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https://en.wikipedia.org/wiki/Entropy%20rate
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In the mathematical theory of probability, the entropy rate or source information rate of a stochastic process is, informally, the time density of the average information in a stochastic process. For stochastic processes with a countable index, the entropy rate is the limit of the joint entropy of members of the process divided by , as tends to infinity:
when the limit exists. An alternative, related quantity is:
For strongly stationary stochastic processes, . The entropy rate can be thought of as a general property of stochastic sources; this is the asymptotic equipartition property. The entropy rate may be used to estimate the complexity of stochastic processes. It is used in diverse applications ranging from characterizing the complexity of languages, blind source separation, through to optimizing quantizers and data compression algorithms. For example, a maximum entropy rate criterion may be used for feature selection in machine learning.
Entropy rates for Markov chains
Since a stochastic process defined by a Markov chain that is irreducible, aperiodic
and positive recurrent has a stationary distribution, the entropy rate is independent of the initial distribution.
For example, for such a Markov chain defined on a countable number of states, given the transition matrix , is given by:
where is the asymptotic distribution of the chain.
A simple consequence of this definition is that an i.i.d. stochastic process has an entropy rate that is the same as the entropy of any individual member of the process.
See also
Information source (mathematics)
Markov information source
Asymptotic equipartition property
Maximal entropy random walk - chosen to maximize entropy rate
References
Cover, T. and Thomas, J. (1991) Elements of Information Theory, John Wiley and Sons, Inc.,
Information theory
Entropy
Markov models
Temporal rates
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https://en.wikipedia.org/wiki/Franz%20Hermann%20Troschel
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Franz Hermann Troschel (10 October 1810 – 6 November 1882) was a German zoologist born in Spandau.
He studied mathematics and natural history at the University of Berlin, where he was awarded his doctorate in 1834. From 1840 to 1849 he was an assistant to Martin Lichtenstein at the Natural History Museum of Berlin. In 1849 he became a professor of zoology and natural history at the University of Bonn. In 1851 he became a member of the Academy of Sciences Leopoldina.
Troschel is remembered for the identification and classification of species in the fields of malacology, ichthyology and herpetology.
Taxon named in his honor
A few of the species that contain his name are
Troschel's sea star (Evasterias troschelii),
Troschel's murex (Murex troschelii), and a
freshwater snail (Bithynia troschelii).
Chlorurus troschelii, commonly known as Troschel's parrotfish, is a species of marine ray-finned fish, a parrotfish from the family Scaridae. It is native to the eastern Indian Ocean, where it lives in coral reefs.
Bibliography
(incomplete)
1842 (With Johannes Peter Müller (1801–1858)).
Über die Bedeutsamkeit des naturgeschichtlichen Unterrichts. Berlin 1845.
Horae ichthyologicae. Berlin 1845–49, 3 volumes (With Johannes Peter Müller).
Handbuch der Zoologie, third to seventh edition, Berlin 1848/1853/1859/1864/1871, (original authors Arend Friedrich August Wiegmann (1802–1841) and Johann Friedrich Ruthe (1788–1859)). Digital 6th edition by the University and State Library Düsseldorf
Troschel F. H. 1856–1879. Das Gebiß der Schnecken zur Begründung einer natürlichen Klassifikation. Berlin 1856–1879, two volumes. Volume 1 and volume 2 were published in parts:
1856. Volume 1, part 1: 1–72, plates 1–4.
(before 30 October) 1857. 1(2): 73–112, plates 5–8.
1858. 1(3): 113–152, plates 9–12.
1861. 1(4): 153–196, plates 13–16.
1863. 1(5): i–viii, 197–252, plates 17–20.
(December) 1865. 2(1): 1–48, plates 1–4.
(December) 1867. 2(2): 49–96, plates 5–8.
1869. 2(3): 97–132, plates 9–12.
1875. 2(4): 133–180, plates 13–16.
(18 September) 1878. 2(5): 181–216, plates 17–20.
(2 September) 1879. 2(6): 217–246, plates 21–24.
Two last parts 2(7) and 2(8) were continued by Johannes Thiele and published in 1891 and 1893.
Troschel was the editor of Archiv für Naturgeschichte from volume 15 (1849) to 48 (1882), Berlin, Nicolaische Verlagsbuchhandlung (Volume 35, part 2, online).
Taxon described by him
See :Category:Taxa named by Franz Hermann Troschel
References
H. von Dechen, 1883. Zur Erinnerung an Dr. Franz Hermann Troschel; Verhandlungen des Naturhistorischen Vereines der Preussischen Rheinlande und Westfalens 40 (Correspondenzblatt 1): p. 35–54 (biography and bibliography).
External links
1810 births
1882 deaths
German malacologists
19th-century German zoologists
Teuthologists
Scientists from Berlin
People from Spandau
Academic staff of the University of Bonn
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https://en.wikipedia.org/wiki/C%C3%A9a%27s%20lemma
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Céa's lemma is a lemma in mathematics. Introduced by Jean Céa in his Ph.D. dissertation, it is an important tool for proving error estimates for the finite element method applied to elliptic partial differential equations.
Lemma statement
Let be a real Hilbert space with the norm Let be a bilinear form with the properties
for some constant and all in (continuity)
for some constant and all in (coercivity or -ellipticity).
Let be a bounded linear operator. Consider the problem of finding an element in such that
for all in
Consider the same problem on a finite-dimensional subspace of so, in satisfies
for all in
By the Lax–Milgram theorem, each of these problems has exactly one solution. Céa's lemma states that
for all in
That is to say, the subspace solution is "the best" approximation of in up to the constant
The proof is straightforward
for all in
We used the -orthogonality of and
which follows directly from
for all in .
Note: Céa's lemma holds on complex Hilbert spaces also, one then uses a sesquilinear form instead of a bilinear one. The coercivity assumption then becomes for all in (notice the absolute value sign around ).
Error estimate in the energy norm
In many applications, the bilinear form is symmetric, so
for all in
This, together with the above properties of this form, implies that is an inner product on The resulting norm
is called the energy norm, since it corresponds to a physical energy in many problems. This norm is equivalent to the original norm
Using the -orthogonality of and and the Cauchy–Schwarz inequality
for all in .
Hence, in the energy norm, the inequality in Céa's lemma becomes
for all in
(notice that the constant on the right-hand side is no longer present).
This states that the subspace solution is the best approximation to the full-space solution in respect to the energy norm. Geometrically, this means that is the projection of the solution onto the subspace in respect to the inner product (see the adjacent picture).
Using this result, one can also derive a sharper estimate in the norm . Since
for all in ,
it follows that
for all in .
An application of Céa's lemma
We will apply Céa's lemma to estimate the error of calculating the solution to an elliptic differential equation by the finite element method.
Consider the problem of finding a function satisfying the conditions
where is a given continuous function.
Physically, the solution to this two-point boundary value problem represents the shape taken by a string under the influence of a force such that at every point between and the force density is (where is a unit vector pointing vertically, while the endpoints of the string are on a horizontal line, see the adjacent picture). For example, that force may be the gravity, when is a constant function (since the gravitational force is the same at all points).
Let the Hilbert space be t
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https://en.wikipedia.org/wiki/Medical%20statistics
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Medical statistics deals with applications of statistics to medicine and the health sciences, including epidemiology, public health, forensic medicine, and clinical research. Medical statistics has been a recognized branch of statistics in the United Kingdom for more than 40 years but the term has not come into general use in North America, where the wider term 'biostatistics' is more commonly used. However, "biostatistics" more commonly connotes all applications of statistics to biology. Medical statistics is a subdiscipline of statistics. "It is the science of summarizing, collecting, presenting and interpreting data in medical practice, and using them to estimate the magnitude of associations and test hypotheses. It has a central role in medical investigations. It not only provides a way of organizing information on a wider and more formal basis than relying on the exchange of anecdotes and personal experience, but also takes into account the intrinsic variation inherent in most biological processes."
Pharmaceutical statistics
Pharmaceutical statistics is the application of statistics to matters concerning the pharmaceutical industry. This can be from issues of design of experiments, to analysis of drug trials, to issues of commercialization of a medicine.
There are many professional bodies concerned with this field including:
European Federation of Statisticians in the Pharmaceutical Industry (EFSPI)
Statisticians In The Pharmaceutical Industry (PSI)
There are also journals including:
Statistics in Medicine
Pharmaceutical Statistics
Clinical biostatistics
Clinical biostatistics is concerned with research into the principles and methodology used in the design and analysis of clinical research and to apply statistical theory to clinical medicine.
There is a society for Clinical Biostatistics with annual conferences since its founding in 1978.
Clinical Biostatistics is taught in postgraduate biostatistical and applied statistical degrees, for example as part of the BCA Master of Biostatistics program in Australia.
Basic concepts
For describing situations
Incidence (epidemiology) vs. Prevalence vs. Cumulative incidence
Many medical tests (such as pregnancy tests) have two possible results: positive or negative. However, tests will sometimes yield incorrect results in the form of false positives or false negatives. False positives and false negatives can be described by the statistical concepts of type I and type II errors, respectively, where the null hypothesis is that the patient will test negative. The precision of a medical test is usually calculated in the form of positive predictive values (PPVs) and negative predicted values (NPVs). PPVs and NPVs of medical tests depend on intrinsic properties of the test as well as the prevalence of the condition being tested for. For example, if any pregnancy test was administered to a population of individuals who were biologically incapable of becoming pregnant, then the test's PPV will b
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https://en.wikipedia.org/wiki/Horocycle
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In hyperbolic geometry, a horocycle (), sometimes called an oricycle, oricircle, or limit circle, is a curve whose normal or perpendicular geodesics all converge asymptotically in the same direction. It is the two-dimensional case of a horosphere (or orisphere).
The centre of a horocycle is the ideal point where all normal geodesics asymptotically converge. Two horocycles who have the same centre are concentric.
Although it appears as if two concentric horocycles cannot have the same length or curvature, in fact any two horocycles are congruent.
A horocycle can also be described as the limit of the circles that share a tangent in a given point, as their radii go towards infinity. In Euclidean geometry, such a "circle of infinite radius" would be a straight line, but in hyperbolic geometry it is a horocycle (a curve).
From the convex side the horocycle is approximated by hypercycles whose distances from their axis go towards infinity.
Properties
Through every pair of points there are 2 horocycles. The centres of the horocycles are the ideal points of the perpendicular bisector of the segment between them.
No three points of a horocycle are on a line, circle or hypercycle.
All horocycles are congruent. (Even concentric horocycles are congruent to each other)
A straight line, circle, hypercycle, or other horocycle cuts a horocycle in at most two points.
The perpendicular bisector of a chord of a horocycle is a normal of that horocycle and the bisector bisects the arc subtended by the chord and is an axis of symmetry of that horocycle.
The length of an arc of a horocycle between two points is:
longer than the length of the line segment between those two points,
longer than the length of the arc of a hypercycle between those two points and
shorter than the length of any circle arc between those two points.
The distance from a horocycle to its center is infinite, and while in some models of hyperbolic geometry it looks like the two "ends" of a horocycle get closer and closer together and closer to its center, this is not true; the two "ends" of a horocycle get further and further away from each other.
A regular apeirogon is circumscribed by either a horocycle or a hypercycle.
If C is the centre of a horocycle and A and B are points on the horocycle then the angles CAB and CBA are equal.
The area of a sector of a horocycle (the area between two radii and the horocycle) is finite.
Standardized Gaussian curvature
When the hyperbolic plane has the standardized Gaussian curvature K of −1:
The length s of an arc of a horocycle between two points is:
where d is the distance between the two points, and sinh and cosh are hyperbolic functions.
The length of an arc of a horocycle such that the tangent at one extremity is limiting parallel to the radius through the other extremity is 1. the area enclosed between this horocycle and the radii is 1.
The ratio of the arc lengths between two radii of two concentric horocycles where the horo
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https://en.wikipedia.org/wiki/Qiudong%20Wang
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Qiudong Wang is a professor at the Department of Mathematics, the University of Arizona. In 1982 he received a B.S. at Nanjing University and in 1994 a Ph.D. at the University of Cincinnati.
Wang is best known for his 1991 paper The global solution of the n-body problem, in which he generalised Karl F. Sundman's results from 1912 to a system of more than three bodies. However, L. K. Babadzanjanz claims to have made the same generalization earlier, in 1979.
References
Chinese emigrants to the United States
Nanjing University alumni
University of Cincinnati alumni
University of Arizona faculty
American astronomers
Living people
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Q.%20Wang
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Q. Wang may refer to:
Qiudong Wang, professor of mathematics
Q. Wang (artist) (born 1962), Chinese-American primitivist painter
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https://en.wikipedia.org/wiki/Langlands%20decomposition
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In mathematics, the Langlands decomposition writes a parabolic subgroup P of a semisimple Lie group as a product of a reductive subgroup M, an abelian subgroup A, and a nilpotent subgroup N.
Applications
A key application is in parabolic induction, which leads to the Langlands program: if is a reductive algebraic group and is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of , extending it to by letting act trivially, and inducing the result from to .
See also
Lie group decompositions
References
Sources
A. W. Knapp, Structure theory of semisimple Lie groups. .
Lie groups
Algebraic groups
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https://en.wikipedia.org/wiki/Infinitesimal%20character
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In mathematics, the infinitesimal character of an irreducible representation ρ of a semisimple Lie group G on a vector space V is, roughly speaking, a mapping to scalars that encodes the process of first differentiating and then diagonalizing the representation. It therefore is a way of extracting something essential from the representation ρ by two successive linearizations.
Formulation
The infinitesimal character is the linear form on the center Z of the universal enveloping algebra of the Lie algebra of G that the representation induces. This construction relies on some extended version of Schur's lemma to show that any z in Z acts on V as a scalar, which by abuse of notation could be written ρ(z).
In more classical language, z is a differential operator, constructed from the infinitesimal transformations which are induced on V by the Lie algebra of G. The effect of Schur's lemma is to force all v in V to be simultaneous eigenvectors of z acting on V. Calling the corresponding eigenvalue
λ = λ(z),
the infinitesimal character is by definition the mapping
z → λ(z).
There is scope for further formulation. By the Harish-Chandra isomorphism, the center Z can be identified with the subalgebra of elements of the symmetric algebra of the Cartan subalgebra a that are invariant under the Weyl group, so an infinitesimal character can be identified with an element of
a*⊗ C/W,
the orbits under the Weyl group W of the space a*⊗ C of complex linear functions on the Cartan subalgebra.
See also
Harish-Chandra isomorphism
Representation theory of Lie groups
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https://en.wikipedia.org/wiki/Box%E2%80%93Behnken%20design
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In statistics, Box–Behnken designs are experimental designs for response surface methodology, devised by George E. P. Box and Donald Behnken in 1960, to achieve the following goals:
Each factor, or independent variable, is placed at one of three equally spaced values, usually coded as −1, 0, +1. (At least three levels are needed for the following goal.)
The design should be sufficient to fit a quadratic model, that is, one containing squared terms, products of two factors, linear terms and an intercept.
The ratio of the number of experimental points to the number of coefficients in the quadratic model should be reasonable (in fact, their designs kept in the range of 1.5 to 2.6).
The estimation variance should more or less depend only on the distance from the centre (this is achieved exactly for the designs with 4 and 7 factors), and should not vary too much inside the smallest (hyper)cube containing the experimental points. (See "rotatability" in "Comparisons of response surface designs".)
Box-Behnken design is still considered to be more proficient and most powerful than other designs such as the three-level full factorial design, central composite design (CCD) and Doehlert design, despite its poor coverage of the corner of nonlinear design space.
The design with 7 factors was found first while looking for a design having the desired property concerning estimation variance, and then similar designs were found for other numbers of factors.
Each design can be thought of as a combination of a two-level (full or fractional) factorial design with an incomplete block design. In each block, a certain number of factors are put through all combinations for the factorial design, while the other factors are kept at the central values. For instance, the Box–Behnken design for 3 factors involves three blocks, in each of which 2 factors are varied through the 4 possible combinations of high and low. It is necessary to include centre points as well (in which all factors are at their central values).
In this table, m represents the number of factors which are varied in each of the blocks.
{| class="wikitable"
!factors||m||no. of blocks||factorial pts. per block||total with 1 centre point||typical total with extra centre points||no. of coefficients in quadratic model
|-
| 3||2|| 3|| 4|| 13||15, 17||10
|-
| 4||2|| 6|| 4|| 25||27, 29||15
|-
| 5||2||10|| 4|| 41||46||21
|-
| 6||3|| 6|| 8|| 49||54||28
|-
| 7||3|| 7|| 8|| 57||62||36
|-
| 8||4||14|| 8||113||120||45
|-
| 9||3||12|| 8|| 97||105||55
|-
|10||4||10||16||161||170||66
|-
|11||5||11||16||177||188||78
|-
|12||4||12||16||193||204||91
|-
|16||4||24||16||385||396||153
|}
The design for 8 factors was not in the original paper. Taking the 9 factor design, deleting one column and any resulting duplicate rows produces an 81 run design for 8 factors, while giving up some "rotatability" (see above). Designs for other numbers of factors have also been invented (at least up to 21). A design for 16 factors exist
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https://en.wikipedia.org/wiki/Samara%20State%20University
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Samara State University () was a classical multi-faculty university and a leading educational institution of higher education in Samara Oblast, Russia. It consists of faculties of Mathematics and Mechanics, Physics, Biology, Chemistry, Philology, History, Sociology, Economy and Management, Psychology, and Law. It is considered to be the most prestigious graduate school in Samara and the area, especially with its highly competitive and demanding programmes in the English Language, Law, Sociology, Political Science, International Relations, and Psychology. SSU is also noted for its postgraduate research in the Sciences and the Humanities.
SSU had its own newspaper, a regularly printed academic journal, as well as journals for students publications. SSU is a centre of teaching and research in Samara Region with a strong reputation nationally and globally. SSU is one of the few Russian universities that issues the European Diploma Supplement to the State Diploma of Higher Education, which confirms educational standards according to international standards (ECTS). In 2015 it was merged with other institutions to form the Samara National Research University. The English language is taught here by high-qualified lecturers.
In 2014 Samara State University celebrated the 95th anniversary since its foundation and the 45th anniversary since its revival. It is a large scientific, educational and cultural centre of the Volga region with a constantly developing infrastructure. The university being of the classical type preserves the highest educational standards and education quality. The university has one regional branch in Togliatti. At present it comprises 10 basic faculties: Physics, Chemistry, Biology, Mechanics and Mathematics, History, Philology, Sociology, Psychology, Law, Economics and Management.
Notable alumni
Notable alumni of Samara State University include:
Violetta Khrapina Bida (born 1994), Olympic épée fencer
Mark Feygin (born 1971), lawyer and human rights activist
Nikolai Mashkin (1900–1950), scholar of Roman history
Mikhail Matveyev (born 1968), member of the State Duma
Dmitry Muratov (born 1961), joint recipient of the 2021 Nobel Peace Prize
Pavel Romanov (1964–2014), sociologist
Evdokia Romanova (born 1990), human rights activist
Svetlana Vanyo (born 1977), Russian-American swimmer and coach
References
External links
Samara State University
Samara State University/Undergraduate
Overview of Samara State University
Education in Samara, Russia
Buildings and structures in Samara Oblast
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https://en.wikipedia.org/wiki/Proviso%20Mathematics%20and%20Science%20Academy
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Proviso Mathematics and Science Academy is a selective enrollment high school in Forest Park, Illinois, United States that opened its doors to 126 freshmen in 2005. It is one of the newest schools in the Proviso Township High Schools District 209. It serves students in many towns in western Cook County including Forest Park, Berkeley, Broadview, Maywood, Melrose Park, Stone Park, Hillside, Bellwood, and Westchester among others. In May 2023, the school was named within the top ten Illinois High Schools, placing 6th in U.S. News & World Report.
Academics and admissions
The academy accepts students through an admissions program including standardized tests, grades, teacher recommendations, and an essay. Proviso's curriculum is informed by that of the Illinois Mathematics and Science Academy. The curriculum focuses on science, math, technology, arts, and foreign languages. They also teach Physics First, which introduces freshmen to physics before studying chemistry and biology.
The school mascot is Monty the Python; school colors are purple and black. As there are no sports facilities at the Academy, students play sports including baseball, basketball, football, track and field, softball, volleyball at either Proviso East or Proviso West. Proviso's extra curricular activities include: Anime Club, Book Club, Comedy Improv, Chorus, Debate, Newspaper, Theater, Robotics, Student Council and Yearbook.
Much of the 5th floor at the campus houses District 209 administration offices which were formerly located at Proviso East High School.
