Growth theorems and coefficient bounds for univalent holomorphic mappings which have parametric representation

Let B be the unit ball in Cn with respect to an arbitrary norm and let f(z,t) be a g-Loewner chain such that e−tf(z,t)−z has a zero of order k+1 at z=0. In this paper, we obtain growth and covering theorems for . Moreover, we consider coefficient bounds and examples of mappings in .     

A spectral condition for odd cycles in graphs

Let G be a graph of sufficiently large order n, and let the largest eigenvalue μ(G) of its adjacency matrix satisfies . Then G contains a cycle of length t for every t?n/320This condition is sharp: the complete bipartite graph T₂(n) with parts of size ⌊n/2⌋ and ⌈n/2⌉ contains no odd cycles and its largest eigenvalue is equal to .This condition is stable: if μ(G) is close to and G fails to contain a cycle of length t for some t?n/321, then G resembles T₂(n).     

Integers with dense divisors 2

Let 1=d₁(n)<d₂(n)<?<d_τ(n)=n be the sequence of all positive divisors of the integer n in increasing order. We say that the divisors of n are t-dense iff max_1?i<τ(n)d_i+1(n)/d_i(n)?t. Let D(x,t) be the number of positive integers not exceeding x whose divisors are t-dense. We show that for x?3, and , we have , where , and d(w) is a continuous function which satisfies d(w)?1/w for w?1. We also consider other counting functions closely related to D(x,t).     

Harmonious order of graphs

We consider the following generalization of the concept of harmonious graphs. Given a graph G=(V,E) and a positive integer t≥|E|, a function is called a t-harmonious labeling of G if is injective for t≥|V| or surjective for t<|V|, and for all distinct edges vw,xy∈E(G). Then the smallest possible t such that G has a t-harmonious labeling is named the harmonious order of G. We determine the harmonious order of some non-harmonious graphs, such as complete graphs K_n (n≥5), complete bipartite graphs K_m,n (m,n>1), even cycles C_n, some powers of paths , disjoint unions of triangles nK₃ (n even). We also present some general results concerning harmonious order of the Cartesian product of two given graphs or harmonious order of the disjoint union of copies of a given graph. Furthermore, we establish an upper bound for harmonious order of trees.     

Carleson embeddings for Sobolev spaces via heat equation

Let u(t,x) be the solution of the heat equation (∂_t-Δ_x)u(t,x)=0 on subject to u(0,x)=f(x) on Rⁿ. The main goal of this paper is to characterize such a nonnegative measure μ on that f(x)?u(t²,x) induces a bounded embedding from the Sobolev space , p∈[1,n) into the Lebesgue space , q∈(0,∞).     

Asymptotic stability of Riemann solutions in BGK approximations to certain multidimensional systems of conservation laws

We prove the asymptotic stability of nonplanar two-states Riemann solutions in BGK approximations of a class of multidimensional systems of conservation laws. The latter consists of systems whose flux-functions in different directions share a common complete system of Riemann invariants, the level surfaces of which are hyperplanes. The asymptotic stability to which the main result refers is in the sense of the convergence as t→∞ in of the space of directions ζ=x/t. That is, the solution z(t,x,ξ) of the perturbed Cauchy problem for the corresponding BGK system satisfies as t→∞, in , where R(ζ) is the self-similar entropy solution of the two-states nonplanar Riemann problem for the system of conservation laws.     

Extremal and Barabanov semi-norms of a semigroup generated by a bounded family of matrices

Let S={Si}_i∈I be an arbitrary family of complex n-by-n matrices, where 1?n<∞. Let denote the joint spectral radius of S, defined as

     

Diagnosability of regular systems

A novel approach aimed at evaluating the diagnosability of regular systems under the PMC model is introduced. The diagnosability is defined as the ability to provide a correct diagnosis, although possibly incomplete. This concept is somehow intermediate between one-step diagnosability and sequential diagnosability. A lower bound to diagnosability is determined by lower bounding the minimum of a “syndrome-dependent” bound t_σ over the set of all the admissible syndromes. In turn, t_σ is determined by evaluating the cardinality of the smallest consistent fault set containing an aggregate of maximum cardinality. The new approach, which applies to any regular system, relies on the “edge-isoperimetric inequalities” of connected components of units declaring each other non-faulty. This approach has been used to derive tight lower bounds to the diagnosability of toroidal grids and hypercubes, which improve the existing bounds for the same structures.     

Infinitely many homoclinic solutions for second order Hamiltonian systems

In this paper we study the existence of infinitely many homoclinic solutions for second order Hamiltonian systems , , where L(t) is unnecessarily positive definite for all t∈R, and W(t,u) is of subquadratic growth as |u|→∞.     

On a problem regarding coefficients of cyclotomic polynomials

Let an(k) be the coefficient of tk in the nth cyclotomic polynomial

     

Weak type estimates on certain Hardy spaces for smooth cone type multipliers

Let ?n∈C^∞(Rd?{0}) be a non-radial homogeneous distance function of degree n∈N satisfying ?n(tξ)=tn?n(ξ). For f∈S(Rd+1) and δ>0, we consider convolution operator associated with the smooth cone type multipliers defined by

     

Extension of Rotfel’d Theorem

Let f(t) be a non-negative concave function on the positive half-line. Given an arbitrary partitioned positive semi-definite matrix, we show that