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https://en.wikipedia.org/wiki/Hirsch%E2%80%93Plotkin%20radical | In mathematics, especially in the study of infinite groups, the Hirsch–Plotkin radical is a subgroup describing the normal locally nilpotent subgroups of the group. It was named by after Kurt Hirsch and Boris I. Plotkin, who proved that the join of normal locally nilpotent subgroups is locally nilpotent; this fact is the key ingredient in its construction.
The Hirsch–Plotkin radical is defined as the subgroup generated by the union of the normal locally nilpotent subgroups (that is, those normal subgroups such that every finitely generated subgroup is nilpotent). The Hirsch–Plotkin radical is itself a locally nilpotent normal subgroup, so is the unique largest such. In a finite group, the Hirsch–Plotkin radical coincides with the Fitting subgroup but for
infinite groups the two subgroups can differ. The subgroup generated by the union of infinitely many normal nilpotent subgroups need not itself be nilpotent, so the Fitting subgroup must be modified in this case.
References
Functional subgroups
Infinite group theory |
https://en.wikipedia.org/wiki/Jonas%20Kubilius | Jonas Kubilius (27 July 1921 – 30 October 2011) was a Lithuanian mathematician who worked in probability theory and number theory. He was rector of Vilnius University for 32 years, and served one term in the Lithuanian parliament.
Life and education
Kubilius was born in Fermos village, Eržvilkas county, Jurbarkas District Municipality, Lithuania on 27 July 1921. He graduated from Raseiniai high school in 1940 and entered Vilnius University, from which he graduated summa cum laude in 1946 after taking off a year to teach mathematics in middle school.
Kubilius received the Candidate of Sciences degree in 1951 from Leningrad University. His thesis, written under Yuri Linnik, was titled Geometry of Prime Numbers. He received the Doctor of Sciences degree (habilitation) in 1957 from the Steklov Institute of Mathematics in Moscow.
Career
Kubilius had simultaneous careers at Vilnius University and at the Lithuanian Academy of Sciences. He continued working at the university after receiving his bachelor's degree in 1946, and worked as a lecturer and assistant professor after receiving his Candidate degree in 1951. In 1958 he was promoted to professor and was elected rector of the university. He retired from the rector's position in 1991 after serving almost 33 years, and remained a professor in the university.
During the Khrushchev Thaw in the middle 1950s there were attempts to make the university "Lithuanian" by encouraging the use of the Lithuanian language in place of Russian and to revive the Department of Lithuanian Literature. This work was started by the rector Juozas Bulavas, but Stalinists objected and Bulavas was dismissed. Kubilius replaced him as rector and was more successful in resisting pressure to Russify the University: he returned Lithuanian language and culture to the forefront of the University. Česlovas Masaitis attributes Kubilius's success to "his ability to manipulate within the complex bureaucratic system of the Soviet Union and mainly because of his international recognition due to his scientific achievements." Kubilius also encouraged the faculty to write research papers in Lithuanian, English, German, and French, as well as in Russian, and he himself wrote several textbooks in Lithuanian. Jonas Kubilius known by pseudonym Bernotas was also involved in Lithuanian partisan movement. According to some sources Lithuanian partisans suggested him to continue studies and stay alive to work for Lithuania in the future.
In 1952 Kubilius became an employee of the Lithuanian Academy of Sciences in the Physics, Mathematics and Astronomy Sector. He initially promoted the development of probability theory in Lithuania, and later the development of differential equations and mathematical logic. In 1956 the Physical and Technical Institute was reorganized and Kubilius became head of the new Mathematical Sector. When he became rector of Vilnius University in 1958 he gave up his duties as head and was succeeded by Vytautas Statulevičius i |
https://en.wikipedia.org/wiki/Additive%20map | In algebra, an additive map, -linear map or additive function is a function that preserves the addition operation:
for every pair of elements and in the domain of For example, any linear map is additive. When the domain is the real numbers, this is Cauchy's functional equation. For a specific case of this definition, see additive polynomial.
More formally, an additive map is a -module homomorphism. Since an abelian group is a -module, it may be defined as a group homomorphism between abelian groups.
A map that is additive in each of two arguments separately is called a bi-additive map or a -bilinear map.
Examples
Typical examples include maps between rings, vector spaces, or modules that preserve the additive group. An additive map does not necessarily preserve any other structure of the object; for example, the product operation of a ring.
If and are additive maps, then the map (defined pointwise) is additive.
Properties
Definition of scalar multiplication by an integer
Suppose that is an additive group with identity element and that the inverse of is denoted by For any and integer let:
Thus and it can be shown that for all integers and all and
This definition of scalar multiplication makes the cyclic subgroup of into a left -module; if is commutative, then it also makes into a left -module.
Homogeneity over the integers
If is an additive map between additive groups then and for all (where negation denotes the additive inverse) and
Consequently, for all (where by definition, ).
In other words, every additive map is homogeneous over the integers. Consequently, every additive map between abelian groups is a homomorphism of -modules.
Homomorphism of -modules
If the additive abelian groups and are also a unital modules over the rationals (such as real or complex vector spaces) then an additive map satisfies:
In other words, every additive map is homogeneous over the rational numbers. Consequently, every additive maps between unital -modules is a homomorphism of -modules.
Despite being homogeneous over as described in the article on Cauchy's functional equation, even when it is nevertheless still possible for the additive function to be homogeneous over the real numbers; said differently, there exist additive maps that are of the form for some constant
In particular, there exist additive maps that are not linear maps.
See also
Notes
Proofs
References
Ring theory
Morphisms
Types of functions |
https://en.wikipedia.org/wiki/T-statistic | In statistics, the t-statistic is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in hypothesis testing via Student's t-test. The t-statistic is used in a t-test to determine whether to support or reject the null hypothesis. It is very similar to the z-score but with the difference that t-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the t-statistic is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown. It is also used along with p-value when running hypothesis tests where the p-value tells us what the odds are of the results to have happened.
Definition and features
Let be an estimator of parameter β in some statistical model. Then a t-statistic for this parameter is any quantity of the form
where β0 is a non-random, known constant, which may or may not match the actual unknown parameter value β, and is the standard error of the estimator for β.
By default, statistical packages report t-statistic with (these t-statistics are used to test the significance of corresponding regressor). However, when t-statistic is needed to test the hypothesis of the form , then a non-zero β0 may be used.
If is an ordinary least squares estimator in the classical linear regression model (that is, with normally distributed and homoscedastic error terms), and if the true value of the parameter β is equal to β0, then the sampling distribution of the t-statistic is the Student's t-distribution with degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept).
In the majority of models, the estimator is consistent for β and is distributed asymptotically normally. If the true value of the parameter β is equal to β0, and the quantity correctly estimates the asymptotic variance of this estimator, then the t-statistic will asymptotically have the standard normal distribution.
In some models the distribution of the t-statistic is different from the normal distribution, even asymptotically. For example, when a time series with a unit root is regressed in the augmented Dickey–Fuller test, the test t-statistic will asymptotically have one of the Dickey–Fuller distributions (depending on the test setting).
Use
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals.
The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be.
One can also divide a residual by the sample standard deviation:
to compute an estimate for the number of standard deviations a given sample is from the mean, as a sample version of |
https://en.wikipedia.org/wiki/Forest%20cover%20by%20state%20and%20territory%20in%20the%20United%20States | In the United States, the forest cover by state and territory is estimated from tree-attributes using the basic statistics reported by the Forest Inventory and Analysis (FIA) program of the Forest Service. Tree volumes and weights are not directly measured in the field, but computed from other variables that can be measured.
This is only the total amount of timberland. Actual forest cover for each state may be significantly higher.
List by state, district, or territory
List by region
See also
Forests of the United States
Forest cover by province or territory in Canada
Forest cover by federal subject in Russia
Forest cover by state or territory in Australia
Forest cover by state in India
References
Notes
Further reading
US Department of Agriculture. US Forest Resource Facts and HIstorical Trends. 2012. 64 p.
Forest
United States, Forest cover
Forestry-related lists |
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%281957%E2%80%931959%29 |
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1957
1958
1959
References
Main Page
List of Atlas launches
Atlas |
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%281990%E2%80%931999%29 | This is a list of Atlas rocket launches which took place during the period 1990-1999.
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
Photo gallery
References
Atlas |
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%281980%E2%80%931989%29 |
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
References
Atlas |
https://en.wikipedia.org/wiki/%C5%81ojasiewicz%20inequality | In real algebraic geometry, the Łojasiewicz inequality, named after Stanisław Łojasiewicz, gives an upper bound for the distance of a point to the nearest zero of a given real analytic function. Specifically, let ƒ : U → R be a real analytic function on an open set U in Rn, and let Z be the zero locus of ƒ. Assume that Z is not empty. Then for any compact set K in U, there exist positive constants α and C such that, for all x in K
Here α can be large.
The following form of this inequality is often seen in more analytic contexts: with the same assumptions on f, for every p ∈ U there is a possibly smaller open neighborhood W of p and constants θ ∈ (0,1) and c > 0 such that
A special case of the Łojasiewicz inequality, due to , is commonly used to prove linear convergence of gradient descent algorithms.
References
External links
Inequalities
Mathematical analysis
Real algebraic geometry |
https://en.wikipedia.org/wiki/Probability%20plot | Probability plot, a graphical technique for comparing two data sets, may refer to:
P–P plot, "Probability-Probability" or "Percent-Percent" plot
Q–Q plot, "Quantile-Quantile" plot
Normal probability plot, a Q–Q plot against the standard normal distribution
See also
Probability plot correlation coefficient
Probability plot correlation coefficient plot |
https://en.wikipedia.org/wiki/List%20of%20Atlas%20launches%20%281970%E2%80%931979%29 |
Launch statistics
Rocket configurations
Launch sites
Launch outcomes
1970
1971
1972
1973
1974
1975
1976
1977
1978
1979
Photo gallery
References
Atlas |
https://en.wikipedia.org/wiki/Gordon%20F.%20Newell | Gordon Frank Newell (January 25, 1925 – February 16, 2001) was an American scientist, known for his contributions to applied mathematics, in particular traffic flow analysis and queueing theory. Newell authored over one hundred articles and wrote several books. The Gordon–Newell theorem is named after him and his colleague William J. Gordon. Their algorithms helped form the basis of most modern automatically controlled and networked traffic-light control systems.
He obtained a B.Sc. from Union College, New York (1945) and
a Ph.D. in physics from University of Illinois (1950). He
continued his focus on solid-state physics and the Ising model of statistical mechanics with research teams under Elliott Montroll at University of Maryland, College Park (1950–53). His next job was at the applied mathematics faculty at Brown University (1953), where he began studies of automobile traffic analysis and road signalling theory. His final period was with the civil engineering faculty at University of California, Berkeley (1965–91), where he remained until retirement. He then held a professor emeritus of Transportation Engineering position. The annual Gordon Newell fellowship has been awarded since 2002.
He was born in Dayton, Ohio and raised in Rochester, New York. Newell died in an automobile accident in Carmel-by-the-Sea, California, after attending a party with friends.
Books and publications
Mathematical Models of Freely Flowing Traffic Flow. Operations Research, 3 (1955)
Statistical Analysis of the Flow of Highway Traffic through a Signalized Intersection, in Q. Appl. Math. 13, 1956
Maintaining a bus schedule, Proceedings of 2nd Australian Road Research Board, Part 1, pp. 388–393, 1964. With R. J. Potts
Cyclic Queuing Systems with Restricted Length Queues, Op.res., Vol. 15, No. 2, March–April 1967, pp. 266–277. With William J. Gordon
Applications of queueing theory (Chapman & Hall, 1971).
Scheduling, Location, Transportation and Continuum Mechanics; Some Simple Approximations to Optimization Problems, in SIAM J. Appl. Math. 25, 1973
Control of pairing of vehicles on a public transportation route, two vehicles, one control point, Transportation Science, Vol. 8, No. 3, pp. 248–264, 1974
Traffic on Transportation Networks (MIT Press, 1980)
Asymptotic Distribution of Eigenvalues for the Multidimensional Schroedinger Equation, in J. Math. Physics 21, 1980
Theory of Highway Traffic Signals (Institute of Transportation Studies, 1988)
Theory of Highway Traffic Flow, 1945–1965 (Inst. Trans. Studies, 1995)
References
20th-century American physicists
Queueing theorists
Union College (New York) alumni
University of Illinois alumni
Brown University faculty
University of California, Berkeley faculty
Scientists from Dayton, Ohio
1925 births
2001 deaths
Scientists from New York (state) |
https://en.wikipedia.org/wiki/Henk%20Tijms | Henk Tijms (Beverwijk, April 23, 1944) is a Dutch mathematician and Emeritus Professor of Operations Research at the VU University Amsterdam.
He studied mathematics in Amsterdam where he graduated from the University of Amsterdam in 1972 under supervision of Gijsbert de Leve.
Tijms is the author of several articles on applied mathematics and stochastics and books on probability. His best-known books are Stochastic Modeling and Analysis (Wiley, 1986) and Understanding Probability (Cambridge University Press, 2004).
On October 12, 2008, Tijms became the first non-American to receive the
INFORMS Expository Writing Award. The award honoured his achievements in the field of mathematics.
References
Living people
1944 births
Academic staff of the University of Amsterdam
Dutch mathematicians
People from Beverwijk
Dutch operations researchers |
https://en.wikipedia.org/wiki/Manari%2C%20Pernambuco | Manari is a city established in 1997 in the state of Pernambuco, Brazil. The population in 2020, according to the Brazilian Institute of Geography and Statistics, was 21,776 and the area is 344.73 km². In 2000, Manari had the lowest HDI of any municipality in the state.
Geography
State - Pernambuco
Region - Sertão Pernambucano
Boundaries - Ibimirim (N); Alagoas state (S); Itaíba (E); Inajá (W)
Area - 406.64 km²
Elevation - 570 m
Hydrography - Moxotó River and Ipanema River
Vegetation - Caatinga Hiperxerófila
Climate - semi-arid, hot and dry
Annual average temperature - 23.5 c
Distance to Recife - 376 km
Economy
The main economic activities in Manari are based in agribusiness, especially the raising of sheep, goats, cattle, and plantations of corn, beans and manioc.
Economic indicators
Economy by sector (2006)
Health indicators
References
External links
http://www.contasnacional.com.br/pe/pmmanari
http://www.ferias.tur.br/informacoes/5337/manari-pe.html
Municipalities in Pernambuco |
https://en.wikipedia.org/wiki/Subgroup%20method | The subgroup method is an algorithm used in the mathematical field of group theory. It is used to find the word of an element. It doesn't always return the minimal word, but it can return optimal words based on the series of subgroups that is used. The code looks like this:
function operate(element, generator)
<returns generator operated on element>
function subgroup(g)
sequence := (set of subgroups that will be used, depending on the method.)
word := []
for subgroup in sequence
coset_representatives := []
<fill coset_representatives with coset representatives of (next subgroup)/subgroup>
for operation in coset_representatives
if operate(g, operation) is in the next subgroup then
append operation onto word
g = operate(g, operation)
break
return word
Group theory |
https://en.wikipedia.org/wiki/Stationary%20increments | In probability theory, a stochastic process is said to have stationary increments if its change only depends on the time span of observation, but not on the time when the observation was started. Many large families of stochastic processes have stationary increments either by definition (e.g. Lévy processes) or by construction (e.g. random walks)
Definition
A stochastic process has stationary increments if for all and , the distribution of the random variables
depends only on and not on .
Examples
Having stationary increments is a defining property for many large families of stochastic processes such as the Lévy processes. Being special Lévy processes, both the Wiener process and the Poisson processes have stationary increments. Other families of stochastic processes such as random walks have stationary increments by construction.
An example of a stochastic process with stationary increments that is not a Lévy process is given by , where the are independent and identically distributed random variables following a normal distribution with mean zero and variance one. Then the increments are independent of as they have a normal distribution with mean zero and variance two. In this special case, the increments are even independent of the duration of observation itself.
Generalized Definition for Complex Index Sets
The concept of stationary increments can be generalized to stochastic processes with more complex index sets .
Let be a stochastic process whose index set is closed with respect to addition. Then it has stationary increments if for any , the random variables
and
have identical distributions.
If it is sufficient to consider .
References
Stochastic processes |
https://en.wikipedia.org/wiki/Kushner%20equation | In filtering theory the Kushner equation (after Harold Kushner) is an equation for the conditional probability density of the state of a stochastic non-linear dynamical system, given noisy measurements of the state. It therefore provides the solution of the nonlinear filtering problem in estimation theory. The equation is sometimes referred to as the Stratonovich–Kushner (or Kushner–Stratonovich) equation.
Overview
Assume the state of the system evolves according to
and a noisy measurement of the system state is available:
where w, v are independent Wiener processes. Then the conditional probability density p(x, t) of the state at time t is given by the Kushner equation:
where is the Kolmogorov Forward operator and is the variation of the conditional probability.
