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Mathematics & Computer Science, Algebra & Number Theory; series representation; power function; monomial; binomial theorem; multinomial theorem; worpitzky identity; stirling numbers of second kind; faulhaber's sum; finite difference; faulhaber's formula; central factorial numbers; binomial coefficients; binomial distribution; binomial transform; bernoulli numbers; oeis; multinomial coefficients
Online: 17 August 2018 (11:10:47 CEST)
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In this paper we discuss a problem of generalization of binomial distributed triangle, that is sequence A287326 in OEIS. The main property of A287326 that it returns a perfect cube n as sum of n-th row terms over k; 0 ≤ k ≤ n−1 or 1 ≤ k ≤ n, by means of its symmetry. In this paper we have derived a similar triangles in order to receive powers m = 5; 7 as row items sum and generalized obtained results in order to receive every odd-powered monomial n2m+1; m ≥ 0 as sum of row terms of corresponding triangle. In other words, in this manuscript are found and discussed the polynomials Dm(n,k) and Um(n,k), such that, when being summed up over k in some range with respect to m and n returns the monomial n2m+1.
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Mathematics & Computer Science, Algebra & Number Theory; partially degenerate Changhee-Genocchi polynomials, degenerate Genocchi polynomials, degenerate Changhee polynomials.
Online: 7 August 2018 (13:16:55 CEST)
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In this paper, we introduce the partially degenerate Changhee-Genocchi polynomials and numbers and investigated some identities of these polynomials. Furthermore, we investigate some explicit identities and properties of the partially degenerate Changhee-Genocchi arising from the nonlinear di erential equations.
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Mathematics & Computer Science, Algebra & Number Theory; q-Bernoulli numbers; q-Bernoulli polynomials; Bernstein polynomials; q-Bernstein polynomials; p-adic integral on Zp
Online: 30 July 2018 (08:17:45 CEST)
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In this paper, we study the p-adic integral representation on Zp of q-Bernoulli numbers arising from two variable q-Bernstein polynomials and investigate some properties for the q-Bernoulli numbers. In addition, we give some new identities of q-Bernoulli numbers.
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Mathematics & Computer Science, Algebra & Number Theory; authentication schemes; resilient maps; gray map
Online: 30 July 2018 (04:26:47 CEST)
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Two new systematic authentication codes based on the Gray map over a Galois ring are introduced. The first introduced code attains optimal impersonation and substitution probabilities. The second code improves space sizes but it does not attain optimal probabilities. Besides it is conditioned to the existence of a special class of bent maps on Galois rings.
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Mathematics & Computer Science, Algebra & Number Theory; matrix algebra; accounting; industry 4.0
Online: 24 July 2018 (05:01:28 CEST)
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The aim of this paper is to show how the mathematical basis is used for a precise treatment of double-entry bookkeeping, which was firstly developed in the nineteenth century by Sir William Rowan Hamilton. This is done by using basic notions of matrix algebra with the roots to the idea of using ordered pairs. We also reveal how the complex numbers and the rationals (fractions) developed in a mainstream of accounting science and became a leading platform for the ongoing processes within the Industry 4.0. The paper concludes with the examples on how accounting operations can be represented by matrix equations with a result of the essential report generating. The author represents a mathematical model of accounting which is independent of the specific existential forms but is capable to undertake the form of any of them and thus has the prospect of being understood and accepted by specialists globally.
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Mathematics & Computer Science, Algebra & Number Theory; RSA, Cryptanalysis, Lattices, LLL algorithm, Coppersmith’s method
Online: 20 July 2018 (11:07:00 CEST)
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In this paper, we gave an attack on RSA when Euler function has small multiplicative inverse modulo "e" and the prime sum p+q is of the form p+q=2^nk_0+k_1 where n is a given positive integer and k_0 and k_1 are two suitably small unknown integers using sublattice reduction techniques and Coppersmith's methods for finding small roots of modular polynomial equations. When we compare this method with an approach using lattice based techniques, this procedure slightly improves the bound and reduces the lattice dimension.
