An answer to Hermann's conjecture on Bleimann–Butzer–Hahn operators

We give a negative answer to the conjecture of Hermann [On the operator of Bleimann, Butzer and Hahn, in: J. Szabados, K. Tandori (Eds.), Approximation Theory, Proc. Conf., Kecskemét/Hung., 1990, North-Holland Publishing Company, Amsterdam, 1991, Colloq. Math. Soc. János Bolyai 58 (1991) 355–360] on Bleimann–Butzer–Hahn operators L_n. Our main result states that for each locally bounded positive function h there exists a continuous positive function f defined on [0,∞) with L_nf→f(n→∞), pointwise on [0,∞), such that

Moreover we construct an explicit counterexample function to Hermann's conjecture.     

On measures of fuzziness of solutions of fuzzy relation equations with generalized connectives

By using a general class of fuzzy connectives of Yager [Fuzzy Sets and Systems4 (1980), 235–242], Pedrycz [Fuzzy relational equations with generalized convectives and their applications, Fuzzy Sets and Systems10 (1983), 185–201] has shown that the classical fuzzy relation equations of Sanchez [in “Fuzzy Automata and Decision Processes” (M. M. Gupta, G. N. Saridis, and B. R. Gaines, Eds.), pp. 221–234, North-Holland, Amsterdam, 1977] can be considered as a particular case of a more extensive class of fuzzy equations. For such types of equations, in this paper the solutions having the greatest energy measure and the smallest possible entropy measure of fuzziness are characterized.     

Maeda-type Decomposition of CJ-generated Algebraic Lattices

We prove necessary and sufficient conditions for the decomposition of an arbitrary CJ-generated algebraic lattice into a direct product of subdirectly irreducible lattices. We generalize earlier results due to F. Maeda, T. Katriák and the present author. New structure theorems for two classes of CJ-generated algebraic lattices are also obtained.AMS Subject Classification (1991) 06B05 06B10Research partially supported by Hungarian National foundation for Scientific Research, Grant No. T029525 and T030243 and by János Bolyai Grant of Hungarian Academy of Science.     

On the relationship between cmp and def for separable metrizable spaces

For each pair of positive integers k and m with k?m there exists a separable metrizable space X(k,m) such that cmpX(k,m)=k and defX(k,m)=m. This solves Problem 6 from [J.M. Aarts, T. Nishiura, Dimension and Extensions, North-Holland, Amsterdam, 1993, p. 71].     

An extension of the concept of gradient semigroups which is stable under perturbation

In this article we introduce the concept of a gradient-like nonlinear semigroup as an intermediate concept between a gradient nonlinear semigroup (those possessing a Lyapunov function, see [J.K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., vol. 25, Amer. Math. Soc., 1989]) and a nonlinear semigroup possessing a gradient-like attractor. We prove that a perturbation of a gradient-like nonlinear semigroup remains a gradient-like nonlinear semigroup. Moreover, for non-autonomous dynamical systems we introduce the concept of a gradient-like evolution process and prove that a non-autonomous perturbation of a gradient-like nonlinear semigroup is a gradient-like evolution process. For gradient-like nonlinear semigroups and evolution processes, we prove continuity, characterization and (pullback and forwards) exponential attraction of their attractors under perturbation extending the results of [A.N. Carvalho, J.A. Langa, J.C. Robinson, A. Suárez, Characterization of non-autonomous attractors of a perturbed gradient system, J. Differential Equations 236 (2007) 570-603] on characterization and of [A.V. Babin, M.I. Vishik, Attractors in Evolutionary Equations, Stud. Math. Appl., vol. 25, North-Holland, Amsterdam, 1992] on exponential attraction.     

Homogenization with respect to Gibbs measures for periodic drift diffusions on lattices

A homogenization problem for infinite dimensional diffusion processes indexed by ${\mathbf{Z}}^d$ having periodic drift coefficients is considered. By an application of the uniform ergodic theorem for the infinite dimensional diffusion processes based on logarithmic Sobolev inequalities, an $L^1$ type homogenization property of the processes with respect to an invariant measure is proved. This is the, so far, best possible analogue in infinite dimensions to a known result in the finite dimensional case (cf. [G. Papanicolaou, S. Varadhan, Boundary value problems with rapidly oscillating random coefficients, Seria Coll. Math. Soc. Janos Bolyai, vol. 27, North-Holland, 1979. [4]]). To cite this article: S. Albeverio et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).     

