Showing papers in "Czechoslovak Mathematical Journal in 2012"

Variable Lebesgue norm estimates for BMO functions

Abstract: In this paper, we are going to characterize the space ${\rm BMO}({\mathbb R}^n)$ through variable Lebesgue spaces and Morrey spaces. There have been many attempts to characterize the space ${\rm BMO}({\mathbb R}^n)$ by using various function spaces. For example, Ho obtained a characterization of ${\rm BMO}({\mathbb R}^n)$ with respect to rearrangement invariant spaces. However, variable Lebesgue spaces and Morrey spaces do not appear in the characterization. One of the reasons is that these spaces are not rearrangement invariant. We also obtain an analogue of the well-known John-Nirenberg inequality which can be seen as an extension to the variable Lebesgue spaces.

The Laplacian spread of graphs

Abstract: The Laplacian spread of a graph is defined as the difference between the largest and second smallest eigenvalues of the Laplacian matrix of the graph. In this paper, bounds are obtained for the Laplacian spread of graphs. By the Laplacian spread, several upper bounds of the Nordhaus-Gaddum type of Laplacian eigenvalues are improved. Some operations on Laplacian spread are presented. Connected c-cyclic graphs with n vertices and Laplacian spread n − 1 are discussed.

Spectral characterization of multicone graphs

Abstract: A multicone graph is defined to be the join of a clique and a regular graph Based on Zhou and Cho’s result [B Zhou, HH Cho, Remarks on spectral radius and Laplacian eigenvalues of a graph, Czech Math J 55 (130) (2005), 781–790], the spectral characterization of multicone graphs is investigated Particularly, we determine a necessary and sufficient condition for two multicone graphs to be cospectral graphs and investigate the structures of graphs cospectral to a multicone graph Additionally, lower and upper bounds for the largest eigenvalue of a multicone graph are given

Stochastic evolution equations driven by Liouville fractional Brownian motion

Abstract: Let H be a Hilbert space and E a Banach space. We set up a theory of stochastic integration of ℒ(H,E)-valued functions with respect to H-cylindrical Liouville fractional Brownian motion with arbitrary Hurst parameter 0 < β < 1. For 0 < β < ½ we show that a function Φ: (0, T) → ℒ(H,E) is stochastically integrable with respect to an H-cylindrical Liouville fractional Brownian motion if and only if it is stochastically integrable with respect to an H-cylindrical fractional Brownian motion. We apply our results to stochastic evolution equations $$dU(t) = AU(t)dt + B dW_H^\beta (t)$$ driven by an H-cylindrical Liouville fractional Brownian motion, and prove existence, uniqueness and space-time regularity of mild solutions under various assumptions on the Banach space E, the operators A: D(A) → E and B: H → E, and the Hurst parameter. As an application it is shown that second-order parabolic SPDEs on bounded domains in ℝ d , driven by space-time noise which is white in space and Liouville fractional in time, admit a mild solution if ¼d < β < 1.

On co-ordinated quasi-convex functions

Abstract: A function f: I → ℝ, where I ⊆ ℝ is an interval, is said to be a convex function on I if $$f(tx + (1 - t)y) \le tf(x) + (1 - t)f(y)$$ holds for all x, y ∈ I and t ∈ [0, 1]. There are several papers in the literature which discuss properties of convexity and contain integral inequalities. Furthermore, new classes of convex functions have been introduced in order to generalize the results and to obtain new estimations.

Exponents for three-dimensional simultaneous Diophantine approximations

Abstract: Let Θ = (θ1,θ2,θ3) ∈ ℝ3. Suppose that 1, θ1, θ2, θ3 are linearly independent over ℤ. For Diophantine exponents $$\begin{gathered} \alpha (\Theta ) = sup\left\{ {\gamma > 0: \mathop {\lim }\limits_{t \to } \mathop {\sup }\limits_{ + \infty } t^\gamma \psi _\Theta (t) 0: \mathop {\lim }\limits_{t \to } \mathop {\inf }\limits_{ + \infty } t^\gamma \psi _\Theta (t) < + \infty } \right\} \hfill \\ \end{gathered}$$ we prove $$\beta (\Theta ) \ge {1 \over 2}\left( {{{\alpha (\Theta )} \over {1 - \alpha (\Theta )}} + \sqrt {{{\left( {{{\alpha (\Theta )} \over {1 - \alpha (\Theta )}}} \right)}^2} + {{4\alpha (\Theta )} \over {1 - \alpha (\Theta )}}} } \right)\alpha (\Theta )$$ .

