Multiplicative Maps With Specified (1, 1)-Entry

In this article, we will show that for many complex-valued maps f over a semigroup G of matrices, there exists a minimum k for the existence of a multiplicative map for which the (1, 1)-entry of } ( A ) is f ( A ). We obtain results on such multiplicative maps, and use them to classify all the multiplicative maps on G such that f ( A ) = ( g ( A )) for any A k G , where f and g are given complex-valued maps.     

General preservers of invariant subspace lattices

Let be the space of all bounded linear operators on a Banach space X and let LatA be the lattice of invariant subspaces of the operator . We characterize some maps with one of the following preserving properties: Lat(Φ(A)+Φ(B))=Lat(A+B), or Lat(Φ(A)Φ(B))=Lat(AB), or Lat(Φ(A)Φ(B)+Φ(B)Φ(A))=Lat(AB+BA), or Lat(Φ(A)Φ(B)Φ(A))=Lat(ABA), or Lat([Φ(A),Φ(B)])=Lat([A,B]).     

All trees are 1-embeddable and all except stars are 2-embeddable

A graph G with n vertices is said to be embeddable (in its complement) if there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))=. It is known that all trees T with n (≥2) vertices and T K_1,n−1 are embeddable. We say that G is 1-embeddable if, for every edge e, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e};and that it is 2-embeddable if,for every pair e₁, e₂ of edges, there is an automorphism φ of K_n such that E(G) ∩ E(φ(G))={e₁, e₂}. We prove here that all trees with n (3) vertices are 1-embeddable; and that all trees T with n (4) vertices and T K_1,n−1 are 2-embeddable. In a certain sense, this result is sharp.     

Strange distributionally chaotic triangular maps II

The notion of distributional chaos was introduced by Schweizer and Smítal [Measures of chaos and a spectral decomposition of dynamical systems on the interval, Trans Am Math Soc 1994;344:737–854] for continuous maps of the interval. For continuous maps of a compact metric space three mutually non-equivalent versions of distributional chaos, DC1–DC3, can be considered. In this paper we study distributional chaos in the class of triangular maps of the square which are monotone on the fibres. The main results: (i) If has positive topological entropy then F is DC1, and hence, DC2 and DC3. This result is interesting since similar statement is not true for general triangular maps of the square [Smítal and Štefánková, Distributional chaos for triangular maps, Chaos, Solitons & Fractals 2004;21:1125–8]. (ii) There are which are not DC3, and such that not every recurrent point of F₁ is uniformly recurrent, while F₂ is Li and Yorke chaotic on the set of uniformly recurrent points. This, along with recent results by Forti et al. [Dynamics of homeomorphisms on minimal sets generated by triangular mappings, Bull Austral Math Soc 1999;59:1–20], among others, make possible to compile complete list of the implications between dynamical properties of maps in , solving a long-standing open problem by Sharkovsky.     

On the relation between interior critical points and parameters for a class of nonlinear problems with Neumann–Robin boundary conditions

We consider boundary value problem

where   0, λ > 0 are parameters and f  C²[0, ∞) such that f(0) < 0. In this paper we study for the cases p  (0, β) and p  (β, θ) (p is the value of the solution at x = 0 and β, θ are such that f(β) = 0, , the relation between λ and the number of interior critical points of the positive solutions of the above system.     

Binding number and Hamiltonian(g, f)-factors in graphs

A(g, f)-factorF of a graphG is called a Hamiltonian(g, f)-factor ifF contains a Hamiltonian cycle. The binding number ofG is defined by $bind(G) = \min \left\{ {\frac{{|N_G (X)|}}{{|X|}}|\not 0 \ne X \subset V(G), N_G (X) \ne V(G)} \right\}$ . Let G be a connected graph, and let a andb be integers such that 4 ≤ a <b. Letg, f be positive integer-valued functions defined onV(G) such that a ≤g(x) < f(x) ≤ b for everyx ∈V(G). In this paper, it is proved that if $bind(G) \geqslant \frac{{(a + b - 5)(n - 1)}}{{(a - 2)n - 3(a + b - 5)}}, \nu (G) \geqslant \frac{{(a + b - 5)^2 }}{{a - 2}}$ and for any nonempty independent subset X ofV(G), thenG has a Hamiltonian(g, f)-factor.     

Nonlinear Strong Commutativity Preserving Maps on Prime Rings

Let 𝒜 be a unital prime ring containing a nontrivial idempotent P. Assume that Φ: 𝒜 → 𝒜 is a nonlinear surjective map. It is shown that Φ preserves strong commutativity if and only if Φ has the form Φ(A) = αA + f(A) for all A ∈ 𝒜, where α ∈ {1, ?1} and f is a map from 𝒜 into 𝒵(𝒜). As an application, a characterization of nonlinear surjective strong commutativity preserving maps on factor von Neumann algebras is obtained.     

Extrapolating the mean-values of multiplicative functions

It is shown that certain commonly occurring conditions may be factored out of sums of multiplicative arithmetic functions.A function is arithmetic if it is defined on the positive integers. Those complex-valued arithmetic functions g which satisfy the relation g(ab) = g(a)g(b) for all coprime pairs of positive integers a, b are here called multiplicative. In this paper g will be a multiplicative function which satisfies |g(n)| ≤ 1 for all positive integers n.     

