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https://en.wikipedia.org/wiki/Bunch%E2%80%93Nielsen%E2%80%93Sorensen%20formula | In mathematics, in particular linear algebra, the Bunch–Nielsen–Sorensen formula, named after James R. Bunch, Christopher P. Nielsen and Danny C. Sorensen, expresses the eigenvectors of the sum of a symmetric matrix and the outer product, , of vector with itself.
Statement
Let denote the eigenvalues of and denote the eigenvalues of the updated matrix . In the special case when is diagonal, the eigenvectors of can be written
where is a number that makes the vector normalized.
Derivation
This formula can be derived from the Sherman–Morrison formula by examining the poles of .
Remarks
The eigenvalues of were studied by Golub.
Numerical stability of the computation is studied by Gu and Eisenstat.
See also
Sherman–Morrison formula
References
External links
Rank-One Modification of the Symmetric Eigenproblem at EUDML
Some Modified Matrix Eigenvalue Problems
A Stable and Efficient Algorithm for the Rank-One Modification of the Symmetric Eigenproblem
Linear algebra |
https://en.wikipedia.org/wiki/Connective%20spectrum | In algebraic topology, a branch of mathematics, a connective spectrum is a spectrum whose homotopy sets of negative degrees are zero.
References
External links
Why are connective spectra called “connective”?
Algebraic topology
Homotopy theory |
https://en.wikipedia.org/wiki/Hyper-Erlang%20distribution | In probability theory, a hyper-Erlang distribution is a continuous probability distribution which takes a particular Erlang distribution Ei with probability pi. A hyper-Erlang distributed random variable X has a probability density function given by
where each pi > 0 with the pi summing to 1 and each of the Eli being an Erlang distribution with li stages each of which has parameter λi.
See also
Phase-type distribution
References
Continuous distributions |
https://en.wikipedia.org/wiki/Platania%2C%20Kozani | Platania () is a village and a community in the Voio municipality of Greece. In the late Ottoman period, it was inhabited by Vallahades; in the 1900 statistics of Vasil Kanchov, where the town appears under its Bulgarian name "Bobusht'"/"Bobushta", it was inhabited by some 300 "Greek Muslims". Before the 2011 local government reform it was part of the municipality of Neapoli, of which it was a municipal district. The 2011 census recorded 93 inhabitants in the village.
References
Populated places in Kozani (regional unit)
Voio (municipality) |
https://en.wikipedia.org/wiki/KAlgebra | KAlgebra is a mathematical graph calculator included in the KDE education package. While it is based on the MathML content markup language, knowledge of MathML is not required for use. The calculator includes numerical, logical, symbolic, and analytical functions, and can plot the results onto a 2D or 3D graph. KAlgebra is free and open source software, licensed under the GPL-2.0-or-later license.
KAlgebra has been mentioned by various media sources as free / open source educational programs.
User interface and syntax
KAlgebra uses an intuitive algebraic syntax, similar to those used on modern graphing calculators. User-entered expressions are converted to MathML in the background, or they can be entered directly. The program is divided into four views, Console, 2D Graph, 3D Graph, and Dictionary. A series of calculations can be performed with user-defined scripts, which are macros that can be reused and shared.
The dictionary includes a comprehensive list of all built-in functions in KAlgebra. Functions can be looked up with parameters, examples, formulas and sample plots. Over 100 functions and operations are currently supported.
Graphing and dictionary
In the 2D and 3D graph views, functions can evaluated and plotted. Currently KAlgebra only supports 3D graphs explicitly dependent only on the x and y. Both views support defining the viewpoint. The user can hover their cursor over a line and find the exact X and Y values for 2D graphs, as well as create a live tangent line.
In the 3D view, the user can control the viewpoint position with the keyboard's arrow keys, and zooming in and out is done with the and keys respectively. The user can also draw lines and make dots on the 3D graph and export the graph in various formats.
References
External links
KAlgebra page on kde.org
KAlgebra page on Kdeapps
The KAlgebra Handbook (manual/documentation)
Free science software
Free software programmed in C++
KDE Applications
Science software for Linux
Science software for macOS
Science software for Windows
Science software that uses Qt |
https://en.wikipedia.org/wiki/GIT%20quotient | In algebraic geometry, an affine GIT quotient, or affine geometric invariant theory quotient, of an affine scheme with an action by a group scheme G is the affine scheme , the prime spectrum of the ring of invariants of A, and is denoted by . A GIT quotient is a categorical quotient: any invariant morphism uniquely factors through it.
Taking Proj (of a graded ring) instead of , one obtains a projective GIT quotient (which is a quotient of the set of semistable points.)
A GIT quotient is a categorical quotient of the locus of semistable points; i.e., "the" quotient of the semistable locus. Since the categorical quotient is unique, if there is a geometric quotient, then the two notions coincide: for example, one has
for an algebraic group G over a field k and closed subgroup H.
If X is a complex smooth projective variety and if G is a reductive complex Lie group, then the GIT quotient of X by G is homeomorphic to the symplectic quotient of X by a maximal compact subgroup of G (Kempf–Ness theorem).
Construction of a GIT quotient
Let G be a reductive group acting on a quasi-projective scheme X over a field and L a linearized ample line bundle on X. Let
be the section ring. By definition, the semistable locus is the complement of the zero set in X; in other words, it is the union of all open subsets for global sections s of , n large. By ampleness, each is affine; say and so we can form the affine GIT quotient
Note that is of finite type by Hilbert's theorem on the ring of invariants. By universal property of categorical quotients, these affine quotients glue and result in
which is the GIT quotient of X with respect to L. Note that if X is projective; i.e., it is the Proj of R, then the quotient is given simply as the Proj of the ring of invariants .
The most interesting case is when the stable locus is nonempty; is the open set of semistable points that have finite stabilizers and orbits that are closed in . In such a case, the GIT quotient restricts to
which has the property: every fiber is an orbit. That is to say, is a genuine quotient (i.e., geometric quotient) and one writes . Because of this, when is nonempty, the GIT quotient is often referred to as a "compactification" of a geometric quotient of an open subset of X.
A difficult and seemingly open question is: which geometric quotient arises in the above GIT fashion? The question is of a great interest since the GIT approach produces an explicit quotient, as opposed to an abstract quotient, which is hard to compute. One known partial answer to this question is the following: let be a locally factorial algebraic variety (for example, a smooth variety) with an action of . Suppose there are an open subset as well as a geometric quotient such that (1) is an affine morphism and (2) is quasi-projective. Then for some linearlized line bundle L on X. (An analogous question is to determine which subring is the ring of invariants in some manner.)
Examples
Finite grou |
https://en.wikipedia.org/wiki/Usha%20Ananthasubramanian | Usha Ananthasubramanian is the former managing director and chief executive officer of the Allahabad Bank,
Education and early career
Ananthasubramanian holds a master's degree in statistics from the University of Madras and a master's degree in Ancient Indian culture from University of Mumbai
Her background in statistics helped her get her first job as a specialist in the actuarial department with Life Insurance Corporation (LIC), while her understanding of Ancient Indian Culture gave her a deeper insight into understanding risk. She was charged by the CBI for the Nirav Modi fraud case during her tenure at PNB
Career in banking
Ananthasubramanian started her career in banking in February 1982, when she joined the Bank of Baroda as a specialist officer in its planning stream.
Prior to joining Bhartiya Mahila Bank, Ananthasubramanian worked at Punjab National Bank and was recognised in the following roles:
Executive director
Member of share transfer committee
Member of audit committee
Member of Committee of Directors to Review Disposal of Vigilance/ Non-Vigilance Disciplinary Action Cases
Member of customer service committee
Member of information technology committee
Member of management committee
Member of power of attorney committee
Member of risk management committee
Member of shareholders/investors grievance committee
Member of special committee of board
She served as MD of Punjab National Bank between August 2015 and May 2017. Prior to that, she served as CEO of Allahabad Bank.
She was ranked the 19th Most Powerful Woman in business by Fortune India First women CHAIRMAN of IBA
Bhartiya Mahila Bank
With a banking career spanning over 31 years, Ananthasubramanian was nominated in 2013 as leader of the core management team constituted by the Ministry of Finance (India) during UPA rule for coordinating the process of establishment of the Women's Bank (Bhartiya Mahila Bank)
Punjab National Bank Scam
Ananthasubramanian was named in CBI chargesheet in association with Punjab National Bank Scam, relating to fraudulent Letters of Undertaking issued to Nirav Modi. CBI alleges that key PNB bank officials, led by then CEO Ananthasubramanian, failed to initiate steps that could have prevented the $2-billion fraud at the lender after the banking watchdog RBI had red-flagged likely gaps in systems controls. It adds that the RBI had instructed that while acknowledging the Caution Advice, they should report the occurrence of such incidents at their banks. If no such incident was observed, a NIL statement should be furnished. RBI revealed that PNB, had not responded to the said caution advice until February 2018, when the fraud came to light. Ananthasubramanian was the MD of PNB when the caution advice was issued.
On 20 August 2018, Usha Ananthasubramanian was granted bail on a surety bond of Rs 1 lakhs by Special CBI court in Mumbai. A week earlier, the government had dismissed Usha on the last day of her work. She was dismissed with imme |
https://en.wikipedia.org/wiki/Equivariant%20topology | In mathematics, equivariant topology is the study of topological spaces that possess certain symmetries. In studying topological spaces, one often considers continuous maps , and while equivariant topology also considers such maps, there is the additional constraint that each map "respects symmetry" in both its domain and target space.
The notion of symmetry is usually captured by considering a group action of a group on and and requiring that is equivariant under this action, so that for all , a property usually denoted by . Heuristically speaking, standard topology views two spaces as equivalent "up to deformation," while equivariant topology considers spaces equivalent up to deformation so long as it pays attention to any symmetry possessed by both spaces. A famous theorem of equivariant topology is the Borsuk–Ulam theorem, which asserts that every -equivariant map necessarily vanishes.
Induced G-bundles
An important construction used in equivariant cohomology and other applications includes a naturally occurring group bundle (see principal bundle for details).
Let us first consider the case where acts freely on . Then, given a -equivariant map , we obtain sections given by , where gets the diagonal action , and the bundle is , with fiber and projection given by . Often, the total space is written .
More generally, the assignment actually does not map to generally. Since is equivariant, if (the isotropy subgroup), then by equivariance, we have that , so in fact will map to the collection of . In this case, one can replace the bundle by a homotopy quotient where acts freely and is bundle homotopic to the induced bundle on by .
Applications to discrete geometry
In the same way that one can deduce the ham sandwich theorem from the Borsuk-Ulam Theorem, one can find many applications of equivariant topology to problems of discrete geometry. This is accomplished by using the configuration-space test-map paradigm:
Given a geometric problem , we define the configuration space, , which parametrizes all associated solutions to the problem (such as points, lines, or arcs.) Additionally, we consider a test space and a map where is a solution to a problem if and only if . Finally, it is usual to consider natural symmetries in a discrete problem by some group that acts on and so that is equivariant under these actions. The problem is solved if we can show the nonexistence of an equivariant map .
Obstructions to the existence of such maps are often formulated algebraically from the topological data of and . An archetypal example of such an obstruction can be derived having a vector space and . In this case, a nonvanishing map would also induce a nonvanishing section from the discussion above, so , the top Stiefel–Whitney class would need to vanish.
Examples
The identity map will always be equivariant.
If we let act antipodally on the unit circle, then is equivariant, since it is an odd function.
Any map is equ |
https://en.wikipedia.org/wiki/Limiting%20point%20%28geometry%29 | In geometry, the limiting points of two disjoint circles A and B in the Euclidean plane are points p that may be defined by any of the following equivalent properties:
The pencil of circles defined by A and B contains a degenerate (radius zero) circle centered at p.
Every circle or line that is perpendicular to both A and B passes through p.
An inversion centered at p transforms A and B into concentric circles.
The midpoint of the two limiting points is the point where the radical axis of A and B crosses the line through their centers. This intersection point has equal power distance to all the circles in the pencil containing A and B. The limiting points themselves can be found at this distance on either side of the intersection point, on the line through the two circle centers. From this fact it is straightforward to construct the limiting points algebraically or by compass and straightedge.
An explicit formula expressing the limiting points as the solution to a quadratic equation in the coordinates of the circle centers and their radii is given by Weisstein.
Inverting one of the two limiting points through A or B produces the other limiting point. An inversion centered at one limiting point maps the other limiting point to the common center of the concentric circles.
References
Circles
Inversive geometry |
https://en.wikipedia.org/wiki/Redshift%20conjecture | In mathematics, more specifically in chromatic homotopy theory, the redshift conjecture states, roughly, that algebraic K-theory has chromatic level one higher than that of a complex-oriented ring spectrum R.
It was formulated by John Rognes in a lecture at Schloss Ringberg, Germany, in January 1999, and made more precise by him in a lecture at Mathematische Forschungsinstitut Oberwolfach, Germany, in September 2000. In July 2022, Burklund, Schlank and Yuan announced a solution of a version of the redshift conjecture for arbitrary -ring spectra, after Hahn and Wilson did so earlier in the case of the truncated Brown-Peterson spectra BP<n>.
References
Notes
Further reading
External links
Algebraic topology
Homotopy theory
Conjectures |
https://en.wikipedia.org/wiki/Yung%20Ta%20Institute%20of%20Technology%20and%20Commerce | Yung Ta Institute of Technology and Commerce (YTIT; ) is a private university located in Linluo Township, Pingtung County, Taiwan.
History
According to statistics compiled by the Ministry of Education in 2013, the Yung Ta Institute of Technology and Commerce had an enrollment of less than 1,000 students, and was considered a potential merger candidate alongside other private educational institutions. The education ministry announced in February 2014 that Yung Ta had been barred from enrolling any new students for one year. Its principal declared in August 2014 that the university would close down soon. However, Yung Ta remained open through 2019. The education ministry stated that if the institute did not close by 8 January 2020, the ministry would force the school to close.
Faculties
Department of Mechanical Engineering
Department of Vehicle Engineering
Department of Electronic Engineering
Department of Electrical Engineering
Department of Architectural Engineering
Department of Industrial and Business Management
Department of Business Administration
Department of Marketing Management
Department of Information Management
Department of Biotechnology
Department of Cosmetic Application and Management
Department of Sports, Health and Leisure
Department of Applied Foreign Languages
Transportation
The school was accessible East of Guilai Station of the Taiwan Railways.
See also
List of universities in Taiwan
References
1967 establishments in Taiwan
Educational institutions established in 1967
Universities and colleges in Pingtung County |
https://en.wikipedia.org/wiki/Central%20Bureau%20of%20Statistics%20%28Nepal%29 | The Central Bureau of Statistics () is the central agency for the collection, consolidation, processing, analysis, publication and dissemination of statistics in Nepal. One of its core tasks is to research and publish censuses of Nepal, the most prominent one being the overall population census and Demographics of Nepal.
History
The Central Bureau of Statistics was established in 1959 under the National Planning Commission of Nepal, which is headed by the Prime Minister of Nepal. Before 2015, different Nepalese governmental organisations gathered statistical information on their own. This led to inconsistencies in statistical information, for which the Bureau called for improvement of the processes, which, however, were not implemented as of 2017.
The Bureau's main functions include providing its government with statistics to help with public policy planning, collecting and analyzing socioeconomic data, and developing methodologies for reliable data collection and implementation.
After the April 2015 Nepal earthquake, the Bureau, in coordination with the United Nations, collected data regarding the damages in order for the Government of Nepal to support and compensate the victims.
References
Government agencies of Nepal
Nepal
1959 establishments in Nepal |
https://en.wikipedia.org/wiki/Snaith%27s%20theorem | In algebraic topology, a branch of mathematics, Snaith's theorem, introduced by Victor Snaith, identifies the complex K-theory spectrum with the localization of the suspension spectrum of away from the Bott element.
References
For a proof, see http://people.fas.harvard.edu/~amathew/snaith.pdf
Victor Snaith, Algebraic Cobordism and K-theory, Mem. Amer. Math. Soc. no 221 (1979)
External links
Theorems in algebraic topology
K-theory |
https://en.wikipedia.org/wiki/Kao%20Yuan%20University | {
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Kao Yuan University (KYU; ) is a private university in Kaohsiung Science Park, Lujhu District, Kaohsiung, Taiwan.
KYU offers a wide range of undergraduate and graduate programs, including programs in engineering, management, design, humanities, and social sciences. Some of the most popular programs at KYU include Industrial Design, Mechanical Engineering, Information Management, and International Business.
The university also offers several graduate programs, including Master's degrees in Business Administration, Industrial Design, and Mechanical Engineering.
History
The university was originally established in 1986 as Private Kao Yuan Junior College of Technology. In 1989, it was approved by the Ministry of Education and subsequently it started recruiting students. In 1991, it was renamed to Kao Yuan Junior College of Technology and Commerce and again to Kao Yuan Institute of Technology in 1998. It was finally renamed to Kao Yuan University in 2005.
Faculties
College of Business and Management
College of Engineering
College of Informatics
College of Mechatronics Engineering
Campuses
The university consists of two campuses. The first campus named Campus One covers an area of 11 hectares and the second campus named Campus Two covers 14.2 hectares.
Library
The university library consists of two joint buildings. The first building was constructed in 1996 and the second one in 2005. The library total space is 13,641 m2. The library has more than 292,000 volume of books and more than 470 title of periodicals.
Transportation
The university is accessible within walking distance South of Luzhu Station of the Taiwan Railways.
See also
List of universities in Taiwan
References
External links
1986 establishments in Taiwan
Universities and colleges established in 1986
Universities and colleges in Kaohsiung
Universities and colleges in Taiwan |
https://en.wikipedia.org/wiki/Gromov%20boundary | In mathematics, the Gromov boundary of a δ-hyperbolic space (especially a hyperbolic group) is an abstract concept generalizing the boundary sphere of hyperbolic space. Conceptually, the Gromov boundary is the set of all points at infinity. For instance, the Gromov boundary of the real line is two points, corresponding to positive and negative infinity.
Definition
There are several equivalent definitions of the Gromov boundary of a geodesic and proper δ-hyperbolic space. One of the most common uses equivalence classes of geodesic rays.
Pick some point of a hyperbolic metric space to be the origin. A geodesic ray is a path given by an isometry such that each segment is a path of shortest length from to .
Two geodesics are defined to be equivalent if there is a constant such that for all . The equivalence class of is denoted .
The Gromov boundary of a geodesic and proper hyperbolic metric space is the set is a geodesic ray in .
Topology
It is useful to use the Gromov product of three points. The Gromov product of three points in a metric space is
. In a tree (graph theory), this measures how long the paths from to and stay together before diverging. Since hyperbolic spaces are tree-like, the Gromov product measures how long geodesics from to and stay close before diverging.
Given a point in the Gromov boundary, we define the sets there are geodesic rays with and . These open sets form a basis for the topology of the Gromov boundary.
These open sets are just the set of geodesic rays which follow one fixed geodesic ray up to a distance before diverging.
This topology makes the Gromov boundary into a compact metrizable space.
The number of ends of a hyperbolic group is the number of components of the Gromov boundary.
Properties of the Gromov boundary
The Gromov boundary has several important properties. One of the most frequently used properties in group theory is the following: if a group acts geometrically on a δ-hyperbolic space, then is hyperbolic group and and have homeomorphic Gromov boundaries.
One of the most important properties is that it is a quasi-isometry invariant; that is, if two hyperbolic metric spaces are quasi-isometric, then the quasi-isometry between them gives a homeomorphism between their boundaries. This is important because homeomorphisms of compact spaces are much easier to understand than quasi-isometries of spaces.
Examples
The Gromov boundary of a tree is a Cantor space.
The Gromov boundary of hyperbolic n-space is an (n-1)-dimensional sphere.
The Gromov boundary of the fundamental group of a compact Riemann surface is the unit circle.