Beyond graduation
Graduates have gone on to four year colleges and universities including University of Chicago, Boston University, Cornell University, Dominican University, Massachusetts Institute of Technology, Northwestern University, University of Illinois at Urbana–Champaign, Harvard University, Case Western Reserve University, and various other prestigious institutes.
References
External links
Official website
School statistics from Interactive Illinois Report Card
Educational institutions established in 2005
Forest Park, Illinois
Public high schools in Cook County, Illinois
Magnet schools in Illinois
2005 establishments in Illinois
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https://en.wikipedia.org/wiki/List%20of%20oldest%20and%20youngest%20Academy%20Award%20winners%20and%20nominees
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This is a list of oldest and youngest Academy Award winners and nominees in the award categories Acting and Directing.
This list is based on "statistics valid through the nomination announcement for the 2015 (88th Academy Awards), announced on January 14, 2016", as documented in The Official Academy Awards Database.
At the 90th Academy Awards, James Ivory became the oldest-ever Oscar winner in any category, at age 89, after receiving the award for Best Adapted Screenplay for his work on Call Me by Your Name. At the 93rd Academy Awards, Ann Roth became the oldest-ever woman to win an Oscar in any category, at age 89, after receiving the award for Best Costume Design for her work on Ma Rainey's Black Bottom.
At the 95th Academy Awards, John Williams became the oldest Oscar nominee in any category, at age 90, after receiving his 53rd Oscar nomination for Best Original Score for his work on The Fabelmans.
There are only three people in Oscar history who have won two Oscars by the age of 30 or younger:
Luise Rainer for The Great Ziegfeld and The Good Earth
Jodie Foster for The Accused and The Silence of the Lambs
Hilary Swank for Boys Don't Cry and Million Dollar Baby
all in the Best Actress in a Leading Role category.
Superlatives
Among the oldest and youngest winners and nominees of Academy Awards in standard competitive categories, the following superlatives emerge:
Best Director
Oldest winners
Source: "Academy Award Statistics: Directing: Oldest/Youngest Directing Nominees/Winners: AMPAS Awards Database
Oldest nominees
Youngest winners
Youngest nominees
Best Actor in a Leading Role
Source: "Academy Award Statistics: Acting: Oldest/Youngest Winners and Nominees for Acting, By Category: Actor [in a Leading Role]", AMPAS Awards Database
Oldest winners
Oldest nominees
Youngest winners
Youngest nominees
Best Actress in a Leading Role
Source: "Academy Award Statistics: Acting: Oldest/Youngest Winners and Nominees for Acting, By Category: Actress [in a Leading Role]", AMPAS Awards Database
Oldest winners
Oldest nominees
Youngest winners
Youngest nominees
Best Actor in a Supporting Role
Source: "Academy Award Statistics: Acting: Oldest/Youngest Winners and Nominees for Acting, By Category: Actor [in a Supporting Role]", AMPAS Awards Database
Oldest winners
Oldest nominees
Youngest winners
Youngest nominees
Best Actress in a Supporting Role
Source: "Academy Award Statistics: Acting: Oldest/Youngest Winners and Nominees for Acting, By Category: Actress [in a Supporting Role]", AMPAS Awards Database
Oldest winners
Oldest nominees
Youngest winners
Youngest nominees
Honorary Awards
Source: "Academy Award Statistics: Acting: Oldest/Youngest Winners and Nominees for Acting, By Category: Acting Honorary Award Winners", AMPAS Awards Database
Academy Honorary Award
Oldest honorees
Oldest acting honorees
Honorary Juvenile Award
Youngest honorees
See also
Academy Award
Academy Juvenile Award
Academy of Motion Pict
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https://en.wikipedia.org/wiki/Integro-differential%20equation
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In mathematics, an integro-differential equation is an equation that involves both integrals and derivatives of a function.
General first order linear equations
The general first-order, linear (only with respect to the term involving derivative) integro-differential equation is of the form
As is typical with differential equations, obtaining a closed-form solution can often be difficult. In the relatively few cases where a solution can be found, it is often by some kind of integral transform, where the problem is first transformed into an algebraic setting. In such situations, the solution of the problem may be derived by applying the inverse transform to the solution of this algebraic equation.
Example
Consider the following second-order problem,
where
is the Heaviside step function. The Laplace transform is defined by,
Upon taking term-by-term Laplace transforms, and utilising the rules for derivatives and integrals, the integro-differential equation is converted into the following algebraic equation,
Thus,
.
Inverting the Laplace transform using contour integral methods then gives
.
Alternatively, one can complete the square and use a table of Laplace transforms ("exponentially decaying sine wave") or recall from memory to proceed:
.
Applications
Integro-differential equations model many situations from science and engineering, such as in circuit analysis. By Kirchhoff's second law, the net voltage drop across a closed loop equals the voltage impressed . (It is essentially an application of energy conservation.) An RLC circuit therefore obeys
where is the current as a function of time, is the resistance, the inductance, and the capacitance.
The activity of interacting inhibitory and excitatory neurons can be described by a system of integro-differential equations, see for example the Wilson-Cowan model.
The Whitham equation is used to model nonlinear dispersive waves in fluid dynamics.
Epidemiology
Integro-differential equations have found applications in epidemiology, the mathematical modeling of epidemics, particularly when the models contain age-structure or describe spatial epidemics. The Kermack-McKendrick theory of infectious disease transmission is one particular example where age-structure in the population is incorporated into the modeling framework.
See also
Delay differential equation
Differential equation
Integral equation
Integrodifference equation
References
Further reading
Vangipuram Lakshmikantham, M. Rama Mohana Rao, “Theory of Integro-Differential Equations”, CRC Press, 1995
External links
Interactive Mathematics
Numerical solution of the example using Chebfun
Differential equations
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https://en.wikipedia.org/wiki/Equipossibility
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Equipossibility is a philosophical concept in possibility theory that is a precursor to the notion of equiprobability in probability theory. It is used to distinguish what can occur in a probability experiment. For example, it is the difference between viewing the possible results of rolling a six sided dice as {1,2,3,4,5,6} rather than {6, not 6}. The former (equipossible) set contains equally possible alternatives, while the latter does not because there are five times as many alternatives inherent in 'not 6' as in 6. This is true even if the die is biased so that 6 and 'not 6' are equally likely to occur (equiprobability).
The Principle of Indifference of Laplace states that equipossible alternatives may be accorded equal probabilities if nothing more is known about the underlying probability distribution. However, it is a matter of contention whether the concept of equipossibility, also called equispecificity (from equispecific), can truly be distinguished from the concept of equiprobability.
In Bayesian inference, one definition of equipossibility is "a transformation group which leaves invariant one's state of knowledge". Equiprobability is then defined by normalizing the Haar measure of this symmetry group. This is known as the principle of transformation groups.
References
External links
Book Chapter by Henry E. Kyburg Jr. on equipossibility, with the 6/not-6 example above
Quotes on equipossibility in classical probability
Probability interpretations
Possibility
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https://en.wikipedia.org/wiki/Equiprobability
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Equiprobability is a property for a collection of events that each have the same probability of occurring. In statistics and probability theory it is applied in the discrete uniform distribution and the equidistribution theorem for rational numbers. If there are events under consideration, the probability of each occurring is
In philosophy it corresponds to a concept that allows one to assign equal probabilities to outcomes when they are judged to be equipossible or to be "equally likely" in some sense. The best-known formulation of the rule is Laplace's principle of indifference (or principle of insufficient reason), which states that, when "we have no other information than" that exactly mutually exclusive events can occur, we are justified in assigning each the probability This subjective assignment of probabilities is especially justified for situations such as rolling dice and lotteries since these experiments carry a symmetry structure, and one's state of knowledge must clearly be invariant under this symmetry.
A similar argument could lead to the seemingly absurd conclusion that the sun is as likely to rise as to not rise tomorrow morning. However, the conclusion that the sun is equally likely to rise as it is to not rise is only absurd when additional information is known, such as the laws of gravity and the sun's history. Similar applications of the concept are effectively instances of circular reasoning, with "equally likely" events being assigned equal probabilities, which means in turn that they are equally likely. Despite this, the notion remains useful in probabilistic and statistical modeling.
In Bayesian probability, one needs to establish prior probabilities for the various hypotheses before applying Bayes' theorem. One procedure is to assume that these prior probabilities have some symmetry which is typical of the experiment, and then assign a prior which is proportional to the Haar measure for the symmetry group: this generalization of equiprobability is known as the principle of transformation groups and leads to misuse of equiprobability as a model for incertitude.
See also
Principle of indifference
Laplacian smoothing
Uninformative prior
A priori probability
Aequiprobabilism
Uniform probability distributions:
Continuous
Discrete
Information gain
References
External links
Quotes on equiprobability in classical probability
Probability interpretations
Philosophy of statistics
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https://en.wikipedia.org/wiki/Isomorphism-closed%20subcategory
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In category theory, a branch of mathematics, a subcategory of a category is said to be isomorphism closed or replete if every -isomorphism with belongs to This implies that both and belong to as well.
A subcategory that is isomorphism closed and full is called strictly full. In the case of full subcategories it is sufficient to check that every -object that is isomorphic to an -object is also an -object.
This condition is very natural. For example, in the category of topological spaces one usually studies properties that are invariant under homeomorphisms—so-called topological properties. Every topological property corresponds to a strictly full subcategory of
References
Category theory
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https://en.wikipedia.org/wiki/Representation%20theory%20of%20SL2%28R%29
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{{DISPLAYTITLE:Representation theory of SL2(R)}}
In mathematics, the main results concerning irreducible unitary representations of the Lie group SL(2,R) are due to Gelfand and Naimark (1946), V. Bargmann (1947), and Harish-Chandra (1952).
Structure of the complexified Lie algebra
We choose a basis H, X, Y for the complexification of the Lie algebra of SL(2,R) so that iH generates the Lie algebra of a compact Cartan subgroup K (so in particular unitary representations split as a sum of eigenspaces of H), and {H,X,Y} is an sl2-triple, which means that they satisfy the relations
One way of doing this is as follows:
corresponding to the subgroup K of matrices
The Casimir operator Ω is defined to be
It generates the center of the universal enveloping algebra of the complexified Lie algebra of SL(2,R). The Casimir element acts on any irreducible representation as multiplication by some complex scalar μ2. Thus in the case of the Lie algebra sl2, the infinitesimal character of an irreducible representation is specified by one complex number.
The center Z of the group SL(2,R) is a cyclic group {I,-I} of order 2, consisting of the identity matrix and its negative. On any irreducible representation, the center either acts trivially, or by the nontrivial character of Z, which represents the matrix -I by multiplication by -1 in the representation space. Correspondingly, one speaks of the trivial or nontrivial central character.
The central character and the infinitesimal character of an irreducible representation of any reductive Lie group are important invariants of the representation. In the case of irreducible admissible representations of SL(2,R), it turns out that, generically, there is exactly one representation, up to an isomorphism, with the specified central and infinitesimal characters. In the exceptional cases there are two or three representations with the prescribed parameters, all of which have been determined.
Finite-dimensional representations
For each nonnegative integer n, the group SL(2,R) has an irreducible representation of dimension n+1, which is unique up to an isomorphism. This representation can be constructed in the space of homogeneous polynomials of degree n in two variables. The case n=0 corresponds to the trivial representation. An irreducible finite-dimensional representation of a noncompact simple Lie group of dimension greater than 1 is never unitary. Thus this construction produces only one unitary representation of SL(2,R), the trivial representation.
The finite-dimensional representation theory of the noncompact group SL(2,R) is equivalent to the representation theory of SU(2), its compact form, essentially because their Lie algebras have the same complexification and they are "algebraically simply connected". (More precisely the group SU(2) is simply connected and SL(2,R) is not, but has no non-trivial algebraic central extensions.) However, in the general infinite-dimensional case, there is no close corr
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https://en.wikipedia.org/wiki/SL2%28R%29
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{{DISPLAYTITLE:SL2(R)}}
In mathematics, the special linear group SL(2, R) or SL2(R) is the group of 2 × 2 real matrices with determinant one:
It is a connected non-compact simple real Lie group of dimension 3 with applications in geometry, topology, representation theory, and physics.
SL(2, R) acts on the complex upper half-plane by fractional linear transformations. The group action factors through the quotient PSL(2, R) (the 2 × 2 projective special linear group over R). More specifically,
PSL(2, R) = SL(2, R) / {±I},
where I denotes the 2 × 2 identity matrix. It contains the modular group PSL(2, Z).
Also closely related is the 2-fold covering group, Mp(2, R), a metaplectic group (thinking of SL(2, R) as a symplectic group).
Another related group is SL±(2, R), the group of real 2 × 2 matrices with determinant ±1; this is more commonly used in the context of the modular group, however.
Descriptions
SL(2, R) is the group of all linear transformations of R2 that preserve oriented area. It is isomorphic to the symplectic group Sp(2, R) and the special unitary group SU(1, 1). It is also isomorphic to the group of unit-length coquaternions. The group SL±(2, R) preserves unoriented area: it may reverse orientation.
The quotient PSL(2, R) has several interesting descriptions, up to Lie group isomorphism:
It is the group of orientation-preserving projective transformations of the real projective line
It is the group of conformal automorphisms of the unit disc.
It is the group of orientation-preserving isometries of the hyperbolic plane.
It is the restricted Lorentz group of three-dimensional Minkowski space. Equivalently, it is isomorphic to the indefinite orthogonal group SO+(1,2). It follows that SL(2, R) is isomorphic to the spin group Spin(2,1)+.
Elements of the modular group PSL(2, Z) have additional interpretations, as do elements of the group SL(2, Z) (as linear transforms of the torus), and these interpretations can also be viewed in light of the general theory of SL(2, R).
Homographies
Elements of PSL(2, R) are homographies on the real projective line :
These projective transformations form a subgroup of PSL(2, C), which acts on the Riemann sphere by Möbius transformations.
When the real line is considered the boundary of the hyperbolic plane, PSL(2, R) expresses hyperbolic motions.
Möbius transformations
Elements of PSL(2, R) act on the complex plane by Möbius transformations:
This is precisely the set of Möbius transformations that preserve the upper half-plane. It follows that PSL(2, R) is the group of conformal automorphisms of the upper half-plane. By the Riemann mapping theorem, it is also isomorphic to the group of conformal automorphisms of the unit disc.
These Möbius transformations act as the isometries of the upper half-plane model of hyperbolic space, and the corresponding Möbius transformations of the disc are the hyperbolic isometries of the Poincaré disk model.
The above formula can be also used t
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https://en.wikipedia.org/wiki/Eugenio%20Elia%20Levi
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Eugenio Elia Levi (18 October 1883 – 28 October 1917) was an Italian mathematician, known for his fundamental contributions in group theory, in the theory of partial differential operators and in the theory of functions of several complex variables. He was a younger brother of Beppo Levi and was killed in action during First World War.
Work
Research activity
He wrote 33 papers, classified by his colleague and friend Mauro Picone according to the scheme reproduced in this section.
Differential geometry
Group theory
He wrote only three papers in group theory: in the first one, discovered what is now called Levi decomposition, which was conjectured by Wilhelm Killing and proved by Élie Cartan in a special case.
Function theory
In the theory of functions of several complex variables he introduced the concept of pseudoconvexity during his investigations on the domain of existence of such functions: it turned out to be one of the key concepts of the theory.
Cauchy and Goursat problems
Boundary value problems
His researches in the theory of partial differential operators lead to the method of the parametrix, which is basically a way to construct fundamental solutions for elliptic partial differential operators with variable coefficients: the parametrix is widely used in the theory of pseudodifferential operators.
Calculus of variations
Publications
The full scientific production of Eugenio Elia Levi is collected in reference .
, reprinted also in , volume I. A a well-known memoir in Group theory: it was presented to the members of the Accademia delle Scienze di Torino during the session of April 2, 1905, by Luigi Bianchi.
. A short note announcing the results of paper .
. An important paper whose results were previously announced in the short note with the same title. It was also translated in Russian by N. D. Ajzenstat, currently available from the All-Russian Mathematical Portal: .
. An important paper in the theory of functions of several complex variables, where the problem of determining what kind of hypersurface can be the boundary of a domain of holomorphy.
. Another important paper in the theory of functions of several complex variables, investigating further the theory started in .
. His "Collected works" in two volumes, collecting all the mathematical papers of Eugenio Elia Levi in a revised typographical form, both amending typographical errors and author's oversights. A collection of all his published papers (in their original typographical form), probably an unordered uncorrected collection of offprints, is available online at the Internet Archive: .
See also
Pseudoconvexity
Levi decomposition
Parametrix
Several complex variables
Notes
References
Biographical and general references
. Wide source of unpublished manuscript documents of and about E.E. Levi. A short presentation could be found on EGEA website
. An ample biographical paper (nearly 40 pages) on Beppo Levi: an earlier version of it was published as . It gives many u
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https://en.wikipedia.org/wiki/Parabolic%20induction
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In mathematics, parabolic induction is a method of constructing representations of a reductive group from representations of its parabolic subgroups.
If G is a reductive algebraic group and is the Langlands decomposition of a parabolic subgroup P, then parabolic induction consists of taking a representation of , extending it to P by letting N act trivially, and inducing the result from P to G.
There are some generalizations of parabolic induction using cohomology, such as cohomological parabolic induction and Deligne–Lusztig theory.
Philosophy of cusp forms
The philosophy of cusp forms was a slogan of Harish-Chandra, expressing his idea of a kind of reverse engineering of automorphic form theory, from the point of view of representation theory. The discrete group Γ fundamental to the classical theory disappears, superficially. What remains is the basic idea that representations in general are to be constructed by parabolic induction of cuspidal representations. A similar philosophy was enunciated by Israel Gelfand, and the philosophy is a precursor of the Langlands program. A consequence for thinking about representation theory is that cuspidal representations are the fundamental class of objects, from which other representations may be constructed by procedures of induction.
According to Nolan Wallach
Put in the simplest terms the "philosophy of cusp forms" says that for each Γ-conjugacy classes of Q-rational parabolic subgroups one should construct automorphic functions (from objects from spaces of lower dimensions) whose constant terms are zero for other conjugacy classes and the constant terms for [an] element of the given class give all constant terms for this parabolic subgroup. This is almost possible and leads to a description of all automorphic forms in terms of these constructs and cusp forms. The construction that does this is the Eisenstein series.
Notes
References
A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples, Princeton Landmarks in Mathematics, Princeton University Press, 2001. .
Representation theory
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https://en.wikipedia.org/wiki/Complementary%20series%20representation
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In mathematics, complementary series representations of a reductive real or p-adic Lie groups are certain irreducible unitary representations that are not tempered and do not appear in the decomposition of the regular representation into irreducible representations.
They are rather mysterious: they do not turn up very often, and seem to exist by accident. They were sometimes overlooked, in fact, in some earlier claims to have classified the irreducible unitary representations of certain groups.
Several conjectures in mathematics, such as the Selberg conjecture, are equivalent to saying that certain representations are not complementary. For examples see the representation theory of SL2(R). Elias M. Stein (1972) constructed some families of them for higher rank groups using analytic continuation, sometimes called the Stein complementary series.
References
, also reprinted as
Representation theory of groups
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https://en.wikipedia.org/wiki/Sum-free%20set
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In additive combinatorics and number theory, a subset A of an abelian group G is said to be sum-free if the sumset A + A is disjoint from A. In other words, A is sum-free if the equation has no solution with .
For example, the set of odd numbers is a sum-free subset of the integers, and the set {N + 1, ..., 2N } forms a large sum-free subset of the set {1, ..., 2N }. Fermat's Last Theorem is the statement that, for a given integer n > 2, the set of all nonzero nth powers of the integers is a sum-free set.
Some basic questions that have been asked about sum-free sets are:
How many sum-free subsets of {1, ..., N } are there, for an integer N? Ben Green has shown that the answer is , as predicted by the Cameron–Erdős conjecture.
How many sum-free sets does an abelian group G contain?
What is the size of the largest sum-free set that an abelian group G contains?
A sum-free set is said to be maximal if it is not a proper subset of another sum-free set.
Let be defined by is the largest number such that any subset of with size n has a sum-free subset of size k. The function is subadditive, and by the Fekete subadditivity lemma, exists. Erdős proved that , and conjectured that equality holds. This was proved by Eberhard, Green, and Manners.
