On a new class of Finsler metrics

In this paper, the geometric meaning of (α,β)-norms is made clear. On this basis, a new class of Finsler metrics called general (α,β)-metrics are introduced, which are defined by a Riemannian metric and a 1-form. These metrics not only generalize (α,β)-metrics naturally, but also include some metrics structured by R. Bryant. The spray coefficients formula of some kinds of general (α,β)-metrics is given and the projective flatness is also discussed.     

Nonlinear dissipative wave equations with space-time dependent potential

We study the long time behavior of solutions for damped wave equations with absorption. These equations are generally accepted as models of wave propagation in heterogeneous media with space-time dependent friction a(t,x)u_t and nonlinear absorption |u|^p−1u (Ikawa (2000) [17]). We consider 1<p<(n+2)/(n−2) and separable a(t,x)=λ(x)η(t) with λ(x)∼(1+|x|)^−α and η(t)∼(1+t)^−β satisfying conditions (A1) or (A2) which are given. The main results are precise decay estimates for the energy, L² and L^p+1 norms of solutions. We also observe the following behavior: if α∈[0,1), β∈(−1,1) and 0<α+β<1, there are three different regions for the decay of solutions depending on p; if α∈(−∞,0) and β∈(−1,1), there are only two different regions for the decay of the solutions depending on p.     

Rough approximations of vague sets in fuzzy approximation space

The combination of the rough set theory, vague set theory and fuzzy set theory is a novel research direction in dealing with incomplete and imprecise information. This paper mainly concerns the problem of how to construct rough approximations of a vague set in fuzzy approximation space. Firstly, the β-operator and its complement operator are introduced, and some new properties are examined. Secondly, the approximation operators are constructed based on β-(complement) operator. Meantime, λ-lower (upper) approximation is firstly proposed, and then some properties of two types of approximation operators are studied. Afterwards, for two different kinds of approximation operators, we introduce two roughness measure methods of the same vague set and discuss a property. Finally, an example is given to illustrate how to calculate the rough approximations and roughness measure of a vague set using the β-(complement) product between two fuzzy matrixes. The results show that the proposed rough approximations and roughness measure of a vague set in fuzzy environment are reasonable.     

More on “Connected (n, m)-graphs with minimum and maximum zeroth-order general Randi? index”

Let G be a graph and d(u) denote the degree of a vertex u in G. The zeroth-order general Randi? index ⁰R_α(G) of the graph G is defined as ∑_u∈V(G)d(u)^α, where the summation goes over all vertices of G and α is an arbitrary real number. In this paper we correct the proof of the main Theorem 3.5 of the paper by Hu et al. [Y. Hu, X. Li, Y. Shi, T. Xu, Connected (n,m)-graphs with minimum and maximum zeroth-order general Randi? index, Discrete Appl. Math. 155 (8) (2007) 1044-1054] and give a more general Theorem. We finally characterize ¹ for α<0 the connected G(n,m)-graphs with maximum value ⁰R_α(G(n,m)), where G(n,m) is a simple connected graph with n vertices and m edges.     

Absolutely convergent Fourier series and function classes. II

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs to one of the Zygmund classes Λ_∗(α) and λ_∗(α) for some 0<α?2. This paper is a natural supplement to our earlier one [F. Móricz, Absolutely convergent Fourier series and function classes, J. Math. Anal. Appl. 324 (2) (2006) 1168-1177] under the same title, and we keep its notations.     

Independence polynomials of k-tree related graphs

An independent set of a graph G is a set of pairwise non-adjacent vertices. Let α(G) denote the cardinality of a maximum independent set and f_s(G) for 0≤s≤α(G) denote the number of independent sets of s vertices. The independence polynomial defined first by Gutman and Harary has been the focus of considerable research recently. Wingard bounded the coefficients f_s(T) for trees T with n vertices: for s≥2. We generalize this result to bounds for a very large class of graphs, maximal k-degenerate graphs, a class which includes all k-trees. Additionally, we characterize all instances where our bounds are achieved, and determine exactly the independence polynomials of several classes of k-tree related graphs. Our main theorems generalize several related results known before.     

Semigroup crossed products and the induced algebras of lattice-ordered groups

Let (G,G₊) be a quasi-lattice-ordered group with positive cone G₊. Laca and Raeburn have shown that the universal C^∗-algebra C^∗(G,G₊) introduced by Nica is a crossed product BG₊α_×G₊ by a semigroup of endomorphisms. The goal of this paper is to extend some results for totally ordered abelian groups to the case of discrete lattice-ordered abelian groups. In particular given a hereditary subsemigroup H₊ of G₊ we introduce a closed ideal IH₊ of the C^∗-algebra BG₊. We construct an approximate identity for this ideal and show that IH₊ is extendibly α-invariant. It follows that there is an isomorphism between C^∗-crossed products and B₊(G/H)β_×G₊. This leads to our main result that B₊(G/H)β_×G₊ is realized as an induced C^∗-algebra .     