The Gromov boundary of most hyperbolic groups is a Menger sponge.
Generalizations
Visual boundary of CAT(0) space
For a complete CAT(0) space X, the visual boundary of X, like the Gromov boundary of δ-hyperbolic space, consists of equivalence class of asymptotic geodesic rays. However, the Gromov product cannot be used to define a topology on it. |
https://en.wikipedia.org/wiki/Opaque%20set | In discrete geometry, an opaque set is a system of curves or other set in the plane that blocks all lines of sight across a polygon, circle, or other shape. Opaque sets have also been called barriers, beam detectors, opaque covers, or (in cases where they have the form of a forest of line segments or other curves) opaque forests. Opaque sets were introduced by Stefan Mazurkiewicz in 1916, and the problem of minimizing their total length was posed by Frederick Bagemihl in 1959.
For instance, visibility through a unit square can be blocked by its four boundary edges, with length 4, but a shorter opaque forest blocks visibility across the square with length . It is unproven whether this is the shortest possible opaque set for the square, and for most other shapes this problem similarly remains unsolved. The shortest opaque set for any bounded convex set in the plane has length at most the perimeter of the set, and at least half the perimeter. For the square, a slightly stronger lower bound than half the perimeter is known. Another convex set whose opaque sets are commonly studied is the unit circle, for which the shortest connected opaque set has length . Without the assumption of connectivity, the shortest opaque set for the circle has length at least and at most .
Several published algorithms claiming to find the shortest opaque set for a convex polygon were later shown to be incorrect. Nevertheless, it is possible to find an opaque set with a guaranteed approximation ratio in linear time, or to compute the subset of the plane whose visibility is blocked by a given system of line segments in polynomial time.
Definitions
Every set in the plane blocks the visibility through a superset of , its coverage . consists of points for which all lines through the point intersect . If a given set forms a subset of the coverage of , then is said to be an opaque set, barrier, beam detector, or opaque cover for . If, additionally, has a special form, consisting of finitely many line segments whose union forms a forest, it is called an opaque forest. There are many possible opaque sets for any given set , including itself, and many possible opaque forests. For opaque forests, or more generally for systems of rectifiable curves, their length can be measured in the standard way. For more general point sets, one-dimensional Hausdorff measure can be used, and agrees with the standard length in the cases of line segments and rectifiable curves.
Most research on this problem assumes that the given set is a convex set. When it is not convex but merely a connected set, it can be replaced by its convex hull without changing its opaque sets. Some variants of the problem restrict the opaque set to lie entirely inside or entirely outside . In this case, it is called an interior barrier or an exterior barrier, respectively. When this is not specified, the barrier is assumed to have no constraints on its location. Versions of the problem in which the opaque set mu |
https://en.wikipedia.org/wiki/List%20of%20Coritiba%20Foot%20Ball%20Club%20records%20and%20statistics | wCoritiba Foot Ball Club is a football club based in Curitiba, Paraná. Coritiba's first trophy was the Campeonato Paranaense (Paraná State Cup), which it won against Britânia in 1916. In 1973, Coritiba won Torneio do Povo (Tournament of the People). In 1985, won the mainly tournament of Brazil, Campeonato Brasileiro.
Titles
Série A: 1
1985
Série B: 2
2007, 2010
Torneio do Povo: 1
1973
Campeonato Paranaense: 38
1916, 1927, 1931, 1933, 1935, 1939, 1941, 1942, 1946, 1947, 1951, 1952, 1954, 1956, 1957, 1959, 1960, 1968, 1969, 1971, 1972, 1973, 1974, 1975, 1976, 1978, 1979, 1986, 1989, 1999, 2003, 2004, 2008, 2010, 2011, 2012, 2013, 2017
Taça Cidade de Curitiba/Taça Clemente Comandulli: 2
1976, 1978
Festival Brasileiro de Futebol: 1
1997
Fita Azul Internacional: 1
1972
Pierre Colon Trophy (Vichy, France): 1
1969
Akwaba Trophy (Africa): 1
1983
Junior Team
Torneio Gradisca (Italia): 2
2013, 2014
Dallas Cup (United States): 2
2012, 2015
Taça Belo Horizonte de Juniores: 1
2010
Club records
First match: Coritibano-Tiro Pontagrossense 0–1 (October 23, 1909)
First official match: Coritiba-Ponta Grossa 5–3 (June 12, 1910)
First goal scorer: Fritz Essenfelter
Biggest win (National Competitions): Coritiba-Ferroviário 7–1 (Couto Pereira, April 16, 1980), Coritiba-Desportiva-ES 7–1 (Couto Pereira, May 4, 1980) & Coritiba-Palmeiras 6–0 (Couto Pereira, May 5, 2011)
Heaviest defeat (national competitions): Grêmio-Coritiba 5–0 (Olímpico, February 29, 1984) & Palmeiras-Coritiba 5–0 (Parque Antártica, August 17, 1996)
Most appearances (any competition): Jairo – 440 (1971–77), (1984–87)
Record goal scorer: Duílio Dias – 202 (1954–64)
Consecutive victories):Coritiba has the worldwide record of consecutive victories (24), achieved between February and May 2011.
References
Coritiba Foot Ball Club |
https://en.wikipedia.org/wiki/Lee%20Young-uk | Lee Young-uk (Hangul:이영욱, Hanja: 李永旭; born August 13, 1980, in Daegu) is a South Korean relief pitcher who plays for the Samsung Lions of the KBO League.
Statistics
External links
Career statistics and player information from Korea Baseball Organization
SSG Landers players
Samsung Lions players
KBO League pitchers
South Korean baseball players
1980 births
Living people
Baseball players from Daegu |
https://en.wikipedia.org/wiki/Schur%20product%20theorem | In mathematics, particularly in linear algebra, the Schur product theorem states that the Hadamard product of two positive definite matrices is also a positive definite matrix.
The result is named after Issai Schur (Schur 1911, p. 14, Theorem VII) (note that Schur signed as J. Schur in Journal für die reine und angewandte Mathematik.)
We remark that the converse of the theorem holds in the following sense. If is a symmetric matrix and the Hadamard product is positive definite for all positive definite matrices , then itself is positive definite.
Proof
Proof using the trace formula
For any matrices and , the Hadamard product considered as a bilinear form acts on vectors as
where is the matrix trace and is the diagonal matrix having as diagonal entries the elements of .
Suppose and are positive definite, and so Hermitian. We can consider their square-roots and , which are also Hermitian, and write
Then, for , this is written as for and thus is strictly positive for , which occurs if and only if . This shows that is a positive definite matrix.
Proof using Gaussian integration
Case of M = N
Let be an -dimensional centered Gaussian random variable with covariance . Then the covariance matrix of and is
Using Wick's theorem to develop we have
Since a covariance matrix is positive definite, this proves that the matrix with elements is a positive definite matrix.
General case
Let and be -dimensional centered Gaussian random variables with covariances , and independent from each other so that we have
for any
Then the covariance matrix of and is
Using Wick's theorem to develop
and also using the independence of and , we have
Since a covariance matrix is positive definite, this proves that the matrix with elements is a positive definite matrix.
Proof using eigendecomposition
Proof of positive semidefiniteness
Let and . Then
Each is positive semidefinite (but, except in the 1-dimensional case, not positive definite, since they are rank 1 matrices). Also, thus the sum is also positive semidefinite.
Proof of definiteness
To show that the result is positive definite requires even further proof. We shall show that for any vector , we have . Continuing as above, each , so it remains to show that there exist and for which corresponding term above is nonzero. For this we observe that
Since is positive definite, there is a for which (since otherwise for all ), and likewise since is positive definite there exists an for which However, this last sum is just . Thus its square is positive. This completes the proof.
References
External links
Bemerkungen zur Theorie der beschränkten Bilinearformen mit unendlich vielen Veränderlichen at EUDML
Linear algebra
Matrix theory |
https://en.wikipedia.org/wiki/List%20of%20urban%20areas%20in%20Finland%20by%20population | This is a list of urban areas in Finland by population, with the 100 largest localities or urban areas in Finland on 31 December 2019. The list is based on data from Statistics Finland that defines an urban area as a cluster of dwellings with at least 200 inhabitants.
See also
Urban areas in Finland
List of cities and towns in Finland
List of Finnish municipalities by population
List of urban areas in the Nordic countries
References
Finland
Cities and towns in Finland
Finland
Urb |
https://en.wikipedia.org/wiki/Eric%20Friedlander | Eric Mark Friedlander (born January 7, 1944 in Santurce, Puerto Rico) is an American mathematician who is working in algebraic topology, algebraic geometry, algebraic K-theory and representation theory.
Friedlander graduated from Swarthmore College with bachelor's degree in 1965 and in 1970 received a Ph.D. from the Massachusetts Institute of Technology, under the supervision of Michael Artin, (Fibrations in Étale Homotopy Theory). He was a postdoctoral instructor at Princeton University: a lecturer in 1971 and assistant professor in 1972. From 1973 to 1974, he was, through the US exchange program, at France, in particular at the Institut des Hautes Études Scientifiques. In 1975, he became an associate professor and in 1980 a professor at Northwestern University, where he was a chairman of the mathematics department from 1987 to 1990 and from 1999 to 2003. In 1999, he became Henry S. Noyes Professor of mathematics. As of 2008, he is Dean's Professor at the University of Southern California.
In 1981 and from 1985 to 1986, he was at the Institute for Advanced Study in Princeton, New Jersey. He received the Humboldt Research Award, while at the University of Heidelberg, from 1996 to 1998. He was also a visiting scholar and visiting professor at ETH Zurich, at the Max Planck Institute for Mathematics in Bonn, at the Mathematical Sciences Research Institute, in Oxford, Cambridge, Paris, at Brown University, the Hebrew University, and at the Institut Henri Poincaré. Since 2000, he has been on the Board of Trustees of the American Mathematical Society.
Friedlander is a co-editor of the Journal of Pure and Applied Algebra. In 1998, he was an invited speaker at the International Congress of Mathematicians in Berlin (Geometry of infinitesimal group schemes). In 2012 he became a fellow of the American Mathematical Society.
Friedlander is married to another mathematician, Susan Friedlander.
Among his students is David A. Cox.
Works
With Andrei Suslin and Vladimir Voevodsky, Cycles, Transfers and Motivic Homology Theories, Annals of mathematical studies, Princeton University Press 2000
With Barry Mazur, Filtration on the Homology of Algebraic Varieties, memoir of the AMS, 1994
Etale Homotopy of Simplicial Schemes, Annals of mathematical studies, Princeton University Press 1982
Editor with Daniel Grayson, Handbook of K-Theory, 2 volumes, Springer Verlag 2005
Editor with Spencer j. Bloch, R. k. Dennis, M.: Applications of algebraic K-theory to algebraic geometry and number theory, contemporary mathematics 55, 1986
Editor with M. Stone, Algebraic K-theory (Evanston 1980), Springer Verlag, lecture notes in mathematics 854, 1981
Editor with Mark Mahowald, Topology and representation theory, contemporary mathematics, volume 158, American Mathematical Society, 1994
With Charles Weibel, An overview over algebraic K-theory, in Algebraic K-theory and its applications, World Scientific 1999, pp. 1–119 (1997 Trieste of lecture notes)
References
External links
H |
https://en.wikipedia.org/wiki/Shoichiro%20Sakai | is a Japanese mathematician.
Life
Sakai studied mathematics at the Tohoku University (Sendai). He there received the B. A. degree in 1953 and a doctorate at the same University in 1961. From 1960 to 1964, he was a faculty member of Waseda University. He then went to the University of Pennsylvania, where he became a professor in 1966 and remained until 1979. He then returned to Japan and went to the Nihon University. In 1992, he received the Japanese Mathematical Society Autumn Prize. He is a fellow of the American Mathematical Society.
Sakai's main field is functional analysis and mathematical physics. His textbook published in the Springer series in C *-algebras and W *-algebras, in which W *-algebras as C *-algebras are introduced with a predual, is widely used. That fact the W *-algebras may be defined in this way is known as a theorem of Sakai (cf. a theorem of Kadison-Sakai.)
Works
C *-algebras and W *-algebras, Springer-Verlag 1971, Ergebnisse der Mathematik und ihrer Grenzgebiete, Volume 60, (republished in 1998 in Classics in Mathematics)
Operator algebras in dynamical systems, the theory of unbounded derivations in C*-algebras, Cambridge University Press (1991),
References
1928 births
Living people
20th-century Japanese mathematicians
21st-century Japanese mathematicians
Fellows of the American Mathematical Society
Functional analysts
Academic staff of Waseda University
University of Pennsylvania faculty |
https://en.wikipedia.org/wiki/Colin%20J.%20Bushnell | Colin John Bushnell (1947 – 1 January 2021) was a British mathematician specialising in number theory and representation theory. He spent most of his career at King's College London, including a stint as the head of the School of Physical Sciences and Engineering, and made several contributions to the representation theory of reductive p-adic groups and the local Langlands correspondence.
Early life and education
Bushnell was born in 1947. He studied mathematics at King's College London, where he received his first class honors undergraduate degree and then a Ph.D. in 1972 under the supervision of Albrecht Fröhlich.
Career
From 1972 to 1975, Bushnell was a lecturer at the University of Illinois at Urbana–Champaign. He returned to King's College London in 1975 as Lecturer, before being promoted to Reader in 1985 and Professor in 1990. From 1988 to 1989, he was a member of the Institute for Advanced Study. From 1996 to 1997, he was a chairman of the mathematics department and from 1997 to 2004 he was the head of the School of Physical Sciences and Engineering. He retired in 2014. He died on 1 January 2021 at the age of 73.
Bushnell has advised doctoral students including Graham Everest.
Research
Bushnell's research included "major contributions to the representation theory of reductive p-adic groups and the study of the local Langlands correspondence."
Awards
In 1994, Bushnell was an invited speaker at the International Congress of Mathematicians in Zurich (Smooth representations of p-adic groups: the role of compact open subgroups).
In 1995, Bushnell was awarded the Senior Whitehead Prize. In 2002, he became a Fellow of King's College London. He was inaugurated in the 2013 class of Fellows of the American Mathematical Society.
Selected publications
With Albrecht Fröhlich, Gauss sums and p-adic division algebras, lecture notes in mathematics, vol. 987, Springer Verlag 1983
With Guy Henniart, The local Langlands conjecture for GL(2), Springer-Verlag, 2006, (Grundlehren der mathematischen Wissenschaften 335)
With Philip Kutzko, The admissible dual of GL(N) via compact open subgroups, Annals of Mathematical Studies 129, Princeton University Press 1993
References
External links
Homepage
1947 births
Alumni of King's College London
20th-century British mathematicians
21st-century British mathematicians
Institute for Advanced Study visiting scholars
Fellows of the American Mathematical Society
Fellows of King's College London
2021 deaths
University of Illinois Urbana-Champaign faculty |
https://en.wikipedia.org/wiki/J%C3%BCrgen%20Jost | Jürgen Jost (born 9 June 1956) is a German mathematician specializing in geometry. He has been a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig since 1996.
Life and work
In 1975, he began studying mathematics, physics, economics and philosophy. In 1980 he received a Dr. rer. nat. from the University of Bonn under the supervision of Stefan Hildebrandt. In 1984 he was at the University of Bonn for the habilitation. After his habilitation, he was at the Ruhr University Bochum, the chair of Mathematics X, Analysis. During this time he was the coordinator of the project "Stochastic Analysis and systems with infinitely many degrees of freedom" July 1987 to December 1996.
For this work he received the 1993 Gottfried Wilhelm Leibniz Prize, awarded by the Deutsche Forschungsgemeinschaft.
Since 1996, he has been director and scientific member at the Max Planck Institute for Mathematics in the Sciences in Leipzig. After more than 10 years of work in Bochum, this he followed: "tackle new research problems in the border area between mathematics and the natural sciences and simultaneously encourage mathematical research in Germany, particularly in the fields of geometry and analysis."
In 1998 he was an honorary professor at the University of Leipzig. In 2002, there, he initiated with two other scientists from the Max Planck Institute, the Interdisciplinary Center for Bioinformatics (IZBI).
In 1986 he was invited speaker at the International Congress of Mathematicians in Berkeley (Two dimensional geometric variational problems). He is a fellow of the American Mathematical Society.
His research focuses are:
Complex dynamical systems
Neural networks
Cognitive structures and theoretical neurobiology, cognitive science, theoretical and mathematical biology
Riemannian geometry; analysis and geometry
Calculus of variations and partial differential equations; mathematical physics
Bibliography
Bosonic Strings: A mathematical treatment, AMS international press, 2001
References
External links
Webpage at Max-Planck-Institut
Interview with Jürgen Jost and Dr. Hans-Joachim Freund
20th-century German mathematicians
Fellows of the American Mathematical Society
Mathematical analysts
Variational analysts
Living people
1956 births
University of Bonn alumni
Academic staff of Ruhr University Bochum
21st-century German mathematicians
Max Planck Institutes researchers
Max Planck Institute directors |
https://en.wikipedia.org/wiki/Roy%20Adler | Roy Lee Adler (February 22, 1931 – July 26, 2016) was an American mathematician.
Adler earned his Ph.D. in 1961 from Yale University under the supervision of Shizuo Kakutani (On some algebraic aspects of measure preserving transformations). He then worked as a mathematician for IBM at the Thomas J. Watson Research Center.
Adler studies dynamical systems, ergodic theory, symbolic and topological dynamics and coding theory. The road coloring problem that was solved by Avraham Trakhtman in 2007 came from him, along with L. W. Goodwyn and Benjamin Weiss.
He was a fellow of the American Mathematical Society. A paper was written on his work and the impact of his work by Bruce Kitchens and others.
Writings
With Brian Marcus: Topological entropy and equivalence of dynamical systems. Memoirs of the American Mathematical Society. 20 (1979), no 219.
With Benjamin Weiss: Similarity of automorphisms of the torus, Memoirs of the American Mathematical Society (1970), no 98.
Symbolic dynamics and Markov partitions, Bulletin of the American Mathematical Society. 35 (1998), no 1, 1–57.
With L. Wayne Goodwyn and Benjamin Weiss: Equivalence of topological Markov shifts, Israel Journal of Mathematics. 27 (1977), 49–63.
With Alan Konheim and M. H. McAndrew: Topological Entropy, Transactions of the American Mathematical Society. 114 (1965), 309–319.
With Tomasz Downarowicz and Michał Misiurewicz: Topological Entropy. Scholarpedia. 3 (2008), no 2, 2200.
With Charles Tresser and Patrick A. Worfolk: Topological conjugacy of linear endomorphisms of the 2-torus, Transactions of the American Mathematical Society. 349 (1997), 1633–1652.
With Benjamin Weiss: Entropy, a complete metric invariant for automorphisms of the torus, Proceeding of the National Academy of Sciences. 57 (1967), 1573–1576.
References
1931 births
2016 deaths
20th-century American mathematicians
21st-century American mathematicians
Jewish American scientists
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
People from Newark, New Jersey
Yale Graduate School of Arts and Sciences alumni
21st-century American Jews |
https://en.wikipedia.org/wiki/Craig%20Huneke | Craig Lee Huneke (born August 27, 1951) is an American mathematician specializing in commutative algebra. He is a professor at the University of Virginia.
Huneke graduated from Oberlin College with a bachelor's degree in 1973 and in 1978 earned a Ph.D. from the Yale University under Nathan Jacobson and David Eisenbud (Determinantal ideal and questions related to factoriality). As a post-doctoral fellow, he was at the University of Michigan. In 1979 he became an assistant professor and was at the Massachusetts Institute of Technology and the University of Bonn (1980). In 1981 he became an assistant professor at Purdue University, where in 1984 he became an associate professor and became a professor in 1987. From 1994 to 1995 he was a visiting professor at the University of Michigan and in 1999 was at the Max Planck Institute for Mathematics in Bonn (as a Fulbright Scholar). In 1999, he was Henry J. Bischoff professor at the University of Kansas. In 2002 he was at MSRI. Since 2012 he has been Marvin Rosenblum professor at the University of Virginia.
