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https://en.wikipedia.org/wiki/Raynaud%27s%20isogeny%20theorem | In mathematics, Raynaud's isogeny theorem, proved by , relates the Faltings heights of two isogeneous elliptic curves.
References
Elliptic curves
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Tate%27s%20isogeny%20theorem | In mathematics, Tate's isogeny theorem, proved by , states that two abelian varieties over a finite field are isogeneous if and only if their Tate modules are isomorphic (as Galois representations).
References
Abelian varieties
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Isogeny%20theorem | In mathematics, isogeny theorem may refer to:
Raynaud's isogeny theorem
Tate's isogeny theorem |
https://en.wikipedia.org/wiki/Siegel%20parabolic%20subgroup | In mathematics, the Siegel parabolic subgroup, named after Carl Ludwig Siegel, is the parabolic subgroup of the symplectic group with abelian radical, given by the matrices of the symplectic group whose lower left quadrant is 0 (for the standard symplectic form).
References
Automorphic forms
Algebraic groups |
https://en.wikipedia.org/wiki/Kronecker%27s%20congruence | In mathematics, Kronecker's congruence, introduced by Kronecker, states that
where p is a prime and Φp(x,y) is the modular polynomial of order p, given by
for j the elliptic modular function and τ running through classes of imaginary quadratic integers of discriminant n.
References
Modular arithmetic
Theorems in number theory |
https://en.wikipedia.org/wiki/Hurwitz%20class%20number | In mathematics, the Hurwitz class number H(N), introduced by Adolf Hurwitz, is a modification of the class number of positive definite binary quadratic forms of discriminant –N, where forms are weighted by 2/g for g the order of their automorphism group, and where H(0) = –1/12.
showed that the Hurwitz class numbers are coefficients of a mock modular form of weight 3/2.
References
Number theory |
https://en.wikipedia.org/wiki/Grothendieck%20existence%20theorem | In mathematics, the Grothendieck existence theorem, introduced by , gives conditions that enable one to lift infinitesimal deformations of a scheme to a deformation, and to lift schemes over infinitesimal neighborhoods over a subscheme of a scheme S to schemes over S.
The theorem can be viewed as an instance of (Grothendieck's) formal GAGA.
See also
Chow's lemma
References
.
.
.
formal GAGA
External links
formal GAGA
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Modular%20unit | In mathematics, modular units are certain units of rings of integers of fields of modular functions, introduced by . They are functions whose zeroes and poles are confined to the cusps (images of infinity).
See also
Cyclotomic unit
Elliptic unit
References
Modular forms |
https://en.wikipedia.org/wiki/Siegel%E2%80%93Weil%20formula | In mathematics, the Siegel–Weil formula, introduced by as an extension of the results of , expresses an Eisenstein series as a weighted average of theta series of lattices in a genus, where the weights are proportional to the inverse of the order of the automorphism group of the lattice.
For the constant terms this is essentially the Smith–Minkowski–Siegel mass formula.
References
Theorems in number theory |
https://en.wikipedia.org/wiki/Manin%E2%80%93Drinfeld%20theorem | In mathematics, the Manin–Drinfeld theorem, proved by and , states that the difference of two cusps of a modular curve has finite order in the Jacobian variety.
References
Modular forms
Theorems in number theory |
https://en.wikipedia.org/wiki/Heegner%27s%20lemma | In mathematics, Heegner's lemma is a lemma used by Kurt Heegner in his paper on the class number problem. His lemma states that if
is a curve over a field with a4 not a square, then it has a solution if it has a solution in an extension of odd degree.
References
Diophantine equations
Lemmas in number theory |
https://en.wikipedia.org/wiki/Mestre%20bound | In mathematics, the Mestre bound is a bound on the analytic rank of an elliptic curve in terms of its conductor, introduced by .
See also
Brumer bound
References
Elliptic curves
Theorems in number theory |
https://en.wikipedia.org/wiki/2006%E2%80%9307%201.%20FC%20N%C3%BCrnberg%20season | The 2006–07 1. FC Nürnberg season was the 107th season in the club's football history.
Match results
Legend
Bundesliga
DFB-Pokal
Player information
Roster and statistics
Transfers
In
Out
Kits
Sources
1. FC Nürnberg seasons
Nuremberg |
https://en.wikipedia.org/wiki/Minimal%20K-type | In mathematics, a minimal K-type is a representation of a maximal compact subgroup K of a semisimple Lie group G that is in some sense the smallest representation of K occurring in a Harish-Chandra module of G. Minimal K-types were introduced by as part of an algebraic description of the Langlands classification.
References
Representation theory |
https://en.wikipedia.org/wiki/Medical%20image%20computing | Medical image computing (MIC) is an interdisciplinary field at the intersection of computer science, information engineering, electrical engineering, physics, mathematics and medicine. This field develops computational and mathematical methods for solving problems pertaining to medical images and their use for biomedical research and clinical care.
The main goal of MIC is to extract clinically relevant information or knowledge from medical images. While closely related to the field of medical imaging, MIC focuses on the computational analysis of the images, not their acquisition. The methods can be grouped into several broad categories: image segmentation, image registration, image-based physiological modeling, and others.
Data forms
Medical image computing typically operates on uniformly sampled data with regular x-y-z spatial spacing (images in 2D and volumes in 3D, generically referred to as images). At each sample point, data is commonly represented in integral form such as signed and unsigned short (16-bit), although forms from unsigned char (8-bit) to 32-bit float are not uncommon. The particular meaning of the data at the sample point depends on modality: for example a CT acquisition collects radiodensity values, while an MRI acquisition may collect T1 or T2-weighted images. Longitudinal, time-varying acquisitions may or may not acquire images with regular time steps. Fan-like images due to modalities such as curved-array ultrasound are also common and require different representational and algorithmic techniques to process. Other data forms include sheared images due to gantry tilt during acquisition; and unstructured meshes, such as hexahedral and tetrahedral forms, which are used in advanced biomechanical analysis (e.g., tissue deformation, vascular transport, bone implants).
Segmentation
Segmentation is the process of partitioning an image into different meaningful segments. In medical imaging, these segments often correspond to different tissue classes, organs, pathologies, or other biologically relevant structures. Medical image segmentation is made difficult by low contrast, noise, and other imaging ambiguities. Although there are many computer vision techniques for image segmentation, some have been adapted specifically for medical image computing. Below is a sampling of techniques within this field; the implementation relies on the expertise that clinicians can provide.
Atlas-Based Segmentation: For many applications, a clinical expert can manually label several images; segmenting unseen images is a matter of extrapolating from these manually labeled training images. Methods of this style are typically referred to as atlas-based segmentation methods. Parametric atlas methods typically combine these training images into a single atlas image, while nonparametric atlas methods typically use all of the training images separately. Atlas-based methods usually require the use of image registration in order to align the atlas imag |
https://en.wikipedia.org/wiki/Heo%20Beom-san | Heo Beom-San (; born 14 September 1989) is a South Korean footballer who plays as a midfielder for Seoul E-Land FC in the K League 2.
Club career statistics
External links
1989 births
Living people
Footballers from Seoul
Men's association football midfielders
South Korean men's footballers
Daejeon Hana Citizen players
Jeju United FC players
Gangwon FC players
Asan Mugunghwa FC players
Seoul E-Land FC players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Andres%20and%20Marzo%27s%20delta | In statistics, Andrés and Marzo's Delta is a measure of an agreement between two observers used in classifying data. It was created by Andres & Marzo in 2004.
Rationale for use
The most commonly used measure of agreement between observers is Cohen's kappa. The value of kappa is not always easy to interpret and it may perform poorly if the values are asymmetrically distributed. It also requires that the data be independent. The delta statistic may be of use when faced with the potential difficulties.
Mathematical formulation
Delta was created with the model of a set of students (C) having to choose correct responses (R) from a set of n questions each with K alternative answers. Then
where the sum is taken over all the answers ( xij ) and xii are the values along the main diagonal of the C x R matrix of answers.
This formula was extended to more complex cases and estimates of the variance of delta were made by Andres and Marzo.
Uses
It has been used in a variety of applications including ecological mapping and alien species identification.
References
Categorical variable interactions |
https://en.wikipedia.org/wiki/N%C3%A9ron%20differential | In mathematics, a Néron differential, named after André Néron, is an almost canonical choice of 1-form on an elliptic curve or abelian variety defined over a local field or global field. The Néron differential behaves well on the Néron minimal models.
For an elliptic curve of the form
the Néron differential is
References
Elliptic curves |
https://en.wikipedia.org/wiki/Genus%20character | In number theory, a genus character of a quadratic number field K is a character of the genus group of K. In other words, it is a real character of the narrow class group of K. Reinterpreting this using the Artin map, the collection of genus characters can also be thought of as the unramified real characters of the absolute Galois group of K (i.e. the characters that factor through the Galois group of the genus field of K).
References
Section 12.5 of
Section 2.3 of
Algebraic number theory |
https://en.wikipedia.org/wiki/Ring%20class%20field | In mathematics, a ring class field is the abelian extension of an algebraic number field K associated by class field theory to the ring class group of some order O of the ring of integers of K.
Properties
Let K be an algebraic number field.
The ring class field for the maximal order O = OK is the Hilbert class field H.
Let L be the ring class field for the order Z[] in the number field K = Q().
If p is an odd prime not dividing n, then p splits completely in L if and only if p splits completely in K.
L = K(a) for a an algebraic integer with minimal polynomial over Q of degree h(−4n), the class number of an order with discriminant −4n.
If O is an order and a is a proper fractional O-ideal (i.e. {x ϵ K * : xa ⊂ a} = O), write j(a) for the j-invariant of the associated elliptic curve. Then K(j(a)) is the ring class field of O and j(a) is an algebraic integer.
References
External links
Ring class fields.
Algebraic number theory |
https://en.wikipedia.org/wiki/Liu%20Lu | Liu Lu (; born 2 April 1989) is a professor of mathematics at Central South University in Changsha, Hunan, where he is China's youngest full university Professor. As a 22-year-old undergraduate student Lu proved that
Ramsey theorem for infinite graphs (the case n = 2) with 2-coloring
does not imply
WKL0 over RCA0,
solving an open problem left by English logician David Seetapun in the 1990s (). For this he was instantly promoted to full professor in the department where he was studying, and awarded a prize of 1 million renminbi. Some established professors were critical of his appointment voicing concern that he was too young, had no teaching experience and that the appointment was mostly designed to get media attention to Liu's university.
References
External links
1989 births
Living people
21st-century Chinese mathematicians
Academic staff of the Central South University
Educators from Liaoning
People from Dalian
Mathematicians from Liaoning |
https://en.wikipedia.org/wiki/Kodaira%E2%80%93Spencer%20map | In mathematics, the Kodaira–Spencer map, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold X, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on X.
Definition
Historical motivation
The Kodaira–Spencer map was originally constructed in the setting of complex manifolds. Given a complex analytic manifold with charts and biholomorphic maps sending gluing the charts together, the idea of deformation theory is to replace these transition maps by parametrized transition maps over some base (which could be a real manifold) with coordinates , such that . This means the parameters deform the complex structure of the original complex manifold . Then, these functions must also satisfy a cocycle condition, which gives a 1-cocycle on with values in its tangent bundle. Since the base can be assumed to be a polydisk, this process gives a map between the tangent space of the base to called the Kodaira–Spencer map.
Original definition
More formally, the Kodaira–Spencer map is
where
is a smooth proper map between complex spaces (i.e., a deformation of the special fiber .)
is the connecting homomorphism obtained by taking a long exact cohomology sequence of the surjection whose kernel is the tangent bundle .
If is in , then its image is called the Kodaira–Spencer class of .
Remarks
Because deformation theory has been extended to multiple other contexts, such as deformations in scheme theory, or ringed topoi, there are constructions of the Kodaira–Spencer map for these contexts.
In the scheme theory over a base field of characteristic , there is a natural bijection between isomorphisms classes of and .
Constructions
Using infinitesimals
Cocycle condition for deformations
Over characteristic the construction of the Kodaira–Spencer map can be done using an infinitesimal interpretation of the cocycle condition. If we have a complex manifold covered by finitely many charts with coordinates and transition functions where Recall that a deformation is given by a commutative diagramwhere is the ring of dual numbers and the vertical maps are flat, the deformation has the cohomological interpretation as cocycles on whereIf the satisfy the cocycle condition, then they glue to the deformation . This can be read asUsing the properties of the dual numbers, namely , we haveandhence the cocycle condition on is the following two rules
Conversion to cocycles of vector fields
The cocycle of the deformation can easily be converted to a cocycle of vector fields as follows: given the cocycle we can form the vector fieldwhich is a 1-cochain. Then the rule for the transition maps of gives this 1-cochain as a 1-cocycle, hence a class .
Using vector fields
One of the original constructions of this map used vector fields in the settings of differential geometry and complex analysis. Given the notation abo |
https://en.wikipedia.org/wiki/Inner%20form | In mathematics, an inner form of an algebraic group over a field is another algebraic group such that there exists an isomorphism between and defined over (this means that is a -form of ) and in addition, for every Galois automorphism the automorphism is an inner automorphism of (i.e. conjugation by an element of ).
Through the correspondence between -forms and the Galois cohomology this means that is associated to an element of the subset where is the subgroup of inner automorphisms of .
Being inner forms of each other is an equivalence relation on the set of -forms of a given algebraic group.
A form which is not inner is called an outer form. In practice, to check whether a group is an inner or outer form one looks at the action of the Galois group on the Dynkin diagram of (induced by its action on , which preserves any torus and hence acts on the roots). Two groups are inner forms of each other if and only if the actions they define are the same.
For example, the -forms of are itself and the unitary groups and . The latter two are outer forms of , and they are inner forms of each other.
References
Algebraic groups |
https://en.wikipedia.org/wiki/Deformation%20ring | In mathematics, a deformation ring is a ring that controls liftings of a representation of a Galois group from a finite field to a local field. In particular for any such lifting problem there is often a universal deformation ring that classifies all such liftings, and whose spectrum is the universal deformation space.
A key step in Wiles's proof of the modularity theorem was to study the relation between universal deformation rings and Hecke algebras.
See also
Deformation (mathematics)
Galois module
References
Number theory |
https://en.wikipedia.org/wiki/Drinfeld%20reciprocity | In mathematics, Drinfeld reciprocity, introduced by , is a correspondence between eigenforms of the moduli space of Drinfeld modules and factors of the corresponding Jacobian variety, such that all twisted L-functions are the same.
References
. English translation in Math. USSR Sbornik 23 (1974) 561–592.
Modular forms |
https://en.wikipedia.org/wiki/Drinfeld%20upper%20half%20plane | In mathematics, the Drinfeld upper half plane is a rigid analytic space analogous to the usual upper half plane for function fields, introduced by .
It is defined to be P1(C)\P1(F∞), where F is a function field of a curve over a finite field, F∞ its completion at ∞, and C the completion of the algebraic closure of F∞.
The analogy with the usual upper half plane arises from the fact that the global function field F is analogous to the rational numbers Q. Then, F∞ is the real numbers R and the algebraic closure of F∞ is the complex numbers C (which are already complete). Finally, P1(C) is the Riemann sphere, so P1(C)\P1(R) is the upper half plane together with the lower half plane.
References
Automorphic forms |
https://en.wikipedia.org/wiki/Jim%20Stasheff | James Dillon Stasheff (born January 15, 1936, New York City) is an American mathematician, a professor emeritus of mathematics at the University of North Carolina at Chapel Hill. He works in algebraic topology and algebra as well as their applications to physics.
Biography
Stasheff did his undergraduate studies in mathematics at the University of Michigan, graduating in 1956.