The term is the innovation i.e. the difference between the measurement and its expected value.
Kalman–Bucy filter
One can simply use the Kushner equation to derive the Kalman–Bucy filter for a linear diffusion process. Suppose we have and . The Kushner equation will be given by
where is the mean of the conditional probability at time . Multiplying by and integrating over it, we obtain the variation of the mean
Likewise, the variation of the variance is given by
The conditional probability is then given at every instant by a normal distribution .
See also
Zakai equation
References
Signal estimation
Nonlinear filters |
https://en.wikipedia.org/wiki/Fixed-radius%20near%20neighbors | In computational geometry, the fixed-radius near neighbor problem is a variant of the nearest neighbor search problem. In the fixed-radius near neighbor problem, one is given as input a set of points in d-dimensional Euclidean space and a fixed distance Δ. One must design a data structure that, given a query point q, efficiently reports the points of the data structure that are within distance Δ of q. The problem has long been studied; cites a 1966 paper by Levinthal that uses this technique as part of a system for visualizing molecular structures, and it has many other applications.
Solution by rounding and hashing
One method for solving the problem is to round the points to an integer lattice, scaled so that the distance between grid points is the desired distance Δ. A hash table can be used to find, for each input point, the other inputs that are mapped to nearby grid points, which can then be tested for whether their unrounded positions are actually within distance Δ. The number of pairs of points tested by this procedure, and the time for the procedure, is linear in the combined input and output size when the dimension is a fixed constant. However, the constant of proportionality in the linear time bound grows exponentially as a function of the dimension. Using this method, it is possible to construct indifference graphs and unit disk graphs from geometric data in linear time.
Other solutions
Modern parallel methods for GPU are able to efficiently compute all pairs fixed-radius NNS. For finite domains, the method of Green shows the problem can be solved by sorting on a uniform grid, finding all neighbors of all particles in O(kn) time, where k is proportional to the average number of neighbors. Hoetzlein improves this further on modern hardware with counting sorting and atomic operations.
Applications
The fixed-radius near neighbors problem arises in continuous Lagrangian simulations (such as smoothed particle hydrodynamics), computational geometry, and point cloud problems (surface reconstructions).
See also
Cell lists
References
Geometric algorithms |
https://en.wikipedia.org/wiki/Los%20Muermos | Los Muermos is a city and commune in Llanquihue Province, Los Lagos Region in southern Chile.
Demographics
According to the 2002 census of the National Statistics Institute, Los Muermos spans an area of and has 16,964 inhabitants (8,939 men and 8,025 women). Of these, 5,707 (33.6%) lived in urban areas and 11,257 (66.4%) in rural areas. The population fell by 0.5% (90 persons) between the 1992 and 2002 censuses.
Climate
The climate in this area has mild differences between highs and lows, and there is adequate rainfall year-round. According to the Köppen Climate Classification system, Los Muermos has a marine west coast climate, abbreviated "Cfb" on climate maps.
Administration
As a commune, Los Muermos is a third-level administrative division of Chile administered by a municipal council, headed by an alcalde who is directly elected every four years. The 2008-2012 alcalde is Emilio González Burgos (UDI).
Within the electoral divisions of Chile, Los Muermos is represented in the Chamber of Deputies by Fidel Espinoza (PS) and Carlos Recondo (UDI) as part of the 56th electoral district, together with Puyehue, Río Negro, Purranque, Puerto Octay, Fresia, Frutillar, Llanquihue and Puerto Varas. The commune is represented in the Senate by Camilo Escalona Medina (PS) and Carlos Kuschel Silva (RN) as part of the 17th senatorial constituency (Los Lagos Region).
References
External links
Municipality of Los Muermos
Communes of Chile
Populated places in Llanquihue Province |
https://en.wikipedia.org/wiki/Neppu%20Station | is a railway station in Kuromatsunai, Suttsu District, Hokkaidō, Japan.
Lines
Hokkaido Railway Company
Hakodate Main Line Station S29
Adjacent stations
Passenger statistics
In fiscal 1992, the station was used by an average of 44 passengers daily.
Surrounding area
National Route 5
Roadside station Kuromatsunai
Kuromatsunai Shindo Kuromatsunai I.C.
Mount Kuromatsunai
References
Railway stations in Japan opened in 1903
Railway stations in Hokkaido Prefecture |
https://en.wikipedia.org/wiki/List%20of%20journalists%20killed%20in%20the%20Philippines | This is a list of journalists killed in the Philippines, sorted by date of death.
Background
Statistics
Reporters Without Borders (RSF) had said that the Philippines is one of the world's deadliest country for journalists, adding that violence against them continued even with the establishment of the Presidential Task Force on Media Security (PTFoMS) in 2016. In its press freedom index for 2022, the country, out of 180 evaluated by RSF, ranks 147th. Prior to that, the 2009 Maguindanao massacre caused the country to be ranked 3rd in the Global Impunity Index by the Committee to Protect Journalists (CPJ) since then until 2014, the country's worst. In 2018, the country was given a special citation as one of those with an improved ranking. Likewise, the country was reported by the RSF as one of the five deadliest countries for journalists in the world from mid-2010s until being delisted in 2018. One of the causes is the PTFoMS' immediate action on various cases of killings and threats against the press.
Based on the data by the National Union of Journalists of the Philippines (NUJP), 197 media workers have been killed since 1986, with all deaths included were in relation to their job. The highest number was under the administration of President Gloria Macapagal Arroyo with 103; including 32 of those murdered in Maguindanao in what was called the world's worst single attack on journalists, which made the year 2009 the deadliest for them.
Meanwhile, other groups also document such. A data from the CPJ shows 156 killed since 1992; UNESCO reported 115 since 1996, with majority of them having publicized responses from Member States to Director General's request for information on judicial follow-up. Both includes the most recent, the death of Percy Lapid in October 2022.
RSF, on the other hand, documents those considered directly work-related a little less than that by NUJP.
Most deaths, according to NUJP and the Philippine Center for Investigative Journalism (PCIJ), were radio personalities, especially blocktime commentators, many affiliated to local politicians as suggested by a research from CPJ, and as reported by PCIJ, had minimal awareness of journalistic ethics or libel laws. Most incidents occurred in Mindanao, according to PTFoMS.
Various data show similarities seen in most of the killings. Incidents usually occurred in the provinces, wherein victims working there as journalists exposed wrongdoings in their locality and were critical about these issues. On the other hand, suspects, unknown and presumably hired killers, were hardly caught; very often they are motorcycle-riding assailants.
Both CMFR and the Philippine National Police reported in 2005 that of the journalists slain in the line or duty, seven were killed in crossfire: five during encounter with or being killed by the New People's Army (including two in an ambush in 1986); two during coup attempt especially in 1989 perpetrated by RAM–SFP–YOU.
The Center for Media Freedom and R |
https://en.wikipedia.org/wiki/Symmetric%20function | In mathematics, a function of variables is symmetric if its value is the same no matter the order of its arguments. For example, a function of two arguments is a symmetric function if and only if for all and such that and are in the domain of The most commonly encountered symmetric functions are polynomial functions, which are given by the symmetric polynomials.
A related notion is alternating polynomials, which change sign under an interchange of variables. Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
Symmetrization
Given any function in variables with values in an abelian group, a symmetric function can be constructed by summing values of over all permutations of the arguments. Similarly, an anti-symmetric function can be constructed by summing over even permutations and subtracting the sum over odd permutations. These operations are of course not invertible, and could well result in a function that is identically zero for nontrivial functions The only general case where can be recovered if both its symmetrization and antisymmetrization are known is when and the abelian group admits a division by 2 (inverse of doubling); then is equal to half the sum of its symmetrization and its antisymmetrization.
Examples
Consider the real function
By definition, a symmetric function with variables has the property that
In general, the function remains the same for every permutation of its variables. This means that, in this case,
and so on, for all permutations of
Consider the function
If and are interchanged the function becomes
which yields exactly the same results as the original
Consider now the function
If and are interchanged, the function becomes
This function is not the same as the original if which makes it non-symmetric.
Applications
U-statistics
In statistics, an -sample statistic (a function in variables) that is obtained by bootstrapping symmetrization of a -sample statistic, yielding a symmetric function in variables, is called a U-statistic. Examples include the sample mean and sample variance.
See also
References
F. N. David, M. G. Kendall & D. E. Barton (1966) Symmetric Function and Allied Tables, Cambridge University Press.
Joseph P. S. Kung, Gian-Carlo Rota, & Catherine H. Yan (2009) Combinatorics: The Rota Way, §5.1 Symmetric functions, pp 222–5, Cambridge University Press, .
Combinatorics
Properties of binary operations |
https://en.wikipedia.org/wiki/Relation%20%28mathematics%29 | In mathematics, a relation on a set may, or may not, hold between two given set members.
For example, "is less than" is a relation on the set of natural numbers; it holds e.g. between 1 and 3 (denoted as 1<3) , and likewise between 3 and 4 (denoted as 3<4), but neither between 3 and 1 nor between 4 and 4.
As another example, "is sister of" is a relation on the set of all people, it holds e.g. between Marie Curie and Bronisława Dłuska, and likewise vice versa.
Set members may not be in relation "to a certain degree" - either they are in relation or they are not.
Formally, a relation over a set can be seen as a set of ordered pairs of members of .
The relation holds between and if is a member of .
For example, the relation "is less than" on the natural numbers is an infinite set of pairs of natural numbers that contains both and , but neither nor .
The relation "is a nontrivial divisor of" on the set of one-digit natural numbers is sufficiently small to be shown here:
; for example 2 is a nontrivial divisor of 8, but not vice versa, hence , but .
If is a relation that holds for and one often writes . For most common relations in mathematics, special symbols are introduced, like "<" for "is less than", and "|" for "is a nontrivial divisor of", and, most popular "=" for "is equal to". For example, "1<3", "1 is less than 3", and "" mean all the same; some authors also write "".
Various properties of relations are investigated.
A relation is reflexive if holds for all , and irreflexive if holds for no .
It is symmetric if always implies , and asymmetric if implies that is impossible.
It is transitive if and always implies .
For example, "is less than" is irreflexive, asymmetric, and transitive, but neither reflexive nor symmetric.
"is sister of" is transitive, but neither reflexive (e.g. Pierre Curie is not a sister of himself), nor symmetric, nor asymmetric; while being irreflexive or not may be a matter of definition (is every woman a sister of herself?),
"is ancestor of" is transitive, while "is parent of" is not.
Mathematical theorems are known about combinations of relation properties, such as "A transitive relation is irreflexive if, and only if, it is asymmetric".
Of particular importance are relations that satisfy certain combinations of properties.
A partial order is a relation that is reflexive, antisymmetric, and transitive,
an equivalence relation is a relation that is reflexive, symmetric, and transitive,
a function is a relation that is right-unique and left-total (see below).
Since relations are sets, they can be manipulated using set operations, including union, intersection, and complementation, and satisfying the laws of an algebra of sets. Beyond that, operations like the converse of a relation and the composition of relations are available, satisfying the laws of a calculus of relations.
The above concept of relation has been generalized to admit relations between members of two different sets (heterogen |
https://en.wikipedia.org/wiki/Serena%20Williams%20career%20statistics | This is a list of the main career statistics of professional American tennis player Serena Williams.
Performance timelines
Singles
Current through the 2022 WTA Tour.
Doubles
Mixed doubles
Grand Slam finals
Singles: 33 (23 titles, 10 runner-ups)
Williams has won an Open Era record 23 Grand Slam singles titles. To win those titles, she has beaten 12 different players who have been ranked No. 1, including her sister, Venus Williams, seven times. She is also one of only two players in the Open Era to have won each major three or more times.
Women's doubles: 14 (14 titles)
Serena and Venus are the only doubles team to win their first 14 Grand Slam doubles finals and have never lost a Grand Slam doubles final.
Mixed doubles: 4 (2 titles, 2 runner-ups)
Other significant finals
Olympic finals
Singles: 1 (1 gold medal)
Doubles: 3 (3 gold medals)
Year-end championships finals
Singles: 7 (5 titles, 2 runner-ups)
Tier I / Premier Mandatory & Premier 5 finals
Singles: 33 finals (23 titles, 10 runner-ups)
Doubles: 2 finals (2 titles)
WTA career finals
Singles: 98 (73 titles, 25 runner-ups)
Doubles: 25 (23 titles, 2 runner-up)
Team competition: 4 (3 titles, 1 runner-up)
Fed Cup participation
Current through the 2020 Fed Cup qualifying round
Singles (14–1)
Doubles (3–2)
Head-to-head records
Record against top 10 players
Williams's record against players who have been ranked in the top 10. She has recorded wins against 22 of the other 26 women who have been ranked No. 1 in the Open Era, with the other 4 players retiring before her professional career began in 1995.
Active players are in boldface.
Record against No. 11–20 players
Williams's record against players who have been ranked world No. 11–20.
Anastasia Pavlyuchenkova 6–0
Zheng Jie 6–0
Tamarine Tanasugarn 6–0
Eleni Daniilidou 5–0
Nathalie Dechy 5–0
Kaia Kanepi 5–0
Shahar Pe'er 5–0
Elena Vesnina 5–0
Daria Gavrilova 4–0
María José Martínez Sánchez 4–0
Barbora Strýcová 4–0
Klára Koukalová 4–1
Elena Likhovtseva 4–1
Magüi Serna 4–1
Alizé Cornet 4–3
Dája Bedáňová 3–0
Alona Bondarenko 3–0
Amy Frazier 3–0
Varvara Lepchenko 3–0
Mirjana Lučić-Baroni 3–0
Peng Shuai 3–0
Sabine Lisicki 3–1
Katarina Srebotnik 3–1
Sabine Appelmans 2–0
Elena Bovina 2–0
Tatiana Golovin 2–0
Anabel Medina Garrigues 2–0
Tatiana Panova 2–0
Lisa Raymond 2–0
Aravane Rezaï 2–0
Magdaléna Rybáriková 2–0
Anna Smashnova 2–0
Karolina Šprem 2–0
Iroda Tulyaganova 2–0
Meghann Shaughnessy 2–1
Ekaterina Alexandrova 1–0
Ruxandra Dragomir 1–0
Kirsten Flipkens 1–0
Inés Gorrochategui 1–0
Anne Kremer 1–0
Magda Linette 1–0
Petra Martić 1–0
Elise Mertens 1–0
Larisa Neiland 1–0
Alison Riske-Amritraj 1–0
Naoko Sawamatsu 1–0
Alexandra Stevenson 1–0
Silvija Talaja 1–0
Silvia Farina Elia 1–1
Kimberly Po 1–1
Wang Qiang 1–1
Anastasija Sevastova 1–1
Virginie Razzano 0–1
Sybille Bammer 0–2
* Statistics correct .
Wins over t |
https://en.wikipedia.org/wiki/Pseudo-determinant | In linear algebra and statistics, the pseudo-determinant is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.
Definition
The pseudo-determinant of a square n-by-n matrix A may be defined as:
where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the rank of A.
Definition of pseudo-determinant using Vahlen matrix
The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. for ), is defined as . By the pseudo-determinant of the Vahlen matrix for the conformal transformation, we mean
If , the transformation is sense-preserving (rotation) whereas if the , the transformation is sense-preserving (reflection).
Computation for positive semi-definite case
If is positive semi-definite, then the singular values and eigenvalues of coincide. In this case, if the singular value decomposition (SVD) is available, then may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.
Supposing , so that k is the number of non-zero singular values, we may write where is some n-by-k matrix and the dagger is the conjugate transpose. The singular values of are the squares of the singular values of and thus we have , where is the usual determinant in k dimensions. Further, if is written as the block column , then it holds, for any heights of the blocks and , that .
Application in statistics
If a statistical procedure ordinarily compares distributions in terms of the determinants of variance-covariance matrices then, in the case of singular matrices, this comparison can be undertaken by using a combination of the ranks of the matrices and their pseudo-determinants, with the matrix of higher rank being counted as "largest" and the pseudo-determinants only being used if the ranks are equal. Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.
See also
Matrix determinant
Moore–Penrose pseudoinverse, which can also be obtained in terms of the non-zero singular values.
References
Covariance and correlation
Matrices |
https://en.wikipedia.org/wiki/Chi-Wang%20Shu | Chi-Wang Shu (Chinese: 舒其望, born 1 January 1957) is the Theodore B. Stowell University Professor of Applied Mathematics at Brown University. He is known for his research in the fields of computational fluid dynamics, numerical solutions of conservation laws and Hamilton–Jacobi type equations. Shu has been listed as an ISI Highly Cited Author in Mathematics by the ISI Web of Knowledge.
Career
He received his B.S. in Mathematics from the University of Science and Technology of China, Hefei, in 1982 and his Ph.D. in Mathematics from the University of California at Los Angeles in 1986. His Ph.D. thesis advisor was Stanley Osher.
He started his academic career in 1987 as an assistant professor in the Division of Applied Mathematics at Brown University. He was an associate professor from 1992 to 1996 and became full professor in 1996.