Absolutely convergent Fourier series and function classes

We study the smoothness property of a function f with absolutely convergent Fourier series, and give best possible sufficient conditions in terms of its Fourier coefficients to ensure that f belongs either to one of the Lipschitz classes Lip(α) and lip(α) for some 0<α?1, or to one of the Zygmund classes Λ_∗(1) and λ_∗(1). Our theorems generalize some of those by Boas [R.P. Boas Jr., Fourier series with positive coefficients, J. Math. Anal. Appl. 17 (1967) 463-483] and one by Németh [J. Németh, Fourier series with positive coefficients and generalized Lipschitz classes, Acta Sci. Math. (Szeged) 54 (1990) 291-304]. We also prove a localized version of a theorem by Paley [R.E.A.C. Paley, On Fourier series with positive coefficients, J. London Math. Soc. 7 (1932) 205-208] on the existence and continuity of the derivative of f.     

Covering graphs with matchings of fixed size

Let m be a positive integer and let G be a graph. We consider the question: can the edge set E(G) of G be expressed as the union of a set M of matchings of G each of which has size exactly m? If this happens, we say that G is [m]-coverable and we call M an [m]-covering of G. It is interesting to consider minimum[m]-coverings, i.e. [m]-coverings containing as few matchings as possible. Such [m]-coverings will be called excessive[m]-factorizations. The number of matchings in an excessive [m]-factorization is a graph parameter which will be called the excessive[m]-index and denoted by . In this paper we begin the study of this new parameter as well as of a number of other related graph parameters.     

Riesz transforms and conjugacy for Laguerre function expansions of Hermite type

Riesz transforms and conjugate Poisson integrals for multi-dimensional Laguerre function expansions of Hermite type with index α are defined and investigated. It is proved that for any multi-index α=(α₁,…,αd) such that αi?−1/2, αi∉(−1/2,1/2), the appropriately defined Riesz transforms , j=1,2,…,d, are Calderón-Zygmund operators, hence their mapping properties follow from a general theory. Similar mapping results are obtained in one dimension, without excluding α∈(−1/2,1/2), by means of a local Calderón-Zygmund theory and weighted Hardy's inequalities. The conjugate Poisson integrals are shown to satisfy a system of Cauchy-Riemann type equations and to recover the Riesz-Laguerre transforms on the boundary. The two specific values of α, (−1/2,…,−1/2) and (1/2,…,1/2), are distinguished since then a connection with Riesz transforms for multi-dimensional Hermite function expansions is established.     

On semi-R-boundedness and its applications

R-Boundedness is a randomized boundedness condition for sets of operators which in recent years has found many applications in the maximal regularity theory of evolution equations, stochastic evolution equations, spectral theory and vector-valued harmonic analysis. However, in some situations additional geometric properties such as Pisier's property (α) are required to guaranty the R-boundedness of a relevant set of operators. In this paper we show that a weaker property called semi-R-boundedness can be used to avoid these geometric assumptions in the context of Schauder decompositions and the H^∞-calculus. Furthermore, we give weaker conditions for stochastic integrability of certain convolutions.     

Determining tension parameters in rational gamma-spline interpolation

Inspired by the classic γ-spline, we propose a method for constructing a G² rational γ-spline curve that interpolates a given set of distinct ordered data-points (planar or spatial). The only input of our method is just these data-points. We also present a procedure to solve the key problem of determining the tension parameters γ_i which are computed in terms of exponential functions that determine the eccentricities of the common conic osculants at the junction points while keeping in geometrical agreement with data-points. This allows the resulting curve to be modified in the close vicinity of each data-point.     

The Independence Number for De Bruijn networks and Kautz networks

Let D be a directed graph; the (l,ω)-Independence Number of graph D, denoted by α_l,ω(D), is an important performance parameter for interconnection networks. De Bruijn networks and Kautz networks, denoted by B(d,n) and K(d,n) respectively, are versatile and efficient topological structures of interconnection networks. For l=1,2,…,n, this paper shows that α_l,d−1(B(d,n))=dⁿ,α_l,d−1(K(d,n))=α_l,d(K(d,n))=dⁿ+dⁿ⁻¹ if d≥3 and n≤d−2. In particular, the paper shows the exact value of the Independence Number for B(d,1) and B(d,2) for any d. For the generalized situation, the paper obtains a lower bound α_l,d−1(B(d,n))≥d² if n≥3 and d≥5.     

Statistical convergence and ideal convergence for sequences of functions

Let I⊂P(N) stand for an ideal containing finite sets. We discuss various kinds of statistical convergence and I-convergence for sequences of functions with values in R or in a metric space. For real valued measurable functions defined on a measure space (X,M,μ), we obtain a statistical version of the Egorov theorem (when μ(X)<∞). We show that, in its assertion, equi-statistical convergence on a big set cannot be replaced by uniform statistical convergence. Also, we consider statistical convergence in measure and I-convergence in measure, with some consequences of the Riesz theorem. We prove that outer and inner statistical convergences in measure (for sequences of measurable functions) are equivalent if the measure is finite.     