Sufficient Optimality Criterion for Linearly Constrained,Separable Concave Minimization Problems

A sufficient optimality criterion for linearly-constrained concave minimization problems is given in this paper. Our optimality criterion is based on the sensitivity analysis of the relaxed linear programming problem. The main result is similar to that of Phillips and Rosen (Ref. 1); however, our proofs are simpler and constructive.In the Phillips and Rosen paper (Ref. 1), they derived a sufficient optimality criterion for a slightly different linearly-constrained concave minimization problem using exponentially many linear programming problems. We introduce special test points and, using these for several cases, we are able to show optimality of the current basic solution.The sufficient optimality criterion described in this paper can be used as a stopping criterion for branch-and-bound algorithms developed for linearly-constrained concave minimization problems.This research was supported by a Bolyai János Research Fellowship BO/00334/00 of the Hungarian Academy of Science and by the Hungarian Scientific Research Foundation, Grant OTKA T038027.     

The minimum number of components in 4-regular perfect systems of difference sets

The number of components m in regular (m, 5, c)-systems is given in the literature to date by the inequality m ? 4c ? 2 (Bermond et al., “Proceedings, 18th Hungarian Combin. Colloq.”, North-Holland, Amsterdam, 1976). The case m = 4c ? 2 is called extremal. It is proved that (4c ? 2, 5, c)-systems do not exist. An example of a (4c, 5, c)-system with c = 2, is given. Since, in a 4-regular system, m must be even, loc. cit., it is concluded that the lower bound on the number of components is given by m >/ 4c.     

Exact constants in generalized inequalities for intermediate derivatives

Consider the Sobolev space W ₂ ⁿ (?₊) on the semiaxis with norm of general form defined by a quadratic polynomial in derivatives with nonnegative coefficients. We study the problem of exact constants A _n,k in inequalities of Kolmogorov type for the values of intermediate derivatives |f ^(k)(0)| ≤ A _n,k ‖f‖. In the general case, the expression for the constants A _n,k is obtained as the ratio of two determinants. Using a general formula, we obtain an explicit expression for the constants A _n,k in the case of the following norms: $$ \left\| f \right\|_1^2 = \left\| f \right\|_{L_2 }^2 + \left\| {f^{(n)} } \right\|_{L_2 }^2 and\left\| f \right\|_2^2 = \sum\limits_{l = 0}^n {\left\| {f^{(l)} } \right\|_{L_2 }^2 } . $$ In the case of the norm ‖ · ‖₁, formulas for the constants A _n,k were obtained earlier by another method due to Kalyabin. The asymptotic behavior of the constants A _n,k is also studied in the case of the norm ‖ · ‖₂. In addition, we prove a symmetry property of the constants A _n,k in the general case.     

Smallest Set-Transversals of k-Partitions

We asymptotically solve an open problem raised independently by Sterboul (Colloq Math Soc J Bolyai 3:1387–1404, 1973), Arocha et al. (J Graph Theory 16:319–326, 1992) and Voloshin (Australas J Combin 11:25–45, 1995). For integers n ≥ k ≥ 2, let f(n, k) denote the minimum cardinality of a family ${\mathcal H}$ of k-element sets over an n-element underlying set X such that every partition ${X_1\cup\cdots\cup X_k=X}$ into k nonempty classes completely partitions some ${H\in\mathcal H}$ ;  that is, ${|H\cap X_i|=1}$ holds for all 1 ≤ i ≤ k. This very natural function—whose defining property for k = 2 just means that ${\mathcal H}$ is a connected graph—turns out to be related to several extensively studied areas in combinatorics and graph theory. We prove general estimates from which ${ f(n,k) = (1+o(1))\, \tfrac{2}{n}\,{n\choose k}}$ follows for every fixed k, and also for all k = o(n ^1/3), as n → ∞. Further, we disprove a conjecture of Arocha et al. (1992). The exact determination of f(n,k) for all n and k appears to be far beyond reach to our present knowledge, since e.g. the equality ${f(n,n-2)={n-2\choose 2}-{\rm ex}(n,\{C_3,C_4\})}$ holds, where the last term is the Turán number for graphs of girth 5.     

Splines on generalized quasi-cross-cut partitions

A new method of obtaining the dimensions of the polynomial splines of degree k and smoothness r f- 1 (r ⩾ 1, and k ⩾ r f- 1 + ξ_Δ) on generalized quasi-cross-cut partitions is presented, where ξ_Δ is a number depending on the structure of the partition.     

