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This is an overview of recent results on the use of 2D Toda $\tau$-functions as generating functions for multiparametric families of weighted Hurwitz numbers. The Bose-Fermi equivalence composed with the characteristic map provides an isomorphism between the zero charge sector of the Fermionic Fock space and the direct sum of the centers of the group algebra of the symmetric groups $S_n$. Specializing the fermionic formula to the case of diagonal group elements gives $\tau$-functions of hypergeometric type, for which the expansion over products of Schur functions is diagonal, with coefficients of {\em content product} type. The corresponding abelian group action on the centre of the $S_n$ group algebra is determined by forming symmetric functions multiplicatively from a weight generating function $G(z)$ and evaluating on the Jucys-Murphy elements of the group algebra. The resulting central elements act diagonally on the basis of orthogonal idempotents and the eigenvalues $r^{G(z)}_\lambda$ are the {\em content product} coefficients appearing in the double Schur function expansion. Both the geometrical meaning of weighted Hurwitz numbers, as weighted sums over $n$-sheeted branched coverings, and the combinatorial one, as weighted enumeration of paths in the Cayley graph of $S_n$ generated by transpositions follow from expansion of the Cauchy-Littlewood generating functions over dual pairs of bases of the algebra of symmetric functions. The coefficients in the resulting $\tau$-function expansion over products of power sum symmetric functions are the weighted Hurwitz numbers. Replacement of the Cauchy-Littlewood generating function by that for Macdonald polynomials provides $(q,t)$-deformations that yield generating functions for quantum weighted Hurwitz numbers.

We prove a conjecture that classifies exceptional numbers. This conjecture arises in two different ways, from cryptography and from coding theory. An odd integer $t\geq 3$ is said to be exceptional if $f(x)=x^t$ is APN (Almost Perfect Nonlinear) over $\mathbb{F}_{2^n}$ for infinitely many values of $n$. Equivalently, $t$ is exceptional if the binary cyclic code of length $2^n-1$ with two zeros $\omega, \omega^t$ has minimum distance 5 for infinitely many values of $n$. The conjecture we prove states that every exceptional number has the form $2^i+1$ or $4^i-2^i+1$.

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

In the paper, the author finds an explicit formula for computing Bell numbers in terms of Kummer confluent hypergeometric functions and Stirling numbers of the second kind.

xvi, 320 p. 24 cm

The purpose of this paper is to contribute to development a general theory of dual-complex numbers. We start by dene the notion of dual-complex and their algebraic properties. In addition, we develop a simple mathematical method based on matrices, simplifying manipulation of dual-complex numbers. Inspired from complex analysis, we generalize the concept of holomorphicity to dual-complex functions. Moreover, a general representation of holomorphic dual-complex functions has been obtained. Finally and as concrete examples, some usual complex functions have been generalized to the algebra of dual-complex numbers.

Book Source: Digital Library of India Item 2015.154739 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T19:40:13Z dc.date.available: 2015-07-06T19:40:13Z dc.date.digitalpublicationdate: 2024-09-03 dc.identifier: CMU dc.identifier.barcode: 1990050020430 dc.identifier.origpath: /data/upload/0020/435 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/154739 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 600 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions Edition 81 dc.rights.holder: D.van Nostrand Co

41 p. 24 cm

Book digitized by Google from the library of the University of California and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of the University of California and uploaded to the Internet Archive by user tpb.

Book digitized by Google from the library of University of California and uploaded to the Internet Archive by user tpb.

6 p.l., [1], 329 (i.e. 337) p. 18 cm

6 p.l., [1], 329 (i.e. 337) p. 18 cm

Book Source: Digital Library of India Item 2015.88267 dc.contributor.author: Baron Von Vega dc.date.accessioned: 2015-06-30T20:40:54Z dc.date.available: 2015-06-30T20:40:54Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1856 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 130049 dc.identifier.origpath: /data7/upload/0173/406 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/88267 dc.description.numberedpages: 575 dc.description.numberedpages: 41 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 616 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Lemcke Amp Buechner. dc.rights: Out_of_copyright dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions

Book Source: Digital Library of India Item 2015.88267 dc.contributor.author: Baron Von Vega dc.date.accessioned: 2015-06-30T20:40:54Z dc.date.available: 2015-06-30T20:40:54Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1856 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 130049 dc.identifier.origpath: /data7/upload/0173/406 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/88267 dc.description.numberedpages: 575 dc.description.numberedpages: 41 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 616 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Lemcke Amp Buechner. dc.rights: Out_of_copyright dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions

