The existence of linear factors in the direct sum of forms and in the tensor product of forms

The tensor product f⊗g of two forms of degree r has a linear factor if and only if both fand g have a linear factor and the sum of the multiplicities of these factors is larger than r. In fact, if (V,ϕ) and (Wθ) are symmetric spaces corresponding to fand g with linear factors U and T respectively the sum of whose multiplicities is larger than r, then the linear factor of (V,ϕ)⊗(Wθ)is U⊗W+V⊗T. Conversely, each linear factor of (V,ϕ)⊗(Wθ)is of this form. Additionally, the direct sum of two nondegenerate symmetric spaces has a linear factor if and only if each is one dimensional and isomorphic to the negative of the other.     

Inverse geophysical problems for some noncompactly supported inhomogeneities

Let[²+k²n(x₁,x₃)]u(x₁,x₂,x₃)=−δ(x₁,y₁δ(x₂,y₂)δ(x₃,y₃) in R³₊. Assume that u(x₁,x₂,x₃=0,y₁,y₂=0,y₃=0,k) is measured at the plane P {x:x₃=0} for all positions of the source on the line y = (y₁,y₂ = 0,y₃ = 0), -∞ < y₁ < ∞, and receiver on the plane(x₁,x₂,x₃ − <x₁,x₂ < ∞, and for low-frequencies 0 < k <k₀, k₀ > 0 is an arbitrary small wave number. Assume thatn(x₁,x₃) is an arbitrary bounded piecewise-continuous function. The basic result is: the above low-frequency surface data determinen(x₁,x₃)uniquely.     

On neighborhood condition for graphs to have [a,b]-factors

Let G be a graph of order n, and let a and b be integers such that 1a<b. Let δ(G) be the minimum degree of G. Then we prove that if δ(G)(k−1)a, n(a+b)(k(a+b)−2)/b, and |N_G(x₁)N_G(x₂)N_G(x_k)|an/(a+b) for any independent subset {x₁,x₂,…,x_k} of V(G), where k2, then G has an [a,b]-factor. This result is best possible in some sense.     

Reconstruction of infinite locally finite connected graphs

Let G be an infinite locally finite connected graph. We study the reconstructibility of G in relation to the structure of its end set . We prove that an infinite locally finite connected graph G is reconstructible if there exists a finite family (Ω_i)_0i (n2) of pairwise finitely separable subsets of such that, for all x,y,x′,y′V(G) and every isomorphism f of G−{x,y} onto G−{x′,y′} there is a permutation π of {0,…,n−1} such that for 0i<n. From this theorem we deduce, as particular consequences, that G is reconstructible if it satisfies one of the following properties: (i) G contains no end-respecting subdivision of the dyadic tree and has at least two ends of maximal order; (ii) the set of thick ends or the one of thin ends of G is finite and of cardinality greater than one. We also prove that if almost all vertices of G are cutvertices, then G is reconstructible if it contains a free end or if it has at least a vertex which is not a cutvertex.     

Profile minimization on triangulated triangles

Given graph G=(V,E) on n vertices, the profile minimization problem is to find a one-to-one function f:V→{1,2,…,n} such that ∑_vV(G){f(v)−min_xN[v] f(x)} is as small as possible, where N[v]={v}{x: x is adjacent to v} is the closed neighborhood of v in G. The trangulated triangle T_l is the graph whose vertices are the triples of non-negative integers summing to l, with an edge connecting two triples if they agree in one coordinate and differ by 1 in the other two coordinates. This paper provides a polynomial time algorithm to solve the profile minimization problem for trangulated triangles T_l with side-length l.     

The chromatic difference sequence of the Cartesian product of graphs

The chromatic difference sequence cds(G) of a graph G with chromatic number n is defined by cds(G) = (a(1), a(2),…, a(n)) if the sum of a(1), a(2),…, a(t) is the maximum number of vertices in an induced t-colorable subgraph of G for t = 1, 2,…, n. The Cartesian product of two graphs G and H, denoted by G□H, has the vertex set V(G□H = V(G) x V(H) and its edge set is given by (x₁, y₁)(x₂, y₂) ε E(G□H) if either x₁ = x₂ and y₁ y₂ ε E(H) or y₁ = y₂ and x₁x₂ ε E(G).