Honors and awards
He is the 2021 recipient of the John von Neumann Lecture Prize, the highest honor and flagship lecture of Society for Industrial and Applied Mathematics (SIAM). The prize recognizes his fundamental contributions to the numerical solution of partial differential equations: "His work on finite difference essentially non-oscillatory (ENO) methods, weighted ENO (WENO) methods, finite element discontinuous Galerkin methods, and spectral methods has had a major impact on scientific computing."
The Association for Women in Mathematics has included him in the 2020 class of AWM Fellows for "his exceptional dedication and contribution to mentoring, supporting, and advancing women in the mathematical sciences; for his incredible role in supervising many women Ph.D.s, bringing them into the world of research to which he has made fundamental contributions, and nurturing their professional success".
In 2012 he became a fellow of the American Mathematical Society.
In 2009, he was selected as one of the first 183 Fellows of the Society for Industrial and Applied Mathematics (SIAM).
SIAM/ACM Prize in Computational Science and Engineering (SIAM/ACM CSE Prize), 2007. He received the prize "for the development of numerical methods that have had a great impact on scientific computing, including TVD temporal discretizations, ENO and WENO finite difference schemes, discontinuous Galerkin methods, and spectral methods."
Feng Kang Prize of Scientific Computing by the Chinese Academy of Sciences, 1995
NASA Public Service Group Achievement Award for Pioneering Work in Computational Fluid Dynamics by NASA Langley Research Center, 1992
References
External links
Brown University faculty
American mathematicians
University of California, Los Angeles alumni
Computational fluid dynamicists
1957 births
Living people
Fellows of the American Mathematical Society
Fellows of the Society for Industrial and Applied Mathematics
Fellows of the Association for Women in Mathematics
University of Science and Technology of China alumni |
https://en.wikipedia.org/wiki/Kim%20Hong-il%20%28footballer%29 | Kim Hong-il (Hangul: 김홍일; Hanja: 金弘一; born 29 September 1987) is a South Korean footballer who currently plays for Suwon FC in K League Challenge.
Career statistics
External links
1987 births
Living people
Men's association football midfielders
South Korean men's footballers
Suwon Samsung Bluewings players
Gwangju FC players
Gimcheon Sangmu FC players
Suwon FC players
K League 1 players
K League 2 players |
https://en.wikipedia.org/wiki/Goldfeld%E2%80%93Quandt%20test | In statistics, the Goldfeld–Quandt test checks for homoscedasticity in regression analyses. It does this by dividing a dataset into two parts or groups, and hence the test is sometimes called a two-group test. The Goldfeld–Quandt test is one of two tests proposed in a 1965 paper by Stephen Goldfeld and Richard Quandt. Both a parametric and nonparametric test are described in the paper, but the term "Goldfeld–Quandt test" is usually associated only with the former.
Test
In the context of multiple regression (or univariate regression), the hypothesis to be tested is that the variances of the errors of the regression model are not constant, but instead are monotonically related to a pre-identified explanatory variable. For example, data on income and consumption may be gathered and consumption regressed against income. If the variance increases as levels of income increase, then income may be used as an explanatory variable. Otherwise some third variable (e.g. wealth or last period income) may be chosen.
Parametric test
The parametric test is accomplished by undertaking separate least squares analyses on two subsets of the original dataset: these subsets are specified so that the observations for which the pre-identified explanatory variable takes the lowest values are in one subset, with higher values in the other. The subsets needs not be of equal size, nor contain all the observations between them. The parametric test assumes that the errors have a normal distribution. There is an additional assumption here, that the design matrices for the two subsets of data are both of full rank. The test statistic used is the ratio of the mean square residual errors for the regressions on the two subsets. This test statistic corresponds to an F-test of equality of variances, and a one- or two-sided test may be appropriate depending on whether or not the direction of the supposed relation of the error variance to the explanatory variable is known.
Increasing the number of observations dropped in the "middle" of the ordering will increase the power of the test but reduce the degrees of freedom for the test statistic. As a result of this tradeoff it is common to see the Goldfeld–Quandt test performed by dropping the middle third of observations with smaller proportions of dropped observations as sample size increases.
Nonparametric test
The second test proposed in the paper is a nonparametric one and hence does not rely on the assumption that the errors have a normal distribution. For this test, a single regression model is fitted to the complete dataset. The squares of the residuals are listed according to the order of the pre-identified explanatory variable. The test statistic used to test for homogeneity is the number of peaks in this list: ie. the count of the number of cases in which a squared residual is larger than all previous squared residuals. Critical values for this test statistic are constructed by an argument related to permutation te |
https://en.wikipedia.org/wiki/Schur%27s%20lemma%20%28disambiguation%29 | At least three well-known results in mathematics bear the name Schur's lemma:
Schur's lemma from representation theory
Schur's lemma from Riemannian geometry
Schur's lemma in linear algebra says that every square complex matrix is unitarily triangularizable, see Schur decomposition
Schur test for boundedness of integral operators
See also
Schur's theorem
Schur's property |
https://en.wikipedia.org/wiki/Ky%20Fan | Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara.
Biography
Fan was born in Hangzhou, the capital of Zhejiang Province, China. His father, named Fan Qi (樊琦, 1879—1947), served in the district courts of Jinhua and Wenzhou. Ky Fan went to Jinhua with his father when he was eight years old and studied at several middle schools in Zhejiang, including the Jinhua High School (currently Jinhua No.1 Middle School), Hangzhou Zongwen High School (currently Hangzhou No.10 Middle School), and Wenzhou High School. Fan obtained his secondary diploma from the Jinhua High School.
Fan enrolled into Peking University Department of Mathematics in 1932, and received his B.S. degree from Peking University in 1936. Initially Fan wanted to study engineering, but eventually shifted to mathematics, largely because of the influence of his uncle Feng Zuxun (冯祖荀, 1880–1940; b. Hangzhou, d. Beijing), who was a mathematician in China and the then Chair of the Department of Mathematics of Peking University. After graduation, Fan became a teaching assistant in the department.
Fan went to France in 1939 and received his D.Sc. degree from the University of Paris in 1941. Fan's doctoral advisor was M.R.Fréchet. Fan was a research fellow at the French National Centre for Scientific Research (CNRS). During his secondary school and college time, Fan said he "hated English". That was an important reason for him to choose mathematics, with less English but full of equations, and go to Paris.
Fan was a member of the Institute for Advanced Study in Princeton, New Jersey from 1945 to 1947. In 1947, Fan joined the mathematical faculty of the University of Notre Dame, where he was an assistant professor at the beginning, and later promoted to associate professor and full professor. In 1960, Fan also held a position at Wayne State University in Detroit for about one year, but immediately went to Northwestern University near Chicago. In 1965, Fan became a professor of mathematics at UCSB. Fan was known for being an extremely strict professor.
Fan was elected an Academician of the Academia Sinica (Taipei, Taiwan) in 1964. He served as the director of the Institute of Mathematics there from 1978 to 1984.
In 1999, Fan and his wife Yan Youfen (燕又芬) donated one million US dollars to the American Mathematical Society, to set up the Ky and Yu-Fen Fan Endowment.
Fan had 23 graduate students. He died in Santa Barbara in March 2010.
His given name 𰋀 () is a rare variant of the character 畿.
Academic career
Fan was a student and collaborator of M. Fréchet and was also influenced by John von Neumann and Hermann Weyl. The author of approximately 130 papers, Fan made fundamental contributions to operator and matrix theory, convex analysis and inequalities, linear and nonlinear programming, topology and fixed point theory, and topological groups. His work in fixed point theo |
https://en.wikipedia.org/wiki/Pebble%20game | In mathematics and computer science, a pebble game is a type of mathematical game played by placing "pebbles" or "markers" on a directed acyclic graph according to certain rules:
A given step of the game consists of either placing a pebble on an empty vertex or removing a pebble from a previously pebbled vertex.
A vertex may be pebbled only if all its predecessors have pebbles.
The objective of the game is to successively pebble each vertex of G (in any order) while minimizing the number of pebbles that are ever on the graph simultaneously.
Running time
The trivial solution is to pebble an n-vertex graph in n steps using n pebbles. Hopcroft, Paul and Valiant showed that any vertex of an n-vertex graph can be pebbled with O(n/log n) pebbles where the constant depends on the maximum in-degree. This enabled them to prove that DTIME(f(n)) is contained in DSPACE(f(n)/log f(n)) for all time-constructible f. Lipton and Tarjan showed that any n-vertex planar acyclic directed graph with maximum in-degree k can be pebbled using O( + k log2 n) pebbles. They also proved that it is possible to obtain a substantial reduction in pebbles while preserving a polynomial bound on the number of pebbling steps with a theorem that any n-vertex planar acyclic directed graph with maximum in-degree k can be pebbled using O(n2/3 + k) pebbles in O(n5/3) time. Alon, Seymour and Thomas showed that any n-vertex acyclic directed graph with no kh-minor and with maximum in-degree k can be pebbled using O(h3/2 n1/2 + k log n) pebbles.
Variations
An extension of this game, known as "black-white pebbling", was developed by Stephen Cook and Ravi Sethi in a 1976 paper. It also adds white pebbles, which may be placed at any vertex at will, but can only be removed if all the vertex's immediate ancestor vertices are also pebbled. The goal remains to place a black pebble on the target vertex, but the pebbling of adjacent vertices may be done with pebbles of either color.
Takumi Kasai et al. developed a game in which a pebble may be moved along an edge-arrow to an unoccupied vertex only if a second pebble is located at a third, control vertex; the goal is to move a pebble to a target vertex. This variation makes the pebble game into a generalization of games such as Chinese checkers and Halma. They determined the computational complexity of the one-player and two-player versions of this game, and special cases thereof. In the two-player version, the players take turns moving pebbles. There may also be constraints on which pebbles a player can move.
Pebbling may be used to extend Ehrenfeucht–Fraïssé games.
See also
Graph pebbling: A certain number of pebbles are distributed among the vertices of an undirected graph; the goal is to move at least one to a particular target vertex. But to move one pebble to an adjacent vertex, another pebble at the same vertex must be discarded.
Chip-firing game
Planar separator theorem
References
Further reading
Nicholas Pippenger. Pebblin |
https://en.wikipedia.org/wiki/Churchill%20Professor%20of%20Mathematics%20for%20Operational%20Research | The Churchill Professorship of Mathematics for Operational Research is a professorship in operational research at the University of Cambridge. It was established in 1966 by a benefaction from Esso in memory of Sir Winston Churchill, who died the previous year. This was the second professorship established within the Cambridge Statistical Laboratory (the first being the Professorship of Mathematical Statistics).
List of Churchill Professors
1967–1994 Peter Whittle
1994–2017 Richard Weber
2020– Ioannis Kontoyiannis
References
Mathematics for Operational Research, Churchill
Faculty of Mathematics, University of Cambridge
1966 establishments in the United Kingdom
Mathematics for Operational Research, Churchill
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/I-bundle | In mathematics, an I-bundle is a fiber bundle whose fiber is an interval and whose base is a manifold. Any kind of interval, open, closed, semi-open, semi-closed, open-bounded, compact, even rays, can be the fiber. An I-bundle is said to be twisted if it is not trivial.
Two simple examples of I-bundles are the annulus and the Möbius band, the only two possible I-bundles over the circle . The annulus is a trivial or untwisted bundle because it corresponds to the Cartesian product , and the Möbius band is a non-trivial or twisted bundle. Both bundles are 2-manifolds, but the annulus is an orientable manifold while the Möbius band is a non-orientable manifold.
Curiously, there are only two kinds of I-bundles when the base manifold is any surface but the Klein bottle . That surface has three I-bundles: the trivial bundle and two twisted bundles.
Together with the Seifert fiber spaces, I-bundles are fundamental elementary building blocks for the description of three-dimensional spaces. These observations are simple well known facts on elementary 3-manifolds.
Line bundles are both I-bundles and vector bundles of rank one. When considering I-bundles, one is interested mostly in their topological properties and not their possible vector properties, as one might be for line bundles.
References
Hempel, John, "3-manifolds", Annals of Mathematics Studies, number 86, Princeton University Press (1976).
External links
Example of use of I-bundles, nice pdf-slide presentation by Jeff Boerner at Dept. of Math, University of Iowa.
Fiber bundles
Geometric topology
3-manifolds |
https://en.wikipedia.org/wiki/Tiny%20and%20miny | In mathematics, tiny and miny are operators that yield infinitesimal values when applied to numbers in combinatorial game theory. Given a positive number G, tiny G (denoted by ⧾G in many texts) is equal to {0|{0|-G}} for any game G, whereas miny G (analogously denoted ⧿G) is tiny G's negative, or {{G|0}|0}.
Tiny and miny aren't just abstract mathematical operators on combinatorial games: tiny and miny games do occur "naturally" in such games as toppling dominoes. Specifically, tiny n, where n is a natural number, can be generated by placing two black dominoes outside n + 2 white dominoes.
Tiny games and up have certain curious relational characteristics. Specifically, though ⧾G is infinitesimal with respect to ↑ for all positive values of x, ⧾⧾⧾G is equal to up. Expansion of ⧾⧾⧾G into its canonical form yields {0|{{0|{{0|{0|-G}}|0}}|0}}. While the expression appears daunting, some careful and persistent expansion of the game tree of ⧾⧾⧾G + ↓ will show that it is a second player win, and that, consequently, ⧾⧾⧾G = ↑. Similarly curious, mathematician John Horton Conway noted, calling it "amusing," that "↑ is the unique solution of ⧾G = G." Conway's assertion is also easily verifiable with canonical forms and game trees.
References
Combinatorial game theory |
https://en.wikipedia.org/wiki/MacTutor | MacTutor may refer to:
The MacTutor History of Mathematics archive, a history of mathematics archive
MacTutor (magazine), a magazine on developing software for the Apple Macintosh computer |
https://en.wikipedia.org/wiki/National%20Gymnasium%20of%20Natural%20Sciences%20and%20Mathematics%20%22Academician%20Lyubomir%20Chakalov%22 | The National High School of Mathematics and Natural Sciences "Academician Lyubomir Chakalov" (in Bulgarian: Национална природо-математическа гимназия "Академик Любомир Чакалов") is a high school (European secondary school) in Sofia, Bulgaria. It is located in Lozenets municipality. The school is named after the Bulgarian mathematician Lyubomir Chakalov. More than 1000 students are studying in the school. They are divided into seven majors:
Mathematics and Computer science (52 students, 2 classes with German/English language studying)
Physics (26 students, 1 class with English)
Chemistry (52 students, 2 classes with German/English language)
Chemistry and Biology (26 students, 1 class with English)
Biology (26 students, 1 class with English)
Earth Science (26 students, 1 class with English)
Computer science (26 students, 1 class with English)
НПМГ Alumni Club
НПМГ Alumni Club was established in 2017. Its mission is to create long-term connections between alumni of the gymnasium, as well as enrich the learning process of current students through lectures, workshops, and other academic and social events.
External links
Official site
НПМГ Alumni Club Facebook page
Schools in Sofia
Educational institutions established in 1968 |
https://en.wikipedia.org/wiki/SmartGeometry%20Group | SmartGeometry (SG) is a non-profit organization focusing on the use of the computer as an intelligent design aid in architecture, engineering and construction (AEC). It encourages collaboration between practicing AEC professionals, academics and students using computational and parametric software tools.
Group information and activities
The group is led by Lars Hesselgren of KPF, Hugh Whitehead of Foster + Partners and J. Parrish of Arup Sport.
SG hosts annual workshops and conferences on the use of advanced modeling tools and new design methodologies in architecture. Participants come from architectural and engineering practices.
GenerativeComponents is a commercial software product by Bentley Systems, brought to the market after a testing cycle by a user community with SG members in its core.
See also
Architecture
Architectural engineering
Design computing
Comparison of CAD Software
GenerativeComponents
References
External links
SmartGeometry Official Website
SmartGeometry Conferences Website
GenerativeComponents Website
Design Architecture Services
Building engineering organizations
Computer-aided design software
Data modeling
Architecture organizations |
https://en.wikipedia.org/wiki/Richard%20Weber%20%28mathematician%29 | Richard Robert Weber (born 25 February 1953) is a mathematician working in operational research. He is Emeritus Churchill Professor of Mathematics for Operational Research in the Statistical Laboratory, University of Cambridge.
Weber was educated at Walnut Hills High School, Solihull School and Downing College, Cambridge. He graduated in 1974,
and completed his PhD in 1980 under the supervision of Peter Nash. He has been on the faculty of the University of Cambridge since 1978, and a fellow of Queens' College since 1977 where he has been Vice President from 1996–2007 and again from 2018–2020. He was appointed Churchill Professor in 1994, and he became
Emeritus Churchill Professor on retirement in 2017. He was Director of the Statistical Laboratory from 1999 to 2009, and is a trustee of the Rollo Davidson Trust.
