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https://en.wikipedia.org/wiki/Laves%20graph | In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.
named this graph after Fritz Laves, who first wrote about it as a crystal structure in 1932. It has also been called the K4 crystal, (10,3)-a network, diamond twin, triamond, and the srs net. The regions of space nearest each vertex of the graph are congruent 17-sided polyhedra that tile space. Its edges lie on diagonals of the regular skew polyhedron, a surface with six squares meeting at each integer point of space.
Several crystalline chemicals have known or predicted structures in the form of the Laves graph. Thickening the edges of the Laves graph to cylinders produces a related minimal surface, the gyroid, which appears physically in certain soap film structures and in the wings of butterflies.
Constructions
From the integer grid
As describes, the vertices of the Laves graph can be defined by selecting one out of every eight points in the three-dimensional integer lattice, and forming their nearest neighbor graph. Specifically, one chooses the points
and all the other points formed by adding multiples of four to these coordinates. The edges of the Laves graph connect pairs of points whose Euclidean distance from each other is the square root of two, , as the points of each pair differ by one unit in two coordinates, and are the same in the third coordinate. The edges meet at 120° angles at each vertex, in a flat plane. All pairs of vertices that are non-adjacent are farther apart, at a distance of at least from each other. The edges of the resulting geometric graph are diagonals of a subset of the faces of the regular skew polyhedron with six square faces per vertex, so the Laves graph is embedded in this skew polyhedron.
It is possible to choose a larger set of one out of every four points of the integer lattice, so that the graph of distance- pairs of this larger set forms two mirror-image copies of the Laves graph, disconnected from each other, with all other pairs of points farther than apart.
As a covering graph
As an abstract graph, the Laves graph can be constructed as the maximal abelian covering graph of the complete graph . Being an abelian covering graph of means that the vertices of the Laves graph can be four-colored such that each vertex has neighbors of the other three colors and so that there are co |
https://en.wikipedia.org/wiki/Chessboard%20detection | Chessboards arise frequently in computer vision theory and practice because their highly structured geometry is well-suited for algorithmic detection and processing. The appearance of chessboards in computer vision can be divided into two main areas: camera calibration and feature extraction. This article provides a unified discussion of the role that chessboards play in the canonical methods from these two areas, including references to the seminal literature, examples, and pointers to software implementations.
Chessboard camera calibration
A classical problem in computer vision is three-dimensional (3D) reconstruction, where one seeks to infer 3D structure about a scene from two-dimensional (2D) images of it. Practical cameras are complex devices, and photogrammetry is needed to model the relationship between image sensor measurements and the 3D world. In the standard pinhole camera model, one models the relationship between world coordinates and image (pixel) coordinates via the perspective transformation
where is the projective space of dimension .
In this setting, camera calibration is the process of estimating the parameters of the matrix of the perspective model. Camera calibration is an important step in the computer vision pipeline because many subsequent algorithms require knowledge of camera parameters as input. Chessboards are often used during camera calibration because they are simple to construct, and their planar grid structure defines many natural interest points in an image. The following two methods are classic calibration techniques that often employ chessboards.
Direct linear transformation
Direct linear transformation (DLT) calibration uses correspondences between world points and camera image points to estimate camera parameters. In particular, DLT calibration exploits the fact that the perspective pinhole camera model defines a set of similarity relations that can be solved via the direct linear transformation algorithm. To employ this approach, one requires accurate coordinates of a non-degenerate set of points in 3D space. A common way to achieve this is to construct a camera calibration rig (example below) built from three mutually perpendicular chessboards. Since the corners of each square are equidistant, it is straightforward to compute the 3D coordinates of each corner given the width of each square. The advantage of DLT calibration is its simplicity; arbitrary cameras can be calibrated by solving a single homogeneous linear system. However, the practical use of DLT calibration is limited by the necessity of a 3D calibration rig and the fact that extremely accurate 3D coordinates are required to avoid numerical instability.
Multiplane calibration
Multiplane calibration is a variant of camera auto-calibration that allows one to compute the parameters of a camera from two or more views of a planar surface. The seminal work in multiplane calibration is due to Zhang. Zhang's method calibrates cameras by so |
https://en.wikipedia.org/wiki/C%27eria%20Ricketts | C'eria Ricketts is a basketball player. She was selected in the 2012 WNBA draft by the Phoenix Mercury, the 24th pick. She played basketball at the University of Arkansas.
Arkansas statistics
Source
References
Year of birth missing (living people)
Living people
American women's basketball players
Arkansas Razorbacks women's basketball players
Basketball players from Louisville, Kentucky
Phoenix Mercury draft picks |
https://en.wikipedia.org/wiki/Janet%20L.%20Norwood | Janet Lippe Norwood (December 11, 1923 – March 27, 2015) was an American statistician and the first female Commissioner of the U.S. Bureau of Labor Statistics (BLS) when she was appointed in 1979 by President Jimmy Carter. She was reappointed twice by President Reagan. She left the Bureau in 1991 and joined the Urban Institute as a Senior Fellow, a position she held until 1999. She was also appointed as the Chair of the Advisory Council on Unemployment Compensation, first by President George H. W. Bush in 1993 and then re-elected by President Bill Clinton. She stepped down from that position in 1996. She received numerous awards including several honorary doctorate degrees from academic institutions, including Harvard University.
Biography
Born in Newark, New Jersey, Janet Norwood grew up in Irvington, and graduated from the New Jersey College for Women (now Douglass College) of Rutgers University. She then earned her doctorate at The Fletcher School of Law and Diplomacy at Tufts University.
After she taught a year of political science at Wellesley College, she and her husband moved to Washington, where he entered Government service and became a U.S. Foreign Service Officer. In the late 1950s and early 1960s, they lived in Luxembourg and Brussels, where she and their two children accompanied him on assignment to the U.S. Mission to the European Communities.
Norwood started at the U.S. Bureau of Labor Statistics as a part-time junior economist in the early 1980s. She rose to head the agency for thirteen years, confirmed by the U.S. Senate for four-year appointments, initially by President Jimmy Carter and twice by President Ronald Reagan. She took on management of the agency soon after the Nixon White House ordered the Bureau to cease holding press conferences on the occasion of the monthly release of employment and unemployment data. Immediately following that order, the Congressional Joint Economic Committee decided to restore the public airing of the data by holding public hearings at which it called the Bureau’s head to testify. Norwood developed a reputation for what the Committee cited, at the time of her retirement from Government service in 1991 and completion of 137 appearances over 13 year before it, her “integrity, professionalism, and impartiality.”
Norwood helped bring recognition to female presence and leadership, in a context where she was frequently the only woman at government agency and professional association meetings. Among the first group of women to be admitted to the Cosmos Club, in Washington, D.C., she became its first female president in 1995. When asked, especially by young women, for guidance about career development, she advised them to have a supportive husband. She had married at the end of her sophomore year of college, when her husband was a private in a World War II Army college training program.
During both her service as Commissioner of Labor Statistics or following government retirement, she served as he |
https://en.wikipedia.org/wiki/Flapit | Flapit is a split-flap display that reveals real-time social media statistics such as Twitter followers or Yelp ratings. The product is designed to show off a bricks-and-mortar company's online community and increase its online presence by letting offline customers interact with the connected counter.
The idea came from a product launched by the retailer C&A called the Fashion Like.
The device can be customised via a web app and API to display any promotional messages, internal stats or discounts. It has 7 digits including numbers, letters and currency symbols
Special messages such as Thank You or Like Us can be displayed on the first flap and are translated into Italian, German, French, Chinese, Japanese, Russian, Portuguese, Spanish and English.
The Flapit counter was officially presented to the press at the CES Las Vegas 2015 and received favorable reviews from major specialised press
References
2014 inventions
Social media |
https://en.wikipedia.org/wiki/Calvo%20%28staggered%29%20contracts | A Calvo contract is the name given in macroeconomics to the pricing model that when a firm sets a nominal price there is a constant probability that a firm might be able to reset its price which is independent of the time since the price was last reset. The model was first put forward by Guillermo Calvo in his 1983 article "Staggered Prices in a Utility-Maximizing Framework". The original article was written in a continuous time mathematical framework, but nowadays is mostly used in its discrete time version. The Calvo model is the most common way to model nominal rigidity in new Keynesian DSGE macroeconomic models.
The Calvo model of pricing
We can define the probability that the firm can reset its price in any one period as h (the hazard rate), or equivalently the probability (1-h) that the price will remain unchanged in that period (the survival rate). The probability h is sometimes called the "Calvo probability" in this context. In the Calvo model the crucial feature is that the price-setter does not know how long the nominal price will remain in place. The probability of the current price lasting for exactly i periods more is
The probability of surviving i subsequent periods thus follows a geometric distribution, with the expected duration of the nominal price from when it is first set is . For example, if the Calvo probability h is 0.25 per period, the expected duration is 4 periods. Since the Calvo probability is constant and does not depend on how long it has been since the price was set, the probability that it will survive i more periods is given by exactly the same geometric distribution for all . Thus if h = 0.25, then however old the price is, it is expected to last another 4 periods.
Calvo pricing and nominal rigidity
With the Calvo model the response of prices to a shock is spread out over time. Suppose a shock hits the economy at time t. A proportion h of prices can respond immediately and the rest (1-h) remain fixed. The next period, there will still be who have remained fixed and not responded to the shock. i periods after the shock this which have shrunk to . After any finite time, there will still be some proportion of prices that have not responded and remained fixed. This contrasts with the Taylor model, where there is a fixed length for contracts - for example 4 periods. After 4 periods, firms will have reset their price.
The Calvo pricing model played a key role in the derivation of the New Keynesian Phillips curve by John Roberts in 1995, and since been used in New Keynesian DSGE models.
where
.
The current expectations of next period's inflation are incorporated as . The coefficient captures the responsiveness of current inflation to current output. The New Keynesian Phillips curve reflects the fact that price-setting is forward looking, and what influences current inflation is not only the level of current demand (as represented by output) but also expected future inflation.
There are differ |
https://en.wikipedia.org/wiki/Modulus%20of%20smoothness | In mathematics, moduli of smoothness are used to quantitatively measure smoothness of functions. Moduli of smoothness generalise modulus of continuity and are used in approximation theory and numerical analysis to estimate errors of approximation by polynomials and splines.
Moduli of smoothness
The modulus of smoothness of order
of a function is the function defined by
and
where the finite difference (n-th order forward difference) is defined as
Properties
1.
2. is non-decreasing on
3. is continuous on
4. For we have:
5. for
6. For let denote the space of continuous function on that have -st absolutely continuous derivative on and
If then
where
Applications
Moduli of smoothness can be used to prove estimates on the error of approximation. Due to property (6), moduli of smoothness provide more general estimates than the estimates in terms of derivatives.
For example, moduli of smoothness are used in Whitney inequality to estimate the error of local polynomial approximation. Another application is given by the following more general version of Jackson inequality:
For every natural number , if is -periodic continuous function, there exists a trigonometric polynomial of degree such that
where the constant depends on
References
Approximation theory
Numerical analysis |
https://en.wikipedia.org/wiki/Milena%20Baji%C4%87 | Milena Bajić (; born 13 November 1996) is Montenegrin college basketball player for the New Jersey Institute of Technology.
Memphis and NJIT statistics
Source
Table Ratios and Totals
Professional career
Personal life
Milena is daughter of Darko and Gordana Bajić, business finance major.
References
External links
NJIT Highlanders bio
Profile at FIBA Europe
Profile at eurobasket.com
1996 births
Living people
Centers (basketball)
Memphis Tigers women's basketball players
Montenegrin expatriate basketball people in the United States
Montenegrin women's basketball players
Power forwards (basketball)
Basketball players from Podgorica |
https://en.wikipedia.org/wiki/Exponential%20map%20%28Lie%20theory%29 | In the theory of Lie groups, the exponential map is a map from the Lie algebra of a Lie group to the group, which allows one to recapture the local group structure from the Lie algebra. The existence of the exponential map is one of the primary reasons that Lie algebras are a useful tool for studying Lie groups.
The ordinary exponential function of mathematical analysis is a special case of the exponential map when is the multiplicative group of positive real numbers (whose Lie algebra is the additive group of all real numbers). The exponential map of a Lie group satisfies many properties analogous to those of the ordinary exponential function, however, it also differs in many important respects.
Definitions
Let be a Lie group and be its Lie algebra (thought of as the tangent space to the identity element of ). The exponential map is a map
which can be defined in several different ways. The typical modern definition is this:
Definition: The exponential of is given by where
is the unique one-parameter subgroup of whose tangent vector at the identity is equal to .
It follows easily from the chain rule that . The map may be constructed as the integral curve of either the right- or left-invariant vector field associated with . That the integral curve exists for all real parameters follows by right- or left-translating the solution near zero.
We have a more concrete definition in the case of a matrix Lie group. The exponential map coincides with the matrix exponential and is given by the ordinary series expansion:
,
where is the identity matrix. Thus, in the setting of matrix Lie groups, the exponential map is the restriction of the matrix exponential to the Lie algebra of .
Comparison with Riemannian exponential map
If G is compact, it has a Riemannian metric invariant under left and right translations, and the Lie-theoretic exponential map for G coincides with the exponential map of this Riemannian metric.
For a general G, there will not exist a Riemannian metric invariant under both left and right translations. Although there is always a Riemannian metric invariant under, say, left translations, the exponential map in the sense of Riemannian geometry for a left-invariant metric will not in general agree with the exponential map in the Lie group sense. That is to say, if G is a Lie group equipped with a left- but not right-invariant metric, the geodesics through the identity will not be one-parameter subgroups of G .
Other definitions
Other equivalent definitions of the Lie-group exponential are as follows:
It is the exponential map of a canonical left-invariant affine connection on G, such that parallel transport is given by left translation. That is, where is the unique geodesic with the initial point at the identity element and the initial velocity X (thought of as a tangent vector).
It is the exponential map of a canonical right-invariant affine connection on G. This is usually different from the canonical left-invarian |
https://en.wikipedia.org/wiki/Exponential%20map | In differential geometry, the exponential map is a generalization of the ordinary exponential function of mathematical analysis. Important special cases include:
exponential map (Riemannian geometry) for a manifold with a Riemannian metric,
exponential map (Lie theory) from a Lie algebra to a Lie group,
More generally, in a manifold with an affine connection, , where is a geodesic with initial velocity X, is sometimes also called the exponential map. The above two are special cases of this with respect to appropriate affine connections.
Euler's formula forming the unit circle in the complex plane. |
https://en.wikipedia.org/wiki/Ruth%20Charney | Ruth Michele Charney (born 1950) is an American mathematician known for her work in geometric group theory and Artin groups. Other areas of research include K-theory and algebraic topology. She holds the Theodore and Evelyn G. Berenson Chair in Mathematics at Brandeis University. She was in the first group of mathematicians named Fellows of the American Mathematical Society. She was in the first group of mathematicians named Fellows of the Association for Women in Mathematics. She served as president of the Association for Women in Mathematics during 2013–2015, and served as president of the American Mathematical Society for the 2021–2023 term.
Life
Charney attended Brandeis University, graduating in mathematics in 1972. She then attended Merce Cunningham Dance Studio for a year, studying modern dance. She received her Ph.D. from Princeton University in 1977 under Wu-Chung Hsiang.
Work
Following her graduation from Princeton, Charney took a postdoctoral position at University of California, Berkeley, followed by an NSF postdoctoral appointment/assistant professor position at Yale University. She worked for Ohio State University until 2003, when she returned to work at Brandeis University.
Charney served as president of the Association for Women in Mathematics during 2013–2015. She emphasized the importance of encouraging young women in mathematics through summer programs, mentorships, and parental involvement.
She has served as an editor of the journal Algebraic and Geometric Topology from 2000 to 2007.
In 2019 she was elected to serve as president of the American Mathematical Society during 2021–2023. She currently serves as the AMS Immediate Past President.
Additionally, she was a member at large for the American Mathematical Society from 1992 to 1994.
Honors
In 2013 Charney was named a Fellow of the American Mathematical Society in the inaugural class.
In 2017 she was selected as a Fellow of the Association for Women in Mathematics in the inaugural class.
Selected publications
Charney, Ruth; Davis, Michael W. Finite K(π,1)s for Artin groups. Prospects in topology (Princeton, NJ, 1994), 110–124, Ann. of Math. Stud., 138, Princeton Univ. Press, Princeton, NJ, 1995. MR1368655
Charney, Ruth Geodesic automation and growth functions for Artin groups of finite type. Math. Ann. 301 (1995), no. 2, 307–324. MR1314589
Charney, Ruth Artin groups of finite type are biautomatic. Math. Ann. 292 (1992), no. 4, 671–683. MR1157320
Charney, Ruth An introduction to right-angled Artin groups. Geom. Dedicata 125 (2007), 141–158. MR2322545
References
External links
Ruth Charney's Author Profile on MathSciNet
American women mathematicians
Brandeis University alumni
Brandeis University faculty
Fellows of the American Mathematical Society
American feminists
Geometers
Group theorists
Topologists
Living people
1950 births
Fellows of the Association for Women in Mathematics
Presidents of the American Mathematical Society |
https://en.wikipedia.org/wiki/Oceania%20Club%20Championship%20and%20OFC%20Champions%20League%20records%20and%20statistics | This page details statistics of the Oceania Club Championship and OFC Champions League.
Performance
Performance by club
Past winners are:
Performance by country
Notes:
is no longer an OFC member.
Performance by city
Participating clubs
Notes:
is no longer an OFC member.
is not an OFC member.
Clubs
Biggest win
Oceania Club Championship era:
Central United 16–0 Lotoha'apai United (20 September 1999)
Wollongong Wolves 16–0 Lotoha'apai United (9 January 2001)
OFC Champions League era:
Auckland City 15–0 Tupapa Maraerenga (27 February 2019)
Highest score
Oceania Club Championship era:
Malaita Eagles 14–2 Konica (1999)
Central United 16–0 Lotoha'apai United (1999)
Wollongong Wolves 16–0 Lotoha'apai United (2001)
OFC Champions League era:
Auckland City 15–0 Tupapa Maraerenga (2019)
Most consecutive wins
Auckland City – 16 (9 April 2016 – 7 April 2018)
Longest undefeated run
Auckland City – 29 (10 May 2014 – 6 April 2019)
Most minutes without conceding a goal
Auckland City – 1079
Players
These records are only of the Champions League era.
All-time top scorers
Preliminary Round goals not included.