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Mathematics & Computer Science, Algebra & Number Theory; Fubini polynomials; Euler numbers; congruence; elementary method
Online: 18 July 2018 (09:31:49 CEST)
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The aim of this paper is to use the elementary method and the properties of the second kind Stirling numbers to study the congruence properties of a new sequence, which is closely related to Fubini polynomials and Euler numbers, and give some interesting congruences for it. This solves a problem proposed by the first author at an unpublished paper.
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Mathematics & Computer Science, Algebra & Number Theory; Zeros at infinity, Jacobian Conjecture
Online: 17 July 2018 (12:04:11 CEST)
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This article contains the theorems concerning the algebraic dependence of polynomial mappings with the constant Jacobian having two zeros at infinity. The work is related to the issues of the classical Jacobian Conjecture. This hypothesis affirm that the polynomial mapping of two complex variables with constant
non-zero Jacobian is invertible. The Jacobian Conjecture is equivalent to the fact that polynomial mappings with constant non-zero Jacobian do not have two zeros at infinity, therefore it is equivalent to the two theorems given in the work. The proofs of these theorems proceeds by induction.
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Mathematics & Computer Science, Algebra & Number Theory; domination, vertex-edge domination, total vertex-edge domination
Online: 11 July 2018 (10:52:10 CEST)
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A novel domination invariant defined by Boutrig and Chellali in the recent: total vertex-edge domination. In this paper we obtain an improved upper bound of total vertex edge-domination number of a tree. If is a connected tree with order , then with and we characterize the trees attaining this upper bound. Furthermore we provide a characterization of trees with
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Mathematics & Computer Science, Algebra & Number Theory; degenerate Bernstein polynomials, degenerate Bernstein operators, degenerate Euler polynomials.
Online: 10 July 2018 (13:56:13 CEST)
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In this paper, we investigate the recently introduced degenerate Bernstein polynomials and operators and derive some of their properties. Also, we give some properties of the degenerate Euler numbers and polynomials and their connection with the degenerate Euler polynomials.
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Mathematics & Computer Science, Algebra & Number Theory; binary system(groupoid); minimum (mutual) covering set; (mutual) shortest distance; (di)frame graph; d/BCK-algebra
Online: 28 June 2018 (04:48:32 CEST)
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In this paper, we observe that if X is a set and (Bin(X), □) is the semigroup of binary systems on X, then its center ZBin(X) consists of the locally-zero-semigroups and that these can be modeled as (simple) graphs and conversely. Using this device we show that we may obtain many results of interest concerning groupoids by reinterpreting graph theoretical properties and at the same time results on graphs G may be obtained by considering them as elements of centers of the semigroups of binary systems (Bin(X), □) where X = V(G), the vertex set of G.
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Mathematics & Computer Science, Algebra & Number Theory; Chebyshev polynomials; sums of finite products; hypergeometric function
Online: 6 June 2018 (09:44:52 CEST)
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Here we consider sums of finite products of Chebyshev polynomials of the third and fourth kinds. Then we represent each of those sums of finite products as linear combinations of the four kinds of Chebyshev polynomials which involve the hypergeometric function 3F2.
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Mathematics & Computer Science, Algebra & Number Theory; double framed T-soft fuzzy set; double framed T-soft fuzzy algebra; double framed B-soft fuzzy algebra
Online: 23 April 2018 (12:15:40 CEST)
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The aim of this article is introduced the concept of double framed T-soft fuzzy set (DFT-soft fuzzy set) which is the combination of soft set and fuzzy set. We also defined the notions and apply this concept in BCK/BCI-algebras. By using example, we also discussed the concept of double framed T-soft fuzzy algebra (DFT-soft fuzzy algebra) and double framed B-soft fuzzy algebra (DFB-soft fuzzy algebra) and also investigated their properties. Each double framed T-soft fuzzy algebra is double framed B-soft fuzzy algebra but by using example, we proved that converse may or may not be possible.