See also
Erdős–Szemerédi theorem
Sum-free sequence
References
Sumsets
Additive combinatorics
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https://en.wikipedia.org/wiki/Weyl%E2%80%93Brauer%20matrices
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In mathematics, particularly in the theory of spinors, the Weyl–Brauer matrices are an explicit realization of a Clifford algebra as a matrix algebra of matrices. They generalize the Pauli matrices to dimensions, and are a specific construction of higher-dimensional gamma matrices. They are named for Richard Brauer and Hermann Weyl, and were one of the earliest systematic constructions of spinors from a representation theoretic standpoint.
The matrices are formed by taking tensor products of the Pauli matrices, and the space of spinors in dimensions may then be realized as the column vectors of size on which the Weyl–Brauer matrices act.
Construction
Suppose that V = Rn is a Euclidean space of dimension n. There is a sharp contrast in the construction of the Weyl–Brauer matrices depending on whether the dimension n is even or odd.
Let = 2 (or 2+1) and suppose that the Euclidean quadratic form on is given by
where (pi, qi) are the standard coordinates on Rn.
Define matrices 1, 1', P, and Q by
.
In even or in odd dimensionality, this quantization procedure amounts to replacing the ordinary p, q coordinates with non-commutative coordinates constructed from P, Q in a suitable fashion.
Even case
In the case when n = 2k is even, let
for i = 1,2,...,k (where the P or Q is considered to occupy the i-th position). The operation is the tensor product of matrices. It is no longer important to distinguish between the Ps and Qs, so we shall simply refer to them all with the symbol P, and regard the index on Pi as ranging from i = 1 to i = 2k. For instance, the following properties hold:
, and for all unequal pairs i and j. (Clifford relations.)
Thus the algebra generated by the Pi is the Clifford algebra of euclidean n-space.
Let A denote the algebra generated by these matrices. By counting dimensions, A is a complete 2k×2k matrix algebra over the complex numbers. As a matrix algebra, therefore, it acts on 2k-dimensional column vectors (with complex entries). These column vectors are the spinors.
We now turn to the action of the orthogonal group on the spinors. Consider the application of an orthogonal transformation to the coordinates, which in turn acts upon the Pi via
.
That is, . Since the Pi generate A, the action of this transformation extends to all of A and produces an automorphism of A. From elementary linear algebra, any such automorphism must be given by a change of basis. Hence there is a matrix S, depending on R, such that
(1).
In particular, S(R) will act on column vectors (spinors). By decomposing rotations into products of reflections, one can write down a formula for S(R) in much the same way as in the case of three dimensions.
There is more than one matrix S(R) which produces the action in (1). The ambiguity defines S(R) up to a nonevanescent scalar factor c. Since S(R) and cS(R) define the same transformation (1), the action of the orthogonal group on spinors is not single-valued, but instead descends to
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https://en.wikipedia.org/wiki/List%20of%20Canisius%20University%20people
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This is a list of notable students, alumni, staff and faculty of Canisius University in Buffalo, New York.
Academia
Sister Marion Beiter, '44 - mathematician, Chairman of Mathematics for Rosary Hill College
H. James Birx - Professor of Anthropology, Canisius University
Paul G. Bulger - President of Buffalo State College
James Demske, S.J., B.A. '47 – President of Canisius College (1966–1993)
John Hurley '78 - 24th president and 1st lay president of Canisius College (2010-2022); president emeritus (2022-pres.)
Mark Huson - Professor of Finance, University of Alberta, Faculty of Business
Paul G. Gassman '57 - former Chair of University of Minnesota Chemistry Department
Thomas J. Lawley '68 - Dean of the Emory University School of Medicine
Steven Seegel, Ph.D. ’99 - Professor of Slavic and Eurasian Studies at University of Texas at Austin
Business
Mary E. Belle '73 – President of Licensing, Jones Apparel Group
Michael Buczkowski '86 - General Manager / Vice President of the Buffalo Bisons
Peter M. Cuviello '69 – Vice President and managing director, Lockheed Martin
Calvin Darden '72 – Senior Vice President, United Parcel Service; named 8th most powerful black executive in the U.S. by Fortune magazine
Gene F. Jankowski '55 – former chairman, CBS Broadcast Group: Chairman and CEO, Jankowski
Gregory R. Maday '70 – Senior Vice President, Warner Bros.
George Mathewson '72 – chairman of the Board, Royal Bank of Scotland
Carl J. Montante '64 - founder, President, and managing director of Uniland Development; named 2010 Buffalo Outstanding Citizen
Charles Moran Jr. - President and COO, Delaware North
John W. Rowe '66 – chairman and CEO, Aetna, Inc.; member of the Academy of Science
Journalism
Anne Burrell '91 – Food Network chef
J. Michael Collins – co-founder of PBS Buffalo; Emmy Award winner for executive producing Reading Rainbow
Elizabeth MacDonald '84 - Gerald Loeb Award and multiple other awards winning financial journalist with The Wall Street Journal, anchorwoman on Fox Business, appeared on NBC's The Today Show, ABC's World News Tonight, Outnumbered, Your World with Neil Cavuto, CBS This Morning, C-SPAN, Court TV, ABC News Radio, NPR, and others
Michael Scheuer '74 – CBS News terrorism analyst; former CIA employee
Elaine Sciolino ’70 – author; Paris Bureau Chief for The New York Times
Cynthia L. Skrzycki '76 – financial columnist for The Washington Post
Adam Zyglis '04 – editorial cartoonist for The Buffalo News, winner of 2015 Pulitzer Prize for Editorial Cartooning
Medicine and science
Sister Marion Beiter, '44 - mathematician, Chairman of Mathematics for Rosary Hill College
Thomas J. Dougherty, Ph.D. '55 – Chief of Radiation/Biology Dept., Roswell Park Comprehensive Cancer Center
Thomas J. Lawley, M.D. '68 – Dean, School of Medicine, Emory University
Mark J. Lema, M.D., Ph.D. '71 – chairman, Critical Care Medicine & Pain Medicine, Roswell Park Comprehensive Cancer Center
Robert J. Lull, M.D. (deceased) '62 – Chief of Nu
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https://en.wikipedia.org/wiki/Extouch%20triangle
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In Euclidean geometry, the extouch triangle of a triangle is formed by joining the points at which the three excircles touch the triangle.
Coordinates
The vertices of the extouch triangle are given in trilinear coordinates by:
or equivalently, where are the lengths of the sides opposite angles respectively,
Related figures
The triangle's splitters are lines connecting the vertices of the original triangle to the corresponding vertices of the extouch triangle; they bisect the triangle's perimeter and meet at the Nagel point. This is shown in blue and labelled "N" in the diagram.
The Mandart inellipse is tangent to the sides of the reference triangle at the three vertices of the extouch triangle.
Area
The area of the extouch triangle, , is given by:
where and are the area and radius of the incircle, is the semiperimeter of the original triangle, and are the side lengths of the original triangle.
This is the same area as that of the intouch triangle.
References
Circles
Objects defined for a triangle
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https://en.wikipedia.org/wiki/Belt%20problem
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The belt problem is a mathematics problem which requires finding the length of a crossed belt that connects two circular pulleys with radius r1 and r2 whose centers are separated by a distance P. The solution of the belt problem requires trigonometry and the concepts of the bitangent line, the vertical angle, and congruent angles.
Solution
Clearly triangles ACO and ADO are congruent right angled triangles, as are triangles BEO and BFO. In addition, triangles ACO and BEO are similar. Therefore angles CAO, DAO, EBO and FBO are all equal. Denoting this angle by (denominated in radians), the length of the belt is
This exploits the convenience of denominating angles in radians that the length of an arc = the radius × the measure of the angle facing the arc.
To find we see from the similarity of triangles ACO and BEO that
For fixed P the length of the belt depends only on the sum of the radius values r1 + r2, and not on their individual values.
Pulley problem
There are other types of problems similar to the belt problem. The pulley problem, as shown, is similar to the belt problem; however, the belt does not cross itself. In the pulley problem the length of the belt is
where r1 represents the radius of the larger pulley, r2 represents the radius of the smaller one, and:
Applications
The belt problem is used in the design of aeroplanes, bicycle gearing, cars, and other items with pulleys or belts that cross each other. The pulley problem is also used in the design of conveyor belts found in airport luggage belts and automated factory lines.
See also
Tangent lines to circles
References
Trigonometry
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https://en.wikipedia.org/wiki/Chow%27s%20lemma
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Chow's lemma, named after Wei-Liang Chow, is one of the foundational results in algebraic geometry. It roughly says that a proper morphism is fairly close to being a projective morphism. More precisely, a version of it states the following:
If is a scheme that is proper over a noetherian base , then there exists a projective -scheme and a surjective -morphism that induces an isomorphism for some dense open
Proof
The proof here is a standard one.
Reduction to the case of irreducible
We can first reduce to the case where is irreducible. To start, is noetherian since it is of finite type over a noetherian base. Therefore it has finitely many irreducible components , and we claim that for each there is an irreducible proper -scheme so that has set-theoretic image and is an isomorphism on the open dense subset of . To see this, define to be the scheme-theoretic image of the open immersion
Since is set-theoretically noetherian for each , the map is quasi-compact and we may compute this scheme-theoretic image affine-locally on , immediately proving the two claims. If we can produce for each a projective -scheme as in the statement of the theorem, then we can take to be the disjoint union and to be the composition : this map is projective, and an isomorphism over a dense open set of , while is a projective -scheme since it is a finite union of projective -schemes. Since each is proper over , we've completed the reduction to the case irreducible.
can be covered by finitely many quasi-projective -schemes
Next, we will show that can be covered by a finite number of open subsets so that each is quasi-projective over . To do this, we may by quasi-compactness first cover by finitely many affine opens , and then cover the preimage of each in by finitely many affine opens each with a closed immersion in to since is of finite type and therefore quasi-compact. Composing this map with the open immersions and , we see that each is a closed subscheme of an open subscheme of . As is noetherian, every closed subscheme of an open subscheme is also an open subscheme of a closed subscheme, and therefore each is quasi-projective over .
Construction of and
Now suppose is a finite open cover of by quasi-projective -schemes, with an open immersion in to a projective -scheme. Set , which is nonempty as is irreducible. The restrictions of the to define a morphism
so that , where is the canonical injection and is the projection. Letting denote the canonical open immersion, we define , which we claim is an immersion. To see this, note that this morphism can be factored as the graph morphism (which is a closed immersion as is separated) followed by the open immersion ; as is noetherian, we can apply the same logic as before to see that we can swap the order of the open and closed immersions.
Now let be the scheme-theoretic image of , and factor as
where is an open immersion and is a closed immersion. Let and
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https://en.wikipedia.org/wiki/Ces%C3%A0ro%20equation
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In geometry, the Cesàro equation of a plane curve is an equation relating the curvature () at a point of the curve to the arc length () from the start of the curve to the given point. It may also be given as an equation relating the radius of curvature () to arc length. (These are equivalent because .) Two congruent curves will have the same Cesàro equation. Cesàro equations are named after Ernesto Cesàro.
Examples
Some curves have a particularly simple representation by a Cesàro equation. Some examples are:
Line: .
Circle: , where is the radius.
Logarithmic spiral: , where is a constant.
Circle involute: , where is a constant.
Cornu spiral: , where is a constant.
Catenary: .
Related parameterizations
The Cesàro equation of a curve is related to its Whewell equation in the following way: if the Whewell equation is then the Cesàro equation is .
References
External links
Curvature Curves at 2dcurves.com.
Curves
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https://en.wikipedia.org/wiki/Joseph%20O%27Rourke
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Joseph O'Rourke may refer to:
Joseph Cornelius O'Rourke (1772–1849), Russian nobleman and military leader
Joseph O'Rourke (professor), researcher in computational geometry
Joseph O'Rourke (activist) (1938–2008), Catholic ex-priest and pro-choice activist
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https://en.wikipedia.org/wiki/Kempner%20function
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In number theory, the Kempner function is defined for a given positive integer to be the smallest number such that divides the For example, the number does not divide , , but does
This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers.
History
This function was first considered by François Édouard Anatole Lucas in 1883, followed by Joseph Jean Baptiste Neuberg in 1887. In 1918, A. J. Kempner gave the first correct algorithm for
The Kempner function is also sometimes called the Smarandache function following Florentin Smarandache's rediscovery of the function
Properties
Since is always at A number greater than 4 is a prime number if and only That is, the numbers for which is as large as possible relative to are the primes. In the other direction, the numbers for which is as small as possible are the factorials: for
is the smallest possible degree of a monic polynomial with integer coefficients, whose values over the integers are all divisible
For instance, the fact that means that there is a cubic polynomial whose values are all zero modulo 6, for instance the polynomial
but that all quadratic or linear polynomials (with leading coefficient one) are nonzero modulo 6 at some integers.
In one of the advanced problems in The American Mathematical Monthly, set in 1991 and solved in 1994, Paul Erdős pointed out that the function coincides with the largest prime factor of for "almost all" (in the sense that the asymptotic density of the set of exceptions is zero).
Computational complexity
The Kempner function of an arbitrary number is the maximum, over the prime powers dividing , of .
When is itself a prime power , its Kempner function may be found in polynomial time by sequentially scanning the multiples of until finding the first one whose factorial contains enough multiples The same algorithm can be extended to any whose prime factorization is already known, by applying it separately to each prime power in the factorization and choosing the one that leads to the largest value.
For a number of the form , where is prime and is less than , the Kempner function of is . It follows from this that computing the Kempner function of a semiprime (a product of two primes) is computationally equivalent to finding its prime factorization, believed to be a difficult problem. More generally, whenever is a composite number, the greatest common divisor of will necessarily be a nontrivial divisor allowing to be factored by repeated evaluations of the Kempner function. Therefore, computing the Kempner function can in general be no easier than factoring composite numbers.
References and notes
Factorial and binomial topics
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https://en.wikipedia.org/wiki/Major%20League%20Soccer%20records%20and%20statistics
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Several Major League Soccer teams and players hold various records and statistics.
MLS Cup and Supporters' Shield winners
MLS Cup finals
Teams with most trophies
Player records (career)
Bold indicates an active player. All statistics are for regular season only.
Goals
As of the end of the 2023 regular season.
Goals from free kicks
Assists
As of the end of the 2022 regular season.
Minutes played
As of the end of the 2023 regular season.
Goals against average
As of the end of the 2020 regular season; minimum 75 matches played.
Player records (single season)
All statistics are for regular season only.
Most goals
Most assists
Most clean sheets
All-time regular season success
Supporters' Shield Standings through 2022 season.
Miami Fusion and Tampa Bay Mutiny folded after completion of the 2001 season.
Chivas USA folded after completion of the 2014 season.
All-time regular season table
Through completion of 2022 regular season.
1 – Ranking based number of points per season.
2 – Includes shoot-out wins from 1996–1999 season.
3 – Includes shoot-out losses from 1996–1999 seasons.
4 – Based on combined conference results before single format for playoff qualification was inaugurated in 2007.
All-time playoffs success
Through 2022 playoffs.
Shows number of best finishes at each playoff level through completion of 2022 playoffs. Does not include the 2020 "Play-In Round."
All-time playoffs table
Through 2020 playoffs.
1 – Ranking based on overall number of points.
2 – Includes shoot-out wins from 1996–1999 seasons. Post-1999 shoot-out wins counted as ties.
3 – Includes shoot-out losses from 1996–1999 seasons. Post-1999 shoot-out losses counted as ties.
Team records (single season)
Points
All stats are from regular season games only.
From 1996 to 1999, a shoot-out was used to determine the winner of a match if it was a tie after 90 minutes.
From 2000 to 2003, a 10-minute golden goal period was used to determine the winner of a match if it was a tie after 90 minutes.
In 2020, teams played a shortened season and unequal number of between 18 and 23 games.
1 – Points earned during shoot-out era
2 – Points earned during shortened 2020 season
3 – Did not win Supporters' Shield
Results
Wins
1 – Includes shoot-out wins in seasons played from 1996–1999.
Losses
1 – Includes shoot-out losses in seasons played from 1996–1999.
Ties
1 – No ties in seasons played from 1996–1999.
Goals
Most for
Fewest for
1 – Goals scored during shortened 2020 season.
Fewest allowed
Most allowed
1 – Goals scored during shortened 2020 season.
Best differential
Worst differential
Single game records
Highest scoring games
Most goals scored by a team
Biggest winning margin
Longest streaks
Consecutive postseason berths
Winning streaks
The longest winning start to a season was achieved by the LA Galaxy in 1996, who opened the campaign with twelve consecutive wins and won their first eight games in regulation.
The longest winning e
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https://en.wikipedia.org/wiki/Casus%20irreducibilis
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In algebra, () is one of the cases that may arise in solving polynomials of degree 3 or higher with integer coefficients algebraically (as opposed to numerically), i.e., by obtaining roots that are expressed with radicals. It shows that many algebraic numbers are real-valued but cannot be expressed in radicals without introducing complex numbers. The most notable occurrence of is in the case of cubic polynomials that have three real roots, which was proven by Pierre Wantzel in 1843.
One can see whether a given cubic polynomial is in so-called by looking at the discriminant, via Cardano's formula.
The three cases of the discriminant
Let
be a cubic equation with . Then the discriminant is given by
It appears in the algebraic solution and is the square of the product
of the differences of the 3 roots .
If , then the polynomial has one real root and two complex non-real roots. is purely imaginary.Although there are cubic polynomials with negative discriminant which are irreducible in the modern sense, casus irreducibilis does not apply.
If , then and there are three real roots; two of them are equal. Whether can be found out by the Euclidean algorithm, and if so, the roots by the quadratic formula. Moreover, all roots are real and expressible by real radicals.All the cubic polynomials with zero discriminant are reducible.
If , then is non-zero and real, and there are three distinct real roots which are sums of two complex conjugates.Because they require complex numbers (in the understanding of the time: cube roots from non-real numbers, i.e. from square roots from negative numbers) to express them in radicals, this case in the 16th century has been termed casus irreducibilis.
Formal statement and proof
More generally, suppose that is a formally real field, and that is a cubic polynomial, irreducible over , but having three real roots (roots in the real closure of ). Then casus irreducibilis states that it is impossible to express a solution of by radicals with radicands .
To prove this, note that the discriminant is positive. Form the field extension . Since this is or a quadratic extension of (depending in whether or not is a square in ), remains irreducible in it. Consequently, the Galois group of over is the cyclic group . Suppose that can be solved by real radicals. Then can be split by a tower of cyclic extensions
At the final step of the tower, is irreducible in the penultimate field , but splits in for some . But this is a cyclic field extension, and so must contain a conjugate of and therefore a primitive 3rd root of unity.
However, there are no primitive 3rd roots of unity in a real closed field. Suppose that ω is a primitive 3rd root of unity. Then, by the axioms defining an ordered field, ω and ω2 are both positive, because otherwise their cube (=1) would be negative. But if ω2>ω, then cubing both sides gives 1>1, a contradiction; similarly if ω>ω2.
Solution in non-real radicals
Cardano's solution
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https://en.wikipedia.org/wiki/William%20Holding%20Echols
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William Holding Echols VI (December 2, 1859 – September 25, 1934), generally called "Reddy" Echols, was a professor of mathematics at the University of Virginia. The Echols Scholars Program is named in his honor.
William Echols was the son of the fifth of the same name who was a Major in Confederate States of America.
Echols attended the university as an undergraduate and received his Bachelor of Science and a civil engineering degree from the university in 1882. Following his graduation, he became an engineering professor, and later director, of the Missouri School of Mines (now Missouri University of Science and Technology). He returned to UVA as adjunct professor of mathematics in 1891, teaching mechanical engineering and serving as the building and grounds supervisor.
On October 27, 1895, a fire started in the Rotunda Annex on the UVA grounds. Echols, in a dramatic attempt to save the Rotunda, attempted to use dynamite to destroy the roofed portico that connected the Annex and the Rotunda and keep the fire from spreading to the historic building. Unfortunately, despite his attempt to hurl 50 pounds of dynamite to the portico from atop the Rotunda dome, the portico held, the fire spread more rapidly than before, and the Rotunda was gutted by the blaze.
In later years, Echols authored a text on elementary calculus. He remained active in University life and was a member of Eli Banana. He died of a heart attack in his home on the East Lawn in 1934 and is buried in the university cemetery.
Selected publications
On a general formula for the expansion of functions in series. Bull. Amer. Math. Soc. 2 (1893) 135–144.
Wronski's expansion. Bull. Amer. Math. Soc. 2 (1893) 178–184.
References
1859 births
1934 deaths
Burials at the University of Virginia Cemetery
Missouri University of Science and Technology faculty
University of Virginia faculty
University of Virginia School of Engineering and Applied Science alumni
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https://en.wikipedia.org/wiki/Proximity%20problems
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Proximity problems is a class of problems in computational geometry which involve estimation of distances between geometric objects.