Stasheff then began his graduate studies at Princeton University; his notes for a 1957 course by John Milnor on characteristic classes first appeared in mimeographed form and later in 1974 in revised form book with Stasheff as a co-author. After his second year at Princeton, he moved to Oxford University on a Marshall Scholarship. Two years later in 1961, with a pregnant wife, needing an Oxford degree to get reimbursed for his return trip to the US, and yet still feeling attached to Princeton, he split his thesis into two parts (one topological, the other algebraic) and earned two doctorates, a D.Phil. from Oxford under the supervision of Ioan James and a Ph.D. later the same year from Princeton under the supervision of John Coleman Moore.
From 1961 to 1962, Stasheff was a C.L.E. Moore instructor at the Massachusetts Institute of Technology. Then in 1962 joined the faculty of University of Notre Dame as an assistant professor; he was promoted to full professor there in 1968. He visited Princeton University from 1968 to 1969 and then stayed there the next year as a Sloan Fellow. Then in 1970 he moved to Temple University, where he held a position until 1978. In 1976, he joined the UNC faculty. He has also visited the Institute for Advanced Study, Lehigh University, Rutgers University, and the University of Pennsylvania.
Stasheff was an editor of the Transactions of the American Mathematical Society from 1978 to 1981, and managing editor from 1979 to 1981. He has been married since 1959 and has two children.
Research
Stasheff's research contributions include the study of associativity in loop spaces and the construction of the associahedron (also called the Stasheff polytope), ideas leading to the theory of operads; homotopy theoretic approaches to Hilbert's fifth problem on the characterization of Lie groups; and the study of Poisson algebras in mathematical physics.
In the 1960s he wrote fundamental papers on higher homotopy theory and homotopy algebras. He introduced , Stasheff algebras and Stasheff polytopes.
In the 1980s he turned to the application of characteristic classes and other topological and algebraic concepts in mathematical physics, first in the algebraic structure of anomalies in quantum field theory, where he worked with among others, Tom Kephart and Paolo Cotta-Ramusino. He referred to the research field as cohomological physics.
Awards and honors
In 2012 he became a fellow of the American Mathematical Society.
Selected publications
References
External links
Homepage
1936 births
Living people
20th-century American mathematicians
21st-century Am |
https://en.wikipedia.org/wiki/List%20of%20Greater%20Western%20Sydney%20Giants%20coaches | The following is a list of the Greater Western Sydney Giants senior coaches in each of their seasons in the Australian Football League.
Key
Coaches
AFL
Statistics are correct to the end of 2023 season.
AFL Women's
''Statistics are correct to the end of the 2018 season
References
Coaches
Sydney-sport-related lists
Lists of Australian Football League coaches by club |
https://en.wikipedia.org/wiki/Silver%20thiocyanate | Silver thiocyanate is the silver salt of thiocyanic acid with the formula AgSCN.
Structure
AgSCN is monoclinic with 8 molecules per unit cell. Each SCN− group has an almost linear molecular geometry, with bond angle 179.6(5)°. Weak Ag—Ag interactions of length 0.3249(2) nm to 0.3338(2) nm are present in the structure.
Production
Silver thiocyanate is produced by the reaction between silver nitrate and potassium thiocyanate.
References
Thiocyanates
Silver compounds |
https://en.wikipedia.org/wiki/Chang%20Lin%20%28footballer%29 | Chang Lin (; born April 17, 1981, in Dalian) is a former Chinese footballer. He currently works for Dalian Yifang as a youth coach.
Career statistics
(Correct as of 2013)
Honors
Dalian Sidelong
China League Two: 2001
Dalian Aerbin
China League Two: 2010
China League One: 2011
References
External links
1981 births
Living people
Chinese men's footballers
Footballers from Dalian
Shanghai Shenhua F.C. players
Zhejiang Professional F.C. players
Dalian Professional F.C. players
Cangzhou Mighty Lions F.C. players
Meizhou Hakka F.C. players
Chinese Super League players
China League One players
Men's association football midfielders
21st-century Chinese people |
https://en.wikipedia.org/wiki/Reuleaux | Reuleaux may refer to:
Franz Reuleaux (1829–1905), German mechanical engineer and lecturer
in geometry:
Reuleaux polygon, a curve of constant width
Reuleaux triangle, a Reuleaux polygon with three sides
Reuleaux heptagon, a Reuleaux polygon with seven sides that provides the shape of some currency coins
Reuleaux tetrahedron, the intersection of four spheres of equal radius centered at the vertices of a regular tetrahedron |
https://en.wikipedia.org/wiki/Chevalley%E2%80%93Iwahori%E2%80%93Nagata%20theorem | In mathematics, the Chevalley–Iwahori–Nagata theorem states that if a linear algebraic group G is acting linearly on a finite-dimensional vector space V, then the map from V/G to the spectrum of the ring of invariant polynomials is an isomorphism if this ring is finitely generated and all orbits of G on V are closed . It is named after Claude Chevalley, Nagayoshi Iwahori, and Masayoshi Nagata.
References
Invariant theory
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Young%E2%80%93Deruyts%20development | In mathematics, the Young–Deruyts development is a method of writing invariants of an action of a group on an n-dimensional vector space V
in terms of invariants depending on at most n–1 vectors .
References
Invariant theory |
https://en.wikipedia.org/wiki/Heinrich%20Schr%C3%B6ter | Heinrich Eduard Schröter (8 January 1829 – 3 January 1892) was a German mathematician, who studied geometry in the tradition of Jakob Steiner.
Life and work
Schröter went to (along with mathematicians Alfred Clebsch, Rudolf Lipschitz, Carl Gottfried Neumann) the Altstädtisches Gymnasium in Königsberg, studying mathematics and physics. After graduating from the Gymnasium in 1845, he entered the University of Königsberg to continue the study of mathematics and physics under Jacobi school's Frederick Richelot (and Franz Ernst Neumann and Otto Hesse). After his volunteer year in the military, he went to the Berlin Friedrich-Wilhelms-University, where he was taught by Peter Gustav Lejeune Dirichlet and Jakob Steiner. In 1854 he received his doctorate in Richelot in Königsberg with a paper on elliptic functions. He then passed the state exam and was qualified as a teacher in 1855 at the University of Breslau (also on elliptic functions). In 1858 he became associate professor in Breslau and in 1861 professor. He died after he fell ill in 1891.
Schröter was influenced by Steiner's lectures, which were available only as note sheets, on synthetic geometry (projective theory of conics) published in 1867. In Die Theorie der Oberflächen (the theory of surfaces) of 1880, one of his major works, he studied second order surfaces and third order space curves, continuing Steiner's work. For this work, he received the Steiner Prize of the Berlin Academy and became its corresponding member. He also investigated the third-order surfaces and fourth-order space curves.
His students included Victor Eberhard, Moritz Pasch, Jakob Rosanes, and Rudolf Sturm.
Writings
1854 Breslau, Philosophische Fakultaet: Inaugural Dissertation: De Aequationibus Modularibus
1855 Breslau, Philosophische Fakultaet : Habilitationsschrift: Entwicklung der Potenzen der elliptischen Transcendenten und die Theilung dieser Funktion. Respondent: A. Grimm, Dr phil.; Opponenten: R. Ladrasch, Gymnasiallehrer; E. Tillich, Cand. phil.; H. Jaschke, Stud. phil.
als Bearbeiter und Herausgeber: Jacob Steiner's Vorlesungen über synthetische Geometrie: Theil 2: Die Theorie der Kegelschnitte, gestützt auf projectivische Eigenschaften. Leipzig 1867, 2. Auflage 1876.
Die Theorie der Oberflächen zweiter Ordnung und der Raumkurven dritter Ordnung als Erzeugnisse projectivischer Gebilde. Leipzig 1880.
Die Theorie der ebenen Curven dritter Ordnung, auf synthetische Weise abgeleitet. Leipzig 1888.
Grundzüge einer rein geometrischen Theorie der Raumcurven vierter Ordnung erster Species. Leipzig 1890.
References
The original article was a translation (Google) of the corresponding article in German Wikipedia.
Rudolf Sturm: Heinrich Schröter. Jahresbericht DMV, Bd. 2, 1893.
1829 births
1892 deaths
Scientists from Königsberg
People from East Prussia
19th-century German mathematicians
University of Königsberg alumni
Humboldt University of Berlin alumni
Academic staff of the University of Breslau |
https://en.wikipedia.org/wiki/Gram%27s%20theorem | In mathematics, Gram's theorem states that an algebraic set in a finite-dimensional vector space invariant under some linear group can be defined by absolute invariants. . It is named after J. P. Gram, who published it in 1874.
References
. Reprinted by Academic Press (1971), .
.
Invariant theory
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/Hilbert%E2%80%93Mumford%20criterion | In mathematics, the Hilbert–Mumford criterion, introduced by David Hilbert and David Mumford, characterizes the semistable and stable points of a group action on a vector space in terms of eigenvalues of 1-parameter subgroups .
Definition of stability
Let G be a reductive group acting linearly on a vector space V, a non-zero point of V is called
semi-stable if 0 is not contained in the closure of its orbit, and unstable otherwise;
stable if its orbit is closed, and its stabilizer is finite. A stable point is a fortiori semi-stable. A semi-stable but not stable point is called strictly semi-stable.
When G is the multiplicative group , e.g. C* in the complex setting, the action amounts to a finite dimensional representation . We can decompose V into a direct sum , where on each component Vi the action is given as . The integer i is called the weight. Then for each point x, we look at the set of weights in which it has a non-zero component.
If all the weights are strictly positive, then , so 0 is in the closure of the orbit of x, i.e. x is unstable;
If all the weights are non-negative, with 0 being a weight, then either 0 is the only weight, in which case x is stabilized by C*; or there are some positive weights beside 0, then the limit is equal to the weight-0 component of x, which is not in the orbit of x. So the two cases correspond exactly to the respective failure of the two conditions in the definition of a stable point, i.e. we have shown that x is strictly semi-stable.
Statement
The Hilbert–Mumford criterion essentially says that the multiplicative group case is the typical situation. Precisely, for a general reductive group G acting linearly on a vector space V, the stability of a point x can be characterized via the study of 1-parameter subgroups of G, which are non-trivial morphisms . Notice that the weights for the inverse are precisely minus those of , so the statements can be made symmetric.
A point x is unstable if and only if there is a 1-parameter subgroup of G for which x admits only positive weights or only negative weights; equivalently, x is semi-stable if and only if there is no such 1-parameter subgroup, i.e. for every 1-parameter subgroup there are both non-positive and non-negative weights;
A point x is strictly semi-stable if and only if there is a 1-parameter subgroup of G for which x admits 0 as a weight, with all the weights being non-negative (or non-positive);
A point x is stable if and only if there is no 1-parameter subgroup of G for which x admits only non-negative weights or only non-positive weights, i.e. for every 1-parameter subgroup there are both positive and negative weights.
Examples and applications
Action of C* on the plane
The standard example is the action of C* on the plane C2 defined as . Clearly the weight in the x-direction is 1 and the weight in the y-direction is -1. Thus by the Hilbert–Mumford criterion, a non-zero point on the x-axis admits 1 as its only weight, and a non-zero point on |
https://en.wikipedia.org/wiki/Bracket%20ring | In mathematics invariant theory, the bracket ring is the subring of the ring of polynomials k[x11,...,xdn] generated by the d-by-d minors of a generic d-by-n matrix (xij).
The bracket ring may be regarded as the ring of polynomials on the image of a Grassmannian under the Plücker embedding.
For given d ≤ n we define as formal variables the brackets [λ1 λ2 ... λd] with the λ taken from {1,...,n}, subject to [λ1 λ2 ... λd] = − [λ2 λ1 ... λd] and similarly for other transpositions. The set Λ(n,d) of size generates a polynomial ring K[Λ(n,d)] over a field K. There is a homomorphism Φ(n,d) from K[Λ(n,d)] to the polynomial ring K[xi,j] in nd indeterminates given by mapping
[λ1 λ2 ... λd] to the determinant of the d by d matrix consisting of the columns of the xi,j indexed by the λ. The bracket ring B(n,d) is the image of Φ. The kernel I(n,d) of Φ encodes the relations or syzygies that exist between the minors of a generic n by d matrix. The projective variety defined by the ideal I is the (n−d)d dimensional Grassmann variety whose points correspond to d-dimensional subspaces of an n-dimensional space.
To compute with brackets it is necessary to determine when an expression lies in the ideal I(n,d). This is achieved by a straightening law due to Young (1928).
See also
Bracket algebra
References
Invariant theory
Algebraic geometry |
https://en.wikipedia.org/wiki/Nullform | In mathematics, a nullform of a vector space acted on linearly by a group is a vector on which all invariants of the group vanish. Nullforms were introduced by . .
References
Invariant theory |
https://en.wikipedia.org/wiki/Abel%E2%80%93Goncharov%20interpolation | In mathematics, Abel–Goncharov interpolation determines a polynomial such that various higher derivatives are the same as those of a given function at given points. It was introduced by and rediscovered by .
References
Interpolation |
https://en.wikipedia.org/wiki/Michel%20Waldschmidt | Michel Waldschmidt (born June 17, 1946 at Nancy, France) is a French mathematician, specializing in number theory, especially transcendental numbers.
Biography
Waldschmidt was educated at Lycée Henri Poincaré and the University of Nancy until 1968. In 1972 he defended his thesis, titled Indépendance algébrique de nombres transcendants (Algebraic independence of transcendental numbers) and directed by Jean Fresnel, the University of Bordeaux, where he was research associate of CNRS in 1971–2. He was then a lecturer at Paris-Sud 11 University in 1972–3, then a lecturer at the University of Paris VI (Pierre et Marie Curie), where he is Professor since 1973. Waldschmidt was also a visiting professor at places including the École normale supérieure. He is a member of the .
Today, Michel Waldschmidt is an expert in the theory of transcendental numbers and diophantine approximations.
He was awarded the Albert Châtelet Prize in 1974, the CNRS Silver Medal in 1978, the Marquet Prize of Academy of Sciences in 1980 and the Special Award of the Hardy–Ramanujan Society in 1986.
From 2001 to 2004 he was president of the Mathematical Society of France. He is a member of several mathematical societies, including the EMS, the AMS and Ramanujan Mathematical Society.
He is interested in exchange programs for researchers and students and was, from 2005 to 2009, Vice President CIMPA (International Centre for Pure and Applied Mathematics), formed in Nice to promote international cooperation. He participated in the coordination of cooperation in mathematics of France with many countries, including India and Middle East.
In 2021 he was awarded the Bertrand Russell Prize by the American Mathematical Society.
Selected publications
Diophantine approximation on linear algebraic groups. Springer, 2000
Nombres transcendants, Lecture Notes in Mathematics, vol. 402, 1974, Springer
Nombres transcendants et groupes algébriques, Astérisque, vol. 69/70, 1979, 2e tirage 1987
Transcendence Methods, Queens Papers in Pure and Applied Mathematics, 1979
With J.-M. Luck, P. Moussa, C. Itzykson (eds.), From Number Theory to Physics, 1995
References
External links
Homepage to Jussieu
on frenchsciencetoday.org
search on author Michel Waldschmidt from Google Scholar
French mathematicians
1946 births
Living people
Number theorists
Academic staff of the University of Paris
Nancy-Université alumni |
https://en.wikipedia.org/wiki/Karl%20Bobek | Karl Joseph Bobek (1855–1899) was a German mathematician working on elliptic functions and geometry.
References
External links
19th-century German mathematicians
1899 deaths
1855 births
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Anton%20von%20Braunm%C3%BChl | Johann Anton Edler von Braunmühl (22 December 1853, Tiflis – 7 March 1908, München) was a German historian of mathematics and mathematician who worked on synthetic geometry and trigonometry.