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Mathematics & Computer Science, Algebra & Number Theory; coarse structure; coarse space; ballean; varieties of coarse spaces
Online: 23 April 2018 (12:13:38 CEST)
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A class of coarse spaces is called a variety if is closed under formation of subspaces, coarse images and products. We classify the varieties of coarse spaces and, in particular, show that if a variety contains an unbounded metric space then is the variety of all coarse spaces.
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Mathematics & Computer Science, Algebra & Number Theory; Fubini polynomials; w-torsion Fubini polynomials; fermionic p-adic integrals; symmetric identities
Online: 23 April 2018 (08:24:37 CEST)
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In this paper, we represent the generating function of w-torsion Fubini polynomials by fermionic p-adic integral on Zp. Then we consider a quotient of such fermionic p-adic integrals on Zp, representing generating functions of three w-torsion Fubini polynomials and derive some new symmetric identities for the w-torsion Fubini and two variable w-torsion Fubini polynomials.
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Mathematics & Computer Science, Algebra & Number Theory; computable upper bound on the heights of rational solutions; computable upper bound on the moduli of integer solutions; Diophantine equation with a finite number of solutions; finite-fold Diophantine representation; single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; single query to an oracle that decides whether or not a given Diophantine equation has a rational solution
Online: 16 April 2018 (05:12:59 CEST)
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Let , and let for every positive integer n. We consider the following hypothesis: if a system S ⊆ {xi · xj = xk : i, j, k ∈ {1, . . . , n}} ∪ {xi + 1 = xk : i, k ∈{1, . . . , n}} has only finitely many solutions in non-negative integers x1, . . . , xn, then each such solution (x1, . . . , xn) satisfies x1, . . . , xn ≤ f (2n). We prove: (1) the hypothesisimplies that there exists an algorithm which takes as input a Diophantine equation, returns an integer, and this integer is greater than the heights of integer (non-negative integer, positive integer, rational) solutions, if the solution set is finite; (2) the hypothesis implies that there exists an algorithm for listing the Diophantine equations with infinitely many solutions in non-negative integers; (3) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many rational solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has a rational solution; (4) the hypothesis implies that the question whether or not a given Diophantine equation has only finitely many integer solutions is decidable by a single query to an oracle that decides whether or not a given Diophantine equation has an integer solution; (5) the hypothesis implies that if a set ⊆ has a finite-fold Diophantine representation, then is computable.
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Mathematics & Computer Science, Algebra & Number Theory; q-calculus; (p,q)-calculus; exponential functions; trigonometric functions; hyperbolic functions
Online: 9 March 2018 (14:49:59 CET)
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This paper includes some new investigations and results for post quantum calculus, denoted by (p,q)-calculus. A chain rule for (p,q)-derivative is developed. Also, a new (p,q)-analogue of the exponential function is introduced and some its properties including the addition property for (p,q)-exponential functions are investigated. Several useful results involving (p,q)-binomial coefficients and (p,q)-antiderivative are discovered. At the final part of this paper, (p,q)-analogue of some elementary functions including trigonometric functions and hyperbolic functions are considered and some properties and relations among them are analyzed extensively.
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Mathematics & Computer Science, Algebra & Number Theory; Hermite-Kampé de Fériet polynomials; Hermite Bernoulli polynomials; poly Bernoulli polynomials; Stirling numbers of second kind; Polylogarithm functions.
Online: 22 February 2018 (15:28:12 CET)
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We introduce the Hermite based poly-Bernoulli polynomials with a q parameter and give someof their basic properties including not only addition property, but also derivative properties and integralrepresentations. We also de.ne the Hermite based λ -Stirling polynomials of the second kind, and thenprovide some relations. Moreover, we derive several correlations and identities including the Hermite-Kampéde Fériet (or Gould-Hopper) family of polynomials, the Hermite based poly-Bernoulli polynomials with a qparameter and the Hermite based λ -Stirling polynomials of the second kind.