Some graphs determined by their (signless) Laplacian spectra

Abstract: Let W n = K 1 ∀ C n−1 be the wheel graph on n vertices, and let S(n, c, k) be the graph on n vertices obtained by attaching n-2c-2k-1 pendant edges together with k hanging paths of length two at vertex υ 0, where υ 0 is the unique common vertex of c triangles. In this paper we show that S(n, c, k) (c ⩾ 1, k ⩾ 1) and W n are determined by their signless Laplacian spectra, respectively. Moreover, we also prove that S(n, c, k) and its complement graph are determined by their Laplacian spectra, respectively, for c ⩾ 0 and k ⩾ 1.

On the $H^p$-$L^q$ boundedness of some fractional integral operators

Abstract: Let A1, …, Am be n × n real matrices such that for each 1 ⩽ i ⩽ m, Ai is invertible and Ai − Aj is invertible for i ≠ j. In this paper we study integral operators of the form $$Tf(x) = \int {{k_1}(x - {A_{1y}}){k_2}(x - {A_{2y}}) \ldots {k_m}(x - {A_{my}})f(y){\rm{d}}y}$$ \({k_i}(y) = \sum\limits_{j \in z} {{2^{jn/{q_i}}}} \varphi i,j({2^j}y),1 \le {q_i} < \infty ,1/{q_1} + 1/q + ... + 1/q = 1 - r,0 \le r < 1, and \varphi i,j\) satisfying suitable regularity conditions. We obtain the boundedness of T: Hp(ℝn) → Lq(ℝn) for 0 < p < 1/r and 1/q = 1/p-r. We also show that we can not expect the Hp-Hq boundedness of this kind of operators.

A characterization of Fuchsian groups acting on complex hyperbolic spaces

Abstract: Let G ⊂ SU(2, 1) be a non-elementary complex hyperbolic Kleinian group. If G preserves a complex line, then G is ℂ-Fuchsian; if G preserves a Lagrangian plane, then G is ℝ-Fuchsian; G is Fuchsian if G is either ℂ-Fuchsian or ℝ-Fuchsian. In this paper, we prove that if the traces of all elements in G are real, then G is Fuchsian. This is an analogous result of Theorem V.G. 18 of B. Maskit, Kleinian Groups, Springer-Verlag, Berlin, 1988, in the setting of complex hyperbolic isometric groups. As an application of our main result, we show that G is conjugate to a subgroup of S(U(1)×U(1, 1)) or SO(2, 1) if each loxodromic element in G is hyperbolic. Moreover, we show that the converse of our main result does not hold by giving a ℂ-Fuchsian group.

Convex domination in the composition and cartesian product of graphs

Abstract: In this paper we characterize the convex dominating sets in the composition and Cartesian product of two connected graphs. The concepts of clique dominating set and clique domination number of a graph are defined. It is shown that the convex domination number of a composition G[H] of two non-complete connected graphs G and H is equal to the clique domination number of G. The convex domination number of the Cartesian product of two connected graphs is related to the convex domination numbers of the graphs involved.

Some remarks on Nagumo's theorem

Abstract: We provide a simpler proof for a recent generalization of Nagumo’s uniqueness theorem by A. Constantin: On Nagumo’s theorem. Proc. Japan Acad., Ser. A 86 (2010), 41–44, for the differential equation x′ = f(t, x), x(0) = 0 and we show that not only is the solution unique but the Picard successive approximations converge to the unique solution. The proof is based on an approach that was developed in Z. S. Athanassov: Uniqueness and convergence of successive approximations for ordinary differential equations. Math. Jap. 35 (1990), 351–367. Some classical existence and uniqueness results for initial-value problems for ordinary differential equations are particular cases of our result.