Braunmühl was born in Tiflis but came from a Bavarian family and his father had gone as an architect to build a palace. The death of his father in 1856 led to the mother and family moving to Munich where he went to school. His mother died in 1866 after which he was taken care of by an uncle. He passed school in 1873 and joined the University of Munich where he studied physics under G. Bauer, L. von Seidel, J. von Lamont, Philip Von Jolly, Friedrich Narr and history under M. Bernays and B. Riehl. He also attended classes in mathematics at the polytechnikum under A. Brill, F. Klein and J.N. Bischoff. He received a doctorate summa cum laude in 1878 and at the same time began to teach at the Realgymnasium. In 1879 he married Franziska Stölzl; they had two daughters. He became a professor in 1892. His teaching were on algebraic analysis, projective geometry, and trigonometry and his students included chemists and architects. In 1893-94 he also began to teach the history of mathematics. This would lead to his comprehensive survey of the history of trigonometry in two volumes, published in 1900/1903. He then took up writing a two-volume history of mathematics but he died before it could be published. His manuscript was worked on by Heinrich Wieleitner.
References
External links
A. von Braunmühl (1903) Vorlesungen über Geschichte der Trigonometrie via Internet Archive
1853 births
1908 deaths
Edlers of Germany
19th-century German mathematicians
20th-century German mathematicians
German historians of mathematics
Academic staff of the Technical University of Munich
19th-century German writers
19th-century German male writers
German male non-fiction writers
Expatriates in the Russian Empire
People from the Kingdom of Bavaria
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Isaak%20Bacharach | Isaak Bacharach (2 December 1854 – 22 September 1942) was a German mathematics professor in Erlangen who proved the Cayley–Bacharach theorem on intersections of cubic curves.
He was murdered at the Theresienstadt concentration camp during The Holocaust.
References
External links
1854 births
1942 deaths
19th-century German mathematicians
20th-century German mathematicians
Algebraic geometers
German people who died in the Theresienstadt Ghetto
People from Seligenstadt
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Otto%20Dersch | Otto Georg Dersch (born March 17, 1848 in Ortenberg, Hesse) was a German mathematician who worked in algebraic geometry. Dersch got his Ph.D. 1873 in Gießen. He was teacher in Groß-Umstadt and Darmstadt and then director of a secondary school in Offenbach am Main, and then became director of a secondary school (Oberrealschule) in Darmstadt until at least 1915.
Publications
References
1848 births
19th-century German mathematicians
Algebraic geometers
Year of death missing
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/W.%20Frahm | W. Frahm was a German mathematician who worked on algebraic geometry.
References
19th-century German mathematicians
Year of birth missing
Year of death missing
Place of birth missing
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Claudio%20Procesi | Claudio Procesi (born 31 March 1941 in Rome) is an Italian mathematician, known for works in algebra and representation theory.
Career
Procesi studied at the Sapienza University of Rome, where he received his degree (Laurea) in 1963. In 1966 he graduated from the University of Chicago advised by Israel Herstein, with a thesis titled "On rings with polynomial identities". From 1966 he was assistant professor at the University of Rome, 1970
associate professor at the University of Lecce, and 1971 at the University of Pisa. From 1973 he was full professor in Pisa and in 1975 ordinary
Professor at the Sapienza University of Rome. He was a visiting scientist at Columbia University (1969–1970), the University of California, Los Angeles (1973/74), at the Instituto Nacional de Matemática Pura e Aplicada, at the Massachusetts Institute of Technology (1991), at the
University of Grenoble, at Brandeis University (1981/2), at the University of Texas at Austin (1984), the Institute for Advanced Study (1994), the Mathematical Sciences Research Institute (1992, etc.), at the International Centre for Theoretical Physics in Trieste, and at the École Normale Supérieure.
Procesi studies noncommutative algebra, algebraic groups, invariant theory, enumerative geometry, infinite dimensional algebras and quantum groups, polytopes, braid groups, cyclic homology, geometry of orbits of compact groups,
arrangements of subspaces and tori.
Procesi proved that
the polynomial invariants of matrices over a field all come from the Hamilton-Cayley theorem, which says that a square matrix satisfies its own characteristic polynomial.
In 1981 he was awarded the Medal of the Accademia dei Lincei, of which he is a member since 1987. In 1986 he received the Feltrinelli Prize in mathematics. In 1978 he was an invited speaker at the International Congress of Mathematicians (ICM) in Helsinki. From 2007 to 2010 he is a vice-president of the International Mathematical Union. He was an editor of the Duke Mathematical Journal, the Journal of Algebra, Communications in Algebra, and Advances in Mathematics. Furthermore, he was on the committee of the Abel Prize and the algebra committee for the ICM 1986–1994.
Works
Articles
with Lieven Le Bruyn:
with Corrado de Concini and George Lusztig:
Books
2017: (with Corrado de Concini) The Invariant Theory of Matrices, American Mathematical Society
2010: (with Corrado de Concini) Topics in Hyperplane Arrangements, Polytopes and Box-Splines, Springer
2006: Lie groups: An approach through invariants and representations, Springer, Universitext
1996: (with Hanspeter Kraft) Classical Invariant Theory
1993: (with Corrado de Concini) Quantum groups, Lecture Notes in Mathematics, Springer
1993: Rings with polynomial identities, Dekker
1983: A primer on invariant theory, Brandeis University
See also
Hessenberg variety
Hodge algebra
Wonderful compactification
References
The original article was a translation (Google) of the corresponding |
https://en.wikipedia.org/wiki/Corrado%20de%20Concini | Corrado de Concini (born 28 July 1949 in Rome) is an Italian mathematician and professor at the Sapienza University of Rome. He studies algebraic geometry, quantum groups, invariant theory, and mathematical physics.
Life and work
He was born in Rome in 1949, the son of Ennio de Concini, a noted screenwriter and film director.
Corrado de Concini received in 1971 the mathematics degree from
Sapienza University of Rome and in 1975 a Ph.D. from the University of Warwick under the supervision of George Lusztig (The mod-2 cohomology of the orthogonal groups over a finite field).
In 1975 he was a lecturer (Professore Incaricato) at the University of Salerno, and in 1976 was associate professor at the University of Pisa. In 1981 he went to the University of Rome, where in 1983 he was a professor of higher algebra. From 1988 to 1996 he was professor at the Scuola Normale Superiore in Pisa, and from 1996 professor at the Sapienza University of Rome.
De Concini was also a visiting scientist at the Brandeis University, the Mittag-Leffler Institute (1981), the Tata Institute of Fundamental Research (1982), Harvard University (1987), the Massachusetts Institute of Technology (1989), the University of Paris VI, the Institut des Hautes Études Scientifiques (1992, 1996), the École Normale Supérieure (2004, Lagrange Michelet Chair), and the Mathematical Sciences Research Institute (2000, 2002).
From 2003 to 2007 he was president of Istituto Nazionale di Alta Matematica Francesco Severi.
In 1986 he was an invited speaker at the International Congress of Mathematicians in Berkeley (Equivariant embeddings of homogeneous spaces). In 1992, he held a plenary lecture on the first European Congress of Mathematicians in Paris (Representations of quantum groups at roots of 1). In 1986 he was awarded the Caccioppoli Prize. Since 1993 he is a corresponding member and since 2009 a full member of
the Accademia dei Lincei (whose gold medal he won in 1990) and since 2005 a corresponding member of the Istituto Lombardo.
Writings
With Claudio Procesi: Topics in Hyperplane Arrangements, Polytopes and Box-Splines, Springer, 2010.
With Claudio Procesi: Quantum groups, in: D-modules, representation theory, and quantum groups (Venice, 1992), 31–140, Lecture Notes in Math., vol. 1565, Springer, Berlin, 1993.
See also
Wonderful compactification
References
External links
1949 births
Living people
Scientists from Rome
20th-century Italian mathematicians
21st-century Italian mathematicians
Group theorists
Algebraic geometers
Topologists
Sapienza University of Rome alumni
Alumni of the University of Warwick
Academic staff of the University of Salerno
Academic staff of the University of Pisa
Academic staff of the Scuola Normale Superiore di Pisa
Academic staff of the Sapienza University of Rome |
https://en.wikipedia.org/wiki/Karl%20Rohn | Karl Friedrich Wilhelm Rohn (January 25, 1855 in Schwanheim – August 4, 1920 in Leipzig) was a German mathematician, who studied geometry.
Life and work
Rohn studied in Darmstadt, Leipzig and Munich, initially engineering but then mathematics by the influence of Alexander von Brill, among the others. In 1878 he received a doctorate under the supervision of Felix Klein in Munich, and in 1879 he habilitated at Leipzig.
The subject of his doctoral thesis and habilitation was the Kummer surfaces of order 4 and their relationship with hyperelliptic functions (with
Riemann surfaces of genus 2). In 1884 he became an associate professor at the University of Leipzig and a year later at the Dresden University of Technology, where in 1887 he was a professor of descriptive geometry. In 1904 he became a professor at Leipzig.
In addition to the Kummer surfaces, he studied algebraic space curves and completed the classification work of Georges Halphen and Max Noether.
In 1913 he was the president of the German Mathematical Society.
Writings
Die verschiedenen Gestalten der Kummer'schen Fläche. In: Mathematische Annalen. 18. Band. Leipzig 1881, S. 99–159. (online)
with Erwin Papperitz: Lehrbuch der Darstellenden Geometrie, 2 Bände, Leipzig 1893, 1896.
with L. Berzolari: Algebraische Raumkurven und abwickelbare Flächen. In: Enzyklopädie der mathematischen Wissenschaften. Erschienen 1926. (online)
References
The original article was a Google translation of the corresponding German article.
Siegfried Gottwald, Hans J. Ilgauds, Karl-Heinz Schlote (Eds): Encyclopedia of important mathematicians. Second Edition. Harri German, Frankfurt am Main 2006, .
Friedrich Schur: Karl Rohn Nachruf. In: Jahresbericht der Deutschen Mathematiker-Vereinigung. 32. Band, Leipzig 1923, S. 201–211
External links
19th-century German mathematicians
1855 births
1920 deaths
Technische Universität Darmstadt alumni
20th-century German mathematicians
Mathematicians from the German Empire |
https://en.wikipedia.org/wiki/Aleksander%20Rajchman | Aleksander Michał Rajchman (13 November 1890 – July or August 1940) was a mathematician of the Warsaw School of Mathematics of the Interwar period. He had origins in the Lwów School of Mathematics and contributed to real analysis, probability and mathematical statistics.
Family Background
Rajchman was born on 13 November 1890 in Warsaw, Congress Poland, a province of the Russian Empire, in the family of assimilated Polish Jews known for contributions to the 20th-century Polish intellectual life. Although the family was partially converted into Roman Catholicism, his parents were agnostic. His father Aleksander Rajchman was a journalist specialized in theatre and music critique, who in the period 1882-1904 was the publisher and editor-in-chief of the artistic weekly Echo Muzyczne, Teatralne i Artystyczne and was co-founder and first director of the National Philharmonic in Warsaw in the years 1901–1904. Mother Melania Amelia Hirszfeld was a socialist and women's rights activist who wrote both critical essays and woman affairs' texts under pseudonyms or anonymously for a few Polish weeklies, organized maternal rallies where she drew attention to the need to improve the household to facilitate women's lives, and was an active member of the secret organization Women's Circle of Polish Crown and Lithuania, and later also the Association of Women's Equality in Warsaw. Rajchmans ran a social salon who hosted many Polish artists of their times, in particular Eliza Orzeszkowa, Maria Konopnicka, and Zenon Pietkiewicz. His older sister a Polish independence activist and historian of education Helena Radlińska was the founder of Polish social pedagogy, his older brother a physician and bacteriologist Ludwik Rajchman was the world leader in social medicine and director of the League of Nations Health Organization, the founder of the United Nations International Children's Emergency Fund (UNICEF) and its first chairman in the years 1946–1950. His nephew a Polish-American electrical engineer Jan A. Rajchman was a computer pioneer who invented logic circuits for arithmetic and magnetic-core memory to result in development of high-speed computer memory systems and whose son John Rajchman is a noted American philosopher of art history, architecture, and continental philosophy. His first cousin a microbiologist and serologist Ludwik Hirszfeld co-discovered the heritability of ABO blood group type and foreseen the serological conflict between mother and child.
Education and Research Work
After his father died in 1904, his mother migrated with rest of the family to Paris in 1909. Alexander studied there and obtained the licencié és sciences degree in 1910. He became a junior assistant at the University of Warsaw in 1919, whereas in 1921 he earned the doctoral degree at the John Casimir University of Lwów under Hugo Steinhaus and became a senior assistant at the University of Warsaw. Next in 1922 he became a professor at the University of Warsaw, and, after his hab |
https://en.wikipedia.org/wiki/Eugene%20Trubowitz | Eugene Trubowitz is an American mathematician who studies analysis and mathematical physics. He is a Global Professor of Mathematics at New York University Abu Dhabi.
Life and work
Trubowitz, who was born in 1951, received his doctorate in 1977 under the supervision of Henry McKean at New York University, with thesis titled The inverse problem for periodic potentials. Since 1983, he wais a full professor of mathematics at the Swiss Federal Institute of Technology Zurich. As of 2016, he has retired from his position at ETH.
Trubowitz studies scattering theory (some with Percy Deift, and inverse scattering theory), integrable systems and their connection to algebraic geometry, mathematical theory of Fermi liquids in the statistical mechanics.
In 1994 he was an invited speaker at the International Congress of Mathematicians in Zürich; his talk was on A rigorous (renormalization group) analysis of superconducting systems.
Writings
with Percy Deift: Inverse scattering on the line, Communications on Pure and Applied Mathematics, vol.32, 1979, pp. 121–251
with Joel Feldman, Horst Knörrer: Riemann Surfaces of Infinite Genus, AMS (American Mathematical Society) 2003
with Feldman, Knörrer: Fermionic functional integrals and the renormalization group, AMS 2002
with D. Gieseker, Knörrer: Geometry of algebraic Fermi curves, Academic Press 1992
with Jürgen Pöschel: Inverse spectral theory, Academic Press 1987
References
The original article was a translation of the corresponding German article.
http://www.gpo.gov/fdsys/pkg/FR-2012-07-27/pdf/2012-18309.pdf
External links
Homepage an der ETH
20th-century American mathematicians
21st-century American mathematicians
Living people
Academic staff of ETH Zurich
Academic staff of New York University Abu Dhabi
New York University alumni
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Rajchman%20measure | In mathematics, a Rajchman measure, studied by , is a regular Borel measure on a locally compact group such as the circle, whose Fourier transform vanishes at infinity.
References
Measures (measure theory) |
https://en.wikipedia.org/wiki/Horst%20Kn%C3%B6rrer | Horst Knörrer (born 31 July 1953, in Bayreuth) is a German mathematician, who studies algebraic geometry and mathematical physics.
Knörrer studied from 1971 at University of Regensburg and University of Erlangen-Nuremberg and received a doctorate in 1978 from the
University of Bonn under the supervision of Egbert Brieskorn (Isolierte Singularitäten von Durchschnitten zweier Quadriken). After that, he was a research assistant until 1985 in Bonn, interrupted by two years 1980 to 1982 at the
Leiden University. In 1985 he completed his habilitation in Bonn and was a Heisenberg fellow the following two years. During 1986/87, he was a department representative at the University of Düsseldorf. Since 1987, he is a full professor of mathematics at the ETH Zurich.
Knörrer studies algebraic geometry and its connection to mathematical physics, for example, for integrable systems, as well as mathematical theory of many-particle systems in statistical mechanics and solid state physics (Fermi liquids). Together with Brieskorn, he wrote an extensive and rich
illustrated textbook on algebraic curves, which also was translated into English.
Writings
References
The original article was a translation of the corresponding German article.
External links
Homepage an der ETH
20th-century German mathematicians
21st-century German mathematicians
1953 births
Living people
Algebraic geometers
Academic staff of ETH Zurich |
https://en.wikipedia.org/wiki/Josef%20Anton%20Gmeiner | Josef Anton Gmeiner (1862-1926) was an Austrian mathematician working in number theory and mathematical analysis.