Top scorers by season
Most goals in a single game
6 goals:
Ross Allen (Team Wellington), 13–0 against Kiwi, group stage, 2019
Most goals in a single season
11 goals:
Ross Allen (2019)
Hat-tricks
Multiple Hat-tricks
Hat-tricks by team
Hat-tricks by nationality
Managers
The winning managers are:
References
External links
OFC website
OFC Champions League
International club association football competition records and statistics |
https://en.wikipedia.org/wiki/Klingen%20Eisenstein%20series | In mathematics, a Klingen Eisenstein series is a Siegel modular form of weight k and degree g depending on another Siegel cusp form f of weight k and degree r<g, given by a series similar to an Eisenstein series. It is a generalization of the Siegel Eisenstein series, which is the special case when the Siegel cusp form is 1. Klingen Eisenstein series is introduced by .
Definition
Suppose that f is a Siegel cusp form of degree r and weight k with k > g + r + 1 an even integer. The Klingen Eisenstein series is
It is a Siegel modular form of weight k and degree g.
Here Pr is the integral points of a certain parabolic subgroup of the symplectic group, and Γr is the group of integral points of the degree g symplectic group. The variable τ is in the Siegel upper half plane of degree g. The function f is originally defined only for elements of the Siegel upper half plane of degree r, but extended to the Siegel upper half plane of degree g by projecting this to the smaller Siegel upper half plane.
The cusp form f is the image of the Klingen Eisenstein series under the operator Φg−r, where Φ is the Siegel operator.
References
Automorphic forms |
https://en.wikipedia.org/wiki/Symposium%20on%20Discrete%20Algorithms | The Annual ACM-SIAM Symposium on Discrete Algorithms (SODA) is an academic conference in the fields of algorithm design and discrete mathematics. It is considered to be one of the top conferences for research in algorithms. SODA has been organized annually since 1990, typically in January. SODA is jointly sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT) and the SIAM Activity Group on Discrete Mathematics, and in format is more similar to a theoretical computer science conference than to a mathematics conference.
History
The first Symposium on Discrete Algorithms was held in 1990 at San Francisco, organized by David Johnson.
In 2012, the ACM Special Interest Group on Algorithms and Computation Theory (ACM SIGACT) and SIAM Activity Group on Discrete Mathematics (SIAG/DM) jointly established SODA Steering Committee to work with SIAM and ACM on organizing SODA.
References
Theoretical computer science conferences
Association for Computing Machinery conferences |
https://en.wikipedia.org/wiki/European%20Union%20Statistics%20on%20Income%20and%20Living%20Conditions | The European Union Statistics on Income and Living Conditions, (also known as European Union Survey on Income and Living Conditions) abbreviated EU-SILC, is a survey department of the EU.
It replaced in 2004 the European Community Household Panel (ECHP), which covered the years 1994–2003. The EU-SILC includes only minor changes relative to its predecessor the ECHP; most importantly, it significantly expands the number of countries included in the sample.
References
Demography
Euthenics
International medical and health organizations |
https://en.wikipedia.org/wiki/Alemdar%20Hasano%C4%9Flu | Alemdar Hasanoğlu is an Azerbaijani mathematician. He was born in 1954.
Biography
He received the degree B.S. in Mathematics from Baku State University, Azerbaijan and received the degrees of M.S., Ph.D. in Computational Mathematics, Dr. Sc. in Mathematical Modeling and Computational Technology in Science from the Special Scientific Committee at the Moscow State University.
Research work
Research interests of Professor Hasanoğlu include nonlinear differential equations, variational methods, inverse problems, mathematical and computational modeling in engineering sciences.
He is an author of more than 100 scientific papers in international journals and conference proceedings, author of 4 books and co-author of 7 books, and conference proceedings, co-editor of special issues. His research projects has been supported by U.S.S.R Academy of Sciences institutions (1982–1989), Kocaeli Governorship, Arcelik A. S., Istanbul Municipality, TUBİTAK, Turkey (1994–2009), Office of Naval Research, USA (2002), INTAS, Brussels (2007–2009), Science for Peace and Security Section, NATO, Brussels (2008–2010).
Awards
Hasanoğlu received numerous honors and awards for his work, including the USSR Academy of Science Medal and Diploma of Honor and the best scientific work "Computational Express-Diagnostics Method for Constructional Medium". He also received various research and teaching awards from Moscow State, Baku State and Kocaeli University, and also from Kocaeli Governorship/Municipality. He is an honorary professor of L.N. Gumilev Eurasian National University, Astana, Kazakhstan.
Page text.
References
1954 births
20th-century Azerbaijani mathematicians
Living people
21st-century Azerbaijani mathematicians |
https://en.wikipedia.org/wiki/Distribution%20learning%20theory | The distributional learning theory or learning of probability distribution is a framework in computational learning theory. It has been proposed from Michael Kearns, Yishay Mansour, Dana Ron, Ronitt Rubinfeld, Robert Schapire and Linda Sellie in 1994 and it was inspired from the PAC-framework introduced by Leslie Valiant.
In this framework the input is a number of samples drawn from a distribution that belongs to a specific class of distributions. The goal is to find an efficient algorithm that, based on these samples, determines with high probability the distribution from which the samples have been drawn. Because of its generality, this framework has been used in a large variety of different fields like machine learning, approximation algorithms, applied probability and statistics.
This article explains the basic definitions, tools and results in this framework from the theory of computation point of view.
Definitions
Let be the support of the distributions of interest. As in the original work of Kearns et al. if is finite it can be assumed without loss of generality that where is the number of bits that have to be used in order to represent any . We focus in probability distributions over .
There are two possible representations of a probability distribution over .
probability distribution function (or evaluator) an evaluator for takes as input any and outputs a real number which denotes the probability that of according to , i.e. if .
generator a generator for takes as input a string of truly random bits and outputs according to the distribution . Generator can be interpreted as a routine that simulates sampling from the distribution given a sequence of fair coin tosses.
A distribution is called to have a polynomial generator (respectively evaluator) if its generator (respectively evaluator) exists and can be computed in polynomial time.
Let a class of distribution over X, that is is a set such that every is a probability distribution with support . The can also be written as for simplicity.
Before defining learnability, it is necessary to define good approximations of a distribution . There are several ways to measure the distance between two distribution. The three more common possibilities are
Kullback-Leibler divergence
Total variation distance of probability measures
Kolmogorov distance
The strongest of these distances is the Kullback-Leibler divergence and the weakest is the Kolmogorov distance. This means that for any pair of distributions , :
Therefore, for example if and are close with respect to Kullback-Leibler divergence then they are also close with respect
to all the other distances.
Next definitions hold for all the distances and therefore the symbol denotes the distance between the distribution and the distribution using one of the distances that we describe above. Although learnability of a class of distributions can be defined using any of these distances, applications refer to |
https://en.wikipedia.org/wiki/Joel%20David%20Hamkins | Joel David Hamkins is an American mathematician and philosopher who is O'Hara Professor of Philosophy and Mathematics at the University of Notre Dame. He has made contributions in mathematical and philosophical logic, set theory and philosophy of set theory (particularly the idea of the set-theoretic multiverse), in computability theory, and in group theory.
Biography
After earning a B.S. in mathematics at the California Institute of Technology, Hamkins earned his Ph.D. in mathematics in 1994 at the University of California, Berkeley under the supervision of W. Hugh Woodin, with a dissertation entitled Lifting and Extending Measures by Forcing; Fragile Measurability. He joined the faculty of the City University of New York in 1995, where he was a member of the doctoral faculties in Mathematics, in Philosophy and in Computer Science at the CUNY Graduate Center and professor of mathematics at the College of Staten Island. He has also held various faculty or visiting fellow positions at University of California at Berkeley, Kobe University, Carnegie Mellon University, University of Münster, Georgia State University, University of Amsterdam, the Fields Institute, New York University and the Isaac Newton Institute.
In September 2018, Hamkins moved to the University of Oxford to become Professor of Logic in the Faculty of Philosophy and Sir Peter Strawson Fellow in Philosophy in University College, Oxford. In January 2022 he moved to the University of Notre Dame as the O'Hara Professor of Philosophy and Mathematics.
Research contributions
Hamkins research work is cited, and he gives talks, including events for the general public. Hamkins was interviewed on his research by Richard Marshall in 2013 for 3:AM Magazine, as part of an ongoing interview series for that magazine of prominent philosophers and public intellectuals, and he is occasionally interviewed by the popular science media about issues in the philosophy of mathematics.
Set theory
In set theory, Hamkins has investigated the indestructibility phenomenon of large cardinals, proving that small forcing necessarily ruins the indestructibility of supercompact and other large cardinals and introducing the lottery preparation as a general method of forcing indestructibility. Hamkins introduced the modal logic of forcing and proved with Benedikt Löwe that if ZFC is consistent, then the ZFC-provably valid principles of forcing are exactly those in the modal theory known as S4.2. Hamkins, Linetsky and Reitz proved that every countable model of Gödel-Bernays set theory has a class forcing extension to a pointwise definable model, in which every set and class is definable without parameters. Hamkins and Reitz introduced the ground axiom, which asserts that the set-theoretic universe is not a forcing extension of any inner model by set forcing. Hamkins proved that any two countable models of set theory are comparable by embeddability, and in particular that every countable model of set theory embed |
https://en.wikipedia.org/wiki/Pertti%20Mattila | Pertti Mattila (born 28 March 1948) is a Finnish mathematician working in geometric measure theory, complex analysis and harmonic analysis. He is Professor of Mathematics in the Department of Mathematics and Statistics at the University of Helsinki, Finland.
He is known for his work on geometric measure theory and in particular applications to complex analysis and harmonic analysis. His work include a counterexample to the general Vitushkin's conjecture and with Mark Melnikov and Joan Verdera he introduced new techniques to understand the geometric structure of removable sets for complex analytic functions which together with other works in the field eventually led to the solution of Painlevé's problem by Xavier Tolsa.
His book Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability is now a widely cited and a standard textbook in this field. Mattila has been the leading figure on creating the geometric measure theory school in Finland and the Mathematics Genealogy Project cites he has supervised so far 15 PhD students in the field.
He obtained his PhD from the University of Helsinki under the supervision of Jussi Väisälä in 1973. He worked at the Institute for Advanced Study at the Princeton University for postdoctoral research in 1979 and from 1988 as Professor of Mathematics at the University of Jyväskylä until appointed as Professor of Mathematics at the University of Helsinki in 2003. In 1998 he was an Invited Speaker of the International Congress of Mathematicians in Berlin. Mattila was the director of the Academy of Finland funded Centre of Excellence of Geometric Analysis and Mathematical Physics from 2002 to 2007 and currently part of the Centre of Excellence in Analysis and Dynamics Research in the University of Helsinki.
References
Bibliography
External links
Personal web page
1948 births
Living people
University of Helsinki alumni
Academic staff of the University of Helsinki
21st-century Finnish mathematicians
20th-century Finnish mathematicians |
https://en.wikipedia.org/wiki/Teana%20Miller | Clifeteana Rena McKiver Miller (born October 5, 1980) is a former professional basketball player in the WNBA.
East Carolina and Tulane statistics
Source
References
External links
STING: Season In Review: Teana Miller
Teana Miller Basketball-Reference.com
1980 births
Living people
American expatriate basketball people in France
American women's basketball players
Basketball players from Wilmington, North Carolina
Charlotte Sting players
East Carolina Pirates women's basketball players
Phoenix Mercury players
People from Duplin County, North Carolina
Tulane Green Wave women's basketball players
Undrafted Women's National Basketball Association players
Centers (basketball) |
https://en.wikipedia.org/wiki/Sirous%20Sangchouli | Sirous Sangchouli (; born 8 December 1982) is an Iranian association football goalkeeper.
Club career
Club career statistics
Honours
Payam Khorasan
Azadegan League : Champion 2007–08
References
Living people
1982 births
Iranian men's footballers
Men's association football goalkeepers
Payam Khorasan F.C. players
Foolad Yazd F.C. players
Fajr Sepasi Shiraz F.C. players
Siah Jamegan F.C. players
Sanat Mes Kerman F.C. players
Place of birth missing (living people) |
https://en.wikipedia.org/wiki/Shahram%20Mehraban | Shahram Mehraban (; born 16 November 1979) is an Iranian association football goalkeeper.
Club career
Club career statistics
References
Living people
1979 births
Iranian men's footballers
Men's association football goalkeepers
Persian Gulf Pro League players
Azadegan League players
F.C. Shahrdari Bandar Abbas players
Rah Ahan Tehran F.C. players
Saba Qom F.C. players
Gahar Zagros F.C. players
Paykan F.C. players
People from Qazvin |
https://en.wikipedia.org/wiki/Introductio | Introductio can refer to:
Introductio in analysin infinitorum, a book on Mathematics by Leonhard Euler
Cosmographiae Introductio, a book on Geography by an unknown author |
https://en.wikipedia.org/wiki/Julia%20Indichova | Julia Indichova is an American reproductive healthcare activist and author. She is best known for her book Inconceivable: A Woman’s Triumph Over Despair and Statistics (2001), which was hailed by Library Journal as “an important consumer health resource…the first such book written from the patient’s point of view.” In 1997 Indichova founded FertileHeart.com, a global, patient driven community, focused on health enhancing approaches to reproductive health.
Indichova's second book, The Fertile Female: How the Power of Longing for a Child Can Save Your Life and Change the World (2007), documents the evolution of The Fertile Heart™ Ovum Practice, an original mind body program that grew out of Indichova's personal experience and her work with women and couples who sought her guidance after the publication of Inconceivable. In the last two decades a growing number of studies and leading reproductive endocrinologists have validated Indichova's whole-person approach to overcoming infertility.
Biography
Julia Indichova was born and raised in Košice, Czechoslovakia in the aftermath of World War II. Her older brother, two grandmothers and an aunt were murdered by the Nazis and both of her parents were concentration camp survivors. Attempting to come to terms with this legacy of violence has been a driving force in her life from an early age.
Indichova started working as a professional actress at the age of ten, and went on to study acting at the Academy of Performing Arts in Bratislava. In June 1969, a year after the 1968 Russian invasion of Czechoslovakia, she immigrated to the United States and in 1972 received her Bachelor of Arts degree in Theater Arts and Russian from Montclair State College.
For the next fifteen years Indichova worked in the post-production film department of the American Broadcasting Company, and later as an actress, dancer, director and producer in New York City and the New York Metropolitan area.
In 1985 Indichova received her MA in TESOL (teaching English to Speakers of Other Languages) from Hunter College in New York. In the following years, she taught English at Columbia University's American Language Program, and at the International English Language Institute at Hunter College. She also taught Russian, German, French and Latin, in the New York City public school system.
Overcoming infertility
In 1992 Indichova was diagnosed with irreversible secondary infertility attributed to soaring Follicle-stimulating hormone (FSH) levels. After a futile quest for a more hopeful prognosis, she embarked on a pilgrimage of research and rigorous self-examination. In 1994, at age 44 she gave birth to a healthy baby girl, conceived naturally, in direct contradiction of all that medical dogma of the day declared possible.
Indichova's desire to share what she learned with women who sought her help led her to create in 1997 The Fertile Heart™ Ovum Mind Body Program, and launched her work as a reproductive health care educator and acti |
https://en.wikipedia.org/wiki/2014%E2%80%9315%20FC%20Utrecht%20season | The 2014–15 FC Utrecht season was the club's 45th season of existence and its 45th season in the Eredivisie.
Competitions
Pre-season friendlies
Eredivisie
Table
Results
KNVB Cup
Statistics
Appearances and goals
Appearances for competitive matches only
|-
|}
Top goalscorers
Top assists
Starting XI
Transfers
In
Out
Loan in
Loan out
References
FC Utrecht seasons
Utrecht Fc |
https://en.wikipedia.org/wiki/All%20models%20are%20wrong | All models are wrong is a common aphorism and anapodoton in statistics; it is often expanded as "All models are wrong, but some are useful". The aphorism acknowledges that statistical models always fall short of the complexities of reality but can still be useful nonetheless. The aphorism originally referred just to statistical models, but it is now sometimes used for scientific models in general.
The aphorism is generally attributed to the statistician George Box. The underlying concept, though, predates Box's writings.
Quotations of George Box
The first record of Box saying "all models are wrong" is in a 1976 paper published in the Journal of the American Statistical Association. The 1976 paper contains the aphorism twice. The two sections of the paper that contain the aphorism state:
Box repeated the aphorism in a paper that was published in the proceedings of a 1978 statistics workshop. The paper contains a section entitled "All models are wrong but some are useful". The section states (p 202-3):
Box repeated the aphorism twice more in his 1987 book, Empirical Model-Building and Response Surfaces (which was co-authored with Norman Draper). The first repetition is on p. 74: "Remember that all models are wrong; the practical question is how wrong do they have to be to not be useful." The second repetition is on p. 424, which is excerpted below.
A second edition of the book was published in 2007, under the title Response Surfaces, Mixtures, and Ridge Analyses. The second edition also repeats the aphorism twice, in contexts identical with those of the first edition (on p. 63 and p. 414).
Box repeated the aphorism two more times in his 1997 book, Statistical Control: By Monitoring and Feedback Adjustment (which was co-authored with Alberto Luceño). The first repetition is on p. 6, which is excerpted below.
The second repetition is on p. 9: "So since all models are wrong, it is very important to know what to worry about;
or, to put it in another way, what models are likely to produce procedures that work in practice (where exact assumptions are never true)".
A second edition of the book was published in 2009, under the title Statistical Control By Monitoring and Adjustment (co-authored with Alberto Luceño and María del Carmen Paniagua-Quiñones). The second edition also repeats the aphorism two times. The first repetition is on p. 61, which is excerpted below.
The second repetition is on p. 63; its context is essentially the same as that of the second repetition in the first edition.
Box's widely cited book Statistics for Experimenters (co-authored with William Hunter) does not include the aphorism in its first edition (published in 1978). The second edition (published in 2005; co-authored with William Hunter and J. Stuart Hunter) includes the aphorism three times: on p. 208, p. 384, and p. 440. On p. 440, the relevant sentence is this: "The most that can be expected from any model is that it can supply a useful approximation to real |
https://en.wikipedia.org/wiki/Nakano%20vanishing%20theorem | In mathematics, specifically in the study of vector bundles over complex Kähler manifolds, the Nakano vanishing theorem, sometimes called the Akizuki–Nakano vanishing theorem, generalizes the Kodaira vanishing theorem. Given a compact complex manifold M with a holomorphic line bundle F over M, the Nakano vanishing theorem provides a condition on when the cohomology groups equal zero. Here, denotes the sheaf of holomorphic (p,0)-forms taking values on F. The theorem states that, if the first Chern class of F is negative,
Alternatively, if the first Chern class of F is positive,
See also
Le Potier's vanishing theorem
References
Original publications
Secondary sources
Theorems in complex geometry
Topological methods of algebraic geometry
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/William%20Abikoff | William Abikoff (born 1944) is an American mathematician. He has been a professor of mathematics at the University of Connecticut since 1981.
