The oscillation of higher-ordernonlinear difference equations

The authors discuss the relation of the oscillation of the following two difference equations,

where m ≥ 2, τ : N → N, N isthe set of integers, |n − τ(n)| ≤ Mfor n N₀, M is a positive integer, is nondecreasing in x, xf(n, x)> 0, as x ≠ 0. Wewill show some relations of the oscillation of the above two equations. Especially, for m to be even, we establish the equivalenceof the oscillation of the above two difference equations.     

Embeddings of the base and bundle isomorphisms

Let π_i :E_i→M, i=1,2, be oriented, smooth vector bundles of rank k over a closed, oriented n-manifold with zero sections s_i :M→E_i. Suppose that U is an open neighborhood of s₁(M) in E₁ and F :U→E₂ a smooth embedding so that π₂Fs₁ :M→M is homotopic to a diffeomorphism f. We show that if k>[(n+1)/2]+1 then E₁ and the induced bundle f^*E₂ are isomorphic as oriented bundles provided that f have degree +1; the same conclusion holds if f has degree −1 except in the case where k is even and one of the bundles does not have a nowhere-zero cross-section. For n≡0(4) and [(n+1)/2]+1<kn we give examples of nonisomorphic oriented bundles E₁ and E₂ of rank k over a homotopy n-sphere with total spaces diffeomorphic with orientation preserved, but such that E₁ and f^*E₂ are not isomorphic oriented bundles. We obtain similar results and counterexamples in the more difficult limiting case where k=[(n+1)/2]+1 and M is a homotopy n-sphere.     

Intersections of nest algebras in finite dimensions

If are maximal nests on a finite-dimensional Hilbert space H, the dimension of the intersection of the corresponding nest algebras is at least dim H. On the other hand, there are three maximal nests whose nest algebras intersect in the scalar operators. The dimension of the intersection of two nest algebras (corresponding to maximal nests) can be of any integer value from n to n(n+1)/2, where n=dim H. For any two maximal nests there exists a basis {f₁,f₂,…,f_n} of H and a permutation π such that and where M_i=  span{f₁,f₂,…,f_i} and N_i= span{f_π(1),f_π(2),…,f_π(i)}. The intersection of the corresponding nest algebras has minimum dimension, namely dim H, precisely when π(j)=n−j+1,1jn. Those algebras which are upper-triangular matrix incidence algebras, relative to some basis, can be characterised as intersections of certain nest algebras.     

Changing cofinalities and collapsing cardinals in models of set theory

If a˜cardinal κ₁, regular in the ground model M, is collapsed in the extension N to a˜cardinal κ₀ and its new cofinality, ρ, is less than κ₀, then, under some additional assumptions, each cardinal λ>κ₁ less than cc(P(κ₁)/[κ₁]^<κ₁) is collapsed to κ₀ as well. If in addition N=M[f], where f : ρ→κ₁ is an unbounded mapping, then N is a˜|λ|=κ₀-minimal extension. This and similar results are applied to generalized forcing notions of Bukovský and Namba.     

How many maxima can there be?

Let C be a planar region. Choose n points p₁,,p_nI.I.D. from the uniform distribution over C. Let M^C_n be the number of these points that are maximal. If C is convex it is known that either E(M^C_n)=Θ(√n)> or E(M^C_n)=O(log n). In this paper we will show that, for general C, there is very little that can be said, a-priori, about E(M^C_n). More specifically we will show that if g is a member of a large class of functions then there is always a region C such that E(M^C_n)=Θ(g(n)). This class contains, for example, all monotically increasing functions of the form g(n)= nln^βn, where 0<<1 and β0. This class also contains nondecreasing functions like g(n)=ln^*n. The results in this paper remain valid in higher dimensions.     