Gmeiner studied physics and mathematics at the University of Innsbruck from 1885. In 1890 he passed the examination qualifying him to teach at Gymnasien. After two years as an assistant at the University of Innsbruck's physical institute, he worked as an auxiliary teacher at secondary schools in various locations, including Graz, Fiume, Klagenfurt and Vienna. He earned his doctorate at the University of Innsbruck in 1895, under the joint supervision of Leopold Gegenbauer and Otto Stolz. He then found employment at the German-language Gymnasium in Pula. He became in 1899 a teacher at a Realschule in Vienna and in 1900 a docent in mathematics at the University of Vienna and at TH Wien. At the German University in Prague he was appointed professor extraordinarius in 1901 and promoted to professor ordinarius in 1904. In 1906 he returned to the University of Innsbruck in the professorial chair vacated by the death of Stolz in 1905.
Selected publications
Gmeiner, J. A. (1890). Die Ergänzungssätze zum bicubischen Reciprocitäts-Gesetze. SBer. Kais. Akad. Wissensch. Wien, 100, 1330-1361.
Theoretische Arithmetik (2 volumes: vol. 1, 1900, vol. 2, 1902) by Otto Stolz & J. A. Gmeiner
References
Austrian mathematicians
1862 births
1926 deaths
20th-century Freikorps personnel
Mathematicians from Austria-Hungary
University of Innsbruck alumni
Academic staff of Charles University
Academic staff of the University of Innsbruck |
https://en.wikipedia.org/wiki/Joseph%20Ehrenfried%20Hofmann | Joseph Ehrenfried Hofmann (* 7 March 1900 in Munich, † 7 May 1973 in Günzburg ) was a German historian of mathematics, known for his research on Gottfried Wilhelm Leibniz.
Life and work
After graduating from high school in 1919 at the Wilhelm Gymnasium in Munich, Hofmann studied at University of Munich with Walther von Dyck and George Faber, gaining Ph.D. in 1927. He was briefly an assistant in Munich and Darmstadt, before he went into the teaching profession in Gunzburg, Nördlingen.
As a student he was drawn to the history of mathematics after observing his mentor Faber publishing works of Euler. Another influence was Henry Wieleitner, with whom he published several works on the history of calculus. As a school teacher, he continued his historical studies. In 1939 he habilitated in the history of mathematics at the University of Berlin.
From 1940 to 1945 he edited an edition of the works of Leibnitz for the Berlin Academy of Sciences. Hofmann returned to secondary education in Gunzburg from 1947 until his retirement in 1963. He also had (in part-time) professor of the History of Mathematics at the Albert Ludwig's University of Freiburg, the Humboldt University of Berlin, the Eberhard Karls University of Tuebingen (honorary professorship in 1950) and the Technical University of Karlsruhe.
Hofmann organized regular symposia on the history of mathematics at the Mathematical Research Institute of Oberwolfach, where he worked right after the war. Hofmann was considered an expert in the development of calculus by Leibniz, whose time in Paris he studied carefully. He recorded how the Newton-Leibniz calculus controversy contributed to the invention of calculus.
He was co-editor of the works of Leibniz, of Nicholas of Cusa, and of Johann Bernoulli as well as a mathematical history of Abraham Gotthelf Kästner. He also wrote about number theory of Leonhard Euler and Pierre de Fermat . He uncovered some new works of Fermat (published 1943).
Out for a morning walk, he was killed by a vehicular hit and run.
Writings
Selected Writings, 2 vols, (Editor Christoph Scriba ), Olms, Hildesheim 1990
with Oskar Becker : History of Mathematics, Bonn, Atheneum Publishing, 1951 (derived from Hofmann Part 2 and 3)
History of Mathematics, 3 volumes, de Gruyter, collection Goschen 1953-1957 (Part 1: From the beginnings to the emergence of Fermat and Descartes, 1953, Part 2: From Fermat and Descartes to the invention of calculus and to the development of new Methods, 2nd edition 1963, Part 3: From the debate over the calculus until the French Revolution, 1957, with detailed bibliography). His history of mathematics has also been translated into Spanish, French and English ( Classical Mathematics, New York, Philosophical Library, 1960, The History of Mathematics, New York, Philosophical Library 1957)
Leibniz in Paris 1672-1676 - his growth to mathematical maturity, Cambridge University Press, 1974
The evolution of Leibniz's Mathematics, Munich, and Leibniz-Verlag, 1 |
https://en.wikipedia.org/wiki/Eberhard%20Knobloch | Eberhard Knobloch (born 6 November 1943, in Görlitz) is a German historian of science and mathematics.
Career
From 1962 to 1967 Knobloch studied classics and mathematics at the University of Berlin and the Technical University of Berlin, after which he passed his state examination as a high school teacher and even as a high school teacher in ancient languages at Goethe began high school in Berlin before 1970 as a research assistant in the history of science back to the TU Berlin was, where he in 1972 with a thesis on Leibniz's combinatorial in Scriba, Christoph received his doctorate.
From 1973 he was professor of mathematics at the College of Education in Berlin . In 1976 he qualified as a professor in Berlin and was a visiting scholar at Oxford, London and Edinburgh. Since 1976 he is head of the math sections of the Academy edition of the works of Gottfried Wilhelm Leibniz (and later the technical-scientific parts). In 1981 he became professor of history of science at the Technical University of Berlin (since 2002 academy professor); retiring in 2009. In 1984 he was a visiting professor at the Russian Academy of Sciences in Leningrad. Since 1999 he has been a regular guest professor at Northwestern Polytechnical University in Xian, China. He also was a visiting professor at the Ecole Normale Supérieure in Paris.
Besides the Leibniz Edition, he also oversaw the Tschirnhaus edition of the Saxon Academy of Sciences and worked at Kepler with edition. He is also director of the Alexander von Humboldt Research Centre of the Berlin-Brandenburg Academy of Sciences. He also dealt with Renaissance technology (such as military engineer Mariano Taccola), the notebooks of Leonhard Euler and Jesuit scholars like Christopher Clavius.
Knobloch assisted the Dieter Lelgemann surveyors to decode and interpret the Ptolemy chart with Susudata.
He is a member of the International Academy of the History of Science in Paris (corresponding member since 1984, member since 1988, 2001 to 2005 as Vice President and later its president). Since 1996, a member of the Leopoldina, corresponding member of the Saxon Academy of Sciences, Member of Academia Scientiarum et Artium Europaea since 1997 and the Berlin-Brandenburg Academy of Sciences . From 2001 to 2005 he was president of the German National Committee for the History of Science. In 2006 he became president of the European Society for the History of Science.
Writings
"The mathematical studies of G.W. Leibniz on combinatorics". Studia Leibnitiana Supplements vol.11, 1973.
"The beginning of the theory of determinants. Leibniz posthumous studies on determinants of calculus". Hildesheim in 1980, Arbor Scientiarum B, Vol.2.
L'art de la guerre : Machines et stratagèmes de Taccola, ingénieur de la Renaissance, coll. "Découvertes Gallimard Albums", Paris: Gallimard, 1992.
with Folkerts, Karin Reich : Mass, number and weight: Mathematics is the key to world understanding and world domination. Wiesbaden 2001.
Johann |
https://en.wikipedia.org/wiki/Walter%20Schnee | Walter Schnee (8 August 1885 in Rawitsch, now Rawicz – 10 June 1958 in Leipzig) was a German mathematician. From 1904 to 1908 he studied mathematics in Berlin. From 1909 to 1917 he worked at the University of Breslau. He then went to the University of Leipzig, where he stayed till 1954. He worked in the field of number theory.
References
Walter Schnee
1885 births
1958 deaths
People from Rawicz
People from the Province of Posen
20th-century German mathematicians |
https://en.wikipedia.org/wiki/Gustavo%20Sannia | Gustavo Sannia (13 May 1875 – 21 December 1930) was an Italian mathematician working in differential geometry, projective geometry, and summation of series. He was the son of Achille Sannia, mathematician and senator of the Kingdom of Italy.
Biography
Gustavo Sannia was born in Naples.
Sannia lived in Turin from 1902 to 1915 and from 1919 to 1922, first as an assistant to D'Ovidio and Fubini and later as a professor. From 1915 to 1919, he taught at the University of Cagliari.
Sannia returned to Naples in 1924, where he would remain until his premature death.
Selected publications
"Deformazioni infinitesime delle curve inestendibili e corrispondenza per ortogonalità di elementi." Rendiconti del Circolo Matematico di Palermo (1884–1940) 21, no. 1 (1906): 229–256.
"Nuova esposizione della geometria infinitesimale délle congruenze rettilinee." Annali di Matematica Pura ed Applicata (1898–1922) 15, no. 1 (1908): 143–185.
"Nuovo metodo per lo studio delle congruenze e dei complessi di raggi." Rendiconti del Circolo Matematico di Palermo (1884–1940) 33, no. 1 (1912): 328–340.
"Osservazioni sulla «Réclamation de priorité» del sig. Zindler." Annali di Matematica Pura ed Applicata (1898–1922) 19, no. 1 (1912): 57–59.
"Su due forme differenziali che individuano una congruenza o un complesso di rette." Rendiconti del Circolo Matematico di Palermo (1884–1940) 33, no. 1 (1912): 67–74.
"Sui differenziali totali di ordine superiore." Rendiconti del Circolo Matematico di Palermo (1884–1940) 36, no. 1 (1913): 305–316.
"Nuovo metodo di sommazione delle serie: Estensione del metodo di Borel." Rendiconti del Circolo Matematico di Palermo (1884–1940) 42, no. 1 (1916): 303–322.
"Riavvicinamento di geometrie differenziali delle superficie: metriche, affine, proiettiva." Annali di Matematica Pura ed Applicata (1898–1922) 31, no. 1 (1922): 165–189.
"Nuova trattazione della geometria proiettivo-differenziale delle curve sghembe." Annali di Matematica Pura ed Applicata 3, no. 1 (1926): 1–25.
References
Bibliography
G. F. Tricomi, Matematici italiani del primo secolo dello stato unitario, Memorie dell'Accademia delle Scienze di Torino. Classe di Scienze fisiche matematiche e naturali, 4th series, vol. 1, 1962.
External links
Gustavo Sannia at mathematica.sns.it
Italian mathematicians
1930 deaths
1875 births |
https://en.wikipedia.org/wiki/Labor%20Research%20Association | The Labor Research Association (LRA) was a left-wing labor statistics bureau established in November 1927 by members of the Workers (Communist) Party of America. The organization published a biannual series of volumes known as the Labor Fact Book; it compiled and produced statistics and information for use by trade unions and political activists. The LRA has been frequently characterized as a front organization of the Communist Party. Jonathan Tasini was the executive director of the Labor Research Association in 2008.
Organizational history
The Labor Research Association (LRA) was established late in 1927 by International Publishers president Alexander Trachtenberg and several individuals formerly associated with the Socialist Party's Rand School of Social Science, including Scott Nearing, Solon DeLeon, and Robert W. Dunn. In addition, founders included the prominent radical intellectuals Anna Rochester and Grace Hutchins.
According to American communist writer Myra Page, her husband John Markey (writing as "John Barnett") began working there in 1930, at which time LRA's directors included Anna Rochester, Bob Dunn, Grace Hutchins, Carl Haessler, and Charlotte Todes Stern (another John Reed Club member, along with her husband Bernhard Stern). Edward Dahlberg contributed writings. Dunn, Hutchins, and Rochester published Labor Fact Book.
Originally conceived and organized by Trachtenberg, LRA was announced at the November 2, 1927 meeting of the Political Committee of the Workers (Communist) Party. The organization's declared task was "to conduct research into economic, social, and political problems in the interest of the American labor movement and to publish its findings in articles, pamphlets and books." To this end, from 1931 to 1963, the LRA published a biannual series of statistical and political yearbooks called The Labor Fact Book. These were produced by International Publishers.
The LRA tried to established connection between the labor movement and the Communist movement.
Labor Fact Book volumes
1931
1934
1936
1938
1941
1943
1945
1947
1949
1951
1953
1955
1957
1959
1961
1963
1965
See also
Bureau of Industrial Research
References
Organizations established in 1927
Communism in the United States
Communist Party USA mass organizations
1927 establishments in the United States |
https://en.wikipedia.org/wiki/Heinrich%20August%20Rothe | Heinrich August Rothe (1773–1842) was a German mathematician, a professor of mathematics at Erlangen. He was a student of Carl Hindenburg and a member of Hindenburg's school of combinatorics.
Biography
Rothe was born in 1773 in Dresden, and in 1793 became a docent at the University of Leipzig. He became an extraordinary professor at Leipzig in 1796, and in 1804 he moved to Erlangen as a full professor, taking over the chair formerly held by Karl Christian von Langsdorf. He died in 1842, and his position at Erlangen was in turn taken by Johann Wilhelm Pfaff, the brother of the more famous mathematician Johann Friedrich Pfaff.
Research
The Rothe–Hagen identity, a summation formula for binomial coefficients, appeared in Rothe's 1793 thesis. It is named for him and for the later work of Johann Georg Hagen. The same thesis also included a formula for computing the Taylor series of an inverse function from the Taylor series for the function itself, related to the Lagrange inversion theorem.
In the study of permutations, Rothe was the first to define the inverse of a permutation, in 1800. He developed a technique for visualizing permutations now known as a Rothe diagram, a square table that has a dot in each cell (i,j) for which the permutation maps position i to position j and a cross in each cell (i,j) for which there is a dot later in row i and another dot later in column j. Using Rothe diagrams, he showed that the number of inversions in a permutation is the same as in its inverse, for the inverse permutation has as its diagram the transpose of the original diagram, and the inversions of both permutations are marked by the crosses. Rothe used this fact to show that the determinant of a matrix is the same as the determinant of the transpose: if one expands a determinant as a polynomial, each term corresponds to a permutation, and the sign of the term is determined by the parity of its number of inversions. Since each term of the determinant of the transpose corresponds to a term of the original matrix with the inverse permutation and the same number of inversions, it has the same sign, and so the two determinants are also the same.
In his 1800 work on permutations, Rothe also was the first to consider permutations that are involutions; that is, they are their own inverse, or equivalently they have symmetric Rothe diagrams. He found the recurrence relation
for counting these permutations, which also counts the number of Young tableaux, and which has as its solution the telephone numbers
1, 2, 4, 10, 26, 76, 232, 764, 2620, 9496, ... .
Rothe was also the first to formulate the q-binomial theorem, a q-analog of the binomial theorem, in an 1811 publication.
Selected publications
Formulae De Serierum Reversione Demonstratio Universalis Signis Localibus Combinatorio-Analyticorum Vicariis Exhibita: Dissertatio Academica, Leipzig, 1793.
"Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das |
https://en.wikipedia.org/wiki/Shohei%20Okada | is a Japanese footballer who plays as a forward for Nankatsu SC.
Club statistics
.
References
External links
Profile at Nankatsu SC
1989 births
Living people
National Institute of Fitness and Sports in Kanoya alumni
Japanese men's footballers
J1 League players
J2 League players
J3 League players
Sagan Tosu players
Shonan Bellmare players
Thespakusatsu Gunma players
Nankatsu SC players
Men's association football forwards
Association football people from Kawasaki, Kanagawa |
https://en.wikipedia.org/wiki/Tatsuro%20Okuda | is a professional Japanese football player. He plays as a goalkeeper for Kochi United.
Club statistics
Updated to 23 February 2018.
References
External links
Profile at Júbilo Iwata
1988 births
Living people
Aichi Gakuin University alumni
Association football people from Nara Prefecture
Japanese men's footballers
J1 League players
J2 League players
Sagan Tosu players
Júbilo Iwata players
V-Varen Nagasaki players
Men's association football goalkeepers |
https://en.wikipedia.org/wiki/2012%20Djurg%C3%A5rdens%20IF%20season | In the 2012 season, Djurgårdens IF competes in the Allsvenskan and Svenska Cupen. Magnus Pehrsson is managing the team for the second year.