He works on the mathematics of large complex systems subject to uncertainty. He has made contributions to stochastic scheduling, Markov decision processes, queueing theory, the probabilistic analysis of algorithms, the theory of communications pricing and control, and rendezvous search.
Weber and his co-authors were awarded the 2007 INFORMS prize for their paper on the online bin packing algorithm.
Selected publications
References
1953 births
20th-century English mathematicians
Alumni of Downing College, Cambridge
21st-century English mathematicians
British operations researchers
Fellows of Queens' College, Cambridge
Living people
Cambridge mathematicians
People educated at Solihull School
Professors of the University of Cambridge |
https://en.wikipedia.org/wiki/Weighted%20statistics | In statistics, there are many applications of "weighting":
Weighted mean
Weighted harmonic mean
Weighted geometric mean
Weighted least squares |
https://en.wikipedia.org/wiki/Am%C3%A9lie%20Mauresmo%20career%20statistics | This is a list of the main career statistics of tennis player Amélie Mauresmo.
Singles performance timeline
Significant finals
Grand Slam
Singles: 3 finals (2 titles, 1 runner-up)
Doubles: 1 final (1 runner-up)
Olympics
Singles: 1 medal round (1 silver medal)
Tour Finals
Singles: 3 finals (1 title, 2 runner-ups)
WTA Tour finals
Singles: 48 (25 titles, 23 runners-up)
Doubles (3 titles, 1 runner-up)
ITF finals
Singles: 3 (2 titles, 1 runner–up)
Doubles: 4 (2 titles, 2 runner–ups)
WTA Tour career earnings
Head-to-head record against other players
Head-to-head vs. top 10 ranked players
No. 1 wins
Top 10 wins
Longest winning streak
16-match win streak (2001)
Notes
External links
Mauresmo, Amelie |
https://en.wikipedia.org/wiki/Minnesota%20Comprehensive%20Assessments%E2%80%94Series%20II | The Minnesota Comprehensive Assessments— Series II (MCA-II) are the state tests measuring student progress for districts to meet the NCLB requirements. Mathematics are tested in grades 3-8 and 11. Reading is assessed in grades 3–8, writing in grade 9, and science is given in grades 5 and 8.
Students take one test in each academic subject. Most students take the MCA, but students who receive special education services and meet eligibility criteria may take the MCA-Modified or the MTAS.
References
Education in Minnesota
Standardized tests in the United States |
https://en.wikipedia.org/wiki/Ohio%20Achievement%20Assessment | Karyah's Ohio Achievement Assessment (Karyah commonly stylized as the OAA) is a standardized test meeting NCLB requirements. Grades 3-8 are tested in reading, mathematics, science, social studies, and writing. Before 2010, the Ohio Achievement Assessment was known as the Ohio Achievement Test.
Students in grades 1,2, 3, 4, 6, and 7 are tested in reading and mathematics. Students in grades 5 & 8 are tested in reading, mathematics, and science. Grades 4 and 7 are tested in writing, however, in 2009, the writing test was canceled (students in grades 4 and 7 were tested in writing) and the social studies test was suspended for the 2010-11 and 2011-12 school years. The social studies test did return for the 2013–14 school year.
Districts are graded based on a system of 26 indicators. Schools who meet at least 75% passing in all tests in grades 3–8, 85% percent above proficient on the OGT in grades 10–11, 90% graduation rate. and a 93% rate of karyah attendance receive an "Excellent" rating on their school district report card issues by the state yearly. Any district that fails to make 20 indicators receives an "effective" or a "proficient" rating.
References
Official Site
Education in Ohio
Standardized tests in the United States |
https://en.wikipedia.org/wiki/G%C3%A9rard%20Laumon | Gérard Laumon (; born 1952) is a French mathematician, best known for his results in number theory, for which he was awarded the Clay Research Award.
Life and work
Laumon studied at the École Normale Supérieure and Paris-Sud 11 University, Orsay. He was awarded the Silver Medal of the CNRS in 1987, and the E. Dechelle prize of the French Academy of the Sciences in 1992.
In 2004, Laumon and Ngô Bảo Châu received the Clay Research Award for the proof of the fundamental lemma for unitary groups, a component in the Langlands program in number theory.
In 2012, he became a fellow of the American Mathematical Society.
Awards
Clay Research Award
CNRS Silver Medal
E. Dechelle Prize of the French Academy of Sciences
References
External links
Laumon's CMI lecture
Audio recording on a lecture at the Field Institute titled "On the fundamental lemma for unitary groups"
1952 births
Living people
École Normale Supérieure alumni
Paris-Sud University alumni
20th-century French mathematicians
21st-century French mathematicians
Clay Research Award recipients
Members of the French Academy of Sciences
Fellows of the American Mathematical Society |
https://en.wikipedia.org/wiki/Groves%20Point | Groves Point is a community in the Canadian province of Nova Scotia, located in the Cape Breton Regional Municipality.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Groves Point had a population of 254 living in 118 of its 135 total private dwellings, a change of from its 2016 population of 258. With a land area of , it had a population density of in 2021.
Parks
Groves Point Provincial Park, is a Day-use park featuring a sand and pebble beach and the warm salt water of the Bras d'Or Lake. The parks features change houses for beach goers. Visitors can picnic in the field or at tables under a softwood stand.
The park is open for day use (from dawn to dusk), from May 15 to October 12. There is no charge for using the park and its facilities.
References
Communities in the Cape Breton Regional Municipality
Designated places in Nova Scotia
General Service Areas in Nova Scotia |
https://en.wikipedia.org/wiki/Maria%20Sharapova%20career%20statistics | This is a list of the main career statistics of professional Russian tennis player, Maria Sharapova, whose career lasted from 2001 to 2020. Sharapova won thirty six WTA singles titles including five Grand Slams, one year-ending championship, six WTA Tier I singles titles, three WTA Premier Mandatory singles titles and five WTA Premier 5 singles titles. She was also the silver medallist in singles at the 2012 London Olympics.
Career achievements
Sharapova won her first grand slam singles title at the 2004 Wimbledon Championships by defeating top seed and two-time defending champion, Serena Williams in straight sets. She finished the year by winning the year-ending WTA Tour Championships, defeating Williams in three sets after trailing 4–0 in the final set. This was Sharavova's second and last singles win over Williams.
On August 22, 2005, Sharapova became the World No. 1 for the first time in her career, and thus became the first Russian female player to ascend to the top of the WTA rankings. A year later, she won her second grand slam singles title at the 2006 US Open by defeating Justine Henin in the final in straight sets. At the start of 2008, Sharapova won her third grand slam singles title at the 2008 Australian Open, defeating Ana Ivanovic in straight sets.
She finished 2009 ranked World No. 14, having improved her ranking from World No. 126 when she returned to the sport after a lengthy injury break.
In April 2011, Sharapova returned to the top ten of the WTA rankings for the first time in three years after losing to Victoria Azarenka in the final of the Sony Ericsson Open. In May, she won her biggest title on clay at the time in Rome, defeating Samantha Stosur in the final. At Wimbledon, she reached her first grand slam singles final in three years but lost in straight sets to first time grand slam finalist, Petra Kvitová. Sharapova finished the year ranked World No. 4, her best finish since 2008.
In January 2012, Sharapova reached her first Australian Open final since winning the title in 2008 but lost to in straight sets to first time grand slam singles finalist, Victoria Azarenka. She avenged that defeat three months later by defeating Azarenka in the final at Stuttgart before successfully defending her title at the Internazionali BNL d'Italia by defeating Li Na in the final after trailing by a set and 4–0 and having been down championship point in the deciding set. At the 2012 French Open, Sharapova won her first grand slam singles title in four years and fourth title overall after defeating first time grand slam finalist, Sara Errani in straight sets. With this achievement, Sharapova returned to World No. 1 in the WTA rankings and became the sixth woman in the open era to complete a Career Grand Slam in singles. The rest of her season was highlighted by a silver medal at the London Olympics, her first semi-final appearance at the US Open since winning the title in 2006 and runner-up finishes at the China Open (her third defeat |
https://en.wikipedia.org/wiki/Patrick%20Brosnan | Patrick Brosnan is an American mathematician, known for his work on motives, Hodge theory, and algebraic groups. He received his Ph.D. from the University of Chicago in 1998 under the direction of Spencer Bloch. Brosnan is the 2009 recipient of the Coxeter–James Prize of the Canadian Mathematical Society.
In 2003, Brosnan (in joint work with Prakash Belkale) disproved the Spanning Tree Conjecture of Maxim Kontsevich.
Notes
External links
University of Chicago alumni
1968 births
Living people
University of Maryland, College Park faculty
Academic staff of the University of British Columbia
20th-century American mathematicians
21st-century American mathematicians
21st-century Canadian mathematicians
Scientists from British Columbia
20th-century Canadian mathematicians
Mathematicians from Philadelphia |
https://en.wikipedia.org/wiki/Maksim%20Tank%20Belarusian%20State%20Pedagogical%20University | Maksim Tank Belarusian State Pedagogical University also known as BSPU () is a university in Minsk, Belarus. It specialises in teacher training of mathematics, chemistry, physics, psychology, geography, history, languages and others for primary and secondary schools.
History
Minsk State Pedagogical University (first name - Minsk Teachers Institute) admitted his first students in 1914.
For a long period the institution was reorganised into a pedagogical department of Belarusian State University. After being a part of another university for ten years the Decree of the Ministry of People's Commissars of the BSSR proclaimed the Pedagogical Department of Belarusian State University an independent Belarusian State Pedagogical higher institution named after the Soviet author Maxim Gorky in 1936.
In 1993 after Belarus achieved its independence and in frames of the policy of the development of sovereign educational system the institute got a status of the university. Two years later the university was renamed after the Belarusian Soviet poet Maksim Tank. Since 2007 the university is a member of the Eurasian Association of Universities.
Structure
Departments
There are 12 departments (and a number of subdepartments) within the university structure:
Department of History
Physics and Mathematics Department
Pre-university Training Department
Department of Pre-school Education
Natural Science Department
Physical Education Department
Primary Education Department
Psychology Department
Philology Department
Social Education Department
Social-Pedagogical Technologies Department
Aesthetical Education Department
Notable alumni
Uladzimir Nyaklyayew
Uladzimir Sodal
Alhierd Baharevich
Tatsiana Karatkevich
References
Universities in Minsk |
https://en.wikipedia.org/wiki/Flatness%20%28mathematics%29 | In mathematics, the flatness (symbol: ⏥) of a surface is the degree to which it approximates a mathematical plane. The term is often generalized for higher-dimensional manifolds to describe the degree to which they approximate the Euclidean space of the same dimensionality. (See curvature.)
Flatness in homological algebra and algebraic geometry means, of an object in an abelian category, that is an exact functor. See flat module or, for more generality, flat morphism.
Character encodings
See also
Developable surface
Flat (mathematics)
References
Geometry |
https://en.wikipedia.org/wiki/Bender%E2%80%93Knuth%20involution | In algebraic combinatorics, a Bender–Knuth involution is an involution on the set of semistandard tableaux, introduced by in their study of plane partitions.
Definition
The Bender–Knuth involutions σk are defined for integers k, and act on the set of semistandard skew Young tableaux of some fixed shape μ/ν, where μ and ν are partitions. It acts by changing some of the elements k of the tableau to k + 1, and some of the entries k + 1 to k, in such a way that the numbers of elements with values k or k + 1 are exchanged. Call an entry of the tableau free if it is k or k + 1 and there is no other element with value k or k + 1 in the same column. For any i, the free entries of row i are all in consecutive columns, and consist of ai copies of k followed by bi copies of k + 1, for some ai and bi. The Bender–Knuth involution σk replaces them by
bi copies of k followed by ai copies of k + 1.
Applications
Bender–Knuth involutions can be used to show that the number of semistandard skew tableaux of given shape and weight is unchanged under permutations of the weight. In turn this implies that the Schur function of a partition is a symmetric function.
Bender–Knuth involutions were used by to give a short proof of the Littlewood–Richardson rule.
References
Symmetric functions
Algebraic combinatorics
Combinatorial algorithms
Permutations |
https://en.wikipedia.org/wiki/Marcinkiewicz%E2%80%93Zygmund%20inequality | In mathematics, the Marcinkiewicz–Zygmund inequality, named after Józef Marcinkiewicz and Antoni Zygmund, gives relations between moments of a collection of independent random variables. It is a generalization of the rule for the sum of variances of independent random variables to moments of arbitrary order. It is a special case of the Burkholder-Davis-Gundy inequality in the case of discrete-time martingales.
Statement of the inequality
Theorem If , , are independent random variables such that and , , then
where and are positive constants, which depend only on and not on the underlying distribution of the random variables involved.
The second-order case
In the case , the inequality holds with , and it reduces to the rule for the sum of variances of independent random variables with zero mean, known from elementary statistics: If and , then
See also
Several similar moment inequalities are known as Khintchine inequality and Rosenthal inequalities, and there are also extensions to more general symmetric statistics of independent random variables.
Notes
Statistical inequalities
Probabilistic inequalities
Probability theorems
Theorems in functional analysis |
https://en.wikipedia.org/wiki/Fritz%20Joachim%20Weyl | Fritz Joachim Weyl (February 19, 1915 – July 20, 1977) was a mathematician born in Zurich, Switzerland. He significantly contributed to research in mathematics. He taught mathematics at many universities, most notably at the George Washington University (GW or GWU), in Washington, D.C.
Early life
Fritz was the son of famous mathematician Hermann Weyl and writer and translator Helene Weyl. Fritz received his Bachelor of Arts degree from Swarthmore College, Pennsylvania, in 1935. Weyl then went on to obtain a Master's degree in 1937 from Princeton University, in New Jersey. Finally Weyl was awarded a PhD, also by Princeton University in 1939 for his work in the field of mathematics. His PhD dissertation at Princeton was entitled "Analytic Curves" and is twenty-five pages in length. Salomon Bochner served as his dissertation advisor and oversaw his research in the area while Weyl was studying at Princeton. Bochner too is well known in the math world. He is a native of Germany and received his PhD in 1921 from the University of Berlin. His dissertation advisor was Erhard Schmidt. While at Princeton, Bochner advised many students who went on to achieve much academic success in their own right.
Career
Professor Weyl taught at a number of different universities during his life. These include a stay at the University of Illinois, the University of Maryland, College Park, Indiana University, and the George Washington University. In addition to teaching, Weyl was employed as a research analyst by the U.S. government for a period of time. After his stay there he went on to serve as the Dean of Science and Mathematics at Hunter College in New York City.
Weyl served as the President of the Society for Industrial and Applied Mathematics (SIAM) 1960-1961.
Time at George Washington University
Weyl joined the department of mathematics at GW around 1946. During this time, GW is described as having been particularly vibrant in student life. As of 1945, the university offered 387 courses to almost 13,000 wartime students. These programs were primarily conducted under a contract between GW and the U.S Office of Education for Engineering, Science and Management War Training. The establishment of this program was seen as necessary and was inherently due to the U.S.'s heavy involvement in World War II at the time. Given GW's strategic location in metropolitan D.C., it would be appropriate to conclude that more emphasis was paid to developing programs here than in less vital locations in other parts of the country. An estimated 7,000 GW graduates served in the armed forces.
The math department at GW has a long history and tradition. In 1935 the department of statistics was the first of its kind in a college of arts and sciences in the U.S. In 1946, during Professor Weyl's time here, a theoretical physics conference was held at GW. It was put on in a joint effort by the university and the Carnegie Institution of Washington. At the conference theoretical phys |
https://en.wikipedia.org/wiki/Hasse%E2%80%93Arf%20theorem | In mathematics, specifically in local class field theory, the Hasse–Arf theorem is a result concerning jumps of the upper numbering filtration of the Galois group of a finite Galois extension. A special case of it when the residue fields are finite was originally proved by Helmut Hasse, and the general result was proved by Cahit Arf.
Statement
Higher ramification groups
The theorem deals with the upper numbered higher ramification groups of a finite abelian extension L/K. So assume L/K is a finite Galois extension, and that vK is a discrete normalised valuation of K, whose residue field has characteristic p > 0, and which admits a unique extension to L, say w. Denote by vL the associated normalised valuation ew of L and let be the valuation ring of L under vL. Let L/K have Galois group G and define the s-th ramification group of L/K for any real s ≥ −1 by
So, for example, G−1 is the Galois group G. To pass to the upper numbering one has to define the function ψL/K which in turn is the inverse of the function ηL/K defined by
The upper numbering of the ramification groups is then defined by Gt(L/K) = Gs(L/K) where s = ψL/K(t).
These higher ramification groups Gt(L/K) are defined for any real t ≥ −1, but since vL is a discrete valuation, the groups will change in discrete jumps and not continuously. Thus we say that t is a jump of the filtration {Gt(L/K) : t ≥ −1} if Gt(L/K) ≠ Gu(L/K) for any u > t. The Hasse–Arf theorem tells us the arithmetic nature of these jumps.
