Acta Mathematica Hungarica最新文献

A (k)-uniform family (mathcal{F}) is called intersecting if (Fcap F'neq emptyset) for all (F,F'in mathcal{F}). The shadow family (partial mathcal{F}) is the family of ((k-1))-element sets that are contained in some members of (mathcal{F}). The shadow degree (or minimum positive co-degree) of (mathcal{F}) is defined as the maximum integer (r) such that every (Ein partial mathcal{F}) is contained in at least (r) members of (mathcal{F}). Balogh, Lemons and Palmer [1] determined the maximum size of an intersecting (k)-uniform family with shadow degree at least (r) for (ngeq n_0(k,r)), where (n_0(k,r)) is doubly exponential in (k) for (4leq rleq k). In the present paper, we present a short proof of this result for (ngeq 2frac{(r+1)^r}{binom{2r-1}{r}}kbinom{2k}{k-1}) and (4leq rleq k).

A set (D) of vertices in a graph (G) is a dominating set, if each vertex of (G) that is not in (D) is adjacent to at least one vertex of (D). The minimum cardinality among all dominating sets in (G) is called the domination number of (G) and is denoted by (gamma(G)). A dominating set (S) such that the induced subgraph by (S) has at least one isolated vertex is called an isolate dominating set. An isolate dominating set of minimum cardinality is called the isolate domination number and is denoted by (gamma_0(G)). We define the isolate bondage number of a graph (G) to be the cardinality of a smallest set (E) of edges for which (gamma_0(G-E)>gamma_0(G)) and is denoted by (b_0(G)). In this paper, we initiate a study on the isolate bondage number.

In this paper we consider a generalized polynomial ( f colon mathbb{R}^N to mathbb{R} ) that satisfies the additional equation ( f(x) f(y) = 0 ) for the pairs ( (x,y) in D ), where ( D subseteq mathbb{R}^{2N} ) has a positive Lebesgue measure or it is a second category Baire set. We prove that ( f(x) = 0 ) for all ( x in mathbb{R}^N ). In fact, the first statement is established in a considerably more general setting. Then we formulate statements concerning the signs of generalized monomials ( g colon mathbb{R} to mathbb{R} ) of even degree that satisfy the inequality ( g(x) g(y) geq 0 ) for the pairs ( (x,y) in E ), where ( E subseteq mathbb{R}^{2} ) has a positive planar Lebesgue measure or it is a second category Baire set.

We consider the conditional Jensen functional equation

for functions (f colon mathcal{G} tomathcal{V}), where ((mathcal{G},+)) and ((mathcal{V},+)) are uniquely 2-divisible groups,with ((mathcal{V},+)) being abelian. Additionally, we investigate the hyperstability of thisfunctional equation.

In a seminal paper, Choquet introduced an integral formula toextend a monotone increasing setfunction on a sigma-algebra to a (nonlinear) functionalon bounded measurable functions. The most important special case is whenthe setfunction is submodular; then this functional is convex (and vice versa). Inthe finite case, an analogous extension was introduced by this author; this is arather special case, but no monotonicity was assumed. In this note we show thatChoquet's integral formula can be applied to all submodular setfunctions, andthe resulting functional is still convex. We extend the construction to submodularsetfunctions defined on a set-algebra (rather than a sigma-algebra). The mainproperty of submodular setfunctions used in the proof is that they have boundedvariation. As a generalization of the convexity of the extension, we show that(under smoothness conditions) a "lopsided" version of Fubini's Theorem holds.

We consider the harmonic series (S(k)=sum^{(k)} m^{-1}) over the integers having (k) occurrences of a given block of (b)-ary digits, of length (p), and relatethem to certain measures on the interval [0, 1). We show that these measures converge weakly to (b^p) times the Lebesgue measure, a fact which allows a new proofof the theorem of Allouche, Hu, and Morin [4] which says (lim S(k)=b^plog(b)).A quantitative error estimate will be given. Combinatorial aspects involve generating series which fall under the scope of the Goulden–Jackson cluster generatingfunction formalism and the work of Guibas–Odlyzko on string overlaps.

In this paper new collectively fixed point results are presentedfor Wu type maps and our ideas generate coincidence results between KKM typemaps and Wu maps. Our arguments are based on Himmelberg’s fixed point theorem and a fixed point result on admissible convex sets in a Hausdorff topologicalvector space.

The enhanced power graph of a group (G) is a graph with vertex set (G), where two distinct vertices (x) and (y) are adjacent if and only if there exists an element (w) in (G) such that both (x) and (y) are powers of (w). Kumar, Ma, Parveen and Singh in [22] found the exact vertex connectivity of the enhanced power graph of finite nilpotent groups whose all except one Sylow subgroups are cyclic. In this paper, we determine the exact vertex connectivity of the enhanced power graph of any finite nilpotent group in full generality, by connecting it to the minimum number of roots of a prime order element in its Sylow subgroups.

Using basic tools of mathematical analysis and elementary probabilitytheory we address several problems on the irrationality of series of distinctunit fractions,(sum_k 1/a_k). In particular, we study subseries of the Lambert series (sum_k 1/(t^k-1)) and two types of irrationality sequences ((a_k)) introduced by PaulErdős and Ronald Graham. Next, we address a question of Erdős, who askedhow rapidly a sequence of positive integers ((a_k)) can grow if both series (sum_k 1/a_k) and (sum_k 1/(a_k+1))have rational sums. Our construction of double exponentiallygrowing sequences ((a_k)) with this property generalizes to any number (d) of series(sum_k 1/(a_k+j)),(j=0,1,2,ldots,d-1),and, in particular, also gives a positive answerto a question of Erdős and Ernst Straus on the interior of the set of (d)-tuples of their sums.Finally, we prove the existence of a sequence ((a_k)) such that all well-defined sums (sum_k 1/(a_k+t)),(tinmathbb{Z}), are rational numbers, giving a negative answer to a conjecture by Kenneth Stolarsky.

We discuss about maximal Tychonoff spaces and minimal paracompact, (T_{2}) countable spaces.