Decompositions of Quotient Rings and m-Power Commuting Maps

Let R be a semiprime ring with symmetric Martindale quotient ring Q, n ≥ 2 and let f(X) = X ⁿ h(X), where h(X) is a polynomial over the ring of integers with h(0) = ±1. Then there is a ring decomposition Q = Q ₁ ⊕ Q ₂ ⊕ Q ₃ such that Q ₁ is a ring satisfying S _2n?2, the standard identity of degree 2n ? 2, Q ₂ ? M _n (E) for some commutative regular self-injective ring E such that, for some fixed q > 1, x ^q  = x for all x ∈ E, and Q ₃ is a both faithful S _2n?2-free and faithful f-free ring. Applying the theorem, we characterize m-power commuting maps, which are defined by linear generalized differential polynomials, on a semiprime ring.     

Completeness of Averaged Scattering Solutions and Inverse Scattering at a Fixed Energy

We prove that the averaged scattering solutions to the Schrödinger equation with short-range electromagnetic potentials (V, A) where V(x) = O(|x|^?ρ), A(x) = O(|x|^?ρ), |x| → ∞, ρ > 1, are dense in the set of all solutions to the Schrödinger equation that are in L ²(K) where K is any connected bounded open set in ? ⁿ ,n ≥ 2, with smooth boundary. We use this result to prove that if two short-range electromagnetic potentials (V ₁, A ₁) and (V ₂, A ₂) in ? ⁿ , n ≥ 3, have the same scattering matrix at a fixed positive energy and if the electric potentials V _j and the magnetic fields F _j : = curl A _j , j = 1, 2, coincide outside of some ball they necessarily coincide everywhere. In a previous paper of Weder and Yafaev the case of electric potentials and magnetic fields that are asymptotic sums of homogeneous terms at infinity was studied. It was proven that all these terms can be uniquely reconstructed from the singularities in the forward direction of the scattering amplitude at a fixed positive energy. The combination of the new uniqueness result of this paper and the result of Weder and Yafaev implies that the scattering matrix at a fixed positive energy uniquely determines electric potentials and magnetic fields that are a finite sum of homogeneous terms at infinity, or more generally, that are asymptotic sums of homogeneous terms that actually converge, respectively, to the electric potential and to the magnetic field.     

Dispersive Estimates of Solutions to the Wave Equation with a Potential in Dimensions n ≥ 4

We prove dispersive estimates for solutions to the wave equation with a real-valued potential V ∈ L ^∞(R ⁿ ), n ≥ 4, satisfying V(x) = O(?x?^?(n+1)/2?ε), ε > 0.     

Prophet Regions for Independent [0, 1]-Valued Random Variables with Random Discounting

Abstract

Let X ₁, X ₂… and B ₁, B ₂… be mutually independent [0, 1]-valued random variables, with EB _j  = β > 0 for all j. Let Y _j  = B ₁ … sB _j?1 X _j for j ≥ 1. A complete comparison is made between the optimal stopping value V(Y ₁,…,Y _n ):=sup{EY _τ:τ is a stopping rule for Y ₁,…,Y _n } and E(max _1≤j≤n Y _j ). It is shown that the set of ordered pairs {(x, y):x = V(Y ₁,…,Y _n ), y = E(max _1≤j≤n Y _j ) for some sequence Y ₁,…,Y _n obtained as described} is precisely the set {(x, y):0 ≤ x ≤ 1, x ≤ y ≤ Ψ _{n, β}(x)}, where Ψ _{n, β}(x) = [(1 ? β)n + 2β]x ? β^?(n?2) x ² if x ≤ β ^n?1, and Ψ _{n, β}(x) = min _j≥1{(1 ? β)jx + β ^j } otherwise. Sharp difference and ratio prophet inequalities are derived from this result, and an analogous comparison for infinite sequences is obtained.     

Dynamics of a family of piecewise-linear area-preserving plane maps III. Cantor set spectra

This paper studies the behavior under iteration of the maps T _ab (x,y) = (F _ab (x) ? y, x) of the plane ?², in which F _ab (x) = ax if x ≥ 0 and bx if x < 0. These maps are area-preserving homeomorphisms of ?² that map rays from the origin to rays from the origin. Orbits of the map correspond to solutions of the nonlinear difference equation x _n+2 = 1/2(a ? b)|x _n+1|+1/2(a+b)x _n+1 ? x _n . This difference equation can be rewritten in an eigenvalue form for a nonlinear difference operator of Schrödinger type ? x _n+2+2x _n+1 ? x _n +V _μ(x _n+1)x _n+1 = Ex _n+1, in which μ = (1/2)(a ? b) is fixed, and V _μ(x) = μ(sgn(x)) is an antisymmetric step function potential, and the energy E = 2 ? 1/2(a+b). We study the set Ω _SB of parameter values where the map T _ab has at least one bounded orbit, which correspond to l _∞-eigenfunctions of this difference operator. The paper shows that for transcendental μ the set Spec_∞[μ] of energy values E having a bounded solution is a Cantor set. Numerical simulations suggest the possibility that these Cantor sets have positive (one-dimensional) measure for all real values of μ.     