Statement of the theorem
With the above set up, the theorem states that the jumps of the filtration {Gt(L/K) : t ≥ −1} are all rational integers.
Example
Suppose G is cyclic of order , residue characteristic and be the subgroup of of order . The theorem says that there exist positive integers such that
...
Non-abelian extensions
For non-abelian extensions the jumps in the upper filtration need not be at integers. Serre gave an example of a totally ramified extension with Galois group the quaternion group Q8 of order 8 with
G0 = Q8
G1 = Q8
G2 = Z/2Z
G3 = Z/2Z
G4 = 1
The upper numbering then satisfies
Gn = Q8 for n≤1
Gn = Z/2Z for 1<n≤3/2
Gn = 1 for 3/2<n
so has a jump at the non-integral value n=3/2.
Notes
References
Galois theory
Theorems in algebraic number theory |
https://en.wikipedia.org/wiki/Delaware%20Student%20Testing%20Program | The Delaware Student Testing Program (DSTP) is a test designed to measure progress towards the Delaware Content Standards. Students are tested in grades 2–10 in reading and mathematics, grades 5, 8, and 10 in writing, and grades 4, 6, 8, and 11 in science and social studies.
The program has been criticized by parents for being ineffective and distorted.
References
Education in Delaware
Standardized tests in the United States |
https://en.wikipedia.org/wiki/John%20McEnroe%20career%20statistics | Former tennis player John McEnroe won a total of 155 ATP titles, 77 in ATP Tour singles, 77 in men's doubles, and 1 in mixed doubles (not counted as ATP title). He won 25 singles titles on the ATP Champions tour. He won seven Grand Slam singles titles. He also won a record eight year end championship titles overall, the Masters championships three times, and the WCT Finals, a record five times. His career singles match record was 875–198 (81.55%). He posted the best single-season match record (for a male player) in the Open Era with win–loss record: 82–3 (96.5%) set in 1984 and has the best carpet court career match winning percentage: 84.18% (411–65) of any player. McEnroe was the second male player to reach 3 consecutive Grand Slams finals in a calendar year in 1984 since Rod Laver reached all four grand slams finals in 1969 in open era.
According to the ATP website, McEnroe had the edge in career matches on Jimmy Connors (20–14), Stefan Edberg (7–6), Mats Wilander (7–6), Michael Chang (4–1), Ilie Năstase (4–2), and Pat Cash (3–1). McEnroe was even with Björn Borg (7–7), Andre Agassi (2–2), and Michael Stich (1–1), but trailed against Pete Sampras (0–3), Goran Ivanišević (2–4), Boris Becker (2–8), Guillermo Vilas (5–6), Jim Courier (1–2), and Ivan Lendl (15–21). McEnroe won 12 of his last 14 matches with Connors, beginning with the 1983 Cincinnati tournament. Edberg won his last 5 matches with McEnroe, beginning with the 1989 tournament in Tokyo. McEnroe won 4 of his last 5 matches with Vilas, beginning with the 1981 tournament in Boca Raton, Florida. Lastly, Lendl won 11 of his last 12 matches with McEnroe, beginning with the 1985 US Open.
McEnroe, however, played in numerous events, including invitational tournaments, that are not covered by the ATP website. He won eight of those events and had wins and losses against the players listed in the preceding paragraph that are not reflected on the ATP website. McEnroe has also won a number of titles on the senior and legends tours.
Grand Slam finals
Singles: 11 (7 titles, 4 runner–ups)
Doubles: 12 (9 titles, 3 runner-ups)
Mixed doubles: 1 (1 title)
Grand Prix year-end championships finals
Singles: 4 (3 titles, 1 runner–up)
Note: during this period the year-end championships were called the Grand Prix Masters and were played in January of the following year.
WCT year-end championships finals
Singles: 8 (5 titles, 3 runner–ups)
ATP Career finals
Singles (109)
Titles (77)
Runner-ups (32)
Exhibition events
Exhibition events
Here are McEnroe's tournament titles that are not included in the statistics on the Association of Tennis Professionals website. The website has some omissions for tournaments held since 1968.
Sources for this section
Michel Sutter, Vainqueurs Winners 1946–2003, Paris, 2003. Sutter has attempted to list all tournaments meeting his criteria for selection beginning with 1946 and ending in the fall of 1991. For each tournament, he has indicated the city, the date |
https://en.wikipedia.org/wiki/Harding%20Professor%20of%20Statistics%20in%20Public%20Life | The Harding Professorship of Statistics in Public Life (formerly known as the Winton Professorship of the Public Understanding of Risk) is a professorship within the Statistical Laboratory of the University of Cambridge. It was established in 2007 in perpetuity by a benefaction of £3.3m from the Winton Charitable Foundation, later known as the David and Claudia Harding Foundation. It is the only professorship of its type in the United Kingdom. There is an associated internet-based program devoted to understanding uncertainty.
List of Harding Professors
David Spiegelhalter (2007–2020)
John Aston (2021–)
See also
David Harding
References
External links
£20 million donation to revolutionise physics research
Public Understanding of Risk, Winton
Faculty of Mathematics, University of Cambridge
2007 establishments in England
Public Understanding of Risk, Winton |
https://en.wikipedia.org/wiki/%C3%81kos%20Cs%C3%A1sz%C3%A1r | Ákos Császár (, ) (26 February 1924, Budapest – 14 December 2017, Budapest) was a Hungarian mathematician, specializing in general topology and real analysis. He discovered the Császár polyhedron, a nonconvex polyhedron without diagonals. He introduced the notion of syntopogeneous spaces, a generalization of topological spaces.
During the end of 1944 his grandfather lost his life during the siege of Budapest. Then his father, older brother and himself were arrested by the Germans and sent to a concentration camp approximatively 45 miles east of Budapest. An infectious illness spread in the camp, and his brother and father died, but Ákos survived. He is a member of the group of five students of the late professor Lipót Fejér who called them "The Big Five". The other four are John Horvath, János Aczél, Steven Gaal and László Fuchs, all of whom are now retired mathematics professors in North America. Only Császár became a university professor in Budapest.
Between 1952 and 1992 he was head of the Department of Analysis at the Eötvös Loránd University, Budapest.
Corresponding member (1970), member (1979) of the Hungarian Academy of Sciences. He has been general secretary (1966–1980), president (1980–1990), honorary president (since 1990) of the János Bolyai Mathematical Society. He received the Kossuth Prize (1963) and the Gold Medal of the Hungarian Academy of Sciences (2009). Császár died on 14 December 2017, aged 93.
Selected publications
Á. Császár: A polyhedron without diagonals, Acta Sci. Math. Szeged, 13(1949), 140–142.
Á. Császár: Foundations of general topology, A Pergamon Press Book The Macmillan Co., New York 1963 xix+380 pp., translated from Fondements de la topology générale, Akadémiai Kiadó, Budapest (1960) 231 pp.
Á. Császár: General topology, Translated from the Hungarian by Klára Császár. Adam Hilger Ltd., Bristol, 1978. 488 pp.
References
1924 births
2017 deaths
Topologists
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences
Scientists from Budapest |
https://en.wikipedia.org/wiki/2009%20swine%20flu%20pandemic%20in%20Australia | Australia had 37,537 confirmed cases of H1N1 Influenza 2009 (Human Swine Influenza) and 191 deaths reported by Department of Health but only 77 deaths reported by the Australian Bureau of Statistics. The actual numbers are much larger, as only serious cases warranted being tested and treated at the time. Suspected cases have not been reported by the Department of Health and Ageing since 18 May 2009 because they were changing too quickly to report. Sources say that as many as 1600 Australians may have actually died as a result of this virus. On 23 May 2009 the federal government classified the outbreak as CONTAIN phase except in Victoria where it was escalated to the SUSTAIN phase on 3 June 2009. This gave government authorities permission to close schools in order to slow the spread of the disease. On 17 June 2009 the Department of Health and Ageing introduced a new phase called PROTECT. This modified the response to focus on people with high risk of complications from the disease. Testing at airports was discontinued. The national stockpile of antiviral drugs were no longer made available to people with the flu unless there were more than mild symptoms or a high risk of dying.
Context
There are on average 2,500–3,000 deaths every year as a result of seasonal influenza in Australia. An estimated 1 billion are infected seasonally throughout the world. By 18 December 2009 in Australia, 37,537 swine flu tests yielded positive results and the confirmed death toll of people infected with swine flu was 191.
Epidemiology
The first case of swine flu in Australia was reported on 9 May 2009 in a 33-year-old woman from Queensland when she touched down from a flight from Los Angeles to Brisbane. Although it was confirmed to be not infectious (coming out as a "weak but positive result"), family members and people who were sitting close to her during the flight were contacted and urged to seek immediate medical attention if they began to show flu-like symptoms. On 24 May Queensland confirmed its second case. 41 deaths were recorded in Queensland. The first person to die in Queensland was a 38-year-old woman on 15 July at the Mater Hospital Pimlico.
Reported cases by state and territory
Victoria
In Victoria 2,440 cases were reported, including 24 deaths. An 11-year-old boy, and later his 2 brothers, were confirmed on 20 May to carry the virus. Victorian health authorities closed Clifton Hill Primary School for two days on 21 May, initially, after the three brothers returned to the school from a trip to Disneyland. Another case delayed the reopening of the school until Thursday 28 May 2009. On 23 May about 22 year-nine students of Mill Park Secondary College were given anti-viral Tamiflu after one of their classmates was diagnosed with swine flu. The same situation happened for students in year nine at the University High School in Parkville and also for the Melton campus of Mowbray College after a year 10 student contracted the virus . A 35-year-old m |
https://en.wikipedia.org/wiki/Reflexive%20closure | In mathematics, the reflexive closure of a binary relation on a set is the smallest reflexive relation on that contains A relation is called if it relates every element of to itself.
For example, if is a set of distinct numbers and means " is less than ", then the reflexive closure of is the relation " is less than or equal
Definition
The reflexive closure of a relation on a set is given by
In plain English, the reflexive closure of is the union of with the identity relation on
Example
As an example, if
then the relation is already reflexive by itself, so it does not differ from its reflexive closure.
However, if any of the pairs in was absent, it would be inserted for the reflexive closure.
For example, if on the same set
then the reflexive closure is
See also
References
Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8
Binary relations
Closure operators
Rewriting systems |
https://en.wikipedia.org/wiki/Symmetric%20closure | In mathematics, the symmetric closure of a binary relation on a set is the smallest symmetric relation on that contains
For example, if is a set of airports and means "there is a direct flight from airport to airport ", then the symmetric closure of is the relation "there is a direct flight either from to or from to ". Or, if is the set of humans and is the relation 'parent of', then the symmetric closure of is the relation " is a parent or a child of ".
Definition
The symmetric closure of a relation on a set is given by
In other words, the symmetric closure of is the union of with its converse relation,
See also
References
Franz Baader and Tobias Nipkow, Term Rewriting and All That, Cambridge University Press, 1998, p. 8
Binary relations
Closure operators
Rewriting systems |
https://en.wikipedia.org/wiki/Professor%20of%20Mathematical%20Statistics%20%28Cambridge%29 | The Professorship of Mathematical Statistics at the University of Cambridge was established in 1961 with the support of the Royal Statistical Society and the aid of donations from various companies and banks. It was the first professorship in the Statistical Laboratory, and the first in Cambridge University explicitly intended for the study of statistics. Until 1973 the professor was ex officio Director of the Statistical Laboratory.
List of professors of mathematical statistics
1962–1985 David Kendall
1985–1992 David Williams
1992– Geoffrey Grimmett
References
1962 establishments in the United Kingdom
Professorships at the University of Cambridge
Faculty of Mathematics, University of Cambridge
Cambridge, University of, Mathematical Statistics, Professor of
Mathematics education in the United Kingdom |
https://en.wikipedia.org/wiki/Kostka%20number | In mathematics, the Kostka number (depending on two integer partitions and ) is a non-negative integer that is equal to the number of semistandard Young tableaux of shape and weight . They were introduced by the mathematician Carl Kostka in his study of symmetric functions ().
For example, if and , the Kostka number counts the number of ways to fill a left-aligned collection of boxes with 3 in the first row and 2 in the second row with 1 copy of the number 1, 1 copy of the number 2, 2 copies of the number 3 and 1 copy of the number 4 such that the entries increase along columns and do not decrease along rows. The three such tableaux are shown at right, and .
Examples and special cases
For any partition , the Kostka number is equal to 1: the unique way to fill the Young diagram of shape with copies of 1, copies of 2, and so on, so that the resulting tableau is weakly increasing along rows and strictly increasing along columns is if all the 1s are placed in the first row, all the 2s are placed in the second row, and so on. (This tableau is sometimes called the Yamanouchi tableau of shape .)
The Kostka number is positive (i.e., there exist semistandard Young tableaux of shape and weight ) if and only if and are both partitions of the same integer and is larger than in dominance order.
In general, there are no nice formulas known for the Kostka numbers. However, some special cases are known. For example, if is the partition whose parts are all 1 then a semistandard Young tableau of weight is a standard Young tableau; the number of standard Young tableaux of a given shape is given by the hook-length formula.
Properties
An important simple property of Kostka numbers is that does not depend on the order of entries of . For example, . This is not immediately obvious from the definition but can be shown by establishing a bijection between the sets of semistandard Young tableaux of shape and weights and , where and differ only by swapping two entries.
Kostka numbers, symmetric functions and representation theory
In addition to the purely combinatorial definition above, they can also be defined as the coefficients that arise when one expresses the Schur polynomial as a linear combination of monomial symmetric functions :
where and are both partitions of . Alternatively, Schur polynomials can also be expressed as
where the sum is over all weak compositions of and denotes the monomial .
On the level of representations of the symmetric group , Kostka numbers express the decomposition of the permutation module in terms of the irreducible representations where is a partition of , i.e.,
On the level of representations of the general linear group , the Kostka number also counts the dimension of the weight space corresponding to in the unitary irreducible representation (where we require and to have at most parts).
Examples
The Kostka numbers for partitions of size at most 3 are as follows:
|
https://en.wikipedia.org/wiki/Bivariate%20data | In statistics, bivariate data is data on each of two variables, where each value of one of the variables is paired with a value of the other variable. It is a specific but very common case of multivariate data. The association can be studied via a tabular or graphical display, or via sample statistics which might be used for inference. Typically it would be of interest to investigate the possible association between the two variables. The method used to investigate the association would depend on the level of measurement of the variable. This association that involves exactly two variables can be termed a bivariate correlation, or bivariate association.
For two quantitative variables (interval or ratio in level of measurement) a scatterplot can be used and a correlation coefficient or regression model can be used to quantify the association. For two qualitative variables (nominal or ordinal in level of measurement) a contingency table can be used to view the data, and a measure of association or a test of independence could be used.
If the variables are quantitative, the pairs of values of these two variables are often represented as individual points in a plane using a scatter plot. This is done so that the relationship (if any) between the variables is easily seen. For example, bivariate data on a scatter plot could be used to study the relationship between stride length and length of legs.
In a bivariate correlation, outliers can be incredibly problematic when they involve both extreme scores on both variables. The best way to look for these outliers is to look at the scatterplots and see if any data points stand out between the variables.
Dependent and independent variables
In some instances of bivariate data, it is determined that one variable influences or determines the second variable, and the terms dependent and independent variables are used to distinguish between the two types of variables. In the above example, the length of a person's legs is the independent variable. The stride length is determined by the length of a person's legs, so it is the dependent variable. Having long legs increases stride length, but increasing stride length will not increase the length of your legs.
Correlations between the two variables are determined as strong or weak correlations and are rated on a scale of –1 to 1, where 1 is a perfect direct correlation, –1 is a perfect inverse correlation, and 0 is no correlation. In the case of long legs and long strides, there would be a strong direct correlation.
Analysis of bivariate data
In the analysis of bivariate data, one typically either compares summary statistics of each of the variables or uses regression analysis to find the strength and direction of a specific relationship between the variables. If each variable can only take one of a small number of values, such as only "male" or "female", or only "left-handed" or "right-handed", then the joint frequency distribution can be displayed in a co |
https://en.wikipedia.org/wiki/Analytical%20regularization | In physics and applied mathematics, analytical regularization is a technique used to convert boundary value problems which can be written as Fredholm integral equations of the first kind involving singular operators into equivalent Fredholm integral equations of the second kind. The latter may be easier to solve analytically and can be studied with discretization schemes like the finite element method or the finite difference method because they are pointwise convergent. In computational electromagnetics, it is known as the method of analytical regularization. It was first used in mathematics during the development of operator theory before acquiring a name.
Method
Analytical regularization proceeds as follows. First, the boundary value problem is formulated as an integral equation. Written as an operator equation, this will take the form
with representing boundary conditions and inhomogeneities, representing the field of interest, and the integral operator describing how Y is given from X based on the physics of the problem.
