Geodetic and Steiner geodetic sets in 3-Steiner distance hereditary graphs

Let G be a connected graph and S⊆V(G). Then the Steiner distance of S, denoted by d_G(S), is the smallest number of edges in a connected subgraph of G containing S. Such a subgraph is necessarily a tree called a Steiner tree for S. The Steiner interval for a set S of vertices in a graph, denoted by I(S) is the union of all vertices that belong to some Steiner tree for S. If S={u,v}, then I(S) is the interval I[u,v] between u and v. A connected graph G is 3-Steiner distance hereditary (3-SDH) if, for every connected induced subgraph H of order at least 3 and every set S of three vertices of H, d_H(S)=d_G(S). The eccentricity of a vertex v in a connected graph G is defined as e(v)=max{d(v,x)|x∈V(G)}. A vertex v in a graph G is a contour vertex if for every vertex u adjacent with v, e(u)?e(v). The closure of a set S of vertices, denoted by I[S], is defined to be the union of intervals between pairs of vertices of S taken over all pairs of vertices in S. A set of vertices of a graph G is a geodetic set if its closure is the vertex set of G. The smallest cardinality of a geodetic set of G is called the geodetic number of G and is denoted by g(G). A set S of vertices of a connected graph G is a Steiner geodetic set for G if I(S)=V(G). The smallest cardinality of a Steiner geodetic set of G is called the Steiner geodetic number of G and is denoted by sg(G). We show that the contour vertices of 3-SDH and HHD-free graphs are geodetic sets. For 3-SDH graphs we also show that g(G)?sg(G). An efficient algorithm for finding Steiner intervals in 3-SDH graphs is developed.     

Some new spaces of double sequences

In this study, we define the double sequence spaces BS, BS(t), CSp, CSbp, CSr and BV, and also examine some properties of those sequence spaces. Furthermore, we show that these sequence spaces are complete paranormed or normed spaces under some certain conditions. We determine the α-duals of the spaces BS, BV, CSbp and the β(?)-duals of the spaces CSbp and CSr of double series. Finally, we give the conditions which characterize the class of four-dimensional matrix mappings defined on the spaces CSbp, CSr and CSp of double series.     

Greedy F-colorings of graphs

Let G=(V,E) be a connected graph. For a symmetric, integer-valued function δ on V×V, where K is an integer constant, N₀ is the set of nonnegative integers, and Z is the set of integers, we define a C-mapping by F(u,v,m)=δ(u,v)+m−K. A coloring c of G is an F-coloring if F(u,v,|c(u)−c(v)|)?0 for every two distinct vertices u and v of G. The maximum color assigned by c to a vertex of G is the value of c, and the F-chromatic number F(G) is the minimum value among all F-colorings of G. For an ordering of the vertices of G, a greedy F-coloring c of s is defined by (1) c(v₁)=1 and (2) for each i with 1?i<n, c(v_i+1) is the smallest positive integer p such that F(v_j,v_i+1,|c(v_j)−p|)?0, for each j with 1?j?i. The greedy F-chromatic number gF(s) of s is the maximum color assigned by c to a vertex of G. The greedy F-chromatic number of G is gF(G)=min{gF(s)} over all orderings s of V. The Grundy F-chromatic number is GF(G)=max{gF(s)} over all orderings s of V. It is shown that gF(G)=F(G) for every graph G and every F-coloring defined on G. The parameters gF(G) and GF(G) are studied and compared for a special case of the C-mapping F on a connected graph G, where δ(u,v) is the distance between u and v and .     

Perfect dexagon triple systems with given subsystems

The graph consisting of the six triples (or triangles) {a,b,c}, {c,d,e}, {e,f,a}, {x,a,y}, {x,c,z}, {x,e,w}, where a,b,c,d,e,f,x,y,z and w are distinct, is called a dexagon triple. In this case the six edges {a,c}, {c,e}, {e,a}, {x,a}, {x,c}, and {x,e} form a copy of K₄ and are called the inside edges of the dexagon triple. A dexagon triple system of order v is a pair (X,D), where D is a collection of edge disjoint dexagon triples which partitions the edge set of 3K_v. A dexagon triple system is said to be perfect if the inside copies of K₄ form a block design. In this note, we investigate the existence of a dexagon triple system with a subsystem. We show that the necessary conditions for the existence of a dexagon triple system of order v with a sub-dexagon triple system of order u are also sufficient.     