Abikoff earned his Ph.D. in 1971 from the New York University Polytechnic School of Engineering under the supervision of Georges Gustave Weill. In 2012, Abikoff became a Fellow of the American Mathematical Society.
Selected works
References
1944 births
Living people
Fellows of the American Mathematical Society
20th-century American mathematicians
21st-century American mathematicians
Polytechnic Institute of New York University alumni
University of Connecticut faculty |
https://en.wikipedia.org/wiki/Alejandro%20Adem | Alejandro Adem is a professor in the Department of Mathematics at the University of British Columbia
and President of the Natural Sciences and Engineering Research Council of Canada. Previously he was Director of the Pacific Institute for the Mathematical Sciences for the period 2008–2015 and during 2015-2019 was the CEO and Scientific Director of Mitacs.
Education and academic career
Alejandro Adem did his undergraduate studies at the National Autonomous University of Mexico, earning a B.S. in 1982. He earned his Ph.D. in 1986 from Princeton University, under the supervision of William Browder. He then worked as Szego Assistant Professor at Stanford University (1986–89) before joining the faculty at the University of Wisconsin–Madison; he moved to the University of British Columbia in 2005.
Adem has held visiting positions at the Institute for Advanced Study in Princeton, the ETH-Zürich, the Max-Planck Institute in Bonn, the University of Paris 7 and at Princeton University. His main areas of research are algebraic topology and group cohomology. Adem has written or co-written over sixty research papers and two research monographs.
From 2013 to 2020 Adem was managing editor of the Memoirs of the American Mathematical Society and the Transactions of the American Mathematical Society. He has served on the scientific and governance boards of the Mathematical Sciences Research Institute, the Banff International Research Station and the Pacific Institute for the Mathematical Sciences.
Honors
Alejandro Adem's awards include an A.P. Sloan Doctoral Dissertation Fellowship (1985), an NSF Young Investigator Award (1992), a Romnes Faculty Fellowship from the Wisconsin Alumni Research Foundation (1995), Canada Research Chair (2004), Fellow of the American Mathematical Society(2012) and the 2015 Jeffery–Williams Prize, which was awarded to him by the Canadian Mathematical Society, with the following citation: "he stands out as one of the few mathematicians who has made important qualitative and calculational contributions to the theory of the cohomology of groups, and applied these results to problems of algebraic topology". In 2016 he was elected corresponding member of the Mexican Academy of Sciences. In 2020 he was made a Fellow of the Canadian Mathematical Society. and in 2021 he was elected a Fellow of the Royal Society of Canada.
Personal life
Adem is the son of geophysicist :es:Julián Adem, who obtained a Ph.D. in applied mathematics from Brown University in 1953. He is also the nephew of noted topologist José Adem (see Adem relations), who obtained a Ph.D. in mathematics from Princeton University in 1952. His father and uncle were both, in their own specialties, elected members of Mexico's exclusive El Colegio Nacional, and were both recipients of Mexico's National Prize for Arts and Sciences.
Books
References
External links
Living people
Fellows of the American Mathematical Society
20th-century Mexican mathematicians
21st-century Mexican |
https://en.wikipedia.org/wiki/Elizabeth%20S.%20Allman | Elizabeth Spencer Allman (born 1965) is an American mathematician. She is a professor of mathematics in the Department of Mathematics and Statistics at the University of Alaska Fairbanks; her research interests range from abstract algebra and algebraic statistics to biomathematics and phylogeny.
Education
Allman earned her Ph.D. in 1995 from the University of California, Los Angeles under the supervision of Murray M. Schacher. Her dissertation, in abstract algebra, was Polynomials Without Roots in Division Algebras.
Contributions
With her Fairbanks colleague John A. Rhodes, Allman is the author of a book on mathematical biology and mathematical modeling, Mathematical Models in Biology: An Introduction (Cambridge University Press, 2004).
Recognition
In 2012, Allman became a Fellow of the American Mathematical Society.
References
External links
Elizabeth S. Allman Home page
Living people
1965 births
American women mathematicians
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
University of California, Los Angeles alumni
University of Alaska Fairbanks faculty
Mathematicians from Alaska
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women scientists
21st-century American women academics |
https://en.wikipedia.org/wiki/Griffiths%20group | In mathematics, more specifically in algebraic geometry, the Griffiths group of a projective complex manifold X measures the difference between homological equivalence and algebraic equivalence, which are two important equivalence relations of algebraic cycles.
More precisely, it is defined as
where denotes the group of algebraic cycles of some fixed codimension k and the subscripts indicate the groups that are homologically trivial, respectively algebraically equivalent to zero.
This group was introduced by Phillip Griffiths who showed that for a general quintic in (projective 4-space), the group is not a torsion group.
Notes
References
Algebraic geometry |
https://en.wikipedia.org/wiki/List%20of%20Santa%20Tecla%20F.C.%20records%20and%20statistics | This article lists various statistics related to Santa Tecla Futbol Club.
All stats accurate as of 23 July 2023.
Honours
As of 23 July 2023, Santa Tecla have won 4 Salvadoran Primera División, 1 Segunda Division and 2 Copa El Salvador
trophies.
Domestic competitions
League
Salvadoran Primera División
Champions (4): Clausura 2015, Apertura 2016, , Clausura 2017, Apertura 2018
Segunda Division
Champions (1): Clausura 2012
Cup
Copa El Salvador and predecessors
Champions (2) : 2016–17, 2018–19
CONCACAF competitions
Official titles
CONCACAF Champions League
Winners (1): TBD
Individual awards
Award winners
Top Goalscorer (-)
The following players have won the Goalscorer while playing for Santa Tecla:
TBD (-) – Clausura 2004
Goalscorers
Most goals scored : 80 - Ricardo Ferreira
Most League goals: 80 - Ricardo Ferreira
Most League goals in a season: 10 - Ricardo Ferreira, Primera Division, Clausura 2014
Most goals scored by a Santa Tecla player in a match: 3 - Ricardo Ferreira v. Juventud Independiente (Santa Tecla 3-1 Juventud Independiente), January 2015 and William Maldonado v. UES (Santa Tecla 5-0 UES), 2016
Most goals scored by a Santa Tecla player in an International match: 1 - Ricardo Ferreira v. C.S.D. Municipal, 25 August 2015, Irvin Herrera v. Real Salt Lake 24 September 2015
Most goals scored in CONCACAF competition: 2 - Gerson Mayen, 2018 CONCACAF Champions League
All-time top goalscorers
<small>Note: Players in bold text are still active with Santa Tecla F.C.</small>
Historical goals
Players
AppearancesCompetitive, professional matches only including substitution, number of appearances as a substitute appears in brackets.Last updated 23 July 2023
Other appearances records
Youngest first-team player: – Henry Quinteros v C.D. Audaz, Primera Division, 22 March 2018
Oldest first-team player: – Sebastián Abreu v UES, Primera Division, 6 August 2016
Most appearances in Primera Division: 223 games – Joel Almeida
Most appearances in International competitions: TBD – TBD
Most appearances in CONCACAF competitions: TBD – TBD
Most appearances in UNCAF competitions: TBD – TBD
Most appearances in CONCACAF Champions League: TBD – TBD
Most appearances in UNCAF Copa: TBD TBD
Most appearances as a foreign player in all competitions: 223 games – Joel Almeida
Most appearances as a foreign player in Primera Division: 223 games – Joel Almeida
Most consecutive League appearances: TBD – TBD – from Month Day, Year at Month Day, Year
Shortest appearance: –
Records
Scorelines
Record League victory: 8-0 v C.D. Pasaquina, Primera division, 14 February 2016
Record League Defeat: 1-7 v Alianza F.C., Primera division, 22 July 2012
Record Cup victory: 6–1 v Escuela de Fútbol de Guazapa, 2017 Copa El Salvador, 22 March 2017
Record Cup loss: 3-5 v Escuela de Fútbol de Guazapa, 2017 Copa El Salvador, 19 October 2016
Record CONCACAF Champions League Victory: 2-1 v Seattle Sounders FC, February 22, 2018
Record CONC |
https://en.wikipedia.org/wiki/Mara%20Neusel | Mara Dicle Neusel (May 14, 1964 – September 5, 2014) was a mathematician, author, teacher and an advocate for women in mathematics. The focus of her mathematical work was on invariant theory, which can be briefly described as the study of group actions and their fixed points.
Life and education
Mara Neusel was born in Stuttgart, Germany, one of two children of Günter and Aylâ (Helvacioglu) Neusel. In 2001 she became the fourth woman to earn the advanced degree venia legendi (Habilitation) in mathematics from the University of Göttingen, following in the footsteps of the first woman mathematician to be awarded the venia legend from Göttingen in 1919, Emmy Noether.
Career
Professor Neusel was the author of a research monograph, an advanced undergraduate text, and a memoir for the American Mathematical Society: Invariant Theory and Finite Groups, Invariant Theory, and Inverse Invariant Theory and Steenrod Operations. The exposition in the text Invariant Theory "stands out by its masterly clarity, comprehensiveness, profundity, and didactical disposition." Neusel served on the editorial boards of Advances in Pure Mathematics and the International Journal of Mathematics and Applied Statistics. A tireless advocate for girls and women in mathematics, Neusel established Emmy Noether High School Mathematics Days in May 2003 which continues to be celebrated with workshops and mathematical competitions. She co-founded the Young Women in Mathematics group at Texas Tech and received a diversity grant to support the group.
Neusel began her career at Texas Tech University in 2002 as an associate professor and was promoted to full professor in 2009. She also held visiting appointments at Yale University, the University of Minnesota, and the University of Notre Dame.
Professor Neusel was had long term memberships in the American Mathematical Society and the Association for Women in Mathematics. Neusel co-organized a special session on "Homological Algebra and Its Applications" at the 2005 Meeting of the American Mathematical Society in Lubbock, Texas as well as a special session on "Commutative Algebra and Algebraic Geometry" at the 40th Anniversary Celebration of the Association for Women in Mathematics at Brown University in 2011.
References
1964 births
2014 deaths
20th-century German mathematicians
American women mathematicians
University of Göttingen alumni
Texas Tech University faculty
20th-century American women scientists
20th-century American scientists
21st-century American women |
https://en.wikipedia.org/wiki/Acarassus | Acarassus or Akarassos was a city in ancient Lycia.
The town, with high probability, was located at the site of today's Elmalı, Antalya Province, Turkey.
Bishopric
Since it was in the Roman province of Lycia, the bishopric of Acarassus was a suffragan of the metropolitan see of Myra, the province's capital. It is listed in all the Notitiae Episcopatuum from the mid-7th century to about 1300. The name of only one of its bishops is known with certainty: Nicolaus attended the Council of Chalcedon in 451 and was one of the signatories of the letter that the Lycian bishops sent in 458 to Byzantine Emperor Leo I the Thracian to protest about the murder of Proterius of Alexandria. Because of the similarity of the names of Acarassus in Lycia and Acrassus in Lydia, it is unclear to which of these two sees two other bishops belonged, one of whom was at the Second Council of Nicaea in 787, the other at the Photian Council of Constantinople (879): Le Quien, Pétridès, and Darrouzès differ in their interpretations.
No longer a residential bishopric, Acarassus is today listed by the Catholic Church as a titular see.
References
Literature
Gustav Hirschfeld, article Akarassos in Paulys Realencyclopädie der Classischen Altertumswissenschaft, vol. I,1, p. Stuttgart 1893, p. 1150. (see German-language Wikisource)
Louis Robert: Hellenica, vol 10, Paris 1955, pp. 207–208.
Hansgerd Hellenkemper, Friedrich Hild: Lykien und Pamphylien (Tabula Imperii Byzantini 8.) Verlag der Österreichischen Akademie der Wissenschaften, Vienna 2004, , pp. 425-426.
External links
Entry in Pleiades database
Populated places in ancient Lycia
Roman towns and cities in Turkey
Former populated places in Turkey
Archaeological sites in Antalya Province
Geography of Antalya Province
Catholic titular sees in Asia
Populated places of the Byzantine Empire
Defunct dioceses of the Ecumenical Patriarchate of Constantinople |
https://en.wikipedia.org/wiki/Jes%C3%BAs%20A.%20De%20Loera | Jesús Antonio De Loera (born January 18, 1966) is a Mexican-American mathematician at the University of California, Davis, specializing in discrete mathematics and discrete geometry.
Education
De Loera did his undergraduate studies at the National Autonomous University of Mexico, earning a B.S. in mathematics in 1989.
After earning a master's degree from Western Michigan University a year later, and another master's degree in 1993 from Cornell University, he finished his doctorate from Cornell in 1995 under the supervision of Bernd Sturmfels.
Career
De Loera joined the UC Davis faculty as an assistant professor in 1998, and was promoted to full professor in 2005. He served on the Scientific Advisory Board for the Institute for Computational and Experimental Research in Mathematics (ICERM) through 2021.
Awards and honors
In 2014, he became a fellow of the American Mathematical Society "for contributions to discrete geometry and combinatorial optimization as well as for service to the profession, including mentoring and diversity".
In 2019 he was named a SIAM Fellow "for contributions to discrete geometry and optimization, polynomial algebra, and mathematical software".
In 2020 he received the INFORMS optimization society's Farkas Prize "in recognition of his pioneering work at the intersection of discrete mathematics, optimization and algebraic geometry". He delivered an invited plenary address at the 2021 National Math Festival. In 2021, he was elected Vice President of the AMS.
References
External links
Home page
1966 births
Living people
20th-century American mathematicians
Mexican mathematicians
Combinatorialists
Researchers in geometric algorithms
National Autonomous University of Mexico alumni
Western Michigan University alumni
Cornell University alumni
University of California, Davis faculty
American academics of Mexican descent
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
Place of birth missing (living people)
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Irena%20Lasiecka | Irena Lasiecka (; born February 4, 1948) is a Polish-American mathematician, a Distinguished University Professor of mathematics and chair of the mathematics department at the University of Memphis. She is also co-editor-in-chief of two academic journals, Applied Mathematics & Optimization and Evolution Equations & Control Theory.
Lasiecka earned her Ph.D. in 1975 from the University of Warsaw under the supervision of Andrzej Wierzbicki. In 2014, she became a fellow of the American Mathematical Society "for contributions to control theory of partial differential equations, mentorship, and service to professional societies."
Her specific areas of study are partial differential equations and related control theory, non-Linear PDEs, the optimization theory, calculus of variations, and boundary stabilization.
Early life and education
Irena Lasiecka was born and raised in Poland, where she received her initial background in mathematics. She studied math for many years at the University of Warsaw, where she earned her Master of Science degree in applied mathematics in 1972. A few years later, she received her PhD from the same university in the same field of study.
Teaching
After receiving her PhD, Lasiecka started to transfer her knowledge of Applied Mathematics to others in addition to more personal studying and research. Her first teaching job was at the Polish Academy of Sciences in 1975, and she later ventured to the United States a few years later, teaching at the University of California, Los Angeles. She has been teaching in the US ever since. The following is a chart listing the institutions in which Lasiecka has been a teaching faculty member of.
Areas of Study in Applied Mathematics
Optimization
Optimization is the mathematical practice of finding the maximum or minimum values for a specific function. It has many real-world uses, and is a common practice for people of many different professions.
The work of Lasiecka involves the optimization differential systems. These involve an optimization problems over functions, with a constraint that relates a function to its derivatives. She has written extensively about this topic in her collaborative work Optimization Methods in Partial Differential Equations.
Control theory
Control Theory is one of Irena Lasiecka's chief areas of study. She begins her book, Mathematical Control Theory of Coupled PDEs, with a description of what Control Theory is. She states, " The classical viewpoint taken in the study of differential equations consisted of the (passive) analysis of the evolution properties displayed by a specific equation, or a class of equations, in response to given data. Control theory, however, injects an active mode of synthesis in the study of differential equations: it seeks to influence their dynamical evolution by selecting and synthesizing suitable data (input functions or control functions) from within a preassigned class, to achieve a predetermined desired outcome or perfo |
https://en.wikipedia.org/wiki/Stacked%20polytope | In polyhedral combinatorics (a branch of mathematics), a stacked polytope is a polytope formed from a simplex by repeatedly gluing another simplex onto one of its facets.
Examples
Every simplex is itself a stacked polytope.
In three dimensions, every stacked polytope is a polyhedron with triangular faces, and several of the deltahedra (polyhedra with equilateral triangle faces) are stacked polytopes
In a stacked polytope, each newly added simplex is only allowed to touch one of the facets of the previous ones. Thus, for instance, the quadaugmented tetrahedron, a shape formed by gluing together five regular tetrahedra around a common line segment is a stacked polytope (it has a small gap between the first and last tetrahedron). However, the similar-looking pentagonal bipyramid is not a stacked polytope, because if it is formed by gluing tetrahedra together, the last tetrahedron will be glued to two triangular faces of previous tetrahedra instead of only one.
Other non-convex stacked deltahedra include:
Combinatorial structure
The undirected graph formed by the vertices and edges of a stacked polytope in d dimensions is a (d + 1)-tree. More precisely, the graphs of stacked polytopes are exactly the (d + 1)-trees in which every d-vertex clique (complete subgraph) is contained in at most two (d + 1)-vertex cliques. For instance, the graphs of three-dimensional stacked polyhedra are exactly the Apollonian networks, the graphs formed from a triangle by repeatedly subdividing a triangular face of the graph into three smaller triangles.
One reason for the significance of stacked polytopes is that, among all d-dimensional simplicial polytopes with a given number of vertices, the stacked polytopes have the fewest possible higher-dimensional faces. For three-dimensional simplicial polyhedra the numbers of edges and two-dimensional faces are determined from the number of vertices by Euler's formula, regardless of whether the polyhedron is stacked, but this is not true in higher dimensions. Analogously, the simplicial polytopes that maximize the number of higher-dimensional faces for their number of vertices are the cyclic polytopes.
References
Polytopes |
https://en.wikipedia.org/wiki/Christiane%20Rousseau | Christiane Rousseau (born March 30, 1954 in Versailles, France) is a French and Canadian mathematician, a professor in the department of mathematics and statistics at the Université de Montréal. She was president of the Canadian Mathematical Society from 2002 to 2004.