Multiplicative properties of generalized matrix functions

We consider scalar-valued matrix functions for n×n matrices A=(a_ij) defined by Where G is a subgroup of S_n the group of permutations on n letters, and χ is a linear character of G. Two such functions are the permanent and the determinant. A function (1) is multiplicative on a semigroup S of n×n matrices if d(AB)=d(A)d(B) AB∈S.

With mild restrictions on the underlying scalar ring we show that every element of a semigroup containing the diagonal matrices on which (1) is multiplicative can have at most one nonzero diagonal(i.e., diagonal with all nonzero entries)and conversely, provided that χ is the principal character(χ≡1).     

The packing of pairs by quadruples

Let X be a finite set of size ν, further let λ be a positive integer and let σ(4,λν) denote the maximum number of quadruples such that each pair of elements of X is contained in at most λ of them. The value of σ(4, 1;ν) has been determined by Brouwer (1979) for all v4. The value of σ(4,λν) has been determined by Billington, Stanton and Stinson (1984) for all ν≡0 (mod 3) and λ>1. In this paper we complete the determination of σ(4,λν) for all ν4 and λ>1.     

Orthogonal Groups 𝒪(n) over GF(2) as Automorphisms

Let (V, Q) be a quadratic vector space over a fixed field. Orthogonal group 𝒪(V, Q) is defined as automorphisms on (V, Q). If Q = I, it is 𝒪(V, I) = 𝒪(n). There is a nice result that 𝒪(n) ? Aut(𝔬(n)) over ? or ?, where 𝔬(n) is the Lie algebra of n × n alternating matrices over the field. How about another field The answer is “Yes” if it is GF(2). We show it explicitly with the combinatorial basis ?. This is a verification of Steinberg's main result in 1961, that is, Aut(𝔬(n)) is simple over the square field, with a nonsimple exception Aut(𝔬(5)) ? 𝒪(5) ? 𝔖₆.     

Extreme convex set functions with finite carrier: General theory

Let Ω={1,…,n} and P={X:SΩ}. A mapping e : P→R⁺ is a convex set function if e()=0 and e(S) + e(T)e(S∩T) + e(S T) for all S. TεP. The set of convex set functions for fixed Ω is a convex cone and the paper is dealing with the extreme points of the base of this cone. To this end a representation theorem is proved: every e ε ¹ can be written as e(·)=max(m¹(·)−α₁…m^t(·)−α_t), where m¹,…,m^t are measures on P and α₁,…,α_t are nonnegative reals. Given additional requirements, the representation is unique and called “canonical”. Fix H {1,…,r},|H| 2. There is a certain subsystem of sets SεP such that m^τ(S)−α_τ=e(S) (τε H}, that is, the subsystem of sets S such that m^τ(S)−α_τ(τεH) is a maximal term in the representation of e by m¹,…,m^τ and α₁,…α_t.e is called nondegenerate is these subsystems determine the measures m¹,…,m^τ uniquely and it turns out that nondegeneracy and extremality are equivalent for e ε ¹. Moreover, it is seen that nondegeneracy is closely related to a generalized version of the problem “represent a given integer λ o by means of integer weights g,…,g_r 0 via σ^r=₁ag=λ such that the integer coefficients a satisfy 0ak (=1,…,r), where k are prescribed integer bounds. Find r such representations with the additional property that the coefficients form a nonsingular matrix.” A solution to the generalized version of this number theoretical problem is given and, finally, a few examples are discussed.     

The deviation, density, and depth of partially ordered sets

Let P be a poset, and let γ be a linear order type with |γ| ≥ 3. The γ-deviation of P, denoted by γ-dev P, is defined inductively as follows: (1) γ-dev P=0, if P contains no chain of order type γ; (2) γ-dev P = , if γ-dev P and each chain C of type γ in P contains elements a and b such that a<b and [a, b] as an interval of P has γ-deviation <. There may be no ordinal such that γ-dev P = ; i.e., γ-dev P does not exist. A chain is γ-dense if each of its intervals contains a chain of order type γ. If P contains a γ-dense chain, then γ-dev P fails to exist. If either (1) P is linearly ordered or (2) a chain of order type γ does not contain a dense interval, then the converse holds. For an ordinal ξ, a special set S(ξ) is used to study ω_ξ-deviation. The depth of P, denoted by δ(P) is the least ordinal β that does not embed in P^*. Then the following statements are equivalent: (1) ω_ξ-dev P does not exist; (2) S(ξ) embeds in P; and (3) P has a subset Q of cardinality _ξ such that δ(Q^*) = ω_{ξ + 1}. Also ω_ξ-dev P = <ω_{ξ + 1} if and only if |δ(P^*)|_ξ; if these equivalent conditions hold, then ω^β_ξ < δ(P^*) ≤ ω ^{+ 1}_ξ for all β < . Applications are made to the study of chains of submodules of a module over an associative ring.     

