Volume 65, Issue 6

We give asymptotically sharp upper bounds on the radius and diameter of (i) a connected graph, (ii) a connected triangle-free graph, (iii) a connected C4-free graph of given order, minimum degree, and maximum degree. We also give better bounds on the radius and diameter for triangle-free graphs with a given order, minimum degree and a given number of distinct terms in the degree sequence of the graph. Our results improve on old classical theorems by Erd˝os, Pach, Pollack and Tuza [Radius, diameter, and minimum degree, J. Combin. Theory Ser. B 47 (1989), 73-79] on radius, diameter and minimum degree.

In this paper, we define the concepts of the radical of an ideal and a primary ideal in posets. Further, the analogue of the first and the second uniqueness theorems regarding primary decomposition of an ideal are obtained. In the last section, we prove that if an ideal in a poset Q has a minimal primary decomposition, then the diameter of the corresponding zero-divisor graph with respect to this ideal is exactly equal to three.

We characterize the set of regular elements of an ordered semigroup S which is a left (right) ideal, an ideal, a bi-ideal or a quasi-ideal of S using the left (resp. right) ideal and the quasi-ideal L E (S)∩R E (S) of S, where E(S) is the set of elements of S for which e ≤ e 2 . Illustrative example is given.

Let (a, a + d, a + 2d) be an arithmetic progression of positive integers. The following statements are proved: (1) If a | 2d, then (a, a + d, a + 2d) ∈ mdeg(Tame(ℂ 3 )). (2) If a ∤ 2d and (a, a + d, a + 2d) ∉ {(4i, 5i, 6i), (4i, 7i, 10i) : i ∈ ℕ + }, then (a, a + d, a + 2d) ∉ mdeg(Tame(ℂ 3 )).

In this paper we prove the following result. Let R be a 2-torsion free semiprime ring and let f : R → R be an additive mapping satisfying the relation f(x)x 2 + x 2 f(x) = 0 for all x ∈ R. In this case f = 0. Any semisimple Banach algebra (for example, C* algebra) is semiprime. Therefore this algebraic result might be of some interest from functional analysis point of view.

Let M n (R) be the ring of all n × n matrices over a unital ring R, let M be a 2-torsion free unital M n (R)-bimodule and let D: M n (R)→M be an additive map. We prove that if D(a)b + aD(b) + D(b)a + bD(a) = 0 whenever a,b ∈ M n (R) are such that ab = ba = 0, then D(a) = δ(a) + aD(1), where δ : M n (R)→M is a derivation and D(1) lies in the centre of M. It is also shown that D is a generalized derivation if and only if D(a)b + aD(b) + D(b)a + bD(a) − aD(1)b − bD(1)a = 0 whenever ab = ba = 0. We apply this results to provide that any (generalized) Jordan derivation from M n (R) into a 2-torsion free M n (R)-bimodule (not necessarily unital) is a (generalized) derivation. Also, we show that if φ: Mn(R) → Mn(R) is an additive map satisfying φ(ab+ba) = aφ(b) + φ(b)a (a,b ∈ M n (R)), then φ(a) = aφ(1) for all a ∈ M n (R), where φ(1) lies in the centre of M n (R). By applying this result we obtain that every Jordan derivation of the trivial extension of M n (R) by M n (R) is a derivation.

In this paper, we prove the existence and uniqueness of solutions of fractional impulsive differential equations with nonlocal boundary conditions by applying the contraction mapping principle.

In this paper we present new characterizations of Banach spaces possessing the Radon-Nikodym property in terms of the limit average range of additive interval functions.

A Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.

In [RUSCHEWEYH, S.-SHEIL-SMALL, T.: Hadamard product of schlicht functions and the Poyla-Schoenberg conjecture, Comment. Math. Helv. 48 (1973), 119-135] the authors proved the P`olya-Schoenberg conjecture that the class of convex univalent functions is preserved under convolution, namely K ∗ K = K. They proved also that the class of starlike functions and the class of close-to-convex functions are closed under convolution with the class K. In this paper we consider similar convolution problems for some classes of functions. Especially we give a new applications of a result [SOKÓŁ, J.: Convolution and subordination in the convex hull of convex mappings, Appl. Math. Lett. 19 (2006), 303-306] on the subordinating relations in the convex hull of convex mappings under convolution. The paper deals with several ideas and techniques used in geometric function theory. Besides being an application to those results it provides interesting corollaries concerning special functions, regions and curves.

Recently, Opps, Saad and Srivastava have given the recursion formulas of Appell’s function F 2 by the contiguous relation of the Gauss hypergeometric series 2 F 1 . Then, Wang has presented various recursion formulas for F 1 , F 2 , F 3 and F 4 Appell’s hypergeometric functions. The aim of this paper is to present various recursion formulas for Srivastava hypergeometric functions by the contiguous relations of hypergeometric series.

Sharp conditions are established for the existence of slowly varying solution and regularly varying solution of index 1 of the half-linear functional differential equation with both retarded and advanced arguments of the form (|x′(t)| α sgn x′(t))′ ± p(t)|x(g(t))| α sgn x(g(t)) ± q(t)|x(h(t))| α sgn x(h(t)) = 0, where α > 0 is a constant, p, q: [a,∞) → (0,∞), a ≧ 0 are continuous functions, g, h are continuous and increasing with g(t) < t, h(t) > t for t ≥ a and .