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Mathematics & Computer Science, Algebra & Number Theory; numbers and polynomials; Bernoulli numbers and polynomials; Genocchi numbers and polynomials; power sums
Online: 19 February 2018 (14:18:47 CET)
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We study on the sums of powers of consequtive integers and alternating sums of power of consequtive integers. We derive many identities and correlations including Bernoulli, Euler and Genocchi polynomials and numbers.
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Mathematics & Computer Science, Algebra & Number Theory; p-adic numbers; p-adic gamma function; p-adic Euler constant; q-Daehee polynomials; Stirling numbers of the first kind
Online: 19 February 2018 (14:12:04 CET)
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In this paper, we investigate p-adic q-integral (q-Volkenborn integral) on Zp of p-adic gamma function via their Mahler expansions. We also derived two q-Volkenborn integrals of p-adic gamma function in terms of q-Daehee polynomials and numbers and q-Daehee polynomials and numbers of the second kind. Moreover, we discover q-Volkenborn integral of the derivative of p-adic gamma function. We acquire the relationship between the p-adic gamma function and Stirling numbers of the first kind. We finally develop a novel and interesting representation for the p-adic Euler constant by means of the q-Daehee polynomials and numbers.
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Mathematics & Computer Science, Algebra & Number Theory; LiDAR; smooth contour line; break line; point cloud; forests
Online: 1 February 2018 (03:37:44 CET)
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A methodology for both accurate and smooth contour line generation from Light Detection and Ranging (LiDAR) point clouds is proposed in this paper. In order to improve the accuracy of contour lines in the area of forests, constrained triangulation networks with break lines are then constructed to generate contour lines. In break line extraction, a bi-threshold method for edge line detection is used to extract both complete and reliable break lines. A point clouds elevation adjustment with constrain of break lines and an interpolator considering a contour interval is proposed to improve the smoothness of contour lines. The proposed interpotator is also can avoid contour line intersection when contour lines are interpolated. Statistical parameters and shape index are then used to evaluate quantitatively the accuracy and smoothness of the resultant contour lines, which fill in the blank of contour lines evaluation in theory. The experiments show that high-quality contours in terms of smoothness and accuracy can be generated from LiDAR point clouds.
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Mathematics & Computer Science, Algebra & Number Theory; quantum chaos; Riemann hypothesis; eigenspectra
Online: 28 December 2017 (10:53:20 CET)
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The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender-Brody-Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results. The holomorphicity of the resulting eigenvalues is examined. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function converges at σ = 1/2.
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Mathematics & Computer Science, Algebra & Number Theory; (p, q)-calculus; apostol-bernoulli polynomials; apostol-euler polynomials; apostol- genocchi polynomials; stirling numbers of second kind; generating function; cauchy product
Online: 21 December 2017 (05:30:46 CET)
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Motivated by Kurt's work [Filomat 30 (4) 921-927, 2016], we first consider a class of a new generating function for (p, q)-analog of Apostol type polynomials of order a including Apostol-Bernoulli, Apostol-Euler and Apostol-Genocchi polynomials of order a. By making use of their generating function, we derive some useful identities. We also introduce (p, q)-analog of Stirling numbers of second kind of order v by which we construct a relation including aforementioned polynomials.
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Mathematics & Computer Science, Algebra & Number Theory; q-umbral calculus; apostol-bernoulli polynomials; modi.ed apostol type q-bernoulli polynomials; q-appell polynomials; generating functions
Online: 18 December 2017 (10:01:24 CET)
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This article aims to identify the generating function of modi
ed Apostol type q-Bernoulli polynomials. With the aid of this generating function, some properties of modi
ed Apostol type q-Bernoulli polynomials are given. It is shown that aforementioned polynomials are q-Appell. Hence, we make use of these polynomials to have applications on q-Umbral calculus. From those applications, we derive some theorems in order to get Apostol type modi
ed q-Bernoulli polynomials as a linear combination of some known polynomials which we stated in the paper.