We obtained four main results: the cds of the product of bipartite graphs, the cds of the product of graphs with cds being nondrop flat and first-drop flat, the non-increasing theorem for powers of graphs and cds of powers of circulant graphs.     

More examples on ordered set reconstruction

We show that there are nonisomorphic ordered sets P and Q that have the same maximal and minimal decks and a rank k such that there is a map B from the elements of rank k in P to the elements of rank k in Q such that P{x} is isomorphic to Q{B(x)} for all x of rank k in P. The examples are preceded by a criterion as to when two nonisomorphic ordered sets will have a rank k and a map B as above.     

Length-bounded disjoint paths in planar graphs

The following problem is considered: given: an undirected planar graph G=(V,E) embedded in , distinct pairs of vertices {r₁,s₁},…,{r_k,s_k} of G adjacent to the unbounded face, positive integers b₁,…,b_k and a function ; find: pairwise vertex-disjoint paths P₁,…,P_k such that for each i=1,…,k, P_i is a r_i−s_i-path and the sum of the l-length of all edges in P_i is at most b_i. It is shown that the problem is NP-hard in the strong sense. A pseudo-polynomial-time algorithm is given for the case of k=2.     

Designs with no three mutually disjoint blocks

Let D(v,b,r,k,λ) be any quasi-symmetric block design with block intersection numbers 0 and y. Suppose D has no three mutually disjoint blocks. We show that for a given value of y, there are only finitely many parameter sets of such designs. Moreover, the ‘extremal’ designs D have one of the following parameter sets: (1) v = 4y, k = 2y, λ = 2y − 1 (y 2) (2) v = y(y²+3y+1), k = y(y+1), λ =y²+y−1(y2) (3) v = (y+1)(y²+2y−1), k = y(y+1), λ =y² (y2) A computer search revealed only three parameter sets in the range 1y199, which are not of the above types.     

Elasticity of factorizations in integral domains

For an atomic integral domain R, define(R)=sup{mn|x₁x_m=y₁y_n, each x_i,y_jεR is irreducible}. We investigate (R), with emphasis for Krull domains R. When R is a Krull domain, we determine lower and upper bounds for (R); in particular,(R)≤max{|Cl(R)| 2, 1}. Moreover, we show that for any real numbers r≥1 or R=∞, there is a Dedekind domain R with torsion class group such that (R)=r.     

Oscillation criteria for second-order nonlinear differential equations with integrable coefficient

In this paper, we consider the second-order nonlinear differential equation

[a(t)|y′(t)|^σ−1y′(t)|′+q(t)f(y(t))=r(t)

where σ > 0 is a constant, a C(R, (0, ∞)), q C(R, R), f C(R, R), xf(x) > 0, f′(x) ≥ 0 for x ≠ 0. Some new sufficient conditions for the oscillation of all solutions of (*) are obtained. Several examples which dwell upon the importance of our results are also included.     

q-distributions and Markov processes

We consider a sequence of integer-valued random variables X_n, n 1, representing a special Markov process with transition probability λ_{n, l}, satisfying P_{n, l} = (1 − λ_{n, l}) P_{n−1, l} + λ_{n, l−1} P_{n−1, l−1}. Whenever the transition probability is given by λ_{n, l} = q^{n + βl + γ} and λ_{n, l} = 1 − q^n+βl+γ, we can find closed forms for the distribution and the moments of the corresponding random variables, showing that they involve functions such as the q-binomial coefficients and the q-Stirling numbers. In general, it turns out that the q-notation, up to now mainly used in the theory of q-hypergeometrical series, represents a powerful tool to deal with these kinds of problems. In this context we speak therefore about q-distributions. Finally, we present some possible, mainly graph theoretical interpretations of these random variables for special choices of , β and γ.     

