New families of exact solitary patterns solutions for the nonlinearly dispersive R(m,n) equations

The R(m,n) equations u_tt+a(uⁿ)_xx+b(u^m)_xxtt=0 (a, b const) are investigated by using some ansatze. As a result, new exact solitary patterns solutions and solitary wave solutions are obtained. These obtained solutions show that not only the nonlinearly dispersive R(m,m) equations (m≠1) but also the linearly dispersive R(1,n) equations (m=1) possess solitary patterns solutions, which has infinite slopes or cusps and solitary wave solutions.     

Stability of Periodic Orbits and Return Trajectories of Continuous Multi-valued Maps on Intervals

Let I be a compact interval of real axis R, and(I, H) be the metric space of all nonempty closed subintervals of I with the Hausdorff metric H and f : I → I be a continuous multi-valued map. Assume that Pn =(x_0, x_1,..., xn) is a return tra jectory of f and that p ∈ [min Pn, max Pn] with p ∈ f(p). In this paper, we show that if there exist k(≥ 1) centripetal point pairs of f(relative to p)in {(x_i; x_i+1) : 0 ≤ i ≤ n-1} and n = sk + r(0 ≤ r ≤ k-1), then f has an R-periodic orbit, where R = s + 1 if s is even, and R = s if s is odd and r = 0, and R = s + 2 if s is odd and r 0. Besides,we also study stability of periodic orbits of continuous multi-valued maps from I to I.     

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.     

On a problem about face polynomials

It is proved that an R-automorphism of polynomial ring R[x₁,…,x_n] is completely determined by its face polynomials, where R is a reduced commutative ring and n≥2. An example is given which shows that the condition R being reduced cannot be weakened.     

ℓ-homology of right-angled Coxeter groups based on barycentric subdivisions

Associated to any finite flag complex L there is a right-angled Coxeter group W_L and a contractible cubical complex Σ_L on which W_L acts properly and cocompactly, and such that the link of each vertex is L. It follows that if L is a triangulation of , then Σ_L is a contractible n-manifold. We establish vanishing (in a certain range) of the reduced ℓ²-homology of Σ_L in the case where L is the barycentric subdivision of a cellulation of a manifold. In particular, we prove the Singer Conjecture (on the vanishing of the reduced ℓ²-homology except in the middle dimension) in the case of Σ_L where L is the barycentric subdivision of a cellulation of , n=6,8.     

Multicolor routing in the undirected hypercube

An undirected routing problem is a pair (G,R) where G is an undirected graph and R is an undirected multigraph such that V(G)=V(R). A solution to an undirected routing problem (G,R) is a collection P of undirected paths of G (possibly containing multiple occurrences of the same path) such that edges of R are in one-to-one correspondence with the paths of P, with the path corresponding to edge {u,v} connecting u and v. We say that a collection of paths P is k-colorable if each path of P can be colored by one of the k colors so that the paths of the same color are edge-disjoint (each edge of G appears at most once in the paths of each single color). In the circuit-switched routing context, and in optical network applications in particular, it is desirable to find a solution to a routing problem that is colorable with as few colors as possible. Let Q_n denote the n-dimensional hypercube, for arbitrary n1. We show that a routing problem (Q_n,R) always admits a 4d-colorable solution where d is the maximum vertex degree of R. This improves over the 16d/2-color result which is implicit in the previous work of Aumann and Rabani [SODA95, pp. 567–576]. Since, for any positive d, there is a multigraph R of degree d such that any solution to (Q_n,R) requires at least d colors, our result is tight up to a factor of four. In fact, when d=1, it is tight up to a factor of two, since there is a graph of degree one (the antipodal matching) that requires two colors.     

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.     

The space localization of unbounded boundary perturbations in nonlinear heat conduction with transfer

We consider the nonlinear parabolic equation u_t = (k(u)u_x)_x + b(u)_x, where u = u(x, t, x ε R¹, t > 0; k(u) ≥ 0, b(u) ≥ 0 are continuous functions as u ≥ 0, b (0) = 0; k, b > 0 as u > 0. At t = 0 nonnegative, continuous and bounded initial value is prescribed. The boundary condition u(0, t) = Ψ(t) is supposed to be unbounded as t → +∞. In this paper, sufficient conditions for space localization of unbounded boundary perturbations are found. For instance, we show that nonlinear equation u_t = (uⁿu_x)_x + (u^β)_x, n ≥ 0, β >; n + 1, exhibits the phenomenon of “inner boundedness,” for arbitrary unbounded boundary perturbations.     

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.     

A parallel computing scheme for minimizing a class of large scale functions

This paper gives a parallel computing scheme for minimizing a twice continuously differentiable function with the form

where x = (x^T₁,…,x^T_m)^T and x_i R^n_i, ∑^m_{i = 1}n_i = n, and n a very big number. It is proved that we may use m parallel processors and an iterative procedure to find a minimizer of ƒ(x). The convergence and convergence rate are given under some conditions. The conditions for finding a global minimizer of ƒ(x by using this scheme are given, too. A similar scheme can also be used parallelly to solve a large scale system of nonlinear equations in the similar way. A more general case is also investigated.     