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Mathematics & Computer Science, Algebra & Number Theory; q-calculus; stirling numbers of the second kind; bernoulli polynomials and numbers; generating function; cauchy product
Online: 18 December 2017 (08:22:41 CET)
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In this paper, we introduce a new generalization of the r-Stirling numbers of the second kind based on the q-numbers via an exponential generating function. We investigate their some properties and derive several relations among q-Bernoulli numbers and polynomials, and newly de
ned (q, r, w)-Stirling numbers of the second kind. We also obtain q-Bernstein polynomials as a linear combination of (q, r, w)-Stirling numbers of the second kind and q-Bernoulli polynomials in w.
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Mathematics & Computer Science, Algebra & Number Theory; Carlitz’s q-Bernoulli polynomials; Stirling numbers of the first kind; Stirling numbers of the second kind; p-adic q-integral
Online: 15 December 2017 (09:31:13 CET)
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The main goal of the present paper is to construct some families of the Carlitz’s q-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and investigate their some explicit properties and identities arising from the bosonic q-Volkenborn integral on ℤp. We then define the modified degenerate Carlitz’s q-Bernoulli polynomials and numbers with weight (α, β) and obtain some recurrence relations and other identities. Moreover, we derive some correlations with the modified Carlitz’s q-Bernoulli polynomials with weight (α, β), the modified degenerate Carlitz’s q-Bernoulli polynomials with weight (α, β), the Stirling numbers of the first kind and second kind.
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Mathematics & Computer Science, Algebra & Number Theory; (p, q)-calculus; Bernoulli polynomials; Euler polynomials; Genocchi polynomials; Frobenius-Euler polynomials; generating function; Cauchy product
Online: 15 December 2017 (09:14:10 CET)
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In the present paper, we introduce (p, q)-extension of Apostol type Frobenius-Euler polynomials and numbers and investigate some basic identities and properties for these polynomials and numbers, including addition theorems, difference equations, derivative properties, recurrence relations and so on. Then, we provide integral representations, explicit formulas and relations for these polynomials and numbers. Moreover, we discover (p, q)-extensions of Carlitz's result [L. Carlitz, Mat. Mag., 32 (1959), 247–260] and Srivastava and Pintér addition theorems in [H. M. Srivastava, A. Pinter, Appl. Math. Lett., 17 (2004), 375–380].
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Mathematics & Computer Science, Algebra & Number Theory; multi-order logarithmic number; multi-order logarithmic polynomial; explicit formula; identity; recurrence relation; inversion theorem; Bell polynomial of the second kind; Stirling number; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity; Faa di Bruno formula
Online: 11 September 2017 (04:22:18 CEST)
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In the paper, the author introduces the notions "multi-order logarithmic numbers" and "multi-order logarithmic polynomials", establishes an explicit formula, an identity, and two recurrence relations by virtue of the Faa di Bruno formula and two identities of the Bell polynomials of the second kind in terms of the Stirling numbers of the fi rst and second kinds, and constructs some determinantal inequalities, product inequalities, logarithmic convexity for multi-order logarithmic numbers and polynomials by virtue of some properties of completely monotonic functions.
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Mathematics & Computer Science, Algebra & Number Theory; inequality; ratio; Bernoulli number; Riemann zeta function; Dirichlet eta function; accuracy
Online: 28 August 2017 (09:30:34 CEST)
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In the paper, by virtue of some properties for the Riemann zeta function, the author finds a double inequality for the ratio of two consecutive Bernoulli numbers with even indexes and analyzes the approximating accuracy of the double inequality.