With Melvin Hochster and others, he developed the theory of tight closure, a device in ring theory that is used to study rings containing a field of characteristic p in which Frobenius endomorphism figures prominently. He also studies linkage theory, Rees algebras, homological theory of modules over Noetherian rings, local cohomology, symbolic powers of ideals, Cohen-Macaulay rings, Gorenstein rings and Hilbert-Kunz functions.
He was an invited speaker at the International Congress of Mathematicians in 1990 in Kyoto (Absolute Integral Closure and Big Cohen-Macaulay Algebras). He is a fellow of the American Mathematical Society.
Huneke's son is historian Samuel Clowes Huneke.
Writings
With Hochster Tightly closed ideals, Bulletin of the American Mathematical Society, volume 18, 1988, pg. 45–48
With Hochster Tight closure, invariant theory, and the Briançon–Skoda theorem, Journal of the American Mathematical Society, volume 3, 1990, pg. 31–116
With Hochster: Phantom Homology, Memoirs American Mathematical Society 1993
Tight closure and its application, American Mathematical Society 1996
With Irena Swanson: Integral closure of ideals, rings, and modules, Cambridge University Press, 2006
With B. Ulrich The structure of linkage, Annals of Mathematics, volume 126, 1987, pg. 277-334
With Hochster Infinite integral extensions and big Cohen-Macaulay algebras, Annals of Mathematics, volume 135, 1992, pg. 53-89
With David Eisenbud, W. Vasconcelos Direct methods for primary decomposition, Inventiones Mathematicae, volume 110, 1992, pg. 207-236
Uniform bounds in noetherian rings, Inventiones Mathematicae, volume 107, 1992, pg. 203-223
With Hochster Comparison of symbolic and ordinary powers of ideals, Inventiones Mathematicae, volume 147, 2002, pg. 349-369
With D. Eisenbud, B. Ulrich The regularity of Tor and graded Betti numbers, American Journal of Mathematics, volume 128, 2006, pg. 573-605
References
Ex |
https://en.wikipedia.org/wiki/Joseph%20Lipman | Joseph Lipman (born June 15, 1938) is a Canadian-American mathematician, working in algebraic geometry.
Lipman graduated from the University of Toronto with a Bachelor's degree in 1960 and then went to Harvard University, receiving his master's degree in 1961. He then earned a Ph.D. there in 1965 under the supervision of Oscar Zariski. In 1965 he was an assistant professor at Queen's University in Kingston and in 1966 was an assistant professor at the Purdue University, where he became professor in 1971. From 1987 to 1992, there, he was head of the mathematics department. He was a member of the MSRI and visiting scholar at the University of Cambridge and the University of Nice and a visiting professor at the Columbia University and Harvard University.
He specializes in singularity theory in algebraic geometry.
In 1958, while studying at the University of Toronto, he became a Putnam Fellow both in the spring and fall William Lowell Putnam Mathematical Competitions. In 1982 he received the Jeffery–Williams Prize. He is a fellow of the American Mathematical Society.
Writings
Collected Papers of Joseph Lipman. Queen's Papers in Pure and Applied Mathematics, vol. 117, Queen's University Press, Kingston, Ontario, 2000.
Editor with Herwig Hauser, Frans Oort, Adolfo Quirós: Resolution of singularities. A research textbook in tribute to Oscar Zariski. Birkhäuser, Basel 2000, . (Progress in Mathematics. volume 181.)
References
External links
Homepage
1938 births
Living people
Canadian mathematicians
Columbia University faculty
Fellows of the American Mathematical Society
Harvard University alumni
Scientists from Toronto
University of Toronto alumni
Putnam Fellows |
https://en.wikipedia.org/wiki/Tommy%20Robredo%20career%20statistics | This is a list of the main career statistics of Spanish professional tennis player Tommy Robredo.
Performance timelines
Singles
Doubles
Major finals
Masters Series
Singles: 1 (1–0)
Doubles: 1 (1–1)
ATP career finals
Singles: 23 (12 titles, 11 runner-ups)
Doubles: 11 (5–6)
Challenger and Futures finals
Singles: 14 (8–6)
Doubles: 5 (3–2)
Wins over top 10 players
He has a record against players who were, at the time the match was played, ranked in the top 10.
ATP Tour career earnings
* Statistics correct .
Notable exhibitions
Team competitions
See also
Spain Davis Cup team
List of Spain Davis Cup team representatives
Tennis in Spain
Sport in Spain
References
External links
Tommy Robredo at the ITF profile
Tennis career statistics |
https://en.wikipedia.org/wiki/Stereoelectronic%20effect | In chemistry, primarily organic and computational chemistry, a stereoelectronic effect is an effect on molecular geometry, reactivity, or physical properties due to spatial relationships in the molecules' electronic structure, in particular the interaction between atomic and/or molecular orbitals. Phrased differently, stereoelectronic effects can also be defined as the geometric constraints placed on the ground and/or transition states of molecules that arise from considerations of orbital overlap. Thus, a stereoelectronic effect explains a particular molecular property or reactivity by invoking stabilizing or destabilizing interactions that depend on the relative orientations of electrons (bonding or non-bonding) in space.
Stereoelectronic effects present themselves in other well-known interactions. These include important phenomena such as the anomeric effect and hyperconjugation. It is important to note that stereoelectronic effects should not be misunderstood as a simple combination of steric effects and electronic effects.
Founded on a few general principles that govern how orbitals interact, the stereoelectronic effect, along with the steric effect, inductive effect, solvent effect, mesomeric effect, and aromaticity, is an important type of explanation for observed patterns of selectivity, reactivity, and stability in organic chemistry. In spite of the relatively straightforward premises, stereoelectronic effects often provide explanations for counterintuitive or surprising observations. As a result, stereoelectronic factors are now commonly considered and exploited in the development of new organic methodology and in the synthesis of complex targets. The scrutiny of stereoelectronic effects has also entered the realms of biochemistry and pharmaceutical chemistry in recent years.
A stereoelectronic effect generally involves a stabilizing donor-acceptor (i.e., filled bonding-empty antibonding, 2-electron 2-orbital) interaction. The donor is usually a higher bonding or nonbonding orbital and the acceptor is often a low-lying antibonding orbital as shown in the scheme below. Whenever possible, if this stereoelectronic effect is to be favored, the donor-acceptor orbitals should have (1) a small energy gap and (2) be geometrically well disposed for interaction. In particular, this means that the shapes of the donor and acceptor orbitals (including π or σ symmetry and size of the interacting lobes) must be well-matched for interaction; an antiperiplanar orientation is especially favorable. Some authors require stereoelectronic effects to be stabilizing. However, destabilizing donor-donor (i.e., filled bonding-filled antibonding, 4-electron 2-orbital) interactions are occasionally invoked and are also sometimes referred to as stereoelectronic effects, although such effects are difficult to distinguish from generic steric repulsion.
Trend of different orbitals
Take the simplest CH2X–CH3 system as an example; the donor orbital is σ(C–H) |
https://en.wikipedia.org/wiki/Emilie%20Martin | Emilie Norton Martin (30 December 1869 – 8 February 1936) was an American mathematician and professor of mathematics at Mount Holyoke College.
Life
Martin earned her bachelor's degree at Bryn Mawr College in 1894 majoring in mathematics and Latin. She continued her graduate studies at Bryn Mawr under the supervision of Charlotte Scott. In 1897-1898 she used a Mary E. Garrett Fellowship from Bryn Mawr to study at the University of Göttingen. In Göttingen, Martin and Virginia Ragsdale attended lectures by Felix Klein and David Hilbert. Although her name and dissertation title were printed in the 1899 commencement program, her Ph.D. wasn't granted until 1901 when her dissertation was published.
In 1903 she became an instructor at Mount Holyoke College. She was later promoted to associate professor and then professor. Although she had earned a doctorate, it did not hasten her promotion to professor. She spent eight years as an instructor and fifteen as an associate professor. This was common for women of her time, who were often unable to pass the associate professorship. Martin's research focused on primitive substitution groups of degree 15 and primitive substitution groups of degree 18. In 1904 she published the index to the first ten volumes of the Bulletin of the American Mathematical Society.
Martin was a member of the American Association for the Advancement of Science, the American Mathematical Society, and the Mathematical Association of America.
References
External links
Emilie Martin on the Mathematics Genealogy Project
Relating To Required Mathematics For Women Students by Emilie Martin for the American Mathematical Monthly
1869 births
1936 deaths
American women mathematicians
19th-century American mathematicians
20th-century American mathematicians
20th-century women mathematicians
20th-century American women
Mount Holyoke College faculty
19th-century American women educators
19th-century American educators
Bryn Mawr College alumni
Mathematicians from New Jersey
People from Elizabeth, New Jersey |
https://en.wikipedia.org/wiki/World%20Economy%20%28disambiguation%29 | The world economy, or global economy, is the economy of the world.
World Economy may also refer to:
The World Economy (journal), an academic journal
The World Economy: Historical Statistics, a book by Angus Maddison |
https://en.wikipedia.org/wiki/Deepak%20Loomba | Deepak Loomba is an Indian businessman.
Education
Deepak followed his schooling in Birla Higher Secondary School Pilani (Rajasthan) with higher studies in Physics and Mathematics in Moscow State University. Deepak, a technopreneur has a number of Intellectual Property Rights (IPR) to his credit. Contributions are in the field of lighting, semiconductors and material processing.
Early career
A serial technopreneur, Deepak started his business career in Moscow, Russia in the early 1990s. The businesses that he/his Companies have been involved included international trading in agro products, chemicals and fertilizers as also metallurgical manufacturing. His first Co. Wazir Inc. in which he sold out his interest, was producing shredded copper from copper scrap and CC Rods for cable industry. Technology developed by Deepak in conjunction with co-authors was world's first to use vortex grinding for grinding shredded copper into copper powder, without taking the copper cathode smelting and water sprinkling route leading to a major breakthrough for production of non-electronic grade copper powder (mainly for paint industry). The company is also a major exporter of wires and cables from Russia.
Through 1996–1997, besides establishing businesses in CIS nations, Deepak also established business for agro imports in Hamburg (Germany) and trading of Chemicals in Basel (Switzerland).
In a complete departure from technology-intensive sector, he set up Resotel Ltd., which was seeded from scratch to become the first integrated supplier of edibles to the hospitality industry in Moscow, before being sold to a local Russian Co.
Return to India
On returning to India, he promoted Transtechnology Consultants, a partnership firm along with Dr. S. K. Agarwal, a physicist, who worked for Solid State Physics Laboratories in Delhi, for providing consultancy services in the field of vacuum technologies, while he worked towards the financial closure of De Core Science & Technologies Ltd., a Company founded by Deepak to establish State-of-the-Art compound semiconductor plant in India. Through its short operation, Transtechnology Consultants executed technology consultancy contracts including one to upgrade and improve an existing Compact Fluorescent assembly line and developed a special Hg-Zn amalgam to replace liquid mercury dosing in gas-discharge lamps leading to substantial reduction in use of mercury.
He co-developed technology for manufacturing of photovoltaic vacuum glass panel for windows. He owns intellectual property in the area of material processing and die architecture of light emitting diodes.
Current occupations
Presently, Loomba is the Chairman and Managing Director of De Core Group which includes De Core Nanosemiconductors Limited Gandhinagar as well as De Core Science & Technologies Ltd., Noida. Both the Companies are public limited companies.
De Core Nanosemiconductors Limited
is South Asia's first compound Nanosemiconductor Fab., producing Li |
https://en.wikipedia.org/wiki/South%20Sudanese%20Australians | South Sudanese Australians are people of South Sudanese ancestry or birth who live in Australia.
Demographics
Following South Sudan's independence in July 2011, the Australian Bureau of Statistics (ABS) included the country amongst the country of birth and ancestry options in the 2011 Census that took place in August. This census recorded 3,487 people born in South Sudan in Australia. However, the ABS note that "South Sudan-born were previously included in the Census count of the Sudan-born, and this is highly likely with a large number in the 2011 Census". Of the 3,487, the largest number were living in the state of Victoria (1,118), followed by Queensland (715), then New South Wales (561) and Western Australia (489). A total of 4,825 people indicated that they were of partial or full South Sudanese ancestry.
The 2016 census recorded 7,699 South Sudan-born people in Australia, with 2,750 living in Victoria, 1,430 in Queensland and 1,201 in Western Australia. 10,755 people indicated that they had partial or full South Sudanese ancestry.
The 2021 census recorded 8,255 people born in South Sudan. 14,273 people indicated that they had South Sudanese ancestry.
Notable South Sudanese Australians
Aweng Ade-Chuol, fashion model
Deng Adel, basketballer
Deng Adut, defence lawyer and New South Wales Australian of the Year for 2017
Adut Akech, international fashion model (April 2018 Vogue Italia and May 2018 British Vogue cover model)
Leek Aleer, Australian rules footballer
Mac Andrew, Australian rules footballer
DyspOra, Adelaide hip hop artist, poet, activist
Aliir Aliir, Australian rules footballer
Kenny Athiu, football player
Bangs, hip hop artist
Nagmeldin 'Peter' Bol, middle distance runner and Olympian
Elijah Buol, lawyer, criminologist and community advocate, 2019 winner of Queensland Local Hero of the Year 2019 Award and Order of Australia medal.
Mabior Chol, Australian rules footballer
Akec Makur Chuot, Australian rules footballer
Majak Daw, Australian rules footballer
Ajak Deng, Australian fashion model (April 2016 Vogue Italia cover model)
Joseph Deng, middle distance runner
Majok Deng, basketballer
Peter Deng, football player
Thomas Deng, football player
Martin Frederick, Australian rules footballer
Michael Frederick, Australian rules footballer
Changkuoth Jiath, Australian rules footballer
Dor Jok, football player
Tom Jok, Australian rules footballer
Gordon Koang, blind popular musician
Subah Koj, Australian fashion model, one of the first two South Sudanese-Australians to walk in the Victoria's Secret Fashion Show
Alou Kuol, football player
Garang Kuol, football player
Jo Lual-Acuil Jr., basketballer
Awer Mabil, football player
Abraham Majok, football player
Ater Majok, basketballer
Thon Maker, basketballer
Mangok Mathiang, basketballer
Majak Mawith, football player
Kot Monoah, Melbourne lawyer, from Oct 2015 chairman of the South Sudanese Community Association of Victoria, previously community liaison officer.
Jackson Morgan, football |
https://en.wikipedia.org/wiki/Toufik%20Mansour | Toufik Mansour is an Israeli mathematician working in algebraic combinatorics. He is a member of the Druze community and is the first Israeli Druze to become a professional mathematician.
Mansour obtained his Ph.D. in mathematics from the University of Haifa in 2001 under Alek Vainshtein. As of 2007, he is a professor of mathematics at the University of Haifa.
He served as chair of the department from 2015 to 2017. He has previously been a faculty member of the Center for Combinatorics at Nankai University from 2004 to 2007, and at The John Knopfmacher Center for Applicable Analysis and Number Theory at the University of the Witwatersrand.
Mansour is an expert on Discrete Mathematics and its applications. In particular, he is interested in permutation patterns, colored permutations, set partitions, combinatorics on words, and compositions. He has written more than 260 research papers, which means that he publishes a paper roughly every 20 days, or that he produces one publication page roughly every day.
Books
.
.
.
See also
List of Israeli Druze
Schröder–Hipparchus number
References
External links
Home page and list of publications
Google Scholar profile
ORCID profile
Israeli mathematicians
1968 births
Living people |
https://en.wikipedia.org/wiki/Pillai%27s%20arithmetical%20function | In number theory, the gcd-sum function,
also called Pillai's arithmetical function, is defined for every by
or equivalently
where is a divisor of and is Euler's totient function.
it also can be written as
where, is the divisor function, and is the Möbius function.
This multiplicative arithmetical function was introduced by the Indian mathematician Subbayya Sivasankaranarayana Pillai in 1933.
References
Arithmetic functions |
https://en.wikipedia.org/wiki/Peter%20Roquette | Peter Jaques Roquette (8 October 1927 – 24 February 2023) was a German mathematician working in algebraic geometry, algebra, and number theory.
Biography
Roquette was born in Königsberg on 8 October 1927. He studied in Erlangen, Berlin, and Hamburg. In 1951 he defended a dissertation at the University of Hamburg under Helmut Hasse, providing a new proof of the Riemann hypothesis for algebraic function fields over a finite field (the first proof was given by André Weil in 1940). In 1951/1952 he was an assistant at the Mathematical Research Institute at Oberwolfach and from 1952 to 1954 at the University of Munich. From 1954 to 1956 he worked at the Institute for Advanced Study in Princeton. In 1954 he was Privatdozent at Munich, and from 1956 to 1959 he worked in the same position at Hamburg. In 1959 he became an associate professor at the University of Saarbrucken and in the same year at the University of Tübingen. From 1967 he was professor at the Ruprecht-Karls-University of Heidelberg, where he retired in 1996.
Roquette worked on number and function fields and especially local p-adic fields. He applied the methods of model theory (nonstandard arithmetic) in number theory, joint with Abraham Robinson, with whom he worked on Mahler's theorem (on the finiteness of integral points on a curve of genus g > 0) using non-standard methods. He authored a number of works on the history of mathematics, in particular on the schools of Helmut Hasse and Emmy Noether. In 1975 Roquette was co-editor of the collected essays by Helmut Hasse.
From 1978, Roquette was a member of the Heidelberg Academy of Sciences and from 1985, the German Academy of Sciences Leopoldina. He has an honorary doctorate from the University of Duisburg-Essen and was an honorary member of the Mathematical Society of Hamburg. In 1958 he was an invited speaker at the International Congress of Mathematicians in Edinburgh (on the topic of "Some fundamental theorems on abelian function fields").
His doctoral students include Gerhard Frey and .
Roquette died in Heidelberg on 24 February 2023, at the age of 95.
Selected publications
Analytic theory of elliptic functions over local fields. Vandenhoeck and Ruprecht 1970.
With Franz Lemmermeyer (Editor): The Correspondence of Helmut Hasse and Emmy Noether 1925-1935 Göttingen State and University Library, 2006..
with Günther Frei (Editor): Emil Artin and Helmut Hasse - correspondence 1923-1934, University of Göttingen Publisher 2008
The Brauer-Hasse-Noether Theorem in Historical Perspective. Mathem. the-Naturwiss writings. Class of the Heidelberg Academy of Sciences, Springer-Verlag, 2005.
Anthony V. Geramita, Paulo Ribenboim (ed.): Collected Papers of Peter Roquette 3 volumes. Queens Papers in Pure and Applied Mathematics Bd.118, Kingston, Ontario, Queen's University, 2002.
With Alexander Prestel: Formally p-adic Fields. Lecture Notes in Mathematics, Springer-Verlag 1984.
Robinson, A.; Roquette, P. On the finiteness theorem of Si |
https://en.wikipedia.org/wiki/Runcicantellated%2024-cell%20honeycomb | In four-dimensional Euclidean geometry, the runcicantellated 24-cell honeycomb is a uniform space-filling honeycomb.
Alternate names
Runcicantellated icositetrachoric tetracomb/honeycomb
Prismatorhombated icositetrachoric tetracomb (pricot)
Great diprismatodisicositetrachoric tetracomb
Related honeycombs
See also
Regular and uniform honeycombs in 4-space:
Tesseractic honeycomb
16-cell honeycomb
24-cell honeycomb
Rectified 24-cell honeycomb
Snub 24-cell honeycomb
5-cell honeycomb
Truncated 5-cell honeycomb
Omnitruncated 5-cell honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 118
o3x3x4o3x - apricot - O118
5-polytopes
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Stericantitruncated%2016-cell%20honeycomb | In four-dimensional Euclidean geometry, the stericantitruncated 16-cell honeycomb is a uniform space-filling honeycomb.