On almost pseudo-conformally symmetric Ricci-recurrent manifolds with applications to relativity

Abstract: The object of the present paper is to study almost pseudo-conformally symmetric Ricci-recurrent manifolds. The existence of almost pseudo-conformally symmetric Ricci-recurrent manifolds has been proved by an explicit example. Some geometric properties have been studied. Among others we prove that in such a manifold the vector field ϱ corresponding to the 1-form of recurrence is irrotational and the integral curves of the vector field ϱ are geodesic. We also study some global properties of such a manifold. Finally, we study almost pseudo-conformally symmetric Ricci-recurrent spacetime. We obtain the Segre’ characteristic of such a spacetime.

M(r,s)-ideals of compact operators

Abstract: We study the position of compact operators in the space of all continuous linear operators and its subspaces in terms of ideals. One of our main results states that for Banach spaces X and Y the subspace of all compact operators K (X, Y) is an M(r1r2, s1s2)-ideal in the space of all continuous linear operators L(X, Y) whenever K (X,X) and K (Y, Y) are M(r1, s1)- and M(r2, s2)-ideals in L(X,X) and L(Y, Y), respectively, with r1 + s1/2 > 1 and r2 +s2/2 > 1. We also prove that the M(r, s)-ideal K (X, Y ) in L(X, Y ) is separably determined. Among others, our results complete and improve some well-known results on M-ideals.

Ideal convergence and divergence of nets in $(\ell)$-groups

Abstract: In this paper we introduce the I- and I*-convergence and divergence of nets in (l)-groups. We prove some theorems relating different types of convergence/divergence for nets in (l)-group setting, in relation with ideals. We consider both order and (D)-convergence. By using basic properties of order sequences, some fundamental properties, Cauchy-type characterizations and comparison results are derived. We prove that I*-convergence/divergence implies I-convergence/divergence for every ideal, admissible for the set of indexes with respect to which the net involved is directed, and we investigate a class of ideals for which the converse implication holds. Finally we pose some open problems.

Integrals and Banach spaces for finite order distributions

Abstract: Let B c denote the real-valued functions continuous on the extended real line and vanishing at −∞. Let B r denote the functions that are left continuous, have a right limit at each point and vanish at −∞. Define A to be the space of tempered distributions that are the nth distributional derivative of a unique function in B c . Similarly with A from B r . A type of integral is defined on distributions in A and A . The multipliers are iterated integrals of functions of bounded variation. For each n ∈ ℕ, the spaces A and A are Banach spaces, Banach lattices and Banach algebras isometrically isomorphic to B c and B r , respectively. Under the ordering in this lattice, if a distribution is integrable then its absolute value is integrable. The dual space is isometrically isomorphic to the functions of bounded variation. The space A 1 is the completion of the L 1 functions in the Alexiewicz norm. The space A 1 contains all finite signed Borel measures. Many of the usual properties of integrals hold: Holder inequality, second mean value theorem, continuity in norm, linear change of variables, a convergence theorem.

An identity with Generalized Derivations on Lie ideals, right ideals and Banach algebras

Abstract: Let R be a prime ring of characteristic different from 2, U the Utumi quotient ring of R, C = Z(U) the extended centroid of R, L a non-central Lie ideal of R, F a non-zero generalized derivation of R. Suppose that [F(u), u]F(u) = 0 for all u e L, then one of the following holds: We also extend the result to the one-sided case. Finally, as an application we obtain some range inclusion results of continuous or spectrally bounded generalized derivations on Banach algebras.

On the composition factors of a group with the same prime graph as $B {n}(5)$

Abstract: Let G be a finite group. The prime graph of G is a graph whose vertex set is the set of prime divisors of |G| and two distinct primes p and q are joined by an edge, whenever G contains an element of order pq. The prime graph of G is denoted by Γ(G). It is proved that some finite groups are uniquely determined by their prime graph. In this paper, we show that if G is a finite group such that Γ(G) = Γ(Bn(5)), where n ⩾ 6, then G has a unique nonabelian composition factor isomorphic to Bn(5) or Cn(5).

The AP-Denjoy and AP-Henstock integrals revisited

Abstract: The note is related to a recently published paper J.M. Park, J. J. Oh, C.-G. Park, D.H. Lee: The AP-Denjoy and AP-Henstock integrals. Czech. Math. J. 57 (2007), 689–696, which concerns a descriptive characterization of the approximate Kurzweil-Henstock integral. We bring to attention known results which are stronger than those contained in the aforementioned work. We show that some of them can be formulated in terms of a derivation basis defined by a local system of which the approximate basis is known to be a particular case. We also consider the relation between the _-finiteness of variational measure generated by a function and the classical notion of the generalized bounded variation.