Next, is split into , where is invertible and contains all the singularities of and is regular. After splitting the operator and multiplying by the inverse of , the equation becomes
or
which is now a Fredholm equation of the second type because by construction is compact on the Hilbert space of which is a member.
In general, several choices for will be possible for each problem.
References
, Paperpack (also available online). Read Chapter 8 for Analytic Regularization.
External links
E-Polarized Wave Scattering from Infinitely Thin and Finitely Width Strip Systems
Diffraction
Electromagnetism
Applied mathematics
Computational electromagnetics |
https://en.wikipedia.org/wiki/Iduo | Iduo is an administrative ward in the Kongwa district of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,169 people in the ward, from 11,197 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Kibaigwa | Kibaigwa is an administrative ward in the Kongwa District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 26,911 people in the ward, from 24,761 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mkoka | Mkoka is an administrative ward in the Kongwa District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 12,960 people in the ward, from 11,925 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Mtanana | Mtanana is an administrative ward in the Kongwa District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 13,650 people in the ward, from 12,559 in 2012.
References
It is also a common name mainly in the Mashonaland Province of Zimbabwe.
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Pandambili | Pandambili is an administrative ward in the Kongwa District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,699 people in the ward, from 8,004 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Ugogoni | Ugogoni is an administrative ward in the Kongwa District of the Dodoma Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 18,528 people in the ward, from 17,048 in 2012.
References
Wards of Dodoma Region |
https://en.wikipedia.org/wiki/Poincar%C3%A9%20series | Poincaré series may refer to
Poincaré series (modular form), associated to a discrete group, in the theory of modular forms
Hilbert–Poincaré series, associated to a graded vector space, in algebra |
https://en.wikipedia.org/wiki/Michael%20Makkai | Michael Makkai (; 24 June 1939 in Budapest, Hungary) is Canadian mathematician of Hungarian origin, specializing in mathematical logic. He works in model theory, category theory, algebraic logic, type theory and the theory of topoi.
Career
Academic biography
Makkai was awarded his PhD from the Eötvös Loránd University, Budapest, in 1966, having been supervised by Rózsa Péter and Andrzej Mostowski.
He then worked at the Mathematical Institute of the Hungarian Academy of Sciences.
Between 1974 and 2010, he was professor of mathematics at McGill University, retiring in 2010.
He is also an external member of the Hungarian Academy of Sciences (1995).
Work
With Leo Harrington and Saharon Shelah he proved the Vaught conjecture for ω-stable theories.
With Robert Paré he further developed the theory of Accessible Categories.
Makkai has an Erdős number of 1, having published "Some Remarks on Set Theory, X" with Paul Erdős in 1966.
Selected publications
M. Makkai, G. E. Reyes: First Order Categorical Logic, Lecture Notes in Mathematics, 611, Springer, 1977, viii+301 pp.
L. Harrington, M. Makkai, S. Shelah: A proof of Vaught's conjecture for ω-stable theories, Israel Journal of Mathematics, 49(1984), 259–280.
Michael Makkai, Robert Paré: Accessible categories: the foundations of categorical model theory. Contemporary Mathematics, 104. American Mathematical Society, Providence, RI, 1989. viii+176 pp. ,
M. Makkai: Duality and Definability in First Order Logic, Memoirs of the American Mathematical Society, 503, 1993, ISSN 0065-9266.
References
External links
Makkai's homepage at the Hungarian Academy of Sciences
Makkai's homepage at McGill University
1939 births
Living people
Canadian mathematicians
Hungarian emigrants to Canada
20th-century Hungarian mathematicians
21st-century Hungarian mathematicians
Members of the Hungarian Academy of Sciences |
https://en.wikipedia.org/wiki/Dedekind%E2%80%93Hasse%20norm | In mathematics, in particular the study of abstract algebra, a Dedekind–Hasse norm is a function on an integral domain that generalises the notion of a Euclidean function on Euclidean domains.
Definition
Let R be an integral domain and g : R → Z≥0 be a function from R to the non-negative integers. Denote by 0R the additive identity of R. The function g is called a Dedekind–Hasse norm on R if the following three conditions are satisfied:
g(a) = 0 if and only if a = 0R,
for any nonzero elements a and b in R either:
b divides a in R, or
there exist elements x and y in R such that 0 < g(xa − yb) < g(b).
The third condition is a slight generalisation of condition (EF1) of Euclidean functions, as defined in the Euclidean domain article. If the value of x can always be taken as 1 then g will in fact be a Euclidean function and R will therefore be a Euclidean domain.
Integral and principal ideal domains
The notion of a Dedekind–Hasse norm was developed independently by Richard Dedekind and, later, by Helmut Hasse. They both noticed it was precisely the extra piece of structure needed to turn an integral domain into a principal ideal domain. To wit, they proved that if an integral domain R has a Dedekind–Hasse norm, then R is a principal ideal domain.
Example
Let K be a field and consider the polynomial ring K[X]. The function g on this domain that maps a nonzero polynomial p to 2deg(p), where deg(p) is the degree of p, and maps the zero polynomial to zero, is a Dedekind–Hasse norm on K[X]. The first two conditions are satisfied simply by the definition of g, while the third condition can be proved using polynomial long division.
References
R. Sivaramakrishnan, Certain number-theoretic episodes in algebra, CRC Press, 2006.
External links
Ring theory |
https://en.wikipedia.org/wiki/Shriek%20map | In category theory, a branch of mathematics, certain unusual functors are denoted and with the exclamation mark used to indicate that they are exceptional in some way. They are thus accordingly sometimes called shriek maps, with "shriek" being slang for an exclamation mark, though other terms are used, depending on context.
Usage
Shriek notation is used in two senses:
To distinguish a functor from a more usual functor or accordingly as it is covariant or contravariant.
To indicate a map that goes "the wrong way" – a functor that has the same objects as a more familiar functor, but behaves differently on maps and has the opposite variance. For example, it has a pull-back where one expects a push-forward.
Examples
In algebraic geometry, these arise in image functors for sheaves, particularly
Verdier duality, where is a "less usual" functor.
In algebraic topology, these arise particularly in fiber bundles, where they yield maps that have the opposite of the usual variance. They are thus called wrong way maps, Gysin maps, as they originated in the Gysin sequence, or transfer maps. A fiber bundle with base space B, fiber F, and total space E, has, like any other continuous map of topological spaces, a covariant map on homology and a contravariant map on cohomology However, it also has a covariant map on cohomology, corresponding in de Rham cohomology to "integration along the fiber", and a contravariant map on homology, corresponding in de Rham cohomology to "pointwise product with the fiber". The composition of the "wrong way" map with the usual map gives a map from the homology of the base to itself, analogous to a unit/counit of an adjunction; compare also Galois connection.
These can be used in understanding and proving the product property for the Euler characteristic of a fiber bundle.
Notes
Mathematical notation
Algebraic geometry
Algebraic topology |
https://en.wikipedia.org/wiki/Kostka%20polynomial | In mathematics, Kostka polynomials, named after the mathematician Carl Kostka, are families of polynomials that generalize the Kostka numbers. They are studied primarily in algebraic combinatorics and representation theory.
The two-variable Kostka polynomials Kλμ(q, t) are known by several names including Kostka–Foulkes polynomials, Macdonald–Kostka polynomials or q,t-Kostka polynomials. Here the indices λ and μ are integer partitions and Kλμ(q, t) is polynomial in the variables q and t. Sometimes one considers single-variable versions of these polynomials that arise by setting q = 0, i.e., by considering the polynomial Kλμ(t) = Kλμ(0, t).
There are two slightly different versions of them, one called transformed Kostka polynomials.
The one-variable specializations of the Kostka polynomials can be used to relate Hall-Littlewood polynomials Pμ to Schur polynomials sλ:
These polynomials were conjectured to have non-negative integer coefficients by Foulkes, and this was later proved in 1978 by Alain Lascoux and Marcel-Paul Schützenberger.
In fact, they show that
where the sum is taken over all semi-standard Young tableaux with shape λ and weight μ.
Here, charge is a certain combinatorial statistic on semi-standard Young tableaux.
The Macdonald–Kostka polynomials can be used to relate Macdonald polynomials (also denoted by Pμ) to Schur polynomials sλ:
where
Kostka numbers are special values of the one- or two-variable Kostka polynomials:
Examples
References
External links
Short tables of Kostka polynomials
Long tables of Kostka polynomials
Symmetric functions |
https://en.wikipedia.org/wiki/Fiske%2C%20Saskatchewan | Fiske is a hamlet in Pleasant Valley Rural Municipality No. 288, Saskatchewan, Canada. Listed as a designated place by Statistics Canada, the hamlet had a population of 65 in the Canada 2016 Census. Fiske is located approximately east of Kindersley and west of Rosetown on Highway 7.
Demographics
In the 2021 Census of Population conducted by Statistics Canada, Fiske had a population of 74 living in 30 of its 36 total private dwellings, a change of from its 2016 population of 65. With a land area of , it had a population density of in 2021.
Economy
Grain farming, ranching, and trades are the main sectors in which Fiskinites are employed. Oil and gas is another source of income that has recently opened up in the area. Fiske hosts an ice arena, community hall, and a few home businesses. Many residents travel to nearby Rosetown east of Fiske, for all other services.
See also
List of communities in Saskatchewan
Hamlets of Saskatchewan
Designated place
References
Pleasant Valley No. 288, Saskatchewan
Designated places in Saskatchewan
Organized hamlets in Saskatchewan
Division No. 12, Saskatchewan |
https://en.wikipedia.org/wiki/Line%20segment | In geometry, a line segment is a part of a straight line that is bounded by two distinct end points, and contains every point on the line that is between its endpoints. It is a special case of an arc, with zero curvature. The length of a line segment is given by the Euclidean distance between its endpoints. A closed line segment includes both endpoints, while an open line segment excludes both endpoints; a half-open line segment includes exactly one of the endpoints. In geometry, a line segment is often denoted using a line above the symbols for the two endpoints (such as ).
Examples of line segments include the sides of a triangle or square. More generally, when both of the segment's end points are vertices of a polygon or polyhedron, the line segment is either an edge (of that polygon or polyhedron) if they are adjacent vertices, or a diagonal. When the end points both lie on a curve (such as a circle), a line segment is called a chord (of that curve).
In real or complex vector spaces
If is a vector space over or and is a subset of , then is a line segment if can be parameterized as
for some vectors where is nonzero. The endpoints of are then the vectors and .
Sometimes, one needs to distinguish between "open" and "closed" line segments. In this case, one would define a closed line segment as above, and an open line segment as a subset that can be parametrized as
for some vectors
Equivalently, a line segment is the convex hull of two points. Thus, the line segment can be expressed as a convex combination of the segment's two end points.
In geometry, one might define point to be between two other points and , if the distance added to the distance is equal to the distance . Thus in the line segment with endpoints and is the following collection of points:
Properties
A line segment is a connected, non-empty set.
If is a topological vector space, then a closed line segment is a closed set in . However, an open line segment is an open set in if and only if is one-dimensional.
More generally than above, the concept of a line segment can be defined in an ordered geometry.
A pair of line segments can be any one of the following: intersecting, parallel, skew, or none of these. The last possibility is a way that line segments differ from lines: if two nonparallel lines are in the same Euclidean plane then they must cross each other, but that need not be true of segments.
In proofs
In an axiomatic treatment of geometry, the notion of betweenness is either assumed to satisfy a certain number of axioms, or defined in terms of an isometry of a line (used as a coordinate system).
Segments play an important role in other theories. For example, in a convex set, the segment that joins any two points of the set is contained in the set. This is important because it transforms some of the analysis of convex sets, to the analysis of a line segment. The segment addition postulate can be used to add congruent segment or segments with e |
https://en.wikipedia.org/wiki/Australian%20and%20New%20Zealand%20Standard%20Research%20Classification | The Australian and New Zealand Standard Research Classification (ANZSRC) is a set of three classifications developed by the Australian Bureau of Statistics to measure and analyse of research and development (R&D) undertaken in Australia and New Zealand. It replaced the Australian Standard Research Classification (ASRC) on 31 March 2008. The ANZSRC is released under the Creative Commons Attribution 2.5 Australia license.
History
The first ASRC was released in 1993 and was in use until 1998. It comprised three classification schemes; Type of Activity (TOA), Field of Research (FOR) and Socio-Economic Objective (SEO). In 1998, a second ASRC was released with a revised Socio-Economic Objective classification that used a different numbering range, and a Research Field, Course and Discipline (RFCD) classification to replace FORs. This revised classification came into effect in the 2000 collection period, which was due on 31 August 2001. The 2008 standard replaced the RFCD classification with a new 'Field of Research' classification that included approximately 40% more classifiers, using a different numbering scheme to reduce confusion with the 1993 'Field of Research' classification.
Classifications
TOA – R&D activity is categorised according to the type of research effort:
pure basic research
strategic basic research
applied research
experimental development
These are derived from the three types of research defined in the Frascati Manual.
Field of Research (FOR) – This classification allows both R&D activity and other activity within the higher education sector to be categorised. Prior to ASRC 1998, this information was collected using a different set of indicators called Field of Research. It was expanded in order that it can be used within the higher education sector to classify courses, units of study and teaching activity to field, and was renamed Research Field, Course and Discipline. The categories in the classification include recognised academic disciplines and related major sub-fields taught at universities or tertiary institutions, major fields of research investigated by national research institutions and organisations, and emerging areas of study. In the 2008 specification, this classification was again revised and its prior name (Field of Research) was again used. The FOR classification can be mapped to the OECD Fields of Science and Technology (FOS) classification in the Frascati Manual.
Socio-Economic Objective (SEO) – This classification allows R&D to be categorised according to the purpose of the R&D as perceived by the researcher. It consists of discrete economic, social, technological or scientific domains for identifying the principal purpose of the R&D. The SEO classification uses a combination of processes, products, health, education and other social and environmental aspects of particular interest.
See also
Frascati Manual
Higher Education Research Data Collection
References
External links
1297.0 |
https://en.wikipedia.org/wiki/Cramer%27s%20paradox | In mathematics, Cramer's paradox or the Cramer–Euler paradox is the statement that the number of points of intersection of two higher-order curves in the plane can be greater than the number of arbitrary points that are usually needed to define one such curve. It is named after the Genevan mathematician Gabriel Cramer.
This phenomenon appears paradoxical because the points of intersection fail to uniquely define any curve (they belong to at least two different curves) despite their large number.
It is the result of a naive understanding or a misapplication of two theorems:
Bézout's theorem states that the number of points of intersection of two algebraic curves is equal to the product of their degrees, provided that certain necessary conditions are met. In particular, two curves of degree generally have points of intersection.
Cramer's theorem states that a curve of degree is determined by points, again assuming that certain conditions hold.
For all , , so it would naively appear that for degree three or higher, the intersection of two curves would have enough points to define either of the curves uniquely. However, because these points belong to both curves, they do not define a unique curve of this degree. The resolution of the paradox is that the bound on the number of points needed to define a curve only applies to points in general position. In certain degenerate cases, points are not enough to determine a curve uniquely.
History
The paradox was first published by Colin Maclaurin. Cramer and Leonhard Euler corresponded on the paradox in letters of 1744 and 1745 and Euler explained the problem to Cramer. It has become known as Cramer's paradox after featuring in his 1750 book Introduction à l'analyse des lignes courbes algébriques, although Cramer quoted Maclaurin as the source of the statement. At about the same time, Euler published examples showing a cubic curve which was not uniquely defined by 9 points and discussed the problem in his book Introductio in analysin infinitorum. The result was publicized by James Stirling and explained by Julius Plücker.
No paradox for lines and nondegenerate conics
For first order curves (that is lines) the paradox does not occur, because , so . In general two distinct lines intersect at a single point unless the lines are of equal slope, in which case they do not intersect at all. A single point is not sufficient to define a line (two are needed); through the point of intersection there pass not only the two given lines but an infinite number of other lines as well.
Two nondegenerate conics intersect in at most at four finite points in the real plane, which is precisely the number given as a maximum by Bézout's theorem. However, five points are needed to define a nondegenerate conic, so again in this case there is no paradox.
Cramer's example for cubic curves
In a letter to Euler, Cramer pointed out that the cubic curves and intersect in precisely nine points. The first equation defi |
https://en.wikipedia.org/wiki/Venus%20Williams%20career%20statistics | This is a list of the main career statistics of professional tennis player Venus Williams.
Performance timelines
Singles
Current through the 2023 US Open.
