes from his career in the world of global finance. In June 2014 he published his second book "The Green Bubble" in Sweden for the think tank Timbro, which argues there is a green bubble in the renewable energy sector.
References
External links
Space.com: Space Travelers Gather in Croatia for Historic Summit
Danish philanthropists
1968 births |
https://en.wikipedia.org/wiki/Victor%20Andreevich%20Toponogov | Victor Andreevich Toponogov (; March 6, 1930 – November 21, 2004) was an outstanding Russian mathematician, noted for his contributions to differential geometry and so-called Riemannian geometry "in the large".
Biography
After finishing secondary school in 1948, Toponogov entered the department of Mechanics and Mathematics at Tomsk State University, graduated with honours in 1953, and continued as a graduate student there until 1956. He moved to an institution in Novosibirsk in 1956 and lived in that city for the rest of his career. Since the institution at Novosibirsk had not yet been fully credentialed, he had defended his Ph.D. thesis at Moscow State University in 1958, on a subject in Riemann spaces. Novosibirsk State University was established in 1959. In 1961 Toponogov became a professor at a newly created Institute of Mathematics and Computing in Novosibirsk affiliated with the state university.
Toponogov's scientific interests were influenced by his advisor Abram Fet, who taught at Tomsk and later at Novosibirsk. Fet was a well-recognized topologist and specialist in variational calculus in the large. Toponogov's work was also strongly influenced by the work of Aleksandr Danilovich Aleksandrov. Later, the class of metric spaces known as CAT(k) spaces would be named after Élie Cartan, Aleksandrov and Toponogov.
Toponogov published over forty papers and some books during his career. His works are concentrated in Riemannian geometry "in the large". A significant number of his students also made notable contributions in this field.
Conjecture on Complete Convex Surfaces
In 1995 Toponogov made the conjecture:
On a complete convex surface S homeomorphic to a plane the following equality holds:
where and are the principal curvatures of S.
In words, it states that every complete convex surface homeomorphic to a plane must have an umbilic point which may lie at infinity. As such, it is the natural open analog of the Carathéodory conjecture for closed convex surfaces.
In the same paper, Toponogov proved the conjecture under either of two assumptions: the integral of the Gauss curvature is less than , or the Gauss curvature and the gradients of the curvatures are bounded on S. The general case remains open.
See also
Toponogov's theorem
References
External links
Toponogov's Biography, including a list of his publications (in English)
1930 births
2004 deaths
20th-century Russian mathematicians
21st-century Russian mathematicians
Moscow State University alumni
Tomsk State University alumni
People from Tomsk
Academic staff of Novosibirsk State University
Differential geometers
Soviet mathematicians
Burials at Yuzhnoye Cemetery (Novosibirsk) |
https://en.wikipedia.org/wiki/Daniel%20Sanchez%20%28French%20footballer%29 | Daniel Sanchez (born 21 November 1953) is a French football manager and former professional player who played as a striker.
Managerial statistics
References
External links
Profile
Profile
1953 births
Living people
French people of Spanish descent
French sportspeople of Moroccan descent
People from Oujda
French men's footballers
Men's association football forwards
OGC Nice players
Paris Saint-Germain F.C. players
FC Mulhouse players
AS Saint-Étienne players
AS Cannes players
Ligue 1 players
French football managers
OGC Nice managers
Tours FC managers
Valenciennes FC managers
Nagoya Grampus managers
Club Africain football managers
Ligue 1 managers
J1 League managers
Tunisian Ligue Professionnelle 1 managers
French expatriate football managers
French expatriate sportspeople in Japan
French expatriate sportspeople in Tunisia
Expatriate football managers in Japan
Expatriate football managers in Tunisia |
https://en.wikipedia.org/wiki/Effective%20method | In logic, mathematics and computer science, especially metalogic and computability theory, an effective method or effective procedure is a procedure for solving a problem by any intuitively 'effective' means from a specific class. An effective method is sometimes also called a mechanical method or procedure.
Definition
The definition of an effective method involves more than the method itself. In order for a method to be called effective, it must be considered with respect to a class of problems. Because of this, one method may be effective with respect to one class of problems and not be effective with respect to a different class.
A method is formally called effective for a class of problems when it satisfies these criteria:
It consists of a finite number of exact, finite instructions.
When it is applied to a problem from its class:
It always finishes (terminates) after a finite number of steps.
It always produces a correct answer.
In principle, it can be done by a human without any aids except writing materials.
Its instructions need only to be followed rigorously to succeed. In other words, it requires no ingenuity to succeed.
Optionally, it may also be required that the method never returns a result as if it were an answer when the method is applied to a problem from outside its class. Adding this requirement reduces the set of classes for which there is an effective method.