Alternate names
Great cellirhombated icositetrachoric tetracomb (gicaricot)
Runcicantic hexadecachoric tetracomb
Related honeycombs
See also
Regular and uniform honeycombs in 4-space:
Tesseractic honeycomb
16-cell honeycomb
24-cell honeycomb
Rectified 24-cell honeycomb
Snub 24-cell honeycomb
5-cell honeycomb
Truncated 5-cell honeycomb
Omnitruncated 5-cell honeycomb
References
Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs
Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
(Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
George Olshevsky, Uniform Panoploid Tetracombs, Manuscript (2006) (Complete list of 11 convex uniform tilings, 28 convex uniform honeycombs, and 143 convex uniform tetracombs) Model 121 (Wrongly named runcinated icositetrachoric honeycomb)
x3x3x4o3x - gicaricot - O130
5-polytopes
Honeycombs (geometry) |
https://en.wikipedia.org/wiki/Ralf%20J.%20Spatzier | Ralf Jürgen Spatzier is a mathematician specialising in differential geometry, dynamical systems, and ergodic theory.
Spatzier received his Ph.D. in Mathematics from the University of Warwick in 1983 under the joint supervision of Caroline Series and Anatole Katok and joined Stony Brook University as an assistant professor. In 1990 he moved to the University of Michigan where he is now a full professor. He is a fellow of the American Mathematical Society.
References
Fellows of the American Mathematical Society
Living people
University of Michigan faculty
20th-century American mathematicians
21st-century American mathematicians
Differential geometers
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Uzawa%20iteration | In numerical mathematics, the Uzawa iteration is an algorithm for solving saddle point problems. It is named after Hirofumi Uzawa and was originally introduced in the context of concave programming.
Basic idea
We consider a saddle point problem of the form
where is a symmetric positive-definite matrix.
Multiplying the first row by and subtracting from the second row yields the upper-triangular system
where denotes the Schur complement.
Since is symmetric positive-definite, we can apply standard iterative methods like the gradient descent
method or the conjugate gradient method to
in order to compute .
The vector can be reconstructed by solving
It is possible to update alongside during the iteration for the Schur complement system and thus obtain an efficient algorithm.
Implementation
We start the conjugate gradient iteration by computing the residual
of the Schur complement system, where
denotes the upper half of the solution vector matching the initial guess for its lower half. We complete the initialization by choosing the first search direction
In each step, we compute
and keep the intermediate result
for later.
The scaling factor is given by
and leads to the updates
Using the intermediate result saved earlier, we can also update the upper part of the solution vector
Now we only have to construct the new search direction by the Gram–Schmidt process, i.e.,
The iteration terminates if the residual has become sufficiently small or if the norm of is significantly smaller than indicating that the Krylov subspace has been almost exhausted.
Modifications and extensions
If solving the linear system exactly is not feasible, inexact solvers can be applied.
If the Schur complement system is ill-conditioned, preconditioners can be employed to improve the speed of convergence of the underlying gradient method.
Inequality constraints can be incorporated, e.g., in order to handle obstacle problems.
References
Further reading
Numerical analysis |
https://en.wikipedia.org/wiki/Minimum%20bottleneck%20spanning%20tree | In mathematics, a minimum bottleneck spanning tree (MBST) in an undirected graph is a spanning tree in which the most expensive edge is as cheap as possible. A bottleneck edge is the highest weighted edge in a spanning tree. A spanning tree is a minimum bottleneck spanning tree if the graph does not contain a spanning tree with a smaller bottleneck edge weight. For a directed graph, a similar problem is known as Minimum Bottleneck Spanning Arborescence (MBSA).
Definitions
Undirected graphs
In an undirected graph and a function , let be the set of all spanning trees Ti. Let B(Ti) be the maximum weight edge for any spanning tree Ti. We define subset of minimum bottleneck spanning trees S′ such that for every and we have for all i and k.
The graph on the right is an example of MBST, the red edges in the graph form a MBST of .
Directed graphs
An arborescence of graph G is a directed tree of G which contains a directed path from a specified node L to each node of a subset V′ of . Node L is called the root of arborescence. An arborescence is a spanning arborescence if . MBST in this case is a spanning arborescence with the minimum bottleneck edge. An MBST in this case is called a Minimum Bottleneck Spanning Arborescence (MBSA).
The graph on the right is an example of MBSA, the red edges in the graph form a MBSA of .
Properties
A MST (or minimum spanning tree) is necessarily a MBST, but a MBST is not necessarily a MST.
Camerini's algorithm for undirected graphs
Camerini proposed an algorithm used to obtain a minimum bottleneck spanning tree (MBST) in a given undirected, connected, edge-weighted graph in 1978. It half divides edges into two sets. The weights of edges in one set are no more than that in the other. If a spanning tree exists in subgraph composed solely with edges in smaller edges set, it then computes a MBST in the subgraph, a MBST of the subgraph is exactly a MBST of the original graph. If a spanning tree does not exist, it combines each disconnected component into a new super vertex, then computes a MBST in the graph formed by these super vertices and edges in the larger edges set. A forest in each disconnected component is part of a MBST in original graph. Repeat this process until two (super) vertices are left in the graph and a single edge with smallest weight between them is to be added. A MBST is found consisting of all the edges found in previous steps.
Pseudocode
The procedure has two input parameters. G is a graph, w is a weights array of all edges in the graph G.
function MBST(graph G, weights w)
E ← the set of edges of G
if | E | = 1 then return E else
A ← half edges in E whose weights are no less than the median weight
B ← E - A
F ← forest of GB
if F is a spanning tree then
return MBST(GB,w)
else
return MBST((GA)η, w) F
In the above (GA)η is the subgraph composed of super vertices (by regarding vertices in a disconnected c |
https://en.wikipedia.org/wiki/Siegfried%20Bosch | Siegfried Bosch is a German mathematician working in arithmetic geometry, focusing in particular on nonarchimedean analytic geometry.
He completed his Ph.D. in 1967 at the University of Göttingen with a dissertation entitled Endliche analytische Homomorphismen (Finite analytic homomorphisms), and received his habilitation degree in 1972. Since 1974 he has been a professor at the University of Münster.
Bosch is the author of several books in algebra and geometry.
Books
References
External links
Homepage
Living people
20th-century German mathematicians
21st-century German mathematicians
University of Göttingen alumni
Academic staff of the University of Münster
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Baha%27%20Faisal | Baha' Faisal Mohammad Seif (; born 30 May 1995) is a Jordanian footballer who plays for the Jordan national football team.
Career statistics
International career
Baha' played his first match with the Jordan national senior team against Bangladesh in the 2018 FIFA World Cup qualification on 24 March 2016, which resulted in an 8–0 win for Jordan.
International goals
With U-17
With U-19
With U-23
Senior team
Scores and results list Jordan's goal tally first.
International statistics
References
External links
1995 births
Living people
Jordanian men's footballers
Jordanian expatriate men's footballers
Jordan men's international footballers
Jordan men's youth international footballers
Men's association football forwards
Al-Wehdat SC players
Kuwait SC players
Al-Shamal SC players
Qatari Second Division players
Qatar Stars League players
Expatriate men's footballers in Kuwait
Expatriate men's footballers in Qatar
Jordanian people of Palestinian descent
2019 AFC Asian Cup players
Jordanian expatriate sportspeople in Kuwait
Kuwait Premier League players
Jordanian expatriate sportspeople in Qatar
Jordanian Pro League players |
https://en.wikipedia.org/wiki/Apotome | Apotome may refer to:
Apotome (mathematics) a mathematical term used by Euclid.
Apotome (music)
Apotome (optics) used for increasing axial resolution of fluorescence microscopy of thick specimens by structured illumination. |
https://en.wikipedia.org/wiki/David%20A.%20Cox | David Archibald Cox (born September 23, 1948 in Washington, D.C.) is a retired American mathematician, working in algebraic geometry.
Cox graduated from Rice University with a bachelor's degree in 1970 and his Ph.D. in 1975 at Princeton University, under the supervision of Eric Friedlander (Tubular Neighborhoods in the Etale Topology). From 1974 to 1975, he was assistant professor at Haverford College and at Rutgers University from 1975 to 1979. In 1979, he became assistant professor and in 1988 professor at Amherst College.
He studies, among other things, étale homotopy theory, elliptic surfaces, computer-based algebraic geometry (such as Gröbner basis), Torelli sets and toric varieties, and history of mathematics. He is also known for several textbooks. He is a fellow of the American Mathematical Society.
From 1987 to 1988 he was a guest professor at Oklahoma State University. In 2012, he received the Lester Randolph Ford Award for Why Eisenstein Proved the Eisenstein Criterion and Why Schönemann Discovered It First.
Writings
With John Little, Donal O'Shea: Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, 3rd. edition, Springer Verlag 2007
David A. Cox, John Little, and Donal O'Shea: Using algebraic geometry, 2nd. edition, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, 2005.
With Sheldon Katz: Mirror Symmetry and Algebraic Geometry, American Mathematical Society 1999
Galois Theory, Wiley/Interscience 2004
With Bernd Sturmfels, Dinesh Manocha (eds.) Applications of computational algebraic geometry, American Mathematical Society 1998
Primes of the form : Fermat, class field theory, and complex multiplication, Wiley 1989
With John Little, Henry Schenck: Toric Varieties, American Mathematical Society 2011
Contributions to Ernst Kunz Residues and duality for projective algebraic varieties, American Mathematical Society 2008
See also
Cox–Zucker machine
Cox ring
References
External links
Homepage
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
1948 births
Living people
Mathematicians from Washington, D.C.
Rice University alumni
Princeton University alumni
Amherst College faculty
Algebraic geometers |
https://en.wikipedia.org/wiki/Scheff%C3%A9%27s%20lemma | In mathematics, Scheffé's lemma is a proposition in measure theory concerning the convergence of sequences of integrable functions. It states that, if is a sequence of integrable functions on a measure space that converges almost everywhere to another integrable function , then if and only if .
In probability theory, almost sure convergence can be weakened to requiring only convergence in probability.
Applications
Applied to probability theory, Scheffe's theorem, in the form stated here, implies that almost everywhere pointwise convergence of the probability density functions of a sequence of -absolutely continuous random variables implies convergence in distribution of those random variables.
History
Henry Scheffé published a proof of the statement on convergence of probability densities in 1947. The result is a special case of a theorem by Frigyes Riesz about convergence in Lp spaces published in 1928.
References
Theorems in measure theory |
https://en.wikipedia.org/wiki/Yuri%20Zhuravlyov%20%28mathematician%29 | Yuri Ivanovich Zhuravlyov (; 14 January 1935 – 14 January 2022) was a Soviet and Russian mathematician specializing in the algebraic theory of algorithms. His research in applied mathematics and computer science was foundational for a number of specialties within discrete mathematics, pattern recognition, and predictive analysis. Zhuravlyov was a full member of the Russian Academy of Sciences and the chairman of its "Applied Mathematics and Informatics" section. He was also the editor-in-chief of the international journal Pattern Recognition and Image Analysis.
Biography
Zhuravlyov was born on 14 January 1935 in Voronezh in the former Soviet Union. In 1952, after finishing high school, he applied and was accepted into the Mathematics Department at Moscow State University. Under the direction of Alexey Lyapunov, he completed his first serious work on the minimization of partially defined boolean functions. The work was published in 1955 and awarded first prize at the All-Soviet student research competition.
In 1957, Zhuravlyov completed his master's thesis on a solution to the problem of finding words in a finite set with consideration for its construction. In 1959, he completed his doctoral work which involved a proof for lack of local unsolvability for constructing the minimal disjunctive normal form.
In 1959, he moved to Novosibirsk, where he pursued government-sponsored research and taught algebra and mathematical logic at the Novosibirsk University. In 1966, he began research into pattern recognition. His first serious work in this field related to the identification of deposits in the area of gold mining. He then developed a model of algorithms for calculating estimates that became foundational for numerous subsequent research and works in the field.
In 1969, Zhuravlyov moved to Moscow to head the Pattern Recognition Lab at the Central Soviet Computing Center. In 1970, he also joined the faculty of the Moscow Institute of Physics and Technology as a full professor.
Throughout the 1970s and 1980s, Zhuravlyov published a series of seminal works in applied mathematics and informatics. In 1991, he founded the journal Pattern Recognition and Image Analysis. In 1992, he was invited to join the Russian Academy of Sciences. In 1997, he became a professor at Moscow State University.
Zhuravlyov died in Moscow on 14 January 2022, on his 87th birthday.
References
External links
Maik journal page
Mathematics Genealogy Project
Springer journal page
1935 births
2022 deaths
20th-century Russian mathematicians
21st-century Russian mathematicians
People from Voronezh
Corresponding Members of the USSR Academy of Sciences
Full Members of the Russian Academy of Sciences
Honorary Members of the Russian Academy of Education
Academic staff of Moscow State University
Academic staff of Novosibirsk State University
Recipients of the Lenin Prize
Recipients of the Order "For Merit to the Fatherland", 3rd class
Recipients of the Order "For Merit to the Father |
https://en.wikipedia.org/wiki/Juraj%20Ceb%C3%A1k | Juraj Cebák (born 29 September 1982) is a Slovak ice hockey defenceman. He is currently a free agent.
Career statistics
Regular season and playoffs
External links
1982 births
Living people
People from Prievidza
Ice hockey people from the Trenčín Region
HC '05 Banská Bystrica players
HC 07 Detva players
HC Bílí Tygři Liberec players
HC Košice players
MHk 32 Liptovský Mikuláš players
HC Vítkovice players
HK Dukla Trenčín players
HK Nitra players
HKM Zvolen players
MHC Martin players
MsHK Žilina players
Slovak ice hockey defencemen
Slovak expatriate ice hockey players in the Czech Republic |
https://en.wikipedia.org/wiki/Mihai%20%C8%9Aurcan%20%28footballer%2C%20born%201989%29 | Mihai Țurcan (born 20 August 1989) is a Moldovan football forward who last played for JK Sillamäe Kalev.
Club statistics
Total matches played in Moldovan First League: 75 matches - 14 goal
References
External links
1989 births
Living people
Men's association football forwards
Moldovan men's footballers
Sportspeople from Bălți
FC Milsami Orhei players
FC Zimbru Chișinău players
CSF Bălți players
FC Tiraspol players
FC Veris Chișinău players
FC Academia Chișinău players
FC Kaisar players
JK Sillamäe Kalev players
Moldovan Super Liga players
Meistriliiga players
Moldovan expatriate men's footballers
Expatriate men's footballers in Estonia
Moldovan expatriate sportspeople in Estonia |
https://en.wikipedia.org/wiki/Quantum%20stochastic%20calculus | Quantum stochastic calculus is a generalization of stochastic calculus to noncommuting variables. The tools provided by quantum stochastic calculus are of great use for modeling the random evolution of systems undergoing measurement, as in quantum trajectories. Just as the Lindblad master equation provides a quantum generalization to the Fokker–Planck equation, quantum stochastic calculus allows for the derivation of quantum stochastic differential equations (QSDE) that are analogous to classical Langevin equations.
For the remainder of this article stochastic calculus will be referred to as classical stochastic calculus, in order to clearly distinguish it from quantum stochastic calculus.
Heat baths
An important physical scenario in which a quantum stochastic calculus is needed is the case of a system interacting with a heat bath. It is appropriate in many circumstances to model the heat bath as an assembly of harmonic oscillators. One type of interaction between the system and the bath can be modeled (after making a canonical transformation) by the following Hamiltonian:
where is the system Hamiltonian, is a vector containing the system variables corresponding to a finite number of degrees of freedom, is an index for the different bath modes, is the frequency of a particular mode, and are bath operators for a particular mode, is a system operator, and quantifies the coupling between the system and a particular bath mode.
In this scenario the equation of motion for an arbitrary system operator is called the quantum Langevin equation and may be written as:
where and denote the commutator and anticommutator (respectively), the memory function is defined as:
and the time dependent noise operator is defined as:
where the bath annihilation operator is defined as:
Oftentimes this equation is more general than is needed, and further approximations are made to simplify the equation.
White noise formalism
For many purposes it is convenient to make approximations about the nature of the heat bath in order to achieve a white noise formalism. In such a case the interaction may be modeled by the Hamiltonian where:
and
where are annihilation operators for the bath with the commutation relation , is an operator on the system, quantifies the strength of the coupling of the bath modes to the system, and describes the free system evolution. This model uses the rotating wave approximation and extends the lower limit of to in order to admit a mathematically simple white noise formalism. The coupling strengths are also usually simplified to a constant in what is sometimes called the first Markov approximation:
Systems coupled to a bath of harmonic oscillators can be thought of as being driven by a noise input and radiating a noise output. The input noise operator at time is defined by:
where , since this operator is expressed in the Heisenberg picture. Satisfaction of the commutation relation allows the model to have a strict |
https://en.wikipedia.org/wiki/Maya%20Jalloul | Maya Jalloul (مايا جلول ; born 16 April 1990) is a Lebanese chess champion.
Education
Jalloul has a bachelor's degree in mathematics from the American University of Beirut and a master's degree in actuarial science from the Saint Joseph University. She is currently pursuing her PhD in economics at the Queen Mary University of London.
Chess career
After winning several local and student competitions, Jalloul won the Arab U-18 women championship in 2007, then the Lebanese Women Championship in 2009 while ranking second in the Arab women championship in 2009.
Jalloul also participated in numerous international chess competitions, including the 39th Chess Olympiad in 2008 in Dresden, Germany, and in the 40th Chess Olympiad in 2012 in Istanbul, Turkey. On the regional level she also competed in the 8th Mediterranean games in Beirut, Lebanon, in 2013, the Arab tournament in Jordan in 2010, the Arab games in Qatar in 2011 and the West Asia chess championship in Qatar in 2013
Jalloul is a Woman FIDE Master (WFM) since 2003.
References
Lebanese chess players
Chess Woman FIDE Masters
American University of Beirut alumni
1990 births
Living people
21st-century Lebanese women
Ukrainian people of Lebanese descent
Russian people of Lebanese descent
Sportspeople of Lebanese descent |
https://en.wikipedia.org/wiki/Rose%20Whelan%20Sedgewick | Rose Whelan Sedgewick ( – 2000) was an American mathematician. She was the first person to earn a PhD in mathematics from Brown University, in 1929. Her subsequent career in mathematics included assistant professorships at the University of Rochester, the University of Connecticut, Hillyer College, and the University of Maryland.
Sedgewick is the namesake of the Rose Whelan Society at Brown, an organization for women and gender minorities who are graduate students, post-doctoral fellows and faculty in pure and applied in mathematics. She was married to fellow mathematician Charles H.W. Sedgewick and had four children. She died on June 7, 2000, at the age of 96.
Professional honors
Mathematical Association of America
American Mathematical Society
Phi Beta Kappa
Sigma Xi
References
American women mathematicians
Brown University alumni
20th-century American mathematicians
2000 deaths
1900s births
20th-century American women scientists
20th-century women mathematicians |
https://en.wikipedia.org/wiki/William%20Reinhardt | William Reinhar(d)t may refer to:
Real people
William Reinhardt (mathematician), see Ackermann set theory
William Reinhart, athlete
Fictional characters
William Reinhardt, character in Hell House (novel)
William Reinhardt (The Passage) |
https://en.wikipedia.org/wiki/Infinite%20loop%20space%20machine | In topology, a branch of mathematics, given a topological monoid X up to homotopy (in a nice way), an infinite loop space machine produces a group completion of X together with infinite loop space structure. For example, one can take X to be the classifying space of a symmetric monoidal category S; that is, . Then the machine produces the group completion . The space may be described by the K-theory spectrum of S.
