Right ideals generated by an idempotent of finite rank

Let R be a K-algebra acting densely on V_D, where K is a commutative ring with unity and V is a right vector space over a division K-algebra D. Let f(X₁,…,X_t) be an arbitrary and fixed polynomial over K in noncommuting indeterminates X₁,…,X_t with constant term 0 such that for some μK occurring in the coefficients of f(X₁,…,X_t). It is proved that a right ideal ρ of R is generated by an idempotent of finite rank if and only if the rank of f(x₁,…,x_t) is bounded above by a same natural number for all x₁,…,x_tρ. In this case, the rank of the idempotent that generates ρ is also explicitly given. The results are then applied to considering the triangularization of ρ and the irreducibility of f(ρ), where f(ρ) denotes the additive subgroup of R generated by the elements f(x₁,…,x_t) for x₁,…,x_tρ.     

Generalized Derivations with Engel Conditions on Polynomials

Let R be a noncommutative prime ring with extended centroid C, and let D: R → R be a nonzero generalized derivation, f(X ₁,…, X _t ) a nonzero polynomial in noncommutative indeterminates X ₁,…, X _t over C with zero constant term, and k ≥ 1 a fixed integer. In this article, D and f(X ₁,…, X _t ) are characterized if the Engel identity is satisfied: [D(f(x ₁,…, x _t )), f(x ₁,…, x _t )] _k  = 0 for all x ₁,…, x _t  ∈ R.     

Selections of bounded variation under the excess restrictions

Let X be a metric space with metric d, c(X) denote the family of all nonempty compact subsets of X and, given F,G∈c(X), let e(F,G)=supx∈Finfy∈Gd(x,y) be the Hausdorff excess of F over G. The excess variation of a multifunction , which generalizes the ordinary variation V of single-valued functions, is defined by where the supremum is taken over all partitions of the interval [a,b]. The main result of the paper is the following selection theorem: If,V₊(F,[a,b])<∞,t₀∈[a,b]andx₀∈F(t₀), then there exists a single-valued functionof bounded variation such thatf(t)∈F(t)for allt∈[a,b],f(t₀)=x₀,V(f,[a,t₀))?V₊(F,[a,t₀))andV(f,[t₀,b])?V₊(F,[t₀,b]). We exhibit examples showing that the conclusions in this theorem are sharp, and that it produces new selections of bounded variation as compared with [V.V. Chistyakov, Selections of bounded variation, J. Appl. Anal. 10 (1) (2004) 1-82]. In contrast to this, a multifunction F satisfying e(F(s),F(t))?C(t−s) for some constant C?0 and all s,t∈[a,b] with s?t (Lipschitz continuity with respect to e(⋅,⋅)) admits a Lipschitz selection with a Lipschitz constant not exceeding C if t₀=a and may have only discontinuous selections of bounded variation if a<t₀?b. The same situation holds for continuous selections of when it is excess continuous in the sense that e(F(s),F(t))→0 as s→t−0 for all t∈(a,b] and e(F(t),F(s))→0 as s→t+0 for all t∈[a,b) simultaneously.     

Cohomology of general coinduced representions

ABSTRACT

Let R be a prime ring with a nonzero derivation d and let f(X ₁,…,X _t ) be a multilinear polynomial over C, the extended centroid of R. Suppose that b[d(f(x ₁,…,x _t )), f(x ₁,…,x _t )] _n  = 0 for all x _i  ∈ R, where 0 ≠ b ∈ R and n is a fixed positive integer. Then f(X ₁,…,X _t ) is centrally valued on R unless char R = 2 and dim _C RC = 4. We prove a more generalized version by replacing R with a left ideal.     

Critically indecomposable graphs

An undirected graph G=(V,E) with a specific subset X⊂V is called X-critical if G and G(X), induced subgraph on X, are indecomposable but G(V−{w}) is decomposable for every w∈V−X. This is a generalization of critically indecomposable graphs studied by Schmerl and Trotter [J.H. Schmerl, W.T. Trotter, Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures, Discrete Mathematics 113 (1993) 191-205] and Bonizzoni [P. Bonizzoni, Primitive 2-structures with the (n−2)-property, Theoretical Computer Science 132 (1994) 151-178], who deal with the case where X is empty.We present several structural results for this class of graphs and show that in every X-critical graph the vertices of V−X can be partitioned into pairs (a₁,b₁),(a₂,b₂),…,(a_m,b_m) such that G(V−{a_j1,b_j1,…,a_jk,b_jk}) is also an X-critical graph for arbitrary set of indices {j₁,…,j_k}. These vertex pairs are called commutative elimination sequence. If G is an arbitrary indecomposable graph with an indecomposable induced subgraph G(X), then the above result establishes the existence of an indecomposability preserving sequence of vertex pairs (x₁,y₁),…,(x_t,y_t) such that x_i,y_i∈V−X. As an application of the commutative elimination sequence of an X-critical graph we present algorithms to extend a 3-coloring (similarly, 1-factor) of G(X) to entire G.     

