Revisiting the hybrid real number system

The hybrid real number system consisting of terminating and nonterminating decimals, dark numbers, dual dark numbers, involving the notions of personal infinities and the impersonal infinity has been discussed. Some algebraic properties of dark numbers and dual dark numbers are discussed.     

About the upper chromatic number of a co-hypergraph

A mixed hypergraph consists of two families of subsets: the edges and the co-edges. In a coloring every co-edge has at least two vertices of the same color, and every edge has at least two vertices of different colors. The largest and smallest possible number of colors in a coloring is termed the upper and lower chromatic numbers, respectively. In this paper we investigate co-hypergraphs i.e., the hypergraphs with only co-edges, with respect to the property of coloring. The relationship between the lower bound of the size of co-edges and the lower bound of the upper chromatic number is explored. The necessary and sufficient conditions for determining the upper chromatic numbers, of a co-hypergraph are provided. And the bounds of the number of co-edges of some uniform co-hypergraphs with certain upper chromatic numbers are given.     

Computation of class numbers of quadratic number fields

We explain how one can dispense with the numerical computation of approximations to the transcendental integral functions involved when computing class numbers of quadratic number fields. We therefore end up with a simpler and faster method for computing class numbers of quadratic number fields. We also explain how to end up with a simpler and faster method for computing relative class numbers of imaginary abelian number fields.

     

A generalization of the formula for the triangular number of the sum and product of natural numbers

This note generalizes the formula for the triangular number of the sum and product of two natural numbers to similar results for the triangular number of the sum and product of r natural numbers. The formula is applied to derive formula for the sum of an odd and an even number of consecutive triangular numbers.     

On the chromatic number of rational five-space

Summary The chromatic number of rational five-space is the chromatic number of the infinite graph whose vertex set is the set of all those five-dimensional vectors with all the coordinates being rational numbers and with two vertices forming an edge iff the Euclidean distance is exactly one. In this paper it is shown that the chromatic number of rational five-space is at least six.     

Graph minors and the crossing number of graphs

There are three general lower bound techniques for the crossing numbers of graphs, all of which can be traced back to Leighton's work on applications of crossing number in VLSI: the Crossing Lemma, the Bisection Method, and the Embedding Method. In this contribution, we sketch their adaptations to the minor crossing number.     

Bounds on the number of numerical semigroups of a given genus

Lower and upper bounds are given for the number n_g of numerical semigroups of genus g. The lower bound is the first known lower bound while the upper bound significantly improves the only known bound given by the Catalan numbers. In a previous work the sequence n_g is conjectured to behave asymptotically as the Fibonacci numbers. The lower bound proved in this work is related to the Fibonacci numbers and so the result seems to be in the direction to prove the conjecture. The method used is based on an accurate analysis of the tree of numerical semigroups and of the number of descendants of the descendants of each node depending on the number of descendants of the node itself.     

Generalized higher order Bernoulli number pairs and generalized Stirling number pairs

From a delta series f(t) and its compositional inverse g(t), Hsu defined the generalized Stirling number pair . In this paper, we further define from f(t) and g(t) the generalized higher order Bernoulli number pair . Making use of the Bell polynomials, the potential polynomials as well as the Lagrange inversion formula, we give some explicit expressions and recurrences of the generalized higher order Bernoulli numbers, present the relations between the generalized higher order Bernoulli numbers of both kinds and the corresponding generalized Stirling numbers of both kinds, and study the relations between any two generalized higher order Bernoulli numbers. Moreover, we apply the general results to some special number pairs and obtain series of combinatorial identities. It can be found that the introduction of generalized Bernoulli number pair and generalized Stirling number pair provides a unified approach to lots of sequences in mathematics, and as a consequence, many known results are special cases of ours.     

An estimate for the number of consecutive quadratic residues

The problem of estimation of the maximal number H of consecutive integer numbers such that they all are either quadratic residues or quadratic nonresidues modulo a prime number p is considered.     

