On a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers

On the basis of the introduced notion of a parameter of a hypercomplex number system, we propose a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers. It is shown that quadruplex (bicomplex) numbers and quaternions can be considered as special cases of the universal system of hypercomplex numbers that correspond to certain values of the indicated parameter. We consider the main algebraic properties of the universal system of hypercomplex numbers and give examples illustrating the applicability of the universal hypercomplex system for certain values of the introduced parameter.     

关于一些级数取整值问题的注记

利用Pell方程的理论,讨论了与广义Lucas数有关的一些级数的取整值问题.在一定条件下,解决了一些级数的取整值问题.     

基于联系数不确定性分析的区间数多属性决策

区间数多属性决策通常是在某一算法下得到方案的排序,很少考虑区间数不确定性的影响,通过区间数转换成联系数,可以客观地反映区间数的不确定性,在此基础上,建立基于联系数的多属性决策模型,从而对不确定性进行较全面的取值分西,使决策结果更为合理.     

Diophantine equations and class numbers

The goals of this paper are to provide: (1) sufficient conditions, based on the solvability of certain diophantine equations, for the non-triviality of the class numbers of certain real quadratic fields; (2) sufficient conditions for the divisibility of the class numbers of certain imaginary quadratic fields by a given integer; and (3) necessary and sufficient conditions for an algebraic integer (which is not a unit) to be the norm of an algebraic integer in a given extension of number fields.     

Totient-numbers of the rectangular association scheme

In this paper we compute the totient numbers of all eigenspaces and of certain sums of eigenspaces of the rectangular association scheme. This is then used to obtain the totient numbers of certain sums of eigenspaces of the hypercubic association schemeH(n, q).     

Periodicity of Balancing Numbers

The balancing numbers originally introduced by Behera and Panda [2] as solutions of a Diophantine equation on triangular numbers possess many interesting properties. Many of these properties are comparable to certain properties of Fibonacci numbers, while some others are more interesting. Wall [14] studied the periodicity of Fibonacci numbers modulo arbitrary natural numbers. The periodicity of balancing numbers modulo primes and modulo terms of certain sequences exhibits beautiful results, again, some of them are identical with corresponding results of Fibonacci numbers, while some others are more fascinating. An important observation concerning the periodicity of balancing numbers is that, the period of this sequence coincides with the modulus of congruence if the modulus is any power of 2. There are three known primes for which the period of the sequence of balancing numbers modulo each prime is equal to the period modulo its square, while for the Fibonacci sequence, till date no such prime is available.     

Approximation durch algebraische Zahlen beschränkten Grades im Körper der formalen laurentreihen

Approximation by Algebraic Numbers of Bounded Degree in the Field of Formal Laurent Series. In this paper results ofWirsing andDavenport-Schmidt concerning the approximation of certain numbers by algebraic numbers of bounded degree are transferred to the field of formal power series connected with a certain non-archimedian valuation.

     

A direct method for evaluating some nice Hankel determinants and proofs of several conjectures

In this paper, we propose a direct method to evaluate Hankel determinants for some generating functions satisfying a certain type of quadratic equations, which cover generating functions of Catalan numbers, Motzkin numbers and Schröder numbers. Additionally, four recent conjectures proposed by Cigler (2011) [3] are proved.     

The moments and statistical distribution of class numbers of quadratic fields with prime discriminant

We establish asymptotic formulas for all the moments of class numbers of quadratic fields with prime discriminant. Further, we study the statistical distribution of these class numbers, introducing a certain random Euler product.     

Nearest interval approximation of a fuzzy number

The problem of the interval approximation of fuzzy numbers is discussed. A new interval approximation operator, which is the best one with respect to a certain measure of distance between fuzzy numbers, is suggested.     

Tropical descendant Gromov–Witten invariants

We define tropical Psi-classes on\({\mathcal{M}_{0,n}(\mathbb{R}^2, d)}\) and consider intersection products of Psi-classes and pull-backs of evaluations on this space. We show a certain WDVV equation which is sufficient to prove that tropical numbers of curves satisfying certain Psi- and evaluation conditions are equal to the corresponding classical numbers. We present an algorithm that generalizes Mikhalkin’s lattice path algorithm and counts rational plane tropical curves satisfying certain Psi- and evaluation conditions.     

Catalan numbers,binary trees,and pointed pseudotriangulations

We study connections among structures in commutative algebra, combinatorics, and discrete geometry, introducing an array of numbers, called Borel’s triangle, that arises in counting objects in each area. By defining natural combinatorial bijections between the sets, we prove that Borel’s triangle counts the Betti numbers of certain Borel-fixed ideals, the number of binary trees on a fixed number of vertices with a fixed number of “marked” leaves or branching nodes, and the number of pointed pseudotriangulations of a certain class of planar point configurations.     

Hodge–de Rham numbers of almost complex 4-manifolds

We introduce and study Hodge–de Rham numbers for compact almost complex 4-manifolds, generalizing the Hodge numbers of a complex surface. The main properties of these numbers in the case of complex surfaces are extended to this more general setting, and it is shown that all Hodge–de Rham numbers for compact almost complex 4-manifolds are determined by the topology, except for one (the irregularity). Finally, these numbers are shown to prohibit the existence of complex structures on certain manifolds, without reference to the classification of surfaces.     

Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine

A proof of a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power series and their (matrix) logarithms.     

Norms extremal with respect to the Mahler measure

In this paper, we introduce and study several norms which are constructed in order to satisfy an extremal property with respect to the Mahler measure. These norms are a natural generalization of the metric Mahler measure introduced by Dubickas and Smyth. We show that bounding these norms on a certain subspace implies Lehmer?s conjecture and in at least one case that the converse is true as well. We evaluate these norms on a class of algebraic numbers that include Pisot and Salem numbers, and for surds. We prove that the infimum in the construction is achieved in a certain finite dimensional space for all algebraic numbers in one case, and for surds in general, a finiteness result analogous to that of Samuels and Jankauskas for the t-metric Mahler measures.     

Normal characteristic numbers

This paper computes the greatest common divisor of certain normal Pontrjagin numbers and Chern numbers. These calculations are of interest in studying singularities.

     

Some exact formulae for the numbers of representations of integers by ternary quadratic forms

In this paper we shall give some exact formulae for the numbers of representations of integers by certain ternary quadratic forms whose genus have class numbers large 1.     

Some post canonical systems in one letter

Post's Canonical systems in one letter are viewed as operations on natural numbers. It is shown that, in a certain sense, these systems are capable of producing arbitrary recursively enumerable sets. Also, certain special cases are examined.     

Series transformation formulas of Euler type,Hadamard product of series,and harmonic number identities

The Hadamard multiplication theorem for series is used to establish several Euler-type series transformation formulas. As applications we obtain a number of binomial identities involving harmonic numbers and an identity for the Laguerre polynomials. We also evaluate in a closed form certain power series with harmonic numbers.     

用逼近方法建立代数无关性(Ⅱ)

设ｆ（ｘ）是以ａｎｘ为元素的连分式,其中系数ａｎ是正代数数．应用［１］中的判别法则,在某些仅与ａｎ有关的条件下证明了对任何绝对值互异的非零实代数数ａ１,…,ａｓ值ｆ（ａ１）代数无关．还对某些实超越数建立了的代数无关性．