The Dichotomy Theorem for evolution bi-families

We prove that the operator G, the closure of the first-order differential operator −d/dt+D(t) on L₂(R,X), is Fredholm if and only if the not well-posed equation u^′(t)=D(t)u(t), t∈R, has exponential dichotomies on R₊ and R₋ and the ranges of the dichotomy projections form a Fredholm pair; moreover, the index of this pair is equal to the Fredholm index of G. Here X is a Hilbert space, D(t)=A+B(t), A is the generator of a bi-semigroup, B(⋅) is a bounded piecewise strongly continuous operator-valued function. Also, we prove some perturbations results and consider various examples of not well-posed problems.     

Convex integer maximization via Graver bases

We present a new algebraic algorithmic scheme to solve convex integer maximization problems of the following form, where c is a convex function on R^d and w₁x,…,w_dx are linear forms on Rⁿ,

max{c(w₁x,…,w_dx):Ax=b,x∈Nⁿ}.     

Hilbert curves of polarized varieties

Let X be a normal Gorenstein complex projective variety. We introduce the Hilbert variety V_X associated to the Hilbert polynomial χ(x₁L₁+?+x_ρL_ρ), where L₁,…,L_ρ is a basis of , ρ being the Picard number of X, and x₁,…,x_ρ are complex variables. After studying general properties of V_X we specialize to the Hilbert curve of a polarized variety (X,L), namely the plane curve of degree dim(X) associated to χ(xK_X+yL). Special emphasis is given to the case of polarized threefolds.     

Homoclinic solutions for a class of the second order Hamiltonian systems

We study the existence of homoclinic orbits for the second order Hamiltonian system , where q∈Rⁿ and V∈C¹(R×Rⁿ,R), V(t,q)=-K(t,q)+W(t,q) is T-periodic in t. A map K satisfies the “pinching” condition b₁|q|²?K(t,q)?b₂|q|², W is superlinear at the infinity and f is sufficiently small in L²(R,Rⁿ). A homoclinic orbit is obtained as a limit of 2kT-periodic solutions of a certain sequence of the second order differential equations.     

Effective computation of singularities of parametric affine curves

Let k be a field of characteristic zero and f(t),g(t) be polynomials in k[t]. For a plane curve parameterized by x=f(t),y=g(t), Abhyankar developed the notion of Taylor resultant (Mathematical Surveys and Monographs, Vol. 35, American Mathematical Society, Providence, RI, 1990) which enables one to find its singularities without knowing its defining polynomial. This concept was generalized as D-resultant by Yu and Van den Essen (Proc. Amer. Math. Soc. 125(3) (1997) 689), which works over an arbitrary field. In this paper, we extend this to a curve in affine n-space parameterized by x₁=f₁(t),…,x_n=f_n(t) over an arbitrary ground field k, where f₁,…,f_n∈k[t]. This approach compares to the usual approach of computing the ideal of the curve first. It provides an efficient algorithm of computing the singularities of such parametric curves using Gröbner bases. Computational examples worked out by symbolic computation packages are included.     

Ordering connected graphs having small degree distances

The concept of degree distance of a connected graph G is a variation of the well-known Wiener index, in which the degrees of vertices are also involved. It is defined by D^′(G)=∑_x∈V(G)d(x)∑_y∈V(G)d(x,y), where d(x) and d(x,y) are the degree of x and the distance between x and y, respectively. In this paper it is proved that connected graphs of order n≥4 having the smallest degree distances are K_1,n−1,BS(n−3,1) and K_1,n−1+e (in this order), where BS(n−3,1) denotes the bistar consisting of vertex disjoint stars K_1,n−3 and K_1,1 with central vertices joined by an edge.     