Education and career
Rousseau earned her Ph.D. from the Université de Montréal in 1977, under the supervision of Dana Schlomiuk. After postdoctoral research at McGill University, she joined the Montréal faculty in 1979, and was promoted to full professor in 1991.
Recognition
She has received the Adrien-Pouliot Prize and the Abel-Gauthier Prize of the Mathematical Association of Québec, and the 2009 Graham Wright Award for Distinguished Service from the Canadian Mathematical Society. She also received the 2014 George Pólya Award of the Mathematical Association of America for her article about a discovery by Inge Lehmann, "How Inge Lehmann Discovered the Inner Core of the Earth". In 2012, she became a fellow of the American Mathematical Society (AMS). In 2017 she became the inaugural recipient of the AMS' Bertrand Russell prize for furthering human values and the common good through mathematics.
In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows.
References
1954 births
Living people
French mathematicians
Canadian mathematicians
Academic staff of the Université de Montréal
People from Versailles
Fellows of the American Mathematical Society
Fellows of the Canadian Mathematical Society
French women mathematicians
Presidents of the Canadian Mathematical Society |
https://en.wikipedia.org/wiki/Nucleus%20%28order%20theory%29 | In mathematics, and especially in order theory, a nucleus is a function on a meet-semilattice such that (for every in ):
Every nucleus is evidently a monotone function.
Frames and locales
Usually, the term nucleus is used in frames and locales theory (when the semilattice is a frame).
Proposition: If is a nucleus on a frame , then the poset of fixed points of , with order inherited from , is also a frame.
References
Order theory |
https://en.wikipedia.org/wiki/Credence%20%28statistics%29 | Credence or degree of belief is a statistical term that expresses how much a person believes that a proposition is true. As an example, a reasonable person will believe with close to 50% credence that a fair coin will land on heads the next time it is flipped (minus the probability that the coin lands on its edge). If the prize for correctly predicting the coin flip is $100, then a reasonable risk-neutral person will wager $49 on heads, but they will not wager $51 on heads.
Credence is a measure of belief strength, expressed as a percentage. Credence values range from 0% to 100%. Credence is closely related to odds, and a person's level of credence is directly related to the odds at which they will place a bet. Credence is especially important in Bayesian statistics.
If a bag contains 4 red marbles and 1 blue marble, and a person withdraws one marble at random, then they should believe with 80% credence that the random marble will be red. In this example, the probability of drawing a red marble is 80%.
Credence values can be based entirely on subjective feelings. For example, if Alice is fairly certain that she saw Bob at the grocery store on Monday, then she might say, "I believe with 90% credence that Bob was at the grocery store on Monday." If the prize for being correct is $100, then Alice will wager $89 that her memory is accurate, but she would not be willing to wager $91 or more. Given that Alice is 90% credent, this level of belief can be expressed as gambling odds in the following ways:
90% credence
1 / 9 fractional odds (1 to 9)
1.11 decimal odds
−900 moneyline odds
The return on a $100 wager is $11.11 (plus the $100 initial wager).
See the article odds for conversion equations.
References
Bayesian statistics |
https://en.wikipedia.org/wiki/Lawrence%20C.%20Washington | Lawrence Clinton Washington (born 1951, Vermont) is an American mathematician at the University of Maryland who specializes in number theory.
Biography
Washington studied at Johns Hopkins University, where in 1971 he received his B.A. and master's degree. In 1974 he earned his PhD at Princeton University under Kenkichi Iwasawa with thesis Class numbers and extensions. He then became an assistant professor at Stanford University and from 1977 at the University of Maryland, where he became in 1981 an associate professor and in 1986 a professor. He held visiting positions at several institutions, including IHES (1980/81), Max-Planck-Institut für Mathematik (1984), the Institute for Advanced Study (1996), and MSRI (1986/87), as well as at the University of Perugia, Nankai University and the State University of Campinas. In 1979–1981 he was a Sloan Fellow.
Recognition
He was named to the 2023 class of Fellows of the American Mathematical Society, "for contributions to number theory, especially cyclotomic fields, and for mentoring at all levels".
Research
Washington wrote a standard work on cyclotomic fields. He also worked on p-adic L-functions. He wrote a treatise with Allan Adler on their discovery of a connection between higher-dimensional analogues of magic squares and p-adic L-functions. Washington has done important work on Iwasawa theory, Cohen-Lenstra heuristics, and elliptic curves and their applications to cryptography.
In Iwasawa theory he proved with Bruce Ferrero in 1979 a conjecture of Kenkichi Iwasawa, that the -invariant vanishes for cyclotomic Zp-extensions of abelian number fields (Theorem of Ferrero-Washington).
More recently, Washington has published on arithmetic dynamics, sums of powers of primes, and Iwasawa invariants of non-cyclotomic Zp extensions
Selected works
Introduction to Cyclotomic Fields, Graduate Texts in Mathematics, Springer, 1982, 2nd edn. 1996
Galois Cohomology in Cornell, Silverman, Stevens (eds.): Modular forms and Fermat's Last Theorem, Springer, 1997
Elliptic Curves: Number theory and cryptography, CRC Press, 2003, 2nd edn. 2008
with James Kraft: An Introduction to Number Theory with Cryptography, CRC Press, 2003, 2nd edn.
with Wade Trappe: Introduction to Cryptography and Coding Theory, Prentice-Hall, 2002, 2nd edn. 2005
Sources
Joseph Oesterlé Travaux de Ferrero et Washington sur le nombre de classes d'idéaux des corps cyclotomiques, Séminaire Bourbaki, Nr. 535, 1978/79
Lawrence C. Washington, Curriculum Vita
References
External links
Homepage
1951 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Modern cryptographers
Number theorists
Johns Hopkins University alumni
Princeton University alumni
University of Maryland, College Park faculty
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Tim%20Cochran | Thomas "Tim" Daniel Cochran (April 7, 1955 – December 16, 2014) was a professor of mathematics at Rice University specializing in topology, especially low-dimensional topology, the theory of knots and links and associated algebra.
Education and career
Tim Cochran was a valedictorian for the Severna Park High School Class of 1973. Later, he was an undergraduate at the Massachusetts Institute of Technology, and received his Ph.D. from the University of California, Berkeley in 1982 (Embedding 4-manifolds in S5). He then returned to MIT as a C.L.E. Moore Postdoctoral Instructor from 1982 to 1984. He was an NSF postdoctoral fellow from 1985 to 1987. Following brief appointments at Berkeley and Northwestern University, he started at Rice University as an associate professor in 1990. He became a full professor at Rice University in 1998. He died unexpectedly, aged 59, on December 16, 2014, while on a year-long sabbatical leave supported by a fellowship from the Simons Foundation.
Research contributions
With his coauthors Kent Orr and Peter Teichner, Cochran defined the solvable filtration of the knot concordance group, whose lower levels encapsulate many classical knot concordance invariants.
Cochran was also responsible for naming the slam-dunk move for surgery diagrams in low-dimensional topology.
Awards and honors
While at Rice, he was named an Outstanding Faculty Associate (1992–93), and received the Faculty Teaching and Mentoring Award from the Rice Graduate Student Association (2014)
He was named a fellow of the American Mathematical Society in 2014, for contributions to low-dimensional topology, specifically knot and link concordance, and for mentoring numerous junior mathematicians.
Selected publications
References
External links
Tim Cochran's home page.
1955 births
2014 deaths
20th-century American mathematicians
21st-century American mathematicians
Topologists
Fellows of the American Mathematical Society
Massachusetts Institute of Technology alumni
Rice University faculty
UC Berkeley College of Letters and Science alumni |
https://en.wikipedia.org/wiki/Idris%20Assani | Idris Assani is a Beninese mathematician, who works as a professor of mathematics at the University of North Carolina at Chapel Hill.
Although born in Niger, Assani is Beninese. He was educated in France, earning a bachelor's degree in commerce from Paris Dauphine University in 1981, a doctorate of the third cycle in mathematics from Pierre and Marie Curie University in 1981, and a doctor of science from Pierre and Marie Curie University in 1986, under the supervision of Antoine Brunel. He joined the UNC mathematics department in 1988 but, allegedly for racist reasons, was turned down for tenure. He appealed through the courts, won his case and gained tenure in 1995, and was promoted to full professor one year later. In doing so he became the first Black tenured associate mathematics professor and the first Black full mathematics professor at UNC, as well as the only mathematician there to be promoted from associate to full so quickly.
Assani's research concerns ergodic theory. He is the author of the research monograph Wiener Wintner Ergodic Theorems (World Scientific, 2003), about mathematics related to the Wiener–Wintner theorem, and is also the editor of several volumes of collected papers. He has made numerous contributions in the area of nonconventional ergodic averages as well as the return times theorem. Some of the highlights of his research contributions include pointwise convergence of averages along cubes, being “the first complete pointwise convergence result obtained in the theory of nonconventional ergodic averages”, as well as the introduction of Wiener-Wintner Dynamical System and, which is a class of dynamical system where one can obtain easier proofs of pointwise convergence results in more general setting (with difficult proofs), such as J. Bourgain’s double recurrence theorem as well as the return times theorem.
In 2012, Assani was named as one of the inaugural fellows of the American Mathematical Society.
References
External links
Home page
Idris Assani Author Profile on MathSciNet.
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
Beninese academics
Paris Dauphine University alumni
Pierre and Marie Curie University alumni
University of North Carolina at Chapel Hill faculty
Fellows of the American Mathematical Society
American people of Beninese descent
20th-century Nigerian mathematicians
African mathematicians |
https://en.wikipedia.org/wiki/David%20G.%20Cantor | David Geoffrey Cantor (April 12, 1935 – November 19, 2012) was an American mathematician, specializing in number theory and combinatorics. The Cantor–Zassenhaus algorithm for factoring polynomials is named after him; he and Hans Zassenhaus published it in 1981.
Biography
Cantor was born on April 12, 1935. He completed his undergraduate studies at the California Institute of Technology, graduating in 1956, and earned his doctorate from the University of California, Los Angeles (UCLA) in 1960, where he was supervised by Basil Gordon and Ernst G. Straus. He became an assistant professor at the University of Washington in 1962, moved back to UCLA in 1964, and retired in 1991. After his retirement, he worked at the Center for Communications Research in La Jolla, California.
Cantor specialized in number theory and combinatorics. The Cantor–Zassenhaus algorithm for factoring polynomials is named after him; he and Hans Zassenhaus published it in 1981. He received the National Science Foundation Postdoctoral Fellowship in 1960 and a Sloan Foundation Fellowship in 1968. In 2012, he became one of the inaugural fellows of the American Mathematical Society. At the time of his death, he had been a member of the American Mathematical Society for 54 years.
Cantor lived in San Diego, California. He died on November 19, 2012, at the age of 77.
See also
Cantor's algorithm
References
1935 births
2012 deaths
20th-century American mathematicians
Combinatorialists
Number theorists
California Institute of Technology alumni
University of California, Los Angeles alumni
University of Washington faculty
University of California, Los Angeles faculty
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Ursula%20Hamenst%C3%A4dt | Ursula Hamenstädt (born 15 January 1961) is a German mathematician who works as a professor at the University of Bonn. Her primary research subject is differential geometry.
Education and career
Hamenstädt earned her PhD from the University of Bonn in 1986, under the supervision of Wilhelm Klingenberg.
Her dissertation, Zur Theorie der Carnot-Caratheodory Metriken und ihren Anwendungen [The theory of Carnot–Caratheodory metrics and their applications], concerned the theory of sub-Riemannian manifolds.
After completing her doctorate, she became a Miller Research Fellow at the University of California, Berkeley and then an assistant professor at the California Institute of Technology before returning to Bonn as a faculty member in 1990.
Honors
Hamenstädt was an invited speaker at the International Congress of Mathematicians in 2010. In 2012 she was elected to the German Academy of Sciences Leopoldina, and in the same year she became one of the inaugural fellows of the American Mathematical Society.
She was the Emmy Noether Lecturer of the German Mathematical Society in 2017.
Selected publications
References
External links
Home page
1961 births
Living people
German women mathematicians
California Institute of Technology faculty
Academic staff of the University of Bonn
Fellows of the American Mathematical Society
University of Bonn alumni
20th-century German mathematicians
21st-century German mathematicians
Differential geometers
20th-century women mathematicians
21st-century women mathematicians
20th-century German women
21st-century German women |
https://en.wikipedia.org/wiki/Multivariate%20testing%20%28disambiguation%29 | Multivariate testing is multivariate hypothesis testing.
Multivariate testing may also refer to:
Multivariate statistics
Multivariate testing in marketing
A/B testing |
https://en.wikipedia.org/wiki/Zak%20transform | In mathematics, the Zak transform (also known as the Gelfand mapping) is a certain operation which takes as input a function of one variable and produces as output a function of two variables. The output function is called the Zak transform of the input function. The transform is defined as an infinite series in which each term is a product of a dilation of a translation by an integer of the function and an exponential function. In applications of Zak transform to signal processing the input function represents a signal and the transform will be a mixed time–frequency representation of the signal. The signal may be real valued or complex-valued, defined on a continuous set (for example, the real numbers) or a discrete set (for example, the integers or a finite subset of integers). The Zak transform is a generalization of the discrete Fourier transform.
The Zak transform had been discovered by several people in different fields and was called by different names. It was called the "Gelfand mapping" because Israel Gelfand introduced it in his work on eigenfunction expansions. The transform was rediscovered independently by Joshua Zak in 1967 who called it the "k-q representation". There seems to be a general consensus among experts in the field to call it the Zak transform, since Zak was the first to systematically study that transform in a more general setting and recognize its usefulness.
Continuous-time Zak transform: Definition
In defining the continuous-time Zak transform, the input function is a function of a real variable. So, let f(t) be a function of a real variable t. The continuous-time Zak transform of f(t) is a function of two real variables one of which is t. The other variable may be denoted by w. The continuous-time Zak transform has been defined variously.
Definition 1
Let a be a positive constant. The Zak transform of f(t), denoted by Za[f], is a function of t and w defined by
.
Definition 2
The special case of Definition 1 obtained by taking a = 1 is sometimes taken as the definition of the Zak transform. In this special case, the Zak transform of f(t) is denoted by Z[f].
.
Definition 3
The notation Z[f] is used to denote another form of the Zak transform. In this form, the Zak transform of f(t) is defined as follows:
.
Definition 4
Let T be a positive constant. The Zak transform of f(t), denoted by ZT[f], is a function of t and w defined by
.
Here t and w are assumed to satisfy the conditions 0 ≤ t ≤ T and 0 ≤ w ≤ 1/T.
Example
The Zak transform of the function
is given by
where denotes the smallest integer not less than (the ceil function).
Properties of the Zak transform
In the following it will be assumed that the Zak transform is as given in Definition 2.
1. Linearity
Let a and b be any real or complex numbers. Then
2. Periodicity
3. Quasi-periodicity
4. Conjugation
5. Symmetry
If f(t) is even then
If f(t) is odd then
6. Convolution
Let denote convolution with respect to the variable t.
Invers |
https://en.wikipedia.org/wiki/Nine-point%20conic | In geometry, the nine-point conic of a complete quadrangle is a conic that passes through the three diagonal points and the six midpoints of sides of the complete quadrangle.
The nine-point conic was described by Maxime Bôcher in 1892. The better-known nine-point circle is an instance of Bôcher's conic. The nine-point hyperbola is another instance.
Bôcher used the four points of the complete quadrangle as three vertices of a triangle with one independent point:
Given a triangle and a point in its plane, a conic can be drawn through the following nine points:
the midpoints of the sides of ,
the midpoints of the lines joining to the vertices, and
the points where these last named lines cut the sides of the triangle.
The conic is an ellipse if lies in the interior of or in one of the regions of the plane separated from the interior by two sides of the triangle, otherwise the conic is a hyperbola. Bôcher notes that when is the orthocenter, one obtains the nine-point circle, and when is on the circumcircle of , then the conic is an equilateral hyperbola.
In 1912 Maud Minthorn showed that the nine-point conic is the locus of the center of a conic through four given points.
References
Fanny Gates (1894) Some Considerations on the Nine-point Conic and its Reciprocal, Annals of Mathematics 8(6):185–8, link from Jstor.
Eric W. Weisstein Nine-point conic from MathWorld.
Michael DeVilliers (2006) The nine-point conic: a rediscovery and proof by computer from International Journal of Mathematical Education in Science and Technology, a Taylor & Francis publication.
Christopher Bradley The Nine-point Conic and a Pair of Parallel Lines from University of Bath.
Further reading
W. G. Fraser (1906) "On relations of certain conics to a triangle", Proceedings of the Edinburgh Mathematical Society 25:38–41.
Thomas F. Hogate (1894) On the Cone of Second Order which is Analogous to the Nine-point Conic, Annals of Mathematics 7:73–6.
P. Pinkerton (1905) "On a nine-point conic, etc.", Proceedings of the Edinburgh Mathematical Society 24:31–3.
External links
Nine-point conic and Euler line generalization at Dynamic Geometry Sketches
Euclidean plane geometry
Projective geometry |
https://en.wikipedia.org/wiki/Michael%20W.%20Campbell | Michael W. Campbell (born 1978) is a Seventh-day Adventist historian, theologian, pastor, and educator. In 2022 he was appointed director of archives, statistics, and research at the North American Division of Seventh-day Adventists.
Biography
Campbell completed a BA at Southern Adventist University, an MA from Andrews University, and in 2008 a PhD from Andrews University. Campbell was ordained as a Seventh-day Adventist minister in 2008 and spent five years in pastoral ministry in Colorado and Kansas. He is the assistant editor of The Ellen G. White Encyclopedia. In 2007 he was one of the organizers of the 50th anniversary conference at Andrews University on Questions on Doctrine. He formerly served as professor of religion at Southwestern Adventist University and before that as professor of church history at the Adventist International Institute of Advanced Studies. He formerly edited the Journal of Asia Adventist Studies, a peer-reviewed academic journal.
Research
Campbell's doctoral dissertation focused on the 1919 Bible Conference. He argues that it was a germinal event in understanding Seventh-day Adventist theology after the death of Adventist prophetess Ellen G. White. Arthur Patrick states that his research helps "push back the horizons of Adventist understanding."