Closed-form expression for Hankel determinants of the Narayana polynomials

Abstract: We considered a Hankel transform evaluation of Narayana and shifted Narayana polynomials. Those polynomials arises from Narayana numbers and have many combinatorial properties. A mainly used tool for the evaluation is the method based on orthogonal polynomials. Furthermore, we provided a Hankel transform evaluation of the linear combination of two consecutive shifted Narayana polynomials, using the same method (based on orthogonal polynomials) and previously obtained moment representation of Narayana and shifted Narayana polynomials.

Numerical range of operators acting on Banach spaces

Abstract: The aim of the paper is to propose a definition of numerical range of an operator on reflexive Banach spaces. Under this definition the numerical range will possess the basic properties of a canonical numerical range. We will determine necessary and sufficient conditions under which the numerical range of a composition operator on a weighted Hardy space is closed. We will also give some necessary conditions to show that when the closure of the numerical range of a composition operator on a small weighted Hardy space has zero.

On the heights of power digraphs modulo n

Abstract: A power digraph, denoted by G(n, k), is a directed graph with ℤ n = {0, 1, &h., n − 1} as the set of vertices and E = {(a, b): a k ≡ b (mod n)} as the edge set. In this paper we extend the work done by Lawrence Somer and Michal Křžek: On a connection of number theory with graph theory, Czech. Math. J. 54 (2004), 465–485, and Lawrence Somer and Michal Křžek: Structure of digraphs associated with quadratic congruences with composite moduli, Discrete Math. 306 (2006), 2174–2185. The heights of the vertices and the components of G(n, k) for n ⩾ 1 and k ⩾ 2 are determined. We also find an expression for the number of vertices at a specific height. Finally, we obtain necessary and sufficient conditions on n such that each vertex of indegree 0 of a certain subdigraph of G(n, k) is at height q ⩾ 1.

ALMOST SURE ASYMPTOTIC BEHAVIOUR OF THE r- NEIGHBOURHOOD SURFACE AREA OF BROWNIAN PATHS

Abstract: We show that whenever the q-dimensional Minkowski content of a subset A ⊂ ℝ d exists and is finite and positive, then the “S-content” defined analogously as the Minkowski content, but with volume replaced by surface area, exists as well and equals the Minkowski content. As a corollary, we obtain the almost sure asymptotic behaviour of the surface area of the Wiener sausage in ℝ d , d ⩾ 3.

Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition

Abstract: In this paper, first we introduce a new notion of commuting condition that φφ1A = Aφ1φ between the shape operator A and the structure tensors φ and φ1 for real hypersurfaces in G2(ℂm+2). Suprisingly, real hypersurfaces of type (A), that is, a tube over a totally geodesic G2(ℂm+1) in complex two plane Grassmannians G2(ℂm+2) satisfy this commuting condition. Next we consider a complete classification of Hopf hypersurfaces in G2(ℂm+2) satisfying the commuting condition. Finally we get a characterization of Type (A) in terms of such commuting condition φφ1A = Aφ1φ.

A note on the number of solutions of the generalized Ramanujan-Nagell equation $x^2-D=p^n$

Abstract: Let D be a positive integer, and let p be an odd prime with p ∤ D. In this paper we use a result on the rational approximation of quadratic irrationals due to M. Bauer, M.A. Bennett: Applications of the hypergeometric method to the generalized Ramanujan-Nagell equation. Ramanujan J. 6 (2002), 209–270, give a better upper bound for N(D, p), and also prove that if the equation U 2 − DV 2 = −1 has integer solutions (U, V), the least solution (u 1, v 1) of the equation u 2 − pv 2 = 1 satisfies p ∤ v 1, and D > C(p), where C(p) is an effectively computable constant only depending on p, then the equation x 2 − D = p n has at most two positive integer solutions (x, n). In particular, we have C(3) = 107.