A subset of these problems stated in terms of points only are sometimes referred to as closest point problems, although the term "closest point problem" is also used synonymously to the nearest neighbor search.
A common trait for many of these problems is the possibility to establish the Θ(n log n) lower bound on their computational complexity by reduction from the element uniqueness problem basing on an observation that if there is an efficient algorithm to compute some kind of minimal distance for a set of objects, it is trivial to check whether this distance equals to 0.
Atomic problems
While these problems pose no computational complexity challenge, some of them are notable because of their ubiquity in computer applications of geometry.
Distance between a pair of line segments. It cannot be expressed by a single formula, unlike, e.g., the distance from a point to a line. Its calculation requires careful enumeration of possible configurations, especially in 3D and higher dimensions.
Bounding box, the minimal axis-aligned hyperrectangle that contains all geometric data
Problems on points
Closest pair of points: Given N points, find two with the smallest distance between them
Closest point query / nearest neighbor query: Given N points, find one with the smallest distance to a given query point
All nearest neighbors problem (construction of the nearest-neighbor graph): Given N points, find a closest one for each of them
Diameter of a point set: Given N points, find two with the largest distance between them
Width of a point set: Given N points, find two (hyper)planes with the smallest distance between them and with all points between them
Minimum spanning tree for a set of points
Euclidean minimum spanning tree
Delaunay triangulation
Voronoi diagram
Smallest enclosing sphere: Given N points, find a smallest sphere (circle) enclosing them all
Largest empty circle: Given N points in the plane, find a largest circle centered within their convex hull and enclosing none of them
Smallest enclosing rectangle: unlike the bounding box problem mentioned above, the rectangle may be of any orientation
Largest empty rectangle
Geometric spanner, a weighted graph over a set of points as its vertices which for every pair of vertices has a path between them of weight at most 'k' times the spatial distance between these points for a fixed 'k'.
Other
Shortest path among obstacles
Distance of closest approach
References
The proximity problems are covered in chapters 6 and 7.
Geometric algorithms
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https://en.wikipedia.org/wiki/Littelmann%20path%20model
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In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras. Its most important application is to complex semisimple Lie algebras or equivalently compact semisimple Lie groups, the case described in this article. Multiplicities in irreducible representations, tensor products and branching rules can be calculated using a coloured directed graph, with labels given by the simple roots of the Lie algebra.
Developed as a bridge between the theory of crystal bases arising from the work of Kashiwara and Lusztig on quantum groups and the standard monomial theory of C. S. Seshadri and Lakshmibai, Littelmann's path model associates to each irreducible representation a rational vector space with basis given by paths from the origin to a weight as well as a pair of root operators acting on paths for each simple root. This gives a direct way of recovering the algebraic and combinatorial structures previously discovered by Kashiwara and Lusztig using quantum groups.
Background and motivation
Some of the basic questions in the representation theory of complex semisimple Lie algebras or compact semisimple Lie groups going back to Hermann Weyl include:
For a given dominant weight λ, find the weight multiplicities in the irreducible representation L(λ) with highest weight λ.
For two highest weights λ, μ, find the decomposition of their tensor product L(λ) L(μ) into irreducible representations.
Suppose that is the Levi component of a parabolic subalgebra of a semisimple Lie algebra . For a given dominant highest weight λ, determine the branching rule for decomposing the restriction of L(λ) to .
(Note that the first problem, of weight multiplicities, is the special case of the third in which the parabolic subalgebra is a Borel subalgebra. Moreover, the Levi branching problem can be embedded in the tensor product problem as a certain limiting case.)
Answers to these questions were first provided by Hermann Weyl and Richard Brauer as consequences of explicit character formulas, followed by later combinatorial formulas of Hans Freudenthal, Robert Steinberg and Bertram Kostant; see . An unsatisfactory feature of these formulas is that they involved alternating sums for quantities that were known a priori to be non-negative. Littelmann's method expresses these multiplicities as sums of non-negative integers without overcounting. His work generalizes classical results based on Young tableaux for the general linear Lie algebra n or the special linear Lie algebra n:
Issai Schur's result in his 1901 dissertation that the weight multiplicities could be counted in terms of column-strict Young tableaux (i.e. weakly increasing to the right along rows, and strictly increasing down columns).
The celebrated Littlewood–Richardson rule that describes both tensor product decompositions and branching from m+n to m n in terms
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https://en.wikipedia.org/wiki/Existentially%20closed%20model
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In model theory, a branch of mathematical logic, the notion of an existentially closed model (or existentially complete model) of a theory generalizes the notions of algebraically closed fields (for the theory of fields), real closed fields (for the theory of ordered fields), existentially closed groups (for the theory of groups), and dense linear orders without endpoints (for the theory of linear orders).
Definition
A substructure M of a structure N is said to be existentially closed in (or existentially complete in) if for every quantifier-free formula φ(x1,…,xn,y1,…,yn) and all elements b1,…,bn of M such that φ(x1,…,xn,b1,…,bn) is realized in N, then φ(x1,…,xn,b1,…,bn) is also realized in M. In other words: If there is a tuple a1,…,an in N such that φ(a1,…,an,b1,…,bn) holds in N, then such a tuple also exists in M. This notion is often denoted .
A model M of a theory T is called existentially closed in T if it is existentially closed in every superstructure N that is itself a model of T. More generally, a structure M is called existentially closed in a class K of structures (in which it is contained as a member) if M is existentially closed in every superstructure N that is itself a member of K.
The existential closure in K of a member M of K, when it exists, is, up to isomorphism, the least existentially closed superstructure of M. More precisely, it is any extensionally closed superstructure M∗ of M such that for every existentially closed superstructure N of M, M∗ is isomorphic to a substructure of N via an isomorphism that is the identity on M.
Examples
Let σ = (+,×,0,1) be the signature of fields, i.e. + and × are binary function symbols and 0 and 1 are constant symbols. Let K be the class of structures of signature σ that are fields. If A is a subfield of B, then A is existentially closed in B if and only if every system of polynomials over A that has a solution in B also has a solution in A. It follows that the existentially closed members of K are exactly the algebraically closed fields.
Similarly in the class of ordered fields, the existentially closed structures are the real closed fields. In the class of linear orders, the existentially closed structures are those that are dense without endpoints, while the existential closure of any countable (including empty) linear order is, up to isomorphism, the countable dense total order without endpoints, namely the order type of the rationals.
References
External links
Encyclopedia of Mathematics article
Model theory
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https://en.wikipedia.org/wiki/Ian%20Wilson%20%28phonetician%29
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Ian Wilson (born in 1966) is a Canadian linguist.
Biography
Wilson has a Bachelor of Mathematics from the University of Waterloo, he has an M.A. in Teaching English as a Foreign/Second Language from the University of Birmingham and a PhD in Linguistics (phonetics) from the University of British Columbia.
He is professor at the University of Aizu in Aizuwakamatsu city, Fukushima prefecture, Japan. His field of research is phonetics, especially articulatory phonetics and articulatory setting. He is one of the first teacher/researchers to use ultrasound in a large-scale ESL classroom as a method of providing direct visual biofeedback to pronunciation learners on the movements of the tongue during speech.
Ian Wilson is a co-author of Articulatory Phonetics, which introduces students to the field of Articulatory Phonetics and Speech Science.
References
External links
Ian Wilson's official website at the University of Aizu
Ian Wilson's Phonetics Laboratory
Wiley-Blackwell's Articulatory Phonetics Link
1966 births
Living people
University of Waterloo alumni
Alumni of the University of Birmingham
University of British Columbia alumni
Phoneticians
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https://en.wikipedia.org/wiki/G.%20W.%20Peck
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G. W. Peck is a pseudonymous attribution used as the author or co-author of a number of published mathematics academic papers. Peck is sometimes humorously identified with George Wilbur Peck, a former governor of the US state of Wisconsin.
Peck first appeared as the official author of a 1979 paper entitled "Maximum antichains of rectangular arrays". The name "G. W. Peck" is derived from the initials of the actual writers of this paper: Ronald Graham, Douglas West, George B. Purdy, Paul Erdős, Fan Chung, and Daniel Kleitman. The paper initially listed Peck's affiliation as Xanadu, but the editor of the journal objected, so Ron Graham gave him a job at Bell Labs. Since then, Peck's name has appeared on some sixteen publications, primarily as a pseudonym of Daniel Kleitman.
In reference to "G. W. Peck", Richard P. Stanley defined a Peck poset to be a graded partially ordered set that is rank symmetric, rank unimodal, and strongly Sperner. The posets in the original paper by G. W. Peck are not quite Peck posets, as they lack the property of being rank symmetric.
See also
Nicolas Bourbaki
Arthur Besse
John Rainwater
Blanche Descartes
Monsieur LeBlanc
References
External links
Imaginary Erdős numbers, Numberphile, Nov 26, 2014. Video interview with Ron Graham in which he tells the story of G. W. Peck.
Academic shared pseudonyms
American mathematicians
Pseudonymous mathematicians
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https://en.wikipedia.org/wiki/Iktaba
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Iktaba () is a Palestinian town located four kilometers Northeast of the city of Tulkarm in the Tulkarm Governorate in the northern West Bank. According to the Palestinian Central Bureau of Statistics (PCBS), the town had a population of 2,665 inhabitants in 2007 and 2,997 by 2017. Refugees make-up 33% of the entire population in 1997.
History
In 1265, after the Mamluks had defeated the Crusaders, Iktaba (Sabahiya) was mentioned among the estates which Sultan Baibars granted his followers. The village was given to the emir Alam al-Din Tardaj al-Amadi.
Ottoman era
Iktaba was incorporated into the Ottoman Empire in 1517 with all of Palestine, and in 1596 it appeared in the tax registers under the name of Staba, being in the Nahiya of Qaqun of the Liwa of Nablus. It had a population of 21 households, all Muslims. The villagers paid a fixed tax rate of 33,3% on various agricultural products, such as wheat, barley, summer crops, olive trees, goats and/or beehives, in addition to "occasional revenues" and a press for olive oil or grapes; a total of 4,100 akçe.
In 1870, the French explorer Victor Guérin noted that village, which he called Astaba, was a "Small hamlet located on a high hill. Ancient cisterns testify to the existence here of an ancient locality. Fig trees and pomegranates grow around the dwellings."
In 1882, the PEF's Survey of Western Palestine described it as: "A place to which a certan effendi of Nablus comes down in spring, a sort of 'Azbeh or spring grazing-place for horses"
British Mandate era
In the 1922 census of Palestine conducted by the British Mandate authorities, Iktaba had a population of 121, all Muslims. In the 1931 census of Palestine, the combined population of Anabta, Iktaba and Nur ash Shams was 2498; 2,457 Muslims, 34 Christians and 1 Druze living in 502 houses.
In the 1945 statistics, the combined population of Anabta and Iktaba was 3,120; 3,080 Muslims and 40 Christians, with a total of 15,445 dunams of land according to an official land and population survey. Of this, a total of 5,908 dunams were plantations and irrigable land, 5,842 were used for cereals, while 84 dunams were built-up (urban) land.
Jordanian era
After the 1948 Arab–Israeli War and the 1949 Armistice Agreements, Iktaba came under Jordanian rule.
In 1961, the population was 372.
Post-1967
After the Six-Day War in 1967, Iktaba has been under Israeli occupation.
References
Bibliography
External links
Welcome To Iktaba
Survey of Western Palestine, Map 11: IAA, Wikimedia commons
Tulkarm Governorate
Villages in the West Bank
Municipalities of the State of Palestine
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https://en.wikipedia.org/wiki/Protestantism%20in%20Colombia
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The National Administrative Department of Statistics (DANE) does not collect religious statistics, and accurate reports are difficult to obtain. However, based on various studies and a survey, about 90% of the population adheres to Christianity, the majority of which (70.9%) are Roman Catholic, while a significant minority (16.7%) adhere to Protestantism (primarily Evangelicalism).
In 2020, figures suggest that Protestants make up 14% of the country's population.
Protestant Christians present in Colombia are Baptists, Lutherans, Mennonites, Nazarenes, Pentecostals and Seventh-day Adventists.
List of Denominations
Alianza Cristiana y Misionera
Assemblies of God
Asociación de Iglesias Hermanos Menonitas de Colombia: 831 (1998)
Church of God Ministry of Jesus Christ International
Church of the Nazarene: 12,860 (1998)
Hermanos en Cristo
Iglesia Cruzada Evangélica
Iglesia de Dios
Iglesia Evangélica Luterana
Misión Evangélica
Mision Indigena
Misión Nuevas Tribus
Presbyterian Church of Colombia
Unión Misionera Evangélica
Seventh-day Adventist Church
Freedom of religion
The constitution provides for freedom of religion and the government generally is in support of this. However, international NGOs have stated that indigenous Protestants face threats, harassment and arbitrary detention in their communities due to their religious beliefs; in particular, Indigenous authorities in the Pizarro and Litoral de San Juan municipalities in the Chocó Department have prohibited the practice of Christianity and stated punishments for Protestants.
In 2023, the country was scored 4 out of 4 for religious freedom.
In the same year, the country was rank as the 22nd most difficulty place in the world to be a Christian.
See also
Religion in Colombia
Christianity in Colombia
Protestantism by country
External links
Adherents.com - Religion by Location
2000 Caribbean, Central & South America Mennonite & Brethren in Christ Churches
World Christian Encyclopedia: Second edition, Volume 1, pp 205 /206
Colombia, International Religious Freedom Report 2006
Union Columbiana Adventista del Séptimo Día
Colombia: nationwide Adventist effort feeds 80,000 in two hours
References
Colombia
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https://en.wikipedia.org/wiki/Fred%20Galvin
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Frederick William Galvin is a mathematician, currently a professor at the University of Kansas. His research interests include set theory and combinatorics.
His notable combinatorial work includes the proof of the Dinitz conjecture. In set theory, he proved with András Hajnal that if ℵω1 is a strong limit cardinal, then
holds. The research on extending this result led Saharon Shelah to the invention of PCF theory. Galvin gave an elementary proof of the Baumgartner–Hajnal theorem (). The original proof by Baumgartner and Hajnal used forcing and absoluteness. Galvin and Shelah also proved the square bracket partition relations and . Galvin also proved the partition relation where η denotes the order type of the set of rational numbers.
Galvin and Karel Prikry proved that every Borel set is Ramsey. Galvin and Komjáth showed that the axiom of choice is equivalent to the statement that every graph has a chromatic number.
Galvin received his Ph.D. in 1967 from the University of Minnesota.
He invented Doublemove Chess in 1957, and Push Chess in 1967.
References
Living people
20th-century American mathematicians
21st-century American mathematicians
Combinatorialists
Set theorists
University of Minnesota alumni
University of Kansas faculty
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/John%20Brillhart
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John David Brillhart (November 13, 1930 – May 21, 2022) was a mathematician who worked in number theory at the University of Arizona.
Early life and education
Brillhart was born on November 13, 1930, in Berkeley, California.
He studied at the University of California, Berkeley, where he received his A.B. in 1953, his M.A. in 1966, and his Ph.D. in 1967. His doctoral thesis in mathematics was supervised by D. H. Lehmer, with assistance from Leonard Carlitz.
Before becoming a mathematician, he served in the United States Army.
Career
Brillhart joined the faculty at the University of Arizona in 1967 and retired in 2001. He advised two Ph.D. students.
Research
Brillart worked in integer factorization. His joint work with Michael A. Morrison in 1975 describes how to implement the continued fraction factorization method originally developed by Lehmer and Ralph Ernest Powers in 1931. One consequence was the first factorization of the Fermat number . Their ideas were influential in the development of the quadratic sieve by Carl Pomerance.
Brillhart was a member of the Cunningham Project, which factors Mersenne, Fermat, and related numbers. He was also a founding member and financial contributor to the Number Theory Foundation started by John L. Selfridge.
References
External links
1930 births
2022 deaths
20th-century American mathematicians
21st-century American mathematicians
Number theorists
University of California, Berkeley alumni
University of Arizona faculty
People from Berkeley, California
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https://en.wikipedia.org/wiki/W.%20B.%20R.%20Lickorish
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William Bernard Raymond Lickorish (born 19 February 1938) is a mathematician. He is emeritus professor of geometric topology in the Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, and also an emeritus fellow of Pembroke College, Cambridge. His research interests include topology and knot theory. He was one of the discoverers of the HOMFLY polynomial invariant of links, and proved the Lickorish-Wallace theorem which states that all closed orientable 3-manifolds can be obtained by Dehn surgery on a link.
Education
Lickorish received his Ph.D from Cambridge in 1964; his thesis was written under the supervision of Christopher Zeeman.
Recognition and awards
In 1991, Lickorish received the Senior Whitehead Prize from the London Mathematical Society. Lickorish and Kenneth Millett won the 1991 Chauvenet Prize for their paper "The New Polynomial Invariants of Knots and Links".
Lickorish was included in the 2019 class of fellows of the American Mathematical Society "for contributions to knot theory and low-dimensional topology".
Selected publications
See also
Lickorish twist theorem
Lickorish–Wallace theorem
References
1935 births
Living people
20th-century British mathematicians
21st-century British mathematicians
Topologists
Fellows of Pembroke College, Cambridge
Cambridge mathematicians
Fellows of the American Mathematical Society
Whitehead Prize winners
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https://en.wikipedia.org/wiki/Mean%20reciprocal%20rank
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The mean reciprocal rank is a statistic measure for evaluating any process that produces a list of possible responses to a sample of queries, ordered by probability of correctness. The reciprocal rank of a query response is the multiplicative inverse of the rank of the first correct answer: 1 for first place, for second place, for third place and so on. The mean reciprocal rank is the average of the reciprocal ranks of results for a sample of queries Q:
where refers to the rank position of the first relevant document for the i-th query.
The reciprocal value of the mean reciprocal rank corresponds to the harmonic mean of the ranks.
Example
For example, suppose we have the following three sample queries for a system that tries to translate English words to their plurals. In each case, the system makes three guesses, with the first one being the one it thinks is most likely correct:
Given those three samples, we could calculate the mean reciprocal rank as (1/3 + 1/2 + 1)/3 = 11/18 or about 0.61.
If none of the proposed results are correct, reciprocal rank is 0. Note that only the rank of the first relevant answer is considered, possible further relevant answers are ignored. If users are interested also in further relevant items, mean average precision is a potential alternative metric.
See also
Information retrieval
Question answering
References
Summary statistics
Information retrieval evaluation
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https://en.wikipedia.org/wiki/Self-avoiding%20walk
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In mathematics, a self-avoiding walk (SAW) is a sequence of moves on a lattice (a lattice path) that does not visit the same point more than once. This is a special case of the graph theoretical notion of a path. A self-avoiding polygon (SAP) is a closed self-avoiding walk on a lattice. Very little is known rigorously about the self-avoiding walk from a mathematical perspective, although physicists have provided numerous conjectures that are believed to be true and are strongly supported by numerical simulations.
In computational physics, a self-avoiding walk is a chain-like path in or with a certain number of nodes, typically a fixed step length and has the property that it doesn't cross itself or another walk. A system of SAWs satisfies the so-called excluded volume condition. In higher dimensions, the SAW is believed to behave much like the ordinary random walk.
SAWs and SAPs play a central role in the modeling of the topological and knot-theoretic behavior of thread- and loop-like molecules such as proteins. Indeed, SAWs may have first been introduced by the chemist Paul Flory in order to model the real-life behavior of chain-like entities such as solvents and polymers, whose physical volume prohibits multiple occupation of the same spatial point.
SAWs are fractals. For example, in the fractal dimension is 4/3, for it is close to 5/3 while for the fractal dimension is . The dimension is called the upper critical dimension above which excluded volume is negligible. A SAW that does not satisfy the excluded volume condition was recently studied to model explicit surface geometry resulting from expansion of a SAW.
The properties of SAWs cannot be calculated analytically, so numerical simulations are employed. The pivot algorithm is a common method for Markov chain Monte Carlo simulations for the uniform measure on -step self-avoiding walks. The pivot algorithm works by taking a self-avoiding walk and randomly choosing a point on this walk, and then applying symmetrical transformations (rotations and reflections) on the walk after the th step to create a new walk.
Calculating the number of self-avoiding walks in any given lattice is a common computational problem. There is currently no known formula, although there are rigorous methods of approximation.