References
J. P. May and R. Thomason The uniqueness of infinite loop space machines
Homotopy theory
Topological spaces
Topology |
https://en.wikipedia.org/wiki/K-theory%20spectrum | In mathematics, given a ring R, the K-theory spectrum of R is an Ω-spectrum whose nth term is given by, writing for the suspension of R,
,
where "+" means the Quillen's + construction. By definition, .
References
Algebraic K-theory |
https://en.wikipedia.org/wiki/Module%20spectrum | In algebra, a module spectrum is a spectrum with an action of a ring spectrum; it generalizes a module in abstract algebra.
The ∞-category of (say right) module spectra is stable; hence, it can be considered as either analog or generalization of the derived category of modules over a ring.
K-theory
Lurie defines the K-theory of a ring spectrum R to be the K-theory of the ∞-category of perfect modules over R (a perfect module being defined as a compact object in the ∞-category of module spectra.)
See also
G-spectrum
References
J. Lurie, Lecture 19: Algebraic K-theory of Ring Spectra
Homotopy theory |
https://en.wikipedia.org/wiki/Thomas%20Barker%20%28mathematician%29 | Thomas Barker (1838–1907) was a Scottish mathematician, professor of pure mathematics at Owens College.
Life
Born 9 September 1838, he was son of Thomas Barker, farmer, of Murcar, Balgonie, near Aberdeen, and of his wife Margaret. Three other children died in infancy. He was educated at Aberdeen Grammar School, and at King's College in the same town, where he graduated in 1857 with distinction in mathematics.
Barker entered Trinity College, Cambridge as minor scholar and subsizar in 1858. He became foundation scholar in 1860, Sheepshanks astronomical exhibitioner in 1861, and came out in the Mathematical Tripos of 1862 as senior wrangler; he was also first Smith's prizeman. He was elected to a fellowship in the autumn of 1862, and was assistant tutor of Trinity till 1865, when he was appointed professor of pure mathematics in Owens College, Manchester. He held this post for twenty years.
Barker was a follower of Augustus De Morgan and George Boole. He was interested in the logical basis rather than in the applications of mathematics, and was an austere teacher. He disliked publication.
After resigning his chair in 1885, Barker lived in retirement, first at Whaley Bridge and then at Buxton. He pursued the study of cryptogamic botany. He died unmarried at Buxton on 20 November 1907, and was buried in Southern Cemetery, Manchester.
Pupils
Barker had a number of distinguished mathematicians and physicists as pupils: they included John Walton Capstick, John Hopkinson, John Henry Poynting, Arthur Schuster, and Joseph John Thomson.
Legacy
By his will Barker provided for the foundation in the University of Manchester of a professorship of cryptogamic botany, and for the endowment of bursaries in mathematics and botany.
Notes
Attribution
1838 births
1907 deaths
Scottish mathematicians
Alumni of the University of Aberdeen
Alumni of Trinity College, Cambridge
Fellows of Trinity College, Cambridge
People from Aberdeenshire
Academics of the Victoria University of Manchester
People from Whaley Bridge
People from Buxton |
https://en.wikipedia.org/wiki/Calvin%20C.%20Moore | Calvin C. Moore (November 2, 1936 - July 26, 2023) was an American mathematician who worked in the theory of operator algebras and topological groups.
Moore graduated from Harvard University with a bachelor's degree in 1958 and with a Ph.D. in 1960 under the supervision of George Mackey (Extensions and cohomology theory of locally compact groups). In 1961 he became assistant professor at the University of California, Berkeley and professor in 1966. From 1977 to 1980, he was director of the Center for Pure and Applied mathematics.
With Shiing-Shen Chern and Isadore Singer, he co-founded Mathematical Sciences Research Institute in 1982. From 1964 to 1965 he was at the Institute for Advanced Study in Princeton, New Jersey.
He was a fellow of the American Academy of Arts and Sciences. From 1965 to 1967 he was a Sloan Fellow. From 1971 to 1979 he was a member of the Board of Trustees of the American Mathematical Society, whose fellow he is. Since 1977, he is co-editor of the Pacific Journal of Mathematics. From 1978 to 1979 he was a Miller research professor at Berkeley.
He has written on a history of mathematics at Berkeley.
His students include Roger Howe, Truman Bewley and .
Writings
With Claude Schochet, Global Analysis on Foliated Spaces, MSRI Publications, Springer Verlag 1988, 2nd ed., Cambridge University Press 2006.
References
External links
Homepage
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
1936 births
2023 deaths
Harvard University alumni
American historians of mathematics
University of California, Berkeley faculty
Scientists from New York City
American founders
Mathematicians from New York (state) |
https://en.wikipedia.org/wiki/Suspension%20of%20a%20ring | In algebra, more specifically in algebraic K-theory, the suspension of a ring R is given by where is the ring of all infinite matrices with coefficients in R having only finitely many nonzero elements in each row or column and is its ideal of matrices having only finitely many nonzero elements. It is an analog of suspension in topology.
One then has: .
References
C. Weibel "The K-book: An introduction to algebraic K-theory"
Algebra |
https://en.wikipedia.org/wiki/Unital%20%28geometry%29 | In geometry, a unital is a set of n3 + 1 points arranged into subsets of size n + 1 so that every pair of distinct points of the set are contained in exactly one subset. This is equivalent to saying that a unital is a 2-(n3 + 1, n + 1, 1) block design. Some unitals may be embedded in a projective plane of order n2 (the subsets of the design become sets of collinear points in the projective plane). In this case of embedded unitals, every line of the plane intersects the unital in either 1 or n + 1 points. In the Desarguesian planes, PG(2,q2), the classical examples of unitals are given by nondegenerate Hermitian curves. There are also many non-classical examples. The first and the only known unital with non prime power parameters, n=6, was constructed by Bhaskar Bagchi and Sunanda Bagchi. It is still unknown if this unital can be embedded in a projective plane of order 36, if such a plane exists.
Unitals
Classical
We review some terminology used in projective geometry.
A correlation of a projective geometry is a bijection on its subspaces that reverses containment. In particular, a correlation interchanges points and hyperplanes.
A correlation of order two is called a polarity.
A polarity is called a unitary polarity if its associated sesquilinear form s with companion automorphism α satisfies
s(u,v) = s(v,u)α for all vectors u, v of the underlying vector space.
A point is called an absolute point of a polarity if it lies on the image of itself under the polarity.
The absolute points of a unitary polarity of the projective geometry PG(d,F), for some d ≥ 2, is a nondegenerate Hermitian variety, and if d = 2 this variety is called a nondegenerate Hermitian curve.
In PG(2,q2) for some prime power q, the set of points of a nondegenerate Hermitian curve form a unital, which is called a classical unital.
Let be a nondegenerate Hermitian curve in for some prime power . As all nondegenerate Hermitian curves in the same plane are projectively equivalent, can be described in terms of homogeneous coordinates as follows:
Ree unitals
Another family of unitals based on Ree groups was constructed by H. Lüneburg. Let Γ = R(q) be the Ree group of type 2G2 of order (q3 + 1)q3(q − 1) where q = 32m+1. Let P be the set of all q3 + 1 Sylow 3-subgroups of Γ. Γ acts doubly transitively on this set by conjugation (it will be convenient to think of these subgroups as points that Γ is acting on.) For any S and T in P, the pointwise stabilizer, ΓS,T is cyclic of order q - 1, and thus contains a unique involution, μ. Each such involution fixes exactly q + 1 points of P. Construct a block design on the points of P whose blocks are the fixed point sets of these various involutions μ. Since Γ acts doubly transitively on P, this will be a 2-design with parameters 2-(q3 + 1, q + 1, 1) called a Ree unital.
Lüneburg also showed that the Ree unitals can not be embedded in projective planes of order q2 (Desarguesian or not) such that the automorphism group Γ is induced |
https://en.wikipedia.org/wiki/Solid%20partition | In mathematics, solid partitions are natural generalizations of partitions and plane partitions defined by Percy Alexander MacMahon. A solid partition of is a three-dimensional array of non-negative integers (with indices ) such that
and
for all
Let denote the number of solid partitions of . As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by Andrews.
Ferrers diagrams for solid partitions
Another representation for solid partitions is in the form of Ferrers diagrams. The Ferrers diagram of a solid partition of is a collection of points or nodes, , with satisfying the condition:
Condition FD: If the node , then so do all the nodes with for all .
For instance, the Ferrers diagram
where each column is a node, represents a solid partition of . There is a natural action of the permutation group on a Ferrers diagram – this corresponds to permuting the four coordinates of all nodes. This generalises the operation denoted by conjugation on usual partitions.
Equivalence of the two representations
Given a Ferrers diagram, one constructs the solid partition (as in the main definition) as follows.
Let be the number of nodes in the Ferrers diagram with coordinates of the form where denotes an arbitrary value. The collection form a solid partition. One can verify that condition FD implies that the conditions for a solid partition are satisfied.
Given a set of that form a solid partition, one obtains the corresponding Ferrers diagram as follows.
Start with the Ferrers diagram with no nodes. For every non-zero , add nodes for to the Ferrers diagram. By construction, it is easy to see that condition FD is satisfied.
For example, the Ferrers diagram with nodes given above corresponds to the solid partition with
with all other vanishing.
Generating function
Let . Define the generating function of solid partitions, , by
The generating functions of integer partitions and plane partitions have simple product formulae, due to Euler and MacMahon, respectively. However, a guess of MacMahon fails to correctly reproduce the solid partitions of 6. It appears that there is no simple formula for the generating function of solid partitions; in particular, there cannot be any formula analogous to the product formulas of Euler and MacMahon.
Exact enumeration using computers
Given the lack of an explicitly known generating function, the enumerations of the numbers of solid partitions for larger integers have been carried out numerically. There are two algorithms that are used to enumerate solid partitions and their higher-dimensional generalizations. The work of Atkin. et al. used an algorithm due to Bratley and McKay. In 1970, Knuth proposed a differ |
https://en.wikipedia.org/wiki/Isador%20Lubin | Isador Lubin (9 June 1896 – 6 July 1978) was the head of the U.S. Bureau of Labor Statistics from 1933 to 1946, and president of the American Statistical Association in 1946.
Career
During the First World War, at the U.S. Food Administration, Lubin analyzed labor and price policy related to food production for the Allied Nations. Later at the War Industries Board’s Price Section, he studied the effect of price shifts on the output of the petroleum and rubber industries.
He was as an instructor at the Institute of Economics and earned a Ph.D. there in 1926. It became part of the Brookings Institution in 1927. Lubin's book Miners' Wages and the Cost of Coal was accepted as a dissertation.
"In 1932, as adviser to Senator Robert M. La Follette Jr., he pioneered the notion of government responsibility for the national accounts."
Lubin was appointed head of the Bureau of Labor Statistics (BLS) by Frances Perkins in July 1933 and stayed in the position until January 1946. For much of this time, Lubin had an office in the White House's West Wing "and served as special statistical adviser to President Franklin Roosevelt." Lubin was sometimes described as a member of President Roosevelt's "brain trust." In 1944 he was elected as a Fellow of the American Statistical Association.
In 1941 Lubin authorized BLS to start a research group at Harvard University directed by Wassily Leontief which constructed the first official table of U.S. industry inputs and outputs." In 1945, Roosevelt appointed Lubin as Minister to the Allied Reparations Commission.
"In his presidential address to the American Statistical Society in January [1947], Lubin emphasized [the role] of statistics in modern economic society and the value to the free world of pertinent data."
Lubin was appointed the Industrial Commissioner of New York state from 1955 to 1958 by Governor W. Averell Harriman.
Personal and legacy
In 1952 Lubin married the former Carol Riegelman (1909-2005), a longtime consultant to the UN/ILO. He had two daughters by a previous marriage to Ann Shumaker Lubin, the editor of "Progressive Education" and co-author of the book, "The Child-Centered School" (1928): Alice Lubin Everitt and Ann Lubin Buttenwieser.
Fellowships named for Dr. Lubin were established at Brandeis University and The New School.
References
Further information
Lewis Lansky. 1976. Isador Lubin: The Ideas and Career of a New Deal Labor Economist. Case Western Reserve University. Dissertation.
American civil servants
20th-century American economists
Bureau of Labor Statistics
Fellows of the American Statistical Association
Presidents of the American Statistical Association
1896 births
1978 deaths
Franklin D. Roosevelt administration personnel
Truman administration personnel |
https://en.wikipedia.org/wiki/Lynne%20Billard | Lynne Billard (born 1943) is an Australian statistician and professor at the University of Georgia, known for her statistics research, leadership, and advocacy for women in science. She has served as president of the American Statistical Association, and the International Biometric Society, one of a handful of people to have led both organizations.
Education
She earned her Bachelor of Science degree in 1966, and Doctoral degree in 1969, both from the University of New South Wales, Australia.
Mathematics Cadetship, University of New South Wales, 1962-1965.
Theory of Statistics II Prize, University of New South Wales, 1964.
Theory of Statistics III Prize, University of New South Wales, 1965.
General Proficiency in Statistics Prize, University of New South Wales, 1965.
First Class Honours in Statistics, University of New South Wales, 1966.
Life and career
In 1975, Billard joined Florida State University, USA as an Associate Professor and in 1980, she moved to the University of Georgia as head of the Department of Statistics and Computer Science. In 1984, when the departments split, she became the first Head of the Department of Statistics at UGA. From 1989 - 1991, she served as an Associate Dean at the University of Georgia, in 1992 she was named a University Professor. Among her other appointments are the following:
Teaching Fellow and Tutor, University of New South Wales, 1/1966-12/1968.
Lecturer, University of Birmingham, U. K., 1/1969-12/1970.
Visiting Assistant Professor, SUNY at Buffalo, 1/1971-8/1971.
Assistant Professor, University of Waterloo, Canada, 9/1971-12/1974.
Visiting Assistant Professor, Stanford University, 1/1974-8/1974.
Visiting Associate Professor, SUNY at Buffalo, 9/1974-6/1975.
Visiting Associate Professor, Stanford University, 6/1974-8/1974.
Associate Professor, Florida State University, 7/1975-8/1980.
Associate Head, Florida State University, 7/1976-6/1978.
Research Fellow, Naval Postgraduate School, 8/1979-9/1979.
Research Fellow, University of California, Berkeley, 9/1979-12/1979.
Professor, Florida State University, 1980-1981, on leave (at University of Georgia).
Professor of Statistics and Head, Department of Computer Science and Statistics, University of Georgia, 9/1980-8/1984.
Professor and Head, Statistics, University of Georgia, 9/1984-3/1989.
Imperial College, London (on leave), 9/1986-12/1986.
Professor and Associate Dean, University of Georgia, 4/1989-8/1991.
Professor, University of Georgia, 9/1991-6/1992.
University Professor, University of Georgia, 7/1992–present.
Adjunct Professor, Australian National University, 7/1997–present.
Honorary Professorial Fellow, University of Melbourne, 9/2009–present.
Research
Lynne Billard has worked to involve statisticians in solving current and applied problems. Her work on the incubation period of AIDS greatly impacted public health education. Overall, her research spans a mix of theoretical and applied work. Most mathematical/theoretical work was motivated by re |
https://en.wikipedia.org/wiki/Alberto%20Pinto%20%28mathematician%29 | Alberto Adrego Pinto is a full professor at the Department of Mathematics, Faculty of Sciences, University of Porto (Portugal). He is a researcher of the Laboratory of Artificial Intelligence and Decision Support, Institute for Systems and Computer Engineering LIAAD, INESC TEC. He is the founder and editor-in-chief of the Journal of Dynamics and Games, published by the American Institute of Mathematical Sciences (AIMS). He is the President of the Portuguese International Center for Mathematics (CIM). Currently, he is also a Special Visiting Researcher from CNPq at Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro, Brazil.
Education
Pinto was an undergraduate student in applied mathematics at University of Porto (1986). He did his MSc with distinction (1998) and his PhD (1991) in mathematics at University of Warwick, UK. He did his Aggregation in Applied Mathematics (2002), passing with unanimous vote, at University of Porto.
Career
Pinto worked with David Rand on his master's thesis (1989) that studied the work of Mitchell Feigenbaum and Dennis Sullivan on scaling functions and he went on to a PhD (1991) on the universality features of other classes of maps that form the boundary between order and chaos.
During this time Pinto met a number of the leaders in dynamical systems, notably Dennis Sullivan and Maurício Peixoto, and this had a great impact on his career. As a result, he and his collaborators have made many important contributions to the study of the fine-scale structure of dynamical systems and this has appeared in leading journals and in his book "Fine Structures of Hyperbolic Diffeomorphisms" (2010) coauthored with Flávio Ferreira and David Rand.
While a postdoc with Dennis Sullivan at the CUNY Graduate Center at City University of New York he met Edson de Faria and through Maurício Peixoto he got in contact with Welington de Melo. With de Melo he proved the rigidity of smooth unimodal maps in the boundary between chaos and order extending the work of Curtis T. McMullen. Furthermore, de Faria, de Melo and Pinto proved the conjecture raised in 1978 in the work of Feigenbaum and Coullet-Tresser which the characterizes the period-doubling boundary between chaos and order for unimodal maps. This appeared in the research article "Global Hyperbolicity of Renormalization for Smooth Unimodal Mappings" published at the journal Annals of Mathematics (2006) and was based in particular in the previous works of Sandy Davie, Dennis Sullivan, Curtis T. McMullen and Mikhail Lyubich.
Since then Pinto has branched out into more applied areas. He has contributed across a remarkably broad area of science including optics, game theory and mathematical economics, finance, immunology, epidemiology, and climate and energy. In these applied areas, he has published widely overpassing more than one hundred scientific articles. He edited two volumes, with Maurício Peixoto and David Rand, entitled "Dynamics and Games I and II" (2011). |
https://en.wikipedia.org/wiki/Karl%20Schaum | Ferdinand Karl Franz Schaum (14 July 1870, Frankfurt am Main – 30 January 1947, Gießen) was a German chemist who specialized in the field of photochemistry.
He studied mathematics and sciences at the Universities of Basel, Berlin, Leipzig and Marburg, earning his doctorate at the latter institution in 1893. Afterwards, he served as an assistant to Theodor Zincke at Marburg and to Wilhelm Ostwald in Leipzig. In 1897 he obtained his habilitation at Marburg with a thesis on types of isometry.
In 1904 he became an associate professor of physical chemistry at the University of Marburg, where he was an important influence towards the career of chemist Max Volmer. In 1908 he relocated to the University of Leipzig as an associate professor of photochemistry and scientific photography. From 1914 to 1935 he worked as a full professor of physical chemistry at the University of Giessen.
Schaum was a member of the Gesellschaft zur Beförderung der gesamten Naturwissenschaften Marburg.
Published works
With zoologist Eugen Korschelt and others, he was co-author of the multi-volume "Handwörterbuch der naturwissenschaften". Also, he was editor of the journal "Zeitschrift für wissenschaftliche Photographie, Photophysik und Photochemie". The following are a few of his principal works:
Die Arten der Isometrie. Eine kritische Studie, 1897 - Types of isometry, a critical study.
Kraft und Leben in der Natur : Lesestücke aus dem Gebiete der Physik und der Chemie, der Botanik und der Zoologie, 1904 - Vitality in nature, Literature in the fields of physics, chemistry, botany and zoology.
Photochemie und Photographie, 1908 - Photochemistry and photography.