Players statistics
Appearances for competitive matches only
|}
Goals
Competitions
Allsvenskan
League table
Matches
Svenska Cupen
References
Djurgarden
Djurgårdens IF Fotboll seasons |
https://en.wikipedia.org/wiki/Abstract%20cell%20complex | In mathematics, an abstract cell complex is an abstract set with Alexandrov topology in which a non-negative integer number called dimension is assigned to each point. The complex is called “abstract” since its points, which are called “cells”, are not subsets of a Hausdorff space as is the case in Euclidean and CW complexes. Abstract cell complexes play an important role in image analysis and computer graphics.
History
The idea of abstract cell complexes (also named abstract cellular complexes) relates to J. Listing (1862) and E. Steinitz (1908). Also A.W Tucker (1933), K. Reidemeister (1938), P.S. Aleksandrov (1956) as well as R. Klette and A. Rosenfeld (2004) have described abstract cell complexes. E. Steinitz has defined an abstract cell complex as where E is an abstract set, B is an asymmetric, irreflexive and transitive binary relation called the bounding relation among the elements of E and dim is a function assigning a non-negative integer to each element of E in such a way that if , then .
V. Kovalevsky (1989) described abstract cell complexes for 3D and higher dimensions. He also suggested numerous applications to image analysis. In his book (2008) he has suggested an axiomatic theory of locally finite topological spaces which are generalization of abstract cell complexes. The book contains among others new definitions of topological balls and spheres independent of metric, a new definition of combinatorial manifolds and many algorithms useful for image analysis.
Basic results
The topology of abstract cell complexes is based on a partial order in the set of its points or cells.
The notion of the abstract cell complex defined by E. Steinitz is related to the notion of an abstract simplicial complex and it differs from a simplicial complex by the property that its elements are no simplices: An n-dimensional element of an abstract complexes must not have n+1 zero-dimensional sides, and not each subset of the set of zero-dimensional sides of a cell is a cell. This is important since the notion of an abstract cell complexes can be applied to the two- and three-dimensional grids used in image processing, which is not true for simplicial complexes. A non-simplicial complex is a generalization which makes the introduction of cell coordinates possible: There are non-simplicial complexes which are Cartesian products of such "linear" one-dimensional complexes where each zero-dimensional cell, besides two of them, bounds exactly two one-dimensional cells. Only such Cartesian complexes make it possible to introduce such coordinates that each cell has a set of coordinates and any two different cells have different coordinate sets. The coordinate set can serve as a name of each cell of the complex which is important for processing complexes.
Abstract complexes allow the introduction of classical topology (Alexandrov-topology) in grids being the basis of digital image processing. This possibility defines the great advantage of abstract cel |
https://en.wikipedia.org/wiki/Residual%20time | In the theory of renewal processes, a part of the mathematical theory of probability, the residual time or the forward recurrence time is the time between any given time and the next epoch of the renewal process under consideration. In the context of random walks, it is also known as overshoot. Another way to phrase residual time is "how much more time is there to wait?".
The residual time is very important in most of the practical applications of renewal processes:
In queueing theory, it determines the remaining time, that a newly arriving customer to a non-empty queue has to wait until being served.
In wireless networking, it determines, for example, the remaining lifetime of a wireless link on arrival of a new packet.
In dependability studies, it models the remaining lifetime of a component.
etc.
Formal definition
Consider a renewal process , with holding times and jump times (or renewal epochs) , and . The holding times are non-negative, independent, identically distributed random variables and the renewal process is defined as . Then, to a given time , there corresponds uniquely an , such that:
The residual time (or excess time) is given by the time from to the next renewal epoch.
Probability distribution of the residual time
Let the cumulative distribution function of the holding times be and recall that the renewal function of a process is . Then, for a given time , the cumulative distribution function of is calculated as:
Differentiating with respect to , the probability density function can be written as
where we have substituted From elementary renewal theory, as , where is the mean of the distribution . If we consider the limiting distribution as , assuming that as , we have the limiting pdf as
Likewise, the cumulative distribution of the residual time is
For large , the distribution is independent of , making it a stationary distribution. An interesting fact is that the limiting distribution of forward recurrence time (or residual time) has the same form as the limiting distribution of the backward recurrence time (or age). This distribution is always J-shaped, with mode at zero.
The first two moments of this limiting distribution are:
where is the variance of and and are its second and third moments.
Waiting time paradox
The fact that (for ) is also known variously as the waiting time paradox, inspection paradox, or the paradox of renewal theory. The paradox arises from the fact that the average waiting time until the next renewal, assuming that the reference time point is uniform randomly selected within the inter-renewal interval, is larger than the average inter-renewal interval . The average waiting is only when , that is when the renewals are always punctual or deterministic.
Special case: Markovian holding times
When the holding times are exponentially distributed with , the residual times are also exponentially distributed. That is because and:
This is a known characteristic of th |
https://en.wikipedia.org/wiki/N.%20Anbuchezhian | N. Anbuchezhian (B.Sc.,(Maths)., MA., (Political Science) is an Indian politician born and brought up from the state (Region) of Tamil Nadu in Sekkapatti village of Madurai District (Dindigul Area now). He was elected Member of Parliament for the Dindigul constituency for the period 1967–1971. Anbuchezhian subsequently served a term in the Legislative Assembly of Tamil Nadu.
Early life
N. Anbuchezhian was born in Sekkapatti village in the Madurai district (now part of Dindigul district), on 1 November 1936. His father, A. Neelamegam Pillai was an agriculturist. He had his school life in his native village. He studied Intermediate Pre-University Course at Virudhunagar Hindu Nadar's Senthikumara Nadar College, Virudhunagar; and then he completed Bachelor's degree Under Graduation in Mathematics at Sir Theagaraya College, Madras(Chennai). He studied Post Graduation Degree MA., Political Science in Pachaiappas College, Chennai.
Politics
Anbuchezhian was a member of the Dravida Munnetra Kazhagam (DMK) and was selected to contest the 1967 election for the 4th Lok Sabha as handpicked candidate directly by C. N. Annadurai, the founder of DMK. He was the first MP for the constituency who was not a member of the Indian National Congress (INC) party. He defeated the sitting Congress MP and Union Minister, T. S. Soundram, with a margin of over 100,000 votes.
Subsequently, between 1971–1976, Anbuchezhian was also the Member of the Legislative Assembly for the Periyakulam state assembly constituency, after defeating Chinnasamy Chettai of the INC by a margin of 9,595 votes. During the same period, he was elected chairman of the Batlagundu in Dindigul District, defeating INC candidate Radhakrishna Chettiar. Anbuchezhian was also elected as Panchayath President in his own village of Sekkapatti.
Anbuchezhian contested Nilakkottai assembly constituency ( General ) during 1962 election in DMK ticket. He lost the election with a slender margin of 944 votes. The election result was declared void by High court. Subsequently, during the reconstitution of assembly constituencies in TN, Nillakottai assembly constituency was categorized as reserved one.
References
Living people
1936 births
Lok Sabha members from Tamil Nadu
Tamil Nadu MLAs 1971–1976
Dravida Munnetra Kazhagam politicians
India MPs 1967–1970
People from Dindigul district |
https://en.wikipedia.org/wiki/Dominant%20functor | In category theory, an abstract branch of mathematics, a dominant functor is a functor F : C → D in which every object of D is a retract of an object of the form F(x) for some object X of C.
References
Functors |
https://en.wikipedia.org/wiki/List%20of%20Preston%20North%20End%20F.C.%20managers |
Managerial history
The following is a list of Preston North End managers and caretaker managers.
Statistics include League, FA Cup, League Cup and Football League Trophy matches. All points averages are calculated using three points for a win.
Caretaker managers are shown in italics.
References
Preston North End
Managers |
https://en.wikipedia.org/wiki/Leah%20Busque | Leah Busque (born November 15, 1979), the founder of TaskRabbit, is an American entrepreneur.
Biography
Busque graduated from Sweet Briar College in 2001, earning a Bachelor of Science in Mathematics and Computer Science. She currently serves on the college's board of directors. Prior to RunMyErrand, Busque was an IBM Corp. engineer.
Busque lives in the San Francisco Bay Area with her husband and three children.
In an interview, she said that, based on "geeky conversations" with her ex-husband she "purchased the domain name runmyerrand.com" and "four months after that" left IBM "to build the first version of the site". When a chance to help the business grow required the family to relocate, they moved from Boston to San Francisco.
TaskRabbit
TaskRabbit is an online and mobile marketplace that connects clients with "taskers" to outsource small jobs and tasks, such as cleaning, performing deliveries, assembling furniture, and more, to others in their neighborhood.
The company was originally founded under the name RunMyErrand.
In 2010, the company was renamed to TaskRabbit.
From 2008 to 2016, Busque served as CEO of TaskRabbit. During this time, she scaled the company to 44 cities and raised more than US$50 million.
In April 2016, Busque transitioned into the role of executive chairwoman.
In September 2017, TaskRabbit was acquired by IKEA.
Entrepreneur
Since the summer of 2017, Busque has been a General Partner at FUEL Capital, where she invests in early-stage companies across the consumer technology, hardware, education, marketplaces, and retail sectors.
References
Living people
1979 births
American computer businesspeople
Sweet Briar College alumni
American women company founders
American company founders
21st-century American women |
https://en.wikipedia.org/wiki/Constant%20strain%20triangle%20element | In numerical mathematics, the constant strain triangle element, also known as the CST element or T3 element, is a type of element used in finite element analysis which is used to provide an approximate solution in a 2D domain to the exact solution of a given differential equation.
The name of this element reflects how the partial derivatives of this element's shape function are linear functions. When applied to plane stress and plane strain problems, this means that the approximate solution obtained for the stress and strain fields are constant throughout the element's domain.
The element provides an approximation for the exact solution of a partial differential equation which is parametrized barycentric coordinate system (mathematics)
FEM elements |
https://en.wikipedia.org/wiki/Catalan%27s%20triangle | In combinatorial mathematics, Catalan's triangle is a number triangle whose entries give the number of strings consisting of n X's and k Y's such that no initial segment of the string has more Y's than X's. It is a generalization of the Catalan numbers, and is named after Eugène Charles Catalan. Bailey shows that satisfy the following properties:
.
.
.
Formula 3 shows that the entry in the triangle is obtained recursively by adding numbers to the left and above in the triangle. The earliest appearance of the Catalan triangle along with the recursion formula is in page 214 of the treatise on Calculus published in 1800 by Louis François Antoine Arbogast.
Shapiro introduces another triangle which he calls the Catalan triangle that is distinct from the triangle being discussed here.
General formula
The general formula for is given by
So
When , the diagonal is the -th Catalan number.
The row sum of the -th row is the -th Catalan number, using the hockey-stick identity and an alternative expression for Catalan numbers.
Table of values
Some values are given by
{| class="wikitable" style="text-align:right;"
|-
!
! width="50" | 0
! width="50" | 1
! width="50" | 2
! width="50" | 3
! width="50" | 4
! width="50" | 5
! width="50" | 6
! width="50" | 7
! width="50" | 8
|-
! 0
| 1 || || || || || || || ||
|-
! 1
| 1 || 1 || || || || || || ||
|-
! 2
| 1 || 2 || 2 || || || || || ||
|-
! 3
| 1 || 3 || 5 || 5 || || || || ||
|-
! 4
| 1 || 4 || 9 || 14 || 14 || || || ||
|-
! 5
| 1 || 5 || 14 || 28 || 42 || 42 || || ||
|-
! 6
| 1 || 6 || 20 || 48 || 90 || 132 || 132 || ||
|-
! 7
| 1 || 7 || 27 || 75 || 165 || 297 || 429 || 429 ||
|-
! 8
| 1 || 8 || 35 || 110 || 275 || 572 || 1001 || 1430 || 1430
|}
Properties
Formula 3 from the first section can be used to prove both
That is, an entry is the partial sum of the above row and also the partial sum of the column to the left (except for the entry on the diagonal).
If , then at some stage there must be more 's than 's, so .
A combinatorial interpretation of the -th value is the number of non-decreasing partitions with exactly parts with maximum part such that each part is less than or equal to its index. So, for example, counts
Generalization
Catalan's trapezoids are a countable set of number trapezoids which generalize Catalan’s triangle. Catalan's trapezoid of order is a number trapezoid whose entries give the number of strings consisting of X-s and Y-s such that in every initial segment of the string the number of Y-s does not exceed the number of X-s by or more. By definition, Catalan's trapezoid of order is Catalan's triangle, i.e., .
Some values of Catalan's trapezoid of order are given by
{| class="wikitable" style="text-align:right;"
|-
!
! width="50" | 0
! width="50" | 1
! width="50" | 2
! width="50" | 3
! width="50" | 4
! width="50" | 5
! width="50" | 6
! width="50" | 7
! width="50" | 8
|-
! 0
| 1 || 1 || || || || || || ||
|-
! 1
| 1 || 2 || 2 || || || || || ||
|-
! 2
| |
https://en.wikipedia.org/wiki/Richard%20Lyons%20%28mathematician%29 | Richard Neil Lyons (born January 22, 1945 in New York City, New York) is an American mathematician, specializing in finite group theory.
Lyons received his PhD in 1970 at the University of Chicago under John Griggs Thompson with a thesis entitled Characterizations of Some Finite Simple Groups with Small 2-Rank. From 1972 to 2017, he was a professor at Rutgers University.
With Daniel Gorenstein and Ronald Solomon he wrote, and is continuing to write, now with Inna Capdeboscq, a series on the second-generation proof of the classification program for finite simple groups. Nine volumes of this series have been published so far. He discovered a sporadic group which Charles Sims constructed and called the Lyons group Ly.
In 2012, he shared the Leroy P. Steele Prize for Mathematical Exposition, awarded by the American Mathematical Society, with Michael Aschbacher, Stephen D. Smith, and Ronald Solomon.
In 2013, he became a fellow of the American Mathematical Society "for contributions to the classification of the finite simple groups, including the discovery of one of the 26 sporadic finite simple groups.".
Works
with Gorenstein: The local structure of finite groups of characteristic 2 type, American Mathematical Society, 1983
with Daniel Gorenstein, Ronald Solomon: The classification of the finite simple groups, American Mathematical Society, 9 Vols., 1994–2021
with Michael Aschbacher, Stephen D. Smith, Ronald Solomon: The classification of finite simple groups: Groups of characteristic 2 type, American Mathematical Society, 2011
References
External links
Homepage
20th-century American mathematicians
21st-century American mathematicians
1945 births
Group theorists
University of Chicago alumni
Rutgers University faculty
Living people
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Telephone%20number%20%28mathematics%29 | In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on vertices, the number of permutations on elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from )
Applications
John Riordan provides the following explanation for these numbers: suppose that people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different patterns of connection are possible? For instance, with three subscribers, there are three ways of forming a single telephone call, and one additional pattern in which no calls are being made, for a total of four patterns. For this reason, the numbers counting how many patterns are possible are sometimes called the telephone numbers.
Every pattern of pairwise connections between people defines an involution, a permutation of the people that is its own inverse. In this permutation, each two people who call each other are swapped, and the people not involved in calls remain fixed in place. Conversely, every possible involution has the form of a set of pairwise swaps of this type. Therefore, the telephone numbers also count involutions. The problem of counting involutions was the original combinatorial enumeration problem studied by Rothe in 1800 and these numbers have also been called involution numbers.
In graph theory, a subset of the edges of a graph that touches each vertex at most once is called a matching. Counting the matchings of a given graph is important in chemical graph theory, where the graphs model molecules and the number of matchings is the Hosoya index. The largest possible Hosoya index of an -vertex graph is given by the complete graphs, for which any pattern of pairwise connections is possible; thus, the Hosoya index of a complete graph on vertices is the same as the th telephone number.