Universality
One of the phenomena associated with self-avoiding walks and statistical physics models in general is the notion of universality, that is, independence of macroscopic observables from microscopic details, such as the choice of the lattice. One important quantity that appears in conjectures for universal laws is the connective constant, defined as follows. Let denote the number of -step self-avoiding walks. Since every -step self avoiding walk can be decomposed into an -step self-avoiding walk and an -step self-avoiding walk, it follows that . Therefore, the sequence is subadditive and we can apply Fekete's lemma to show that the following limit exists:
is called the con
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https://en.wikipedia.org/wiki/Lorentz%20curve
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Lorentz curve may refer to
the Cauchy–Lorentz distribution, a probability distribution
the Lorenz curve, a graphical representation of the inequality in a quantity's distribution
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https://en.wikipedia.org/wiki/Partial%20linear%20space
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A partial linear space (also semilinear or near-linear space) is a basic incidence structure in the field of incidence geometry, that carries slightly less structure than a linear space.
The notion is equivalent to that of a linear hypergraph.
Definition
Let an incidence structure, for which the elements of are called points and the elements of are called lines. S is a partial linear space, if the following axioms hold:
any line is incident with at least two points
any pair of distinct points is incident with at most one line
If there is a unique line incident with every pair of distinct points, then we get a linear space.
Properties
The De Bruijn–Erdős theorem shows that in any finite linear space which is not a single point or a single line, we have .
Examples
Projective space
Affine space
Polar space
Generalized quadrangle
Generalized polygon
Near polygon
References
.
Lynn Batten: Combinatorics of Finite Geometries. Cambridge University Press 1986, , p. 1-22
Lynn Batten and Albrecht Beutelspacher: The Theory of Finite Linear Spaces. Cambridge University Press, Cambridge, 1992.
Eric Moorhouse: Incidence Geometry. Lecture notes (archived)
External links
partial linear space at the University of Kiel
partial linear space at PlanetMath
Geometry
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https://en.wikipedia.org/wiki/Simplicial%20volume
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In the mathematical field of geometric topology, the simplicial volume (also called Gromov norm) is a certain measure of the topological complexity of a manifold. More generally, the simplicial norm measures the complexity of homology classes.
Given a closed and oriented manifold, one defines the simplicial norm by minimizing the sum of the absolute values of the coefficients over all singular chains homologous to a given cycle. The simplicial volume is the simplicial norm of the fundamental class.
It is named after Mikhail Gromov, who introduced it in 1982. With William Thurston, he proved that the simplicial volume of a finite volume hyperbolic manifold is proportional to the hyperbolic volume.
The simplicial volume is equal to twice the Thurston norm
Thurston also used the simplicial volume to prove that hyperbolic volume decreases under hyperbolic Dehn surgery.
References
Michael Gromov. Volume and bounded cohomology. Inst. Hautes Études Sci. Publ. Math. 56 (1982), 5–99.
External links
Simplicial volume at the Manifold Atlas.
Homology theory
Norms (mathematics)
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https://en.wikipedia.org/wiki/Trace%20operator
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In mathematics, the trace operator extends the notion of the restriction of a function to the boundary of its domain to "generalized" functions in a Sobolev space. This is particularly important for the study of partial differential equations with prescribed boundary conditions (boundary value problems), where weak solutions may not be regular enough to satisfy the boundary conditions in the classical sense of functions.
Motivation
On a bounded, smooth domain , consider the problem of solving Poisson's equation with inhomogeneous Dirichlet boundary conditions:
with given functions and with regularity discussed in the application section below. The weak solution of this equation must satisfy
for all .
The -regularity of is sufficient for the well-definedness of this integral equation. It is not apparent, however, in which sense can satisfy the boundary condition on : by definition, is an equivalence class of functions which can have arbitrary values on since this is a null set with respect to the n-dimensional Lebesgue measure.
If there holds by Sobolev's embedding theorem, such that can satisfy the boundary condition in the classical sense, i.e. the restriction of to agrees with the function (more precisely: there exists a representative of in with this property). For with such an embedding does not exist and the trace operator presented here must be used to give meaning to . Then with is called a weak solution to the boundary value problem if the integral equation above is satisfied. For the definition of the trace operator to be reasonable, there must hold for sufficiently regular .
Trace theorem
The trace operator can be defined for functions in the Sobolev spaces with , see the section below for possible extensions of the trace to other spaces. Let for be a bounded domain with Lipschitz boundary. Then there exists a bounded linear trace operator
such that extends the classical trace, i.e.
for all .
The continuity of implies that
for all
with constant only depending on and . The function is called trace of and is often simply denoted by . Other common symbols for include and .
Construction
This paragraph follows Evans, where more details can be found, and assumes that has a -boundary. A proof (of a stronger version) of the trace theorem for Lipschitz domains can be found in Gagliardo. On a -domain, the trace operator can be defined as continuous linear extension of the operator
to the space . By density of in such an extension is possible if is continuous with respect to the -norm. The proof of this, i.e. that there exists (depending on and ) such that
for all
is the central ingredient in the construction of the trace operator. A local variant of this estimate for -functions is first proven for a locally flat boundary using the divergence theorem. By transformation, a general -boundary can be locally straightened to reduce to this case, where the -regularity of the transform
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https://en.wikipedia.org/wiki/Berger%27s%20sphere
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In Riemannian geometry, a Berger sphere, named after Marcel Berger, is a standard 3-sphere with Riemannian metric from a one-parameter family, which can be obtained from the standard metric by shrinking along fibers of a Hopf fibration. It is interesting in that it is one of the simplest examples of Gromov collapse.
More precisely, one first considers the Lie algebra spanned by generators x1, x2, x3 with Lie bracket [xi,xj] = −2εijkxk. This is well known to correspond to the simply connected Lie group S3. Denote by ω1, ω2, ω3 the left invariant 1-forms on S3 which equal the dual covectors to x1, x2, x3. Then the standard metric on S3 is ω12+ω22+ω32. The Berger metric is βω12+ω22+ω32, for any constant β>0.
There are also higher-dimensional analogues of Berger spheres.
References
Riemannian geometry
Spheres
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https://en.wikipedia.org/wiki/Automorphic%20factor
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In mathematics, an automorphic factor is a certain type of analytic function, defined on subgroups of SL(2,R), appearing in the theory of modular forms. The general case, for general groups, is reviewed in the article 'factor of automorphy'.
Definition
An automorphic factor of weight k is a function
satisfying the four properties given below. Here, the notation and refer to the upper half-plane and the complex plane, respectively. The notation is a subgroup of SL(2,R), such as, for example, a Fuchsian group. An element is a 2×2 matrix
with a, b, c, d real numbers, satisfying ad−bc=1.
An automorphic factor must satisfy:
For a fixed , the function is a holomorphic function of .
For all and , one has for a fixed real number k.
For all and , one has Here, is the fractional linear transform of by .
If , then for all and , one has Here, I denotes the identity matrix.
Properties
Every automorphic factor may be written as
with
The function is called a multiplier system. Clearly,
,
while, if , then
which equals when k is an integer.
References
Robert Rankin, Modular Forms and Functions, (1977) Cambridge University Press . (Chapter 3 is entirely devoted to automorphic factors for the modular group.)
Modular forms
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https://en.wikipedia.org/wiki/Schouten%E2%80%93Nijenhuis%20bracket
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In differential geometry, the Schouten–Nijenhuis bracket, also known as the Schouten bracket, is a type of graded Lie bracket defined on multivector fields on a smooth manifold extending the Lie bracket of vector fields. There are two different versions, both rather confusingly called by the same name. The most common version is defined on alternating multivector fields and makes them into a Gerstenhaber algebra, but there is also another version defined on symmetric multivector fields, which is more or less the same as the Poisson bracket on the cotangent bundle. It was invented by Jan Arnoldus Schouten (1940, 1953) and its properties were investigated by his student Albert Nijenhuis (1955). It is related to but not the same as the Nijenhuis–Richardson bracket and the Frölicher–Nijenhuis bracket.
Definition and properties
An alternating multivector field is a section of the exterior algebra ∧∗TM over the tangent bundle of a manifold M. The alternating multivector fields form a graded supercommutative ring with the product of a and b written as ab (some authors use a∧b). This is dual to the usual algebra of differential forms Ω∗M by the pairing on homogeneous elements:
The degree of a multivector A in is defined to be |A| = p.
The skew symmetric Schouten–Nijenhuis bracket is the unique extension of the Lie bracket of vector fields to a graded bracket on the space of alternating multivector fields that makes the alternating multivector fields into a Gerstenhaber algebra.
It is given in terms of the Lie bracket of vector fields by
for vector fields ai, bj and
for vector fields and smooth function , where is the common interior product operator.
It has the following properties.
|ab| = |a| + |b| (The product has degree 0)
|[a,b]| = |a| + |b| − 1 (The Schouten–Nijenhuis bracket has degree −1)
(ab)c = a(bc), ab = (−1)|a||b|ba (the product is associative and (super) commutative)
[a, bc] = [a, b]c + (−1)|b|(|a| − 1)b[a, c] (Poisson identity)
[a,b] = −(−1)(|a| − 1)(|b| − 1) [b,a] (Antisymmetry of Schouten–Nijenhuis bracket)
[[a,b],c] = [a,[b,c]] − (−1)(|a| − 1)(|b| − 1)[b,[a,c]] (Jacobi identity for Schouten–Nijenhuis bracket)
If f and g are functions (multivectors homogeneous of degree 0), then [f,g] = 0.
If a is a vector field, then [a,b] = Lab is the usual Lie derivative of the multivector field b along a, and in particular if a and b are vector fields then the Schouten–Nijenhuis bracket is the usual Lie bracket of vector fields.
The Schouten–Nijenhuis bracket makes the multivector fields into a Lie superalgebra if the grading
is changed to the one of opposite parity (so that the even and odd subspaces are switched), though
with this new grading it is no longer a supercommutative ring. Accordingly, the Jacobi identity may also be expressed in the symmetrical form
Generalizations
There is a common generalization of the Schouten–Nijenhuis bracket for alternating multivector fields and the Frölicher–Nijenhuis bracket du
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https://en.wikipedia.org/wiki/Quasi-birth%E2%80%93death%20process
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In queueing models, a discipline within the mathematical theory of probability, the quasi-birth–death process describes a generalisation of the birth–death process. As with the birth-death process it moves up and down between levels one at a time, but the time between these transitions can have a more complicated distribution encoded in the blocks.
Discrete time
The stochastic matrix describing the Markov chain has block structure
where each of A0, A1 and A2 are matrices and A*0, A*1 and A*2 are irregular matrices for the first and second levels.
Continuous time
The transition rate matrix for a quasi-birth-death process has a tridiagonal block structure
where each of B00, B01, B10, A0, A1 and A2 are matrices. The process can be viewed as a two dimensional chain where the block structure are called levels and the intra-block structure phases. When describing the process by both level and phase it is a continuous-time Markov chain, but when considering levels only it is a semi-Markov process (as transition times are then not exponentially distributed).
Usually the blocks have finitely many phases, but models like the Jackson network can be considered as quasi-birth-death processes with infinitely (but countably) many phases.
Stationary distribution
The stationary distribution of a quasi-birth-death process can be computed using the matrix geometric method.
References
Queueing theory
Markov processes
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https://en.wikipedia.org/wiki/Bracket%20%28mathematics%29
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In mathematics, brackets of various typographical forms, such as parentheses ( ), square brackets [ ], braces { } and angle brackets ⟨ ⟩, are frequently used in mathematical notation. Generally, such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Sometimes, for the clarity of reading, different kinds of brackets are used to express the same meaning of precedence in a single expression with deep nesting of sub-expressions.
Historically, other notations, such as the vinculum generally, were similarly used for grouping. In present-day use, these notations all have specific meanings. The earliest use of brackets to indicate aggregation (i.e. grouping) was suggested in 1608 by Christopher Clavius, and in 1629 by Albert Girard.
Symbols for representing angle brackets
A variety of different symbols are used to represent angle brackets. In e-mail and other ASCII text, it is common to use the less-than (<) and greater-than (>) signs to represent angle brackets, because ASCII does not include angle brackets.
Unicode has pairs of dedicated characters; other than less-than and greater-than symbols, these include:
and
and
and
and
and , which are deprecated
In LaTeX the markup is \langle and \rangle: .
Non-mathematical angled brackets include:
and , used in East-Asian text quotation
and , which are dingbats
There are additional dingbats with increased line thickness, and some angle quotation marks and deprecated characters.
Algebra
In elementary algebra, parentheses ( ) are used to specify the order of operations. Terms inside the bracket are evaluated first; hence 2×(3 + 4) is 14, is 2 and (2×3) + 4 is 10. This notation is extended to cover more general algebra involving variables: for example . Square brackets are also often used in place of a second set of parentheses when they are nested—so as to provide a visual distinction.
In mathematical expressions in general, parentheses are also used to indicate grouping (i.e., which parts belong together) when necessary to avoid ambiguities and improve clarity. For example, in the formula , used in the definition of composition of two natural transformations, the parentheses around serve to indicate that the indexing by is applied to the composition , and not just its last component .
Functions
The arguments to a function are frequently surrounded by brackets: . With some standard function when there is little chance of ambiguity, it is common to omit the parentheses around the argument altogether (e.g., ). Note that this is never done with a general function , in which case the parenthesis are always included
Coordinates and vectors
In the Cartesian coordinate system, brackets are used to specify the coordinates of a point. For example, (2,3) denotes the point with x-coordinate 2 and y-coordinate 3.
The inner product of two vectors is com
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https://en.wikipedia.org/wiki/SPQR%20tree
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In graph theory, a branch of mathematics, the triconnected components of a biconnected graph are a system of smaller graphs that describe all of the 2-vertex cuts in the graph. An SPQR tree is a tree data structure used in computer science, and more specifically graph algorithms, to represent the triconnected components of a graph. The SPQR tree of a graph may be constructed in linear time and has several applications in dynamic graph algorithms and graph drawing.
The basic structures underlying the SPQR tree, the triconnected components of a graph, and the connection between this decomposition and the planar embeddings of a planar graph, were first investigated by ; these structures were used in efficient algorithms by several other researchers prior to their formalization as the SPQR tree by .
Structure
An SPQR tree takes the form of an unrooted tree in which for each node x there is associated an undirected graph or multigraph Gx. The node, and the graph associated with it, may have one of four types, given the initials SPQR:
In an S node, the associated graph is a cycle graph with three or more vertices and edges. This case is analogous to series composition in series–parallel graphs; the S stands for "series".
In a P node, the associated graph is a dipole graph, a multigraph with two vertices and three or more edges, the planar dual to a cycle graph. This case is analogous to parallel composition in series–parallel graphs; the P stands for "parallel".
In a Q node, the associated graph has a single real edge. This trivial case is necessary to handle the graph that has only one edge. In some works on SPQR trees, this type of node does not appear in the SPQR trees of graphs with more than one edge; in other works, all non-virtual edges are required to be represented by Q nodes with one real and one virtual edge, and the edges in the other node types must all be virtual.
In an R node, the associated graph is a 3-connected graph that is not a cycle or dipole. The R stands for "rigid": in the application of SPQR trees in planar graph embedding, the associated graph of an R node has a unique planar embedding.
Each edge xy between two nodes of the SPQR tree is associated with two directed virtual edges, one of which is an edge in Gx and the other of which is an edge in Gy. Each edge in a graph Gx may be a virtual edge for at most one SPQR tree edge.
An SPQR tree T represents a 2-connected graph GT, formed as follows. Whenever SPQR tree edge xy associates the virtual edge ab of Gx with the virtual edge cd of Gy, form a single larger graph by merging a and c into a single supervertex, merging b and d into another single supervertex, and deleting the two virtual edges. That is, the larger graph is the 2-clique-sum of Gx and Gy. Performing this gluing step on each edge of the SPQR tree produces the graph GT; the order of performing the gluing steps does not affect the result. Each vertex in one of the graphs Gx may be associated in this way with
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https://en.wikipedia.org/wiki/Nijenhuis%20bracket
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In mathematics there are four different but related brackets named after Albert Nijenhuis, giving Lie superalgebra structures to various spaces of tensors:
Frölicher–Nijenhuis bracket (defined on vector valued forms, extending the Lie bracket of vector fields)
Nijenhuis–Richardson bracket (defined on vector valued forms; this has a different degree to the Frölicher-Nijenhuis bracket)
Schouten–Nijenhuis bracket (2 versions, defined on either symmetric or antisymmetric multivectors)
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https://en.wikipedia.org/wiki/Spectrum%20bias
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In biostatistics, spectrum bias refers to the phenomenon that the performance of a diagnostic test may vary in different clinical settings because each setting has a different mix of patients. Because the performance may be dependent on the mix of patients, performance at one clinic may not be predictive of performance at another clinic. These differences are interpreted as a kind of bias. Mathematically, the spectrum bias is a sampling bias and not a traditional statistical bias; this has led some authors to refer to the phenomenon as spectrum effects, whilst others maintain it is a bias if the true performance of the test differs from that which is 'expected'. Usually the performance of a diagnostic test is measured in terms of its sensitivity and specificity and it is changes in these that are considered when referring to spectrum bias. However, other performance measures such as the likelihood ratios may also be affected by spectrum bias.
Generally spectrum bias is considered to have three causes. The first is due to a change in the case-mix of those patients with the target disorder (disease) and this affects the sensitivity. The second is due to a change in the case-mix of those without the target disorder (disease-free) and this affects the specificity. The third is due to a change in the prevalence, and this affects both the sensitivity and specificity. This final cause is not widely appreciated, but there is mounting empirical evidence as well as theoretical arguments which suggest that it does indeed affect a test's performance.
Examples where the sensitivity and specificity change between different sub-groups of patients may be found with the carcinoembryonic antigen test and urinary dipstick tests.
Diagnostic test performances reported by some studies may be artificially overestimated if it is a case-control design where a healthy population ('fittest of the fit') is compared with a population with advanced disease ('sickest of the sick'); that is two extreme populations are compared, rather than typical healthy and diseased populations.
If properly analyzed, recognition of heterogeneity of subgroups can lead to insights about the test's performance in varying populations.
See also
Simpson's paradox
Biased sample
Reporting bias
Reference class problem
References
Biostatistics
Bias
Design of experiments
Medical statistics
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https://en.wikipedia.org/wiki/Albert%20Nijenhuis
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Albert Nijenhuis (November 21, 1926 – February 13, 2015) was a Dutch-American mathematician who specialized in differential geometry and the theory of deformations in algebra and geometry, and later worked in combinatorics.
His high school studies at the gymnasium in Arnhem were interrupted by the evacuation of Arnhem by the Nazis after the failure of Operation Market Garden by the Allies. He continued his high school mathematical studies by himself on his grandparents’ farm, and then took state exams in 1945.
His university studies were carried out at the University of Amsterdam, where he received the degree of Candidaat (equivalent to a Bachelor of Science) in 1947, and a Doctorandus (equivalent to a Masters in Science) in 1950, cum laude. He was a Medewerker (associate) at the Mathematisch Centrum (now the Centrum Wiskunde & Informatica) in Amsterdam 1951–1952. He obtained a Ph.D. in mathematics in 1952, cum laude (Theory of the geometric object). His thesis advisor was Jan Arnoldus Schouten.
He came to the United States in 1952 as a Fulbright fellow (1952–1953) at Princeton University. He then studied at the Institute for Advanced Study in Princeton, New Jersey 1953–1955, after which he spent a year as an Instructor in mathematics at the University of Chicago. He then moved to the University of Washington in Seattle, first as an assistant professor and then a professor of mathematics, departing in 1963 for the University of Pennsylvania, where he was a professor of mathematics until his retirement in 1987. He was a Fulbright Professor at the University of Amsterdam in 1963–1964, and a visiting professor at the University of Geneva in 1967–1968, and at Dartmouth College in 1977–1978. following his retirement, he was a professor emeritus of the University of Pennsylvania and an Affiliate Professor at the University of Washington.
In 1958 he was an invited speaker at the International Mathematical Congress in Edinburgh. He was a J.S. Guggenheim Fellow in 1961–1962, again studying at the Institute for Advanced Study. In 1966 he became a correspondent member of the Royal Netherlands Academy of Arts and Sciences, and in 2012 he became a fellow of the American Mathematical Society.
Career
His early work was in the area of differential geometry. He developed the Nijenhuis tensor in 1951, during his Ph.D studies at the University of Amsterdam. It was also during this time that he explored the properties of the Schouten-Nijenhuis bracket, although his work was not published until 1955. In a lecture at the American Mathematical Society Summer Institute in Differential Geometry (1956) in Seattle he was the first to mention deformations of complex structures and their exact relationship to cohomology.
With Alfred Frölicher, he developed the Frölicher-Nijenhuis bracket (1955). Further work in this area with Roger Richardson yielded the Nijenhuis–Richardson bracket (1964).