A method of finding automorphism groups of endomorphism monoids of relational systems

For any set X and any relation ρ on X, let T(X,ρ) be the semigroup of all maps a:X→X that preserve ρ. Let S(X) be the symmetric group on X. If ρ is reflexive, the group of automorphisms of T(X,ρ) is isomorphic to N_S(X)(T(X,ρ)), the normalizer of T(X,ρ) in S(X), that is, the group of permutations on X that preserve T(X,ρ) under conjugation. The elements of N_S(X)(T(X,ρ)) have been described for the class of so-called dense relations ρ. The paper is dedicated to applications of this result.     

Factorization of Q(h(T)(x)) over a finite field where Q(x) is irreducible and h(T)(x) is linear—I

Let GF(q) be the finite field of order q, let Q(x) be an irreducible polynomial in GF(q)(x), and let h(T)(x) be a linear polynomial in GF(q)[x], where T:x→x^q. We use properties of the linear operator h(T) to give conditions for Q(h(T)(x)) to have a root of arbitrary degree k over GF(q), and we describe how to count the irreducible factors of Q(h(T)(x)) of degree k over GF(q). In addition we compare our results with those Ore which count the number of irreducible factors belonging to a linear polynomial having index k.     

Intersection problems and integral equivalence

Let A and B be matrices over a principal ideal domain, Π. Necessary conditions, involving the invariant factors of A and B, are given for B to be a submatrix of A or a principal submatrix of A.If a given nonnegative integral matrix, B, is the intersection matrix of a pair of families of subsets of an n-set, and n is the smallest integer for which this is true, we say that the content of B is n. In that event, B is a submatrix of K(n), the intersection matrix of all subsets of an n-set. More refined results are obtained in certain cases by considering S(n, k, l), the intersection matrix of the k-subsets of an n-set versus its l-subsets. The invariant factors of K(n) and S(n, k, l) are calculated and it is shown how this information may be used to get lower bounds for the content of B. In the more widely studied symmetric version of the content problem, B must be a principal submatrix of K(n) or, possibly, S(n, k) = S(n, k, k). In this case, the invariant factors of K(n) ? xI or S(n, k) ? xI also provide relevant information.     

Normal coverings of finite symmetric and alternating groups

In this paper we investigate the minimum number of maximal subgroups Hi, i=1,…,k of the symmetric group Sn (or the alternating group An) such that each element in the group Sn (respectively An) lies in some conjugate of one of the Hi. We prove that this number lies between a?(n) and bn for certain constants a,b, where ?(n) is the Euler phi-function, and we show that the number depends on the arithmetical complexity of n. Moreover in the case where n is divisible by at most two primes, we obtain an upper bound of 2+?(n)/2, and we determine the exact value for Sn when n is odd and for An when n is even.     

Asymptotic estimates of sums involving the Moebius function

Let p(n) denote the smallest prime factor of an integer n>1 and let p(1)=∞. We study the asymptotic behavior of the sum M(x,y)=Σ_{1≤n≤x,p(n)>y}μ(n) and use this to estimate the size of A(x)=max_|f|≤1|Σ_2≤n<xμ(n)f(p(n))|, where μ(n) is the Moebius function. Applications of bounds for A(x), M(x,y) and similar quantities are discussed.     

Inverse cyclotomic polynomials

Let Ψn(x) be the monic polynomial having precisely all non-primitive nth roots of unity as its simple zeros. One has Ψn(x)=(xn−1)/Φn(x), with Φn(x) the nth cyclotomic polynomial. The coefficients of Ψn(x) are integers that like the coefficients of Φn(x) tend to be surprisingly small in absolute value, e.g. for n<561 all coefficients of Ψn(x) are ?1 in absolute value. We establish various properties of the coefficients of Ψn(x), especially focusing on the easiest non-trivial case where n is composed of 3 distinct odd primes.     

Convergence of Galerkin solutions and continuous dependence on data in spectrally-hyperviscous models of 3D turbulent flow

We obtain results on the convergence of Galerkin solutions and continuous dependence on data for the spectrally-hyperviscous Navier-Stokes equations. Let uN denote the Galerkin approximates to the solution u, and let wN=u−uN. Then our main result uses the decomposition wN=PnwN+QnwN where (for fixed n) Pn is the projection onto the first n eigenspaces of A=−Δ and Qn=I−Pn. For assumptions on n that compare well with those in related previous results, the convergence of ‖QnwN(t)Hβ_‖ as N→∞ depends linearly on key parameters (and on negative powers of λn), thus reflective of Kolmogorov-theory predictions that in high wavenumber modes viscous (i.e. linear) effects dominate. Meanwhile ‖PnwN(t)Hβ_‖ satisfies a more standard exponential estimate, with positive, but fractional, dependence on λn. Modifications of these estimates demonstrate continuous dependence on data.     