Growth conditions, compact perturbations and operator subdecomposability, with applications to generalized Cesàro operators

We adapt recent results of Albrecht and Ricker to obtain conditions under which growth constraints on the left resolvent of a Banach space operator are preserved under suitable perturbations. As an application, we establish Bishop's property (β) for certain generalized Cesàro operators on the classical Hardy spaces Hp, 1<p<∞. Our methods also apply to unilateral weighted shifts whose weight sequence converges sufficiently rapidly as well as to perturbations of restrictions of a class of generalized scalar operators.     

On island sequences of labelings with a condition at distance two

An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that |f(x)−f(y)|≥2 if d(x,y)=1, and |f(x)−f(y)|≥1 if d(x,y)=2, where d(x,y) denotes the distance between the pair of vertices x,y. The lambda number of G, denoted λ(G), is the minimum range of labels used over all L(2,1)-labelings of G. An L(2,1)-labeling of G which achieves the range λ(G) is referred to as a λ-labeling. A hole of an L(2,1)-labeling is an unused integer within the range of integers used. The hole index of G, denoted ρ(G), is the minimum number of holes taken over all its λ-labelings. An island of a given λ-labeling of G with ρ(G) holes is a maximal set of consecutive integers used by the labeling. Georges and Mauro [J.P. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2,1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223] inquired about the existence of a connected graph G with ρ(G)≥1 possessing two λ-labelings with different ordered sequences of island cardinalities. This paper provides an infinite family of such graphs together with their lambda numbers and hole indices. Key to our discussion is the determination of the path covering number of certain 2-sparse graphs, that is, graphs containing no pair of adjacent vertices of degree greater than 2.     

Ramsey numbers for a disjoint union of good graphs

We give the Ramsey number for a disjoint union of some G-good graphs versus a graph G generalizing the results of Stahl (1975) [5] and Baskoro et al. (2006) [1] and the previous result of the author Bielak (2009) [2]. Moreover, a family of G-good graphs with s(G)>1 is presented.     

Minimum monopoly in regular and tree graphs

In this paper we consider a graph optimization problem called minimum monopoly problem, in which it is required to find a minimum cardinality set S⊆V, such that, for each u∈V, |N[u]∩S|?|N[u]|/2 in a given graph G=(V,E). We show that this optimization problem does not have a polynomial-time approximation scheme for k-regular graphs (k?5), unless P=NP. We show this by establishing two L-reductions (an approximation preserving reduction) from minimum dominating set problem for k-regular graphs to minimum monopoly problem for 2k-regular graphs and to minimum monopoly problem for (2k-1)-regular graphs, where k?3. We also show that, for tree graphs, a minimum monopoly set can be computed in linear time.     

Identifying codes and locating-dominating sets on paths and cycles

Let G=(V,E) be a graph and let r≥1 be an integer. For a set D⊆V, define N_r[x]={y∈V:d(x,y)≤r} and D_r(x)=N_r[x]∩D, where d(x,y) denotes the number of edges in any shortest path between x and y. D is known as an r-identifying code (r-locating-dominating set, respectively), if for all vertices x∈V (x∈V?D, respectively), D_r(x) are all nonempty and different. Roberts and Roberts [D.L. Roberts, F.S. Roberts, Locating sensors in paths and cycles: the case of 2-identifying codes, European Journal of Combinatorics 29 (2008) 72-82] provided complete results for the paths and cycles when r=2. In this paper, we provide results for a remaining open case in cycles and complete results in paths for r-identifying codes; we also give complete results for 2-locating-dominating sets in cycles, which completes the results of Bertrand et al. [N. Bertrand, I. Charon, O. Hudry, A. Lobstein, Identifying and locating-dominating codes on chains and cycles, European Journal of Combinatorics 25 (2004) 969-987].     

Maximum transversal in partial Latin squares and rainbow matchings

In a partial Latin square P a set of distinct entries, such that no two of which are in the same row or column is called a transversal. By the size of a transversal T, we mean the number of its entries. We define a duplex to be a partial Latin square of order n containing 2n entries such that exactly two entries lie in each row and column and each of n symbols occurs exactly twice. We show that determining the maximum size of a transversal in a given duplex is an NP-complete problem. This problem relates to independent sets in certain subfamilies of cubic graphs. Generalizing the concept of transversals in edge coloring of graphs we are led to introduce the concept of rainbow matching. We show that if each color appears at most twice then it is a polynomial time problem to know whether there exists a rainbow matching of size at least ⌊n/2⌋-t for each fixed t, where n is the order of the graph. As an application we show that for any fixed t, there is a polynomial time algorithm which decides whether α(G)?n-t, for any graph G on 2n vertices containing a perfect matching. At the end we mention some other applications of rainbow matching.