Books published
2023. We Stand on Their Shoulders
2022. 1922: The Rise of Adventist Fundamentalism
2020. The Pocket Dictionary for Understanding Adventism
2019. 1919: The Untold Story of Adventism's Struggle with Fundamentalism
2018. The Ellen G. White Pocket Dictionary (with Jud Lake)
2017. Here We Stand: Luther, the Reformation, and Seventh-day Adventism (co-editor)
2013. Discovering Ellen G. White
2013. Discovering Our Adventist Past
See also
History of the Seventh-day Adventist Church
Seventh-day Adventist Church
Questions on Doctrine
References
External links
Adventist International Institute of Advanced Studies profile
Seventh-day Adventist Periodical Index (SDAPI) search for articles by Campbell
Southern Adventist University alumni
Seventh-day Adventist religious workers
Protestant writers
1978 births
Living people
American Seventh-day Adventists
Andrews University alumni |
https://en.wikipedia.org/wiki/1977%E2%80%9378%20VfL%20Bochum%20season | The 1977–78 VfL Bochum season was the 40th season in club history.
Review and events
Matches
Legend
Bundesliga
DFB-Pokal
Squad
Squad and statistics
Squad, appearances and goals scored
Transfers
Summer
In:
Out:
Sources
External links
1977–78 VfL Bochum season at Weltfussball.de
1977–78 VfL Bochum season at kicker.de
1977–78 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Michael%20T.%20Anderson | Michael T. Anderson (born November 18, 1950 in Boulder, Colorado) is an American mathematician. He is a professor of mathematics at the State University of New York at Stony Brook. His research concerns differential geometry including Ricci curvature and minimal surfaces.
After doing his undergraduate studies at the University of California, Santa Barbara,
Anderson received his Ph.D. from the University of California, Berkeley in 1981 under the supervision of H. Blaine Lawson.
In 2012, Anderson became a fellow of the American Mathematical Society.
Major publications
Anderson, Michael T.; Schoen, Richard. Positive harmonic functions on complete manifolds of negative curvature. Ann. of Math. (2) 121 (1985), no. 3, 429–461.
Anderson, Michael T. Ricci curvature bounds and Einstein metrics on compact manifolds. J. Amer. Math. Soc. 2 (1989), no. 3, 455–490.
Anderson, Michael T. Convergence and rigidity of manifolds under Ricci curvature bounds. Invent. Math. 102 (1990), no. 2, 429–445.
References
1950 births
Living people
Fellows of the American Mathematical Society
Stony Brook University faculty
University of California, Santa Barbara alumni
University of California, Berkeley alumni
20th-century American mathematicians
21st-century American mathematicians |
https://en.wikipedia.org/wiki/Todd%20Arbogast | Todd Arbogast is an American mathematician. He is a professor of mathematics at the University of Texas at Austin, where he is a W. A. "Tex" Moncrief, Jr. Simulation-Based Engineering and Sciences Professor and Frank E. Gerth III Faculty Fellow. His research concerns the numerical analysis of partial differential equations.
Arbogast did his undergraduate studies at the University of Minnesota, in mathematics and physics, graduating in 1981.
He received his Ph.D. in mathematics from the University of Chicago in 1987 under the supervision of Jim Douglas, Jr.
In 2012, Arbogast became a Fellow of the American Mathematical Society.
See also
Louis François Antoine Arbogast
References
External links
Home page
Living people
Fellows of the American Mathematical Society
University of Texas at Austin faculty
University of Chicago alumni
University of Minnesota College of Science and Engineering alumni
20th-century American mathematicians
21st-century American mathematicians
Year of birth missing (living people) |
https://en.wikipedia.org/wiki/Matthias%20Aschenbrenner | Matthias Aschenbrenner (born 1972 in Bad Kötzting) is a German-American mathematician. He is a professor of mathematics and director of the logic group at the University of Vienna. His research interests include differential algebra and model theory.
Career
Aschenbrenner earned his "Vordiplom" at the University of Passau in 1996. In 2001, he received his Ph.D. from the University of Illinois at Urbana–Champaign, where he was a student of Lou van den Dries. For his dissertation, he was awarded the 2001 Sacks Prize by the Association for Symbolic Logic. After a visiting position at the University of California, Berkeley, Aschenbrenner joined the faculty at the University of Illinois at Chicago in 2003, moving to the University of California, Los Angeles in 2007. In 2012, Aschenbrenner became a Fellow of the American Mathematical Society. He was jointly awarded the 2018 Karp Prize with Lou van den Dries and Joris van der Hoeven "for their work in model theory, especially on asymptotic differential algebra and the model theory of transseries". In 2018, Aschenbrenner was an invited speaker at the International Congress of Mathematicians in Rio de Janeiro. Aschenbrenner moved to the University of Vienna in 2020, where he is also director of the logic group.
References
1972 births
Living people
German emigrants to the United States
Fellows of the American Mathematical Society
Academic staff of the University of Vienna
21st-century American mathematicians
Model theorists
University of Passau alumni
People from Bad Kötzting
University of Illinois College of Liberal Arts and Sciences alumni |
https://en.wikipedia.org/wiki/Andrea%20Paroni | Andrea Paroni (born 14 October 1989) is an Italian footballer who plays as a goalkeeper for club Virtus Entella.
Career statistics
References
1989 births
Living people
People from San Vito al Tagliamento
Footballers from Friuli Venezia Giulia
Italian men's footballers
Men's association football goalkeepers
Serie B players
Serie C players
Lega Pro Seconda Divisione players
Serie D players
Udinese Calcio players
Virtus Entella players |
https://en.wikipedia.org/wiki/G-prior | In statistics, the g-prior is an objective prior for the regression coefficients of a multiple regression. It was introduced by Arnold Zellner.
It is a key tool in Bayes and empirical Bayes variable selection.
Definition
Consider a data set , where the are Euclidean vectors and the are scalars.
The multiple regression model is formulated as
where the are random errors.
Zellner's g-prior for is a multivariate normal distribution with covariance matrix proportional to the inverse Fisher information matrix for , similar to a Jeffreys prior.
Assume the are i.i.d. normal with zero mean and variance . Let be the matrix with th row equal to .
Then the g-prior for is the multivariate normal distribution with prior mean a hyperparameter and covariance matrix proportional to , i.e.,
where g is a positive scalar parameter.
Posterior distribution of beta
The posterior distribution of is given as
where and
is the maximum likelihood (least squares) estimator of . The vector of regression coefficients can be estimated by its posterior mean under the g-prior, i.e., as the weighted average of the maximum likelihood estimator and ,
Clearly, as g →∞, the posterior mean converges to the maximum likelihood estimator.
Selection of g
Estimation of g is slightly less straightforward than estimation of .
A variety of methods have been proposed, including Bayes and empirical Bayes estimators.<ref
References
Bayesian statistics
Regression analysis |
https://en.wikipedia.org/wiki/Patricia%20E.%20Bauman | Patricia E. Bauman is an American mathematician who studies partial differential equations that model the behavior of liquid crystals and superconductors. She is a professor of mathematics at Purdue University.
Education and career
Bauman received her Ph.D. from the University of Minnesota in 1982, under the supervision of Eugene Fabes; her dissertation was Properties of Non-Negative Solutions of Second-Order Elliptic Equations and their Adjoints. She was a postdoctoral researcher at the Courant Institute of Mathematical Sciences and a C. L. E. Moore instructor of mathematics at the Massachusetts Institute of Technology before joining the Purdue University faculty.
Recognition
Bauman was given an AMS Centennial Fellowship for 1994 to 1995. In 2012, Bauman became a fellow of the American Mathematical Society.
She was elected chair of the Society for Industrial and Applied Mathematics Activity Group on Mathematical Aspects of Materials Science (SIAG/MS) for 2017–2019. She is a former AMS Council member at large.
References
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Fellows of the American Mathematical Society
University of Minnesota alumni
Purdue University faculty
20th-century women mathematicians
21st-century women mathematicians
Year of birth missing (living people)
20th-century American women academics
Massachusetts Institute of Technology School of Science faculty
21st-century American women academics |
https://en.wikipedia.org/wiki/Marilyn%20Breen | Marilyn Janet Breen (born 1944) is a professor of mathematics at the University of Oklahoma. Her research involves geometry, including visibility and orthogonal polygons.
Life and work
Breen graduated in 1966 from Agnes Scott College, and received her Ph.D. from Clemson University in 1970 under the supervision of William Ray Hare Jr. She joined the Oklahoma faculty in 1971 and was promoted to full professor in 1982.
Awards and honors
In her time at the University of Oklahoma, Breen won several awards for teaching and research, including an "outstanding teacher" award.
In 2012, Breen became a fellow of the American Mathematical Society.
Selected publications
References
Living people
1944 births
Women mathematicians
Fellows of the American Mathematical Society
Agnes Scott College alumni
Clemson University alumni
University of Oklahoma faculty |
https://en.wikipedia.org/wiki/Laura%20DeMarco | Laura Grace DeMarco is a professor of mathematics at Harvard University, whose research concerns dynamical systems and complex analysis.
Career
DeMarco received her Ph.D. from Harvard University in 2002 under the supervision of Curtis T. McMullen. She held an NSF Postdoctoral Fellowship and was an L. E. Dickson Instructor at the University of Chicago from September 2002 to August 2005. She was also an assistant professor at the University of Chicago, and then she moved to the University of Illinois at Chicago, where she was tenured and promoted to professor. She moved to Northwestern University in 2014, and was promoted to Henry S. Noyes Professor of Mathematics in 2019, before she moved to Harvard University in 2020.
DeMarco is an organizer of GROW (Graduate Research Opportunities for Women) undergraduate conference.
Awards and honors
In 2013, DeMarco became a fellow of the American Mathematical Society in the inaugural class of fellows. In 2017, she received the AMS Ruth Lyttle Satter Prize in Mathematics in Mathematics for her contributions to complex dynamics, potential theory, and the emerging field of arithmetic dynamics. In 2020, DeMarco was elected a member of the National Academy of Sciences.
She was an invited speaker at the 2018 International Congress of Mathematicians, speaking in the section on Dynamical Systems and Ordinary Differential Equations. She is the 2023 AWM-AMS Emmy Noether Lecturer in recognition of her "fundamental and influential contributions to complex dynamics, arithmetic dynamics, and arithmetic geometry."
Her work with Holly Krieger and Hexi Ye, "Uniform Manin–Mumford for a family of genus 2 curves", published in the Annals of Mathematics, won the 2020 Alexanderson Award of the American Institute of Mathematics.
Further reading
References
Year of birth missing (living people)
Living people
Women mathematicians
Dynamical systems theorists
Fellows of the American Mathematical Society
Harvard University alumni
American women mathematicians
Northwestern University faculty
University of Chicago faculty
University of Illinois Chicago faculty
Members of the United States National Academy of Sciences
Harvard University Department of Mathematics faculty
Harvard University faculty
21st-century American women |
https://en.wikipedia.org/wiki/Shelly%20Harvey | Shelly Lynn Harvey is a professor of Mathematics at Rice University. Her research interests include knot theory, low-dimensional topology, and group theory.
Early life
Harvey grew up in Rancho Cucamonga, California and graduated California Polytechnic State University in 1997. She received her Ph.D. from Rice University in 2002 under the supervision of Tim Cochran. After postdoctoral studies at the University of California, San Diego and the Massachusetts Institute of Technology, she returned to Rice University in 2005 as the first female tenure-track mathematician there.
Recognitions
Harvey was a Sloan Fellow in 2006. In 2012, she became one of the inaugural fellows of the American Mathematical Society.
Selected publications
.
.
.
References
External links
Shelly Harvey's official website
Living people
American women mathematicians
Fellows of the American Mathematical Society
California Polytechnic State University alumni
Rice University alumni
Rice University faculty
21st-century American mathematicians
Topologists
21st-century women mathematicians
Year of birth missing (living people)
21st-century American women scientists |
https://en.wikipedia.org/wiki/Jane%20M.%20Hawkins | Jane Margaret Hawkins is an American mathematician who works as a professor of mathematics at University of North Carolina at Chapel Hill. Her research concerns dynamical systems and complex dynamics, including cellular automata and Julia sets. More recent research has included work on cellular automata models for the spread of HIV, Hepatitis C and Ebola.
Education and career
Hawkins earned a B.S. in mathematics from the College of the Holy Cross, a Jesuit college in Worcester, Massachusetts, and received a Marshall Scholarship for graduate studies in the U.K. She then moved to England for graduate studies, receiving her Ph.D. from University of Warwick in 1981 under the supervision of Klaus Schmidt. Her dissertation, Type III Diffeomorphisms of Manifolds, concerned dynamical systems and ergodic theory.
She taught at the State University of New York, Stony Brook from 1980 to 1987, before moving to the University of North Carolina at Chapel Hill.
Her book Ergodic Dynamics was published by Springer in 2021.
Service
In 2004, Hawkins presented testimony to a subcommittee of the House Committee on Appropriations in support of grants to mathematical research, and again in 2012. She served as Treasurer of the American Mathematical Society, chair of the AMS Investment Committee, and Board of Trustees Member from 2011 until February 1, 2021. She is also a former AMS Council member at large. Hawkins has done a lot of outreach for women in mathematics, most notably participating in the George Washington University's Summer Program for Women in Math (SPWM) as a faculty member, guest lecturer, and panel moderator between 1999 and its final year in 2013.
Recognition
Hawkins was the first female valedictorian at the College of the Holy Cross. In 2012, Hawkins became an inaugural fellow of the American Mathematical Society.
Personal
Hawkins is married to mathematician Michael E. Taylor, whom she credits with encouraging her with continuing in research.
References
External links
Home page
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Cellular automatists
College of the Holy Cross alumni
Alumni of the University of Warwick
University of North Carolina at Chapel Hill faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
Year of birth missing (living people)
20th-century American women
21st-century American women
Marshall Scholars |
https://en.wikipedia.org/wiki/Ellen%20Kirkman | Ellen Elizabeth Kirkman is professor of mathematics at Wake Forest University. Her research interests include noncommutative algebra, representation theory, and homological algebra.
Education
She received her Ph.D. in Mathematics and M.A. in Statistics from Michigan State University in 1975.
Her doctoral dissertation, On the Characterization of Inertial Coefficient Rings, was supervised by Edward C. Ingraham.
Professional activities
Kirkman is on the board of directors of Enhancing Diversity in Graduate Education (EDGE), a program to help women pursuing studies in the mathematical sciences. From 2012 - 2020 she served as treasurer of the Association for Women in Mathematics (AWM).
Kirkman's professional activities include serving on the American Mathematical Society (AMS) Nominating Committee 2009–11, as a Mathematical Association of America (MAA) Governor 2006–8, on the Joint Data Committee of AMS-ASA-MAA-IMS-SIAM (2000– 2007 and 2009–present) and directing the CBMS 2010 survey of undergraduate mathematical sciences programs. She is an associate editor of Communications in Algebra.
Kirkman served as treasurer and was on the Executive Committee of the Association for Women in Mathematics from 2012 to 2020.
Recognition
In 2012, Kirkman became a fellow of the American Mathematical Society.
She is part of the 2019 class of fellows of the Association for Women in Mathematics.
She has also received service awards from Wake Forest University and the Southeastern Section of the MAA. She received the 2022 AWM Service Award "for her eight years of service (2012 – 2020) as AWM Treasurer and Chair of the Financial Oversight and Investment Committee, for her service on the Membership Portfolio Committee, and for her role as an organizer and a research leader in the WINART (Women in Noncommutative Algebra and Representation Theory) Research Network."
References
External links
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Michigan State University alumni
Wake Forest University faculty
Fellows of the American Mathematical Society
Fellows of the Association for Women in Mathematics
Algebraists
20th-century women mathematicians
21st-century women mathematicians
Year of birth missing (living people)
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Barbara%20L.%20Osofsky | Barbara L. Osofsky (born 1937) is a retired professor of mathematics at Rutgers University. Her research concerns abstract algebra. Osofsky's contributions to mathematics include her characterization of semisimple rings in terms of properties of cyclic modules. Osofsky also established a logical equivalence between the continuum hypothesis and statements about the global dimension of associative rings.
Career
Osofsky received her Ph.D. from Rutgers University in 1964. She then worked at Rutgers until 2004, when she retired. She served as acting chair of the Rutgers mathematics department in 1978.
Awards and honors
In 1973, Osofsky addressed a national meeting of the AMS. She was the first woman in 50 years to do so. She became the first female editor of an AMS journal in 1974 when she became the editor of Proceedings of the American Mathematical Society.
From 2000 to 2002, Osofsky served as First Vice-President of the Mathematical Association of America. In 2005, she was awarded the MAA meritorious service award.
In 2012, Osofsky became a fellow of the American Mathematical Society.
Selected publications
Osofsky, B. L. A generalization of quasi-Frobenius rings. J. Algebra 4 1966 373–387.
Osofsky, B. L. Rings all of whose finitely generated modules are injective. Pacific J. Math. 14 1964 645–650.
Osofsky, Barbara L.; Smith, Patrick F. Cyclic modules whose quotients have all complement submodules direct summands. J. Algebra 139 (1991), no. 2, 342–354.
References
Living people
1937 births
American women mathematicians
20th-century American mathematicians
21st-century American mathematicians
Algebraists
Fellows of the American Mathematical Society
Rutgers University alumni
Rutgers University faculty
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Alice%20Silverberg | Alice Silverberg (born 1958) is professor of Mathematics and Computer Science at the University of California, Irvine. She was faculty at the Ohio State University from 1984 through 2004. She has given over 300 lectures at universities around the world, and she has brought attention to issues of sexism and discrimination through her blog Alice's Adventures in Numberland.
Research
Silverberg's research concerns number theory and cryptography. With Karl Rubin, she introduced the CEILIDH system for torus-based cryptography in 2003, and she currently holds 10 patents related to cryptography. She is also known for her work on theoretical aspects of abelian varieties.
Education and career
Silverberg graduated from Harvard University in 1979, and received her Ph.D. from Princeton University in 1984 under the supervision of Goro Shimura. She began her academic career at Ohio State University in 1984 and became a full professor in 1996. She moved to the University of California at Irvine in 2004 as Professor of Mathematics and Computer Science, and in 2018 she was awarded the title of Distinguished Professor. Over the past 25 years she has organized or co-organized more than ten conferences in mathematics and cryptography, and has served on the program committees of more than twenty others. Silverberg has a long record of service with the American Mathematical Society and is currently a member of their nominating committee. She has served as an editor for the Association for Women in Mathematics since 2008, and recently joined the board of the Number Theory Foundation.
Honors
In 2012, Silverberg became a fellow of the American Mathematical Society. She was elected to the 2019 class of fellows of the Association for Women in Mathematics "For her outstanding research in number theory and deep commitment to the promotion of fairness and equal opportunity evidenced by her service and outreach efforts", also citing her many invited lectures and her blog.
