Euler-Seidel method for certain combinatorial numbers and a new characterization of Fibonacci sequence

In this paper we use the Euler-Seidel method for deriving new identities for hyperharmonic and r-Stirling numbers. The exponential generating function is determined for hyperharmonic numbers, which result is a generalization of Gosper’s identity. A classification of second order recurrence sequences is also given with the help of this method.      

Mean value in invexity analysis

In this paper, a generalization of the mean value theorem is considered in the case of functions defined on an invex set with respect to η (which is not necessarily connected).     

Rationals and decimals as required in the school curriculum: Part 2: From rationals to decimals

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out jointly with his wife, Nadine, in her classroom at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national program with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the second portion of that experiment.     

A New Construction of Central Relative (p a , p a , p a , 1)-Difference Sets

Semiregular relative difference sets (RDS) in a finite group E which avoid a central subgroup C are equivalent to orthogonal cocycles. For example, every abelian semiregular RDS must arise from a symmetric orthogonal cocycle, and vice versa. Here, we introduce a new construction for central (p ^a , p ^a , p ^a , 1)-RDS which derives from a novel type of orthogonal cocycle, an LP cocycle, defined in terms of a linearised permutation (LP) polynomial and multiplication in a finite presemifield. The construction yields many new non-abelian (p ^a , p ^a , p ^a , 1)-RDS. We show that the subset of the LP cocycles defined by the identity LP polynomial and multiplication in a commutative semifield determines the known abelian (p ^a , p ^a , p ^a , 1)-RDS, and give a second new construction using presemifields.We use this cohomological approach to identify equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS with elementary abelian C and E/C. We show that for p = 2, a 3 and p = 3, a 2, every central (p ^a , p ^a , p ^a , 1)-RDS is equivalent to one arising from an LP cocycle, and list them all by equivalence class. For p = 2, a = 4, we list the 32 distinct equivalence classes which arise from field multiplication. We prove that, for any p, there are at least a equivalence classes of central (p ^a , p ^a , p ^a , 1)-RDS, of which one is abelian and a – 1 are non-abelian.     

On the maximum of r-Stirling numbers

Determining the location of the maximum of Stirling numbers is a well-developed area. In this paper we give the same results for the so-called r-Stirling numbers which are natural generalizations of Stirling numbers.     

Cycles of polynomial mappings in two variables over rings of integers in quadratic fields

We find all possible cycle-lengths for polynomial mappings in two variables over rings of integers in quadratic extensions of rationals.     

Sets of Type (a, b) From Subgroups of ΓL(1, pR)

In this paper k-sets of type (a, b) with respect to hyperplanes are constructed in finite projective spaces using powers of Singer cycles. These are then used to construct further examples of sets of type (a, b) using various disjoint sets. The parameters of the associated strongly regular graphs are also calculated. The construction technique is then related to work of Foulser and Kallaher classifying rank three subgroups of AL(1, p ^R). It is shown that the sets of type (a, b) arising from the Foulser and Kallaher construction in the case of projective spaces are isomorphic to some of those constructed in the present paper.     

q-Ideals and a-Ideals in BCI-Algebras

In this note we define the notions of q-ideals and a-ideals in BCI-algebras. We give several characterizations and the extensive theorems about q-ideals and a-ideals. We show that a non-empty subset of a BCI-algebra is a-ideal if and only if it is both q-ideal and p-ideal. Finally, we give four characterizations of associative BCI-algebras by a-ideals and eight characterizations of quasi-associative BCI-algebras by q-ideals.Supported by Fujian Education Committee Foundation, China.AMS Subject Classification (2000), 03G25, 06F35     

Convergence of self-normalized generalizedU-statistics

Conditions are given for almost certain and distribution convergence of self-normalized generalizedU-statistics composed of random variables without particular probabilistic structure. The set of almost certain limit points of some classicalU-statistics is obtained. A variant of theU-statistic involving squares of some of the random variables is also treated. Applications include Martingale differences, stationary sequences, and the classical i.i.d. case where a Marcinkiewicz-Zygmund-type strong law is obtained.     