Algorithms
An effective method for calculating the values of a function is an algorithm. Functions for which an effective method exists are sometimes called effectively calculable.
Computable functions
Several independent efforts to give a formal characterization of effective calculability led to a variety of proposed definitions (general recursive functions, Turing machines, λ-calculus) that later were shown to be equivalent. The notion captured by these definitions is known as recursive or effective computability.
The Church–Turing thesis states that the two notions coincide: any number-theoretic function that is effectively calculable is recursively computable. As this is not a mathematical statement, it cannot be proven by a mathematical proof.
See also
Decidability (logic)
Decision problem
Function problem
Effective results in number theory
Recursive set
Undecidable problem
References
S. C. Kleene (1967), Mathematical logic. Reprinted, Dover, 2002, , pp. 233 ff., esp. p. 231.
Metalogic
Computability theory
Theory of computation |
https://en.wikipedia.org/wiki/Serial%20relation | In set theory a serial relation is a homogeneous relation expressing the connection of an element of a sequence to the following element. The successor function used by Peano to define natural numbers is the prototype for a serial relation.
Bertrand Russell used serial relations in The Principles of Mathematics (1903) as he explored the foundations of order theory and its applications. The term serial relation was also used by B. A. Bernstein for an article showing that particular common axioms in order theory are nearly incompatible: connectedness, irreflexivity, and transitivity.
A serial relation R is an endorelation on a set U. As stated by Russell, where the universal and existential quantifiers refer to U. In contemporary language of relations, this property defines a total relation. But a total relation may be heterogeneous. Serial relations are of historic interest.
For a relation R, let {y: xRy } denote the "successor neighborhood" of x. A serial relation can be equivalently characterized as a relation for which every element has a non-empty successor neighborhood. Similarly, an inverse serial relation is a relation in which every element has non-empty "predecessor neighborhood".
In normal modal logic, the extension of fundamental axiom set K by the serial property results in axiom set D.
Russell's series
Relations are used to develop series in The Principles of Mathematics. The prototype is Peano's successor function as a one-one relation on the natural numbers. Russell's series may be finite or generated by a relation giving cyclic order. In that case, the point-pair separation relation is used for description. To define a progression, he requires the generating relation to be a connected relation. Then ordinal numbers are derived from progressions, the finite ones are finite ordinals. Distinguishing open and closed series results in four total orders: finite, one end, no end and open, and no end and closed.
Contrary to other writers, Russell admits negative ordinals. For motivation, consider the scales of measurement using scientific notation, where a power of ten represents a decade of measure. Informally, this parameter corresponds to orders of magnitude used to quantify physical units. The parameter takes on negative as well as positive values.
Stretch
Russell adopted the term stretch from Meinong, who had contributed to the theory of distance. Stretch refers to the intermediate terms between two points in a series, and the "number of terms measures the distance and divisibility of the whole." To explain Meinong, Russell refers to the Cayley-Klein metric, which uses stretch coordinates in anharmonic ratios which determine distance by using logarithm.
References
External links
Here: page 416.
.
Binary relations
Order theory |
https://en.wikipedia.org/wiki/Planar | Planar is an adjective meaning "relating to a plane (geometry)".
Planar may also refer to:
Science and technology
Planar (computer graphics), computer graphics pixel information from several bitplanes
Planar (transmission line technologies), transmission lines with flat conductors
Planar, the structure resulting from the planar process used in the manufacture of semiconductor devices, such as planar transistors
Planar graph, graph that can be drawn in the plane so that no edges cross
Planar mechanism, a system of parts whose motion is constrained to a two-dimensional plane
Planar Systems, an Oregon-headquartered manufacturer of digital displays
Zeiss Planar, photographic lens designed by Paul Rudolph at Carl Zeiss in 1896
See also
List of planar symmetry groups
Planarity, a computer puzzle game
Plane (disambiguation)
Planer (disambiguation) |
https://en.wikipedia.org/wiki/Homogeneous%20relation | In mathematics, a homogeneous relation (also called endorelation) on a set X is a binary relation between X and itself, i.e. it is a subset of the Cartesian product . This is commonly phrased as "a relation on X" or "a (binary) relation over X". An example of a homogeneous relation is the relation of kinship, where the relation is between people.
Common types of endorelations include orders, graphs, and equivalences. Specialized studies of order theory and graph theory have developed understanding of endorelations. Terminology particular for graph theory is used for description, with an ordinary (undirected) graph presumed to correspond to a symmetric relation, and a general endorelation corresponding to a directed graph. An endorelation R corresponds to a logical matrix of 0s and 1s, where the expression xRy corresponds to an edge between x and y in the graph, and to a 1 in the square matrix of R. It is called an adjacency matrix in graph terminology.