Additional work
In 2017, Silverberg began a blog entitled Alice's Adventures in Numberland, which humorously discusses issues surrounding sexism in academia. This is a topic which she has previously discussed in interviews, and has been quoted on.
References
External links
Home page
Alice's Adventures in Numberland
Living people
1958 births
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Number theorists
Harvard University alumni
Princeton University alumni
Ohio State University faculty
University of California, Irvine faculty
Fellows of the American Mathematical Society
Fellows of the Association for Women in Mathematics
20th-century women mathematicians
21st-century women mathematicians
Scientists from California
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Problems%2C%20Problems%2C%20Problems | Problems, Problems, Problems is a series of educational mathematics textbooks ranging from grade 7-12. The math questions are from previous math contests held by the Centre for Education in Mathematics and Computing at the University of Waterloo. The series is a resource to complement the Canadian high school curriculum and for self-development, although it is also used to study for the CEMC Math Contests. Each combined grade level has several book volumes. Each book contains around 300 questions, and is categorized into various sections. Each question has a reference number in the form "year-contest-question number".
The Problems, Problems, Problems series collection is a National Library of Canada Cataloging in Publication (CIP).
References
Series of mathematics books |
https://en.wikipedia.org/wiki/Government%20Statistical%20Service | The Government Statistical Service (GSS) is the community of all civil servants in the United Kingdom who work in the collection, production and communication of official statistics. It includes not only statisticians, but also economists, social researchers, IT professionals, and secretarial and clerical staff. Members of the GSS work in the Office for National Statistics, most UK Government departments, and the devolved administrations. The GSS publishes around 2,000 sets of statistics each year, as well as providing professional advice and
analysis to decision-makers.
The National Statistician is the Head of the GSS.
The Government Statistician Group (GSG) is a subset of the GSS, and is the community of professional government statisticians who meet the standards for statistical qualifications and competence set by the National Statistician's Office.
History
The GSS was formed in 1968, in response to a series of recommendations made by Claus Moser, director of the Central Statistical Office, who recognised that 'society was going through radical changes, and social and economic policy was being made on incorrect and out of date statistics. Data was not being shared efficiently across government delaying its use in decisions. There was duplication of work in some areas and gaps in others. It lacked a joined-up approach across government.'
References
External links
GSS website
GSS homepage on gov.uk
GSS overview on ONS website
Statistical organisations in the United Kingdom
Civil Service (United Kingdom)
1968 establishments in the United Kingdom |
https://en.wikipedia.org/wiki/List%20of%20USM%20Alger%20records%20and%20statistics | USM Alger is an Algerian sports club based in Algiers which is best known for its professional association football team. They played their first match in 1937, but only won their first trophy in 1963, to become the first Algerian champions after independent.
USM Alger are one of the most successful clubs in Algeria, having domestically won the Algerian Ligue Professionnelle 1 7 times, the Algerian Cup 8 times and the Algerian Super Cup 2 times. Internationally, USM Alger has won the UAFA Club Championship once, winning the last edition in 2013. The club also reached the final match of the 2015 CAF Champions League but he lost to TP Mazembe.
This list includes the major honours won by USM Alger and all-time statistics and records set by the club, its players and its coaches. The players section includes the club's top goalscorers and those who have made most appearances in first-team competitive matches. It also displays international achievements by players representing USM Alger, and the highest transfer fees paid and received by the club.
Honours
As of the 2023–24 season, USM Alger have won a total of 20 titles (regional competitions not considered), of which 18 were achieved domestically and 2 were obtained in international competitions. The club's most recent honour is the 2023 CAF Super Cup.
Players
Appearances
Most appearances: Lamine Zemmamouche – 405 (2004–2009, 2011–2023);
Most appearances in a season: Mohamed Hamdoud (2003–04) – 42;
Oldest title winner: Lamine Zemmamouche – 38 years, 2 months and 15 days (2022–23 CAF Confederation Cup, 3 June 2023);
Oldest league title winner: Abdelaziz Ben Tifour – 35 years, 11 months and 9 days (1962–63 Algerian Championnat National, 16 June 1963);
Youngest league title winner: Mohamed Madani – 18 years and 23 days (1962–63 Algerian Championnat National, 16 June 1963);
Most league appearances: Lamine Zemmamouche – 298 (2004–2009, 2011–2023);
Most Algerian Cup appearances: Lamine Zemmamouche – 33 (2004–2009, 2011–2023);
Most African Cup appearances: Lamine Zemmamouche – 49 (2004–2009, 2011–2023);
Most league appearances by a non-Algerian player: Carolus Andriamatsinoro – 97 (2012–2017);
Most consecutive appearances in international club competitions: 19 – Lamine Zemmamouche;
Youngest debutant:
Youngest starter in the league: Zineddine Mekkaoui – 18 years, 4 months and 22 days (against NA Hussein Dey, 2004–05 Algerian Championnat National, 2 June 2005);
Youngest league debutant: Hocine Metref – 17 years, 10 months and 1 day (against MO Constantine, 2001–02 Algerian Championnat National, 2 November 2001);
Youngest debutant in the African Cup / CAF Champions League: Michael Eneramo – 19 years, 3 months and 8 days (against Olympic Azzaweya, 2005 CAF Champions League First round, first leg, 6 March 2005);
Youngest captain in the African Cup / CAF Champions League: Billel Dziri – 25 years, 1 months and 16 days (against CD Travadores, 1997 CAF Champions League First round, first leg, 8 March 1997);
Y |
https://en.wikipedia.org/wiki/Andrej%20Dujella | Andrej Dujella (born May 21, 1966 in Pula) is a Croatian professor of mathematics at the University of Zagreb and a fellow of the Croatian Academy of Sciences and Arts.
Life
Born in Pula, a native of Zadar, Dujella took part in the International Mathematical Olympiad, where he won a bronze medal in 1984.
He received his M.Sc. and Ph.D. in mathematics from the University of Zagreb with a dissertation titled "Generalized Diophantine–Davenport problem". His main area of research is number theory, in particular Diophantine equations, elliptic curves, and applications of number theory in cryptography. Dujella is author of the monograph "Number Theory" (translated from Croatian).
Dujella's main contribution to number theory is in connection to Diophantine m-tuples.
Dujella has shown that there exists no Diophantine 6-tuple and that there exist at most a finite number of Diophantine 5-tuples. He applied Diophantine tuples to construct elliptic curves with high rank. In 1998, Dujella and Attila Pethő introduced congruence method to obtain lower bound for number of Diophantine 5-tuples.
References
1966 births
Croatian mathematicians
Members of the Croatian Academy of Sciences and Arts
Living people
Number theorists
International Mathematical Olympiad participants
Faculty of Science, University of Zagreb alumni
Academic staff of the University of Zagreb |
https://en.wikipedia.org/wiki/Nicolas%20Careca | Nicolas Morês da Cruz (born 18 May 1997) commonly known as Nicolas Careca, is a Brazilian footballer who plays for Paysandu Sport Club, as a forward.
Career statistics
Honours
CRB
Campeonato Alagoano: 2022
References
External links
1997 births
Living people
Brazilian men's footballers
Brazilian expatriate men's footballers
Men's association football forwards
Campeonato Brasileiro Série A players
Campeonato Brasileiro Série B players
Ukrainian Premier League players
Liga Portugal 2 players
Grêmio Foot-Ball Porto Alegrense players
Figueirense FC players
Oeste Futebol Clube players
FC Vorskla Poltava players
G.D. Estoril Praia players
Clube de Regatas Brasil players
Guarani FC players
Brazilian expatriate sportspeople in Ukraine
Brazilian expatriate sportspeople in Portugal
Expatriate men's footballers in Ukraine
Expatriate men's footballers in Portugal
Footballers from Brasília |
https://en.wikipedia.org/wiki/Brooke%20Shipley | Brooke Elizabeth Shipley is an American mathematician. She works as a professor at the University of Illinois at Chicago, where she was head of the Department of Mathematics, Statistics and Computer Science from 2014 to 2022. Her research concerns homotopy theory and homological algebra.
Education and career
Shipley graduated from Harvard University in 1990. She earned her Ph.D. in 1995 from the Massachusetts Institute of Technology, under the supervision of Haynes Miller, for her work on the convergence of the homology spectral sequence of a cosimplicial space.
Shipley then was awarded a NSF Postdoctoral Research Fellowship. After postdoctoral studies at the University of Notre Dame and the University of Chicago, she joined the faculty of Purdue University in 1998 and earned tenure there in 2002. She then moved to University of Illinois at Chicago in 2003.
In 2009, Shipley became Co-Principal Investigator on UIC's National Science Foundation's ADVANCE grant to support the Women in Science and Engineering System Transformation (WISEST) program. She served as the director of WISEST from 2012 to 2013. She served as an American Mathematical Society Council member at large from 2018 to 2020.
Recognition
In 2014, she was elected as a fellow of the American Mathematical Society "for contributions to homotopy theory and homological algebra as well as for service to the mathematical community." Then in 2016, she became a representative of the Committee of Academic Sponsors at the Mathematical Sciences Research Institute
She and John Greenlees were the joint winners of the 2022 Senior Berwick Prize for their paper "An algebraic model for rational torus-equivariant spectra".
Selected Papers
Brooke Shipley, HZ-algebra spectra are differential graded algebras, American Journal of Mathematics, 129(2):351–379, 2007.
Awards
NSF Career Award (2002-2009)
Purdue University School of Science Outstanding Assistant Professor (2001)
References
External links
Home page
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Harvard University alumni
Massachusetts Institute of Technology alumni
Purdue University faculty
University of Illinois Chicago faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Ruth%20J.%20Williams | Ruth Jeannette Williams is an Australian-born American mathematician at the University of California, San Diego where she holds the Charles Lee Powell Chair as a Distinguished Professor of Mathematics. Her research concerns probability theory and stochastic processes.
Early life and education
Williams was born in Australia and moved to the United States in 1978.
Williams graduated from the University of Melbourne with a Bachelor of Sciences, with honors, in 1976 and a Master of Science in mathematics in 1978. Williams went on to earn her Ph.D. from Stanford University in 1983, under the supervision of Chung Kai-lai.
Recognition
Williams was president of the Institute of Mathematical Statistics from 2011 to 2012.
Williams is a member of the National Academy of Sciences and a fellow of the American Academy of Arts and Sciences, the American Association for the Advancement of Science, the American Mathematical Society, the Institute of Mathematical Statistics, the Institute for Operations Research and the Management Sciences, and the Society for Industrial and Applied Mathematics. In 1998 she was an Invited Speaker of the International Congress of Mathematicians in Berlin. Williams was an American Mathematical Society (AMS) Council member at large.
Her other awards and honors include:
Alfred P. Sloan Fellow (1988)
Guggenheim Fellow (2001)
Best Publication Award of the INFORMS Applied Probability Society (2007), jointly with Amber L. Puha and H. Christian Gromoll
John von Neumann Theory Prize (2016), jointly with Martin I. Reiman for "seminal research contributions over the past several decades, to the theory and applications of stochastic networks/systems and their heavy traffic approximations".
Honorary Doctorate from the University of Melbourne (2018)
Award for the Advancement of Women in Operations Research and the Management Sciences (2017), annual INFORMS meeting
Honorary Doctor of Science degree from La Trobe University in Australia
National Science Foundation Presidential Young Investigator
References
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
American statisticians
Women statisticians
Stanford University alumni
University of California, San Diego faculty
Fellows of the American Academy of Arts and Sciences
Fellows of the American Association for the Advancement of Science
Fellows of the American Mathematical Society
Fellows of the Institute of Mathematical Statistics
Fellows of the Institute for Operations Research and the Management Sciences
Fellows of the Society for Industrial and Applied Mathematics
Members of the United States National Academy of Sciences
John von Neumann Theory Prize winners
20th-century women mathematicians
21st-century women mathematicians
Probability theorists
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Katrin%20Wendland | Katrin Wendland (born 1970) is a German mathematical physicist who works as a professor at Trinity College Dublin.
Wendland earned a diploma in mathematics from the University of Bonn in 1996, and a PhD in physics from the University of Bonn in 2000, under the supervision of Werner Nahm. After being a lecturer and then senior lecturer at the University of Warwick from 2002 to 2006, she returned to Germany as a professor at the University of Augsburg, where she held the Chair for Analysis and Geometry. She moved to Albert-Ludwigs-Universität Freiburg in 2011 and then to Trinity College Dublin in 2022.
In 2009, Wendland was given the Medal for special merits for Bavaria in a united Europe by the Bavarian government. In 2010 she was an invited speaker at the International Congress of Mathematicians, with a talk entitled "On the geometry of singularities in quantum field theories". In 2012 she became one of the inaugural fellows of the American Mathematical Society. She was elected to the Akademie der Wissenschaften und der Literatur in 2013. In 2023 she was elected a fellow of Trinity College Dublin.
References
21st-century German mathematicians
21st-century German physicists
21st-century German women scientists
Academics of Trinity College Dublin
Fellows of the American Mathematical Society
Fellows of Trinity College Dublin
German women mathematicians
German women physicists
Living people
University of Bonn alumni
Academic staff of the University of Freiburg
1970 births
21st-century women mathematicians |
https://en.wikipedia.org/wiki/Ioana%20Dumitriu | Ioana Dumitriu (born July 6, 1976) is a Romanian-American mathematician who works as a professor of mathematics at the University of California, San Diego. Her research interests include the theory of random matrices, numerical analysis, scientific computing, and game theory.
Life
Dumitriu is the daughter of two Romanian electrical engineering professors from Bucharest. Early in her life she was identified as having mathematical talent, and at age 11 won a national mathematics contest. She entered mathematics training camps in preparation for participation on the Romanian team at the International Mathematical Olympiad, although her highest level of participation in the olympiad was the national semifinal.
As a 19-year-old freshman at NYU, Dumitriu already was taking graduate-level classes in mathematics. She graduated summa cum laude from NYU in 1999 with a B.A. in mathematics and a minor in computer science. She earned her Ph.D. in 2003 from the Massachusetts Institute of Technology under the supervision of Alan Edelman, with a thesis on Eigenvalue statistics for beta-ensembles. After postdoctoral research as a Miller Research Fellow at the University of California, Berkeley, she joined the faculty of the University of Washington in 2006, moving to UC San Diego in 2019.
Awards and honors
Dumitriu won the Alice T. Schafer prize for excellence in mathematics by an undergraduate woman in 1996. Also in 1996, as a sophomore at New York University, Dumitriu became the first woman to become a Putnam Fellow, meaning that she earned one of the top five scores at the William Lowell Putnam Mathematical Competition. In 1995, 1996, and 1997 she won the Elizabeth Lowell Putnam Award that is given to the top woman in the contest, a record that was not matched until ten years later when Alison Miller also won the same award in three consecutive years.
She won the Leslie Fox Prize for Numerical Analysis (given to a young numerical analysis researcher who excels both mathematically and in presentation skills) in 2007. In 2009 she received a CAREER Award from the National Science Foundation. In 2012, she became one of the inaugural fellows of the American Mathematical Society.
References
Selected publications
External links
1976 births
Living people
Numerical analysts
Romanian mathematicians
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Romanian emigrants to the United States
Courant Institute of Mathematical Sciences alumni
University of Washington faculty
Putnam Fellows
Fellows of the American Mathematical Society
Massachusetts Institute of Technology alumni
University of California, Berkeley people
20th-century women mathematicians
21st-century women mathematicians
University of California, San Diego faculty
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Judy%20L.%20Walker | Judy Leavitt Walker is an American mathematician. She is the Aaron Douglas Professor of Mathematics at the University of Nebraska–Lincoln, where she chaired the mathematics department from 2012 through 2016 and currently serves as the Associate Vice Chancellor for Faculty Affairs. Her research is in the area of algebraic coding theory.
In 2012, Walker became one of the inaugural Fellows of the American Mathematical Society.
Throughout her career, Walker has worked on issues related to women in mathematics. She is co-founder of the University of Nebraska's All Girls/All Math program for high school girls; she represented her department at the White House when they were recognized with a 1998 Presidential Award for Excellence in Science, Mathematics, and Engineering mentoring; and she is co-founder of the Nebraska Conference for Undergraduate Women in Mathematics.
Education
Walker graduated from the University of Michigan in 1990, and went on to graduate studies at the University of Illinois at Urbana–Champaign, earning her Ph.D. in 1996 under the supervision of Nigel Boston.
Career
She joined the University of Nebraska–Lincoln faculty in 1996, was promoted to full professor in 2006 and took up the Aaron Douglas Professorship in 2012. She held a visiting professorship at École Polytechnique Fédérale de Lausanne (EPFL) during fall 2011.
Academic interests
Walker's research is in coding theory, i.e., the study of error-correcting codes. Her primary contributions have been in algebraic geometry codes and low-density parity check codes.
Selected awards
Louise Hay Award for Contributions to Mathematics Education (Association for Women in Mathematics, 2016)
Outstanding Teaching and Instructional Creativity Award (University of Nebraska system, 2014)
George Pólya Lecturer (Mathematical Association of America, 2009-2011)
Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching (Mathematical Association of America, 2006)
Irving Reiner Memorial Award (University of Illinois Department of Mathematics, 1996)
Fellow of the Association for Women in Mathematics (2019)
2021 class of Fellows of the American Association for the Advancement of Science.
Selected publications
C. Curto, V. Itskov, K. Morrison, Z. Roth, J. Walker (2013), "Combinatorial neural codes from a mathematical coding theory perspective", Neural Computation, 25, No. 7: 1891-1925. Online pdf
N. Axvig, D.T. Dreher, K. Morrison, E. Psota, L.C. Pérez, J. Walker (2009), "Analysis of connections between pseudocodewords", IEEE Transactions on Information Theory, 55: 4099-4107. Online pdf
. Online PDF
. Online pdf
J. Walker (2000), "Codes and Curves", Student Mathematical Library, IAS/Park City Mathematical Subseries, 7. American Mathematical Society, Providence, RI. Online pdf
J.-F. Voloch, J. Walker (2000), "Euclidean weights of codes from elliptic curves over rings", Trans. Amer. Math. Soc., 352: 5063–5076. Online pdf
J. Walker (1999), "Algebraic geometr |
https://en.wikipedia.org/wiki/Abigail%20Thompson | Abigail A. Thompson (born 1958 in Norwalk, Connecticut) is an American mathematician. She works as a professor of mathematics at the University of California, Davis, where she specializes in knot theory and low-dimensional topology.