The forcing hull and forcing geodetic numbers of graphs

For every pair of vertices u,v in a graph, a u-v geodesic is a shortest path from u to v. For a graph G, let I_G[u,v] denote the set of all vertices lying on a u-v geodesic. Let S⊆V(G) and I_G[S] denote the union of all I_G[u,v] for all u,v∈S. A subset S⊆V(G) is a convex set of G if I_G[S]=S. A convex hull [S]_G of S is a minimum convex set containing S. A subset S of V(G) is a hull set of G if [S]_G=V(G). The hull number h(G) of a graph G is the minimum cardinality of a hull set in G. A subset S of V(G) is a geodetic set if I_G[S]=V(G). The geodetic number g(G) of a graph G is the minimum cardinality of a geodetic set in G. A subset F⊆V(G) is called a forcing hull (or geodetic) subset of G if there exists a unique minimum hull (or geodetic) set containing F. The cardinality of a minimum forcing hull subset in G is called the forcing hull number f_h(G) of G and the cardinality of a minimum forcing geodetic subset in G is called the forcing geodetic number f_g(G) of G. In the paper, we construct some 2-connected graph G with (f_h(G),f_g(G))=(0,0),(1,0), or (0,1), and prove that, for any nonnegative integers a, b, and c with a+b≥2, there exists a 2-connected graph G with (f_h(G),f_g(G),h(G),g(G))=(a,b,a+b+c,a+2b+c) or (a,2a+b,a+b+c,2a+2b+c). These results confirm a conjecture of Chartrand and Zhang proposed in [G. Chartrand, P. Zhang, The forcing hull number of a graph, J. Combin. Math. Combin. Comput. 36 (2001) 81-94].     

Some bounds on the p-domination number in trees

Let p be a positive integer and G=(V,E) a graph. A subset S of V is a p-dominating set if every vertex of V-S is dominated at least p times, and S is a p-dependent set of G if the subgraph induced by the vertices of S has maximum degree at most p-1. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G) and the maximum cardinality of a p-dependent set of G is the p-dependence number β_p(G). For every positive integer p?2, we show that for a bipartite graph G, γ_p(G) is bounded above by (|V|+|Y_p|)/2, where Y_p is the set of vertices of G of degree at most p-1, and for every tree T, γ_p(T) is bounded below by β_p-1(T). Moreover, we characterize the trees achieving equality in each bound.     

Some exact distributions of the number of one-sided deviations and the time of the last such deviation in the simple random walk

For every positive integer n, let S_n be the n-th partial sum of a sequence of independent and identically distributed random variables, each assuming the values +1 and −1 with respective probabilities p (0<p<1)) and q (= 1 −p) and having mean μ = p − q. For a fixed positive real number λ, let N⁺[N1] be the total number of values of n for which S_n > (μ + λ)n [S_n⩾(μ + λ)n] and let L⁺[L1] be the supremum of the values of n for which S_n > (μ + λ)n [S_n⩾(μ + λ)n], where sup Oslash; = 0. Explicit expressions for the exact distributions of N⁺, N1, L⁺ and L1 are given when μ + λ = ±k/(k + 2) for any nonnegative integer k.     

One step semi-explicit methods based on the Cayley transform for solving isospectral flows

This note deals with the numerical solution of the matrix differential system Y′ = [B(t,Y), Y], Y(0) = Y₀, t ⩾ 0, where Y₀ is a real constant symmetric matrix, B maps symmetric into skew-symmetric matrices, and [B(t,Y),Y] is the Lie bracket commutator of B(t,Y) and Y, i.e. [B(t,Y),Y] = B(t,Y)Y − YB(t,Y). The unique solution of (1) is isospectral, that is the matrix Y(t) preserves the eigenvalues of Y₀ and is symmetric for all t (see [1, 5]). Isospectral methods exploit the Flaschka formulation of (1) in which Y(t) is written as Y(t) = U(t)Y₀U^T(t), for t ⩾ 0, where U(t) is the orthogonal solution of the differential system U′ = B(t, UY₀U^T)U, U(0) = I, t ⩾ 0, (see [5]). Here a numerical procedure based on the Cayley transform is proposed and compared with known isospectral methods.     