Linear transformations on tensor spaces

Let U⊗V denote the tensor product of two finite dimensional vector spaces U and V over an infinite field. Let k be a positive integer such that k≤dim U and k≤ dim V Let D_k denote the set of all non-zero elements of U⊗V of rank less than k. In this paper we study linear transformations T on U⊗V such that (TD_k)⊆D_k.     

k-Lucas numbers and associated bipartite graphs

For a positive integer k2, the k-Fibonacci sequence {g_n^(k)} is defined as: g₁^(k)==g_k−2^(k)=0, g_k−1^(k)=g_k^(k)=1 and for n>k2, g_n^(k)=g_n−1^(k)+g_n−2^(k)++g_n−k^(k). Moreover, the k-Lucas sequence {l_n^(k)} is defined as l_n^(k)=g_n−1^(k)+g_n+k−1^(k) for n1. In this paper, we consider the relationship between g_n^(k) and l_n^(k) and 1-factors of a bipartite graph.     

Algorithmic complexity of list colorings

Given a graph G = (V,E) and a finite set L(v) at each vertex v ε V, the List Coloring problem asks whether there exists a function f:V → _vεVL(V) such that (i) f(v)εL(v) for each vεV and (ii) f(u) ≠f(v) whenever u, vεV and uvεE. One of our results states that this decision problem remains NP-complete even if all of the followingconditions are met: (1) each set L(v) has at most three elements, (2) each “color” xε_vεVL(v) occurs in at most three sets L(v), (3) each vertex vεV has degree at most three, and (4) G is a planar graph. On the other hand, strengthening any of the assumptions (1)–(3) yields a polynomially solvable problem. The connection between List Coloring and Boolean Satisfiability is discussed, too.     

Cycle decompositions of crowns

For an integer n3, the crown S_n⁰ is defined to be the graph with vertex set {x₀,x₁,…,x_n−1,y₀,y₁,…,y_n−1} and edge set {x_iy_j: 0i,jn−1, i≠j}. In this paper we give some sufficient conditions for the edge decomposition of the crown into isomorphic cycles.     

On the surface viscosity at the boundary between phases

The equations of motion of the interphase boundary are considered. It is shown that the conditions at the surface separating the phases obtained in /1, 2/ by different methods, are identical. The study of the dynamics of the fluid-fluid interface was initiated by Bussinesq /3/ who postulated a linear relationship between the surface stress tensor T_β and the strain rate tensor S_β, assigning two viscosity coefficients to the surface, the dilatation coefficient k (the analog of volume viscosity) and the two-dimensional shear viscosity . In the three-dimensional coordinate system two of whose axes u¹ and usu2 coincide with the axes of any coordinate system at the surface and whose third axis u³ is perpendicular to the surface, his results have the form Tβ = [γ + (k - )θ]a_β + S_β , θ = a^βS_β, V_{, β} = r_{. β} — v_sb_β,      

Boundedness and asymptotic behavior of the solutions of a fuzzy difference equation

In this paper we study the existence, the uniqueness, the boundedness and the asymptotic behavior of the positive solutions of the fuzzy difference equation x_n+1=∑_i=0^kA_i/x_n−i^p_i, where k{1,2,…,}, A_i, i{0,1,…,k}, are positive fuzzy numbers, p_i, i{0,1,…,k}, are positive constants and x_i, i{−k,−k+1,…,0}, are positive fuzzy numbers.     

A note on a property of a weak self-similar perfect set

Let S be a compact, weak self-similar perfect set based on a system of weak contractions f_j, j=1,…,m each of which is characterized by a variable contraction coefficient _j(l) as d(f_j(x),f_j(y)) _j(l)d(x,y), d(x,y)<l, l>0. If the relation ∑^m_j=1j(l₀)<1 holds at at least one point l₀, then every nonempty compact metric space is a continuous image of the set S.     

Strong equality of domination parameters in trees

We study the concept of strong equality of domination parameters. Let P₁ and P₂ be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P₂ also has property P₁. Let ψ₁(G) and ψ₂(G), respectively, denote the minimum cardinalities of sets with properties P₁ and P₂, respectively. Then ψ₁(G)ψ₂(G). If ψ₁(G)=ψ₂(G) and every ψ₁(G)-set is also a ψ₂(G)-set, then we say ψ₁(G) strongly equals ψ₂(G), written ψ₁(G)≡ψ₂(G). We provide a constructive characterization of the trees T such that γ(T)≡i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T)=γ_t(T), where γ_t(T) denotes the total domination number of T, is also presented.