Soon thereafter his interests shifted to combinatorics. Much o
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https://en.wikipedia.org/wiki/Dirk%20Kreimer
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Dirk Kreimer (born 12 July 1960) is a German physicist who pioneered the Hopf-algebraic approach to perturbative quantum field theory with Alain Connes and other co-authors. He is currently Humboldt professor at the department of mathematics of Humboldt University in Berlin, where he teaches the courses of Quantum Field Theory (I and II) and Hopf Algebras and the Renormalization Group.
References
External links
Living people
1960 births
21st-century German physicists
Boston University faculty
Academic staff of the Humboldt University of Berlin
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https://en.wikipedia.org/wiki/Patrick%20Hughes%20%28cricketer%29
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Lewis Patrick Hughes (10 April 1943 – 19 February 2022), also known as Podge Hughes, was an Irish cricketer and maths teacher at Mount Temple Comprehensive School. A right-handed batsman and right-arm fast-medium bowler, he played thirteen times for the Ireland cricket team between 1965 and 1978 including five first-class matches.
Playing career
Hughes made his debut for Ireland against Hampshire in September 1965 in a first-class match. It was very much a poor start to his international career, as he scored a duck in each innings and only took one wicket, and he did not play for Ireland again for almost three years. In June 1972 he played his final first-class match against Scotland and was again absent from the team for an extended period, this time for over five years, returning for a match against Sussex in July 1977. His last match for Ireland was against the MCC at Eglinton, County Londonderry in June 1978.
Statistics
In all matches for Ireland, he scored 159 runs at an average of 10.60. He took 15 wickets at an average of 50.87. In first-class cricket, he scored 55 runs at an average of 11.00 and took nine wickets at an average of 48.77.
Personal life and death
Hughes died on 19 February 2022, at the age of 78. His cousin Alan also represented Ireland at cricket. In February 2023, the hockey pitch in Mount Temple School was renamed in his memory, in partnership with Clontarf Hockey Club.
References
1943 births
2022 deaths
Cricketers from County Dublin
Irish cricketers
Mount Temple Comprehensive School
People from Blackrock, Dublin
Sportspeople from Dún Laoghaire–Rathdown
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https://en.wikipedia.org/wiki/Category%20of%20modules
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In algebra, given a ring R, the category of left modules over R is the category whose objects are all left modules over R and whose morphisms are all module homomorphisms between left R-modules. For example, when R is the ring of integers Z, it is the same thing as the category of abelian groups. The category of right modules is defined in a similar way.
One can also define the category of bimodules over a ring R but that category is equivalent to the category of left (or right) modules over the enveloping algebra of R (or over the opposite of that).
Note: Some authors use the term module category for the category of modules. This term can be ambiguous since it could also refer to a category with a monoidal-category action.
Properties
The categories of left and right modules are abelian categories. These categories have enough projectives and enough injectives. Mitchell's embedding theorem states every abelian category arises as a full subcategory of the category of modules of some ring.
Projective limits and inductive limits exist in the categories of left and right modules.
Over a commutative ring, together with the tensor product of modules ⊗, the category of modules is a symmetric monoidal category.
Objects
A monoid object of the category of modules over a commutative ring R is exactly an associative algebra over R.
See also: compact object (a compact object in the R-mod is exactly a finitely presented module).
Category of vector spaces
The category K-Vect (some authors use VectK) has all vector spaces over a field K as objects, and K-linear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use ModR), the category of left R-modules.
Much of linear algebra concerns the description of K-Vect. For example, the dimension theorem for vector spaces says that the isomorphism classes in K-Vect correspond exactly to the cardinal numbers, and that K-Vect is equivalent to the subcategory of K-Vect which has as its objects the vector spaces Kn, where n is any cardinal number.
Generalizations
The category of sheaves of modules over a ringed space also has enough injectives (though not always enough projectives).
See also
Algebraic K-theory (the important invariant of the category of modules.)
Category of rings
Derived category
Module spectrum
Category of graded vector spaces
Category of abelian groups
Category of representations
Change of rings
References
Bibliography
External links
http://ncatlab.org/nlab/show/Mod
Vector spaces
Linear algebra
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https://en.wikipedia.org/wiki/Germany%20Billie%20Jean%20King%20Cup%20team
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The Germany women's national tennis team represents Germany in Billie Jean King Cup tennis competition and are governed by Deutscher Tennis Bund.
Current team
Statistics correct as of 16 October 2023.
History
Germany competed in its first Fed Cup in 1963. They won the Cup in 1987 and 1992, and finished as runners-up five times.
Finals
Players
Player records
Results
1963–1969
1970–1979
1980–1989
1990–1999
2000–2009
2010–2019
2020–
See also
Billie Jean King Cup
Tennis in Germany
Germany Davis Cup team
Germany at the Hopman Cup
External links
Billie Jean King Cup teams
Fed Cup
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https://en.wikipedia.org/wiki/Scree%20plot
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In multivariate statistics, a scree plot is a line plot of the eigenvalues of factors or principal components in an analysis. The scree plot is used to determine the number of factors to retain in an exploratory factor analysis (FA) or principal components to keep in a principal component analysis (PCA). The procedure of finding statistically significant factors or components using a scree plot is also known as a scree test. Raymond B. Cattell introduced the scree plot in 1966.
A scree plot always displays the eigenvalues in a downward curve, ordering the eigenvalues from largest to smallest. According to the scree test, the "elbow" of the graph where the eigenvalues seem to level off is found and factors or components to the left of this point should be retained as significant.
Etymology
The scree plot is named after the elbow's resemblance to a scree in nature.
Criticism
This test is sometimes criticized for its subjectivity. Scree plots can have multiple "elbows" that make it difficult to know the correct number of factors or components to retain, making the test unreliable. There is also no standard for the scaling of the and axes, which means that different statistical programs can produce different plots from the same data.
The test has also been criticized for producing too few factors or components for factor retention.
As the "elbow" point has been defined as point of maximum curvature, as maximum curvature captures the leveling off effect operators use to identify knees, this has led to the creation of a Kneedle algorithm.
See also
Biplot
Parallel analysis
Elbow method
Determining the number of clusters in a data set
References
Statistical charts and diagrams
Factor analysis
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https://en.wikipedia.org/wiki/Lai%20Yee%20Hing
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Lai Yee Hing () is a Singaporean organic chemistry professor based in the National University of Singapore. He was the principal of NUS High School of Mathematics and Science and held this position from July 2004 to 30 August 2007.
Lai Yee Hing graduated with a B.Sc from the former Nanyang University (now National University of Singapore) in 1976 and received his PhD from the University of Victoria in 1980. Following that he spent 2 years in University of California Berkeley as a post-doctoral fellow, working with Peter Vollhardt. His research interests include novel aromatic compounds, conjugated organic materials and macrocycles.
External links
https://archive.today/20121225035151/http://www.highsch.nus.edu.sg/
https://web.archive.org/web/20070518050314/http://www.chemistry.nus.edu.sg/ourpeople/academic_staff/laiyh.htm
https://web.archive.org/web/20070705010029/http://staff.science.nus.edu.sg/~chmlaiyh/index.htm
Living people
Singaporean chemists
Academic staff of the National University of Singapore
Year of birth missing (living people)
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https://en.wikipedia.org/wiki/Brian%20Rotman
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Brian Rotman is a British-born professor who works in the United States. Trained as a mathematician and now an established philosopher, Rotman has blended semiotics, mathematics and the history of writing in his work and teaching throughout his career.
He is currently a distinguished humanities professor in the department of comparative studies at Ohio State University, has also taught at Stanford and given invited lectures at universities throughout the United States including Berkeley, MIT, Brown, Stanford, Duke, Notre Dame, Penn State, Minnesota, and Cornell. Rotman’s best known books include Signifying Nothing: The Semiotics of Zero which provides a wide-ranging exploration of the zero sign, Ad Infinitum... The Ghost in Turing’s Machine,
and Theory of Sets and Transfinite Numbers (written jointly with G. T. Kneebone)
Life
Rotman grew up above and inside his father’s sweet and tobacco shop in Brick Lane in the East End of London. He studied mathematics at the University of Nottingham, after which he taught the subject at a grammar school, a technical college and then for 20 years at Bristol University, along the way obtaining an M.Sc in the foundations of mathematics and a Ph.D in combinatorial mathematics. During this time he wrote, with G. T. Kneebone, a graduate textbook on set theory, The Theory of Sets and Transfinite Numbers, as well as numerous papers on ordered structures and Boolean algebras, and in 1977 published Jean Piaget: Psychologist of the Real an exposition and critique of the ideas behind the work of the Swiss child psychologist.
In 1979 he co-founded Mouth and Trousers, a London fringe theatre company based at the York and Albany pub in Camden Town, which operated for nearly four years during which time he wrote several stage plays. In 1984 he left Bristol and mathematics teaching and worked in London as a free-lance copy writer until the stock market crash of 1987 put an end to such work. In that year his essay Signifying Nothing: the Semiotics of Zero on the cultural significance of the mathematical zero sign was published. In 1990 he and his wife Lesley Ferris, an American theatre director and academic, and their two daughters, emigrated to the United States and lived in Memphis, Tennessee for 6 years. During this time, he gained expertise in the classroom training the young and spirited minds of calculus students at Memphis University School (a distinguished "school for boys") during the 1995-1996 school year.
In 1991 he published Ad Infinitum ... the Ghost in Turing’s Machine – a polemic against the ‘naturalness’ of the natural numbers. In the following years he received fellowships from Stanford Humanities Center, the National Endowment for the Humanities, and the American Council of Learned Societies. From 1996 to 1998 he was a professor of interdisciplinary studies in the English department at Louisiana State University in Baton Rouge, Louisiana and in 1998 moved to Columbus, Ohio to join the faculty of the
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https://en.wikipedia.org/wiki/Statistical%20model%20specification
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In statistics, model specification is part of the process of building a statistical model: specification consists of selecting an appropriate functional form for the model and choosing which variables to include. For example, given personal income together with years of schooling and on-the-job experience , we might specify a functional relationship as follows:
where is the unexplained error term that is supposed to comprise independent and identically distributed Gaussian variables.
The statistician Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".
Specification error and bias
Specification error occurs when the functional form or the choice of independent variables poorly represent relevant aspects of the true data-generating process. In particular, bias (the expected value of the difference of an estimated parameter and the true underlying value) occurs if an independent variable is correlated with the errors inherent in the underlying process. There are several different possible causes of specification error; some are listed below.
An inappropriate functional form could be employed.
A variable omitted from the model may have a relationship with both the dependent variable and one or more of the independent variables (causing omitted-variable bias).
An irrelevant variable may be included in the model (although this does not create bias, it involves overfitting and so can lead to poor predictive performance).
The dependent variable may be part of a system of simultaneous equations (giving simultaneity bias).
Additionally, measurement errors may affect the independent variables: while this is not a specification error, it can create statistical bias.
Note that all models will have some specification error. Indeed, in statistics there is a common aphorism that "all models are wrong". In the words of Burnham & Anderson, "Modeling is an art as well as a science and is directed toward finding a good approximating model ... as the basis for statistical inference".
Detection of misspecification
The Ramsey RESET test can help test for specification error in regression analysis.
In the example given above relating personal income to schooling and job experience, if the assumptions of the model are correct, then the least squares estimates of the parameters and will be efficient and unbiased. Hence specification diagnostics usually involve testing the first to fourth moment of the residuals.
Model building
Building a model involves finding a set of relationships to represent the process that is generating the data. This requires avoiding all the sources of misspecification mentioned above.
One approach is to start with a model in general form that relies on a theoretical understanding of the data-generating process. Then the model can be fit to the data and checked for the various sources of misspecification, in a task called stati
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https://en.wikipedia.org/wiki/Pitteway%20triangulation
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In computational geometry, a Pitteway triangulation is a point set triangulation in which the nearest neighbor of any point p within the triangulation is one of the vertices of the triangle containing p.
Alternatively, it is a Delaunay triangulation in which each internal edge crosses its dual Voronoi diagram edge. Pitteway triangulations are named after Michael Pitteway, who studied them in 1973. Not every point set supports a Pitteway triangulation. When such a triangulation exists it is a special case of the Delaunay triangulation, and consists of the union of the Gabriel graph and convex hull.
History
The concept of a Pitteway triangulation was introduced by . See also , who writes "An optimal partition
is one in which, for any point within any triangle, that point lies at least
as close to one of the vertices of that triangle as to any other data point." The name "Pitteway triangulation" was given by .
Counterexamples
points out that not every point set supports a Pitteway triangulation. For instance, any triangulation of a regular pentagon includes a central isosceles triangle such that a point p near the midpoint of one of the triangle sides has its nearest neighbor outside the triangle.
Relation to other geometric graphs
When a Pitteway triangulation exists, the midpoint of each edge interior to the triangulation must have the two edge endpoints as its nearest neighbors, for any other neighbor would violate the Pitteway property for nearby points in one of the two adjacent triangles. Thus, a circle having that edge as diameter must be empty of vertices, so the Pitteway triangulation consists of the Gabriel graph together with the convex hull of the point set. Conversely, when the Gabriel graph and convex hull together form a triangulation, it is a Pitteway triangulation.
Since all Gabriel graph and convex hull edges are part of the Delaunay triangulation, a Pitteway triangulation, when it exists, is unique for points in general position and coincides with the Delaunay triangulation. However point sets with no Pitteway triangulation will still have a Delaunay triangulation.
In the Pitteway triangulation, each edge pq either belongs to the convex hull or crosses the edge of the Voronoi diagram that separates the cells containing p and q. In some references this property is used to define a Pitteway triangulation, as a Delaunay triangulation in which all internal Delaunay edges cross their dual Voronoi edges. However, a Pitteway triangulation may include convex hull edges that do not cross their duals.
Notes
References
.
.
.
.
Triangulation (geometry)
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https://en.wikipedia.org/wiki/PMG%20%22Ekzarh%20Antim%20I%22
|
PPMG "Exarch Antim I" () is a public high school in Vidin, Bulgaria. The school specializes in maths and science. It is attended by students from the fifth to the twelfth grade.
Building
The school building was built in 1891 with donations provided by the Exarch Antim I.
See also
Education in Bulgaria
External links
http://pmg-vd.org/
Schools in Bulgaria
Buildings and structures in Vidin
1891 establishments in Bulgaria
Educational institutions established in 1891
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https://en.wikipedia.org/wiki/List%20of%20women%20in%20mathematics
|
This is a list of women who have made noteworthy contributions to or achievements in mathematics. These include mathematical research, mathematics education, the history and philosophy of mathematics, public outreach, and mathematics contests.
A
Karen Aardal (born 1961), Norwegian and Dutch applied mathematician, theoretical computer scientist, and operations researcher
Hanan Mohamed Abdelrahman, Egyptian and Norwegian mathematics educator
Izabela Abramowicz (1889–1973), Polish mathematician and mathematics educator
Louise Doris Adams (1889–1965), British mathematics reformer, president of the Mathematical Association
Rachel Blodgett Adams (1894–1982), American mathematician, one of the earliest mathematics doctorates from Radcliffe College
Tatyana Afanasyeva (1876–1964), Russian-Dutch researcher in statistical mechanics, randomness, and geometry education
Amandine Aftalion (born 1973), French applied mathematician, studies superfluids and the mathematics of footracing
Maria Gaetana Agnesi (1718–1799), Italian mathematician and philosopher, possibly the first female mathematics professor
Ilka Agricola (born 1973), German expert on differential geometry and its applications in mathematical physics
Nkechi Agwu (born 1962), African American ethnomathematician
Dorit Aharonov (born 1970), Israeli specialist in quantum computing
Beatrice Aitchison (1908–1997), American topologist who became a transportation economist in the US civil service
Noreen Sher Akbar, Pakistani fluid dynamicist
Shabnam Akhtari, Iranian number theorist
Asuman Aksoy, Turkish-American functional analyst
Meike Akveld, Swiss knot theorist and mathematics educator
Fatiha Alabau (born 1961), French expert in control of partial differential equations, president of French applied mathematics society
Mara Alagic, Serbian mathematics educator, editor-in-chief of Journal of Mathematics and the Arts
Lara Alcock, British mathematics educator and author
Helen Popova Alderson (1924–1972), Russian and British mathematician and translator, wrote on quasigroups and reciprocity laws
Grace Alele-Williams (1932–2022), first woman to lead a Nigerian university
Aldona Aleškevičienė-Statulevičienė (1936–2017), Lithuanian probability theorist
Stephanie B. Alexander, American differential geometer
Florence Eliza Allen (1876–1960), second female and fourth overall mathematics PhD from the University of Wisconsin
Linda J. S. Allen, American mathematician and mathematical biologist
Elizabeth S. Allman (born 1965), American mathematical biologist
Ann S. Almgren, American applied mathematician who works on computational simulations of supernovae and white dwarfs
Melania Alvarez, Mexican-Canadian mathematics educator, organizer of summer mathematics camps for indigenous students
Yvette Amice (1936–1993), French expert on p-adic analysis who became president of the French mathematical society
Divsha Amirà (1899–1966), Israeli geometer and mathematics educator
T. A. Sarasvati Amm
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https://en.wikipedia.org/wiki/Covariance%20and%20contravariance
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Covariance and contravariance may refer to:
Covariance and contravariance of vectors, in mathematics and theoretical physics
Covariance and contravariance of functors, in category theory
Covariance and contravariance (computer science), whether a type system preserves the ordering ≤ of types
See also
Covariance, in probability theory and statistics, the measure of how much two random variables vary together
Covariance (disambiguation)
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https://en.wikipedia.org/wiki/Quantum%20graph
|
In mathematics and physics, a quantum graph is a linear, network-shaped structure of vertices connected on edges (i.e., a graph) in which each edge is given a length and where a differential (or pseudo-differential) equation is posed on each edge. An example would be a power network consisting of power lines (edges) connected at transformer stations (vertices); the differential equations would then describe the voltage along each of the lines, with boundary conditions for each edge provided at the adjacent vertices ensuring that the current added over all edges adds to zero at each vertex.
Quantum graphs were first studied by Linus Pauling as models of free electrons in organic molecules in the 1930s. They also arise in a variety of mathematical contexts, e.g. as model systems in quantum chaos, in the study of waveguides, in photonic crystals and in Anderson localization, or as limit on shrinking thin wires. Quantum graphs have become prominent models in mesoscopic physics used to obtain a theoretical understanding of nanotechnology. Another, more simple notion of quantum graphs was introduced by Freedman et al.
Aside from actually solving the differential equations posed on a quantum graph for purposes of concrete applications, typical questions that arise are those of controllability (what inputs have to be provided to bring the system into a desired state, for example providing sufficient power to all houses on a power network) and identifiability (how and where one has to measure something to obtain a complete picture of the state of the system, for example measuring the pressure of a water pipe network to determine whether or not there is a leaking pipe).
Metric graphs
A metric graph
is a graph consisting of a set of vertices and
a set of edges where each edge has been associated
with an interval so that is the coordinate on the
interval, the vertex corresponds to and
to or vice versa. The choice of which vertex lies at zero is
arbitrary with the alternative corresponding to a change of coordinate on the
edge.
The graph has a natural metric: for two
points on the graph, is
the shortest distance between them
where distance is measured along the edges of the graph.
Open graphs: in the combinatorial graph model
edges always join pairs of vertices however in a quantum graph one may also
consider semi-infinite edges. These are edges associated with the interval
attached to a single vertex at .
A graph with one or more
such open edges is referred to as an open graph.
Quantum graphs
Quantum graphs are metric graphs equipped with a differential
(or pseudo-differential) operator acting on functions on the graph.
A function on a metric graph is defined as the -tuple of functions
on the intervals.
The Hilbert space of the graph is
where the inner product of two functions is
may be infinite in the case of an open edge. The simplest example of an operator on a metric graph is the Laplace operator. Th
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https://en.wikipedia.org/wiki/Stemm
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Stemm may refer to:
STEMM, American metal band
STEMM, abbreviation for Science, technology, engineering, mathematics, and medicine
Stemm, Indiana, a community in the US
See also
Stem (disambiguation)
Stemme, a German light aircraft manufacturer
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https://en.wikipedia.org/wiki/New%20York%20State%20Sportswriters%20Association
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The New York State Sportswriters Association (NYSSWA), founded in 1967, is a source of reference information and statistics about scholastic athletics in the state. Begun by sportswriters Larry Serrell of the Schenectady Daily Gazette and Chuck Korbar of the Buffalo Evening News, NYSSWA membership grew from 12 in 1967 to 246 in 1971 and over 500 annual subscribers by 1995, according to the organization.