Education and career
Thompson graduated from Wellesley College in 1979, and earned her Ph.D. in 1986 from Rutgers University under the joint supervision of Martin Scharlemann and Julius L. Shaneson. After visiting positions at the Hebrew University of Jerusalem and the University of California, Berkeley, she joined the University of California Davis faculty in 1988. Thompson had a postdoctoral fellowship with the National Science Foundation from 1988 to 1991 and a Sloan Foundation Fellowship from 1991 to 1993. She was a member of the Institute for Advanced Study in 1990-1991, 2000-2001, and 2015-2016. She became the Chair of the Department of Mathematics at UC Davis in 2017. She is one of the current vice presidents of the American Mathematical Society; her term is February 1, 2019 to January 31, 2022.
Research
Thompson extended David Gabai's concept of thin position from knots to 3-manifolds and Heegaard splittings.
Education reform
Thompson has also been an activist for reform of primary and secondary school mathematics education. She has publicly attacked the Mathland-based curriculum in use in the mid-1990s when the oldest of her three children began studying mathematics in school, claiming that it provided an inadequate foundation in basic mathematical skills, left no opportunity for independent work, and was based on poorly written materials. As an alternative, she founded a program at UC Davis to improve teacher knowledge of mathematics, and became the director of the California State Summer School for Mathematics and Science, a month-long summer mathematics camp for high school students.
Recognition
Thompson won the 2003 Ruth Lyttle Satter Prize in Mathematics for her research on thin position and Heegard splittings. In 2013, she became one of the inaugural fellows of the American Mathematical Society.
In February 2020, Abigail Thompson has been recognized by the American Council of Trustees and Alumni (ACTA) as a "Hero of Intellectual Freedom." The award is due to an op-ed Thompson published in The Wall Street Journal on December 19, 2019, denouncing the use of mandatory diversity statements in faculty hiring practices in the University of California system. Thompson will deliver the keynote address at ACTA’s ATHENA Roundtable Conference on November 13, 2020. In December of 2019 she published a similar opinion piece under the heading "A word from... Abigail Thompson" in the Notices of the American Mathematical Society, of which she was one of the Vice Presidents at the time. Both opinion pieces generated a lot of discussion within the mathematics community and the academy in general, with official responses from the Association for Women in Mathematics, and the UC Davis Chancellor and Vice Chancellor |
https://en.wikipedia.org/wiki/1978%E2%80%9379%20VfL%20Bochum%20season | The 1978–79 VfL Bochum season was the 41st season in club history.
Review and events
Matches
Legend
Bundesliga
DFB-Pokal
Intertoto Cup
Squad
Squad and statistics
Squad, appearances and goals scored
Transfers
Summer
In:
Out:
Winter
In:
Out:
Sources
External links
1978–79 VfL Bochum season at Weltfussball.de
1978–79 VfL Bochum season at kicker.de
1978–79 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Susan%20Loepp | Susan Renee Loepp (born 1967) is an American mathematician who works as a professor of mathematics at Williams College. Her research concerns commutative algebra.
Professional career
Loepp graduated from Bethel College (Kansas) in 1989, and earned her Ph.D. in 1994 from the University of Texas at Austin, under the supervision of Raymond Heitmann. After postdoctoral studies at the University of Nebraska she took her present faculty position at Williams. She has publications in Journal of Algebra and Journal of Pure and Applied Algebra.
Book
With William Wootters, she is the co-author of the book Protecting Information: From Classical Error Correction to Quantum Cryptography (Cambridge University Press, 2006). The book covers topics in quantum cryptography and quantum computing and the potential impacts of quantum physics. These potential impacts include quantum computers which, if built, could crack our currently used public-key cryptosystems, and quantum cryptography which promises to provide an alternative to these cryptosystems.
Awards and honors
In 2007, Loepp won the Young Alumnus Award from Bethel College. In 2010, she won the Northeastern Section of the Mathematical Association of America’s Teaching Award. In 2012, she won the Deborah and Franklin Tepper Haimo Award for Distinguished College or University Teaching of Mathematics of the Mathematical Association of America, which honors “college or university teachers who have been widely recognized as extraordinarily successful and whose teaching effectiveness has been shown to have had influence beyond their own institutions.” In 2013, she was elected as one of the inaugural fellows of the American Mathematical Society.
Loepp was an American Mathematical Society (AMS) Council member at large from 2019-2021.
References
1967 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Bethel College (Kansas) alumni
University of Texas at Austin College of Natural Sciences alumni
Williams College faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Jane%20Cronin%20Scanlon | Jane Smiley Cronin Scanlon (July 17, 1922 – June 19, 2018) was an American mathematician and an emeritus professor of mathematics at Rutgers University. Her research concerned partial differential equations and mathematical biology.
Education and career
Scanlon earned a bachelor's degree in mathematics from Wayne University (now Wayne State University). She completed her Ph.D. in mathematics at the University of Michigan in 1949, under the supervision of Erich Rothe. Her dissertation was Branch Points of Solutions of Equations in Banach Space.
After working for the United States Air Force and the American Optical Company, she returned to academia as a lecturer at Wheaton College (Massachusetts) and then Stonehill College.
She moved to the Polytechnic Institute of Brooklyn in 1957, and to Rutgers in 1965. In 1974 Scanlon was elected as an AMS Member at Large and held the position until 1976. She retired in 1991. During her twenty-six years at Rutgers, she supervised seven doctoral students.
She died in June 2018 at the age of 95.
Recognition
Scanlon was a Noether Lecturer in 1985, and Pi Mu Epsilon J. Sutherland Frame Lecturer in 1989. Her talks concerned "entrainment of frequency" and the application of this principle to mathematical models of the Purkinje fibers in the heart. In 2012, she became one of the inaugural fellows of the American Mathematical Society.
Personal life
She married the physicist Joseph Scanlon in 1953. The two divorced in 1979. Upon her death, she was survived by four children and seven grandchildren.
Selected publications
Articles
Books
Advanced Calculus, Boston, Heath 1967
Differential equations: Introduction and Qualitative Theory, Dekker 1980, 2nd edition 1994, 3rd edition CRC/Chapman and Hall 2008
Fixed points and topological degrees in nonlinear analysis, American Mathematical Society 1964; 1995 pbk edition of 1972 reprint with corrections
Mathematical aspects of the Hodgkin-Huxley neural theory, Cambridge University Press 1987
Mathematics of Cell Electrophysiology, Dekker 1981
as editor
References
External links
1922 births
2018 deaths
20th-century American mathematicians
American women mathematicians
Wayne State University alumni
University of Michigan alumni
Wheaton College (Massachusetts) faculty
Rutgers University faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
20th-century American women
United States Air Force civilians
21st-century American women
Educators from Manhattan |
https://en.wikipedia.org/wiki/Katherine%20Flegal | Katherine Mayhew Flegal is an American epidemiologist and senior scientist at the Centers for Disease Control and Prevention's National Center for Health Statistics. She is one of the most highly cited scientists in the field of the epidemiology of obesity according to Thomson Reuters and has been called "one of the great epidemiologists" by former FDA Commissioner David A. Kessler.
Education and career
Flegal was born in Berkeley, California, where she grew up. She earned a bachelor's degree in anthropology from the University of California, Berkeley (UC) in 1967 and lived in Turkey for 7 months. Following that she had a job as a programmer trainee at the Alameda County Data Processing Center and became responsible for its food stamp program. This gave her extensive experience in data management, coding, and record keeping. After three years and some community college courses in chemistry and biology, Flegal returned to UC to earn a second bachelor’s degree, in food and nutrition.
Next Flegal completed Master's and Ph.D. degrees at Cornell University, expanding her interests in statistics and epidemiology and receiving her Ph.D. in 1982. When a post-doctoral position at the University of Pittsburgh offered little opportunity for research, Flegal completed a second master's degree there, in public health (MPH). In 1987, after working at the University of Michigan's biostatistics department, Flegal began working at the Centers for Disease Control (CDC).
Research
Flegal is well known for a series of influential and highly cited articles on the prevalence of obesity in U.S. children and adults.
In 1994, Flegal and her CDC co-authors were among the first to publish data indicating that the percentage of overweight people in the United States had been increasing from the 1980s onwards. Their paper appeared in the Journal of the American Medical Association (JAMA).
In addition, she was a major contributor to the development of the 2000 CDC growth charts, used in the U.S. to assess the growth patterns of infants and children.
In 2005 Flegal and co-authors from CDC and NIH published a study in JAMA which found that being overweight was associated with lower mortality than normal weight and that obesity was associated with slightly higher mortality. The study received considerable opposition, in part because its conclusions differed from those of another paper published by senior CDC authors in March 2004. After considerable discussion, the CDC accepted Flegal's figures as correct. Flegal herself has stated "Our paper was straightforward and defensible, used only publicly available data, and corrected the errors in several previous papers on the topic". Flegal's article received CDC's highest science award, the Charles C. Shepard award, in 2006.
In 2013, Flegal was the lead author of a systematic review and meta-analysis published in JAMA regarding the association of overweight and obesity with mortality. In a large sample, drawn from other countri |
https://en.wikipedia.org/wiki/Orientation%20of%20a%20vector%20bundle | In mathematics, an orientation of a real vector bundle is a generalization of an orientation of a vector space; thus, given a real vector bundle π: E →B, an orientation of E means: for each fiber Ex, there is an orientation of the vector space Ex and one demands that each trivialization map (which is a bundle map)
is fiberwise orientation-preserving, where Rn is given the standard orientation. In more concise terms, this says that the structure group of the frame bundle of E, which is the real general linear group GLn(R), can be reduced to the subgroup consisting of those with positive determinant.
If E is a real vector bundle of rank n, then a choice of metric on E amounts to a reduction of the structure group to the orthogonal group O(n). In that situation, an orientation of E amounts to a reduction from O(n) to the special orthogonal group SO(n).
A vector bundle together with an orientation is called an oriented bundle. A vector bundle that can be given an orientation is called an orientable vector bundle.
The basic invariant of an oriented bundle is the Euler class. The multiplication (that is, cup product) by the Euler class of an oriented bundle gives rise to a Gysin sequence.
Examples
A complex vector bundle is oriented in a canonical way.
The notion of an orientation of a vector bundle generalizes an orientation of a differentiable manifold: an orientation of a differentiable manifold is an orientation of its tangent bundle. In particular, a differentiable manifold is orientable if and only if its tangent bundle is orientable as a vector bundle. (note: as a manifold, a tangent bundle is always orientable.)
Operations
To give an orientation to a real vector bundle E of rank n is to give an orientation to the (real) determinant bundle of E. Similarly, to give an orientation to E is to give an orientation to the unit sphere bundle of E.
Just as a real vector bundle is classified by the real infinite Grassmannian, oriented bundles are classified by the infinite Grassmannian of oriented real vector spaces.
Thom space
From the cohomological point of view, for any ring Λ, a Λ-orientation of a real vector bundle E of rank n means a choice (and existence) of a class
in the cohomology ring of the Thom space T(E) such that u generates as a free -module globally and locally: i.e.,
is an isomorphism (called the Thom isomorphism), where "tilde" means reduced cohomology, that restricts to each isomorphism
induced by the trivialization . One can show, with some work, that the usual notion of an orientation coincides with a Z-orientation.
See also
The integration along the fiber
Orientation bundle (or orientation sheaf) - this is used to formulate the Thom isomorphism for non-oriented bundles.
References
J.P. May, A Concise Course in Algebraic Topology. University of Chicago Press, 1999.
Linear algebra
Analytic geometry
Orientation (geometry) |
https://en.wikipedia.org/wiki/Cobordism%20ring | In mathematics, the oriented cobordism ring is a ring where elements are oriented cobordism classes of manifolds, the multiplication is given by the Cartesian product of manifolds and the addition is given as the disjoint union of manifolds. The ring is graded by dimensions of manifolds and is denoted by
where consists of oriented cobordism classes of manifolds of dimension n. One can also define an unoriented cobordism ring, denoted by . If O is replaced U, then one gets the complex cobordism ring, oriented or unoriented.
In general, one writes for the cobordism ring of manifolds with structure B.
A theorem of Thom says:
where MO is the Thom spectrum.
Notes
References
External links
bordism ring in nLab
The unoriented cobordism ring, a blog post by Akhil Mathew
Algebraic topology |
https://en.wikipedia.org/wiki/Atiyah%E2%80%93Bott%20formula | In algebraic geometry, the Atiyah–Bott formula says the cohomology ring
of the moduli stack of principal bundles is a free graded-commutative algebra on certain homogeneous generators. The original work of Michael Atiyah and Raoul Bott concerned the integral cohomology ring of .
See also
Borel's theorem, which says that the cohomology ring of a classifying stack is a polynomial ring.
Notes
References
Theorems in algebraic geometry |
https://en.wikipedia.org/wiki/1945%E2%80%9346%20Rochester%20Royals%20season | The 1945–46 Rochester Royals season was the franchise's first season in the National Basketball League.
Season standings
Eastern Division
Western Division
Statistics
Regular season
Playoffs
Awards and records
NBL Rookie of the Year: Red Holzman
All-NBL First Team: George Glamack, Red Holzman
All-NBL Second Team: Al Cervi
Transactions
References
External links
Sacramento Kings History Timeline | Sacramento Kings
Sacramento Kings seasons
Rochester
National Basketball League (United States) championship seasons
Rochester Royals
Rochester Royals |
https://en.wikipedia.org/wiki/Prosper%20Kasim | Prosper Kasim (born 15 December 1996), often written as Kasim Prosper, is a Ghanaian professional footballer who plays as a forward for USL Championship club Birmingham Legion.
Career statistics
References
External links
(archive)
IFK Göteborg profile
1996 births
Living people
Ghanaian men's footballers
Ghana men's under-20 international footballers
Men's association football midfielders
IFK Göteborg players
Norrby IF players
Mjällby AIF players
Birmingham Legion FC players
Allsvenskan players
Superettan players
Ettan Fotboll players
Ghanaian expatriate men's footballers
Expatriate men's footballers in Sweden
Expatriate men's soccer players in the United States
Ghanaian expatriate sportspeople in Sweden
International Allies F.C. players
USL Championship players |
https://en.wikipedia.org/wiki/Tankosiq | Tankosiq is a village in Ferizaj Municipality, Kosovo. According to the Kosovo Agency of Statistics (KAS) estimate from the 2011 census, there were 2,096 people residing in Tankosiq, with Albanians constituting the majority of the population.
History
During World War II, the village's Serb population was subjected to expulsions, murders and pillaging by Albanian paramilitaries.
References
Villages in Ferizaj |
https://en.wikipedia.org/wiki/Sheaf%20of%20modules | In mathematics, a sheaf of O-modules or simply an O-module over a ringed space (X, O) is a sheaf F such that, for any open subset U of X, F(U) is an O(U)-module and the restriction maps F(U) → F(V) are compatible with the restriction maps O(U) → O(V): the restriction of fs is the restriction of f times that of s for any f in O(U) and s in F(U).
The standard case is when X is a scheme and O its structure sheaf. If O is the constant sheaf , then a sheaf of O-modules is the same as a sheaf of abelian groups (i.e., an abelian sheaf).
If X is the prime spectrum of a ring R, then any R-module defines an OX-module (called an associated sheaf) in a natural way. Similarly, if R is a graded ring and X is the Proj of R, then any graded module defines an OX-module in a natural way. O-modules arising in such a fashion are examples of quasi-coherent sheaves, and in fact, on affine or projective schemes, all quasi-coherent sheaves are obtained this way.
Sheaves of modules over a ringed space form an abelian category. Moreover, this category has enough injectives, and consequently one can and does define the sheaf cohomology as the i-th right derived functor of the global section functor .
Examples
Given a ringed space (X, O), if F is an O-submodule of O, then it is called the sheaf of ideals or ideal sheaf of O, since for each open subset U of X, F(U) is an ideal of the ring O(U).
Let X be a smooth variety of dimension n. Then the tangent sheaf of X is the dual of the cotangent sheaf and the canonical sheaf is the n-th exterior power (determinant) of .
A sheaf of algebras is a sheaf of module that is also a sheaf of rings.
Operations
Let (X, O) be a ringed space. If F and G are O-modules, then their tensor product, denoted by
or ,
is the O-module that is the sheaf associated to the presheaf (To see that sheafification cannot be avoided, compute the global sections of where O(1) is Serre's twisting sheaf on a projective space.)
Similarly, if F and G are O-modules, then
denotes the O-module that is the sheaf . In particular, the O-module
is called the dual module of F and is denoted by . Note: for any O-modules E, F, there is a canonical homomorphism
,
which is an isomorphism if E is a locally free sheaf of finite rank. In particular, if L is locally free of rank one (such L is called an invertible sheaf or a line bundle), then this reads:
implying the isomorphism classes of invertible sheaves form a group. This group is called the Picard group of X and is canonically identified with the first cohomology group (by the standard argument with Čech cohomology).
If E is a locally free sheaf of finite rank, then there is an O-linear map given by the pairing; it is called the trace map of E.
For any O-module F, the tensor algebra, exterior algebra and symmetric algebra of F are defined in the same way. For example, the k-th exterior power
is the sheaf associated to the presheaf . If F is locally free of rank n, then is called the determinant line |
https://en.wikipedia.org/wiki/List%20of%20Prva%20HNL%20players | This is a list of Prva HNL players who have made 200 or more appearances in the Croatian First Football League. Statistics are updated as of 16 April 2022.
Key
Players with this background and symbol are still playing in the Prva HNL. Bold indicates current club.
The name used for each club is the name they had when player most recently played a league match for them.
Seasons = number of seasons a player had participated in Prva HNL; Years = a span between the first and most recent year a player had played a match in Prva HNL
List of players
References
General
Prva HNL players
Association football player non-biographical articles |
https://en.wikipedia.org/wiki/Ana%20Caraiani | Ana Caraiani (born 1985) is a Romanian-American mathematician, who is a Royal Society University Research Fellow and Hausdorff Chair at the University of Bonn. Her research interests include algebraic number theory and the Langlands program.
Education
She was born in Bucharest and studied at Mihai Viteazul High School. In 2001, Caraiani became the first Romanian female competitor in 15 years at the International Mathematical Olympiad, where she won a silver medal. In the following two years, she won two gold medals.
After graduating high school in 2003, she pursued her studies in the United States. As an undergraduate student at Princeton University, Caraiani was a two-time Putnam Fellow (the only female competitor at the William Lowell Putnam Mathematical Competition to win more than once) and Elizabeth Lowell Putnam Award winner. Caraiani graduated summa cum laude from Princeton in 2007, with an undergraduate thesis on Galois representations supervised by Andrew Wiles.