Almost automorphic solutions for differential equations with piecewise constant argument in a Banach space

Using spectral theory we obtain sufficient conditions for the almost automorphy of bounded solutions to differential equations with piecewise constant argument of the form x^′(t)=A(t)x([t])+f(t),t∈R, where A(t) is an almost automorphy operator, f(t) is an X-valued almost automorphic function and X is a finite dimensional Banach space.     

Second order estimates for large solutions of elliptic equations

This paper deals with the second term asymptotic behavior of large solutions to the problems Δu=b(x)f(u), x∈Ω, subject to the singular boundary condition u(x)=∞, x∈∂Ω, where Ω is a smooth bounded domain in R^N, and b(x) is a non-negative weight function. The absorption term f is regularly varying at infinite with index ρ>1 (that is lim_u→∞f(ξu)/f(u)=ξ^ρ for every ξ>0) and the mapping f(u)/u is increasing on (0,+∞). Our analysis relies on the Karamata regular variation theory.     

ADDITIVE SUBGROUPS GENERATED BY POLYNOMIAL VALUES ON RIGHT IDEALS

Let R be a prime ring with extended centroid C, ρ a nonzero right ideal of R and f(X ₁,…,X_t ) a nonzero polynomial over C. We determine the additive subgroup of RC generated by all elements f(x ₁,…,x_t ) for x ₁,…,x_t ∈ ρ C and obtain a result concerning an independence property of polynomials in prime rings.     

The set of periodic scalar differential equations with cubic nonlinearities

We study the structure induced by the number of periodic solutions on the set of differential equations x^′=f(t,x) where f∈C³(R²) is T-periodic in t, fx³(t,x)<0 for every (t,x)∈R², and f(t,x)→?∞ as x→∞, uniformly on t. We find that the set of differential equations with a singular periodic solution is a codimension-one submanifold, which divides the space into two components: equations with one periodic solution and equations with three periodic solutions. Moreover, the set of differential equations with exactly one periodic singular solution and no other periodic solution is a codimension-two submanifold.     

Vanishing power values of commutators with derivations

Let R be a prime ring of char R ≠ 2, let d be a nonzero derivation of R, and let ρ be a nonzero right ideal of R such that [[d(x)x ⁿ , d(y)] _m , [y, x] _s ] ^t = 0 for all x, y ? ρ, where n ≥ 1, m ≥ 0, s ≥ 0, and t ≥ 1 are fixed integers. If [ρ, ρ]ρ ≠ 0 then d(ρ)ρ = 0.     

Derivations with power central values on multilinear polynomials in prime rings

Let R be a prime ring of char R ≠ = 2 with center Z(R) and with extended centroid C, d a nonzero derivation of R and f(x ₁, ..., x _n ) a nonzero multilinear polynomial over C. Suppose that x ^s d(x)x ^t ∈ Z(R) for all x ∈ {d(f(x ₁, ..., x _n ))|x ₁, ..., x _n ∈ ρ}, where ρ is a nonzero right ideal of R and s ≥ 0, t ≥ 0 are fixed integers. If d(ρ)ρ ≠ = 0, then ρ C = eRC for some idempotent e in the socle of RC and f(x ₁, ..., x _n ) ^N is central-valued in eRCe, where N = s + t + 1.      

Lower quasicontinuity, joint continuity and related concepts

Let X and Y be topological spaces, let Z be a metric space, and let f:X×Y→Z be a mapping. It is shown that when Y has a countable base B, then under a rather general condition on the set-valued mappings X∋x→fx(B)∈Z², B∈B, there is a residual set R⊂X such that for every (a,b)∈R×Y, f is jointly continuous at (a,b) if (and only if) fa:Y→Z is continuous at b. Several new results are also established when the notion of continuity is replaced by that of quasicontinuity or by that of cliquishness. Our approach allows us to unify and improve various results from the literature.     

The signed domatic number of some regular graphs

Let G be a finite and simple graph with vertex set V(G), and let f:V(G)→{−1,1} be a two-valued function. If ∑_x∈N[v]f(x)≥1 for each v∈V(G), where N[v] is the closed neighborhood of v, then f is a signed dominating function on G. A set {f₁,f₂,…,f_d} of signed dominating functions on G with the property that for each x∈V(G), is called a signed dominating family (of functions) on G. The maximum number of functions in a signed dominating family on G is the signed domatic number on G. In this paper, we investigate the signed domatic number of some circulant graphs and of the torus C_p×C_q.