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Leonard Carlitz
Leonard Carlitz (December 26, 1907 – September 17, 1999) was an American mathematician. Carlitz supervised 44 doctorates at Duke University and published over 770 papers. Chronology * 1907 Born Philadelphia, PA, USA * 1927 BA, University of Pennsylvania * 1930 PhD, University of Pennsylvania, 1930 under Howard Mitchell, who had studied under Oswald Veblen at Princeton * 1930–31 at Caltech with E. T. Bell * 1931 married Clara Skaler * 1931–32 at Cambridge with G. H. Hardy * 1932 Joined the faculty of Duke University where he served for 45 years * 1938 to 1973 Editorial Board Duke Mathematical Journal (Managing Editor from 1945.) * 1939 Birth of son Michael * 1940 Supervision of his first doctoral student E. F. Canaday, awarded 1940 * 1945 Birth of son Robert * 1964 First James B. Duke Professor in Mathematics * 1977 Supervised his 44th and last doctoral student, Jo Ann Lutz, awarded 1977 * 1977 Retired * 1990 Death of wife Clara, after 59 years of marriage * 1999 Sep ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maillet's Determinant
In mathematics, Maillet's determinant ''D''''p'' is the determinant of the matrix introduced by whose entries are ''R''(''s/r'') for ''s'',''r'' = 1, 2, ..., (''p'' – 1)/2 ∈ Z/''p''Z for an odd prime ''p'', where and ''R''(''a'') is the least positive residue of ''a'' mod ''p'' . calculated the determinant ''D''''p'' for ''p'' = 3, 5, 7, 11, 13 and found that in these cases it is given by (–''p'')(''p'' – 3)/2, and conjectured that it is given by this formula in general. showed that this conjecture is incorrect; the determinant in general is given by ''D''''p'' = (–''p'')(''p'' – 3)/2''h''−, where ''h''− is the first factor of the class number of the cyclotomic field In number theory, a cyclotomic field is a number field obtained by adjoining a complex root of unity to , the field of rational numbers. Cyclotomic fields played a crucial role in the development of modern algebra and number theory because of th . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Al-Salam–Carlitz Polynomials
In mathematics, Al-Salam–Carlitz polynomials ''U''(''x'';''q'') and ''V''(''x'';''q'') are two families of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by . give a detailed list of their properties. Definition The Al-Salam–Carlitz polynomials are given in terms of basic hypergeometric function In mathematics, basic hypergeometric series, or ''q''-hypergeometric series, are ''q''-analogue generalizations of generalized hypergeometric series, and are in turn generalized by elliptic hypergeometric series. A series ''x'n'' is called h ...s by : U_n^(x;q) = (-a)^nq^_2\phi_1(q^, x^;0;q,qx/a) : V_n^(x;q) = (-a)^nq^_2\phi_0(q^, x;-;q,q^n/a) References * * Further reading * Wang, M. (2009). q-integral representation of the Al-Salam–Carlitz polynomials. Applied Mathematics Letters, 22(6), 943-945. * Askey, R., & Suslov, S. K. (1993). The q-harmonic oscillator and the Al-Salam and Carlitz polynomials. Letters in Mathematical Physics, 29(2), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Caltech
The California Institute of Technology (branded as Caltech or CIT)The university itself only spells its short form as "Caltech"; the institution considers other spellings such a"Cal Tech" and "CalTech" incorrect. The institute is also occasionally referred to as "CIT", most notably in its alma mater, but this is uncommon. is a private university, private research university in Pasadena, California. Caltech is ranked among the best and most selective academic institutions in the world, and with an enrollment of approximately 2400 students (acceptance rate of only 5.7%), it is one of the world's most selective universities. The university is known for its strength in science and engineering, and is among a small group of Institute of Technology (United States), institutes of technology in the United States which is primarily devoted to the instruction of pure and applied sciences. The institution was founded as a preparatory and vocational school by Amos G. Throop in 1891 and began ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlitz Polynomial (other)
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In mathematics, Carlitz polynomial, named for Leonard Carlitz, may refer to: *Al-Salam–Carlitz polynomials *Tricomi–Carlitz polynomials In mathematics, the Tricomi–Carlitz polynomials or (Carlitz–)Karlin–McGregor polynomials are polynomials studied by and and , related to random walks on the positive integers. They are given in terms of Laguerre polynomials In mathematics, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Carlitz Exponential
In mathematics, the Carlitz exponential is a characteristic ''p'' analogue to the usual exponential function studied in real and complex analysis. It is used in the definition of the Carlitz module – an example of a Drinfeld module. Definition We work over the polynomial ring F''q'' 'T''of one variable over a finite field F''q'' with ''q'' elements. The completion C∞ of an algebraic closure of the field F''q''((''T''−1)) of formal Laurent series in ''T''−1 will be useful. It is a complete and algebraically closed field. First we need analogues to the factorials, which appear in the definition of the usual exponential function. For ''i'' > 0 we define : := T^ - T, \, :D_i := \prod_ and ''D''0 := 1. Note that the usual factorial is inappropriate here, since ''n''! vanishes in F''q'' 'T''unless ''n'' is smaller than the characteristic of F''q'' 'T'' Using this we define the Carlitz exponential ''e''''C'':C∞ → C∞ by the convergent sum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bateman Polynomials
