Additive Maps Preserving Nilpotent Perturbation of Scalars

Let X be a Banach space over F(= R or C) with dimension greater than 2. Let N(X) be the set of all nilpotent operators and B_0(X) the set spanned by N(X). We give a structure result to the additive maps on FI + B_0(X) that preserve rank-1 perturbation of scalars in both directions. Based on it, a characterization of surjective additive maps on FI + B_0(X) that preserve nilpotent perturbation of scalars in both directions are obtained. Such a map Φ has the form either Φ(T) = cAT A~(-1)+ φ(T)I for all T ∈ FI + B_0(X) or Φ(T) = cAT*A~(-1)+ φ(T)I for all T ∈ FI + B_0(X), where c is a nonzero scalar,A is a τ-linear bijective transformation for some automorphism τ of F and φ is an additive functional.In addition, if dim X = ∞, then A is in fact a linear or conjugate linear invertible bounded operator.     

Oscillation theorems for second-order half-linear differential equations

Oscillation criteria for the second-order half-linear differential equation

[r(t)|ξ′(t)|⁻¹ ξ′(t)]′ + p(t)|ξ(t)|⁻¹ξ(t)=0, t t₀

are established, where > 0 is a constant and exists for t [t₀, ∞). We apply these results to the following equation:

where , D = (D₁,…, D_N), Ω_a = x ^N : |x| ≥ a} is an exterior domain, and c C([a, ∞), ), n > 1 and N ≥ 2 are integers. Here, a > 0 is a given constant.     

On meromorphic solutions of a certain type of nonlinear differential equations

We consider transcendental meromorphic solutions with N(r,f) = S(r,f) of the following type of nonlinear differential equations:f~n + Pn-2(f) = p1(z)e~(α1(z)) +p2(z)e~(α2(z)),where n≥ 2 is an integer, Pn-2(f) is a differential polynomial in f of degree not greater than n-2 with small functions of f as its coefficients, p1(z), p2(z) are nonzero small functions of f, and α1(z), α2(z)are nonconstant entire functions. In particular, we give out the conditions for ensuring the existence of meromorphic solutions and their possible forms of the above equation. Our results extend and improve some known results obtained most recently.     

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.     

On the degree of the inverse of an automorphism of the affine space

We study the degree of the inverse of an automorphism f of the affine n-space over a -algebra k, in terms of the degree d of f and of other data. For n = 1, we obtain a sharp upper bound for deg (f^{− 1}) in terms of d and of the nilpotency index of the ideal generated by the coefficients of f^′'. For n = 2 and arbitrary d≥ 3, we construct a (triangular) automorphism f of Jacobian one such that deg(f^{− 1}) > d. This answers a question of A. van den Essen (see [3]) and enables us to deduce that some schemes introduced by authors to study the Jacobian conjecture are not reduced. Still for n = 2, we give an upper bound for deg (f^{− 1}) when f is triangular. Finally, in the case d = 2 and any n, we complete a result of G. Meisters and C. Olech and use it to give the sharp bound for the degree of the inverse of a quadratic automorphism, with Jacobian one, of the affine 3-space.     

Integral methods for computing solutions of a class of singular two-point boundary value problems

In this paper, we establish three iteration methods to compute solutions for a class of (weakly) singular two-point boundary value problems (xy^′)^′=f(x,y), where x(0,1) and <2. We obtain the sufficient conditions for existence of a unique solution on . Finally, we given some numerical examples.     

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.     

L2-approximation of real-valued functions with interpolatory constraints

We construct the polynomial p_m,n^* of degree m which interpolates a given real-valued function f L²[a, b] at pre-assigned n distinct nodes and is the best approximant to f in the L₂-sense over all polynomials of degree m with the same interpolatory character. It is shown that the L₂-error p_m,n^* − f → 0 as m → ∞ if f C[a, b].     

Diagonal flips in pseudo-triangulations on closed surfaces

Negami has already shown that there is a natural number N(F²) for any closed surface F² such that two triangulations on F² with n vertices can be transformed into each other by a sequence of diagonal flips if nN(F²). We investigate the same theorem for pseudo-triangulations with or without loops, estimating the length of a sequence of diagonal flips. Our arguments will be applied to simple triangulations to obtain a linear upper bound for N(F²) with respect to the genus of F².     