Almost perfect matchings in random uniform hypergraphs

We consider the following model H_r(n, p) of random r-uniform hypergraphs. The vertex set consists of two disjoint subsets V of size | V | = n and U of size | U | = (r − 1)n. Each r-subset of V × (_r−1^U) is chosen to be an edge of H ε H_r(n, p) with probability p = p(n), all choices being independent. It is shown that for every 0 < < 1 if P = (C ln n)/n^r−1 with C = C() sufficiently large, then almost surely every subset V₁ V of size | V₁ | = (1 − )n is matchable, that is, there exists a matching M in H such that every vertex of V₁ is contained in some edge of M.     

An efficient algorithm for the parametric resource allocation problem

The parametric resource allocation problem asks to minimize the sum of separable single-variable convex functions containing a parameter λ, Σ_{i = 1}ⁿ (ƒ_i(x_i + λg_i(x_i)), under simple constraints Σ_{i = 1}ⁿ x_i = M, l_i≤x_i≤u_i and x_i: nonnegative integers for i = 1, 2, …, n, where M is a given positive integer, and l_i and u_i are given lower and upper bounds on x_i. This paper presents an efficient algorithm for computing the sequence of all optimal solutions when λ is continuously changed from 0 to ∞. The required time is O(G√Mlog² n + n log n + n log(M/n)), where G = Σ_{i = 1}ⁿ u_i − Σ_{i = 1n} l_i and an evaluation of ƒ_i(·) or g_i(·) is assumed to be done in constant time.     

Computing the face lattice of a polytope from its vertex-facet incidences

We give an algorithm that constructs the Hasse diagram of the face lattice of a convex polytope P from its vertex-facet incidences in time O(min{n,m}··), where n is the number of vertices, m is the number of facets, is the number of vertex-facet incidences, and  is the total number of faces of P. This improves results of Fukuda and Rosta [Computational Geometry 4 (4) (1994) 191–198], who described an algorithm for enumerating all faces of a d-polytope in O(min{n,m}·d·²) steps. For simple or simplicial d-polytopes our algorithm can be specialized to run in time O(d··). Furthermore, applications of the algorithm to other atomic lattices are discussed, e.g., to face lattices of oriented matroids.     

Commutativity preserving linear maps and Lie automorphisms of strictly triangular matrix space

In this paper we classify linear maps preserving commutativity in both directions on the space N(F) of strictly upper triangular (n+1)×(n+1) matrices over a field F. We show that for n3 a linear map on N(F) preserves commutativity in both directions if and only if =^′+f where ^′ is a product of standard maps on N(F) and f is a linear map of N(F) into its center.     

ContributionsChromatic Ramsey numbers

Suppose G is a graph. The chromatic Ramsey number r_c(G) of G is the least integer m such that there exists a graph F of chromatic number m for which the following is true: for any 2-colouring of the edges of F there is a monochromatic subgraph isomorphic to G. Let M_n = min[r_c(G): χ(G) = n]. It was conjectured by Burr et al. (1976) that M_n = (n − 1)² + 1. This conjecture has been confirmed previously for n 4. In this paper, we shall prove that the conjecture is true for n = 5. We shall also improve the upper bounds for M₆ and M₇.     

The thermal equilibrium state of semiconductor devices

The thermal equilibrium state of two oppositely charged gases confined to a bounded domain , m = 1,2 or m = 3, is entirely described by the gases' particle densities p, n minimizing the total energy (p, n). it is shown that for given P, N > 0 the energy functional admits a unique minimizer in {(p, n) ε L²(Ω) x L ²(Ω) : p, n ≥ 0, _Ωp = P, _Ωn = N} and that p, n ε C(Ω) ∩ L^∞(Ω).

The analysis is applied to the hydrodynamic semiconductor device equations. These equations in general possess more than one thermal equilibrium solution, but only the unique solution of the corresponding variational problem minimizes the total energy. It is equivalent to prescribe boundary data for electrostatic potential and particle densities satisfying the usual compatibility relations and to prescribe V^e and P, N for the variational problem.     

On modules with the Kulikov property and pure semisimple modules and rings

For an R-module M let σ[M] denote the category of submodules of M-generated modules. M has the Kulikov property if submodules of pure projective modules in σ[M] are pure projective. The following is proved: Assume M is a locally noetherian module with the Kulikov property and there are only finitely many simple modules in σ[M]. Then, for every n ε , there are only finitely many indecomposable modules of length n in σ[M].