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Mathematics & Computer Science, Algebra & Number Theory; Bell number; Bell polynomial; generalization; explicit formula; inversion formula; inversion theorem; Stirling number; Bell polynomial of the second kind; determinantal inequality; product inequality; completely monotonic function; logarithmic convexity
Online: 26 August 2017 (09:12:05 CEST)
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In the paper, the authors present unified generalizations for the Bell numbers and polynomials, establish explicit formulas and inversion formulas for these generalizations in terms of the Stirling numbers of the first and second kinds with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem connected with the Stirling numbers of the first and second kinds, construct determinantal and product inequalities for these generalizations with aid of properties of the completely monotonic functions, and derive the logarithmic convexity for the sequence of these generalizations.
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Mathematics & Computer Science, Algebra & Number Theory; Bell polynomial; Bell number; Bell polynomial of the second kind; higher order derivative; generating function; Faa di Bruno formula; inversion theorem; Stirling number of the first kind; Stirling number of the second kind; explicit formula; inversion formula; logarithmically absolute monotonicity; logarithmically complete monotonicity; determinantal inequality; product inequality
Online: 25 August 2017 (08:41:30 CEST)
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In the paper, the author (1) presents an explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials, with the help of the Faà di Bruno formula, properties of the Bell polynomials of the second kind, and the inversion theorem for the Stirling numbers of the first and second kinds; (2) recovers an explicit formula and its inversion formula for the Bell polynomials in terms of the Stirling numbers of the first and second kinds, with the aid of the above explicit formula and its inversion formula for higher order derivatives of generating functions of the Bell polynomials; (3) constructs some determinantal and product inequalities and deduces the logarithmic convexity of the Bell polynomials, with the assistance of the complete monotonicity of generating functions of the Bell polynomials. These inequalities are main results of the paper.
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Mathematics & Computer Science, Algebra & Number Theory; identity; inverse matrix; explicit formula; generating function; Chebyshev polynomials of the second kind; Catalan number; triangular matrix; classical hypergeometric function; integral representation
Online: 7 August 2017 (15:49:11 CEST)
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In the paper, the authors establish two identities to express the generating function of the Chebyshev polynomials of the second kind and its higher order derivatives in terms of the generating function and its derivatives each other, deduce an explicit formula and an identities for the Chebyshev polynomials of the second kind, derive the inverse of an integer, unit, and lower triangular matrix, present several identities of the Catalan numbers, and give some remarks on the closely related results including connections of the Catalan numbers respectively with the Chebyshev polynomials, the central Delannoy numbers, and the Fibonacci polynomials.
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Mathematics & Computer Science, Algebra & Number Theory; prime graph; simple group; orthogonal groups; quasirecognition
Online: 10 July 2017 (09:01:04 CEST)
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Let G be a finite group. The prime graph Γ(G) of G is defined as follows: The set of vertices of Γ(G) is the set of prime divisors of |G| and two distinct vertices p and p' are connected in Γ(G), whenever G has an element of order pp'. A non-abelian simple group P is called recognizable by prime graph if for any finite group G with Γ(G)=Γ(P), G has a composition factor isomorphic to P. In [4] proved finite simple groups 2Dn(q), where n ≠ 4k are quasirecognizable by prime graph. Now in this paper we discuss the quasirecognizability by prime graph of the simple groups 2D2k(q), where k ≥ 9 and q is a prime power less than 105.
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Mathematics & Computer Science, Algebra & Number Theory; closed expression; Fibonacci number; Fibonacci polynomial; tridiagonal determinant; Hessenberg determinant
Online: 28 March 2017 (03:11:06 CEST)
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In the paper, the authors nd a new closed expression for the Fibonacci polynomials and, consequently, for the Fibonacci numbers, in terms of a tridiagonal determinant.
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Mathematics & Computer Science, Algebra & Number Theory; explicit formula; recursive formula; generalized Motzkin number; Motzkin number; restricted hexagonal number; Catalan number; generating function
Online: 27 March 2017 (11:02:41 CEST)
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In the paper, the authors find two explicit formulas and recover a recursive formula for the generalized Motzkin numbers. Consequently, the authors deduce two explicit formulas and a recursive formula for the Motzkin numbers, the Catalan numbers, and the restricted hexagonal numbers respectively.