Particular homogeneous relations
Some particular homogeneous relations over a set X (with arbitrary elements , ) are:
Empty relation
; that is, holds never;
Universal relation
; that is, holds always;
Identity relation (see also identity function)
}; that is, holds if and only if .
Example
Fifteen large tectonic plates of the Earth's crust contact each other in a homogeneous relation. The relation can be expressed as a logical matrix with 1 indicating contact and 0 no contact. This example expresses a symmetric relation.
Properties
Some important properties that a homogeneous relation over a set may have are:
for all , . For example, ≥ is a reflexive relation but > is not.
(or ) for all , not . For example, > is an irreflexive relation, but ≥ is not.
for all , if then . For example, the relation over the integers in which each odd number is related to itself is a coreflexive relation. The equality relation is the only example of a both reflexive and coreflexive relation, and any coreflexive relation is a subset of the identity relation.
for all , if then .
for all , if then .
for all , if then and . A relation is quasi-reflexive if, and only if, it is both left and right quasi-reflexive.
The previous 6 alternatives are far from being exhaustive; e.g., the binary relation defined by is neither irreflexive, nor coreflexive, nor reflexive, since it contains the pair , and , but not , respectively. The latter two facts also rule out (any kind of) quasi-reflexivity.
for all , if then . For example, "is a blood relative of" is a symmetric relation, because is a blood relative of if and only if is a blood relative of .
for all , if and then . For example, ≥ is an antisymmetric relation; so is >, but vacuously (the condition in the definition is always false).
for all , if then not . A relation is asymmetric if and only if it is both antisymmetric and irreflexive. For example, > is an asymmetric relation, but ≥ is not.
Again, the previous 3 alternatives are |
https://en.wikipedia.org/wiki/Non-positive%20curvature | In mathematics, spaces of non-positive curvature occur in many contexts and form a generalization of hyperbolic geometry. In the category of Riemannian manifolds, one can consider the sectional curvature of the manifold and require that this curvature be everywhere less than or equal to zero. The notion of curvature extends to the category of geodesic metric spaces, where one can use comparison triangles to quantify the curvature of a space; in this context, non-positively curved spaces are known as (locally) CAT(0) spaces.
Riemann Surfaces
If is a closed, orientable Riemann surface then it follows from the Uniformization theorem that may be endowed with a complete Riemannian metric with constant Gaussian curvature of either , or . As a result of the Gauss–Bonnet theorem one can determine that the surfaces which have a Riemannian metric of constant curvature i.e. Riemann surfaces with a complete, Riemannian metric of non-positive constant curvature, are exactly those whose genus is at least . The Uniformization theorem and the Gauss–Bonnet theorem can both be applied to orientable Riemann surfaces with boundary to show that those surfaces which have a non-positive Euler characteristic are exactly those which admit a Riemannian metric of non-positive curvature. There is therefore an infinite family of homeomorphism types of such surfaces whereas the Riemann sphere is the only closed, orientable Riemann surface of constant Gaussian curvature .
The definition of curvature above depends upon the existence of a Riemannian metric and therefore lies in the field of geometry. However the Gauss–Bonnet theorem ensures that the topology of a surface places constraints on the complete Riemannian metrics which may be imposed on a surface so the study of metric spaces of non-positive curvature is of vital interest in both the mathematical fields of geometry and topology. Classical examples of surfaces of non-positive curvature are the Euclidean plane and flat torus (for curvature ) and the hyperbolic plane and pseudosphere (for curvature ). For this reason these metrics as well as the Riemann surfaces which on which they lie as complete metrics are referred to as Euclidean and hyperbolic respectively.
Generalizations
The characteristic features of the geometry of non-positively curved Riemann surfaces are used to generalize the notion of non-positive beyond the study of Riemann surfaces. In the study of manifolds or orbifolds of higher dimension, the notion of sectional curvature is used wherein one restricts one's attention to two-dimensional subspaces of the tangent space at a given point. In dimensions greater than the Mostow–Prasad rigidity theorem ensures that a hyperbolic manifold of finite area has a unique complete hyperbolic metric so the study of hyperbolic geometry in this setting is integral to the study of topology.
In an arbitrary geodesic metric space the notions of being Gromov hyperbolic or of being a CAT(0) space generalise the n |
https://en.wikipedia.org/wiki/Folded%20spectrum%20method | In mathematics, the folded spectrum method (FSM) is an iterative method for solving large eigenvalue problems.
Here you always find a vector with an eigenvalue close to a search-value . This means you can get a vector in the middle of the spectrum without solving the matrix.
, with and the Identity matrix.