Sensitivity analysis of M-estimates

Bahadur representation of the difference of estimators of regression coefficients for the full data set and for the set from which one observation was deleted is given for the M-estimators which are generated by a continuous -function. The representation is invariant with respect to the scale of residuals and it indicates that the bound of the norm of the difference is proportional to the gross error sensitivity. Then for the -function which corresponds to the median it is shown that the difference of the estimates for the full data and for data without one observation, although being bounded in probability, can be much larger than indicated by the gross error sensitivity.     

The Cambridge Mathematical Journal and its descendants: the linchpin of a research community in the early and mid-Victorian Age

The Cambridge Mathematical Journal and its successors, the Cambridge and Dublin Mathematical Journal, and the Quarterly Journal of Pure and Applied Mathematics, were a vital link in the establishment of a research ethos in British mathematics in the period 1837–1870. From the beginning, the tension between academic objectives and economic viability shaped the often precarious existence of this line of communication between practitioners. Utilizing archival material, this paper presents episodes in the setting up and maintenance of these journals during their formative years.     

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.     

Asymptotic results in robust quasi-bayesian estimation

Let Θ be a real random variable with a known (a priori) distribution. Let the observations given Θ be iid with a common distributionF₀(·)=F(·−0)F is known to belong to some family of distributions on the real line. We find estimators of Θ which are asymptotically minimax for two types of loss functions.     

Rationals and decimals as required in the school curriculum: Part 1: Rationals as measurement

In the late seventies, Guy Brousseau set himself the goal of verifying experimentally a theory he had been building up for a number of years. The theory, consistent with what was later named (non-radical) constructivism, was that children, in suitable carefully arranged circumstances, can build their own knowledge of mathematics. The experiment, carried out jointly with his wife, Nadine, in her classroom at the École Jules Michelet, was to teach all of the material on rational and decimal numbers required by the national program with a carefully structured, tightly woven and interdependent sequence of “situations.” This article describes and discusses the first portion of that experiment.     

Some Triple Block Intersection Numbers of Paley 2-Designs of QN-type

We determine here some possible values for thecardinality of the intersection of three blocks from Paley2-(2q+1, q, (q-1)/2) designs where qis a prime power such that (mod 4).     

Absolutely continuous functions of several variables and diffeomorphisms

In [4], a class of absolutely continuous functions of d-variables, motivated by applications to change of variables in an integral, has been introduced. The main result of this paper states that absolutely continuous functions in the sense of [4] are not stable under diffeomorphisms. We also show an example of a function which is absolutely continuous with respect cubes but not with respect to balls.     

Some morphisms of modules over the ring of pseudorational numbers

We study various morphisms of modules over the ring of pseudorational numbers R. We obtain a criterion for a quasi-isomorphism between finitely generated R-modules, introduce the concept of a pseudohomomorphism, and prove that the Krull-Remak-Schmidt theorem holds in the category of pseudohomomorphisms of finitely generated R-modules.     

Asymptotic Behavior of Loss Probability in GI/M/1/K Queue as K Tends to Infinity

We obtain an asymptotic behavior of the loss probability for the GI/M/1/K queue as K for cases of <1, >1 and =1.     

The role of restriction theorems in harmonic analysis on harmonic NA groups

We shall obtain inequalities for Fourier transform via moduli of continuity on NA groups. These results in particular settle the conjecture posed in a recent paper by W.O. Bray and M. Pinsky in the context of noncompact rank one symmetric spaces. These problems naturally demand versions of Fourier restriction theorem on these spaces which we shall prove. We shall also elaborate on the connection between the restriction theorem and the Kunze-Stein phenomena on NA groups. For noncompact Riemannian symmetric spaces of rank one analogues of all the results follow the same way.     

Explicit weight two motivic cohomology complexes and algebraicK-theory

Beilinsonet al.'s motivic cohomology complex is related to a motivic complex (considered earlier by Lichtenbaum) formed using groups related to the scissors congruence groups. This involves an algebraic study of configurations of lines in projective spaces. One thereby obtains a more precise relationship between the cohomology of Beilinson's complex and lower dimensionalK-groups. Most of the proofs omitted in Beilinsonet al. are supplied here. There is also a list of certain torsional elements of the thirdK-group for fields between the rationals and the real cyclotomics.