The boundary value problems for linear and nonlinear singular degenerate differential-operator equations with variable coefficients are studied. Degenerate linear problems are considered on the moving domain. The uniform maximal regularity properties of the singular degenerate linear problem with parameters and the existence and uniqueness result for the degenerate nonlinear problem is shown. This problem occur in fluid mechanics and environmental engineering.

The paper is devoted to the interior Euler-Lagrange operator in field theory, representing an important tool for constructing the variational sequence. We give a new invariant definition of this operator by means of a natural decomposition of spaces of differential forms, appearing in the sequence, which defines its basic properties. Our definition extends the well-known cases of the Euler-Lagrange class (Euler-Lagrange form) and the Helmholtz class (Helmholtz form). This linear operator has the property of a projector, and its kernel consists of contact forms. The result generalizes an analogous theorem valid for variational sequences over 1-dimensional manifolds and completes the known heuristic expressions by explicit characterizations and proofs.

We present the results on the stability of some set-valued functional equations with applications to the gamma-type equations or linear equations.

The paper is sacrificed to discussing the property of some special subfamily F of the family of all permutations of ℕ, especially in the context of the fundamental properties of families C and D of the, so called, convergent and divergent permutations, in other words, permutations preserving or not preserving the convergence of the rearranged real series, respectively. Family F is the subgroup of the group G generated by C. In the paper we will prove, among others, that F ⊂ C −1 ◦ C and F \ (C ◦ C −1 ) ≠ ∅. So, in particular we have G ≠ C◦C −1 . The family F is neither the subset nor superset of any of the families C, D and C −1 . By using the permutations belonging to the family F we receive the strengthening of some Kronrod’s theorems (which are the generalizations of the Riemann Derangement Theorem).

In this paper, using an A-summation process we give a theorem of the Korovkin type for double sequences of positive linear operators acting from a weighted space C p1 into B p2 . We also give a quantitative version of our main theorem with the help of the weighted modulus of continuity.

In this paper, we extend some results concerning statistical convergence in locally solid Riesz spaces by introducing a more general idea of convergence, namely I λ -statistical convergence. We investigate the ideas of I λ -statistical-τ-convergence, I λ -statistical-τ-boundedness and I λ -statistical-τ-Cauchy condition of sequences in a locally solid Riesz space endowed with a topology τ and investigate some of their consequences.

Let B be an accretive operator on a complex Hilbert space and let V be the classical Volterra operator on L 2 (0, 1). We study a contraction property for (I −mB)(I +B) −1 , m ∈ ℝ. In particular, we show that ‖(I −mV)(I+V) −1 ‖ = 1 only for m ∈ [−1, 1].

In this paper by using the Suzuki’s method we shall present some results on absolute retractivity of the fixed points set of KS-multifunctions.

Let ℤ 2 be equipped with the Khalimsky topology κ, it is a T 0 -Alexandroff topology which has some specific properties concerning continuity and connectivity. We define digital-arcs and the geodesics; this enables us to define D-convexity on the digital plane (ℤ 2 , κ). First, we prove a theorem dealing with the relationship between D-convexity and connectivity. The second result links together the convexity in ℝ 2 and the D-convexity in ℤ 2 . For this purpose, we suggest the continuous digitization of the real line segment and thus prove that the digitization of a convex subset of ℝ 2 is a D-convex subset of (ℤ 2 , κ).

In this paper we obtain natural boundary conditions for a large class of variational problems with free boundary values. In comparison with the already existing examples, our framework displays complete freedom concerning the topology of Y - the manifold of dependent and independent variables underlying a given problem - as well as the order of its Lagrangian. Our result follows from the natural behavior, under boundary-friendly transformations, of an operator, similar to the Euler map, constructed in the context of relative horizontal forms on jet bundles (or Grassmann fibrations) over Y . Explicit examples of natural boundary conditions are obtained when Y is an (n + 1)-dimensional domain in ℝ n+1 , and the Lagrangian is first-order (in particular, the hypersurface area).

In the paper, several precise exponential inequalities for the sums of bounded or semi-bounded random variables are established, which involve independent random variables, martingale difference sequence, negatively associated random variables, Markov chains.

Let {X n | n ≥ 1} be a strictly stationary sequence of φ-mixing random variables with zero mean and finite variance. Set we show the exact rates in the law of iterated logarithm of a kind of weighted infinite series of as ε ↘ 0.

The technique of covariant differentiation is being implemented into the single integral variational calculus with second derivatives in Riemannian manifolds and into the Grässer-Rund-Weyssenhoff homogeneous generalized canonical framework for the parameter-invariant variational problem. As an example of physical nature we relate to each other the equations of motion with higher derivatives that describe the physical phenomena known as ‘Zitterbewegung’, the electromagnetic self-interaction, and spherical top. Although two of these phenomena, the quiver (the ‘Zitterbewegung’) and the radiation friction, have been known only within the framework of Special Relativity, our approach allows to generalize the corresponding equations of motion to the curved spacetime. From the geometrical point of view these generalizations depend neither on the signature of the Riemannian space nor on its dimension.

S. Gudder and, later, S. Pulmanová and E. Vinceková, have studied in two recent papers a certain ordering of bounded self-adjoint operators on a Hilbert space. We present some further results on this ordering and show that some structure theorems of the ordered set of operators can be obtained in a more abstract setting of posets having the upper bound property and equipped with a certain orthogonality relation.