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Mathematics & Computer Science, Algebra & Number Theory; homomorphism; Grassmannian; generalized geometry; cathelineau's complex
Online: 15 March 2017 (08:04:30 CET)
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In this article, a generalization of the geometry of Grassmannian chain complex of free abelian groups generated by the projective configuration of points and Cathelineau's infinitesimal complex of polylogarithmic groups is proposed. Firstly, homomorphisms for weight n =2 up to weight n=6 will be introduced to connect sub-complexes of Grassmannian and Cathelineau. Lately, generalization of these morphisms will be shown for weight n=N. The associated diagrams will also be proven to be commutative and bi-complex.
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Mathematics & Computer Science, Algebra & Number Theory; viewpoint; ordinary differential equation; solution; derivative polynomial; identity; Stirling numbers; Bernoulli number; Bernoulli polynomial; Frobenius-Euler polynomial
Online: 12 October 2016 (11:47:48 CEST)
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In the paper, the authors view some ordinary differential equations and their solutions from the angle of (the generalized) derivative polynomials and simplify some known identities for the Bernoulli numbers and polynomials, the Frobenius-Euler polynomials, the Euler numbers and polynomials, in terms of the Stirling numbers of the first and second kinds.
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Mathematics & Computer Science, Algebra & Number Theory; derangement number; closed form; Hessenberg determinant; tridiagonal determinant; generating function; recurrence relation; derivative
Online: 11 October 2016 (10:53:07 CEST)
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In the paper, the authors find closed forms for derangement numbers in terms of the Hessenberg determinants, discover a recurrence relation of derangement numbers, present a formula for any higher order derivative of the exponential generating function of derangement numbers, and compute some related Hessenberg and tridiagonal determinants.
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Mathematics & Computer Science, Algebra & Number Theory; determinantal expression; recurrence relation; Euler polynomial; Euler number; Hessenberg determinant
Online: 11 October 2016 (10:40:02 CEST)
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In the paper, by a very simple approach, the author establishes an expression in terms of a lower Hessenberg determinant for the Euler polynomials. By the determinantal expression, the author finds a recurrence relation for the Euler polynomials. By the way, the author derives the corresponding expression and recurrence relation for the Euler numbers.
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Mathematics & Computer Science, Algebra & Number Theory; integral form; infinite series; modulus function; transformation function
Online: 26 August 2016 (12:57:10 CEST)
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According to the fact that the low convergence level on the complete elliptic integral of the first kind for the modulus which having values approach to one. In this paper we propose novelty of the complete elliptic integral consists of the new infinite series. We apply the scheme of iteration by substituting the common modulus with own modulus function into the new infinite series. We obtained so many new exact formulas of the complete elliptic integral derived from this method correspond to the number of iteration order. On the other hand, it has been also obtained a lot of new modulus functions rather than common used previously. The calculation results show that the number of significant figures of the new infinite series of the complete elliptic integral of the first kind is increased more and more. It means more fastly convergent would be obtained comparison values with the previous infinite series.
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Mathematics & Computer Science, Algebra & Number Theory; Riemann hypothesis, Farey sequence, Gauss sum associated with a Dirichlet character
Online: 10 August 2016 (11:35:25 CEST)
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Relationships between the Farey sequence and the Riemann hypothesis other than the Franel-Landau theorem are discussed. Whether a function similar to Chebyshev’s second function is square-root close to a line having a slope different from 1 is discussed. The nontrivial zeros of the Riemann zeta function can be used to approximate many functions in analytic number theory. For example, it could be said that the nontrival zeta function zeros and the Möbius function generate in essence the same function - the Mertens function. A different approach is to start with a sequence that is analogous to the nontrivial zeros of the zeta function and follow the same procedure with both this sequence and the nontrivial zeros of the zeta function to generate in essence the same function. A procedure for generating such a function is given.