Generalized Sidon sets of perfect powers

The Ramanujan Journal - For $$h \ge 2$$ and an infinite set of positive integers A, let $$R_{A,h}(n)$$ denote the number of representations of the positive integer n as the sum of h distinct terms...     

Bounds for the generalized Marcum function of the second kind

The Ramanujan Journal - This paper provides elementary proofs for several types of congruences involving multipartitions and self-convolutions of the divisor function. Our computations use...     

Asymptotic Analysis of Finite-Source M/GI/1 Retrial Queueing Systems with Collisions and Server Subject to Breakdowns and Repairs

Methodology and Computing in Applied Probability - This paper deals with a retrial queuing system with a finite number of sources and collision of the customers, where the server is subject to...     

Study of the influence of physical aging on macroradical decay in poly(methyl methacrylate)

The macroradical decay in poly(methyl methacrylate) samples with different thermal histories was investigated in the temperature interval 20–100 °C using ESR spectroscopy and the second order kinetic model. The rate constants exhibit two different regimes with the transitions atT _tr=68±1°C which are independent of thermal treatment. ForT<T _tr andT>T _tr the rate constants as well as the corresponding activation parameters are sensitive to history because of different physical microstructures. The compensation law, i.e., the linear relation between lnk _{o, eff} andE _eff, was analyzed in terms of the so-called compensation quantitiesk _c andT _c and a proximity betweenT _c=T _tr andT _o=53±3 °C — Vogel temperature for -segmental dynamics was found. A comparison of kinetic and dynamic data suggests that the decay of terminal macroradicals in the low-temperature region is controlled by secondary relaxations and that the -mobility contributes to a more rapid decay at higher temperatures belowT _g.     

Identities of symmetric and skew-symmetric matrices in characteristicp

It is well known how the Kostant-Rowen Theorem extends the validity of the famous Amitsur-Levitzki identity to skew-symmetric matrices. Here we give a general method, based on a graph theoretic approach, for deriving extensions of known permanental-type identities to skew-symmetric and symmetric matrices over a commutative ring of prime characteristic. Our main result has a typical Kostant-Rowen flavour: IfM≥p[n+1/2] then $C_M (X,Y) = \sum\limits_{\alpha ,\beta \in Sym(M)} {x_{\alpha (1)} y_{\beta (1)} x_{\alpha (2)} y_{\beta (2)} } ...x_{\alpha (M)} y_{\beta (M)} = 0$ is an identity onM _n ^? (Ω), the set ofnxn skew-symmetric matrices over a commutative ring Ω withp1_Ω=0 (provided that $P > \sqrt {[n + 1/2)} $ ). Otherwise, the stronger conditionM≥pn implies thatC _M(X,Y)=0 is an identity on the full matrix ringM _n(Ω).     

Remarks on power structures

Chris Brink, in his paper,Power Structures, describes the construction ofgeneralized quotient algebras. IfA is any algebra, the corresponding generalized quotient algebraA/R is defined for every value-preserving relation . As we will see in the present paper, in order to make the corresponding operations on the setA/R well-defined, it is not enough forR to be value-preserving. The only redundancy in the definition of the usual quotient algebra is symmetry.Presented by G. Grätzer.     

Smallest maximal snakes of translates of convex domains

LetK be a convex domain. A maximal snake of sizen is a set of non-overlapping translatesK ₁, ...,K _N ofK, whereK _i touchesK _j if and only if |i –j|=1 and no translate ofK can touchK ₁ orK _n without intersecting an additionalK _i ,i=1, ...,n. The size of the smallest maximal snake is proved to be 11 ifK is a parallelogram and to be 10 otherwise.     

A note on packing paths in planar graphs

Seymour (1981) proved that the cut criterion is necessary and sufficient for the solvability of the edge-disjoint paths problem when the union of the supply graph and the demand graph is planar and Eulerian. When only planarity is required, Middendorf and Pfeiffer (1993) proved the problem to be NP-complete. For this case, Korach and Penn (1992) proved that the cut criterion is sufficient for the existence of a near-complete packing of paths. Here we generalize this result by showing how a natural strengthening of the cut criterion yields better packings of paths.Analogously to Seymour's approach, we actually prove a theorem on packing cuts in an arbitrary graph and then the planar edge-disjoint paths case is obtained by planar dualization. The main result is derived from a theorem of Seb (1990) on the structure of ±1 weightings of a bipartite graph with no negative circuits.Research partially supported by the Hungarian National Foundation for Scientific Research Grants OTKA 2118 and 4271.     

Lines of curvature near umbilical points on surfaces immersed in R4

LetM be a compact orientable surface,I:M R⁴ an immersion of classC ^r and a normal unitary smooth vector field. We prove that in the space of pairs (I, ) the family of pairs all whose umbilical points are Darbouxian is generic.     

On the capacity of boolean graph formulæ

In a recent paper we have solved several well-known combinatorial problems treating them as special cases of our generalization of Shannon's notion of graph capacity. We present a new simple formalism to deal with all such problems in a unified manner, considering graphs or families of graphs as special formulæ the variables of which are pairs of vertices of their common vertex sets. In all of these problems, the maximum size of a set ofn-length sequences from a fixed alphabet is to be determined under various restrictions on the element pairs present in the same coordinate of any two sequences from the set. For sufficiently homogeneous formulæ capacity becomes computable.New applications include generalizations of our result on the maximum number of pairwise qualitatively independentk-partitions of ann-set from independence to various forms of qualitative dependence.Work partially supported by the Italian Ministry of the University and Scientific Research, Project: Algoritmi, Modelli di Calcolo e Strutture Informative.