Some paranormed sequence spaces of non-absolute type derived by weighted mean

The sequence spaces ?_∞(p), c(p) and c₀(p) were introduced and studied by Maddox [I.J. Maddox, Paranormed sequence spaces generated by infinite matrices, Proc. Cambridge Philos. Soc. 64 (1968) 335-340]. In the present paper, the sequence spaces λ(u,v;p) of non-absolute type which are derived by the generalized weighted mean are defined and proved that the spaces λ(u,v;p) and λ(p) are linearly isomorphic, where λ denotes the one of the sequence spaces ?_∞, c or c₀. Besides this, the β- and γ-duals of the spaces λ(u,v;p) are computed and the basis of the spaces c₀(u,v;p) and c(u,v;p) is constructed. Additionally, it is established that the sequence space c₀(u,v) has AD property and given the f-dual of the space c₀(u,v;p). Finally, the matrix mappings from the sequence spaces λ(u,v;p) to the sequence space μ and from the sequence space μ to the sequence spaces λ(u,v;p) are characterized.     

Radius-edge-invariant and diameter-edge-invariant graphs

The eccentricity e(v) of v is the distance to a farthest vertex from v. The radius r(G) is the minimum eccentricity among the vertices of G and the diameter d(G) is the maximum eccentricity. For graph G−e obtained by deleting edge e in G, we have r(G−e)?r(G) and d(G−e)?d(G). If for all e in G, r(G−e)=r(G), then G is radius-edge-invariant. Similarly, if for all e in G, d(G−e)=d(G), then G is diameter-edge-invariant. In this paper, we study radius-edge-invariant and diameter-edge-invariant graphs and obtain characterizations of radius-edge-invariant graphs and diameter-edge-invariant graphs of diameter two.     

On the Fourier transforms of retarded Lorentz-invariant functions

In this article we evaluate the Fourier transforms of retarded Lorentz-invariant functions (and distributions) as limits of Laplace transforms. Our method works generally for any retarded Lorentz-invariant functions φ(t) (t?Rⁿ) which is, besides, a continuous function of slow growth. We give, among others, the Fourier transform of G_R(t, α, m², n) and G_A(t, α, m², n), which, in the particular case α = 1, are the characteristic functions of the volume bounded by the forward and the backward sheets of the hyperboloid u = m² and by putting α = ?k are the derivatives of k-order of the retarded and the advanced-delta on the hyperboloid u = m². We also obtain the Fourier transform of the function W(t, α, m², n) introduced by M. Riesz (Comm. Sem. Mat. Univ. Lund4 (1939)). We finish by evaluating the Fourier transforms of the distributional functions G_R(t, α, m², n), G_A(t, α, m², n) and W(t, α, m², n) in their singular points.     

The Erd?s-Ko-Rado properties of set systems defined by double partitions

Let F be a family of subsets of a finite set V. The star ofFatv∈V is the sub-family {A∈F:v∈A}. We denote the sub-family {A∈F:|A|=r} by F^(r).A double partitionP of a finite set V is a partition of V into large sets that are in turn partitioned into small sets. Given such a partition, the family F(P)induced byP is the family of subsets of V whose intersection with each large set is either contained in just one small set or empty.Our main result is that, if one of the large sets is trivially partitioned (that is, into just one small set) and 2r is not greater than the least cardinality of any maximal set of F(P), then no intersecting sub-family of F(P)^(r) is larger than the largest star of F(P)^(r). We also characterise the cases when every extremal intersecting sub-family of F(P)^(r) is a star of F(P)^(r).     

The multiple originator broadcasting problem in graphs

Given a graph G and a vertex subset S of V(G), the broadcasting time with respect toS, denoted by b(G,S), is the minimum broadcasting time when using S as the broadcasting set. And the k-broadcasting number, denoted by b_k(G), is defined by b_k(G)=min{b(G,S)|S⊆V(G),|S|=k}.Given a graph G and two vertex subsets S, S^′ of V(G), define , d(S,S^′)=min{d(u,v)|u∈S, v∈S^′}, and for all v∈V(G). For all k, 1?k?|V(G)|, the k-radius of G, denoted by r_k(G), is defined as r_k(G)=min{d(G,S)|S⊆V(G), |S|=k}.In this paper, we study the relation between the k-radius and the k-broadcasting numbers of graphs. We also give the 2-radius and the 2-broadcasting numbers of the grid graphs, and the k-broadcasting numbers of the complete n-partite graphs and the hypercubes.