Caraiani did her graduate studies at Harvard University under the supervision of Wiles' student Richard Taylor, earning her Ph.D. in 2012 with a dissertation concerning local-global compatibility in the Langlands correspondence.
Career
After spending a year as an L.E. Dickson Instructor at the University of Chicago, she returned to Princeton and the Institute for Advanced Study as a Veblen Instructor and NSF Postdoctoral Fellow. In 2016, she moved to the Hausdorff Center for Mathematics as a Bonn Junior Fellow. She moved to Imperial College London in 2017 as a Royal Society University Research Fellow and Senior Lecturer. In 2019, she became a Royal Society University Research Fellow and Reader at Imperial College London. As of 2021, Caraiani is a full professor at Imperial College London. She rejoined the University of Bonn in 2022 as Hausdorff Chair.
Research
Caraiani's research work includes the papers "Patching and the p-adic local Langlands correspondence" (2016), "On the generic part of the cohomology of compact unitary Shimura varieties" (2017) with Peter Scholze, and "Potential automorphy over CM fields" (2023). These three papers all happen to be directly related to the Langlands program, but she does have other interests.
Caraiani discusses the Langlands program from a more general perspective in the survey article "New frontiers in Langlands reciprocity".
Recognition
In 2007, the Association for Women in Mathematics awarded Caraiani their Alice T. Schafer Prize. In 2018, she was one of the winners of the Whitehead Prize of the London Mathematical Society.
She was elected as a Fellow of the American Mathematical Society in the 2020 Class, for "contributions to arithmetic geometry and number theory, in particular the -adic Langlands program". She is one of the 2020 winners of the EMS Prize. In September 2022 she was awarded the 2023 New Horizons in Mathematics Prize.
References
External links
Caraiani's scores at the IMO
Professional home page
Personal home pag |
https://en.wikipedia.org/wiki/1980%E2%80%9381%20VfL%20Bochum%20season | The 1980–81 VfL Bochum season was the 43rd season in club history.
Review and events
Matches
Legend
Bundesliga
DFB-Pokal
Intertoto Cup
Squad
Squad and statistics
Squad, appearances and goals scored
Transfers
Summer
In:
Out:
Sources
External links
1980–81 VfL Bochum season at Weltfussball.de
1980–81 VfL Bochum season at kicker.de
1980–81 VfL Bochum season at Fussballdaten.de
Bochum
VfL Bochum seasons |
https://en.wikipedia.org/wiki/Cotangent%20sheaf | In algebraic geometry, given a morphism f: X → S of schemes, the cotangent sheaf on X is the sheaf of -modules that represents (or classifies) S-derivations in the sense: for any -modules F, there is an isomorphism
that depends naturally on F. In other words, the cotangent sheaf is characterized by the universal property: there is the differential such that any S-derivation factors as with some .
In the case X and S are affine schemes, the above definition means that is the module of Kähler differentials. The standard way to construct a cotangent sheaf (e.g., Hartshorne, Ch II. § 8) is through a diagonal morphism (which amounts to gluing modules of Kähler differentials on affine charts to get the globally-defined cotangent sheaf.) The dual module of the cotangent sheaf on a scheme X is called the tangent sheaf on X and is sometimes denoted by .
There are two important exact sequences:
If S →T is a morphism of schemes, then
If Z is a closed subscheme of X with ideal sheaf I, then
The cotangent sheaf is closely related to smoothness of a variety or scheme. For example, an algebraic variety is smooth of dimension n if and only if ΩX is a locally free sheaf of rank n.
Construction through a diagonal morphism
Let be a morphism of schemes as in the introduction and Δ: X → X ×S X the diagonal morphism. Then the image of Δ is locally closed; i.e., closed in some open subset W of X ×S X (the image is closed if and only if f is separated). Let I be the ideal sheaf of Δ(X) in W. One then puts:
and checks this sheaf of modules satisfies the required universal property of a cotangent sheaf (Hartshorne, Ch II. Remark 8.9.2). The construction shows in particular that the cotangent sheaf is quasi-coherent. It is coherent if S is Noetherian and f is of finite type.
The above definition means that the cotangent sheaf on X is the restriction to X of the conormal sheaf to the diagonal embedding of X over S.
Relation to a tautological line bundle
The cotangent sheaf on a projective space is related to the tautological line bundle O(-1) by the following exact sequence: writing for the projective space over a ring R,
(See also Chern class#Complex projective space.)
Cotangent stack
For this notion, see § 1 of
A. Beilinson and V. Drinfeld, Quantization of Hitchin’s integrable system and Hecke eigensheaves
There, the cotangent stack on an algebraic stack X is defined as the relative Spec of the symmetric algebra of the tangent sheaf on X. (Note: in general, if E is a locally free sheaf of finite rank, is the algebraic vector bundle corresponding to E.)
See also: Hitchin fibration (the cotangent stack of is the total space of the Hitchin fibration.)
Notes
See also
cotangent complex
References
External links
Algebraic geometry |
https://en.wikipedia.org/wiki/Dual%20module | In mathematics, the dual module of a left (respectively right) module M over a ring R is the set of module homomorphisms from M to R with the pointwise right (respectively left) module structure. The dual module is typically denoted M∗ or .
If the base ring R is a field, then a dual module is a dual vector space.
Every module has a canonical homomorphism to the dual of its dual (called the double dual). A reflexive module is one for which the canonical homomorphism is an isomorphism. A torsionless module is one for which the canonical homomorphism is injective.
Example: If is a finite commutative group scheme represented by a Hopf algebra A over a commutative ring k, then the Cartier dual is the Spec of the dual k-module of A.
References
Module theory |
https://en.wikipedia.org/wiki/Agata%20Smoktunowicz | Agata Smoktunowicz FRSE (born 12 October 1973) is a Polish mathematician who works as a professor at the University of Edinburgh. Her research is in abstract algebra.
Contributions
Smoktunowicz's contributions to mathematics include constructing noncommutative nil rings, solving a "famous problem" formulated in 1970 by Irving Kaplansky.
She proved the Artin–Stafford gap conjecture according to which the Gelfand–Kirillov dimension of a graded domain cannot fall within the open interval (2,3). She also found an example of a nil ideal of a ring R that does not lift to a nil ideal of the polynomial ring R[X], disproving a conjecture of Amitsur and hinting that the Köthe conjecture might be false.
Awards and honours
Smoktunowicz was an invited speaker at the International Congress of Mathematicians in 2006. She won the Whitehead Prize of the London Mathematical Society in 2006, the European Mathematical Society Prize in 2008, and the Sir Edmund Whittaker Memorial Prize of the Edinburgh Mathematical Society in 2009. In 2009, she was elected as a fellow of the Royal Society of Edinburgh, and in 2012, she became one of the inaugural fellows of the American Mathematical Society. She also won the Polish Academy of Sciences annual research prize in 2018.
Education and career
Smoktunowicz earned a master's degree from the University of Warsaw in 1997, a PhD in 1999 from the Institute of Mathematics of the Polish Academy of Sciences, and a habilitation in 2007, again from the Polish Academy of Sciences. After temporary positions at Yale University and the University of California, San Diego, she joined the University of Edinburgh in 2005, and was promoted to professor there in 2007.
Selected publications
.
.
.
.
References
1973 births
Living people
Polish women mathematicians
University of Warsaw alumni
Academics of the University of Edinburgh
Whitehead Prize winners
Fellows of the American Mathematical Society
Fellows of the Royal Society of Edinburgh
20th-century Polish mathematicians
21st-century Polish mathematicians
Algebraists
Polish emigrants to the United Kingdom
20th-century women mathematicians
21st-century women mathematicians
Sir Edmund Whittaker Memorial Prize winners |
https://en.wikipedia.org/wiki/Rebecca%20A.%20Herb | Rebecca A. Herb (born 1948) is an American mathematician, a professor emerita at the University of Maryland. Her research involves abstract algebra and Lie groups.
In 2012, Herb became one of the inaugural fellows of the American Mathematical Society.
In 2013, she was one of ten recipients of the first Service Awards of the Association for Women in Mathematics “for her service as AWM Treasurer (2004–2012), and her help during AWM’s transition from its headquarters at the University of Maryland to the management company STAT.”
Herb earned her Ph.D. in 1974 from the University of Washington under the supervision of Garth William Warner, Jr. From 2004 until 2012 (when she was succeeded by Ellen Kirkman) Herb was treasurer of the Association for Women in Mathematics.
Herb was an American Mathematical Society (AMS) Council member at large.
References
1948 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
University of Washington alumni
University of Maryland, College Park faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Rekha%20R.%20Thomas | Rekha Rachel Thomas is a mathematician and operations researcher. She works as a professor of mathematics at the University of Washington, and was the Robert R. and Elaine F. Phelps Professor there from 2008 until 2012. Her research interests include mathematical optimization and computational algebra.
Thomas earned a PhD in operations research from Cornell University in 1994, supervised by Bernd Sturmfels; her dissertation concerned Gröbner bases and integer programming. Prior to joining the University of Washington in 2000, she did postdoctoral studies at Yale University and the Zuse Institute Berlin, and held a faculty position at Texas A&M University beginning in 1995.
Thomas is the author of the textbook Lectures in Geometric Combinatorics (Student Mathematical Library, 33, American Mathematical Society, 2006). She was a plenary speaker at the 21st International Symposium on Mathematical Programming in 2012.
In 2013 she became one of the inaugural fellows of the American Mathematical Society.
References
External links
Year of birth missing (living people)
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
Cornell University alumni
University of Texas at Austin faculty
University of Washington faculty
Fellows of the American Mathematical Society
20th-century women mathematicians
21st-century women mathematicians
Combinatorialists
Operations researchers
20th-century American women
21st-century American women
American academics of Indian descent |
https://en.wikipedia.org/wiki/Dialgebra | In abstract algebra, a dialgebra is the generalization of both algebra and coalgebra. The notion was originally introduced by Lambek as "subequalizers". Many algebraic notions have previously been generalized to dialgebras. Dialgebra also attempts to obtain Lie algebras from associated algebras.
See also
F-algebra
References
Further reading
dialgebra in nLab
Algebra |
https://en.wikipedia.org/wiki/Gigliola%20Staffilani | Gigliola Staffilani (born March 24, 1966) is an Italian-American mathematician who works as the Abby Rockefeller Mauze Professor of Mathematics at the Massachusetts Institute of Technology. Her research concerns harmonic analysis and partial differential equations, including the Korteweg–de Vries equation and Schrödinger equation.
Education and career
Staffilani grew up on a farm in Martinsicuro in central Italy, speaking only the local dialect, and with no books until her older brother brought some back from his school. Her father died when she was 10, and her mother decided that she did not need to continue on to high school, but her brother helped her change her mother's mind. She came to love mathematics at her school, and was encouraged by her teachers and brother to continue her studies, with the idea that she could return to Martinsicuro as a mathematics teacher. She earned a fellowship to study at the University of Bologna, where she earned a laurea in mathematics in 1989 with an undergraduate thesis on Green's functions for elliptic partial differential equations.
At the suggestion of one of her professors at Bologna, she moved to the University of Chicago for her graduate studies, to study with Carlos Kenig. This was a big change in her previous plans, because it would mean that she could not return to Martinsicuro. When she arrived at Chicago, still knowing very little English and not having taken the Test of English as a Foreign Language, she had the wrong type of visa to obtain the teaching fellowship she had been promised. She almost returned home, but remained after Paul Sally intervened and loaned her enough money to get by until the issue could be resolved. At Chicago, she studied dispersive partial differential equations with Kenig, earning a master's degree in 1991 and a Ph.D. in 1995.
After postdoctoral studies at the Institute for Advanced Study, Stanford University, and Princeton University, Staffilani took a tenure-track faculty position at Stanford in 1999, and earned tenure there in 2001. While at Stanford, she met her husband, Tomasz Mrowka, a mathematics professor at MIT, and after a year and a half found a faculty position closer to him at Brown University. She moved to MIT in 2002, where, in 2006 she became the second female full professor of mathematics. She served as an American Mathematical Society Council member at large from 2018 to 2020.
Collaboration
Staffilani is a frequent collaborator with James Colliander, Markus Keel, Hideo Takaoka, and Terence Tao, forming a group known as the "I-team". The name of this group has been said to come from the notation for a mollification operator used in the team's method of almost conserved quantities, or as an abbreviation for "interaction", referring both to the teamwork of the group and to the interactions of light waves with each other. The group's work was featured prominently in Fefferman's 2006 Fields Medal citations for group member Tao.
Awards and honors
|
https://en.wikipedia.org/wiki/Irene%20M.%20Gamba | Irene Martínez Gamba (born 1957) is an Argentine–American mathematician. She works as a professor of mathematics at the University of Texas at Austin, where she holds the W.A. Tex Moncrief, Jr. Chair in Computational Engineering and Sciences and is head of the Applied Mathematics Group in the Institute for Computational Engineering and Sciences.
Education and career
Gamba graduated from the University of Buenos Aires in 1981. She went to the University of Chicago for her graduate studies, earning a master's degree in 1985 and a Ph.D. in 1989, under the supervision of Jim Douglas, Jr.
After postdoctoral studies at Purdue University and the Courant Institute of Mathematical Sciences of New York University, she became an Assistant Professor at NYU in 1994 and Associate Professor in 1996. She became a Professor at the University of Texas at Austin in 1997. At the University of Texas, she was the Joe B. and Louise Cook Professor from 2007 to 2013, the John T. Stuart III Centennial Professor from 2013 to 2014, and the W.A. Tex Moncreif, Jr. Chair in Computational Sciences and Engineering III since 2014.
Recognition
In 2012, Gamba became a fellow of the Society for Industrial and Applied Mathematics, and one of the inaugural fellows of the American Mathematical Society. The Association for Women in Mathematics selected her as their 2014 Sonia Kovalevsky Lecturer. She is also a former AMS Council Member at large.
References
External links
Home page
1957 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Argentine mathematicians
Argentine women mathematicians
American women mathematicians
University of Buenos Aires alumni
University of Chicago alumni
New York University faculty
University of Texas at Austin faculty
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/James%20McClennon | James William McClennon (16 December 1900 – 1971) was an English professional footballer who played in the Football League for Brentford and Grimsby Town as a full back.
Career statistics
References
1900 births
Sportspeople from Tynemouth
Footballers from Tyne and Wear
English men's footballers
English Football League players
Men's association football fullbacks
Brentford F.C. players
Walker Celtic F.C. players
Grimsby Town F.C. players
1971 deaths
West Stanley F.C. players |
https://en.wikipedia.org/wiki/Linda%20Preiss%20Rothschild | Linda Preiss Rothschild ( Preiss; born February 28, 1945) is a professor emeritus of mathematics at the University of California, San Diego. Her thesis research concerned Lie groups, but subsequently her interests broadened to include also polynomial factorization, partial differential equations, harmonic analysis, and the theory of several complex variables.
Education and career
The daughter of Philadelphia fur merchants, she was unable to attend the best high school of the city, which were then restricted to boys. She graduated from the University of Pennsylvania in 1966. Rejected from graduate study at Princeton University because it was also male-only, she earned her Ph.D. in 1970 from the Massachusetts Institute of Technology under the supervision of Isadore Singer.
She held temporary positions at MIT, Tufts University, Columbia University, the Institute for Advanced Study, and Princeton University before landing an associate professorship at the University of Wisconsin in 1976. She moved to San Diego in 1983, and retired in 2011.
Rothschild was a Member at Large at the AMS from 1977 to 1979. She served as president of the Association for Women in Mathematics from 1983 to 1985, and vice president of the American Mathematical Society from 1985 to 1987. She has been co-editor-in-chief of the journal Mathematical Research Letters since 1994.
The mother of two sons by her first marriage, her
second husband, M. Salah Baouendi, was a distinguished professor of mathematics at UC San Diego; he died in 2011.
Awards and honors
Rothschild was awarded a Sloan Fellowship in 1976.
In 1997, Rothschild gave the Noether Lecture of the Association for Women in Mathematics, on the subject "How do Real Manifolds live in Complex Space?", and she was an invited speaker at the International Congress of Mathematicians in 2006. She was elected as a fellow of the American Academy of Arts and Sciences in 2005, and in 2012 she became one of the inaugural fellows of the American Mathematical Society. A conference in her honor was held in 2008 at the University of Fribourg in Switzerland. In 2017, she was selected as a fellow of the Association for Women in Mathematics in the inaugural class.
She and her husband were jointly awarded the Stefan Bergman Prize of the American Mathematical Society in 2003.
References
External links
Official Homepage
Profile, researchgate.net
1945 births
Living people
20th-century American mathematicians
21st-century American mathematicians
American women mathematicians
University of Pennsylvania alumni
Massachusetts Institute of Technology School of Science alumni
University of Wisconsin–Madison faculty
University of California, San Diego faculty
Fellows of the American Academy of Arts and Sciences
Fellows of the American Mathematical Society
Fellows of the Association for Women in Mathematics
20th-century women mathematicians
21st-century women mathematicians
20th-century American women
21st-century American women |
https://en.wikipedia.org/wiki/Second%20covariant%20derivative | In the math branches of differential geometry and vector calculus, the second covariant derivative, or the second order covariant derivative, of a vector field is the derivative of its derivative with respect to another two tangent vector fields.
Definition
Formally, given a (pseudo)-Riemannian manifold (M, g) associated with a vector bundle E → M, let ∇ denote the Levi-Civita connection given by the metric g, and denote by Γ(E) the space of the smooth sections of the total space E. Denote by T*M the cotangent bundle of M. Then the second covariant derivative can be defined as the composition of the two ∇s as follows:
For example, given vector fields u, v, w, a second covariant derivative can be written as
by using abstract index notation. It is also straightforward to verify that
Thus
When the torsion tensor is zero, so that , we may use this fact to write Riemann curvature tensor as
Similarly, one may also obtain the second covariant derivative of a function f as
Again, for the torsion-free Levi-Civita connection, and for any vector fields u and v, when we feed the function f into both sides of
we find
.
This can be rewritten as
so we have
That is, the value of the second covariant derivative of a function is independent on the order of taking derivatives.
Notes
Tensors in general relativity
Riemannian geometry |
https://en.wikipedia.org/wiki/Horace%20Jones%20%28footballer%29 | Horace Jones was an English professional footballer who played as a wing half in the Football League for Brentford.
Career statistics
References
Footballers from Stafford
English men's footballers
Year of birth missing
English Football League players
Men's association football wing halves
Brentford F.C. players
Ton Pentre F.C. players
Hednesford Town F.C. players
Year of death missing
Southern Football League players |
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