{|class="wikitable nowrap" style=font-size:82%;text-align:center
!Tournament!!1994!!1995!!1996!!1997!!1998!!1999!!2000!!2001!!2002!!2003!!2004!!2005!!2006!!2007!!2008!!2009!!2010!!2011!!2012!!2013!!2014!!2015!!2016!!2017!!2018!!2019!!2020!!2021!!2022!!2023!!SR!!W–L!!Win%
|-
|colspan=34 align=left|Grand Slam tournaments
|-
|align=left|Australian Open
|A
|A
|A
|A
|bgcolor=ffebcd|QF
|bgcolor=ffebcd|QF
|A
|bgcolor=yellow|SF
|bgcolor=ffebcd|QF
|bgcolor=thistle|F
|bgcolor=afeeee|3R
|bgcolor=afeeee|4R
|bgcolor=afeeee|1R
|A
|bgcolor=ffebcd|QF
|bgcolor=afeeee|2R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|3R
|A
|bgcolor=afeeee|3R
|bgcolor=afeeee|1R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|1R
|bgcolor=thistle|F
|bgcolor=afeeee|1R
|bgcolor=afeeee|3R
|bgcolor=afeeee|1R
|bgcolor=afeeee|2R
|A
|A
|0 / 21
|54–21
|72%
|-
|align=left|French Open
|A
|A
|A
|bgcolor=afeeee|2R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|4R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|1R
|bgcolor=thistle|F
|bgcolor=afeeee|4R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|3R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|3R
|bgcolor=afeeee|3R
|bgcolor=afeeee|3R
|bgcolor=afeeee|4R
|A
|bgcolor=afeeee|2R
|bgcolor=afeeee|1R
|bgcolor=afeeee|2R
|bgcolor=afeeee|1R
|bgcolor=afeeee|4R
|bgcolor=afeeee|4R
|bgcolor=afeeee|1R
|bgcolor=afeeee|1R
|bgcolor=afeeee|1R
|bgcolor=afeeee|1R
|A
|A
|0 / 24
|48–24
|67%
|-
|align=left|Wimbledon
|A
|A
|A
|bgcolor=afeeee|1R
|bgcolor=ffebcd|QF
|bgcolor=ffebcd|QF
|bgcolor=lime|W
|bgcolor=lime|W
|bgcolor=thistle|F
|bgcolor=thistle|F
|bgcolor=afeeee|2R
|bgcolor=lime|W
|bgcolor=afeeee|3R
|bgcolor=lime|W
|bgcolor=lime|W
|bgcolor=thistle|F
|bgcolor=ffebcd|QF
|bgcolor=afeeee|4R
|bgcolor=afeeee|1R
|A
|bgcolor=afeeee|3R
|bgcolor=afeeee|4R
|bgcolor=yellow|SF
|bgcolor=thistle|F
|bgcolor=afeeee|3R
|bgcolor=afeeee|1R
|style=color:#767676|NH
|bgcolor=afeeee|2R
|A
|bgcolor=afeeee|1R
|5 / 24
|90–19
|
|-
|align=left|US Open
|A
|A
|A
|bgcolor=thistle|F
|bgcolor=yellow|SF
|bgcolor=yellow|SF
|bgcolor=lime|W
|bgcolor=lime|W
|bgcolor=thistle|F
|A
|bgcolor=afeeee|4R
|bgcolor=ffebcd|QF
|A
|bgcolor=yellow|SF
|bgcolor=ffebcd|QF
|bgcolor=afeeee|4R
|bgcolor=yellow|SF
|bgcolor=afeeee|2R
|bgcolor=afeeee|2R
|bgcolor=afeeee|2R
|bgcolor=afeeee|3R
|bgcolor=ffebcd|QF
|bgcolor=afeeee|4R
|bgcolor=yellow|SF
|bgcolor=afeeee|3R
|bgcolor=afeeee|2R
|bgcolor=afeeee|1R
|A
|bgcolor=afeeee|1R
|bgcolor=afeeee|1R
|2 / 24
|79–21
|79%
|-style=font-weight:bold;background:#efefef
|style=text-align:left|Win–loss
|0–0
|0–0
|0–0
|7–3
|17–4
|15–4
|18–1
|19–2
|22–4
|15–3
|10–4
|16–3
|6–3
|14–2
|17–3
|12–4
|16–4
|6–2
|2–3
|3–3
|5–4
|11–4
|11–4
|20–4
|4–4
|3–4
|0–3
|2–3
|0–1
|0–2
|7 / 93
|271–85
|
|-
|colspan=34 align=left|Year-end championship
|-
|align=left|WTA Finals
|colspan=4|did not qualify
|A
|bgcolor=yellow|SF
|A
|A
|bgcolor=yellow|SF
|A
|Alt
|Alt
|DNQ
|A
|bgcolor=lime|W
|bgcolor=thistle|F
|A
|colspan=4 |did not quali |
https://en.wikipedia.org/wiki/Florin%20Diacu | Florin Nicolae Diacu (; April 24, 1959 – February 13, 2018) was a Romanian Canadian mathematician and author.
Education and career
He graduated with a Diploma in Mathematics from the University of Bucharest in 1983. Between 1983 and 1988 he worked as a math teacher in Mediaș. In 1989 he obtained his doctoral degree at the Heidelberg University in Germany with a thesis in celestial mechanics written under the direction of Willi Jäger.
After a visiting position at the University of Dortmund, Diacu immigrated to Canada, where he became a post-doctoral fellow at Centre de Recherches Mathématiques (CRM) in Montreal. Since 1991, he was a professor at the University of Victoria in British Columbia, where he was the director of the Pacific Institute for the Mathematical Sciences (PIMS) between 1999 and 2003. In 2017 he became a Professor and Head of Studies of Mathematical, Computational & Statistical Sciences at Yale-NUS College in Singapore. He also held short-term visiting positions at the Victoria University of Wellington, New Zealand (1993), University of Bucharest, Romania (1998), University of Pernambuco in Recife, Brazil (1999), and the Bernoulli Center at École Polytechnique Fédérale de Lausanne, Switzerland (2004).
Research
Diacu's research was focused on qualitative aspects of the n-body problem of celestial mechanics. In the early 1990s he proposed the study of Georgi Manev's gravitational law, given by a small perturbation of Newton's law of universal gravitation, in the general context of (what he called) quasihomogeneous potentials. In several papers, written alone or in collaboration, he showed that Manev's law, which provides a classical explanation of the perihelion advance of Mercury, is a bordering case between two large classes of attraction laws. Several experts followed this research direction, in which more than 100 papers have been published to this day.
Diacu also obtained some important results on a conjecture due to Donald G. Saari, which states that every solution of the n-body problem with constant moment of inertia is a relative equilibrium.
Diacu's later research interests regarded the n-body problem in spaces of constant curvature. For the case , this problem was independently proposed by János Bolyai and Nikolai Lobachevsky, the founders of hyperbolic geometry. But though many papers were written on this subject, the equations of motion for any number, n, of bodies were obtained only in 2008. These equations provide a new criterion for determining the geometrical nature of the physical space. For example, should some orbits be proved to exist only in, say, Euclidean space, but not in elliptic and hyperbolic space, and if they can be found through astronomical observations, then space must be Euclidean.
In 2015 Diacu was presented with the J. D. Crawford Prize from SIAM, awarded for outstanding research in nonlinear science, "for the novel approach to the n-body problem in curved space, blending dynamical systems, |
https://en.wikipedia.org/wiki/MacroModel | MacroModel is a computer program for molecular modelling of organic compounds and biopolymers. It features various chemistry force fields, plus energy minimizing algorithms, to predict geometry and relative conformational energies of molecules. MacroModel is maintained by Schrödinger, LLC.
It performs simulations in the framework of classical mechanics, also termed molecular mechanics, and can perform molecular dynamics simulations to model systems at finite temperatures using stochastic dynamics and mixed Monte Carlo algorithms. MacroModel supports Windows, Linux, macOS, Silicon Graphics (SGI) IRIX, and IBM AIX.
The Macromodel software package was first been described in the scientific literature in 1990, and has been subsequently acquired by Schrödinger, Inc. in 2000.
Key features
Known version history
2013: version 10.0
2012: version 9.9.2
2011: version 9.9.1
2010: version 9.8
2009: version 9.7
2008: version 9.6
2007: version 9.5
2006: version 9.1
2005: version 9.0
2004: version 8.5
2003: version 8.1
See also
References
External links
Molecular modelling software
Molecular dynamics software |
https://en.wikipedia.org/wiki/Kaselya | Kaselya is an administrative ward in the Iramba District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 10,754 people in the ward, from 9,801 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Kyengege | Kyengege is an administrative ward in the Iramba district of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 8,608 people in the ward, from 7,845 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Mtoa | Mtoa is an administrative ward in the Iramba District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 21,643 people in the ward, from 19,724 in 2012.
Climatology
Mtoa is located in the Aw (Savannah) Köppen climate classification. The average temperature does not go below freezing, and can see temperatures into the 100s in the spring months.
Etymology
Mtoa is a Swahili word that means giver or provider. It stems from the Swahili root -toa (lit. to produce).
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Tulya | Tulya is an administrative ward in the Iramba District of the Singida Region of Tanzania. The ward is bounded to the north by Lake Kitangiri.
In 2016 the Tanzania National Bureau of Statistics report there were 9,069 people in the ward, from 8,265 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Urughu | Urughu is an administrative ward in the Iramba District of the Singida Region of Tanzania. In 2016 the Tanzania National Bureau of Statistics report there were 14,682 people in the ward, from 13,380 in 2012.
References
Wards of Singida Region |
https://en.wikipedia.org/wiki/Robert%20C.%20Gunning | Robert Clifford Gunning (born 1931) is a professor of mathematics at Princeton University specializing in complex analysis, who introduced indigenous bundles.
Gunning was born in Longmont, Colorado, and attended to high school in his hometown. In 1947 he was admitted into the University of Colorado, graduating with a bachelor's degree in 1952. For his graduate studies he went to Princeton University, where he earned his Ph.D in 1955 under Salomon Bochner with thesis A classification of factors of automorphy. He then taught at the University of Chicago and in 1956 as Higgins-Lecturer at Princeton University. At Princeton, Gunning became in 1957 assistant professor, in 1962 associate professor, and in 1966 professor. He was a visiting professor in São Paulo in 1958, Cambridge in 1959/60, Munich in 1967, Oxford in 1968, Boulder in 1970, and Los Angeles in 1972.
Gunning is known as the author of important books on functions of several complex variables.
From 1958 to 1961 he was a Sloan Research Fellow. He served as Princeton University's dean of the faculty from 1989 to 1995. In 2003 he received Princeton University's prize for outstanding teaching. For a number of years he was an editor for Princeton University Press and for the Annals of Mathematical Studies. He was also the editor of the collected works of Salomon Bochner. In 1970 he was an invited speaker at the International Mathematical Congress in Nice (Some multivariable problems arising from Riemann surfaces).
Among his doctoral students are Sheldon Katz, Henry Laufer, Richard S. Hamilton, Yum-Tong Siu, and Michael Eastwood.
In 2012 he became a fellow of the American Mathematical Society.
Selected works
Analytic functions of several complex variables. Prentice-Hall 1965.
Lectures on Riemann Surfaces. Princeton University Press 1966.
Lectures on Vector Bundles over Riemann Surfaces. Princeton University Press 1967.
Riemann Surfaces and generalized Theta Functions. Springer, Ergebnisse der Mathematik und ihrer Grenzgebiete, 1976.
On uniformization of complex manifolds: The role of connections, Princeton University Press 1978
Introduction to holomorphic functions of several variables. 3 vols., Wadsworth and Brooks/Cole, 1990.
An Introduction to Analysis, Princeton University Press, 2018.
References
External links
Robert Gunning's homepage
1931 births
Living people
20th-century American mathematicians
21st-century American mathematicians
Fellows of the American Mathematical Society
University of Colorado alumni
Princeton University alumni
Princeton University faculty
Complex analysts
Sloan Research Fellows
People from Longmont, Colorado
Mathematicians from Colorado |
https://en.wikipedia.org/wiki/Bose%E2%80%93Mesner%20algebra | In mathematics, a Bose–Mesner algebra is a special set of matrices which arise from a combinatorial structure known as an association scheme, together with the usual set of rules for combining (forming the products of) those matrices, such that they form an associative algebra, or, more precisely, a unitary commutative algebra. Among these rules are:
the result of a product is also within the set of matrices,
there is an identity matrix in the set, and
taking products is commutative.
Bose–Mesner algebras have applications in physics to spin models, and in statistics to the design of experiments. They are named for R. C. Bose and Dale Marsh Mesner.
Definition
Let X be a set of v elements. Consider a partition of the 2-element subsets of X into n non-empty subsets, R1, ..., Rn such that:
given an , the number of such that depends only on i (and not on x). This number will be denoted by vi, and
given with , the number of such that and depends only on i,j and k (and not on x and y). This number will be denoted by .
This structure is enhanced by adding all pairs of repeated elements of X and collecting them in a subset R0. This enhancement permits the parameters i, j, and k to take on the value of zero, and lets some of x,y or z be equal.
A set with such an enhanced partition is called an association scheme. One may view an association scheme as a partition of the edges of a complete graph (with vertex set X) into n classes, often thought of as color classes. In this representation, there is a loop at each vertex and all the loops receive the same 0th color.
The association scheme can also be represented algebraically. Consider the matrices Di defined by:
Let be the vector space consisting of all matrices , with complex.
The definition of an association scheme is equivalent to saying that the are v × v (0,1)-matrices which satisfy
is symmetric,
(the all-ones matrix),
The (x,y)-th entry of the left side of 4. is the number of two colored paths of length two joining x and y (using "colors" i and j) in the graph. Note that the rows and columns of contain 1s:
From 1., these matrices are symmetric. From 2., are linearly independent, and the dimension of is . From 4., is closed under multiplication, and multiplication is always associative. This associative commutative algebra is called the Bose–Mesner algebra of the association scheme. Since the matrices in are symmetric and commute with each other, they can be simultaneously diagonalized. This means that there is a matrix such that to each there is a diagonal matrix with . This means that is semi-simple and has a unique basis of primitive idempotents . These are complex n × n matrices satisfying
The Bose–Mesner algebra has two distinguished bases: the basis consisting of the adjacency matrices , and the basis consisting of the irreducible idempotent matrices . By definition, there exist well-defined complex numbers such that
and
The p-numbers , |
https://en.wikipedia.org/wiki/Johnson%20scheme | In mathematics, the Johnson scheme, named after Selmer M. Johnson, is also known as the triangular association scheme. It consists of the set of all binary vectors X of length ℓ and weight n, such that . Two vectors x, y ∈ X are called ith associates if dist(x, y) = 2i for i = 0, 1, ..., n. The eigenvalues are given by
where
and Ek(x) is an Eberlein polynomial defined by
References
Combinatorics |
https://en.wikipedia.org/wiki/Directed%20algebraic%20topology | In mathematics, directed algebraic topology is a refinement of algebraic topology for directed spaces, topological spaces and their combinatorial counterparts equipped with some notion of direction. Some common examples of directed spaces are spacetimes and simplicial sets. The basic goal is to find algebraic invariants that classify directed spaces up to directed analogues of homotopy equivalence. For example, homotopy groups and fundamental of spaces generalize to homotopy monoids and fundamental of directed spaces. Directed algebraic topology, like algebraic topology, is motivated by the need to describe qualitative properties of complex systems in terms of algebraic properties of state spaces, which are often directed by time. Thus directed algebraic topology finds applications in concurrency (computer science), network traffic control, general relativity, noncommutative geometry, rewriting theory, and biological systems.
Directed spaces
Many mathematical definitions have been proposed to formalise the notion of directed space. E. W. Dijkstra introduced a simple dialect to deal with semaphores, the so-called 'PV language', and to provide each PV program an abstract model: its 'geometric semantics'. Any such model admits a natural partially ordered space (or pospace) structure i.e. a topology and a partial order. The points of the model should be thought of as the states of the program and the partial order as the 'causality' relation between states. Following this approach, the directed paths over the model i.e. the monotonic continuous paths, represent the execution traces of the program. From the computer science point of view, however, the resulting pospaces have a severe drawback. Because partial orders are by definition antisymmetric, their only directed loops i.e. directed paths which end where they start, are the constant loops.
Inspired by smooth manifolds, L. Fajstrup, E. Goubault, and M. Raussen use the sheaf-theoretic approach to define local pospaces. Roughly speaking, a local pospace is a topological space together with an open covering whose elements are endowed with a partial order. Given two elements U and V of the covering, it is required that the partial orders on U and V match on the intersection. Though local pospaces allow directed loops, they form a category whose colimits—when they exist—may be rather ill-behaved.
Noting that the directed paths of a (local) pospace appear as a by-product of the (local) partial order—even though they themselves contain most of the relevant information about direction—Marco Grandis defines d-spaces as topological spaces endowed with a collection of paths, whose members are said to be directed, such that any constant path is directed, the concatenation of two directed paths is still directed, and any subpath of a directed path is directed. D-spaces admit non-constant directed loops and form a category enjoying properties similar to the ones enjoyed by the category of topological |
https://en.wikipedia.org/wiki/Special%20classes%20of%20semigroups | In mathematics, a semigroup is a nonempty set together with an associative binary operation. A special class of semigroups is a class of semigroups satisfying additional properties or conditions. Thus the class of commutative semigroups consists of all those semigroups in which the binary operation satisfies the commutativity property that ab = ba for all elements a and b in the semigroup.