With our techniques we provide simple proofs for some results on left pure semisimple rings obtained by Prest and Zimmermann-Huisgen and Zimmermann with different methods.     

Neighborhood unions and cyclability of graphs

A graph G is said to be cyclable if for each orientation of G, there exists a set S of vertices such that reversing all the arcs of with one end in S results in a hamiltonian digraph. Let G be a 3-connected graph of order n36. In this paper, we show that if for any three independent vertices x₁, x₂ and x₃, |N(x₁)N(x₂)|+|N(x₂)N(x₃)|+|N(x₃)N(x₁)|2n+1, then G is cyclable.     

On the Equation n_1n_2= n_3n_4 Restricted to Factor Closed Sets

We study the number of solutions N(B,F) of the diophantine equation n_1n_2 = n_3 n_4,where 1 ≤ n_1 ≤ B,1 ≤ n_3 ≤ B,n_2,n_4 ∈ F and F[1,B] is a factor closed set.We study more particularly the case when F={m = p_1~(ε1)···p_k~(εk),ε_j∈{0,1},1 ≤ j ≤ k},p_1,...,p_k being distinct prime numbers.     

Asymptotic bounds for some bipartite graph: complete graph Ramsey numbers

The Ramsey number r(H,K_n) is the smallest integer N so that each graph on N vertices that fails to contain H as a subgraph has independence number at least n. It is shown that r(K_2,m,K_n)(m−1+o(1))(n/log n)² and r(C_2m,K_n)c(n/log n)^m/(m−1) for m fixed and n→∞. Also r(K_2,n,K_n)=Θ(n³/log² n) and .     

Discriminant preserving linear maps

Consider the n-square matrices over an infiniie field Kas an n²-dimcnsional vector space M( nK). We determine all linear maps Ton M(nK) such that discriminant TX- discriminant Xfor all Xin M(nK)     

A generalized π2-diffeomorphism finiteness theorem

The π₂-diffeomorphism finiteness result of F. Fang-X. Rong and A. Petrunin-W. Tuschmann (independently) asserts that the diffeomorphic types of compact n-manifolds M with vanishing first and second homotopy groups can be bounded above in terms of n; and upper bounds on the absolute value of sectional curvature and diameter of M: In this paper, we will generalize this π₂-diffeomorphism finiteness by removing the condition that π₁(M) = 0 and asserting the diffeomorphism finiteness on the Riemannian universal cover of M:     

Asymptotic behavior of a nonlinear delay difference equation

This paper considers a class of nonlinear difference equations

Δ₃y_n + ƒ(n, y_n, y_n−r) = 0, n N (n₀)

. A necessary and sufficient condition for the existence of a bounded nonoscillatory solution is given.     

Linear time algorithms for the weighted tailored 2-partition problem and the weighted 2-center problem under l∞-distance

In this paper, the weighted tailored 2-partition problem and the weighted 2-center problem under l_∞-distance are considered. An O(2^d−1·d·n) algorithm to solve the weighted tailored 2-partition problem and an O(d²·n + d²·log*d) time algorithm to solve the weighted 2-center problem in the d-dimensional case are presented.     

Linear operators preserving unitarily invariant norms of matrices

Let F_{m × n} be the set of all m × n matrices over the field F = C or R Denote by U_n(F) the group of all n × n unitary or orthogonal matrices according as F = C or F-R. A norm N() on Fm ×n, is unitarily invariant if N(UAV) = N(A): for all A ∈ F _m×n U ∈ U_m(F). and V ∈ U_n(F). We characterize those linear operators TF_{m × n} → F_{m × n}which satisfy N (T(A)) = N(A)for all A ∈ F_{m × n}

for a given unitarily invariant norm N(). It is shown that the problem is equivalent to characterizing those operators which preserve certain subsets in F_{m × n} To develop the theory we prove some results concerning unitary operators on F_{m × n} which are of independent interest.