For $1\le c\le p-1$, let $E_1,E_2,\dots ,E_m$ be fixed numbers of the set $\{0,1\}$, and let $a_1, a_2,\dots , a_m$ $(1\le a_i\le p$, $i=1,2,\dots , m)$ be of opposite parity with $E_1,E_2,\dots ,E_m$ respectively such that $a_1a_2\dots a_m\equiv c\pmod p$. Let \begin {equation*} N(c,m,p)=\frac {1}{2^{m-1}}\mathop {\mathop {\sum }_{a_1=1}^{p-1} \mathop {\sum }_{a_2=1}^{p-1}\dots \mathop {\sum }_{a_m=1}^{p-1}} _{a_1a_2\dots a_m\equiv c\pmod p} (1-(-1)^{a_1+E_1})(1-(-1)^{a_2+E_2})\dots (1-(-1)^{a_m+E_m}). \end {equation*} \endgraf We are interested in the mean value of the sums \begin {equation*} \sum _{c=1}^{p-1}E^2(c,m,p), \end {equation*} where $ E(c,m,p)=N(c,m,p)-({(p-1)^{m-1}})/({2^{m-1}})$ for the odd prime $p$ and any integers $m\ge 2$. When $m=2$, $c=1$, it is the Lehmer problem. In this paper, we generalize the Lehmer problem and use analytic method to give an interesting asymptotic formula of the generalized Lehmer problem.
Investigated are possibilistic distributions taking as their values sequences from the infinite Cartesian product of identical copies of a fixed finite subset of the unit interval of real numbers. Uniform and lexicographic partial orderings on the space of these sequences are defined and the related complete lattices introduced. Lattice-valued entropy function is defined in the common pattern for both the orderings, naturally leading to different entropy values for the particular ordering applied in the case under consideration. The mappings on possibilistic distributions with uniform partial ordering under which the corresponding entropy values are conserved as well as approximations of possibilistic distributions with respect to this entropy function are also investigated.
We give a representation of the class of all n-dimensional copulas such that, for a fixed m∈N, 2≤m<n, all their m-dimensional margins are equal to the independence copula. Such an investigation originated from an open problem posed by Schweizer and Sklar.
One of numerical invariants concerning domination in graphs is the $k$-subdomination number $\gamma ^{-11}_{kS}(G)$ of a graph $G$. A conjecture concerning it was expressed by J. H. Hattingh, namely that for any connected graph $G$ with $n$ vertices and any $k$ with $\frac{1}{2} n < k \leqq n$ the inequality $\gamma ^{-11}_{kS}(G) \leqq 2k - n$ holds. This paper presents a simple counterexample which disproves this conjecture. This counterexample is the graph of the three-dimensional cube and $k=5$.
The symbol K(B,C) denotes a directed graph with the vertex set B∪C for two (not necessarily disjoint) vertex sets B,C in which an arc goes from each vertex of B into each vertex of C. A subdigraph of a digraph D which has this form is called a bisimplex in D. A biclique in D is a bisimplex in D which is not a proper subgraph of any other and in which B ≠ ∅ and C ≠ ∅. The biclique digraph C→ (D) of D is the digraph whose vertex set is the set of all bicliques in D and in which there is an arc from K(B1, C1) into K(B2, C2) if and only if C1 ∩ B2 = ∅. The operator which assigns C→ (D) to D is the biclique operator C→ . The paper solves a problem of E. Prisner concerning the periodicity of C→ .
In this note we deal with a question concerning monounary algebras which is analogous to an open problem for partially ordered sets proposed by Duffus and Rival.
In the paper by Hilout and Piétrus (2006) a semilocal convergence analysis was given for the secant-like method to solve generalized equations using Hölder-type conditions introduced by the first author (for nonlinear equations). Here, we show that this convergence analysis can be refined under weaker hypothesis, and less computational cost. Moreover finer error estimates on the distances involved and a larger radius of convergence are obtained.