Some algebraic constructions in rational homotopy theory

In this note we describe constructions in the category of differential graded commutative algebras over the rational numbers Q which are analogs of the space F(X, Y) of continuous maps of X to Y, the component F(X, Y,ƒ) containing ƒ ε F(X, Y), fibrations, induced fibrations, the space Γ(π) of sections of a fibration π: E → X, and the component Γ(π,σ) containing σ ε Γ (π). As a focus, we address the problem of expressing π^*(F(X, Y, ƒ)) = Hom(π_*(F(X,Y, ƒ)),Q) in terms of differential graded algebra models for X and Y.     

Octrees with near optimal cost for ray-shooting

Predicting and optimizing the performance of ray shooting is a very important problem in computer graphics due to the severe computational demands of ray tracing and other applications, e.g., radio propagation simulation. Aronov and Fortune were the first to guarantee an overall performance within a constant factor of optimal in the following model of computation: build a triangulation compatible with the scene, and shoot rays by locating origin and traversing until hit is found. Triangulations are not a very popular model in computer graphics, but space decompositions like kd-trees and octrees are used routinely. Aronov et al. in [B. Aronov, H. Brönnimann, A.Y. Chang, Y.-J. Chiang, Cost prediction for ray shooting, in: Proc. 18th Annu. ACM Sympos. Comput. Geom., ACM, New York, 2002, pp. 293–302; Computational Geometry, submitted for publication] developed a cost measure for such decompositions, and proved it to reliably predict the average cost of ray shooting.

In this paper, we address the corresponding optimization problem on octrees with the same cost measure as the optimizing criterion. More generally, we solve the generalization for generalized octrees in any d dimensions with scenes made up of (d−1)-dimensional simplices. We give a construction of trees which yields cost O(M), where M is the infimum of the cost measure on all trees. Sometimes, a balance condition is important (informally, balanced trees ensures that adjacent leaves have similar size): we also show that rebalancing does not affect the cost by more than a constant multiplicative factor. These are the first and only known results that provide performance guarantees on the approximation factor for 3-dimensional ray shooting with this realistic model of computation. Our results have been validated experimentally by Aronov et al. in [B. Aronov, H. Brönnimann, A.Y. Chang, Y.-J. Chiang, Cost-driven octree construction schemes: an experimental study, in: Proc. of 19th Annu. ACM Sympos. Comput. Geom., ACM, New York, 2003, pp. 227–236; Computational Geometry 21 (1–2) (2005) 127–148].     

Doubly stochastic graph matrices, II

If G is a graph on n vertices, its Laplacian matrix L(G) = D(G) - A(G) is the difference of the diagonal matrix of vertex degrees and the adjacency matrix. The main purpose of this note is to continue the study of the positive definite, doubly stochastic graph matrix (I_n + L(G))-¹= ω(G) = (w_ij). If, for example, w(G) = min w_ij, then w(G)≥0 with equality if and only if G is disconnected and w(G) ≤ l/(n + 1) with equality if and only if G = K_n. If i¦j, then w_ii ≥2wij, with equality if and only if the ith vertex has degree n - 1. In a sense made precise in the note, max w,, identifies most remote vertices of G. Relations between these new graph invariants and the algebraic connectivity emerge naturally from the fact that the second largest eigenvalue of ω(G) is 1/(1 + a(G)).     

Multiplicative maps on ℬ(X)

A structural feature of multiplicative maps on ℬ(X) which sends some rank-1 operator to an operator of rank not greater than 1 is given, and based on it, some characterizations of automorphism of ℬ(X) are obtained and some multiplicative preserver problems are answered. Project supported by the National Natural Science Foundation of China (Grant No.19671055) and Provincial Natural Science Foundation of Shanxi.     

一类非线性奇异微分方程正解的存在性定理

设(i) f(t,u): (0,1)×(0,+∞)→[0,+∞)连续，关于u 单调增加; (ii) 存在函数g:[1,+∞)→(0,+∞),g(b)0,G(t,s)是相应问题的Green函数。     

S-Noetherian Rings of the Forms 𝒜[X] and 𝒜[[X]]

Let 𝒜 = (A _n ) _n≥0 be an ascending chain of commutative rings with identity, S ? A ₀ a multiplicative set of A ₀, and let 𝒜[X] (respectively, 𝒜[[X]]) be the ring of polynomials (respectively, power series) with coefficient of degree i in A _i for each i ∈ ?. In this paper, we give necessary and sufficient conditions for the rings 𝒜[X] and 𝒜[[X]] to be S ? Noetherian.     

The chromatic number of the product of two 4-chromatic graphs is 4

For any graphG and numbern≧1 two functionsf, g fromV(G) into {1, 2, ...,n} are adjacent if for all edges (a, b) ofG, f(a) ≠g(b). The graph of all such functions is the colouring graph ℒ(G) ofG. We establish first that χ(G)=n+1 implies χ(ℒ(G))=n iff χ(G ×H)=n+1 for all graphsH with χ(H)≧n+1. Then we will prove that indeed for all 4-chromatic graphsG χ(ℒ(G))=3 which establishes Hedetniemi’s [3] conjecture for 4-chromatic graphs. This research was supported by NSERC grant A7213