The active readership includes newspapers, radio and television stations in the state, as well as high school coaches and administrators, college sports coaches and the parents of athletes. The organization's biggest undertaking is the eight-page newsletter that is published 50 times a year. The NYSSWA publishes weekly team rankings in all major sports and also selects all-state teams in soccer, basketball, football, baseball, and softball. The NYSSWA newsletter has been edited and distributed for over 40 years by Neil Kerr of the Syracuse Post-Standard.
See also
National Sports Media Association
New Jersey Sports Writers Association
Philadelphia Sports Writers Association
External links
Official website
Sports in New York (state)
High school sports in New York (state)
College sports in New York (state)
American sports journalism organizations
Journalism-related professional associations
Non-profit organizations based in New York (state)
Sports organizations established in 1967
1967 establishments in New York (state)
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https://en.wikipedia.org/wiki/Home%20runs%20per%20nine%20innings
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In baseball statistics, home runs allowed per 9 innings pitched (HR/9IP or HR/9) or home runs allowed per nine innings (denoted by HR/9) is the average number of home runs given up by a pitcher per nine innings pitched. It is determined by multiplying the number of home runs allowed by nine and dividing by the number of innings pitched. Pitchers with high fly ball rates are more likely than pitchers with high ground ball rates to have high HR/9 rates.
Leaders
The career leaders in HR/9IP through 2018 were Jim Devlin (0.0448), Al Spalding (0.0468), and Reb Russell (0.0488).
There were 87 single-season leaders in HR/9IP through 2018 who had pitched a season without giving up a home run. All played prior to 1927.
The active leaders in HR/9IP through 2018 were Clayton Kershaw (0.6225), Adam Wainwright (0.6755), and Charlie Morton (0.7682).
References
Pitching statistics
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https://en.wikipedia.org/wiki/Norman%20Zada
|
Norman Zada (born Norman Askar Zadeh) is a former adjunct mathematics professor and an entrepreneur. He is the founder of Perfect 10, an adult magazine focusing on women without cosmetic surgery, and runs the United States Investing Competition. Zada is the son of Lotfi Zadeh, the creator of fuzzy logic.
Education and early career
Zada obtained a PhD in Operations Research at the University of California, Berkeley and worked at IBM. He was an adjunct mathematics Professor at Stanford University, Columbia University, UCLA and University of California, Irvine, writing articles on applied mathematics as well as the 2020 book Hold'em Poker Super Strategy. and the 1974 book Winning Poker Systems.
After teaching, he won both backgammon and sports handicapping championships and later became a money manager. In the 1980s he ran a number of financial competitions, including the United States Investing Championship. Zada made headlines in 1996 when he offered $400,000 for anyone successfully refuting his claim that balancing the United States federal budget over a multi-year period without an accompanying substantial trade surplus would be effectively mathematically impossible.
Perfect 10 magazine
Zada launched Perfect 10 magazine, an adult magazine focusing on women without cosmetic surgery, after a friend was rejected from Playboy magazine because her proportions did not fit the magazine's tastes. He estimates losing approximately $46 million on Perfect 10 since 1996, when the magazine was first published. It has been claimed that these losses have been borne by Zada because of the deductions this allows against gains made in the money market.
His magazine was the plaintiff in Perfect 10 v. Google, Inc., a lawsuit charging contributory copyright infringement through the search engine displaying thumbnails of Perfect 10 images hosted at unauthorized third-party sites. Other lawsuits Zada filed involved adult verification system supplier Cybernet Ventures, from which he received a confidential settlement, and Visa and MasterCard, where he alleged that these credit card companies benefited from fees charged to access unauthorized material at third-party pay sites. The company also sued Giganews, Inc.
It has been claimed that Zada spends minimal time – 40 to 50 hours a year – creating content for the site, but "8 hours a day, 365 days a year" on litigation, leading some to call Perfect 10 little more than a copyright troll – by 2015, the company had filed 20 to 30 lawsuits.
Personal life
Zada is the son of Lotfi A. Zadeh, a computer scientist who coined the term fuzzy logic.
Zada at one time owned a large mansion in Beverly Park that he sold in 2010 for $16.5 million.
Zada apparently uses the name "Dr. Norman Zadeh" in connection with the United States Investing Championship. In a press release for the competition issued on December 28, 2018, Zada is referred to by that name.
References
External links
Perfect 10 website
Living people
Year of birt
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https://en.wikipedia.org/wiki/Lie%20algebra%20bundle
|
In mathematics, a weak Lie algebra bundle
is a vector bundle over a base space X together with a morphism
which induces a Lie algebra structure on each fibre .
A Lie algebra bundle is a vector bundle in which
each fibre is a Lie algebra and for every x in X, there is an open set containing x, a Lie algebra L and a homeomorphism
such that
is a Lie algebra isomorphism.
Any Lie algebra bundle is a weak Lie algebra bundle, but the converse need not be true in general.
As an example of a weak Lie algebra bundle that is not a strong Lie algebra bundle, consider the total space over the real line . Let [.,.] denote the Lie bracket of and deform it by the real parameter as:
for and .
Lie's third theorem states that every bundle of Lie algebras can locally be integrated to a bundle of Lie groups. In general globally the total space might fail to be Hausdorff. But if all fibres of a real Lie algebra bundle over a topological space are mutually isomorphic as Lie algebras, then it is a locally trivial Lie algebra bundle. This result was proved by proving that the real orbit of a real point under an algebraic group is open in the real part of its complex orbit. Suppose the base space is Hausdorff and fibers of total space are isomorphic as Lie algebras then there exists a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle. Every semi simple Lie algebra bundle is locally trivial. Hence there exist a Hausdorff Lie group bundle over the same base space whose Lie algebra bundle is isomorphic to the given Lie algebra bundle.
See also
Algebra bundle
Adjoint bundle
References
Algebraic topology
Complex analysis
Differential topology
Vector bundles
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https://en.wikipedia.org/wiki/Matrix%20determinant%20lemma
|
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT.
Statement
Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that
Here, uvT is the outer product of two vectors u and v.
The theorem can also be stated in terms of the adjugate matrix of A:
in which case it applies whether or not the square matrix A is invertible.
Proof
First the proof of the special case A = I follows from the equality:
The determinant of the left hand side is the product of the determinants of the three matrices. Since the first and third matrix are triangular matrices with unit diagonal, their determinants are just 1. The determinant of the middle matrix is our desired value. The determinant of the right hand side is simply (1 + vTu). So we have the result:
Then the general case can be found as:
Application
If the determinant and inverse of A are already known, the formula provides a numerically cheap way to compute the determinant of A corrected by the matrix uvT. The computation is relatively cheap because the determinant of A + uvT does not have to be computed from scratch (which in general is expensive). Using unit vectors for u and/or v, individual columns, rows or elements of A may be manipulated and a correspondingly updated determinant computed relatively cheaply in this way.
When the matrix determinant lemma is used in conjunction with the Sherman–Morrison formula, both the inverse and determinant may be conveniently updated together.
Generalization
Suppose A is an invertible n-by-n matrix and U, V are n-by-m matrices. Then
In the special case this is the Weinstein–Aronszajn identity.
Given additionally an invertible m-by-m matrix W, the relationship can also be expressed as
See also
The Sherman–Morrison formula, which shows how to update the inverse, A−1, to obtain (A + uvT)−1.
The Woodbury formula, which shows how to update the inverse, A−1, to obtain (A + UCVT)−1.
The binomial inverse theorem for (A + UCVT)−1.
References
Lemmas in linear algebra
Matrix theory
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https://en.wikipedia.org/wiki/Brian%20Hartley
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Brian Hartley (15 May 1939 – 8 October 1994) was a British mathematician specialising in group theory.
Education
Hartley's PhD thesis was completed in 1964 at the University of Cambridge under Philip Hall's supervision.
Career and research
Hartley spent a year at the University of Chicago, and another at MIT before being appointed as a lecturer at the newly established University of Warwick in 1966, and was promoted to reader in 1973. He moved to a chair at the School of Mathematics at the University of Manchester in 1977 where he served as head of the Mathematics department between 1982 and 1984.
He published more than 100 papers, mostly on group theory, and collaborated widely with other mathematicians. His main interest was locally finite groups where he used his wide knowledge of finite groups to prove properties of infinite groups which shared some of the features of finite groups. One recurrent theme appearing in his work was the relationship between the structure of groups and their subgroups consisting of elements fixed by particular automorphisms.
Hartley is perhaps best known by undergraduates for his book Rings, Modules and Linear Algebra, with Trevor Hawkes.
Personal life
Hartley was a keen hill walker, and it was while descending Helvellyn in the English Lake District that he collapsed with a heart attack and died.
Awards and honours
The 'Brian Hartley Room' at the School of Mathematics at Manchester is named in his honour.
References
1939 births
1994 deaths
20th-century British mathematicians
Group theorists
Academics of the Victoria University of Manchester
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https://en.wikipedia.org/wiki/Harish-Chandra%20character
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In mathematics, the Harish-Chandra character, named after Harish-Chandra, of a representation of a semisimple Lie group G on a Hilbert space H is a distribution on the group G that is analogous to the character of a finite-dimensional representation of a compact group.
Definition
Suppose that π is an irreducible unitary representation of G on a Hilbert space H.
If f is a compactly supported smooth function on the group G, then the operator on H
is of trace class, and the distribution
is called the character (or global character or Harish-Chandra character) of the representation.
The character Θπ is a distribution on G that is invariant under conjugation, and is an eigendistribution of the center of
the universal enveloping algebra of G, in other words an invariant eigendistribution, with eigenvalue the infinitesimal character of the representation π.
Harish-Chandra's regularity theorem states that any invariant eigendistribution, and in particular any character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function.
References
A. W. Knapp, Representation Theory of Semisimple Groups: An Overview Based on Examples.
Representation theory of Lie groups
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https://en.wikipedia.org/wiki/Harish-Chandra%27s%20regularity%20theorem
|
In mathematics, Harish-Chandra's regularity theorem, introduced by , states that every invariant eigendistribution on a semisimple Lie group, and in particular every character of an irreducible unitary representation on a Hilbert space, is given by a locally integrable function. proved a similar theorem for semisimple p-adic groups.
had previously shown that any invariant eigendistribution is analytic on the regular elements of the group, by showing that on these elements it is a solution of an elliptic differential equation. The problem is that it may have singularities on the singular elements of the group; the regularity theorem implies that these singularities are not too severe.
Statement
A distribution on a group G or its Lie algebra is called invariant if it is invariant under conjugation by G.
A distribution on a group G or its Lie algebra is called an eigendistribution if it is an eigenvector of the center of the universal enveloping algebra of G (identified with the left and right invariant differential operators of G).
Harish-Chandra's regularity theorem states that any invariant eigendistribution on a semisimple group or Lie algebra is a locally integrable function.
The condition that it is an eigendistribution can be relaxed slightly to the condition that its image under the center of the universal enveloping algebra is finite-dimensional. The regularity theorem also implies that on each Cartan subalgebra the distribution can be written as a finite sum of exponentials divided by a function Δ that closely resembles the denominator of the Weyl character formula.
Proof
Harish-Chandra's original proof of the regularity theorem is given in a sequence of five papers .
gave an exposition of the proof of Harish-Chandra's regularity theorem for the case of SL2(R), and sketched its generalization to higher rank groups.
Most proofs can be broken up into several steps as follows.
Step 1. If Θ is an invariant eigendistribution then it is analytic on the regular elements of G. This follows from elliptic regularity, by showing that the center of the universal enveloping algebra has an element that is "elliptic transverse to an orbit of G" for any regular orbit.
Step 2. If Θ is an invariant eigendistribution then its restriction to the regular elements of G is locally integrable on G. (This makes sense as the non-regular elements of G have measure zero.) This follows by showing that ΔΘ on each Cartan subalgebra is a finite sum of exponentials, where Δ is essentially the denominator of the Weyl denominator formula, with 1/Δ locally integrable.
Step 3. By steps 1 and 2, the invariant eigendistribution Θ is a sum S+F where F is a locally integrable function and S has support on the singular elements of G. The problem is to show that S vanishes. This is done by stratifying the set of singular elements of G as a union of locally closed submanifolds of G and using induction on the codimension of the strata. While it is possible for an eigenf
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https://en.wikipedia.org/wiki/Positively%20separated%20sets
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In mathematics, two non-empty subsets A and B of a given metric space (X, d) are said to be positively separated if the infimum
(Some authors also specify that A and B should be disjoint sets; however, this adds nothing to the definition, since if A and B have some common point p, then d(p, p) = 0, and so the infimum above is clearly 0 in that case.)
For example, on the real line with the usual distance, the open intervals (0, 2) and (3, 4) are positively separated, while (3, 4) and (4, 5) are not. In two dimensions, the graph of y = 1/x for x > 0 and the x-axis are not positively separated.
References
Metric geometry
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https://en.wikipedia.org/wiki/Metric%20outer%20measure
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In mathematics, a metric outer measure is an outer measure μ defined on the subsets of a given metric space (X, d) such that
for every pair of positively separated subsets A and B of X.
Construction of metric outer measures
Let τ : Σ → [0, +∞] be a set function defined on a class Σ of subsets of X containing the empty set ∅, such that τ(∅) = 0. One can show that the set function μ defined by
where
is not only an outer measure, but in fact a metric outer measure as well. (Some authors prefer to take a supremum over δ > 0 rather than a limit as δ → 0; the two give the same result, since μδ(E) increases as δ decreases.)
For the function τ one can use
where s is a positive constant; this τ is defined on the power set of all subsets of X. By Carathéodory's extension theorem, the outer measure can be promoted to a full measure; the associated measure μ is the s-dimensional Hausdorff measure. More generally, one could use any so-called dimension function.
This construction is very important in fractal geometry, since this is how the Hausdorff measure is obtained. The packing measure is superficially similar, but is obtained in a different manner, by packing balls inside a set, rather than covering the set.
Properties of metric outer measures
Let μ be a metric outer measure on a metric space (X, d).
For any sequence of subsets An, n ∈ N, of X with
and such that An and A \ An+1 are positively separated, it follows that
All the d-closed subsets E of X are μ-measurable in the sense that they satisfy the following version of Carathéodory's criterion: for all sets A and B with A ⊆ E and B ⊆ X \ E,
Consequently, all the Borel subsets of X — those obtainable as countable unions, intersections and set-theoretic differences of open/closed sets — are μ-measurable.
References
Measures (measure theory)
Metric geometry
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https://en.wikipedia.org/wiki/Jeff%20Paris%20%28mathematician%29
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Jeffrey Bruce Paris (; born 15 November 1944) is a British mathematician and Professor of Logic in the School of Mathematics at the University of Manchester.
Education
Paris gained his doctorate supervised by Robin Gandy at Manchester in 1969 with a dissertation on Large Cardinals and the Generalized Continuum Hypothesis.
Research and career
Paris is known for his work on mathematical logic, in particular provability in arithmetic, uncertain reasoning and inductive logic with an emphasis on rationality and common sense principles.
Awards and honours
Paris was awarded the Whitehead Prize in 1983 and elected a Fellow of the British Academy (FBA) in 1999.
Personal life
Paris was married to Malvyn Loraine Blackburn until 1983 when he married Alena Vencovská. He has three sons and three daughters including runner Jasmin Paris.
References
20th-century British mathematicians
21st-century British mathematicians
British logicians
Living people
Academics of the University of Manchester
Fellows of the British Academy
Whitehead Prize winners
1944 births
Alumni of the University of Manchester
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https://en.wikipedia.org/wiki/Problem%20of%20points
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The problem of points, also called the problem of division of the stakes, is a classical problem in probability theory. One of the famous problems that motivated the beginnings of modern probability theory in the 17th century, it led Blaise Pascal to the first explicit reasoning about what today is known as an expected value.
The problem concerns a game of chance with two players who have equal chances of winning each round. The players contribute equally to a prize pot, and agree in advance that the first player to have won a certain number of rounds will collect the entire prize. Now suppose that the game is interrupted by external circumstances before either player has achieved victory. How does one then divide the pot fairly? It is tacitly understood that the division should depend somehow on the number of rounds won by each player, such that a player who is close to winning will get a larger part of the pot. But the problem is not merely one of calculation; it also involves deciding what a "fair" division actually is.
Early solutions
Luca Pacioli considered such a problem in his 1494 textbook Summa de arithmetica, geometrica, proportioni et proportionalità. His method was to divide the stakes in proportion to the number of rounds won by each player, and the number of rounds needed to win did not enter his calculations at all.
In the mid-16th century Niccolò Tartaglia noticed that Pacioli's method leads to counterintuitive results if the game is interrupted when only one round has been played. In that case, Pacioli's rule would award the entire pot to the winner of that single round, though a one-round lead early in a long game is far from decisive. Tartaglia constructed a method that avoids that particular problem by basing the division on the ratio between the size of the lead and the length of the game. This solution is still not without problems, however; in a game to 100 it divides the stakes in the same way for a 65–55 lead as for a 99–89 lead, even though the former is still a relatively open game whereas in the latter situation victory for the leading player is almost certain. Tartaglia himself was unsure whether the problem was solvable at all in a way that would convince both players of its fairness: "in whatever way the division is made there will be cause for litigation".
Pascal and Fermat
The problem arose again around 1654 when Chevalier de Méré posed it to Blaise Pascal. Pascal discussed the problem in his ongoing correspondence with Pierre de Fermat. Through this discussion, Pascal and Fermat not only provided a convincing, self-consistent solution to this problem, but also developed concepts that are still fundamental to probability theory.
The starting insight for Pascal and Fermat was that the division should not depend so much on the history of the part of the interrupted game that actually took place, as on the possible ways the game might have continued, were it not interrupted. It is intuitively clear that a pl
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https://en.wikipedia.org/wiki/Quartile%20coefficient%20of%20dispersion
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In statistics, the quartile coefficient of dispersion is a descriptive statistic which measures dispersion and is used to make comparisons within and between data sets. Since it is based on quantile information, it is less sensitive to outliers than measures such as the coefficient of variation. As such, it is one of several robust measures of scale.
The statistic is easily computed using the first (Q1) and third (Q3) quartiles for each data set. The quartile coefficient of dispersion is:
Example
Consider the following two data sets:
A = {2, 4, 6, 8, 10, 12, 14}
n = 7, range = 12, mean = 8, median = 8, Q1 = 4, Q3 = 12, quartile coefficient of dispersion = 0.5
B = {1.8, 2, 2.1, 2.4, 2.6, 2.9, 3}
n = 7, range = 1.2, mean = 2.4, median = 2.4, Q1 = 2, Q3 = 2.9, quartile coefficient of dispersion = 0.18
The quartile coefficient of dispersion of data set A is 2.7 times as great (0.5 / 0.18) as that of data set B.
See also
Robust measures of scale
Coefficient of variation
Interquartile range
Median absolute deviation
References
Statistical deviation and dispersion
Statistical ratios
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https://en.wikipedia.org/wiki/Implicit%20curve
|
In mathematics, an implicit curve is a plane curve defined by an implicit equation relating two coordinate variables, commonly x and y. For example, the unit circle is defined by the implicit equation . In general, every implicit curve is defined by an equation of the form
for some function F of two variables. Hence an implicit curve can be considered as the set of zeros of a function of two variables. Implicit means that the equation is not expressed as a solution for either x in terms of y or vice versa.
If is a polynomial in two variables, the corresponding curve is called an algebraic curve, and specific methods are available for studying it.
Plane curves can be represented in Cartesian coordinates (x, y coordinates) by any of three methods, one of which is the implicit equation given above. The graph of a function is usually described by an equation in which the functional form is explicitly stated; this is called an explicit representation. The third essential description of a curve is the parametric one, where the x- and y-coordinates of curve points are represented by
two functions both of whose functional forms are explicitly stated, and which are dependent on a common parameter
Examples of implicit curves include:
a line:
a circle:
the semicubical parabola:
Cassini ovals (see diagram),
(see diagram).
The first four examples are algebraic curves, but the last one is not algebraic. The first three examples possess simple parametric representations, which is not true for the fourth and fifth examples. The fifth example shows the possibly complicated geometric structure of an implicit curve.
The implicit function theorem describes conditions under which an equation can be solved implicitly for x and/or y – that is, under which one can validly write or . This theorem is the key for the computation of essential geometric features of the curve: tangents, normals, and curvature. In practice implicit curves have an essential drawback: their visualization is difficult. But there are computer programs enabling one to display an implicit curve. Special properties of implicit curves make them essential tools in geometry and computer graphics.
An implicit curve with an equation can be considered as the level curve of level 0 of the surface (see third diagram).