References
External links
1870 births
1947 deaths
Academic staff of the University of Giessen
Academic staff of the University of Marburg
Academic staff of Leipzig University
20th-century German chemists
Scientists from Frankfurt |
https://en.wikipedia.org/wiki/Nonlinear%20modelling | In mathematics, nonlinear modelling is empirical or semi-empirical modelling which takes at least some nonlinearities into account. Nonlinear modelling in practice therefore means modelling of phenomena in which independent variables affecting the system can show complex and synergetic nonlinear effects. Contrary to traditional modelling methods, such as linear regression and basic statistical methods, nonlinear modelling can be utilized efficiently in a vast number of situations where traditional modelling is impractical or impossible. The newer nonlinear modelling approaches include non-parametric methods, such as feedforward neural networks, kernel regression, multivariate splines, etc., which do not require a priori knowledge of the nonlinearities in the relations. Thus the nonlinear modelling can utilize production data or experimental results while taking into account complex nonlinear behaviours of modelled phenomena which are in most cases practically impossible to be modelled by means of traditional mathematical approaches, such as phenomenological modelling.
Contrary to phenomenological modelling, nonlinear modelling can be utilized in processes and systems where the theory is deficient or there is a lack of fundamental understanding on the root causes of most crucial factors on system. Phenomenological modelling describes a system in terms of laws of nature. Nonlinear modelling can be utilized in situations where the phenomena are not well understood or expressed in mathematical terms. Thus nonlinear modelling can be an efficient way to model new and complex situations where relationships of different variables are not known.
Statistical models |
https://en.wikipedia.org/wiki/Polar%20point%20group | In geometry, a polar point group is a point group in which there is more than one point that every symmetry operation leaves unmoved. The unmoved points will constitute a line, a plane, or all of space.
While the simplest point group, C1, leaves all points invariant, most polar point groups will move some, but not all points. To describe the points which are unmoved by the symmetry operations of the point group, we draw a straight line joining two unmoved points. This line is called a polar direction. The electric polarization must be parallel to a polar direction. In polar point groups of high symmetry, the polar direction can be a unique axis of rotation, but if the symmetry operations do not allow any rotation at all, such as mirror symmetry, there can be an infinite number of such axes: in that case the only restriction on the polar direction is that it must be parallel to any mirror planes.
A point group with more than one axis of rotation or with a mirror plane perpendicular to an axis of rotation cannot be polar.
Polar crystallographic point group
Of the 32 crystallographic point groups, 10 are polar:
The space groups associated with a polar point group do not have a discrete set of possible origin points that are unambiguously determined by symmetry elements.
When materials having a polar point group crystal structure are heated or cooled, they may temporarily generate a voltage called pyroelectricity.
Molecular crystals which have symmetry described by one of the polar space groups, such as sucrose, may exhibit triboluminescence.
References
Symmetry
Crystallography
Group theory |
https://en.wikipedia.org/wiki/VNS3 | VNS3 is a software-only virtual appliance that allows users to control access and network topology and secure data in motion across public and private clouds. VNS3 is a virtual router, switch, firewall, protocol re-distributor, and SSL/IPSec VPN concentrator. The Network Virtualization Software creates a customer-controlled overlay network over top of the underlying network backbone.
Uses
VNS3 is a network routing and security virtual appliance that lets extend networks into public, private, and hybrid clouds VNS3 lets enterprise data center administrators "create encrypted LAN between virtual machines in a private cloud, as well as encrypted WAN across multiple public clouds."
History
Developers Cohesive Networks first named their multi-sourced infrastructure concept "v-cube-v." The software ran in internal production starting in 2007. The company named the early commercial version of VNS3 "VPN3" or "VPN-Cubed" and later renamed the software to VNS3 in 2012.
Amazon Web Services users first began downloading VPN-Cubed from the partner directory on 5 December 2008. VNS3 gained popularity (as VPN-Cubed) as part of the Amazon Web Services public cloud ecosystem and with independent reviews from ZDNet, High Scalability, InfoQ, Chris Hoff, and CloudAve.
In 2012, developers Cohesive Networks released a major version update. The release updated the software to 3.0 and rebranded it as VNS3 (VNS-Cubed). 451 Research analyst William Fellows wrote "VNS[3] is not only for VPNs – hence the name change – since overlays can be within a cloud, between clouds, between a private datacenter and a cloud (or clouds), or between multiple datacenters."
In 2013, Cohesive Networks released a 3.0.1 version of the product, as well as a free edition of VNS3 in Amazon Web Services. VNS3 was recognized in the 6th Annual International Datacenters Awards as the winner of the Public Cloud Services & Infrastructure award In early 2014 VNS3 3.5 was released with major software updates and a new integration with Docker Docker's open-source virtualization platform added the ability to run other networking applications as containers inside VNS3 virtual machines. Users can create an overlay network "as a substrate for layer 4-7 network application services – things like proxy, reverse proxy, SSL termination, content caching and network intrusion detection" William Fellows writes.
In early 2015 the company renamed to Cohesive Networks to emphasize the networking capabilities of VNS3 and to spin off the less successful part of the business. The company later announced a new line of VNS3-based products, including VNS3:turret application segmentation controller, the VNS3:ms network management platform, and the VNS3:ha - high availability add on.
After 2008, VNS3 became available in more public cloud providers and geographic regions, including Amazon Web Services EC2, GoGrid, Flexiant, IBM SoftLayer, Google Compute Engine, HP Cloud Services, Mircorsoft Azure, and CenturyLink |
https://en.wikipedia.org/wiki/Hendecagrammic%20prism | In geometry, a hendecagrammic prism is a star polyhedron made from two identical regular hendecagrams connected by squares. The related hendecagrammic antiprisms are made from two identical regular hendecagrams connected by equilateral triangles.
Hendecagrammic prisms and bipyramids
There are 4 hendecagrammic uniform prisms, and 6 hendecagrammic uniform antiprisms. The prisms are constructed by 4.4.11/q vertex figures, Coxeter diagram. The hendecagrammic bipyramids, duals to the hendecagrammic prisms are also given.
Hendecagrammic antiprisms
The antiprisms with 3.3.3.3.11/q vertex figures, . Uniform antiprisms exist for p/q>3/2, and are called crossed for p/q<2. For hendecagonal antiprism, two crossed antiprisms can not be constructed as uniform (with equilateral triangles): 11/8, and 11/9.
Hendecagrammic trapezohedra
The hendecagrammic trapezohedra are duals to the hendecagrammic antiprisms.
See also
Prismatic uniform polyhedron
References
External links
Polyhedra |
https://en.wikipedia.org/wiki/Mannheim%20School%20of%20Computer%20Science%20and%20Mathematics | The Mannheim School of Computer Science and Mathematics (MSCM) is among the younger of the five schools comprising the University of Mannheim, located in Mannheim, Baden-Württemberg, Germany. The School of Computer Science and Mathematics, established in 1967, covers the fields of Computer Science, Business Informatics and Mathematics. The Department of Computer Science at the University of Mannheim is considered as a leading public institution for computer science/business informatics in Germany. It has been consistently ranked among the top computer science programs in recent years.
In the past 15 years, researchers from University of Mannheim Department of Computer Science have made developments in the fields of algorithms, computer networks, distributed systems, parallel processing, programming languages, robotics, language technologies, human-computer interaction and software engineering.
Department of Computer Science
The Institute of Computer Science and Business Informatics consists of eleven Chairs and Professorships dedicated to Data Management, Software Development, Dependable Distributed Systems, Web Technologies, Cryptography, Process Modelling, Artificial Intelligence and Mobile and Visual Media. Their common point of interest is the management of complex data material for society and economy. The institute is mainly located in the building A 5,6. Together with the business informatics group that are part of the Business School, the Institute of Computer Science recently founded the Center for Business Informatics to ensure that research and teaching standards in this area remain at the highest level. The department has exchange agreements with renowned universities like the Carnegie Mellon University in the United States, the Seoul National University in South Korea or the National University of Singapore and maintains a highly international student body with more than 27% of students coming from abroad.
Department of Mathematics
The Institute of Mathematics consists of eleven Chairs and 22 Professorships that focus on classical mathematical disciplines as well as on economic and practical-oriented fields of mathematics. The main areas of research include Algebra, Analysis, Geometry, Stochastics and Mathematical Statistics as well as Mathematics in Finance and Insurance.
Through its successful focus on business mathematics in research and teaching, the Institute of Mathematics is constantly expanding its close cooperation with the University's Department of Economics and the Business School.
See also
University of Mannheim
Mannheim Business School
Education in Germany
List of universities in Germany
References
External links
Department of Computer Science
Department of Mathematics
Institute for Technical Informatics and IT Security
University of Mannheim
Mannheim
Educational institutions established in 1967
1967 establishments in Germany |
https://en.wikipedia.org/wiki/Weyr%20canonical%20form | In mathematics, in linear algebra, a Weyr canonical form (or, Weyr form or Weyr matrix) is a square matrix which (in some sense) induces "nice" properties with matrices it commutes with. It also has a particularly simple structure and the conditions for possessing a Weyr form are fairly weak, making it a suitable tool for studying classes of commuting matrices. A square matrix is said to be in the Weyr canonical form if the matrix has the structure defining the Weyr canonical form. The Weyr form was discovered by the Czech mathematician Eduard Weyr in 1885. The Weyr form did not become popular among mathematicians and it was overshadowed by the closely related, but distinct, canonical form known by the name Jordan canonical form. The Weyr form has been rediscovered several times since Weyr’s original discovery in 1885. This form has been variously called as modified Jordan form, reordered Jordan form, second Jordan form, and H-form. The current terminology is credited to Shapiro who introduced it in a paper published in the American Mathematical Monthly in 1999.
Recently several applications have been found for the Weyr matrix. Of particular interest is an application of the Weyr matrix in the study of phylogenetic invariants in biomathematics.
Definitions
Basic Weyr matrix
Definition
A basic Weyr matrix with eigenvalue is an matrix of the following form: There is a partition
of with
such that, when is viewed as an block matrix , where the block is an matrix, the following three features are present:
The main diagonal blocks are the scalar matrices for .
The first superdiagonal blocks are full column rank matrices in reduced row-echelon form (that is, an identity matrix followed by zero rows) for .
All other blocks of W are zero (that is, when ).
In this case, we say that has Weyr structure .
Example
The following is an example of a basic Weyr matrix.
In this matrix, and . So has the Weyr structure . Also,
and
General Weyr matrix
Definition
Let be a square matrix and let be the distinct eigenvalues of . We say that is in Weyr form (or is a Weyr matrix) if has the following form:
where is a basic Weyr matrix with eigenvalue for .
Example
The following image shows an example of a general Weyr matrix consisting of three basic Weyr matrix blocks. The basic Weyr matrix in the top-left corner has the structure (4,2,1) with eigenvalue 4, the middle block has structure (2,2,1,1) with eigenvalue -3 and the one in the lower-right corner has the structure (3, 2) with eigenvalue 0.
Relation between Weyr and Jordan forms
The Weyr canonical form is related to the Jordan form by a simple permutation for each Weyr basic block as follows: The first index of each Weyr subblock forms the largest Jordan chain. After crossing out these rows and columns, the first index of each new subblock forms the second largest Jordan chain, and so forth.
The Weyr form is canonical
That the Weyr form is a canonical |
https://en.wikipedia.org/wiki/Girth%20%28geometry%29 | In three-dimensional geometry, the girth of a geometric object, in a certain direction, is the perimeter of its parallel projection in that direction. For instance, the girth of a unit cube in a direction parallel to one of the three coordinate axes is four: it projects to a unit square, which has four as its perimeter.
Surfaces of constant girth
The girth of a sphere in any direction equals the circumference of its equator, or of any of its great circles. More generally,
if is a surface of constant width , then every projection of is a curve of constant width, with the same width . All curves of constant width have the same perimeter, the same value as the circumference of a circle with that width (this is Barbier's theorem). Therefore, every surface of constant width is also a surface of constant girth: its girth in all directions is the same number . Hermann Minkowski proved, conversely, that every convex surface of constant girth is also a surface of constant width.
Projection versus cross-section
For a prism or cylinder, its projection in the direction parallel to its axis is the same as its cross section, so in these cases the girth also equals the perimeter of the cross section. In some application areas such as shipbuilding this alternative meaning, the perimeter of a cross section, is taken as the definition of girth.
Application
Girth is sometimes used by postal services and delivery companies as a basis for pricing. For example, Canada Post requires that an item's length plus girth not exceed a maximum allowed value. For a rectangular box, the girth is 2 * (height + width), i.e. the perimeter of a projection or cross section perpendicular to its length.
References
Euclidean solid geometry |
https://en.wikipedia.org/wiki/Eduard%20Weyr | Eduard Weyr (June 22, 1852 – July 23, 1903) was a Czech mathematician now chiefly remembered as the discoverer of a certain canonical form for square matrices over algebraically closed fields. Weyr presented this form briefly in a paper published in 1885. He followed it up with a more elaborate treatment in a paper published in 1890. This particular canonical form has been named as the Weyr canonical form in a paper by Shapiro published in The American Mathematical Monthly in 1999. Previously, this form has been variously called as modified Jordan form, reordered Jordan form, second Jordan form, and H-form.
Weyr's father was a mathematician at a secondary school in Prague, and his older brother, Emil Weyr, was also a mathematician. Weyr studied at Prague Polytechnic and Charles-Ferdinand University in Prague. He received his doctorate from the University of Göttingen in 1873 with dissertation Über algebraische Raumcurven. After a short spell in Paris studying under Charles Hermite and Joseph Alfred Serret, he returned to Prague where he eventually became a professor at Charles-Ferdinand University. Weyr also published research in geometry, in particular projective and differential geometry. In 1893 in Chicago, his paper Sur l'équation des lignes géodésiques was read (but not by him) at the International Congress of Mathematicians held in connection with the World's Columbian Exposition.
Weyr canonical form
The image shows an example of a general Weyr matrix consisting of two blocks each of which is a basic Weyr matrix. The basic Weyr matrix in the top-left corner has the structure (4,2,1) and the other one has the structure (2,2,1,1).
References
19th-century Czech people
Mathematicians from Austria-Hungary
Czech mathematicians
Mathematicians from Prague
Charles University alumni
1852 births
1903 deaths
University of Göttingen alumni
Academic staff of Charles University
Linear algebraists |
https://en.wikipedia.org/wiki/Yuri%20Luchko | Yuri Luchko is a German professor of mathematics at the Berlin University of Applied Sciences and Technology. His 90 works were peer-reviewed and appeared in such journals as the Fractional Calculus and Applied Analysis and Journal of Mathematical Analysis and Applications, among others.
References
21st-century German mathematicians
Living people
20th-century births
Scientists from Berlin
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Claude-Curdin%20Paschoud | Claude-Curdin Paschoud (born April 3, 1994) is a Swiss professional ice hockey defenceman who currently plays for HC Davos in the Swiss National League (NL).
Career statistics
Regular season and playoffs
International
References
External links
1994 births
Living people
Sportspeople from Davos
HC Davos players
Swiss ice hockey defencemen
Ice hockey people from Graubünden |
https://en.wikipedia.org/wiki/Michael%20Struwe | Michael Struwe (born 6 October 1955 in Wuppertal) is a German mathematician who specializes in calculus of variations and nonlinear partial differential equations. He won the 2012 Cantor medal from the Deutsche Mathematiker-Vereinigung for "outstanding achievements in the field of geometric analysis, calculus of variations and nonlinear partial differential equations".
He studied mathematics at the University of Bonn, gaining his PhD in 1980 with the title Infinitely Many Solutions for Superlinear, Anticoercive Elliptic Boundary Value Problems without Oddness. He took research positions in Paris and at ETH Zürich before gaining his habilitation in Bonn in 1984. Since 1986, he has been working at ETH Zürich, initially as an assistant professor, becoming a full professor in 1993. His specialisms included nonlinear partial differential equations and calculus of variations.
He is joint editor of the journals Calculus of Variations, Commentarii Mathematici Helvetici, International Mathematical Research Notices and Mathematische Zeitschrift.
His publications include the book Variational methods (Applications to nonlinear PDE and Hamiltonian systems) (Springer-Verlag, 1990), which was praised by Jürgen Jost as "very useful" with an "impressive range of often difficult examples".
Struwe was awarded a Gauss Lecture by the German Mathematical Society in 2011. In 2012, Struwe was selected as one of the inaugural fellows of the American Mathematical Society.
Major publications
Struwe, Michael. On the evolution of harmonic maps in higher dimensions. J. Differential Geom. 28 (1988), no. 3, 485–502.
Struwe, Michael; Tarantello, Gabriella. On multivortex solutions in Chern-Simons gauge theory. Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), no. 1, 109–121.
References
External links
page at ETH
1955 births
20th-century German mathematicians
Academic staff of ETH Zurich
Living people
Fellows of the American Mathematical Society
21st-century German mathematicians |
https://en.wikipedia.org/wiki/K-trivial%20set | In mathematics, a set of natural numbers is called a K-trivial set if its initial segments viewed as binary strings are easy to describe: the prefix-free Kolmogorov complexity is as low as possible, close to that of a computable set. Solovay proved in 1975 that a set can be K-trivial without being computable.
The Schnorr–Levin theorem says that random sets have a high initial segment complexity. Thus the K-trivials are far from random. This is why these sets are studied in the field of algorithmic randomness, which is a subfield of Computability theory and related to algorithmic information theory in computer science.
At the same time, K-trivial sets are close to computable. For instance, they are all superlow, i.e. sets whose Turing jump is computable from the Halting problem, and form a Turing ideal, i.e. class of sets closed under Turing join and closed downward under Turing reduction.
Definition
Let K be the prefix-free Kolmogorov Complexity, i.e. given a string x, K(x) outputs the least length of the input string under a prefix-free universal machine. Such a machine, intuitively, represents a universal programming language with the property that no valid program can be obtained as a proper extension of another valid program. For more background of K, see e.g. Chaitin's constant.
We say a set A of the natural numbers is K-trivial via a constant b ∈ if
.
A set is K-trivial if it is K-trivial via some constant.
Brief history and development
In the early days of the development of K-triviality, attention was paid to separation of K-trivial sets and computable sets.
Chaitin in his 1976 paper mainly studied sets such that there exists b ∈ with
where C denotes the plain Kolmogorov complexity. These sets are known as C-trivial sets. Chaitin showed they coincide with the computable sets. He also showed that the K-trivials are computable in the halting problem. This class of sets is commonly known as sets in arithmetical hierarchy.
Robert M. Solovay was the first to construct a noncomputable K-trivial set, while construction of a computably enumerable such A was attempted by Calude, Coles and other unpublished constructions by Kummer of a K-trivial, and Muchnik junior of a low for K set.
Developments 1999–2008
In the context of computability theory, a cost function is a computable function
For a computable approximation of set A, such a function measures the cost c(n,s) of changing the approximation to A(n) at stage s. The first cost function construction was due to Kučera and Terwijn. They built a computably enumerable set that is low for Martin-Löf-randomness but not computable. Their cost function was adaptive, in that the definition of the cost function depends on the computable approximation of the set being built.
A cost function construction of a K-trivial computably enumerable noncomputable set first appeared in Downey et al.
We say a set A obeys a cost function c if there exists a comput |
https://en.wikipedia.org/wiki/Combinant | In the mathematical theory of probability, the combinants cn of a random variable X are defined via the combinant-generating function G(t), which is defined from the moment generating function M(z) as
which can be expressed directly in terms of a random variable X as
wherever this expectation exists.
The nth combinant can be obtained as the nth derivatives of the logarithm of combinant generating function evaluated at –1 divided by n factorial:
Important features in common with the cumulants are:
the combinants share the additivity property of the cumulants;
for infinite divisibility (probability) distributions, both sets of moments are strictly positive.
References
Google Books
Theory of probability distributions |
https://en.wikipedia.org/wiki/Glen%20Bredon | Glen Eugene Bredon (August 24, 1932 in Fresno, California – May 8, 2000, in North Fork, California) was an American mathematician who worked in the area of topology.