Two-valued number pyramids

Number pyramids are common in elementary school mathematics. Trying to express the value of the top block in terms of the values at the base leads to the binomial coefficients. It also seems natural to ask for the maximal number of odd numbers in a number pyramid of a given size. The answer is easy to state, but the proof is nontrivial: A \(k\) step number pyramid can have at most \(\left\lfloor\frac{k(k+1)+1}{3}\right\rfloor\) odd numbers, which equals two thirds of the number of blocks rounded to the nearest integer. All maximal and almost maximal solutions are given explicitly. To this end, we rephrase the question in terms of colored tilings. In the outlook we present relations to other—mostly geometric—subjects and problems.

     

Riordan arrays and r-Stirling number identities

By using the theory of Riordan arrays, we establish four pairs of general r-Stirling number identities, which reduce to various identities on harmonic numbers, hyperharmonic numbers, the Stirling numbers of the first and second kind, the r-Stirling numbers of the first and second kind, and the r-Lah numbers. We further discuss briefly the connections between the r-Stirling numbers and the Cauchy numbers, the generalized hyperharmonic numbers, and the poly-Bernoulli polynomials. Many known identities are shown to be special cases of our results, and the combinatorial interpretations of several particular identities are also presented as supplements.     

Carmichael numbers in number rings

We generalize Carmichael numbers to ideals in number rings and prove a generalization of Korselt's Criterion for these Carmichael ideals. We investigate when Carmichael numbers in the integers generate Carmichael ideals in the algebraic integers of abelian number fields. In particular, we show that given any composite integer n, there exist infinitely many quadratic number fields in which n is not Carmichael. Finally, we show that there are infinitely many abelian number fields K with discriminant relatively prime to n such that n is not Carmichael in K.     

Efficiency test of pseudorandom number generators using random walks

Markov chain Monte Carlo methods and computer simulations usually use long sequences of random numbers generated by deterministic rules, so-called pseudorandom number generators. Their efficiency depends on the convergence rate to the stationary distribution and the quality of random numbers used for simulations. Various methods have been employed to measure the convergence rate to the stationary distribution, but the effect of random numbers has not been much discussed. We present how to test the efficiency of pseudorandom number generators using random walks.     

Subgraphs of large connectivity and chromatic number in graphs of large chromatic number

For each pair k, m of natural numbers there exists a natural number f(k, m) such that every f(k, m)-chromatic graph contains a k-connected subgraph of chromatic number at least m.     

Transcendence of some power series for Liouville number arguments

In this paper, we prove that some power series with rational coefficients take either values of rational numbers or transcendental numbers for the arguments from the set of Liouville numbers under certain conditions in the field of complex numbers. We then apply this result to an algebraic number field. In addition, we establish the p-adic analogues of these results and show that these results have analogues in the field of p-adic numbers.     

A novel dynamic model of pseudo random number generator

An interesting hierarchy of random number generators is introduced in this paper based on the review of random numbers characteristics and chaotic functions theory. The main objective of this paper is to produce an ergodic dynamical system which can be implemented in random number generators. In order to check the efficacy of pseudo random number generators based on this map, we have carried out certain statistical tests on a series of numbers obtained from the introduced hierarchy. The results of the tests were promising, as the hierarchy passed the tests satisfactorily, and offers a great capability to be employed in a pseudo random number generator.     

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.     

On the number of hexagons in a map

A lower bound is given for the number of hexagonal faces in a simple map on a closed surface whose graph is 3-connected depending on the numbers of i-gonal faces, i ≠ 6, and the genus of the surface.     

About one - an inquiry about the meanings and uses of the number one

The story of the number one is told. The long struggle to recognize one as a number is described. The role of the number one in philosophy, religion, mathematics and science is briefly discussed. Finally, the relationship between one and some other numbers is presented.     

完全二部图K3,3与星Sn的积图的交叉数

确定图的交叉数是NP-完全问题．目前有关完全二部图与星图的积图的交叉数结果并不多．引入了一些新的收缩技巧,建立了积图K3,3□Sn与完全三部图K3,3□Sn之间的交叉数关系．从而,为进一步完全确定积图K3,3□Sn的交叉数提供了一条新途径．