In mathematics, the Bateman polynomials are a family ''F''''n'' of orthogonal polynomials introduced by . The Bateman–Pasternack polynomials are a generalization introduced by . Bateman polynomials can be defined by the relation :F_n\left(\frac\right)\operatorname(x) = \operatorname(x)P_n(\tanh(x)). where ''P''''n'' is a Legendre polynomial. In terms of generalized hypergeometric functions, they are given by :F_n(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+1)\\ 1,~1 \end; 1\right). generalized the Bateman polynomials to polynomials ''F'' with :F_n^m\left(\frac\right)\operatorname^(x) = \operatorname^(x)P_n(\tanh(x)) These generalized polynomials also have a representation in terms of generalized hypergeometric functions, namely :F_n^m(x)=_3F_2\left(\begin-n,~n+1,~\tfrac12(x+m+1)\\ 1,~m+1 \end; 1\right). showed that the polynomials ''Q''''n'' studied by , see Touchard polynomials, are the same as Bateman polynomials up to a change of variable: more precisely : Q_n(x)=(-1)^n2^nn! ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Acta Arithmetica
''Acta Arithmetica'' is a scientific journal of mathematics publishing papers on number theory. It was established in 1935 by Salomon Lubelski and Arnold Walfisz. The journal is published by the Institute of Mathematics of the Polish Academy of Sciences The Institute of Mathematics of the Polish Academy of Sciences is a research institute of the Polish Academy of Sciences.Online archives (Library of Science, Issues: 1935–2000) 1935 establishments in Poland [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Daqing Wan
Daqing Wan (born 1964 in China) is a Chinese mathematician working in the United States. He received his Ph.D. from the University of Washington in Seattle in 1991, under the direction of Neal Koblitz. Since 1997, he has been on the faculty of mathematics at the University of California at Irvine; he has also held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, Pennsylvania State University, the University of Rennes, the Mathematical Sciences Research Institute in Berkeley, California, and the Chinese Academy of Sciences in Beijing. His primary interests include number theory and arithmetic algebraic geometry, particularly zeta functions over finite fields. He is known for his proof of Dwork's conjecture that the p-adic unit root zeta function attached to a family of varieties over a finite field of characteristic p is p-adic meromorphic In the mathematical field of complex analysis, a meromorphic function on an open subset ''D'' of the comple ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bicentric Quadrilateral
In Euclidean geometry, a bicentric quadrilateral is a convex quadrilateral that has both an incircle and a circumcircle. The radii and center of these circles are called ''inradius'' and ''circumradius'', and ''incenter'' and ''circumcenter'' respectively. From the definition it follows that bicentric quadrilaterals have all the properties of both tangential quadrilaterals and cyclic quadrilaterals. Other names for these quadrilaterals are chord-tangent quadrilateral and inscribed and circumscribed quadrilateral. It has also rarely been called a ''double circle quadrilateral'' and ''double scribed quadrilateral''. If two circles, one within the other, are the incircle and the circumcircle of a bicentric quadrilateral, then every point on the circumcircle is the vertex of a bicentric quadrilateral having the same incircle and circumcircle. This is a special case of Poncelet's porism, which was proved by the French mathematician Jean-Victor Poncelet (1788–1867). Special cases ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Polynomials
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series :y_n(x)=\sum_^n\frac\,\left(\frac\right)^k. Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials :\theta_n(x)=x^n\,y_n(1/x)=\sum_^n\frac\,\frac. The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is :y_3(x)=15x^3+15x^2+6x+1 while the third-degree reverse Bessel polynomial is :\theta_3(x)=x^3+6x^2+15x+15. The reverse Bessel polynomial is used in the design of Bessel electronic filters. Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name. :y_n(x)=\,x^\theta_n(1/x)\, :y_n(x)=\sqrt\,e^K_(1/x) :\theta_n(x)=\sqrt\,x^e^K_(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernoulli Numbers
In mathematics, the Bernoulli numbers are a sequence of rational numbers which occur frequently in analysis. The Bernoulli numbers appear in (and can be defined by) the Taylor series expansions of the tangent and hyperbolic tangent functions, in Faulhaber's formula for the sum of ''m''-th powers of the first ''n'' positive integers, in the Euler–Maclaurin formula, and in expressions for certain values of the Riemann zeta function. The values of the first 20 Bernoulli numbers are given in the adjacent table. Two conventions are used in the literature, denoted here by B^_n and B^_n; they differ only for , where B^_1=-1/2 and B^_1=+1/2. For every odd , . For every even , is negative if is divisible by 4 and positive otherwise. The Bernoulli numbers are special values of the Bernoulli polynomials B_n(x), with B^_n=B_n(0) and B^+_n=B_n(1). The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jacob Bernoulli, after whom they are named, and inde ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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