Book Source: Digital Library of India Item 2015.88267 dc.contributor.author: Baron Von Vega dc.date.accessioned: 2015-06-30T20:40:54Z dc.date.available: 2015-06-30T20:40:54Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1856 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 130049 dc.identifier.origpath: /data7/upload/0173/406 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/88267 dc.description.numberedpages: 575 dc.description.numberedpages: 41 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 616 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Lemcke Amp Buechner. dc.rights: Out_of_copyright dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions

Book Source: Digital Library of India Item 2015.88267 dc.contributor.author: Baron Von Vega dc.date.accessioned: 2015-06-30T20:40:54Z dc.date.available: 2015-06-30T20:40:54Z dc.date.digitalpublicationdate: 2004-11-05 dc.date.citation: 1856 dc.identifier: RMSC, IIIT-H dc.identifier.barcode: 130049 dc.identifier.origpath: /data7/upload/0173/406 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/88267 dc.description.numberedpages: 575 dc.description.numberedpages: 41 dc.description.scanningcentre: RMSC, IIIT-H dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 616 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Universal Digital Library dc.publisher: Lemcke Amp Buechner. dc.rights: Out_of_copyright dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions

"Symmetric Functions Formed by Systems of Elements of a Finite Algebra and their Connection with Fermat's Quotient and Bernoulli's Numbers" is an article from The Annals of Mathematics, Volume 18 . View more articles from The Annals of Mathematics . View this article on JSTOR . View this article's JSTOR metadata . You may also retrieve all of this items metadata in JSON at the following URL: https://archive.org/metadata/jstor-2007115

This paper is in the public domain in USA. Metadata comes from the CrossRef API, see full record in the source URL below.

scan of book Functions and limits Unit 4 Block 2 Functions and numbers M101 Mathematics Foundation Course 2nd edition by The Mathemetics Foundation Course Team [1982] {515--oclc} Contents material:- Tangents Iteration and convergence Derived functions Limits

We consider the inversion enumerator I_n(q), which counts labeled trees or, equivalently, parking functions. This polynomial has a natural extension to generalized parking functions. Substituting q = -1 into this generalized polynomial produces the number of permutations with a certain descent set. In the classical case, this result implies the formula I_n(-1) = E_n, the number of alternating permutations. We give a combinatorial proof of these formulas based on the involution principle. We also give a geometric interpretation of these identities in terms of volumes of generalized chain polytopes of ribbon posets. The volume of such a polytope is given by a sum over generalized parking functions, which is similar to an expression for the volume of the parking function polytope of Pitman and Stanley.

We review recent results in the theory of numbers and single-valued functions on the complex plane which arise in quantum field theory.

We propose a new method for computing the $\varphi$-functions of large sparse matrices with low rank or fast decaying singular values. The key is to reduce the computation of $\varphi_{\ell}$-functions of a large matrix to $\varphi_{\ell+1}$-functions of some $r$-by-$r$ matrices, where $r$ is the numerical rank of the large matrix in question. Some error analysis on the new method is given. Furthermore, we propose two novel strategies for estimating 2-norm condition numbers of the $\varphi$-functions. Numerical experiments illustrate the numerical behavior of the new algorithms and show the effectiveness of our theoretical results.

We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection between matchings with no 2-nestings and inversion tables with no subsequence of decreasing consecutive numbers. We then prove using the involution principle that inversion tables with no subsequence of decreasing consecutive numbers have the same generating function as the Fishburn numbers. The Fishburn numbers have previously been shown by Bousquet-M\'elou, Claesson, Dukes and Kitaev to enumerate $\textbf{(2+2)}$-avoiding posets, matchings with no left- or right-nestings, permutations avoiding a particular pattern, and so-called ascent sequences, and by Dukes and Parviainen to enumerate upper triangular matrices with non-negative entries and no empty rows or columns. Claesson and Linusson conjectured they also enumerated matchings with no 2-nestings. Using these new interpretations of the Fishburn numbers and another version of the involution, we prove the conjectured equality (also proven using matrices by Jel\'{\i}nek and by Yan) of two refinements by Remmel and Kitaev of the Fishburn generating function. In an appendix, we state and prove another conjecture of Claesson and Linusson giving the distribution of left-nestings over the set of all matchings.