Education and career
Bredon received a bachelor's degree from Stanford University in 1954 and a master's degree from Harvard University in 1955. In 1958, he wrote his PhD thesis at Harvard (Some theorems on transformation groups) under the supervision of Andrew M. Gleason. Starting in 1960, he worked as a professor at the University of California, Berkeley and since 1969 at Rutgers University, until he retired in 1993, after which he moved to North Fork, California.
From 1958 to 1960 and 1966/67 he was at the Institute for Advanced Study.
The Bredon cohomology of topological spaces under action of a topological group is named after him.
In the late 1980s, he wrote the program DOS.MASTER for Apple II computers. He is the author of the programs Merlin (a macro assembler) and ProSel for Apple machines.
Personal life
In 1963, while at Berkeley, he married folk singer Anne Bredon (née Loeb), with whom he had two children, Aaron and Joelle.
Apple software
Bredon is the author of the programs Merlin (a macro assembler) and ProSel for Apple machines.
DOS.MASTER (also: DOS Master) is a program for Apple II computers which allows Apple DOS 3.3 programs to be placed on a hard drive or 3½" floppy disk and run from ProDOS. Bredon wrote it as a commercial program during the late 1980s where it experienced widespread success; it was released into the public domain by his family after the author's death.
DOS.MASTER was created as a result of Apple Computer's abandonment of the DOS 3.3 operating system and its subsequent replacement by ProDOS. Apple provided a program to copy files from DOS 3.3 volumes to new ProDOS volumes; however, programs written for DOS 3.3 did not run on ProDOS volumes. DOS.MASTER enabled a widely installed base of previously ProDOS incompatible programs written for DOS 3.3 to be run under ProDOS. DOS.MASTER took a large ProDOS partition, formatted it as a file, and then created a series of DOS 3.3 volumes within that file. The program allowed the user to create one of four DOS 3.3 volume sizes: 140 KB (the standard capacity of an Apple II 5¼" floppy disk), 160 KB, 200 KB, or 400 KB (the maximum that DOS 3.3 could address). Up to 255 of these volumes could be created on the larger ProDOS partition, space allowing, essentially simulating a very large stack of virtual floppy disk drives.
Works
Equivariant Cohomology Theories, Lecture Notes in Mathematics, Springer Verlag, 1967
Topology and Geometry, Graduate Texts in Mathematics, Springer Verlag 1993, 1996
References
External links
About Glen Bredon – In Memoriam
1932 births
2000 deaths
People from Fresno, California
Mathematicians from California
20th-century American mathematicians
Stanford University alumni
Harvard University alumni
University of California, Berkeley College of Letters and Science faculty
Rutge |
https://en.wikipedia.org/wiki/Kolmogorov%27s%20two-series%20theorem | In probability theory, Kolmogorov's two-series theorem is a result about the convergence of random series. It follows from Kolmogorov's inequality and is used in one proof of the strong law of large numbers.
Statement of the theorem
Let be independent random variables with expected values and variances , such that converges in ℝ and converges in ℝ. Then converges in ℝ almost surely.
Proof
Assume WLOG . Set , and we will see that with probability 1.
For every ,
Thus, for every and ,
While the second inequality is due to Kolmogorov's inequality.
By the assumption that converges, it follows that the last term tends to 0 when , for every arbitrary .
References
Durrett, Rick. Probability: Theory and Examples. Duxbury advanced series, Third Edition, Thomson Brooks/Cole, 2005, Section 1.8, pp. 60–69.
M. Loève, Probability theory, Princeton Univ. Press (1963) pp. Sect. 16.3
W. Feller, An introduction to probability theory and its applications, 2, Wiley (1971) pp. Sect. IX.9
Probability theorems |
https://en.wikipedia.org/wiki/Kernel%20embedding%20of%20distributions | In machine learning, the kernel embedding of distributions (also called the kernel mean or mean map) comprises a class of nonparametric methods in which a probability distribution is represented as an element of a reproducing kernel Hilbert space (RKHS). A generalization of the individual data-point feature mapping done in classical kernel methods, the embedding of distributions into infinite-dimensional feature spaces can preserve all of the statistical features of arbitrary distributions, while allowing one to compare and manipulate distributions using Hilbert space operations such as inner products, distances, projections, linear transformations, and spectral analysis. This learning framework is very general and can be applied to distributions over any space on which a sensible kernel function (measuring similarity between elements of ) may be defined. For example, various kernels have been proposed for learning from data which are: vectors in , discrete classes/categories, strings, graphs/networks, images, time series, manifolds, dynamical systems, and other structured objects. The theory behind kernel embeddings of distributions has been primarily developed by Alex Smola, Le Song , Arthur Gretton, and Bernhard Schölkopf. A review of recent works on kernel embedding of distributions can be found in.
The analysis of distributions is fundamental in machine learning and statistics, and many algorithms in these fields rely on information theoretic approaches such as entropy, mutual information, or Kullback–Leibler divergence. However, to estimate these quantities, one must first either perform density estimation, or employ sophisticated space-partitioning/bias-correction strategies which are typically infeasible for high-dimensional data. Commonly, methods for modeling complex distributions rely on parametric assumptions that may be unfounded or computationally challenging (e.g. Gaussian mixture models), while nonparametric methods like kernel density estimation (Note: the smoothing kernels in this context have a different interpretation than the kernels discussed here) or characteristic function representation (via the Fourier transform of the distribution) break down in high-dimensional settings.
Methods based on the kernel embedding of distributions sidestep these problems and also possess the following advantages:
Data may be modeled without restrictive assumptions about the form of the distributions and relationships between variables
Intermediate density estimation is not needed
Practitioners may specify the properties of a distribution most relevant for their problem (incorporating prior knowledge via choice of the kernel)
If a characteristic kernel is used, then the embedding can uniquely preserve all information about a distribution, while thanks to the kernel trick, computations on the potentially infinite-dimensional RKHS can be implemented in practice as simple Gram matrix operations
Dimensionality-independent |
https://en.wikipedia.org/wiki/Minimum%20overlap%20problem | In number theory and set theory, the minimum overlap problem is a problem proposed by Hungarian mathematician Paul Erdős in 1955.
Formal statement of the problem
Let and be two complementary subsets, a splitting of the set of natural numbers , such that both have the same cardinality, namely . Denote by the number of solutions of the equation , where is an integer varying between . is defined as:
The problem is to estimate when is sufficiently large.
History
This problem can be found amongst the problems proposed by Paul Erdős in combinatorial number theory, known by English speakers as the Minimum overlap problem. It was first formulated in the 1955 article Some remarks on number theory (in Hebrew) in Riveon Lematematica, and has become one of the classical problems described by Richard K. Guy in his book Unsolved problems in number theory.
Partial results
Since it was first formulated, there has been continuous progress made in the calculation of lower bounds and upper bounds of , with the following results:
Lower
Upper
J. K. Haugland showed that the limit of exists and that it is less than 0.385694. For his research, he was awarded a prize in a young scientists competition in 1993. In 1996, he improved the upper bound to 0.38201 using a result of Peter Swinnerton-Dyer. This has now been further improved to 0.38093. In 2022, the lower bound was shown to be at least 0.379005 by E. P. White.
The first known values of
The values of for the first 15 positive integers are the following:
It is just the Law of Small Numbers that it is
References
Additive number theory
Unsolved problems in number theory |
https://en.wikipedia.org/wiki/Differentiable%20stack | A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.
Differentiable stacks are particularly useful to handle spaces with singularities (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory.
Definition
Definition 1 (via groupoid fibrations)
Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category together with a functor to the category of differentiable manifolds such that
is a fibred category, i.e. for any object of and any arrow of there is an arrow lying over ;
for every commutative triangle in and every arrows over and over , there exists a unique arrow over making the triangle commute.
These properties ensure that, for every object in , one can define its fibre, denoted by or , as the subcategory of made up by all objects of lying over and all morphisms of lying over . By construction, is a groupoid, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of descent.
Any manifold defines its slice category , whose objects are pairs of a manifold and a smooth map ; then is a groupoid fibration which is actually also a stack. A morphism of groupoid fibrations is called a representable submersion if
for every manifold and any morphism , the fibred product is representable, i.e. it is isomorphic to (for some manifold ) as groupoid fibrations;
the induce smooth map is a submersion.
A differentiable stack is a stack together with a special kind of representable submersion (every submersion described above is asked to be surjective), for some manifold . The map is called atlas, presentation or cover of the stack .
Definition 2 (via 2-functors)
Recall that a prestack (of groupoids) on a category , also known as a 2-presheaf, is a 2-functor , where is the 2-category of (set-theoretical) groupoids, their morphisms, and the natural transformations between them. A stack is a prestack satisfying further glueing properties (analogously to the glueing properties satisfied by a sheaf). In order to state such properties precisely, one needs to define (pre)stacks on a site, i.e. a category equipped with a Grothendieck topology.
Any object defines a stack , which associated to another object the groupoid of morphisms from to . A stack is called geometric if there is an object and a morphism of stacks (often called atlas, presentation or cover of the stack ) such that
the morphism is representable, i.e. for every object in and any morphism the fibred product is isomorphic to (for some object ) as stacks;
the induces mor |
https://en.wikipedia.org/wiki/Volkenborn%20integral | In mathematics, in the field of p-adic analysis, the Volkenborn integral is a method of integration for p-adic functions.
Definition
Let : be a function from the p-adic integers taking values in the p-adic numbers. The Volkenborn integral is defined by the limit, if it exists:
More generally, if
then
This integral was defined by Arnt Volkenborn.
Examples
where is the k-th Bernoulli number.
The above four examples can be easily checked by direct use of the definition and Faulhaber's formula.
The last two examples can be formally checked by expanding in the Taylor series and integrating term-wise.
with the p-adic logarithmic function and the p-adic digamma function.
Properties
From this it follows that the Volkenborn-integral is not translation invariant.
If then
See also
P-adic distribution
References
Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen I. In: Manuscripta Mathematica. Bd. 7, Nr. 4, 1972,
Arnt Volkenborn: Ein p-adisches Integral und seine Anwendungen II. In: Manuscripta Mathematica. Bd. 12, Nr. 1, 1974,
Henri Cohen, "Number Theory", Volume II, page 276
Integrals |
https://en.wikipedia.org/wiki/Ryan%20Taylor%20%28soccer%29 | Ryan Taylor (born June 15, 1990) is an American professional soccer player.
Career statistics
References
External links
USL Profile
1990 births
Living people
People from Midlothian, Virginia
American men's soccer players
Fredericksburg Hotspur players
Richmond Kickers players
Radford Highlanders men's soccer players
Men's association football goalkeepers
Soccer players from Virginia
USL League Two players
USL Championship players |
https://en.wikipedia.org/wiki/Integral%20of%20inverse%20functions | In mathematics, integrals of inverse functions can be computed by means of a formula that expresses the antiderivatives of the inverse of a continuous and invertible function in terms of and an antiderivative of This formula was published in 1905 by Charles-Ange Laisant.
Statement of the theorem
Let and be two intervals of
Assume that is a continuous and invertible function. It follows from the intermediate value theorem that is strictly monotone. Consequently, maps intervals to intervals, so is an open map and thus a homeomorphism. Since and the inverse function are continuous, they have antiderivatives by the fundamental theorem of calculus.
Laisant proved that if is an antiderivative of then the antiderivatives of are:
where is an arbitrary real number. Note that it is not assumed that is differentiable.
In his 1905 article, Laisant gave three proofs. First, under the additional hypothesis that is differentiable, one may differentiate the above formula, which completes the proof immediately. His second proof was geometric. If and the theorem can be written:
The figure on the right is a proof without words of this formula. Laisant does not discuss the hypotheses necessary to make this proof rigorous, but this can be proved if is just assumed to be strictly monotone (but not necessarily continuous, let alone differentiable). In this case, both and are Riemann integrable and the identity follows from a bijection between lower/upper Darboux sums of and upper/lower Darboux sums of The antiderivative version of the theorem then follows from the fundamental theorem of calculus in the case when is also assumed to be continuous. Laisant's third proof uses the additional hypothesis that is differentiable. Beginning with one multiplies by and integrates both sides. The right-hand side is calculated using integration by parts to be and the formula follows.
Nevertheless, it can be shown that this theorem holds even if or is not differentiable: it suffices, for example, to use the Stieltjes integral in the previous argument. On the other hand, even though general monotonic functions are differentiable almost everywhere, the proof of the general formula does not follow, unless is absolutely continuous.
It is also possible to check that for every in the derivative of the function is equal to In other words:
To this end, it suffices to apply the mean value theorem to between and taking into account that is monotonic.
Examples
Assume that hence The formula above gives immediately
Similarly, with and
With and
History
Apparently, this theorem of integration was discovered for the first time in 1905 by Charles-Ange Laisant, who "could hardly believe that this theorem is new", and hoped its use would henceforth spread out among students and teachers. This result was published independently in 1912 by an Italian engineer, Alberto Caprilli, in an opuscule entitled "Nuove formole d'integrazione". It wa |
https://en.wikipedia.org/wiki/Timeline%20of%20women%20in%20mathematics%20in%20the%20United%20States | There is a long history of women in mathematics in the United States. All women mentioned here are American unless otherwise noted.
Timeline
19th Century
1829: The first public examination of an American girl in geometry was held.
1886: Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned from Columbia University.
20th Century
1913: Mildred Sanderson earned her PhD for a thesis that included an important theorem about modular invariants.
1927: Anna Pell-Wheeler became the first woman to present a lecture at the American Mathematical Society Colloquium.
1943: Euphemia Haynes became the first African-American woman to earn a Ph.D. in mathematics, which she earned from Catholic University of America.
1949: Gertrude Mary Cox became the first woman elected into the International Statistical Institute.
1956: Gladys West began collecting data from satellites at the Naval Surface Warfare Center Dahlgren Division. Her calculations directly impacted the development of accurate GPS systems.
1962: Mina Rees became the first woman to win the Mathematical Association of America's highest honor, the Yueh-Gin Gung and Dr. Charles Y. Hu Award for Distinguished Service to Mathematics.
1966: Mary L. Boas published Mathematical Methods in the Physical Sciences, which was still widely used in college classrooms as of 1999.
1970s
1970: Mina Rees became the first female president of the American Association for the Advancement of Science.
1971:
Mary Ellen Rudin constructed the first Dowker space.
The Association for Women in Mathematics (AWM) was founded. It is a professional society whose mission is to encourage women and girls to study and to have active careers in the mathematical sciences, and to promote equal opportunity for and the equal treatment of women and girls in the mathematical sciences. It is incorporated in the state of Massachusetts.
The American Mathematical Society established its Joint Committee on Women in the Mathematical Sciences (JCW), which later became a joint committee of multiple scholarly societies.
1973: Jean Taylor published her dissertation on "Regularity of the Singular Set of Two-Dimensional Area-Minimizing Flat Chains Modulo 3 in R3" which solved a long-standing problem about length and smoothness of soap-film triple function curves.
1974: Joan Birman published the book Braids, Links, and Mapping Class Groups. It has become a standard introduction, with many of today's researchers having learned the subject through it.
1975–1977: Marjorie Rice, who had no formal training in mathematics beyond high school, discovered three new types of tessellating pentagons and more than sixty distinct tessellations by pentagons.
1975: Julia Robinson became the first female mathematician elected to the National Academy of Sciences.
1979:
Dorothy Lewis Bernstein became the first female president of the Mathematical Association of America.
Mary Ellen Rudin became the first woma |
https://en.wikipedia.org/wiki/List%20of%20Ravan%20Baku%20FK%20records%20and%20statistics | Ravan Baku is an Azerbaijani professional football club based in Baku.
This list encompasses the major records set by the club and their players in the Azerbaijan Premier League. The player records section includes details of the club's goalscorers and those who have made more than 50 appearances in first-team competitions.
Player
Most appearances
Players played over 50 competitive, professional matches only. Appearances as substitute (goals in parentheses) included in total.
Overall scorers
Competitive, professional matches only, appearances including substitutes appear in brackets.
Team
Record wins
Record win: 6–2 v Turan Tovuz, 2012-13 Azerbaijan Premier League, 30 March 2013
Record League win: 6–2 v Turan Tovuz, 2012-13 Azerbaijan Premier League, 30 March 2013
Record Azerbaijan Cup win: 5–2 v Qaradağ, 4 December 2013
Record away win: 6–2 v Turan Tovuz, 2012-13 Azerbaijan Premier League, 30 March 2013
Record home win 5–1 v Kəpəz, 2011-12 Azerbaijan Premier League, 28 April 2012
Record defeats
Record defeat: 0–5
v Baku, 2013-14 Azerbaijan Premier League, 18 August 2013
Record League defeat: 0–4
v Baku, 2013-14 Azerbaijan Premier League, 18 August 2013
Record away defeat: 0–5
v Baku, 2013-14 Azerbaijan Premier League, 18 August 2013
Record Azerbaijan Cup defeat: 0–5
v Khazar Lankaran, Quarterfinals 2nd leg, 7 March 2013
Record home defeat: 3–5
v AZAL, Azerbaijan Premier League, 14 May 2013
Goals
Most Premier League goals scored in a season: 46 – 2012–13
Fewest League goals scored in a season: 30 – 2013-14
Most League goals conceded in a season: 53 – 2012–13
Fewest League goals conceded in a season: 22 – 2013-14
Points
Most points in a season:
41 in 32 matches, Azerbaijan Premier League, 2011–12
Fewest points in a season:
22 in 38 matches, Azerbaijan Premier League, 2013–14
International representatives
Current Ravan players
Former Ravan players
References
Ravan Baku FK
Ravan Baku FK |
https://en.wikipedia.org/wiki/Cyclic%20subspace | In mathematics, in linear algebra and functional analysis, a cyclic subspace is a certain special subspace of a vector space associated with a vector in the vector space and a linear transformation of the vector space. The cyclic subspace associated with a vector v in a vector space V and a linear transformation T of V is called the T-cyclic subspace generated by v. The concept of a cyclic subspace is a basic component in the formulation of the cyclic decomposition theorem in linear algebra.
Definition
Let be a linear transformation of a vector space and let be a vector in . The -cyclic subspace of generated by , denoted , is the subspace of generated by the set of vectors . In the case when is a topological vector space, is called a cyclic vector for if is dense in . For the particular case of finite-dimensional spaces, this is equivalent to saying that is the whole space .
There is another equivalent definition of cyclic spaces. Let be a linear transformation of a topological vector space over a field and be a vector in . The set of all vectors of the form , where is a polynomial in the ring of all polynomials in over , is the -cyclic subspace generated by .
The subspace is an invariant subspace for , in the sense that .
Examples
For any vector space and any linear operator on , the -cyclic subspace generated by the zero vector is the zero-subspace of .
If is the identity operator then every -cyclic subspace is one-dimensional.
is one-dimensional if and only if is a characteristic vector (eigenvector) of .
Let be the two-dimensional vector space and let be the linear operator on represented by the matrix relative to the standard ordered basis of . Let . Then . Therefore and so . Thus is a cyclic vector for .
Companion matrix
Let be a linear transformation of a -dimensional vector space over a field and be a cyclic vector for . Then the vectors
form an ordered basis for . Let the characteristic polynomial for be
.
Then
Therefore, relative to the ordered basis , the operator is represented by the matrix
This matrix is called the companion matrix of the polynomial .
See also
Companion matrix
Krylov subspace
External links
PlanetMath: cyclic subspace
References
Linear algebra |
https://en.wikipedia.org/wiki/Zden%C4%9Bk%20%C5%A0vestka | Zdeněk Švestka (30 September 1925 – 2 March 2013) was a Czech astronomer. For several decades he was the world's leading expert on solar flares. He studied mathematics and physics at Charles University, Prague, until graduating in 1948. Together with Cornelis de Jager, he was the co-founder and editor of the journal Solar Physics. For 38 years, from the establishment of the journal in 1967 until his retirement in 2005, he handled all papers on solar flares, while De Jager took care of everything else. The minor planet 17805 Švestka was named after him.