A Ferrers diagram is a geometric shape formed by a collection of squares in the plane, grouped into a polyomino with a horizontal top edge, a vertical left edge, and a single monotonic chain of edges from top right to bottom left. A standard Young tableau is formed by placing the numbers from 1 to into these squares in such a way that the numbers increase from left to right and from top to bottom throughout the tableau.
According to the Robinson–Schensted correspondence, permutations correspond one-for-one with ordered pairs of standard Young tableaux. Inverting a permutation c |
https://en.wikipedia.org/wiki/Alberto%20Gonz%C3%A1lez%20Dom%C3%ADnguez | Alberto González Domínguez (11 April 1904 in Buenos Aires – 14 September 1982 in Buenos Aires) was an Argentine mathematician working on analysis, probability theory and quantum field theory.
González Domínguez received his Ph.D. from the University of Buenos Aires in 1939 under the direction of Julio Rey Pastor. That same year, González Domínguez received a Guggenheim Fellowship and worked for two years with Jacob Tamarkin at Brown University. González Domínguez spent most of his career as a professor at the University of Buenos Aires.
References
External links
Biographical Sketch
Full curriculum vitae including the list of publications
1904 births
1982 deaths
20th-century Argentine mathematicians
20th-century philologists
Mathematical analysts
Brown University alumni
University of Buenos Aires alumni
Academic staff of the University of Buenos Aires
University of California, Berkeley faculty
University of Hawaiʻi faculty
University of Illinois Chicago faculty
Argentine philologists
Argentine translators
English–Spanish translators
French–Spanish translators
German–Spanish translators
Italian–Spanish translators
Argentine writers in French
Argentine writers in German |
https://en.wikipedia.org/wiki/Nicholas%20Felton%20%28graphic%20designer%29 | Nicholas Felton is an infographic designer. He is the author of Personal Annual Reports that weave measurements into a tapestry of graphs, maps and statistics to reflect the year's activities. He is the co-founder of Daytum.com, and was a member of the product design team at Facebook. His work has been profiled in publications including the New York Times, Wall Street Journal, Wired and Good Magazine and has been recognized as one of the 50 most influential designers in America by Fast Company. He is credited for influencing the design of Facebook's timeline. Nicholas is currently a Human Interface designer at Apple.
Work
His work focuses on "translating quotidian data into meaningful objects and experiences". His most famous art project is the Feltron Project (personal annual reports starting in 2005 until 2014), where he registers the minutiae of his life, including data regarding the places he visited, the music he listened to, and his everyday activities in general (gathered from his own memory, calendar, photos, and Last.fm data) and transforms it into a series of artistic charts. His purpose is not only analytical but also aesthetic, playing between the realms of self-quantification, design and art.
In the same vein he created in 2009 Daytum together with Ryan Case, an app destined to track personalized every-day data, which could be shared between friends. Even though it was very popular, it had some issues regarding design, so its reception was not completely positive. They are also responsible for the creation of Facebook's timeline, which was a challenge due to the impossibility to predict the user's content. That is why their design focused on being flexible enough to allow for a wide variety of content.
Felton's most recent work is the app Reporter, which via short surveys gathers data about the user and tries to "illuminate aspects of your life that might be otherwise unmeasurable".
Even though it might seem that Felton's main concern is self-quantification, he no longer quantifies his own life. After the 2014 edition, he paused his Annual Reports, stating that: "The world of personal data has changed considerably since the project began in 2005 and this edition [2014 edition] attempts to capture its current state". Regarding this last edition, he commented that: "While previous editions have relied on custom solutions to gather ethereal personal data, this edition is based entirely on commercially available applications and devices". As a consequence, he no longer gathered data manually, which generated a problem of context: devices cannot connect raw data in order to tell a story. Nonetheless Felton is confident that data extracted from these devices, if put into relation, can lead us to modify our habits to live a happier and healthier life.
He is now focused on exploring new ways to present the data rather than in the gathering itself, especially through the use of photography, as we can see in his book Photoviz: Visualizin |
https://en.wikipedia.org/wiki/Ralph%20Duncan%20James | Ralph Duncan James (8 February 1909, Liverpool, England – 19 May 1979, Salt Spring Island, British Columbia, Canada) was a Canadian mathematician working on number theory and mathematical analysis.
Born in Liverpool, Ralph moved with his parents to Vancouver, British Columbia when he was 10 years old. After graduating from high school, Ralph attended University of British Columbia. After graduating, he continued in mathematics, writing a master’s thesis on Tangential Coordinates. Proceeding to University of Chicago, he studied number theory and Waring's problem under L. E. Dickson. In 1932 he was a awarded a Ph.D. on the strength of his dissertation Analytical Investigations of Waring's Theorem. He continued post-graduate study, first with E. T. Bell at California Institute of Technology, then in 1934 with G. H. Hardy at Cambridge University. He published in the Transactions of the American Mathematical Society and extended some work of Viggo Brun in 1938.
Ralph James was a professor of mathematics at University of California, Berkeley from 1934 to 1939. He was then called to University of Saskatchewan where he became Head of the mathematics department. In 1943 he began his long tenure at University of British Columbia, becoming Head of the department in 1948. James made contributions to the theory of the Perron integral and to solution of Goldbach's conjecture.
Since 1978, the Canadian Mathematical Society have awarded the Coxeter–James Prize in his honor.
Papers
Ralph Duncan James published the following papers in the course of his career:
1934: Mathematische Annalen "On the representation of integers as sums of pyramidal numbers"
1934: Transactions of the American Mathematical Society 36(2):395–444 "The value of the number g(k) in Waring's problem"
1938: Transactions of the American Mathematical Society 43(2):296–302 "A problem in additive number theory"
1939: Duke Mathematical Journal 5:948–62 "Integers which are not represented by certain ternary quadratic forms"
1942: (with Hermann Weyl ) American Journal of Mathematics 64:539–52 "Elementary note on prime number problems of Vinogradoff’s type"
1943: Bulletin of the American Mathematical Society 49:422–32 "On the sieve method of Viggo Brun"
1946: (with Walter Gage ) Transactions of the Royal Society of Canada Section III(3) 40:25–35 "A generalized integral"
1949: Bulletin of the AMS 55:246–60 "Recent progress on the Goldbach problem"
1950: Canadian Journal of Mathematics 2:297–306 "A generalized integral II"
1954: (with Ivan Niven) Proceedings of the American Mathematical Society 5:834–8 "Unique factorization in multiplicative systems"
1955: Bulletin of the AMS 61:1–15 "Integrals and summable trigonometric series"
1956: Pacific Journal of Mathematics 6:99–110 "Summable trigonometric series"
1968: Canadian Mathematical Bulletin 11:733–5 "The factors of a square-free integer"
1970: American Mathematical Monthly 77(1):94 "Review: Studies in Number Theory"
Refere |
https://en.wikipedia.org/wiki/Ordinal%20regression | In statistics, ordinal regression, also called ordinal classification, is a type of regression analysis used for predicting an ordinal variable, i.e. a variable whose value exists on an arbitrary scale where only the relative ordering between different values is significant. It can be considered an intermediate problem between regression and classification. Examples of ordinal regression are ordered logit and ordered probit. Ordinal regression turns up often in the social sciences, for example in the modeling of human levels of preference (on a scale from, say, 1–5 for "very poor" through "excellent"), as well as in information retrieval. In machine learning, ordinal regression may also be called ranking learning.
Linear models for ordinal regression
Ordinal regression can be performed using a generalized linear model (GLM) that fits both a coefficient vector and a set of thresholds to a dataset. Suppose one has a set of observations, represented by length- vectors through , with associated responses through , where each is an ordinal variable on a scale . For simplicity, and without loss of generality, we assume is a non-decreasing vector, that is, . To this data, one fits a length- coefficient vector and a set of thresholds with the property that . This set of thresholds divides the real number line into disjoint segments, corresponding to the response levels.
The model can now be formulated as
or, the cumulative probability of the response being at most is given by a function (the inverse link function) applied to a linear function of . Several choices exist for ; the logistic function
gives the ordered logit model, while using the probit function gives the ordered probit model. A third option is to use an exponential function
which gives the proportional hazards model.
Latent variable model
The probit version of the above model can be justified by assuming the existence of a real-valued latent variable (unobserved quantity) , determined by
where is normally distributed with zero mean and unit variance, conditioned on . The response variable results from an "incomplete measurement" of , where one only determines the interval into which falls:
Defining and , the above can be summarized as if and only if .
From these assumptions, one can derive the conditional distribution of as
where is the cumulative distribution function of the standard normal distribution, and takes on the role of the inverse link function . The log-likelihood of the model for a single training example , can now be stated as
(using the Iverson bracket .) The log-likelihood of the ordered logit model is analogous, using the logistic function instead of .
Alternative models
In machine learning, alternatives to the latent-variable models of ordinal regression have been proposed. An early result was PRank, a variant of the perceptron algorithm that found multiple parallel hyperplanes separating the various ranks; its output is a weight vector a |
https://en.wikipedia.org/wiki/Statistical%20data%20type | In statistics, groups of individual data points may be classified as belonging to any of various statistical data types, e.g. categorical ("red", "blue", "green"), real number (1.68, -5, 1.7e+6), odd number (1,3,5) etc. The data type is a fundamental component of the semantic content of the variable, and controls which sorts of probability distributions can logically be used to describe the variable, the permissible operations on the variable, the type of regression analysis used to predict the variable, etc. The concept of data type is similar to the concept of level of measurement, but more specific: For example, count data require a different distribution (e.g. a Poisson distribution or binomial distribution) than non-negative real-valued data require, but both fall under the same level of measurement (a ratio scale).
Various attempts have been made to produce a taxonomy of levels of measurement. The psychophysicist Stanley Smith Stevens defined nominal, ordinal, interval, and ratio scales. Nominal measurements do not have meaningful rank order among values, and permit any one-to-one transformation. Ordinal measurements have imprecise differences between consecutive values, but have a meaningful order to those values, and permit any order-preserving transformation. Interval measurements have meaningful distances between measurements defined, but the zero value is arbitrary (as in the case with longitude and temperature measurements in degree Celsius or degree Fahrenheit), and permit any linear transformation. Ratio measurements have both a meaningful zero value and the distances between different measurements defined, and permit any rescaling transformation.
Because variables conforming only to nominal or ordinal measurements cannot be reasonably measured numerically, sometimes they are grouped together as categorical variables, whereas ratio and interval measurements are grouped together as quantitative variables, which can be either discrete or continuous, due to their numerical nature. Such distinctions can often be loosely correlated with data type in computer science, in that dichotomous categorical variables may be represented with the Boolean data type, polytomous categorical variables with arbitrarily assigned integers in the integral data type, and continuous variables with the real data type involving floating point computation. But the mapping of computer science data types to statistical data types depends on which categorization of the latter is being implemented.
Other categorizations have been proposed. For example, Mosteller and Tukey (1977) distinguished grades, ranks, counted fractions, counts, amounts, and balances. Nelder (1990) described continuous counts, continuous ratios, count ratios, and categorical modes of data. See also Chrisman (1998), van den Berg (1991).
The issue of whether or not it is appropriate to apply different kinds of statistical methods to data obtained from different kinds of measurement proced |
https://en.wikipedia.org/wiki/Olof%20Thorin | G. Olof Thorin (23 February 1912, Halmstad – 14 February 2004, Danderyd Hospital) was a Swedish mathematician working on analysis and probability, who introduced the Riesz–Thorin theorem.
References
Swedish mathematicians
1912 births
2004 deaths |
https://en.wikipedia.org/wiki/Monika%20Bohge | Monika Bohge (Lüdenscheid, 1947) is a German writer.
Biography
She studied to become a teacher of mathematics and religion and worked in several centres for people with disabilities. She has authored many spiritual chants and was a member of the band TAKT.
Works
Ich frage mich. Strube-Verlag 1988 (Mus.: Herbert Beuerle)
Die Geschichte von Zachäus. Strube-Verlag 1991 (Mus.: Joachim Schwarz)
Rede nicht von deinem Glauben. 1995 (Mus.: Hartmut Reußwig)
Du bist dabei. Strube-Verlag 2001 (Mus.: Rolf Schweizer)
Begegnung mit dem Propheten. Strube-Verlag 2004 (Mus.: Rolf Schweizer)
External links and references
1947 births
Living people
German women writers
Women religious writers |
https://en.wikipedia.org/wiki/Linear%20predictor%20function | In statistics and in machine learning, a linear predictor function is a linear function (linear combination) of a set of coefficients and explanatory variables (independent variables), whose value is used to predict the outcome of a dependent variable. This sort of function usually comes in linear regression, where the coefficients are called regression coefficients. However, they also occur in various types of linear classifiers (e.g. logistic regression, perceptrons, support vector machines, and linear discriminant analysis), as well as in various other models, such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights".
Definition
The basic form of a linear predictor function for data point i (consisting of p explanatory variables), for i = 1, ..., n, is
where , for k = 1, ..., p, is the value of the k-th explanatory variable for data point i, and are the coefficients (regression coefficients, weights, etc.) indicating the relative effect of a particular explanatory variable on the outcome.
Notations
It is common to write the predictor function in a more compact form as follows:
The coefficients β0, β1, ..., βp are grouped into a single vector β of size p + 1.
For each data point i, an additional explanatory pseudo-variable xi0 is added, with a fixed value of 1, corresponding to the intercept coefficient β0.
The resulting explanatory variables xi0(= 1), xi1, ..., xip are then grouped into a single vector xi of size p + 1.
Vector Notation
This makes it possible to write the linear predictor function as follows:
using the notation for a dot product between two vectors.
Matrix Notation
An equivalent form using matrix notation is as follows:
where and are assumed to be a (p+1)-by-1 column vectors, is the matrix transpose of (so is a 1-by-(p+1) row vector), and indicates matrix multiplication between the 1-by-(p+1) row vector and the (p+1)-by-1 column vector, producing a 1-by-1 matrix that is taken to be a scalar.
Linear regression
An example of the usage of a linear predictor function is in linear regression, where each data point is associated with a continuous outcome yi, and the relationship written
where is a disturbance term or error variable — an unobserved random variable that adds noise to the linear relationship between the dependent variable and predictor function.
Stacking
In some models (standard linear regression, in particular), the equations for each of the data points i = 1, ..., n are stacked together and written in vector form as
where
The matrix X is known as the design matrix and encodes all known information about the independent variables. The variables are random variables, which in standard linear regression are distributed according to a standard normal distribution; they express the influence of any unknown factors on the outcome.
This makes it possible to find optimal coefficients through the method of least |
https://en.wikipedia.org/wiki/Multiplication%20and%20repeated%20addition | In mathematics education, there was a debate on the issue of whether the operation of multiplication should be taught as being a form of repeated addition. Participants in the debate brought up multiple perspectives, including axioms of arithmetic, pedagogy, learning and instructional design, history of mathematics, philosophy of mathematics, and computer-based mathematics.
Background of the debate
In the early 1990s Leslie Steffe proposed the counting scheme children use to assimilate multiplication into their mathematical knowledge. Jere Confrey contrasted the counting scheme with the splitting conjecture. Confrey suggested that counting and splitting are two separate, independent cognitive primitives. This sparked academic discussions in the form of conference presentations, articles and book chapters.
The debate originated with the wider spread of curricula that emphasized scaling, zooming, folding and measuring mathematical tasks in the early years. Such tasks both require and support models of multiplication that are not based on counting or repeated addition. Debates around the question, "Is multiplication really repeated addition?" appeared on parent and teacher discussion forums in the mid-1990s.
Keith Devlin wrote a Mathematical Association of America column titled, "It Ain't No Repeated Addition" that followed up on his email exchanges with teachers, after he mentioned the topic briefly in an earlier article. The column linked the academic debates with practitioner debates. It sparked multiple discussions in research and practitioner blogs and forums. Keith Devlin has continued to write on this topic.
Pedagogical perspectives
From counting to multiplication
In typical mathematics curricula and standards, such as the Common Core State Standards Initiative, the meaning of the product of real numbers steps through a series of notions generally beginning with repeated addition and ultimately residing in scaling.