We show that two classes of combinatorial objects--inversion tables with no subsequence of decreasing consecutive numbers and matchings with no 2-nestings--are enumerated by the Fishburn numbers. In particular, we give a simple bijection between matchings with no 2-nestings and inversion tables with no subsequence of decreasing consecutive numbers. We then prove using the involution principle that inversion tables with no subsequence of decreasing consecutive numbers have the same generating function as the Fishburn numbers. The Fishburn numbers have previously been shown by Bousquet-M\'elou, Claesson, Dukes and Kitaev to enumerate $\textbf{(2+2)}$-avoiding posets, matchings with no left- or right-nestings, permutations avoiding a particular pattern, and so-called ascent sequences, and by Dukes and Parviainen to enumerate upper triangular matrices with non-negative entries and no empty rows or columns. Claesson and Linusson conjectured they also enumerated matchings with no 2-nestings. Using these new interpretations of the Fishburn numbers and another version of the involution, we prove the conjectured equality (also proven using matrices by Jel\'{\i}nek and by Yan) of two refinements by Remmel and Kitaev of the Fishburn generating function. In an appendix, we state and prove another conjecture of Claesson and Linusson giving the distribution of left-nestings over the set of all matchings.

We study the properties of Eisenstein-Kronecker numbers, which are related to special values of Hecke $L$-function of imaginary quadratic fields. We prove that the generating function of these numbers is a reduced (normalized or canonical in some literature) theta function associated to the Poincare bundle of an elliptic curve. We introduce general methods to study the algebraic and $p$-adic properties of reduced theta functions for CM abelian varieties. As a corollary, when the prime $p$ is ordinary, we give a new construction of the two-variable $p$-adic measure interpolating special values of Hecke $L$-functions of imaginary quadratic fields, originally constructed by Manin-Vishik and Katz. Our method via theta functions also gives insight for the case when $p$ is supersingular. The method of this paper will be used in subsequent papers to study the precise $p$-divisibility of critical values of Hecke $L$-functions associated to Hecke characters of quadratic imaginary fields for supersingular $p$, as well as explicit calculation in two-variables of the $p$-adic elliptic polylogarithm for CM elliptic curves.

We prove bounds for the covering numbers of classes of convex functions and convex sets in Euclidean space. Previous results require the underlying convex functions or sets to be uniformly bounded. We relax this assumption and replace it with weaker integral constraints. Existing results can be recovered as special cases of our results.

We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formul\ae\ of a new type for the generating series of intersection numbers of $\psi$-classes as well as of mixed $\psi$- and $\kappa$-classes in full genera.

We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formul\ae\ of a new type for the generating series of intersection numbers of $\psi$-classes as well as of mixed $\psi$- and $\kappa$-classes in full genera.

We derive an explicit generating function of correlations functions of an arbitrary tau-function of the KdV hierarchy. In particular applications, our formulation gives closed formul\ae\ of a new type for the generating series of intersection numbers of $\psi$-classes as well as of mixed $\psi$- and $\kappa$-classes in full genera.

We consider the connection of functional decompositions of rational functions over the real and complex numbers, and a question about curves on a Riemann sphere which are invariant under a rational function.

Translation of: Logarithmisch-trigonometrisches Handbuch

xxviii, 575 p. 24 cm

xxviii, 575 p. 24 cm

575 p. ; 24 cm

Book Source: Digital Library of India Item 2015.154738 dc.contributor.author: W.l.f. Fischer dc.contributor.other: D.van Nostrand Co dc.date.accessioned: 2015-07-06T19:40:12Z dc.date.available: 2015-07-06T19:40:12Z dc.date.copyrightexpirydate: 0000-00-00 dc.date.digitalpublicationdate: 2024-09-03 dc.date.citation: 1916 dc.identifier: CMU dc.identifier.barcode: 1990050020429 dc.identifier.origpath: /data/upload/0020/434 dc.identifier.copyno: 1 dc.identifier.uri: http://www.new.dli.ernet.in/handle/2015/154738 dc.description.scanningcentre: AKCE, Universal Digital Library dc.description.main: 1 dc.description.tagged: 0 dc.description.totalpages: 605 dc.format.mimetype: application/pdf dc.language.iso: English dc.publisher.digitalrepublisher: Akce Universal Digital Library dc.publisher: D.van Nostrand Co dc.rights: Out_of_copyright dc.source.library: Cmu dc.subject.classification: Logarithmic Tables dc.title: Logarithmic Tables Of Numbers And Trigonometrical Functions dc.rights.holder: D.van Nostrand Co

Document number CIA-RDP60-00213A000100030090-3 declassified and released through the CIA's CREST database. Previously available only on four computers located outside of Washington D.C., the Agency was successfully pressured into putting the files online as a result of a MuckRock lawsuit and the efforts of Emma Best. The metadata was collected by Data.World, and the files are now being archived and made text searchable by the Internet Archive.

