The intricacy of avoiding arrays is 2

Let A be any n×n array on the symbols [n], with at most one symbol in each cell. An n×n Latin square L avoids A if all entries in L differ from the corresponding entries in A. If A is split into two arrays B and C in a special way, there are Latin squares L_B and L_C avoiding B and C, respectively. In other words, the intricacy of avoiding arrays is 2, the number of arrays into which A has to be split.     

On some families of cooperative fuzzy games

In a fuzzy cooperative game the players may choose to partially participate in a coalition. A fuzzy coalition consists of a group of participating players along with their participation level. The characteristic function of a fuzzy game specifies the worth of each such coalition. This paper introduces well-known properties of classical cooperative games to the theory of fuzzy games, and studies their interrelations. It deals with convex games, exact games, games with a large core, extendable games and games with a stable core.     

On some families of subsemigroups of a numerical semigroup

Semigroup Forum - To a given numerical semigroup S we associate a family of subsemigroups $${partial ^nS}$$ , $$nin {mathbb {N}}$$ , that permits us to understand some of the structure of S....     

On the metric dimension of some families of graphs

The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [G. Chartrand, D. Erwin, G. L. Johns and P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34; C. Poisson and P. Zhang, The metric dimension of unicyclic graphs, J. Comb. Math Comb. Comput. 40 (2002) 17–32], Robotic Navigation [S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Disc. Appl. Math. 70 (1996) 217–229; B. Shanmukha, B. Sooryanarayana and K. S. Harinath, Metric dimension of wheels, Far East J. Appl. Math. 8 (3) (2002) 217–229] and Combinatorial Search and Optimization [A. Sebö and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2) (2004) 383–393]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study its behavior with respect to the join and the cartesian product of graphs.     

On support points for some families of univalent functions

Given a closed subset of the familyS* (α) of functions starlike of order α, a continuous Fréchet differentiable functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α). The support points ofS* (α) is completely characterized and shown to coincide with the extreme points of its convex hulls. Given any finite collection of support points ofS* (α), a continuous linear functional,J, is constructed with this collection as the solution set to the extremal problem ReJ(f) overS* (α).     

On the analytical solution of some families of transcendental equations

The problem of finding the exact analytical closed-form solution of some families of transcendental equations is studied, in some detail, by the Special Trans Function Theory (STFT). The mathematical genesis of the analytical closed-form solution is presented, and the structure of the theoretical derivation, proofs and numerical results confirm the validity and base principle of the STFT. Undoubtedly, the proposed analytical approach implies the qualitative improvement of the conventional analytical and numerical methods.     

On the number of distinct elliptic curves in some families

We give explicit formulas for the number of distinct elliptic curves over a finite field (up to isomorphism over the algebraic closure of the ground field) in several families of curves of cryptographic interest such as Edwards curves and their generalization due to D. J. Bernstein and T. Lange as well as the curves introduced by C. Doche, T. Icart and D. R. Kohel.     

On some properties of min-wise independent families and groups of permutations

Informally, a family of permutations is k-restricted min-wise independent if for any with |X| ⩽ k, each x ∈ X has an equal chance of being mapped to the minimum among π(X). In the second section of this paper, the connection of min-wise independent families of permutations and independence on lth minimum is studied. In the third section, we present a method of constructing a (k + 1)-restricted min-wise independent family from a k-restricted min-wise independent family when k is odd. As a corollary, we improve the existing upper bound on the minimum size of a 4-restricted min-wise independent family. In the last section, we consider min-wise independent groups of permutations. Bibliography: 12 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 316, 2004, pp. 30–41.     

On some families of multi-point iterative methods for solving nonlinear equations

Some semi-discrete analogous of well known one-point family of iterative methods for solving nonlinear scalar equations dependent on an arbitrary constant are proposed. The new families give multi-point iterative processes with the same or higher order of convergence. The convergence analysis and numerical examples are presented.     

On the existence of a cyclic vector for some families of operators

Under certain restrictions, it is proved that a family of self-adjoint commuting operatorsA=(A ) _where is a nuclear space, possesses a cyclic vector iff there exists a Hubert spaceH of full operator-valued measureE, where is the space dual to andE is the joint resolution of the identity of the familyA.Published in Ukrainskii Matematicheskii Zhurnal, Vol.45, No. 10, pp. 1362–1370, October, 1993.     

On multiple pattern avoiding set partitions

We study classes of set partitions determined by the avoidance of multiple patterns, applying a natural notion of partition containment that has been introduced by Sagan. We say that two sets S and T of patterns are equivalent if for each n the number of partitions of size n avoiding all the members of S is the same as the number of those that avoid all the members of T.     

On cyclic strings avoiding a pattern

We count the number of cyclic strings over an alphabet that avoid a single pattern under two different assumptions. In the first case, we assume that the symbols of the alphabet are on numbered positions on a circle, while in the second case we assume that the symbols can be freely rotated on the circle (i.e., we deal with necklaces). In each case, we provide a generating function, and we explain how these two cases are related. For the situation of avoiding more than one pattern, we formulate a general conjecture for the first case, and a conditional result for the second case. We also explain the differences between our theory and the theory of Edlin and Zeilberger (2000) by emphasizing how we modified the definition of the enumeration of cyclic strings that avoid one or more patterns when their lengths are less than the length of the longest pattern.