Once the natural (or whole) numbers have been defined and understood as a means to count, a child is introduced to the basic operations of arithmetic, in this order: addition, subtraction, multiplication and division. These operations, although introduced at a very early stage of a child's mathematics education, have a lasting impact on the development of number sense in students as advanced numeric abilities.
In these curricula, multiplication is introduced immediately after posing questions related to repeated addition, such as: "There are 3 bags of 8 apples each. How many apples are there in all? A student can do:
or choose the alternative
This approach is supported for several years of teaching and learning, and sets up the perception that multiplication is just a more efficient way of adding. Once 0 is brought in, it affects no significant change because
which is 0, and the commutative property would lead us also to define
Thus, repeated addition extends to the whole numbers (0, 1, 2, 3, 4, ...). The |
https://en.wikipedia.org/wiki/Karin%20Reich | Karin Anna Reich is a German historian of mathematics.
Career
From 1967 to 1973 Reich was a scientific assistant at the Research Institute of the Deutsches Museum in Munich and the Institute for the History of Mathematics and Natural Sciences at the Ludwig Maximilian University of Munich, where in 1973 she graduated under supervision of Helmuth Gericke. In 1980 she completed her time in Munich, publishing The development of tensor calculus, in 1994 in a revised form as a book.
In 1980 she became Professor of the History of Natural Science and Engineering at the Stuttgart College of Librarianship. In 1980/81 and 1981/82 she had a teaching assignment for the History of Mathematics at the University of Heidelberg. In 1981 she represented the Department of History of Science at the University of Hamburg. In 1982, she became associate professor and in 1988 Professor for History of Mathematics at the University of Stuttgart. From 1994 until her retirement she was a professor at the Institute for the History of Natural Science, Mathematics and Engineering at the University of Hamburg, where she succeeded Christoph J. Scriba as director.
Recognition
Reich is a corresponding member of the Göttingen Academy of Sciences and Humanities.
Selected publications
Reich's publications include biographies of Carl Friedrich Gauss, Michael Stifel and François Viète. With Gericke, Reich produced an annotated translation of Viète's Analyticam In artem Isagoge from 1591. She wrote a history of vector-and tensor and differential geometry. With Kurt Vogel, Gericke and Reich reissued John Tropfke's history of elementary mathematics.
Reich's books include:
Maß, Zahl und Gewicht: Mathematik als Schlüssel zu Weltverständnis und Weltbeherrschung [Measure, number and weight: Mathematics as key to understanding and mastering the world] (with Menso Folkerts and Eberhard Knobloch, VCH, Acta Humaniora, Weinheim, 1989)
Die Entwicklung des Tensorkalküls: Vom absoluten Differentialkalkül zur Relativitätstheorie [The development of tensor calculus: From the absolute differential calculus to relativity theory] (Birkhäuser, 1994)
Im Umfeld der "Theoria motus": Gauß' Briefwechsel mit Perthes, Laplace, Delambre und Legendre [On matters having to do with the "Theoria motus": Gauss' correspondence with Perthes, Laplace, Delambre and Legendre] (Vandenhoeck & Ruprecht, 2001)
Carl Friedrich Gauß und Russland: Sein Briefwechsel mit in Russland wirkenden Wissenschaftlern [Carl Friedrich Gauss and Russia: His correspondence with scientists working in Russia] (with Elena Roussanova, De Gruyter, 2012)
Carl Friedrich Gauß und Christopher Hansteen: Der Briefwechsel beider Gelehrten im historischen Kontext [Carl Friedrich Gauss and Christopher Hansteen:A correspondence between two scholars in historical context] (with Elena Roussanova, De Gruyter, 2015)
References
Further reading
Gudrun Wolf Schmidt (eds.): "There is no particular way for kings to geometry". Festschrift for Karin Reich . Rauner, |
https://en.wikipedia.org/wiki/Helmuth%20Gericke | Paul Fritz Helmuth Gericke (1909–2007) was a German mathematician and an historian of mathematics.
Life
Gericke was born in Aachen on 7 May 1909. From 1926 to 1931 he studied physics and mathematics at the universities of Greifswald, Marburg and Göttingen. In 1931, he obtained his doctorate with a thesis on the Volta effect. In 1934, he was an assistant to Wilhelm Süss in Freiburg. With Süss, he attained his habilitation in pure mathematics in 1941.
After 1945, he helped Süss to further develop the Mathematical Research Institute Oberwolfach. His interest in the history of mathematics was aroused by the work of Joseph Ehrenfried Hofmann, whom he had met in Oberwolfach in 1945 and 1946. In 1947, he began to hold lectures in Freiburg on topics related to the history of mathematics. He also received support from Heinrich Behnke, which enabled him to publish his work.
In 1952 he was appointed associate professor at the University of Freiburg. He took a professorship at the University of Munich in 1963, where he was appointed as the first Professor of the History of Science. There he founded the Institute for the History of Science. In 1964, against his stated will, he was chosen as deputy chairman of the German Society for the History of Medicine, Science and Technology. In 1977, he became professor emeritus. He began his professional career working on differential geometry and the body of complex numbers, but from 1947 he devoted himself to subjects in the history of mathematics, publishing several books in this field. His focus was on the development of mathematics in ancient Greece and the mathematics of the 19th century.
He died in Freiburg on 15 August 2007 at the age of 98.
Writings
Über den Volta-Effekt, Coburg (1932)
9 Sonderabdrucke 1935 – 1953 (Contents: Über die größte Kugel in einer konvexen Punktmenge (1935) – Zur Arbeit von P. Ganapathi : A Note on the Oval (1935), Einige kennzeichnende Eigenschaften des Kreises (1935), Über ein Konvergenzkriterium (1937), Über eine Ungleichung für gemischte Volumina (1937), Stützbare Bereiche in komplexer Fourier-Darstellung (1940), Algebraische Betrachtungen zu den Aristotelischen Syllogismen (1952), Einige Grundgedanken der modernen Algebra (1952), Über den Begriff der algebraischen Struktur (1953))
Zur Geschichte der Mathematik an der Universität Freiburg im Breisgau (with E. Albert, 1955)
Theorie der Verbände, Mannheim (1963)
Die Entwicklung physikalischer Begriffe bei den Griechen, Göttingen (1965)
Geschichte des Zahlbegriffs, Mannheim (1970)
50 Jahre GAMM (Gesellschaft für Angewandte Mathematik und Mechanik) as editor, Munich (1972)
Aus der Chronik der Deutschen Mathematiker-Vereinigung, Stuttgart (1980)
Mathematik in Antike und Orient, Berlin (1984)
Mathematik im Abendland. Von den römischen Feldmessern bis zu Descartes, Berlin (1990)
Mathematik in Antike, Orient und Abendland'', Wiesbaden (2003; reprint of the individual volumes [1984] and [1990])
Sources
Folkerts: Obituary fo |
https://en.wikipedia.org/wiki/Leopold%20Schmetterer | Leopold Karl Schmetterer (8 November 1919 in Vienna – 23 August 2004 in Gols) was an Austrian mathematician working on analysis, probability, and statistics.
Decorations and awards
1973: Fellow of the American Statistical Association
1975: Austrian Cross of Honour for Science and Art, 1st class
1976: Science Award of the City of Vienna
1981: Austrian State Prize for Science Policy (Ludwig Boltzmann Prize)
1984: Erwin Schrödinger Prize of the Austrian Academy of Sciences
1969: Gibble award
References
Friedrich Pukelsheim: Leopold Schmetterer 8.11.1919–24.8.2004, Jahrbuch der Bayerischen Akademie der Wissenschaften 2004, 317–320.
Mathematicians from Vienna
1919 births
2004 deaths
Recipients of the Austrian Cross of Honour for Science and Art, 1st class
Recipients of the Austrian State Prize
Members of the German Academy of Sciences at Berlin
Fellows of the American Statistical Association
Probability Theory and Related Fields editors |
https://en.wikipedia.org/wiki/Zygmunt%20Zalcwasser | Zygmunt Zalcwasser (1898 – 1943) was a Polish mathematician from the Warsaw School of Mathematics in the period between the World Wars collaborating especially in the fields of logic, set theory, general topology and real analysis. Zalcwasser, who worked on the Fourier series, introduced the Zalcwasser rank [Za] measuring the uniform convergence of sequences of continuous functions on the unit interval. Zalcwasser received his Ph.D. at the Warsaw University in 1928. He served as professor at the Wolna Wszechnica Polska in 1933–34 and after the invasion of Poland in 1939 lived in the Warsaw Ghetto. He was murdered in the gas chambers of the Treblinka extermination camp in 1943 during the Holocaust in Poland.
References
20th-century Polish mathematicians
1898 births
1943 deaths
Polish people who died in Treblinka extermination camp
Polish Jews who died in the Holocaust |
https://en.wikipedia.org/wiki/Waraszkiewicz%20spiral | In mathematics, Waraszkiewicz spirals are subsets of the plane introduced by . Waraszkiewicz spirals give an example of an uncountable family of pairwise incomparable continua, meaning that there is no continuous map from one onto another.
References
Topology |
https://en.wikipedia.org/wiki/Clifford%20S.%20Gardner | Clifford Spear Gardner (January 14, 1924 – September 25, 2013) was an American mathematician specializing in applied mathematics.
Career
Gardner studied at Phillips Academy and Harvard, where he earned his baccalaureate in 1944. In 1953 he earned a PhD from New York University, under the supervision of Fritz John. Thereafter he worked at NASA in Langley Field, the Courant Institute of Mathematical Sciences of NYU, Lawrence Livermore National Laboratory and the Princeton Plasma Physics Laboratory. He was a mathematics professor at the University of Texas at Austin from 1967 to 1990, when he retired as professor emeritus..
In 1985 he won the Norbert Wiener Prize for his contributions to supersonic aerodynamics and plasma physics. In 2006 he received with Martin Kruskal, Robert M. Miura, and John M. Greene the Leroy P. Steele Prize for their work on the inverse scattering transformation method for the solution of nonlinear differential equations (special soliton modeling equations similar to the Korteweg de Vries equation). They developed a systematic approach to solving many nonlinear partial differential equations in a way similar to Fourier analysis for linear PDEs.
Gardner died September 25, 2013, in Austin, Texas.
References
External links
1924 births
20th-century American mathematicians
21st-century American mathematicians
Harvard University alumni
New York University alumni
2013 deaths
University of Texas at Austin faculty
Applied mathematicians
Phillips Academy alumni |
https://en.wikipedia.org/wiki/Alexander%20Schrijver | Alexander (Lex) Schrijver (born 4 May 1948 in Amsterdam) is a Dutch mathematician and computer scientist, a professor of discrete mathematics and optimization at the University of Amsterdam and a fellow at the Centrum Wiskunde & Informatica in Amsterdam. Since 1993 he has been co-editor in chief of the journal Combinatorica.
Biography
Schrijver earned his Ph.D. in 1977 from the Vrije Universiteit in Amsterdam, under the supervision of Pieter Cornelis Baayen. He worked for the Centrum Wiskunde & Informatica (under its former name as the Mathematisch Centrum) in pure mathematics from 1973 to 1979, and was a professor at Tilburg University from 1983 to 1989. In 1989 he rejoined the Centrum Wiskunde & Informatica, and in 1990 he also became a professor at the University of Amsterdam. In 2005, he stepped down from management at CWI and instead became a CWI Fellow.
Awards and honors
Schrijver was one of the winners of the Delbert Ray Fulkerson Prize of the American Mathematical Society in 1982 for his work with Martin Grötschel and László Lovász on applications of the ellipsoid method to combinatorial optimization; he won the same prize in 2003 for his research on minimization of submodular functions. He won the INFORMS Frederick W. Lanchester Prize in 1986 for his book Theory of Linear and Integer Programming, and again in 2004 for his book Combinatorial Optimization: Polyhedra and Efficiency. He was an Invited Speaker of the International Congress of Mathematicians (ICM) in 1986 in Berkeley and of the ICM in 1998 in Berlin. In 2003, he won the George B. Dantzig Prize of the Mathematical Programming Society and SIAM for "deep and fundamental research contributions to discrete optimization". In 2006, he was a joint winner of the INFORMS John von Neumann Theory Prize with Grötschel and Lovász for their work in combinatorial optimization, and in particular for their joint work in the book Geometric Algorithms and Combinatorial Optimization showing the polynomial-time equivalence of separation and optimization. In 2008, his work with Adri Steenbeek on scheduling the Dutch train system was honored with INFORMS' Franz Edelman Award for Achievement in Operations Research and the Management Sciences. He won the SIGMA prize of the Dutch SURF foundation in 2008, for a mathematics education project. In 2015 he won the EURO Gold Medal, the highest distinction within Operations Research in Europe.
In 2005 Schrijver won the Spinoza Prize of the NWO, the highest scientific award in the Netherlands, for his research in combinatorics and algorithms. Later in the same year he became a Knight of the Order of the Netherlands Lion. In 2002, Schrijver received an honorary doctorate from the University of Waterloo in Canada, and in 2011 he received another one from Eötvös Loránd University in Hungary.
Schrijver became a member of the Royal Netherlands Academy of Arts and Sciences in 1995. He became a corresponding member of the North Rhine-Westphalia Academy for Science |
https://en.wikipedia.org/wiki/List%20of%20people%20from%20Bath%2C%20Maine | The following list includes notable people who were born or have lived in Bath, Maine.
Authors and academics
Robert Jaffe, physicist
McDonald Clarke, poet
Eleanor P. Cushing, mathematics professor at Smith College
Alice May Douglas, poet and author
George F. Magoun, first president of Iowa College (now Grinnell College)
Edward Page Mitchell, editorial and short story writer
William Maxwell Reed, author of children's science books
Susan Marr Spalding (1841–1908), poet
Geoffrey Wolff, novelist, essayist, biographer, and travel writer; lives in Bath
Glenn Cummings, economist, politician and University of Southern Maine President
Business
Charles W. Morse, businessman
Media and arts
Georgia Cayvan, stage actress
Claude Demetrius, songwriter
Emma Eames, singer
Chad Finn, sportswriter
John Adams Jackson, sculptor
William Zorach, sculptor
Military
Charles Frederick Hughes, US Navy admiral
William Smith, US Army private; Medal of Honor recipient
Silas Soule, abolitionist and Civil War era soldier
Politics
Nathaniel S. Berry, 28th governor of New Hampshire
Samuel Davis, US congressman
Thomas W. Hyde, US senator; Union Army general and Metal of Honor recipient; founder of Bath Iron Works
William King, first governor of Maine
Arthur Mayo, state legislator
Freeman H. Morse, US congressman and mayor
Amos Nourse, physician and US senator
William LeBaron Putnam, lawyer and politician
Harold M. Sewall, last United States Minister to Hawaii
Sumner Sewall, 58th governor of Maine
Mary Small, politician
Francis B. Stockbridge, US senator
Peleg Tallman, US congressman
Science and engineering
Edward Davis, buccaneer and engineer
Francis H. Fassett, architect
Henry Gannett, geographer
George Edward Harding, architect
Robert Jaffe, physicist
References
Bath, Maine
Bath |
https://en.wikipedia.org/wiki/Hesse%20configuration | In geometry, the Hesse configuration is a configuration of 9 points and 12 lines with three points per line and four lines through each point. It can be realized in the complex projective plane as the set of inflection points of an elliptic curve, but it has no realization in the Euclidean plane. It was introduced by Colin Maclaurin and studied by , and is also known as Young's geometry, named after the later work of John Wesley Young on finite geometry.
Description
The Hesse configuration has the same incidence relations as the lines and points of the affine plane over the field of 3 elements. That is, the points of the Hesse configuration may be identified with ordered pairs of numbers modulo 3, and the lines of the configuration may correspondingly be identified with the triples of points satisfying a linear equation . Alternatively, the points of the configuration may be identified by the squares of a tic-tac-toe board, and the lines may be identified with the lines and broken diagonals of the board.