Log-convexity and log-concavity of hypergeometric-like functions

We find sufficient conditions for log-convexity and log-concavity for the functions of the forms a?∑fkk(a)xk, a?∑fkΓ(a+k)xk and a?∑fkxk/k(a). The most useful examples of such functions are generalized hypergeometric functions. In particular, we generalize the Turán inequality for the confluent hypergeometric function recently proved by Barnard, Gordy and Richards and log-convexity results for the same function recently proved by Baricz. Besides, we establish a reverse inequality which complements naturally the inequality of Barnard, Gordy and Richards. Similar results are established for the Gauss and the generalized hypergeometric functions. A conjecture about monotonicity of a quotient of products of confluent hypergeometric functions is made.     

Inequalities between best polynomial approximations and some smoothness characteristics in the space <Emphasis Type="Italic">L</Emphasis> <Subscript>2</Subscript> and widths of classes of functions

We obtain exact constants in Jackson-type inequalities for smoothness characteristics Λ_k(f), k ∈ N, defined by averaging the kth-order finite differences of functions f ∈ L₂. On the basis of this, for differentiable functions in the classes L₂^r, r ∈ N, we refine the constants in Jackson-type inequalities containing the kth-order modulus of continuity ω_k. For classes of functions defined by their smoothness characteristics Λ_k(f) and majorants Φ satisfying a number of conditions, we calculate the exact values of certain n-widths.     

Variants of the Hajnal-Szemerédi Theorem

The Hajnal-Szemerédi Theorem [Hajnal & Szemerédi, Colloq Math Soc J Bolyai, 1970] states that a graph with hk vertices and minimum degree at least (h − 1)k contains k vertex disjoint copies of K_h. Its proof is not algorithmic. Here, we present an algorithm that, for a fixed h, finds in such a graph k − C(h) vertex disjoint copies of K_h in polynomial time in k, C(h) being a constant depending on h only. The proof suggests a variant of the theorem for h-partite graphs, which is conjectured here and proven in a slightly weaker form in some special cases. © 1999 John Wiley & Sons, Inc. J Graph Theory 31: 275–282, 1999     

A projected subgradient method for solving generalized mixed variational inequalities

We consider the projected subgradient method for solving generalized mixed variational inequalities. In each step, we choose an ε_k-subgradient u^k of the function f and w^k in a set-valued mapping T, followed by an orthogonal projection onto the feasible set. We prove that the sequence is weakly convergent.     

On Constantive Simple and Order-Primal Algebras

A finite algebra is said to be order-primal if its clone of all term operations is the set of all operations defined on A which preserve a given partial order ≤ on A. In this paper we study algebraic properties of order-primal algebras for connected ordered sets (A; ≤). Such order-primal algebras are constantive, simple and have no non-identical automorphisms. We show that in this case cannot have only unary fundamental operations or only one at least binary fundamental operation. We prove several properties of the varieties and the quasi-varieties generated by constantive and simple algebras and apply these properties to order-primal algebras. Further, we use the properties of order-primal algebras to formulate new primality criteria for finite algebras.* Research supported by the Hungarian research grant No. TO34137 and by the János Bolyai grant.** Research supported by the Thailand Research Fund.     

Global stability in difference equations satisfying the generalized Yorke condition

We present several conditions sufficient for global stability of the zero solution of nonautonomous difference equation xn+1=qxn+fn(xn,…,xn−k), n∈Z, when the nonlinearities fn satisfy a sort of negative feedback condition. Moreover, for every k∈N, we indicate qk such that one of our stability conditions is sharp if q∈(0,qk]. We apply our results to discrete versions of Nicholson's blowflies equation, the Mackey-Glass equations, and the Wazewska and Lasota equation.     

Jackson-Stechkin type inequalities for special moduli of continuity and widths of function classes in the space L 2

We obtain sharp Jackson-Stechkin type inequalities for moduli of continuity of kth order Ω _k in which, instead of the shift operator T _h f, the Steklov operator S _h (f) is used. Similar smoothness characteristic of functions were studied earlier in papers of Abilov, Abilova, Kokilashvili, and others. For classes of functions defined by these characteristics, we calculate the exact values of certain n-widths.     

A generalized mean-value theorem

We give another proof of Seymour and Zaslavsky's theorem: For every familyf ₁,f ₂,...,f _n of continous functions defined on [0, 1], there exists a finite setF[0, 1] such that the average sum off _k overF coincides with the integral off _k for everyk=1, 2,...,n.