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.     

Subsets of an interval whose product is a power

We form squares from the product of integers in a short interval [n, n + t_n], where we include n in the product. If p is prime, p|n, and (₂^p) > n, we prove that p is the minimum t_n. If no such prime exists, we prove t_n √5n when n> 32. If n = p(2p − 1) and both p and 2p − 1 are primes, then t_n = 3p> 3 √n/2. For n(n + u) a square > n², we conjecture that a and b exist where n < a < b < n + u and nab is a square (except n = 8 and N = 392). Let g₂(n) be minimal such that a square can be formed as the product of distinct integers from [n, g₂(n)] so that no pair of consecutive integers is omitted. We prove that g₂(n) 3n − 3, and list or conjecture the values of g₂(n) for all n. We describe the generalization to kth powers and conjecture the values for large n.     

Circular permutation graph family with applications

In a circular permutation diagram, there are two sets of terminals on two concentric circles: C_in and C_out. Given a permutation Π = [π₁, π₂, …, π_n], terminal i on C_in and terminal π_i on C_out are connected by a wire. The intersection graph G_c of a circular permutation diagram D_c is called a circular permutation graph of a permutation Π corresponding to the diagram D_c. The set of all circular permutation graphs of a permutation Π is called the circular permutation graph family of permutation Π. In this paper, we propose the following: (1) an O(V + E) time algorithm to check if a labeled graph G = (V, E) is a labeled circular permutation graph. (2) An O(n log n + nt) time algorithm to find a maximum independent set of a family, where n = Π and t is the cardinality of the output. (Number t in the worst case is O(n). However, if Π is uniformly distributed (and independent from i), its expected value is O(√n).) (3) An O(min(δV_clog logV_c,V_clogV_c) + E_c) time algorithm for finding a maximum independent set of a circular permutation diagram D_c, where δ is the minimum degree of vertices in the intersection graph G_c = (V_c,E_c) of D_c. (4) An O(n log log n) time algorithm for finding a maximum clique and the chromatic number of a circular permutation diagram, where n is the number of wires in the diagram.     

Extremal properties of strong quadrature weights and maximal mass results for truncated strong moment problems

A bisequence of complex numbers {μ_n}_−∞^∞ determines a strong moment functional satisfying L[xⁿ] = μ_n. If is positive-definite on a bounded interval (a,b) R{0}, then has an integral representation , n=0, ±1, ±2,…, and quadrature rules {w_ni,x_ni} exist such that μ_k = ∑_i=i^n_ns_ni^kw_ni. This paper is concerned with establishing certain extremal properties of the weights w_ni and using these properties to obtain maximal mass results satisfied by distributions ψ(x) representing when only a finite bisequence of moments {μ_k}_k=−nⁿ⁻¹ is given.     

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.     

Upper bounds for Ramsey numbers

The Ramsey number R(G₁,G₂,…,G_k) is the least integer p so that for any k-edge coloring of the complete graph K_p, there is a monochromatic copy of G_i of color i. In this paper, we derive upper bounds of R(G₁,G₂,…,G_k) for certain graphs G_i. In particular, these bounds show that R(9,9)6588 and R(10,10)23556 improving the previous best bounds of 6625 and 23854.     

π-Armendariz rings relative to a monoid

Let Mbe a monoid. A ring Ris called M-π-Armendariz if whenever α = a₁g₁+ a₂g₂+ · · · + a_ng_n, β = b₁h₁+ b₂h₂+ · · · + b_mh_m ∈ R[M] satisfy αβ ∈ nil(R[M]), then aibj ∈ nil(R) for all i, j. A ring R is called weakly 2-primal if the set of nilpotent elements in R coincides with its Levitzki radical. In this paper, we consider some extensions of M-π-Armendariz rings and further investigate their properties under the condition that R is weakly 2-primal. We prove that if R is an M-π-Armendariz ring then nil(R[M]) = nil(R)[M]. Moreover, we study the relationship between the weak zip-property (resp., weak APP-property, nilpotent p.p.-property, weak associated prime property) of a ring R and that of the monoid ring R[M] in case R is M-π-Armendariz.     

Limiting values of the variance and the moments of the dimension of a sum or intersection of random vector subspaces

Let W be an n-dimensional vector space over a field F; for each positive integer m, let the m-tuples (U₁, …, U_m) of vector subspaces of W be uniformly distributed; and consider the statistics X_m,1 dim_F(∑_i=1^m U_i) and X_m,2 dim_F (∩_i=1^m U_i). If F is finite of cardinality q, we determine lim E(X_m,1^k), and lim E(X_m,2^k), and hence, lim var(X_m,1) and lim var(X_m,2), for any k > 0, where the limits are taken as q → ∞ (for fixed n). Further, we determine whether these, and other related, limits are attained monotonically. Analogous issues are also addressed for the case of infinite F.     

A bijective proof of an identity for noncrossing graphs

We give a bijective proof for the identity a_n+2=8b_n, where a_n is the number of noncrossing simple graphs with n (possibly isolated) vertices and b_n is the number of noncrossing graphs without isolated vertices and with n (possibly multiple) edges.