|
Logarithmically Concave Sequence
In mathematics, a sequence = of nonnegative real numbers is called a logarithmically concave sequence, or a log-concave sequence for short, if holds for . ''Remark:'' some authors (explicitly or not) add two further conditions in the definition of log-concave sequences: * is non-negative * has no internal zeros; in other words, the support of is an interval of . These conditions mirror the ones required for log-concave functions. Sequences that fulfill the three conditions are also called Pólya Frequency sequences of order 2 (PF2 sequences). Refer to chapter 2 of for a discussion on the two notions. For instance, the sequence satisfies the concavity inequalities but not the internal zeros condition. Examples of log-concave sequences are given by the binomial coefficients along any row of Pascal's triangle and the elementary symmetric means of a finite sequence of real numbers. References * See also *Unimodality *Logarithmically concave function In convex analys ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle 5
Pascal, Pascal's or PASCAL may refer to: People and fictional characters * Pascal (given name), including a list of people with the name * Pascal (surname), including a list of people and fictional characters with the name ** Blaise Pascal, French mathematician, physicist, inventor, philosopher, writer and theologian Places * Pascal (crater), a lunar crater * Pascal Island (Antarctica) * Pascal Island (Western Australia) Science and technology * Pascal (unit), the SI unit of pressure * Pascal (programming language), a programming language developed by Niklaus Wirth ** Microsoft Pascal **Turbo Pascal * PASCAL (database), a bibliographic database maintained by the Institute of Scientific and Technical Information * Pascal (microarchitecture), codename for a microarchitecture developed by Nvidia Other uses * (1895–1911) * (1931–1942) * Pascal and Maximus, fictional characters in ''Tangled'' * Pascal blanc, a French white wine grape * Pascal College, secondary education ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Support (mathematics)
In mathematics, the support of a real-valued function f is the subset of the function domain of elements that are not mapped to zero. If the domain of f is a topological space, then the support of f is instead defined as the smallest closed set containing all points not mapped to zero. This concept is used widely in mathematical analysis. Formulation Suppose that f : X \to \R is a real-valued function whose domain is an arbitrary set X. The of f, written \operatorname(f), is the set of points in X where f is non-zero: \operatorname(f) = \. The support of f is the smallest subset of X with the property that f is zero on the subset's complement. If f(x) = 0 for all but a finite number of points x \in X, then f is said to have . If the set X has an additional structure (for example, a topology), then the support of f is defined in an analogous way as the smallest subset of X of an appropriate type such that f vanishes in an appropriate sense on its complement. The notion of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmically Concave Function
In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^ for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) for all and . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is '' log-convex'' if it satisfies the reverse inequality : f(\theta x + (1 - \theta) y) \leq f(x)^ f(y)^ for all and . Properties * A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. * Every concave function that is nonnegative on it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
American Mathematical Society
The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, advocacy and other programs. The society is one of the four parts of the Joint Policy Board for Mathematics and a member of the Conference Board of the Mathematical Sciences. History The AMS was founded in 1888 as the New York Mathematical Society, the brainchild of Thomas Fiske, who was impressed by the London Mathematical Society on a visit to England. John Howard Van Amringe became the first president while Fiske became secretary. The society soon decided to publish a journal, but ran into some resistance over concerns about competing with the '' American Journal of Mathematics''. The result was the ''Bulletin of the American Mathematical Society'', with Fiske as editor-in-chief. The de facto journal, as intended, was influentia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binomial Coefficient
In mathematics, the binomial coefficients are the positive integers that occur as coefficients in the binomial theorem. Commonly, a binomial coefficient is indexed by a pair of integers and is written \tbinom. It is the coefficient of the term in the polynomial expansion of the binomial power ; this coefficient can be computed by the multiplicative formula : \binom nk = \frac, which using factorial notation can be compactly expressed as : \binom = \frac. For example, the fourth power of is : \begin (1 + x)^4 &= \tbinom x^0 + \tbinom x^1 + \tbinom x^2 + \tbinom x^3 + \tbinom x^4 \\ &= 1 + 4x + 6 x^2 + 4x^3 + x^4, \end and the binomial coefficient \tbinom =\tfrac = \tfrac = 6 is the coefficient of the term. Arranging the numbers \tbinom, \tbinom, \ldots, \tbinom in successive rows for gives a triangular array called Pascal's triangle, satisfying the recurrence relation : \binom = \binom + \binom . The binomial coefficients occur in many areas of mathematics, and espe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pascal's Triangle