Generalized Knopp identities for homogeneous Hardy sums and Cochrane-Hardy sums

Abstract: Let q, h, a, b be integers with q > 0. The classical and the homogeneous Dedekind sums are defined by $$s(h,q) = \sum\limits_{j = 1}^q {\left( {\left( {{j \over q}} \right)} \right)\left( {\left( {{{hj} \over q}} \right)} \right),{\rm{ }}s(a,b,q) = \sum\limits_{j = 1}^q {\left( {\left( {{{aj} \over q}} \right)} \right)\left( {\left( {{{bj} \over q}} \right)} \right),} } $$ respectively, where $$((x)) = \left\{ \begin{gathered} x - [x] - \tfrac{1} {2},if x is not an integer; \hfill \\ 0,if x is an integer. \hfill \\ \end{gathered} \right. $$ The Knopp identities for the classical and the homogeneous Dedekind sum were the following: $$\sum\limits_{d|n} {\sum\limits_{r = 1}^d {s\left( {{n \over d}a + rq,dq} \right) = \sigma (n)s(a,q),} } $$ $$\sum\limits_{d|n} {\sum\limits_{{r_1} = 1}^d {\sum\limits_{{r_2} = 1}^d s \left( {{n \over d}a + {r_1}q,{n \over d}b + {r_2}q,dq} \right) = n\sigma (n)s(a,b,q),} } $$ where σ(n) =Σd|nd.

Differences of weighted composition operators from hardy space to weighted-type spaces on the unit ball

Abstract: In this paper, we limit our analysis to the difference of the weighted composition operators acting from the Hardy space to weighted-type space in the unit ball of ℂN, and give some necessary and sufficient conditions for their boundedness or compactness. The results generalize the corresponding results on the single weighted composition operators and on the differences of composition operators, for example, M. Lindstrom and E. Wolf: Essential norm of the difference of weighted composition operators. Monatsh. Math. 153 (2008), 133–143.

Conditions under which the least compactification of a regular continuous frame is perfect

Abstract: We characterize those regular continuous frames for which the least compactification is a perfect compactification. Perfect compactifications are those compactifications of frames for which the right adjoint of the compactification map preserves disjoint binary joins. Essential to our characterization is the construction of the frame analog of the two-point compactification of a locally compact Hausdorff space, and the concept of remainder in a frame compactification. Indeed, one of the characterizations is that the remainder of the regular continuous frame in each of its compactifications is compact and connected.

Decomposition of ℓ-group-valued measures

Abstract: We deal with decomposition theorems for modular measures µ: L → G defined on a D-lattice with values in a Dedekind complete l-group. Using the celebrated band decomposition theorem of Riesz in Dedekind complete l-groups, several decomposition theorems including the Lebesgue decomposition theorem, the Hewitt-Yosida decomposition theorem and the Alexandroff decomposition theorem are derived. Our main result—also based on the band decomposition theorem of Riesz—is the Hammer-Sobczyk decomposition for l-group-valued modular measures on D-lattices. Recall that D-lattices (or equivalently lattice ordered effect algebras) are a common generalization of orthomodular lattices and of MV-algebras, and therefore of Boolean algebras. If L is an MV-algebra, in particular if L is a Boolean algebra, then the modular measures on L are exactly the finitely additive measures in the usual sense, and thus our results contain results for finitely additive G-valued measures defined on Boolean algebras.

On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Abstract: Consider the homogeneous equation $$u'(t) = l(u)(t){\rm{ for a}}{\rm{.e}}{\rm{. }}t \in [a,b]$$ where l: C([a, b];ℝ) → L([a, b];ℝ) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.

The Kurzweil-Henstock theory of stochastic integration

Abstract: The Kurzweil-Henstock approach has been successful in giving an alternative definition to the classical Ito integral, and a simpler and more direct proof of the Ito Formula. The main advantage of this approach lies in its explicitness in defining the integral, thereby reducing the technicalities of the classical stochastic calculus. In this note, we give a unified theory of stochastic integration using the Kurzweil-Henstock approach, using the more general martingale as the integrator. We derive Henstock’s Lemmas, absolute continuity property of the primitive process, integrability of stochastic processes and convergence theorems for the Kurzweil-Henstock stochastic integrals. These properties are well-known in the classical (non-stochastic) integration theory but have not been explicitly derived in the classical stochastic integration.