On oscillation of second order neutral type delay differential equations

Oscillation criteria are obtained by using the so called H-method for the second order neutral type delay differential equations of the form

(r(t)ψ(x(t))z^′(t))^′+q(t)f(x(σ(t)))=0, tt₀,

where z(t)=x(t)+p(t)x(τ(t)), r, p, q, τ, σ, C([t₀,∞),R) and f,ψC(R,R).

The results of the paper contains several results obtained previously as special cases. Furthermore, we are also able to fix an error in a recent paper related to the oscillation of second order nonneutral delay differential equations.     

Optimal fault-tolerant routings with small routing tables for k-connected graphs

We study the problem of designing fault-tolerant routings with small routing tables for a k-connected network of n processors in the surviving route graph model. The surviving route graph R(G,ρ)/F for a graph G, a routing ρ and a set of faults F is a directed graph consisting of nonfaulty nodes of G with a directed edge from a node x to a node y iff there are no faults on the route from x to y. The diameter of the surviving route graph could be one of the fault-tolerance measures for the graph G and the routing ρ and it is denoted by D(R(G,ρ)/F). We want to reduce the total number of routes defined in the routing, and the maximum of the number of routes defined for a node (called route degree) as least as possible. In this paper, we show that we can construct a routing λ for every n-node k-connected graph such that n2k², in which the route degree is , the total number of routes is O(k²n) and D(R(G,λ)/F)3 for any fault set F (|F|<k). In particular, in the case that k=2 we can construct a routing λ′ for every biconnected graph in which the route degree is , the total number of routes is O(n) and D(R(G,λ′)/{f})3 for any fault f. We also show that we can construct a routing ρ₁ for every n-node biconnected graph, in which the total number of routes is O(n) and D(R(G,ρ₁)/{f})2 for any fault f, and a routing ρ₂ (using ρ₁) for every n-node biconnected graph, in which the route degree is , the total number of routes is and D(R(G,ρ₂)/{f})2 for any fault f.     

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 .     

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.     

Invariants of linear maps on matrix algebras

Let L be a linear map on the space of n×n matrices over a field. We determine the structure of the maps L that preserve commutativity. We also determine the structure of all linear maps on complex matrices that preserve the higher numerical range. The main tool is the Fundamental Theorem of Projective Geometry.     

The game of n-times nim

The following game is considered. The first player can take any number of stones, but not all the stones, from a single pile of stones. After that, each player can take at most n-times as many as the previous one. The player first unable to move loses and his opponent wins. Let f₁,f₂,… be an initial sequence of stones in increasing order, such that the second player has a winning strategy when play begins from a pile of size f_i. It is proved that there exist constants c=c(n) and k₀=k₀(n) such that f_k+1=f_k+f_k−c for all k>k₀, and lim_n→∞ c(n)/(nlogn)=1.     

Partitioning Boolean lattices into antichains

Let f(n) be the smallest integer t such that a poset obtained from a Boolean lattice with n atoms by deleting both the largest and the smallest elements can be partitioned into t antichains of the same size except for possibly one antichain of a smaller size. In this paper, it is shown that f(n)b n²/log n. This is an improvement of the best previously known upper bound for f(n).     

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.     

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.     

Degrees in a digraph whose nodes are graphs

By an f-graph we mean an unlabeled graph having no vertex of degree greater than f. Let D(n, f) denote the digraph whose node set is the set of f-graphs of order n and such that there is an arc from the node corresponding to graph H to the node corresponding to the graph K if and only if K is obtainable from H by the addition of a single edge. In earlier work, algorithms were developed which produce exact results about the structure of D(n, f), nevertheless many open problems remain. For example, the computation of the order and size of D(n, f) for a number of values of n and f have been obtained. Formulas for the order and size for f = 2 have also been derived. However, no closed form formulas have been determined for the order and size of D(n, f) for any value of f. Here we focus on questions concerning the degrees of the nodes in D(n,n − 1) and comment on related questions for D(n,f) for 2 f < n − 1.     

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.