In contrast to the Conjugate gradient method, here the gradient calculates by twice multiplying matrix
Literature
https://web.archive.org/web/20070806144253/http://www.sst.nrel.gov/topics/nano/escan.html
Numerical linear algebra |
https://en.wikipedia.org/wiki/Flip%20%28mathematics%29 | In algebraic geometry, flips and flops are codimension-2 surgery operations arising in the minimal model program, given by blowing up along a relative canonical ring. In dimension 3 flips are used to construct minimal models, and any two birationally equivalent minimal models are connected by a sequence of flops. It is conjectured that the same is true in higher dimensions.
The minimal model program
The minimal model program can be summarised very briefly as follows: given a variety , we construct a sequence of contractions , each of which contracts some curves on which the canonical divisor is negative. Eventually, should become nef (at least in the case of nonnegative Kodaira dimension), which is the desired result. The major technical problem is that, at some stage, the variety may become 'too singular', in the sense that the canonical divisor is no longer a Cartier divisor, so the intersection number with a curve is not even defined.
The (conjectural) solution to this problem is the flip. Given a problematic as above, the flip of is a birational map (in fact an isomorphism in codimension 1) to a variety whose singularities are 'better' than those of . So we can put , and continue the process.
Two major problems concerning flips are to show that they exist and to show that one cannot have an infinite sequence of flips. If both of these problems can be solved, then the minimal model program can be carried out.
The existence of flips for 3-folds was proved by . The existence of log flips, a more general kind of flip, in dimension three and four were proved by
whose work was fundamental to the solution of the existence of log flips and other problems in higher dimension.
The existence of log flips in higher dimensions has been settled by . On the other hand, the problem of termination—proving that there can be no infinite sequence of flips—is still open in dimensions greater than 3.
Definition
If is a morphism, and K is the canonical bundle of X, then the relative canonical ring of f is
and is a sheaf of graded algebras over the sheaf of regular functions on Y.
The blowup
of Y along the relative canonical ring is a morphism to Y. If the relative canonical ring is finitely generated (as an algebra over ) then the morphism is called the flip of if is relatively ample, and the flop of if K is relatively trivial. (Sometimes the induced birational morphism from to is called a flip or flop.)
In applications, is often a small contraction of an extremal ray, which implies several extra properties:
The exceptional sets of both maps and have codimension at least 2,
and only have mild singularities, such as terminal singularities.
and are birational morphisms onto Y, which is normal and projective.
All curves in the fibers of and are numerically proportional.
Examples
The first example of a flop, known as the Atiyah flop, was found in .
Let Y be the zeros of in , and let V be the blowup of Y at the origin.
|
https://en.wikipedia.org/wiki/Flat%20%28geometry%29 | In geometry, a flat or affine subspace is a subset of an affine space that is itself an affine space (of equal or lower dimension). In the case the parent space is Euclidean, a flat is a Euclidean subspace which inherits the notion of distance from its parent space.
The flats in a plane (two-dimensional space) are points, lines, and the plane itself; the flats in three-dimensional space are points, lines, planes, and the space itself. In an -dimensional space, there are -flats of every dimension from 0 to ; subspaces one dimension lower than the parent space, -flats, are called hyperplanes.
Flats occur in linear algebra, as geometric realizations of solution sets of systems of linear equations.
A flat is a manifold and an algebraic variety, and is sometimes called a linear manifold or linear variety to distinguish it from other manifolds or varieties.
Descriptions
By equations
A flat can be described by a system of linear equations. For example, a line in two-dimensional space can be described by a single linear equation involving and :
In three-dimensional space, a single linear equation involving , , and defines a plane, while a pair of linear equations can be used to describe a line. In general, a linear equation in variables describes a hyperplane, and a system of linear equations describes the intersection of those hyperplanes. Assuming the equations are consistent and linearly independent, a system of equations describes a flat of dimension .
Parametric
A flat can also be described by a system of linear parametric equations. A line can be described by equations involving one parameter:
while the description of a plane would require two parameters:
In general, a parameterization of a flat of dimension would require parameters .
Operations and relations on flats
Intersecting, parallel, and skew flats
An intersection of flats is either a flat or the empty set.
If each line from one flat is parallel to some line from another flat, then these two flats are parallel. Two parallel flats of the same dimension either coincide or do not intersect; they can be described by two systems of linear equations which differ only in their right-hand sides.
If flats do not intersect, and no line from the first flat is parallel to a line from the second flat, then these are skew flats. It is possible only if sum of their dimensions is less than dimension of the ambient space.
Join
For two flats of dimensions and there exists the minimal flat which contains them, of dimension at most . If two flats intersect, then the dimension of the containing flat equals to minus the dimension of the intersection.
Properties of operations
These two operations (referred to as meet and join) make the set of all flats in the Euclidean -space a lattice and can build systematic coordinates for flats in any dimension, leading to Grassmann coordinates or dual Grassmann coordinates. For example, a line in three-dimensional space is determined by two distin |
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