References
Czech astronomers
2013 deaths
1925 births
Scientists from Prague
Charles University alumni |
https://en.wikipedia.org/wiki/Gon%C3%A7alo%20Abecasis | Gonçalo Rocha Abecasis (born 1976) is a Portuguese American biomedical researcher at the University of Michigan and was chair of the Department of Biostatistics in the School of Public Health. He leads a group at the Center for Statistical Genetics in the Department of Biostatistics, where he is also the Felix E. Moore Collegiate Professor of Biostatistics and director of the Michigan Genomic Initiative. His group develops statistical tools to analyze the genetics of human disease.
Education and early life
Abecasis is the oldest of 12 children in his family, born in Moura to José Manuel and Maria Teresa. He grew up in Portugal and Macau (then Portuguese Macau). After learning computer programming in a high school club, he received a Bachelor of Science degree in Genetics from the University of Leeds in 1997, working with Mary Anne Shaw, and a D. Phil in human genetics from the University of Oxford in 2001, working with William Cookson and Lon Cardon.
Research
Abecasis works on statistical and computational approaches to human genetic disease, including psoriasis and cardiovascular diseases. His group develops tools to analyze and visualize biomedical data, often using C++. He first applied his programming knowledge during his Ph.D. studies to develop tools to analyze the data on asthma susceptibility that his project was generating. He continued as a biostatistician when he moved to Michigan in 2001, where he was recruited and mentored by Michael Boehnke. He is a proponent of data sharing. His work has included the 1000 Genomes Project and a collaboration with Oxford researchers.
Awards and honours
Abecasis won an Excellence in Research Award from the University of Michigan School of Public Health in 2008, and he became a professor at the University of Michigan in 2009. He won the 2013 Overton Prize from the International Society for Computational Biology, giving a keynote speech at the ISMB, and was named a Pew Scholar by the Pew Charitable Trusts in 2005. His work was cited in 2010 by US Vice-President Joe Biden in a speech on the importance of biomedical research regarding the Recovery Act Innovation Report.
His published work is influential - as of 2018, he had an h-index of 145 and 144,018 citations. Thomson Reuters noted that he had 10 'hot papers' (in the top 0.1% by citations) in 2009. He was also noted for his 'hot papers' in 2011 and 2012.
In October 2014, he was awarded the Curt Stern Award in Human Genetics by the American Society of Human Genetics (jointly with Mark J. Daly).
He was elected to the Institute of Medicine in October 2014.
Personal life
Goncalo is married to University of Michigan scientist Cristen Willer. They have five children.
References
Portuguese biologists
University of Michigan faculty
American bioinformaticians
Statistical geneticists
Genetic epidemiologists
Alumni of the University of Leeds
Alumni of the University of Oxford
Portuguese emigrants to the United States
1976 births
Living people
People fr |
https://en.wikipedia.org/wiki/Conditional%20probability%20table | In statistics, the conditional probability table (CPT) is defined for a set of discrete and mutually dependent random variables to display conditional probabilities of a single variable with respect to the others (i.e., the probability of each possible value of one variable if we know the values taken on by the other variables). For example, assume there are three random variables where each has states. Then, the conditional probability table of provides the conditional probability values – where the vertical bar means “given the values of” – for each of the K possible values of the variable and for each possible combination of values of This table has cells. In general, for variables with states for each variable the CPT for any one of them has the number of cells equal to the product
A conditional probability table can be put into matrix form. As an example with only two variables, the values of with k and j ranging over K values, create a K×K matrix. This matrix is a stochastic matrix since the columns sum to 1; i.e. for all j. For example, suppose that two binary variables x and y have the joint probability distribution given in this table:
Each of the four central cells shows the probability of a particular combination of x and y values. The first column sum is the probability that x =0 and y equals any of the values it can have – that is, the column sum 6/9 is the marginal probability that x=0. If we want to find the probability that y=0 given that x=0, we compute the fraction of the probabilities in the x=0 column that have the value y=0, which is 4/9 ÷ 6/9 = 4/6. Likewise, in the same column we find that the probability that y=1 given that x=0 is 2/9 ÷ 6/9 = 2/6. In the same way, we can also find the conditional probabilities for y equalling 0 or 1 given that x=1. Combining these pieces of information gives us this table of conditional probabilities for y:
With more than one conditioning variable, the table would still have one row for each potential value of the variable whose conditional probabilities are to be given, and there would be one column for each possible combination of values of the conditioning variables.
Moreover, the number of columns in the table could be substantially expanded to display the probabilities of the variable of interest conditional on specific values of only some, rather than all, of the other variables.
References
Conditional probability |
https://en.wikipedia.org/wiki/Timeline%20of%20women%20in%20mathematics | This is a timeline of women in mathematics.
Timeline
Early Common Era
Before 350: Pandrosion, a Greek Alexandrine mathematician known for an approximate solution to doubling the cube and a simplified exact solution to the construction of the geometric mean.
c. 350–370 until 415: The lifetime of Hypatia, a Greek Alexandrine Neoplatonist philosopher in Egypt who was the first well-documented woman in mathematics.
18th Century
1748: Italian mathematician Maria Agnesi published the first book discussing both differential and integral calculus, called Instituzioni analitiche ad uso della gioventù italiana.
1759: French mathematician Émilie du Châtelet's translation and commentary on Isaac Newton’s work Principia Mathematica was published posthumously; it is still considered the standard French translation.
c. 1787 – 1797: Self-taught Chinese astronomer Wang Zhenyi published at least twelve books and multiple articles on astronomy and mathematics.
19th Century
1827: French mathematician Sophie Germain saw her theorem, known as Germain's Theorem, published in a footnote of a book by the mathematician Adrien-Marie Legendre. In this theorem Germain proved that if x, y, and z are integers and if x5 + y5 = z5 then either x, y, or z must be divisible by 5. Germain's theorem was a major step toward proving Fermat's Last Theorem for the case where n equals 5.
1829: The first public examination of an American girl in geometry was held.
1858: Florence Nightengale became the first female member of the Royal Statistical Society.
1873: Sarah Woodhead of Britain became the first woman to take, and to pass, the Cambridge Mathematical Tripos Exam.
1874: Russian mathematician Sofia Kovalevskaya became the first woman in modern Europe to gain a doctorate in mathematics, which she earned from the University of Göttingen in Germany.
1880: Charlotte Angas Scott of Britain obtained special permission to take the Cambridge Mathematical Tripos Exam, as women were not normally allowed to sit for the exam. She came eighth on the Tripos of all students taking them, but due to her sex, the title of "eighth wrangler," a high honour, went officially to a male student. At the ceremony, however, after the seventh wrangler had been announced, all the students in the audience shouted her name. Because she could not attend the award ceremony, Scott celebrated her accomplishment at Girton College where there were cheers and clapping at dinner, and a special evening ceremony where the students sang "See the Conquering Hero Comes", and she received an ode written by a staff member, and was crowned with laurels.
1885: Charlotte Angas Scott became the first British woman to receive a doctorate in mathematics, which she received from the University of London.
1886: Winifred Edgerton Merrill became the first American woman to earn a PhD in mathematics, which she earned from Columbia University.
1888: The Kovalevskaya top, one of a brief list of known examples of integrable |
https://en.wikipedia.org/wiki/Q-construction | In algebra, Quillen's Q-construction associates to an exact category (e.g., an abelian category) an algebraic K-theory. More precisely, given an exact category C, the construction creates a topological space so that is the Grothendieck group of C and, when C is the category of finitely generated projective modules over a ring R, for , is the i-th K-group of R in the classical sense. (The notation "+" is meant to suggest the construction adds more to the classifying space BC.) One puts
and call it the i-th K-group of C. Similarly, the i-th K-group of C with coefficients in a group G is defined as the homotopy group with coefficients:
.
The construction is widely applicable and is used to define an algebraic K-theory in a non-classical context. For example, one can define equivariant algebraic K-theory as of of the category of equivariant sheaves on a scheme.
Waldhausen's S-construction generalizes the Q-construction in a stable sense; in fact, the former, which uses a more general Waldhausen category, produces a spectrum instead of a space. Grayson's binary complex also gives a construction of algebraic K-theory for exact categories. See also module spectrum#K-theory for a K-theory of a ring spectrum.
The construction
Let C be an exact category; i.e., an additive full subcategory of an abelian category that is closed under extension. If there is an exact sequence in C, then the arrow from M′ is called an admissible mono and the arrow from M is called an admissible epi.
Let QC be the category whose objects are the same as those of C and morphisms from X to Y are isomorphism classes of diagrams such that the first arrow is an admissible epi and the second admissible mono and two diagrams are isomorphic if they differ only at the middle and there is an isomorphism between them. The composition of morphisms is given by pullback.
Define a topological space by where is a loop space functor and is the classifying space of the category QC (geometric realization of the nerve). As it turns out, it is uniquely defined up to homotopy equivalence (so the notation is justified.)
Operations
Every ring homomorphism induces and thus where is the category of finitely generated projective modules over R. One can easily show this map (called transfer) agrees with one defined in Milnor's Introduction to algebraic K-theory. The construction is also compatible with the suspension of a ring (cf. Grayson).
Comparison with the classical K-theory of a ring
A theorem of Daniel Quillen states that, when C is the category of finitely generated projective modules over a ring R, is the i-th K-group of R in the classical sense for . The usual proof of the theorem (cf. ) relies on an intermediate homotopy equivalence. If S is a symmetric monoidal category in which every morphism is an isomorphism, one constructs (cf. Grayson) the category that generalizes the Grothendieck group construction of a monoid. Let C be an exact category in which every exact se |
https://en.wikipedia.org/wiki/Spectral%20invariants | In symplectic geometry, the spectral invariants are invariants defined for the group of Hamiltonian diffeomorphisms of a symplectic manifold, which is closed related to Floer theory and Hofer geometry.
Arnold conjecture and Hamiltonian Floer homology
If (M, ω) is a symplectic manifold, then a smooth vector field Y on M is a Hamiltonian vector field if the contraction ω(Y, ·) is an exact 1-form (i.e., the differential of a Hamiltonian function H). A Hamiltonian diffeomorphism of a symplectic manifold (M, ω) is a diffeomorphism Φ of M which is the integral of a smooth path of Hamiltonian vector fields Yt. Vladimir Arnold conjectured that the number of fixed points of a generic Hamiltonian diffeomorphism of a compact symplectic manifold (M, ω) should be bounded from below by some topological constant of M, which is analogous to the Morse inequality. This so-called Arnold conjecture triggered the invention of Hamiltonian Floer homology by Andreas Floer in the 1980s.
Floer's definition adopted Witten's point of view on Morse theory. He considered spaces of contractible loops of M and defined an action functional AH associated to the family of Hamiltonian functions, so that the fixed points of the Hamiltonian diffeomorphism correspond to the critical points of the action functional. Constructing a chain complex similar to the Morse–Smale–Witten complex in Morse theory, Floer managed to define a homology group, which he also showed to be isomorphic to the ordinary homology groups of the manifold M.
The isomorphism between the Floer homology group HF(M) and the ordinary homology groups H(M) is canonical. Therefore, for any "good" Hamiltonian path Ht, a homology class α of M can be represented by a cycle in the Floer chain complex, formally a linear combination
where ai are coefficients in some ring and xi are fixed points of the corresponding Hamiltonian diffeomorphism. Formally, the spectral invariants can be defined by the min-max value
Here the maximum is taken over all the values of the action functional AH on the fixed points appeared in the linear combination of αH, and the minimum is taken over all Floer cycles that represent the class α.
Symplectic geometry |
https://en.wikipedia.org/wiki/Torus%20action | In algebraic geometry, a torus action on an algebraic variety is a group action of an algebraic torus on the variety. A variety equipped with an action of a torus T is called a T-variety. In differential geometry, one considers an action of a real or complex torus on a manifold (or an orbifold).
A normal algebraic variety with a torus acting on it in such a way that there is a dense orbit is called a toric variety (for example, orbit closures that are normal are toric varieties).
Linear action of a torus
A linear action of a torus can be simultaneously diagonalized, after extending the base field if necessary: if a torus T is acting on a finite-dimensional vector space V, then there is a direct sum decomposition:
where
is a group homomorphism, a character of T.
, T-invariant subspace called the weight subspace of weight .
The decomposition exists because the linear action determines (and is determined by) a linear representation and then consists of commuting diagonalizable linear transformations, upon extending the base field.
If V does not have finite dimension, the existence of such a decomposition is tricky but one easy case when decomposition is possible is when V is a union of finite-dimensional representations ( is called rational; see below for an example). Alternatively, one uses functional analysis; for example, uses a Hilbert-space direct sum.
Example: Let be a polynomial ring over an infinite field k. Let act on it as algebra automorphisms by: for
where
= integers.
Then each is a T-weight vector and so a monomial is a T-weight vector of weight . Hence,
Note if for all i, then this is the usual decomposition of the polynomial ring into homogeneous components.
Białynicki-Birula decomposition
The Białynicki-Birula decomposition says that a smooth algebraic T-variety admits a T-stable cellular decomposition.
It is often described as algebraic Morse theory.
See also
Sumihiro's theorem
GKM variety
Equivariant cohomology
monomial ideal
References
A. Bialynicki-Birula, "Some Theorems on Actions of Algebraic Groups," Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480–497
M. Brion, C. Procesi, Action d'un tore dans une variété projective, in Operator algebras, unitary representations, and invariant theory (Paris 1989), Prog. in Math. 92 (1990), 509–539.
Algebraic geometry
Algebraic groups |
https://en.wikipedia.org/wiki/Abdul%20Qayum%20Wardak | Abdul Qayum Wardak (c. 1923–1999) was a politician from Afghanistan. Obtained B.S. degree in mathematics, University of Illinois, 1952; and M.A. degree in Nuclear Physics, Georgetown University, 1954. Returned to Afghanistan, 1955. Graduate studies in the Soviet Union. Attended School of Nuclear Science, Lemont, Illinois, 1957. Physics Teacher, Kabul Military Academy, 1955. Member, Faculty of Science, Kabul University, 1960. Dean of Science Faculty, Kabul University, 1972. Minister of Mines and Industries, 1973–75. Minister of Education, 1974. Minister of Tribal Affaires, 1976. Married to Masuma Esmati Wardak (former Member of Parliament and Teacher at Pashto Academy).
References
External links
Famous Wardaks
Pashtun people
1999 deaths
Year of birth uncertain
1920s births |
https://en.wikipedia.org/wiki/Change%20of%20rings | In algebra, a change of rings is an operation of changing a coefficient ring to another.
Constructions
Given a ring homomorphism , there are three ways to change the coefficient ring of a module; namely, for a right R-module M and a right S-module N, one can form
, the induced module, formed by extension of scalars,
, the coinduced module, formed by co-extension of scalars, and
, formed by restriction of scalars.
They are related as adjoint functors:
and
This is related to Shapiro's lemma.
Operations
Restriction of scalars
Throughout this section, let and be two rings (they may or may not be commutative, or contain an identity), and let be a homomorphism. Restriction of scalars changes S-modules into R-modules. In algebraic geometry, the term "restriction of scalars" is often used as a synonym for Weil restriction.
Definition
Suppose that is a module over . Then it can be regarded as a module over where the action of is given via
where denotes the action defined by the -module structure on .
Interpretation as a functor
Restriction of scalars can be viewed as a functor from -modules to -modules. An -homomorphism automatically becomes an -homomorphism between the restrictions of and . Indeed, if and , then
.
As a functor, restriction of scalars is the right adjoint of the extension of scalars functor.
If is the ring of integers, then this is just the forgetful functor from modules to abelian groups.
Extension of scalars
Extension of scalars changes R-modules into S-modules.
Definition
Let be a homomorphism between two rings, and let be a module over . Consider the tensor product , where is regarded as a left -module via . Since is also a right module over itself, and the two actions commute, that is for , (in a more formal language, is a -bimodule), inherits a right action of . It is given by for , . This module is said to be obtained from through extension of scalars.
Informally, extension of scalars is "the tensor product of a ring and a module"; more formally, it is a special case of a tensor product of a bimodule and a module – the tensor product of an R-module with an -bimodule is an S-module.
Examples
One of the simplest examples is complexification, which is extension of scalars from the real numbers to the complex numbers. More generally, given any field extension K < L, one can extend scalars from K to L. In the language of fields, a module over a field is called a vector space, and thus extension of scalars converts a vector space over K to a vector space over L. This can also be done for division algebras, as is done in quaternionification (extension from the reals to the quaternions).
More generally, given a homomorphism from a field or commutative ring R to a ring S, the ring S can be thought of as an associative algebra over R, and thus when one extends scalars on an R-module, the resulting module can be thought of alternatively as an S-module, or as an R-module with an algebra representa |
https://en.wikipedia.org/wiki/Bani%20K.%20Mallick | Bani K. Mallick is a Distinguished Professor and Susan M. Arseven `75 Chair in Data Science and Computational Statistics in the Department of Statistics at Texas A&M University in College Station. He is the Director of the Center for Statistical Bioinformatics. Mallick is well known for his contribution to the theory and practice of Bayesian semiparametric methods and uncertainty quantification. Mallick is an elected fellow of American Association for the Advancement of Science, American Statistical Association, Institute of Mathematical Statistics, International Statistical Institute and the Royal Statistical Society. He received the Distinguished research award from Texas A&M University and the Young Researcher award from the International Indian Statistical Association.
Mallick's areas of research include semiparametric classification and regression, hierarchical spatial modeling, inverse problem, uncertainty quantification and Bioinformatics. He is renowned for his ability to do major collaborative research with scientists from myriad fields beyond his own, including nuclear engineering, petroleum engineering, industrial engineering, traffic mapping. He has coauthored or co-edited six books and more than 200 research publications.
Mallick earned his undergraduate from the Presidency University, Kolkata, MS from the Calcutta University and Ph.D. from the University of Connecticut.
Bibliography
Dey, D., Ghosh, S. and Mallick, B. (1999) Bayesian Generalized Linear Model, Marcel Dekker
Denison, D., Holmes, C., Mallick, B. and Smith, AFM (2002) Bayesian nonlinear Classification and Regression, Wiley, London
Denison, D., Hanson, M., Holmes, C., Mallick, B. and Yu, B. (2003) Nonlinear Classification and Regression, Springer Verlag, New York
Mallick, B., Gold, D., and Baladandanayak, V. (2010) Bayesian analysis of gene expression data, Wiley International.
Biegler, L., Ghattas, O., Keyes, D., Mallick, B., Tenorio, L. and Wilcox, K. (2011) Large scale inverse problems and quantification of Uncertainty, Wiley, International
Dey, D., Ghosh, S. and Mallick, B. (2011) Bayesian Modeling issues in Bioinformatics and Biostatistics, chapman and hall.
References
Living people
Indian statisticians
Texas A&M University faculty
University of Connecticut alumni
Fellows of the American Association for the Advancement of Science
Fellows of the American Statistical Association
Fellows of the Institute of Mathematical Statistics
Fellows of the Royal Statistical Society
Elected Members of the International Statistical Institute
Scientists from Kolkata
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/William%20T.%20Trotter | William Thomas Trotter Jr. is an American mathematician, who is on the faculty of the Department of Mathematics at the Georgia Institute of Technology. His main expertise is partially ordered sets, but he has also done significant work in other areas of combinatorics, such as the Szemerédi–Trotter theorem and Chvátal-Rödl-Szemerédi-Trotter theorem.
Trotter is the author of the book Combinatorics and partially ordered sets: dimension theory (Johns Hopkins University Press, 1992). With Mitchel Keller, he is also the author of a self-published textbook, Applied Combinatorics (2017).
References
External links
William T. (Tom) Trotter
Living people
20th-century American mathematicians
21st-century American mathematicians
Georgia Tech faculty
Year of birth missing (living people) |
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