In mathematics, Pascal's triangle is an infinite triangular array of the binomial coefficients which play a crucial role in probability theory, combinatorics, and algebra. In much of the Western world, it is named after the French mathematician Blaise Pascal, although other mathematicians studied it centuries before him in Persia, India, China, Germany, and Italy. The rows of Pascal's triangle are conventionally enumerated starting with row n = 0 at the top (the 0th row). The entries in each row are numbered from the left beginning with k = 0 and are usually staggered relative to the numbers in the adjacent rows. The triangle may be constructed in the following manner: In row 0 (the topmost row), there is a unique nonzero entry 1. Each entry of each subsequent row is constructed by adding the number above and to the left with the number above and to the right, treating blank entries as 0. For example, the initial number of row 1 (or any other row) is 1 (the sum of 0 and 1), whereas ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newton's Inequalities
In mathematics, the Newton inequalities are named after Isaac Newton. Suppose ''a''1, ''a''2, ..., ''a''''n'' are non-negative real numbers and let e_k denote the ''k''th elementary symmetric polynomial in ''a''1, ''a''2, ..., ''a''''n''. Then the elementary symmetric means, given by :S_k = \frac, satisfy the inequality :S_S_ \le S_k^2. Equality holds if and only if all the numbers ''a''''i'' are equal. It can be seen that ''S''1 is the arithmetic mean, and ''S''''n'' is the ''n''-th power of the geometric mean In mathematics, the geometric mean is a mean or average which indicates a central tendency of a finite collection of positive real numbers by using the product of their values (as opposed to the arithmetic mean which uses their sum). The geometri .... See also * Maclaurin's inequality References * * *D.S. Bernstein ''Matrix Mathematics: Theory, Facts, and Formulas'' (2009 Princeton) p. 55 * * * {{Isaac Newton Isaac Newton ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Annals Of The New York Academy Of Sciences
The ''Annals of the New York Academy of Sciences'' is an academic journal published by Wiley-Blackwell on behalf of the New York Academy of Sciences. It is one of the oldest science journals still being published, having been founded in 1823. The editor-in-chief is Douglas Braaten. Each issue is of substantial length and explores a single topic with a multidisciplinary approach. A review published on '' Ulrichsweb'' states the scope is enormous and describes the journal as highly respected and the articles as penetrating. Abstracting and indexing The journal is abstracted and indexed in: According to the ''Journal Citation Reports'', the journal has a 2019 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 4.728, ranking it 13th out of 71 journals in the categor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unimodality
In mathematics, unimodality means possessing a unique mode. More generally, unimodality means there is only a single highest value, somehow defined, of some mathematical object. Unimodal probability distribution In statistics, a unimodal probability distribution or unimodal distribution is a probability distribution which has a single peak. The term "mode" in this context refers to any peak of the distribution, not just to the strict definition of mode which is usual in statistics. If there is a single mode, the distribution function is called "unimodal". If it has more modes it is "bimodal" (2), "trimodal" (3), etc., or in general, "multimodal". Figure 1 illustrates normal distributions, which are unimodal. Other examples of unimodal distributions include Cauchy distribution, Student's ''t''-distribution, chi-squared distribution and exponential distribution. Among discrete distributions, the binomial distribution and Poisson distribution can be seen as unimodal, thoug ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmically Concave Function
In convex analysis, a non-negative function is logarithmically concave (or log-concave for short) if its domain is a convex set, and if it satisfies the inequality : f(\theta x + (1 - \theta) y) \geq f(x)^ f(y)^ for all and . If is strictly positive, this is equivalent to saying that the logarithm of the function, , is concave; that is, : \log f(\theta x + (1 - \theta) y) \geq \theta \log f(x) + (1-\theta) \log f(y) for all and . Examples of log-concave functions are the 0-1 indicator functions of convex sets (which requires the more flexible definition), and the Gaussian function. Similarly, a function is '' log-convex'' if it satisfies the reverse inequality : f(\theta x + (1 - \theta) y) \leq f(x)^ f(y)^ for all and . Properties * A log-concave function is also quasi-concave. This follows from the fact that the logarithm is monotone implying that the superlevel sets of this function are convex. * Every concave function that is nonnegative on it ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithmically Concave Measure
In mathematics, a Borel measure ''μ'' on ''n''-dimensional Euclidean space \mathbb^ is called logarithmically concave (or log-concave for short) if, for any compact subsets ''A'' and ''B'' of \mathbb^ and 0 < ''λ'' < 1, one has : \mu(\lambda A + (1-\lambda) B) \geq \mu(A)^\lambda \mu(B)^, where ''λ'' ''A'' + (1 − ''λ'') ''B'' denotes the Minkowski sum of ''λ'' ''A'' and (1 − ''λ'') ''B''. Examples The Brunn–Minkowski inequality asserts that the Lebesgue measure is log-concave. The restriction of the Lebesgue measure to any convex set is also log-concave. By a theorem of Borell, a probability measure on R^d is log-concave if and only if it has a density with respect to the Lebesgue measure on some affine hyperplane, and this density is a logarithmically concave function. Thus, any Gaussian measure is log-concave. The Prékopa–Leindler inequality shows that a convolution of log-concave m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