Slope and curvature
In general, implicit curves fail the vertical line test (meaning that some values of x are associated with more than one value of y) and so are not necessarily graphs of functions. However, the implicit function theorem gives conditions under which an implicit curve locally is given by the graph of a function (so in particular it has no self-intersections). If the defining relations are sufficiently smooth then, in such regions, implicit curves have well defined slopes, tangent lines, normal vectors, and curvature.
There are several possible ways to compute these quantities for a given implicit curve. One method is to use implicit differentia
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https://en.wikipedia.org/wiki/Mims
|
Mims or MIMS may refer to:
Education
Manchester Institute for Mathematical Sciences, School of Mathematics, University of Manchester, England
Mandarin Immersion Magnet School, Houston, Texas, United States
Mandya Institute of Medical Sciences, Mandya, Karnataka, India
MediCiti Institute of Medical Sciences, near Hyderabad, Telengana, India
People
In politics
John Mims (1815–1856), mayor of Atlanta, Georgia, US
Livingston Mims (1833–1906), later mayor of Atlanta, Georgia
Sam Mims V (born 1972), of the Mississippi House of Representatives
Sam Mims Jr. (1880-1946), Mississippi state senator
William C. Mims (born 1957), Virginia judge, state senator and attorney general
Mims Davies (born 1975), British member of Parliament
In sport
David Mims (offensive tackle) (born 1988), American football player
David Mims (wide receiver) (born 1970), American football player
Denzel Mims (born 1997), American football wide receiver
Donna Mae Mims (1927–2009), American racecar driver
Jordan Mims (born 1999), American football player
Marvin Mims (born 2002), American football player
Ralph Mims (born 1985), American basketball player
Other people
D. Jeffrey Mims, American artist
Edwin Mims (1872–1959), American professor of English literature
Forrest Mims (born 1944), American amateur scientist and author
William Mims (1927–1991), American actor
Other uses
Mims, Florida, United States
Membrane-introduction mass spectrometry
Monthly Index of Medical Specialities, a guide to pharmaceuticals
Municipal Infrastructure Management System, used in Canada
See also
Mimms (disambiguation)
|
https://en.wikipedia.org/wiki/Sure
|
Sure may refer to:
Seemingly unrelated regressions
Series of Unsurprising Results in Economics (SURE), an economics academic journal
Sure, as probability, see certainty
Sure (brand), a brand of antiperspirant deodorant
Sure (company), a telecommunications company operating in British Crown Dependencies and Overseas Territories
Stein's unbiased risk estimate (SURE), in estimation theory
The river Sauer
In music
"Sure" (Every Little Thing song), from the album Eternity
"Sure" (Take That song), from the album Nobody Else
See also
Shure
|
https://en.wikipedia.org/wiki/Alar%20Toomre
|
Alar Toomre (born 5 February 1937, in Rakvere) is an American astronomer and mathematician. He is a professor of applied mathematics at the Massachusetts Institute of Technology. Toomre's research is focused on the dynamics of galaxies. He is a 1984 MacArthur Fellow.
Career
Following the Soviet occupation of Estonia in 1944, Toomre and his family fled to Germany; they emigrated to the United States in 1949. He received an undergraduate degree in Aeronautical Engineering and Physics from MIT in 1957 and then studied at the University of Manchester on a Marshall Scholarship where he obtained a Ph.D. in fluid mechanics.
Toomre returned to MIT to teach after completing his Ph.D. and remained there for two years. After spending a year at the Institute for Advanced Study, he returned again to MIT as part of the faculty, where he stayed. Toomre was appointed an Associate Professor of Mathematics at MIT in 1965, and Professor in 1970.
Scientific accomplishments
In 1964, Toomre devised a local gravitational stability criterion for differentially rotating disks. It is known as the Toomre stability criterion, which is usually measured by a parameter denoted as Q. The Q parameter measures the relative
importance of vorticity and internal velocity dispersion (large values of which stabilise) versus the disk surface density (large values of which destabilise). The parameter is constructed so that Q<1 implies instability.
Toomre collaborated with Peter Goldreich in 1969 on the subject of polar wander, developing the theory of polar wander. Whether true polar wander has been observed on earth, or apparent polar wander is accountable for all the observations of paleomagnetism remains a controversial issue.
Toomre conducted the first computer simulations of galaxy mergers in the 1970s with his brother Jüri, an astrophysicist and solar physicist. Although the small number of particles in the simulations obscured many processes in galactic collisions, Toomre and Toomre were able to identify tidal tails in his simulations, similar to those seen in the Antennae Galaxies and the Mice. The brothers attempted to reproduce specific galaxy mergers in their simulations, and it was their reproduction of the Antennae galaxies that gave them the greatest pleasure. In 1977 Toomre suggested that elliptical galaxies are the remnants of the major mergers of spiral galaxies. He further showed that based on the local galaxy merger rate, over a Hubble time the observed number of elliptical galaxies are produced if the universe begins with only spiral galaxies. This idea remained controversial and widely debated for some time.
From this work, the Toomre brothers identified the process of collision evolution as the Toomre sequence. The sequence begins with two well separated spiral galaxies and follows them (as for the Antennae) through collisional disruption until they settle into a single elliptical galaxy.
Awards and honors
In 1993, Toomre received the Dirk B
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https://en.wikipedia.org/wiki/Monotonically%20normal%20space
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In mathematics, specifically in the field of topology, a monotonically normal space is a particular kind of normal space, defined in terms of a monotone normality operator. It satisfies some interesting properties; for example metric spaces and linearly ordered spaces are monotonically normal, and every monotonically normal space is hereditarily normal.
Definition
A topological space is called monotonically normal if it satisfies any of the following equivalent definitions:
Definition 1
The space is T1 and there is a function that assigns to each ordered pair of disjoint closed sets in an open set such that:
(i) ;
(ii) whenever and .
Condition (i) says is a normal space, as witnessed by the function .
Condition (ii) says that varies in a monotone fashion, hence the terminology monotonically normal.
The operator is called a monotone normality operator.
One can always choose to satisfy the property
,
by replacing each by .
Definition 2
The space is T1 and there is a function that assigns to each ordered pair of separated sets in (that is, such that ) an open set satisfying the same conditions (i) and (ii) of Definition 1.
Definition 3
The space is T1 and there is a function that assigns to each pair with open in and an open set such that:
(i) ;
(ii) if , then or .
Such a function automatically satisfies
.
(Reason: Suppose . Since is T1, there is an open neighborhood of such that . By condition (ii), , that is, is a neighborhood of disjoint from . So .)
Definition 4
Let be a base for the topology of .
The space is T1 and there is a function that assigns to each pair with and an open set satisfying the same conditions (i) and (ii) of Definition 3.
Definition 5
The space is T1 and there is a function that assigns to each pair with open in and an open set such that:
(i) ;
(ii) if and are open and , then ;
(iii) if and are distinct points, then .
Such a function automatically satisfies all conditions of Definition 3.
Examples
Every metrizable space is monotonically normal.
Every linearly ordered topological space (LOTS) is monotonically normal. This is assuming the Axiom of Choice, as without it there are examples of LOTS that are not even normal.
The Sorgenfrey line is monotonically normal. This follows from Definition 4 by taking as a base for the topology all intervals of the form and for by letting . Alternatively, the Sorgenfrey line is monotonically normal because it can be embedded as a subspace of a LOTS, namely the double arrow space.
Any generalised metric is monotonically normal.
Properties
Monotone normality is a hereditary property: Every subspace of a monotonically normal space is monotonically normal.
Every monotonically normal space is completely normal Hausdorff (or T5).
Every monotonically normal space is hereditarily collectionwise normal.
The image of a monotonically normal space under a continuous closed map is monotonically normal.
A compact H
|
https://en.wikipedia.org/wiki/Leslie%20Fox
|
Leslie Fox (30 September 1918 – 1 August 1992) was a British mathematician noted for his contribution to numerical analysis.
Overview
Fox studied mathematics as a scholar of Christ Church, Oxford graduating with a first in 1939 and continued to undertake research in the engineering department. While working on his D.Phil. in computational and engineering mathematics under the supervision of Sir Richard Southwell he was also engaged in highly secret war work. He worked on the numerical solution of partial differential equations at a time when numerical linear algebra was performed on a desk calculator. Computational efficiency and accuracy was thus even more important than in the days of electronic computers. Some of this work was published after the end of the Second World War jointly with his supervisor Richard Southwell.
On gaining his doctorate in 1942, Fox joined the Admiralty Computing service. Following World War II in 1945, he went to work in the mathematics division of the National Physical Laboratory. He left the National Physical Laboratory in 1956 and spent a year at the University of California. In 1957 Fox took up an appointment at Oxford University where he set up the Oxford University Computing Laboratory. In 1963, Fox was appointed as Professor of Numerical Analysis at Oxford and Fellow of Balliol College, Oxford.
Fox's laboratory at Oxford was one of the founding organisations of the Numerical Algorithms Group (NAG), and Fox was also a dedicated supporter of the Institute of Mathematics and its Applications (IMA). The Leslie Fox Prize for Numerical Analysis of the IMA is named in his honour.
Mathematical work
A detailed description of Fox's mathematical research can be found in obituaries
and is summarised here. His early work with Southwell was concerned with the numerical solution of partial differential equations arising in engineering problems that, due to the complexity of their geometry, did not have analytical solutions. Southwell's group developed efficient and accurate relaxation methods, which could be implemented on desk calculators. Fox's contributions were particularly notable because he combined practical skills with theoretical advances in relaxation methods, which were to become important areas of research in numerical analysis. During the 1950 automatic electronic computers were replacing manual electro-mechanical devices. This led to different problems in the implementation of numerical algorithms; however, the approach of approximating a partial differential equation by finite difference method and thus reducing the problem to a system of linear equations was the same. Careful analysis of the errors was a theme of many of Fox's early papers. His work at the Admiralty Computing Service and the National Physical Laboratory led to an interest in the computation of special functions, and his calculations were used in published tables. The techniques applied to the computation of special functions had much wid
|
https://en.wikipedia.org/wiki/Meusnier%27s%20theorem
|
In differential geometry, Meusnier's theorem states that all curves on a surface passing through a given point p and having the same tangent line at p also have the same normal curvature at p and their osculating circles form a sphere. The theorem was first announced by Jean Baptiste Meusnier in 1776, but not published until 1785.
At least prior to 1912, several writers in English were in the habit of calling the result Meunier's theorem, although there is no evidence that Meusnier himself ever spelt his name in this way.
This alternative spelling of Meusnier's name also appears on the Arc de Triomphe in Paris.
References
Further references
Meusnier's theorem Johannes Kepler University Linz, Institute for Applied Geometry
Meusnier's theorem in Springer Online
Theorems in differential geometry
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https://en.wikipedia.org/wiki/Chinese%20hypothesis
|
In number theory, the Chinese hypothesis is a disproven conjecture stating that an integer n is prime if and only if it satisfies the condition that is divisible by n—in other words, that an integer n is prime if and only if . It is true that if n is prime, then (this is a special case of Fermat's little theorem), however the converse (if then n is prime) is false, and therefore the hypothesis as a whole is false. The smallest counterexample is n = 341 = 11×31. Composite numbers n for which is divisible by n are called Poulet numbers. They are a special class of Fermat pseudoprimes.
History
Once, and sometimes still, mistakenly thought to be of ancient Chinese origin, the Chinese hypothesis actually originates in the mid-19th century from the work of Qing dynasty mathematician Li Shanlan (1811–1882). He was later made aware his statement was incorrect and removed it from his subsequent work but it was not enough to prevent the false proposition from appearing elsewhere under his name; a later mistranslation in the 1898 work of Jeans dated the conjecture to Confucian times and gave birth to the ancient origin myth.
References
Bibliography
Pseudoprimes
Conjectures about prime numbers
Disproved conjectures
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https://en.wikipedia.org/wiki/Brocard%27s%20problem
|
Brocard's problem is a problem in mathematics that seeks integer values of such that is a perfect square, where is the factorial. Only three values of are known — 4, 5, 7 — and it is not known whether there are any more.
More formally, it seeks pairs of integers and such thatThe problem was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa Ramanujan.
Brown numbers
Pairs of the numbers that solve Brocard's problem were named Brown numbers by Clifford A. Pickover in his 1995 book Keys to Infinity, after learning of the problem from Kevin S. Brown. As of October 2022, there are only three known pairs of Brown numbers:
based on the equalities
Paul Erdős conjectured that no other solutions exist. Computational searches up to one quadrillion have found no further solutions.
Connection to the abc conjecture
It would follow from the abc conjecture that there are only finitely many Brown numbers.
More generally, it would also follow from the abc conjecture that
has only finitely many solutions, for any given integer , and that
has only finitely many integer solutions, for any given polynomial of degree at least 2 with integer coefficients.
References
Further reading
External links
Diophantine equations
Srinivasa Ramanujan
Unsolved problems in number theory
Factorial and binomial topics
|
https://en.wikipedia.org/wiki/Discrepancy
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Discrepancy may refer to:
Mathematics
Discrepancy of a sequence
Discrepancy theory in structural modelling
Discrepancy of hypergraphs, an area of discrepancy theory
Discrepancy (algebraic geometry)
Statistics
Discrepancy function in the context of structural equation models
Deviance (statistics)
Deviation (statistics)
Divergence (statistics)
See also
Deviance (disambiguation)
Deviation (disambiguation)
|
https://en.wikipedia.org/wiki/Vertex%20%28geometry%29
|
In geometry, a vertex (: vertices or vertexes) is a point where two or more curves, lines, or edges meet or intersect. As a consequence of this definition, the point where two lines meet to form an angle and the corners of polygons and polyhedra are vertices.
Definition
Of an angle
The vertex of an angle is the point where two rays begin or meet, where two line segments join or meet, where two lines intersect (cross), or any appropriate combination of rays, segments, and lines that result in two straight "sides" meeting at one place.
Of a polytope
A vertex is a corner point of a polygon, polyhedron, or other higher-dimensional polytope, formed by the intersection of edges, faces or facets of the object.
In a polygon, a vertex is called "convex" if the internal angle of the polygon (i.e., the angle formed by the two edges at the vertex with the polygon inside the angle) is less than π radians (180°, two right angles); otherwise, it is called "concave" or "reflex". More generally, a vertex of a polyhedron or polytope is convex, if the intersection of the polyhedron or polytope with a sufficiently small sphere centered at the vertex is convex, and is concave otherwise.
Polytope vertices are related to vertices of graphs, in that the 1-skeleton of a polytope is a graph, the vertices of which correspond to the vertices of the polytope, and in that a graph can be viewed as a 1-dimensional simplicial complex the vertices of which are the graph's vertices.
However, in graph theory, vertices may have fewer than two incident edges, which is usually not allowed for geometric vertices. There is also a connection between geometric vertices and the vertices of a curve, its points of extreme curvature: in some sense the vertices of a polygon are points of infinite curvature, and if a polygon is approximated by a smooth curve, there will be a point of extreme curvature near each polygon vertex. However, a smooth curve approximation to a polygon will also have additional vertices, at the points where its curvature is minimal.
Of a plane tiling
A vertex of a plane tiling or tessellation is a point where three or more tiles meet; generally, but not always, the tiles of a tessellation are polygons and the vertices of the tessellation are also vertices of its tiles. More generally, a tessellation can be viewed as a kind of topological cell complex, as can the faces of a polyhedron or polytope; the vertices of other kinds of complexes such as simplicial complexes are its zero-dimensional faces.
Principal vertex
A polygon vertex of a simple polygon is a principal polygon vertex if the diagonal intersects the boundary of only at and . There are two types of principal vertices: ears and mouths.
Ears
A principal vertex of a simple polygon is called an ear if the diagonal that bridges lies entirely in . (see also convex polygon) According to the two ears theorem, every simple polygon has at least two ears.
Mouths
A principal vertex of a simple polygon
|
https://en.wikipedia.org/wiki/Microphone%20Mathematics
|
"Microphone Mathematics" is the second single by Quasimoto, the rapping alter ego of Madlib. These tracks later appeared on his debut album The Unseen. On the album, however, "Discipline 99" was split into 2 tracks. Part 0 featured "Mr. Herb," while part 1 featured Wildchild of the Lootpack.
Track listing
Side A
Microphone Mathematics
Discipline #99 (feat. The Lootpack)
Low Class Conspiracy (feat. Madlib)
Side B
Microphone Mathematics (Instrumental)
Discipline #99 (feat. The Lootpack) (Instrumental)
Low Class Conspiracy (feat. Madlib) (Instrumental)
1999 singles
1999 songs
Madlib songs
Song recordings produced by Madlib
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https://en.wikipedia.org/wiki/Mathematics%20%28Cherry%20Ghost%20song%29
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"Mathematics" is the debut single from Manchester band Cherry Ghost. It was released as a digital download on March 26, 2007 and on CD and 7" vinyl on April 9, 2007. It went to #57 on the UK singles chart. "Mathematics" acquired the title "song of the week" on BBC Radio 2 in early 2007, and Zane Lowe of BBC Radio 1 declared the song "the hottest record in the world" in February 2007. Jimi Goodwin of Doves plays bass and drums on the single. The B-side "Junebug" is a Sparklehorse cover.
The song's inspiration likely stemmed from songwriter Simon Aldred's Bachelor's degree in Pure Mathematics from the University of Leeds.
Two music videos were made for the song. The first, a self-produced video featuring a man in a bird costume, was posted in late 2006. The second, featuring Simon Aldred's family home movies, appeared on Heavenly Records' website in early 2008.
Track listings
All songs written by Simon Aldred except where noted.
Promo CD (HVN167CDRP):
Released in March 2007
"Mathematics" (Edit) – 3:58
"Mathematics" (Album Version) – 4:34
CD (HVN167CD):
"Mathematics" – 4:34
"Throw Me to the Dogs" – 3:44
"I Need You" – 4:56
7" vinyl (HVN167):
"Mathematics" – 4:34
"Junebug" (Linkous) – 2:01
Digital download (UK iTunes only):
"Mathematics" – 4:34
"Throw Me to the Dogs" – 3:44
References
Heavenly Recordings singles
2007 songs
Songs written by Cherry Ghost
2007 singles
Cherry Ghost songs
|
https://en.wikipedia.org/wiki/Dimension%20function
|
In mathematics, the notion of an (exact) dimension function (also known as a gauge function) is a tool in the study of fractals and other subsets of metric spaces. Dimension functions are a generalisation of the simple "diameter to the dimension" power law used in the construction of s-dimensional Hausdorff measure.
Motivation: s-dimensional Hausdorff measure
Consider a metric space (X, d) and a subset E of X. Given a number s ≥ 0, the s-dimensional Hausdorff measure of E, denoted μs(E), is defined by
where
μδs(E) can be thought of as an approximation to the "true" s-dimensional area/volume of E given by calculating the minimal s-dimensional area/volume of a covering of E by sets of diameter at most δ.
As a function of increasing s, μs(E) is non-increasing. In fact, for all values of s, except possibly one, Hs(E) is either 0 or +∞; this exceptional value is called the Hausdorff dimension of E, here denoted dimH(E). Intuitively speaking, μs(E) = +∞ for s < dimH(E) for the same reason as the 1-dimensional linear length of a 2-dimensional disc in the Euclidean plane is +∞; likewise, μs(E) = 0 for s > dimH(E) for the same reason as the 3-dimensional volume of a disc in the Euclidean plane is zero.
The idea of a dimension function is to use different functions of diameter than just diam(C)s for some s, and to look for the same property of the Hausdorff measure being finite and non-zero.
Definition
Let (X, d) be a metric space and E ⊆ X. Let h : [0, +∞) → [0, +∞] be a function. Define μh(E) by
where
Then h is called an (exact) dimension function (or gauge function) for E if μh(E) is finite and strictly positive. There are many conventions as to the properties that h should have: Rogers (1998), for example, requires that h should be monotonically increasing for t ≥ 0, strictly positive for t > 0, and continuous on the right for all t ≥ 0.
Packing dimension
Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one "packs" E from inside with pairwise disjoint balls of diameter at most δ. Just as before, one can consider functions h : [0, +∞) → [0, +∞] more general than h(δ) = δs and call h an exact dimension function for E if the h-packing measure of E is finite and strictly positive.
Example
Almost surely, a sample path X of Brownian motion in the Euclidean plane has Hausdorff dimension equal to 2, but the 2-dimensional Hausdorff measure μ2(X) is zero. The exact dimension function h is given by the logarithmic correction
I.e., with probability one, 0 < μh(X) < +∞ for a Brownian path X in R2. For Brownian motion in Euclidean n-space Rn with n ≥ 3, the exact dimension function is
References
Dimension theory
Fractals
Metric geometry
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