Each point belongs to four lines: in the tic tac toe interpretation of the configuration, one line is horizontal, one vertical, and two are diagonals or broken diagonals. Each line contains three points. In the language of configurations the Hesse configuration has the notation 94123, meaning that there are 9 points, 4 lines per point, 12 lines, and 3 points per line.
The Hesse configuration has 216 symmetries (its automorphism group has order 216). The group of its symmetries is known as the Hessian group.
Related configurations
Removing any one point and its four incident lines from the Hesse configuration produces another configuration of type 8383, the Möbius–Kantor configuration.
In the Hesse configuration, the 12 lines may be grouped into four triples of parallel (non-intersecting) lines. Removing from the Hesse configuration the three lines belonging to a single triple produces a configuration of type 9393, the Pappus configuration.
The Hesse configuration may in turn be augmented by adding four points, one for each triple of non-intersecting lines, and one line containing the four new points, to form a configuration of type 134134, the set of points and lines of the projective plane over the three-element field.
Realizability
The Hesse configuration can be realized in the complex projective plane as the 9 inflection points of an elliptic curve and the 12 lines through triples of inflection points. If a given set of nine points in the complex plane is the set of inflections of an elliptic curve C, it is also the set of inflections of every curve in a pencil of curves generated by C and by the Hessian curve of C, the Hesse pencil.
The Hessian polyhedron is a representation of the Hesse configuration in the complex plane.
The Hesse configuration shares with the Möbius–Kantor configuration the property of having a complex realization but not being realizable by points and straight lines in the Euclidean plane. In the Hesse configur |
https://en.wikipedia.org/wiki/Hesse%20pencil | In mathematics, the syzygetic pencil or Hesse pencil, named for Otto Hesse, is a pencil (one-dimensional family) of cubic plane elliptic curves in the complex projective plane, defined by the equation
Each curve in the family is determined by a pair of parameter values () (not both zero) and consists of the points in the plane whose homogeneous coordinates satisfy the equation for those parameters. Multiplying both and by the same scalar does not change the curve, so there is only one degree of freedom in selecting a curve from the pencil, but the two-parameter form given above allows either or (but not both) to be set to zero.
Each curve in the pencil passes through the nine points of the complex projective plane whose homogeneous coordinates are some permutation of 0, –1, and a cube root of unity. There are three roots of unity, and six permutations per root, giving 18 choices for the homogeneous coordinates of each point, but they are equivalent in pairs giving only nine points. The family of cubics through these nine points forms the Hesse pencil. More generally, one can replace the complex numbers by any field containing a cube root of unity and define the Hesse pencil over this field to be the family of cubics through these nine points.
The nine common points of the Hesse pencil are the inflection points of each of the cubics in the pencil. Any line that passes through at least two of these nine points passes through exactly three of them; the nine points and twelve lines through triples of points form the Hesse configuration.
Every elliptic curve is birationally equivalent to a curve of the Hesse pencil; this is the Hessian form of an elliptic curve. However, the parameters () of the Hessian form may belong to an extension field of the field of definition of the original curve.
References
Elliptic curves |
https://en.wikipedia.org/wiki/Hessian%20group | In mathematics, the Hessian group is a finite group of order 216, introduced by who named it for Otto Hesse. It may be represented as the group of affine transformations with determinant 1 of the affine plane over the field of 3 elements. It has a normal subgroup that is an elementary abelian group of order 32, and the quotient by this subgroup is isomorphic to the group SL2(3) of order 24. It also acts on the Hesse pencil of elliptic curves, and forms the automorphism group of the Hesse configuration of the 9 inflection points of these curves and the 12 lines through triples of these points.
The triple cover of this group is a complex reflection group, 3[3]3[3]3 or of order 648, and the product of this with a group of order 2 is another complex reflection group, 3[3]3[4]2 or of order 1296.
References
External links
Finite groups |
https://en.wikipedia.org/wiki/List%20of%20the%20busiest%20airports%20in%20the%20Caribbean | This is a list of the busiest airports in the Caribbean region by passenger traffic. Statistics are available for almost all the airstrips taken into account. The present list intends to include all the international airports located in the area geographically defined as the Caribbean.
Given that each country has a different body to control these statistics, the compilation of data is difficult and not homogeneously distributed. The information presented here, represents the best available data from different Internet sources. The list contains statistics for different years, since each country authority does not have strong regulations reporting passengers traffic. The ranking is ordered according to total passenger traffic (unless the footnotes indicate the contrary). Information on aircraft movements or cargo movements is not available for all of the airports.
In graph
Ranking of airports
Note: Although there are more than fifteen international airports in the Caribbean area, statistics were not available for each one of them. Any additional information could improve the article.
Notes
See also
List of the busiest airports in Central America
List of the busiest airports in Latin America
List of the busiest airports in South America
References
Caribbean
Airports
Caribbean
Busiest |
https://en.wikipedia.org/wiki/Mateo%20M%C3%ADguez | Mateo Míguez Adán (born 11 May 1987 in Redondela, Galicia), known simply as Mateo, is a Spanish footballer who plays for Coruxo FC as an attacking midfielder.
Career statistics
Club
References
External links
Celta de Vigo biography
1987 births
Living people
Spanish men's footballers
Footballers from Redondela
Men's association football midfielders
Segunda División players
Segunda División B players
Segunda Federación players
RC Celta Fortuna players
RC Celta de Vigo players
SD Ponferradina players
CD Guadalajara (Spain) footballers
Coruxo FC players
Veikkausliiga players
Ykkönen players
PK-35 Vantaa (men) players
Spanish expatriate men's footballers
Expatriate men's footballers in Finland
Spanish expatriate sportspeople in Finland |
https://en.wikipedia.org/wiki/Alafi%20Mahmud | Mohd Alafi bin Mahmud (born 29 April 1985) is a Malaysian professional footballer who plays for Malaysia M3 League side Imigresen.
Career statistics
Club
Honours
Club
PDRM
Malaysia Premier League: 2014
References
External links
1985 births
Living people
Malaysian men's footballers
Sri Pahang FC players
Perlis F.A. players
Penang F.C. players
Negeri Sembilan FC players
Men's association football forwards |
https://en.wikipedia.org/wiki/Perles%20configuration | In geometry, the Perles configuration is a system of nine points and nine lines in the Euclidean plane for which every combinatorially equivalent realization has at least one irrational number as one of its coordinates. It can be constructed from the diagonals and symmetry lines of a regular pentagon, omitting one of the symmetry lines. In turn, it can be used to construct higher-dimensional convex polytopes that cannot be given rational coordinates, having the fewest vertices of any known example. All of the realizations of the Perles configuration in the projective plane are equivalent to each other under projective transformations.
Construction
One way of constructing the Perles configuration is to start with a regular pentagon and its five diagonals. These diagonals form the sides of a smaller inner pentagon nested inside the outer pentagon. Each vertex of the outer pentagon is situated opposite from a vertex of the inner pentagon. The nine points of the configuration consist of four out of the five vertices of each pentagon and the shared center of the two pentagons. Two opposite vertices are omitted, one from each pentagon.
The nine lines of the configuration consist of the five lines that are diagonals of the outer pentagon and sides of the inner pentagon, and the four lines that pass through the center and through opposite pairs of vertices from the two pentagons.
Projective invariance and irrationality
A realization of the Perles configuration is defined to consist of any nine points and nine lines with the same intersection pattern. That means that a point and line intersect each other in the realization, if and only if they intersect in the configuration constructed from the regular pentagon. Every realization of this configuration in the Euclidean plane or, more generally, in the real projective plane is equivalent, under a projective transformation, to a realization constructed in this way from a regular pentagon.
Because the cross-ratio, a number defined from any four collinear points, does not change under projective transformations, every realization has four points having the same cross-ratio as the cross-ratio of the four collinear points in the realization derived from the regular pentagon. But, these four points have as their cross-ratio, where is the golden ratio, an irrational number. Every four collinear points with rational coordinates have a rational cross ratio, so the Perles configuration cannot be realized by rational points. Branko Grünbaum has conjectured that every configuration that can be realized by irrational but not rational numbers has at least nine points; if so, the Perles configuration would be the smallest possible irrational configuration of points and lines.
Application in polyhedral combinatorics
Perles used his configuration to construct an eight-dimensional convex polytope with twelve vertices that can similarly be realized with real coordinates but not with rational coordinates. The points of |
https://en.wikipedia.org/wiki/John%20Bonnycastle | John Bonnycastle (baptized 29 December 1751 in Hardwick or Whitchurch, England – 15 May 1821 in Woolwich, England) was an English teacher of mathematics and author.
Life
John Bonnycastle was born in Buckinghamshire, in about 1750.
Nothing is known of his family or early life, but he went to London where he established an Academy.
He became a tutor to the two sons of the Earl of Pontefract at Easton in Northumberland.
Between 1782 and 1785, he was appointed Professor of Mathematics at the Royal Military Academy, Woolwich, where he remained until his death on 15 May 1821.
He was a prolific writer, and wrote for the early volumes of Rees's Cyclopædia, about algebra, analysis and astronomy.
Family
At the age of 19, he married a Miss Rolt, but she died young. On Oct.7th, 1786 he married Brigette Newell with whom he had six children Charlotte, William, Mary, Sir Richard (Royal Engineer/Author), Humphrey and Charles.
His son Richard Henry Bonnycastle settled in Canada, where the family became quite well known in Winnipeg and Calgary.
His son, Charles Bonnycastle (1796-1840) became Professor of Mathematics at the University of Virginia.
Writings
The Scholar's guide to Arithmetic, 1780
Introduction to Algebra, 1782
Introduction to Astronomy, 1786 (7th edition 1816)
Euclid's 'Elements' with notes, 1789
A Treatise on Plane and Spherical Geometry, 1806
A Treatise of Algebra, 1813
Notes
References
Gentleman's Magazine, 1821, i, 472, 482
John Bonnycastle
1750s births
1821 deaths
Academics of the Royal Military Academy, Woolwich
British textbook writers
English mathematicians
Schoolteachers from Buckinghamshire
Mathematics educators |
https://en.wikipedia.org/wiki/Micha%20Perles | Micha Asher Perles is an Israeli mathematician working in geometry, a professor emeritus at the Hebrew University. He earned his Ph.D. in 1964 from the Hebrew University, under the supervision of Branko Grünbaum.
His contributions include:
The Perles configuration, a set of nine points in the Euclidean plane whose collinearities can be realized only by using irrational numbers as coordinates. Perles used this configuration to prove the existence of irrational polytopes in higher dimensions.
The Perles–Sauer–Shelah lemma, a result in extremal set theory whose proof was credited to Perles by Saharon Shelah.
The pumping lemma for context-free languages, a widely used method for proving that a language is not context-free that Perles discovered with Yehoshua Bar-Hillel and Eli Shamir.
Notable students of Perles include Noga Alon, Gil Kalai, and Nati Linial.
References
External links
Micha Asher Perles' home page
Micha A. Perles' publication list at DBLP
Micha A. Perles' online publications at arXiv
Israeli mathematicians
Living people
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Franklin%20H.%20Westervelt | Franklin Herbert Westervelt ( – ) was an American engineer, computer scientist, and educator at the University of Michigan and Wayne State University. Westervelt received degrees in Mathematics, Mechanical and Electrical Engineering from the College of Engineering at the University of Michigan. He attained his PhD in 1961. He was a Professor of Mechanical Engineering at the University of Michigan and an Associate Director at the U-M Computing Center. He was involved in early studies on how to use computers in engineering education.
Biography
He was born in Benton Harbor, Michigan on to Herbert Oleander Westervelt and Dorothy Ulbright.
From 1965 to 1970 he was Project Director for the ARPA sponsored CONCOMP (Research in Conversational Use of Computers) Project. He was involved in the design of the architecture and negotiations with IBM over the virtual memory features that would be included in what became the IBM S/360 Model 67 computer. When IBM's TSS/360 time-sharing operating system for the S/360-67 was not available, the CONCOMP project supported the initial development of Michigan Terminal System (MTS) in cooperation with the staff of the University of Michigan Computing Center. This included David L. Mills development of the original PDP-8 Data Concentrator with its interface to an IBM S/360 Input/Output channel, the first such interface to be built outside of IBM. CONCOMP also developed the integration for the IBM 7772 based Audio Response Unit (ARU) as an MTS I/O device, the MAD/I compiler, mini-computer based graphics terminals, and the Set-Theoretic Data Structure model that was later used in ILIR:MICRO.
ARPANET program manager Larry Roberts asked Frank to explore the questions of message size and contents for the ARPANET, and to write a position paper on the intercomputer communication protocol including “conventions for character and block transmission, error checking and re transmission, and computer and user identification." Frank also served as a representative to the statewide Michigan Inter-university Committee on Information Systems (MICIS) and was involved in establishing the MERIT Computer Network.
Fred Gibbons, a successful entrepreneur and venture capitalist, said that the University of Michigan College of Engineering, where he earned his BSE and MSE degrees in the late 1960s and early 1970s when computers were unknown or a novelty in most classrooms and the school didn’t even offer a formal computer major, "... was at the forefront of technology that turned out to be very important to me personally, and I got early exposure to it from a couple of great guys–professors Frank Westervelt and Bernard Galler."
U-M Vice President for Research Geoffrey Norman, writing in 1976, gave special credit to the triumvirate of Michigan computer specialists who contributed greatly to the future of computing at Michigan and in the nation as a whole. "Bartels, Arden, and Westervelt," Norman has said, "were a team that we took great care |
https://en.wikipedia.org/wiki/Scaled%20correlation | In statistics, scaled correlation is a form of a coefficient of correlation applicable to data that have a temporal component such as time series. It is the average short-term correlation. If the signals have multiple components (slow and fast), scaled coefficient of correlation can be computed only for the fast components of the signals, ignoring the contributions of the slow components. This filtering-like operation has the advantages of not having to make assumptions about the sinusoidal nature of the signals.
For example, in the studies of brain signals researchers are often interested in the high-frequency components (beta and gamma range; 25–80 Hz), and may not be interested in lower frequency ranges (alpha, theta, etc.). In that case scaled correlation can be computed only for frequencies higher than 25 Hz by choosing the scale of the analysis, s, to correspond to the period of that frequency (e.g., s = 40 ms for 25 Hz oscillation).
Definition
Scaled correlation between two signals is defined as the average correlation computed across short segments of those signals. First, it is necessary to determine the number of segments that can fit into the total length of the signals for a given scale :
Next, if is Pearson's coefficient of correlation for segment , the scaled correlation across the entire signals is computed as
Efficiency
In a detailed analysis, Nikolić et al. showed that the degree to which the contributions of the slow components will be attenuated depends on three factors, the choice of the scale, the amplitude ratios between the slow and the fast component, and the differences in their oscillation frequencies. The larger the differences in oscillation frequencies, the more efficiently will the contributions of the slow components be removed from the computed correlation coefficient. Similarly, the smaller the power of slow components relative to the fast components, the better will scaled correlation perform.
Application to cross-correlation
Scaled correlation can be applied to auto- and cross-correlation in order to investigate how correlations of high-frequency components change at different temporal delays. To compute cross-scaled-correlation for every time shift properly, it is necessary to segment the signals anew after each time shift. In other words, signals are always shifted before the segmentation is applied. Scaled correlation has been subsequently used to investigate synchronization hubs in the visual cortex. Scaled correlation can be also used to extract functional networks.
Advantages over filtering methods
Scaled correlation should be in many cases preferred over signal filtering based on spectral methods. The advantage of scaled correlation is that it does not make assumptions about the spectral properties of the signal (e.g., sinusoidal shapes of signals). Nikolić et al. have shown that the use of Wiener–Khinchin theorem to remove slow components is inferior to results obtained by scaled correlation. |
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