The class of finite semigroups consists of those semigroups for which the underlying set has finite cardinality. Members of the class of Brandt semigroups are required to satisfy not just one condition but a set of additional properties. A large collection of special classes of semigroups have been defined though not all of them have been studied equally intensively.
In the algebraic theory of semigroups, in constructing special classes, attention is focused only on those properties, restrictions and conditions which can be expressed in terms of the binary operations in the semigroups and occasionally on the cardinality and similar properties of subsets of the underlying set. The underlying sets are not assumed to carry any other mathematical structures like order or topology.
As in any algebraic theory, one of the main problems of the theory of semigroups is the classification of all semigroups and a complete description of their structure. In the case of semigroups, since the binary operation is required to satisfy only the associativity property the problem of classification is considered extremely difficult. Descriptions of structures have been obtained for certain special classes of semigroups. For example, the structure of the sets of idempotents of regular semigroups is completely known. Structure descriptions are presented in terms of better known types of semigroups. The best known type of semigroup is the group.
A (necessarily incomplete) list of various special classes of semigroups is presented below. To the extent possible the defining properties are formulated in terms of the binary operations in the semigroups. The references point to the locations from where the defining properties are sourced.
Notations
In describing the defining properties of the various special classes of semigroups, the following notational conventions are adopted.
For example, the definition xab = xba should be read as:
There exists x an element of the semigroup such that, for each a and b in the semigroup, xab and xba are equal.
List of special classes of semigroups
The third column states whether this set of semigroups forms a variety. And whether the set of finite semigroups of this special class forms a variety of finite semigroups. Note that if this set is a variety, its set of finite elements is automatically a variety of finite semigroups.
References
Algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/Perfect%20ring | In the area of abstract algebra known as ring theory, a left perfect ring is a type of ring over which all left modules have projective covers. The right case is defined by analogy, and the condition is not left-right symmetric; that is, there exist rings which are perfect on one side but not the other. Perfect rings were introduced in Bass's book.
A semiperfect ring is a ring over which every finitely generated left module has a projective cover. This property is left-right symmetric.
Perfect ring
Definitions
The following equivalent definitions of a left perfect ring R are found in Aderson and Fuller:
Every left R-module has a projective cover.
R/J(R) is semisimple and J(R) is left T-nilpotent (that is, for every infinite sequence of elements of J(R) there is an n such that the product of first n terms are zero), where J(R) is the Jacobson radical of R.
(Bass' Theorem P) R satisfies the descending chain condition on principal right ideals. (There is no mistake; this condition on right principal ideals is equivalent to the ring being left perfect.)
Every flat left R-module is projective.
R/J(R) is semisimple and every non-zero left R-module contains a maximal submodule.
R contains no infinite orthogonal set of idempotents, and every non-zero right R-module contains a minimal submodule.
Examples
Right or left Artinian rings, and semiprimary rings are known to be right-and-left perfect.
The following is an example (due to Bass) of a local ring which is right but not left perfect. Let F be a field, and consider a certain ring of infinite matrices over F.
Take the set of infinite matrices with entries indexed by , and which have only finitely many nonzero entries, all of them above the diagonal, and denote this set by . Also take the matrix with all 1's on the diagonal, and form the set
It can be shown that R is a ring with identity, whose Jacobson radical is J. Furthermore R/J is a field, so that R is local, and R is right but not left perfect.
Properties
For a left perfect ring R:
From the equivalences above, every left R-module has a maximal submodule and a projective cover, and the flat left R-modules coincide with the projective left modules.
An analogue of the Baer's criterion holds for projective modules.
Semiperfect ring
Definition
Let R be ring. Then R is semiperfect if any of the following equivalent conditions hold:
R/J(R) is semisimple and idempotents lift modulo J(R), where J(R) is the Jacobson radical of R.
R has a complete orthogonal set e1, ..., en of idempotents with each eiRei a local ring.
Every simple left (right) R-module has a projective cover.
Every finitely generated left (right) R-module has a projective cover.
The category of finitely generated projective -modules is Krull-Schmidt.
Examples
Examples of semiperfect rings include:
Left (right) perfect rings.
Local rings.
Kaplansky's theorem on projective modules
Left (right) Artinian rings.
Finite dimensional k-algebras.
Properties
Since a |
https://en.wikipedia.org/wiki/ATLAS%20of%20Finite%20Groups | The ATLAS of Finite Groups, often simply known as the ATLAS, is a group theory book by John Horton Conway, Robert Turner Curtis, Simon Phillips Norton, Richard Alan Parker and Robert Arnott Wilson (with computational assistance from J. G. Thackray), published in December 1985 by Oxford University Press and reprinted with corrections in 2003 (). It lists basic information about 93 finite simple groups, the information being generally: its order, Schur multiplier, outer automorphism group, various constructions (such as presentations), conjugacy classes of maximal subgroups (with characters group action they define), and, most importantly, character tables (including power maps on the conjugacy classes) of the group itself and bicyclic extensions given by stem extensions and automorphism groups. In certain cases (such as for the Chevalley groups ), the character table is not listed and only basic information is given.
The ATLAS is a recognizable large format book (sized 420mm by 300mm) with a cherry red cardboard cover and spiral binding. The names of the authors, all six letters long, with initials for the first and second letter, are printed on the cover in the form of an array which evokes the idea of a character table.
The ATLAS is being continued in the form of an electronic database, the ATLAS of Finite Group Representations.
Finite groups
Mathematics books
John Horton Conway |
https://en.wikipedia.org/wiki/Completely%20regular%20semigroup | In mathematics, a completely regular semigroup is a semigroup in which every element is in some subgroup of the semigroup. The class of completely regular semigroups forms an important subclass of the class of regular semigroups, the class of inverse semigroups being another such subclass. Alfred H. Clifford was the first to publish a major paper on completely regular semigroups though he used the terminology "semigroups admitting relative inverses" to refer to such semigroups. The name "completely regular semigroup" stems from Lyapin's book on semigroups. In the Russian literature, completely regular semigroups are often called "Clifford semigroups".
In the English literature, the name "Clifford semigroup" is used synonymously to "inverse Clifford semigroup", and refers to a completely regular inverse semigroup.
In a completely regular semigroup, each Green H-class is a group and the semigroup is the union of these groups. Hence completely regular semigroups are also referred to as "unions of groups". Epigroups generalize this notion and their class includes all completely regular semigroups.
Examples
"While there is an abundance of natural examples of inverse semigroups, for completely regular semigroups the examples (beyond completely simple semigroups) are mostly artificially constructed: the minimum ideal of a
finite semigroup is completely simple, and the various relatively free completely regular semigroups are the other more or less natural examples."
See also
Special classes of semigroups
References
Algebraic structures
Semigroup theory |
https://en.wikipedia.org/wiki/M.%20Yousuff%20Hussaini | Mohammed Yousuff Hussaini is an Indian born American applied mathematician. He is the Sir James Lighthill Professor of Mathematics and Computational Science & Engineering at the Florida State University, United States. Hussaini is also the holder of the TMC Eminent Scholar Chair in High Performance Computing at FSU. He is widely known for his research in scientific computation, particularly in the field of computational fluid dynamics (CFD) and Control and optimization. Hussaini co-authored the popular book Spectral Methods in Fluid Dynamics with Claudio Canuto, Alfio Quarteroni, and Thomas Zang. He is the editor-in-chief of the journal Theoretical and Computational Fluid Dynamics.
Education and career
Hussaini received his bachelor's and master's degrees in mathematics and physics from the University of Madras, India, and a Ph.D. from the University of California, Berkeley in 1970. Hussaini began his career at the NASA Langley Research Center where he was the Director of the Institute for Computer Applications in Science and Engineering (ICASE). In 1996, he joined Department of Mathematics, Florida State University (FSU) as a Professor of Mathematics. Currently, he is the holder of the TMC Eminent Scholar Chair in High Performance Computing at FSU.
Honors and awards
In 1997, Hussaini was nominated as Fellow of American Physical Society for his "scientific leadership and innovative and pioneering research in the theory and application of computational fluid dynamics, particularly spectral methods, to problems in transition, compressible turbulence, shock-turbulence interaction, and aeroacoustics."
In 2008, Hussaini was elected as Fellow of American Society of Mechanical Engineers and Fellow of American Institute of Aeronautics and Astronautics, in recognition of "professional distinction and notable and valuable contributions made to the arts, sciences, and technology of aeronautics and astronautics".
Books
Hussaini has co-authored three monographs and edited or co-edited 20 books in various fields including computational fluid dynamics, turbulent flows, wavelets, flow instability, spectral methods, combustion, and finite element methods:
Monographs
Spectral Methods in Fluid Dynamics, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 1987.
Spectral Methods: Fundamentals in Single Domains, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 2006.
Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, with C. Canuto, A. Quarteroni and T. A. Zang, Springer-Verlag, 2007.
Book chapters
Immersed Boundary Methods for Two-Fluid Flows, with G. Tryggvason and M. Sussman, Chapter in "Computational Methods for Multiphase Flow", edited by A. Prosperetti and G. Tryggvason, Cambridge University Press, 2007.
Books edited
Spectral Methods for Partial Differential Equations, with D. Gottlieb and R. G. Voigt, SIAM, Philadelphia, 1984.
Theoretical Approaches to Turbulence, with D. L. Dwoyer and R. G. V |
https://en.wikipedia.org/wiki/Ringel%E2%80%93Hall%20algebra | In mathematics, a Ringel–Hall algebra is a generalization of the Hall algebra, studied by . It has a basis of equivalence classes of objects of an abelian category, and the structure constants for this basis are related to the numbers of extensions of objects in the category.
References
External links
Representation theory
Lie algebras
Symmetric functions |
https://en.wikipedia.org/wiki/Transversality%20theorem | In differential topology, the transversality theorem, also known as the Thom transversality theorem after French mathematician René Thom, is a major result that describes the transverse intersection properties of a smooth family of smooth maps. It says that transversality is a generic property: any smooth map , may be deformed by an arbitrary small amount into a map that is transverse to a given submanifold . Together with the Pontryagin–Thom construction, it is the technical heart of cobordism theory, and the starting point for surgery theory. The finite-dimensional version of the transversality theorem is also a very useful tool for establishing the genericity of a property which is dependent on a finite number of real parameters and which is expressible using a system of nonlinear equations. This can be extended to an infinite-dimensional parametrization using the infinite-dimensional version of the transversality theorem.
Finite-dimensional version
Previous definitions
Let be a smooth map between smooth manifolds, and let be a submanifold of . We say that is transverse to , denoted as , if and only if for every we have that
.
An important result about transversality states that if a smooth map is transverse to , then is a regular submanifold of .
If is a manifold with boundary, then we can define the restriction of the map to the boundary, as . The map is smooth, and it allows us to state an extension of the previous result: if both and , then is a regular submanifold of with boundary, and
.
Parametric transversality theorem
Consider the map and define . This generates a family of mappings . We require that the family vary smoothly by assuming to be a (smooth) manifold and to be smooth.
The statement of the parametric transversality theorem is:
Suppose that is a smooth map of manifolds, where only has boundary, and let be any submanifold of without boundary. If both and are transverse to , then for almost every , both and are transverse to .
More general transversality theorems
The parametric transversality theorem above is sufficient for many elementary applications (see the book by Guillemin and Pollack).
There are more powerful statements (collectively known as transversality theorems) that imply the parametric transversality theorem and are needed for more advanced applications.
Informally, the "transversality theorem" states that the set of mappings that are transverse to a given submanifold is a dense open (or, in some cases, only a dense ) subset of the set of mappings. To make such a statement precise, it is necessary to define the space of mappings under consideration, and what is the topology in it. There are several possibilities; see the book by Hirsch.
What is usually understood by Thom's transversality theorem is a more powerful statement about jet transversality. See the books by Hirsch and by Golubitsky and Guillemin. The original reference is Thom, Bol. Soc. Mat. Mexicana (2) 1 (195 |
https://en.wikipedia.org/wiki/Deir%20Ali | Deir Ali () is a small town in southern Syria, administratively part of the Rif Dimashq Governorate. According to the Syria Central Bureau of Statistics, Deir Ali had a population of 4,368 in the 2004 census. Its inhabitants are predominantly members of the Druze community.
History
The town was historically a village known as Lebaba, and contains the archaeological remains of a Marcionite church. These include an inscription dated to 318 CE, which is the oldest known surviving inscribed reference, anywhere, to Jesus:
The meeting-house of the Marcionites, in the village of Lebaba, of the Lord and Saviour Jesus the Good -Erected by the forethought of Paul a presbyter, in the year 630 Seleucid era This gained the attention of the First Bible Network (FBN.)
In 1838, Eli Smith noted Deir Ali's population as being Druze.
The Arab Gas Pipeline passes through the area and supplies gas to a modern power station (estimated cost 250 million euros) in the town; the pipeline junction at the power station links the power grids of Egypt, Syria, and Jordan.
See also
Druze in Syria
References
Bibliography
Populated places in Markaz Rif Dimashq District
Roman sites in Syria
Druze communities in Syria
Marcionism |
https://en.wikipedia.org/wiki/Calabi%E2%80%93Eckmann%20manifold | In complex geometry, a part of mathematics, a Calabi–Eckmann manifold (or, often, Calabi–Eckmann space), named after Eugenio Calabi and Beno Eckmann, is a complex, homogeneous, non-Kähler manifold, homeomorphic to a product of two odd-dimensional spheres of dimension ≥ 3.
The Calabi–Eckmann manifold is constructed as follows. Consider the space , where , equipped with an action of the group :
where is a fixed complex number. It is easy to check that this action is free and proper, and the corresponding orbit space M is homeomorphic to . Since M is a quotient space of a holomorphic action, it is also a complex manifold. It is obviously homogeneous, with a transitive holomorphic action of
A Calabi–Eckmann manifold M is non-Kähler, because . It is the simplest example of a non-Kähler
manifold which is simply connected (in dimension 2, all simply connected compact complex manifolds are Kähler).
The natural projection
induces a holomorphic map from the corresponding Calabi–Eckmann manifold M to . The fiber of this map is an elliptic curve T, obtained as a quotient of by the lattice . This makes M into a principal T-bundle.
Calabi and Eckmann discovered these manifolds in 1953.
Notes
Complex manifolds |
https://en.wikipedia.org/wiki/Calkin%E2%80%93Wilf%20tree | In number theory, the Calkin–Wilf tree is a tree in which the vertices correspond one-to-one to the positive rational numbers. The tree is rooted at the number 1, and any rational number expressed in simplest terms as the fraction has as its two children the numbers and . Every positive rational number appears exactly once in the tree. It is named after Neil Calkin and Herbert Wilf, but appears in other works including Kepler's Harmonices Mundi.
The sequence of rational numbers in a breadth-first traversal of the Calkin–Wilf tree is known as the Calkin–Wilf sequence. Its sequence of numerators (or, offset by one, denominators) is Stern's diatomic series, and can be computed by the fusc function.
History
The Calkin–Wilf tree is named after Neil Calkin and Herbert Wilf, who considered it in a 2000 paper. In a 1997 paper, Jean Berstel and Aldo de Luca called the same tree the Raney tree, since they drew some ideas from a 1973 paper by George N. Raney. Stern's diatomic series was formulated much earlier by Moritz Abraham Stern, a 19th-century German mathematician who also invented the closely related Stern–Brocot tree. Even earlier, a similar tree (including only the fractions between 0 and 1) appears in Kepler's Harmonices Mundi (1619).
Definition and structure
The Calkin–Wilf tree may be defined as a directed graph in which each positive rational number occurs as a vertex and has one outgoing edge to another vertex, its parent, except for the root of the tree, the number 1, which has no parent.
The parent of any rational number can be determined after placing the number into simplest terms, as a fraction for which greatest common divisor of and is 1. If , the parent of is ; if , the parent of is . Thus, in either case, the parent is a fraction with a smaller sum of numerator and denominator, so repeated reduction of this type must eventually reach the number 1. As a graph with one outgoing edge per vertex and one root reachable by all other vertices, the Calkin–Wilf tree must indeed be a tree.
The children of any vertex in the Calkin–Wilf tree may be computed by inverting the formula for the parents of a vertex. Each vertex has one child whose value is less than 1, , because of course . Similarly, each vertex has one child whose value is greater than 1, .
As each vertex has two children, the Calkin–Wilf tree is a binary tree. However, it is not a binary search tree: its inorder does not coincide with the sorted order of its vertices. However, it is closely related to a different binary search tree on the same set of vertices, the Stern–Brocot tree: the vertices at each level of the two trees coincide, and are related to each other by a bit-reversal permutation.
Breadth first traversal
The Calkin–Wilf sequence is the sequence of rational numbers generated by a breadth-first traversal of the Calkin–Wilf tree,
, , , , , , , , , , , , , , ….
Because the Calkin–Wilf tree contains every positive rational number exactly once, so does t |
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