One purpose of this paper is to construct twisted q-Euler numbers by using p-adic invariant integral on Zp in the sense of fermionic. Finally, we consider twisted Euler q-zeta function and q-l-series which interpolate twisted q-Euler numbers and polynomials.

In [3] Bege introduced the generalized Apostol's Mobius functions. In this paper we are presenting new properties of this functions. By introducing the special set of k-free numbers we have obtained some asymptotic formulas for the partial sums of these functions.

In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.

In this paper we give the q-extension of Euler numbers which can be viewed as interpolating of the q-analogue of Euler zeta function ay negative integers, in the same way that Riemann zeta function interpolates Bernoulli numbers at negative integers. Finally we woll treat some identities of the q-extension of the euler numbers by using fermionic p-adic q-integration on Z_p.

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.

We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.

The fundamental aim of this paper is to describe q-Analogue of p-adic log gamma functions with weight alpha and beta. Moreover, we give relationship between p-adic q-log gamma funtions with weight ({\alpha}, {\beta}) and q-extension of Genocchi numbers with weight alpha and beta and modified q Euler numbers with weight {\alpha}

Explicitly orbital-dependent approximations to the exchange-correlation energy functional of density functional theory typically not only depend on the single-particle Kohn-Sham orbitals but also on their occupation numbers in the ground state Slater determinant. The variational calculation of the corresponding exchange-correlation potentials with the Optimized Effective Potential (OEP) method therefore also requires a variation of the occupation numbers with respect to a variation in the effective single-particle potential, which is usually not taken into account. Here it is shown under which circumstances this procedure is justified.

Explicitly orbital-dependent approximations to the exchange-correlation energy functional of density functional theory typically not only depend on the single-particle Kohn-Sham orbitals but also on their occupation numbers in the ground state Slater determinant. The variational calculation of the corresponding exchange-correlation potentials with the Optimized Effective Potential (OEP) method therefore also requires a variation of the occupation numbers with respect to a variation in the effective single-particle potential, which is usually not taken into account. Here it is shown under which circumstances this procedure is justified.

We use the circle method to evaluate the behavior of limit-periodic functions on primes. For those limit-periodic functions that satisfy a kind of Barban-Davenport-Halberstam condition and whose singular series converge fast enough, we can evaluate their average value on primes. As an application, this result is used to show how tuples of different k-free numbers behave when linear shifts are applied.

We use the circle method to evaluate the behavior of limit-periodic functions on primes. For those limit-periodic functions that satisfy a kind of Barban-Davenport-Halberstam condition and whose singular series converge fast enough, we can evaluate their average value on primes. As an application, this result is used to show how tuples of different k-free numbers behave when linear shifts are applied.

A multiparametric family of 2D Toda $\tau$-functions of hypergeometric type is shown to provide generating functions for composite, signed Hurwitz numbers that enumerate certain classes of branched coverings of the Riemann sphere and paths in the Cayley graph of $S_n$. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(\mu, \nu)$ in their series expansion over products $P_\mu P'_\nu$ of power sum symmetric functions in the two sets of Toda flow parameters and powers of the $l+m$ auxiliary parameters are shown to enumerate $|\mu|=|\nu|=n$ fold branched covers of the Riemann sphere with specified ramification profiles $ \mu$ and $\nu$ at a pair of points, and two sets of additional branch paints, satisfying certain additional conditions on their ramification profile lengths. The first group consists of $l$ branch points, with ramification profile lengths fixed to be the numbers $(n-c_1, ..., n- c_l)$; the second consists of $m$ further groups of "coloured" branch points, of variable number, for which the sums of the complements of the ramification profile lengths within the groups are fixed to equal the numbers $(d_1, ..., d_m)$. The latter are counted with sign determined by the parity of the total number of such branch points. The coefficients $F^{c_1, ..., c_l}_{d_1, ..., d_m}(\mu, \nu)$ are also shown to enumerate paths in the Cayley graph of the symmetric group $S_n$ generated by transpositions, starting, as in the usual double Hurwitz case, at an element in the conjugacy class of cycle type $\mu$ and ending in the class of type $\nu$, but with the first $l$ consecutive subsequences of $(c_1, ..., c_l)$ transpositions strictly monotonically increasing, and the subsequent subsequences of $(d_1, ..., d_m)$ transpositions weakly increasing.