Inverse Problems for Time-Dependent Singular Heat Conductivities: Multi-Dimensional Case

We consider an inverse boundary value problem for the heat equation ? _t u = div (γ? _x u) in (0, T) × Ω, u = f on (0, T) × ?Ω, u| _t=0 = u ₀, in a bounded domain Ω ? ? ⁿ , n ≥ 2, where the heat conductivity γ(t, x) is piecewise constant and the surface of discontinuity depends on time: γ(t, x) = k ² (x ∈ D(t)), γ(t, x) = 1 (x ∈ Ω?D(t)). Fix a direction e* ∈ 𝕊 ^n?1 arbitrarily. Assuming that ?D(t) is strictly convex for 0 ≤ t ≤ T, we show that k and sup {e*·x; x ∈ D(t)} (0 ≤ t ≤ T), in particular D(t) itself, are determined from the Dirichlet-to-Neumann map : f → ?_ν u(t, x)|_{(0, T)×?Ω}. The knowledge of the initial data u ₀ is not used in the proof. If we know min_0≤t≤T (sup _x∈D(t) x·e*), we have the same conclusion from the local Dirichlet-to-Neumann map. Numerical examples of stationary and moving circles inside the unit disk are shown. The results have applications to nondestructive testing. Consider a physical body consisting of homogeneous material with constant heat conductivity except for a moving inclusion with different conductivity. Then the location and shape of the inclusion can be monitored from temperature and heat flux measurements performed at the boundary of the body. Such a situation appears for example in blast furnaces used in ironmaking.     

On the Centralizers of Derivations with Engel Conditions

Let R be a noncommutative prime ring and d, δ two nonzero derivations of R. If δ([d(x), x] _n ) = 0 for all x ∈ R, then char R = 2, d ² = 0, and δ = αd, where α is in the extended centroid of R. As an application, if char R ≠ 2, then the centralizer of the set {[d(x), x] _n  | x ∈ R} in R coincides with the center of R.     

Generalized Derivations with Engel Conditions on One-Sided Ideals

Let R be a noncommutative prime ring and I a nonzero left ideal of R. Let g be a generalized derivation of R such that [g(r ^k ), r ^k ] _n  = 0 for all r ∈ I, where k, n are fixed positive integers. Then there exists c ∈ U, the left Utumi quotient ring of R, such that g(x) = xc and I(c ? α) = 0 for a suitable α ∈ C. In particular we have that g(x) = α x, for all x ∈ I.     

Zeros of semilinear systems with applications to nonlinear partial difference equations on graphs†

Let Q be a m × m real matrix and f _j  : ? → ?, j = 1, …, m, be some given functions. If x and f(x) are column vectors whose j-coordinates are x _j and f _j (x _j ), respectively, then we apply the finite dimensional version of the mountain pass theorem to provide conditions for the existence of solutions of the semilinear system Qx = f(x) for Q symmetric and positive semi-definite. The arguments we use are a simple adaptation of the ones used by Neuberger. An application of the above concerns partial difference equations on a finite, connected simple graph. A derivation of a graph 𝒢 is just any linear operator D:C ⁰(𝒢) → C ⁰(𝒢), where C ⁰(𝒢) is the real vector space of real maps defined on the vertex set V of the graph. Given a derivation D and a function F:V × ? → ?, one has associated a partial difference equation Dμ = F(v,μ), and one searches for solutions μ ∈ C ⁰(𝒢). Sufficient conditions in order to have non-trivial solutions of partial difference equations on any finite, connected simple graph for D symmetric and positive semi-definite derivation are provided. A metric (or weighted) graph is a pair (𝒢, d), where 𝒢 is a connected finite degree simple graph and d is a positive function on the set of edges of the graph. The metric d permits to consider some classical derivations, such as the Laplacian operator ?₂. In (Neuberger, Elliptic partial difference equations on graphs, Experiment. Math. 15 (2006), pp. 91–107) was considered the nonlinear elliptic partial difference equations ?₂ u = F(u), for the metric d = 1.     

Weighted Least Square Convergence of Lagrange Interpolation on the Unit Circle

In the paper, a result of Walsh and Sharma on least square convergence of Lagrange interpolation polynomials based on the n-th roots of unity is extended to Lagrange interpolation on the sets obtained by projecting vertically the zeros of (1-x)²=P ^(a,β) _n(x),a>0,β>0,(1-x)P^(a,β) _n(x),a>0,β>-1,(1+x)P P^(a,β) _n(x),a>-1,β0 and P^(a,β) _n(x),a>-1,β>-1, respectively, onto the unit circle, where P^(a,β) _n(x),a>-1,β>-1, stands for the n-th Jacobi polynomial. Moreover, a result of Saff and Walsh is also extended.     

On the Approximation of Positive Functions by Power Series, II

We consider positive functionsh=h(x) defined forxR⁺₀. Conditions for the existence of a power seriesN(x)=∑ c_nxⁿ,c_n0, with the propertyd₁h(x)/N(x)d₂, x0,for some constantsd₁, d₂R⁺, are investigated in [J. Clunie and T. Kövari,Canad. J. Math.20(1968), 7–20; P. Erd s and T. Kövari,Acta Math. Acad. Sci. Hung.7(1956), 305–316; U. Schmid,Complex Variables18(1992), 187–192; U. Schmid, J.Approx. Theory83(1995), 342–346]. In this paper, methods are discussed which allow for a given functionhthe construction of the coefficientsc_n,n ₀, for the above defined power seriesNand to find suitable constantsd₁andd₂. We also study the power seriesH(x)=∑ xⁿ/u_n, where we setu_n=sup{xⁿ/h(x), x0}, forn ₀, and the relation betweenhandHconcerning the above stated inequalities.     

Generalized Skew Derivations with Nilpotent Values in Prime Rings

Let ? be a prime ring, 𝒞 the extended centroid of ?, ? a Lie ideal of ?, F be a nonzero generalized skew derivation of ? with associated automorphism α, and n ≥ 1 be a fixed integer. If (F(xy) ? yx) ⁿ  = 0 for all x, y ∈ ?, then ? is commutative and one of the following statements holds:

(1) Either ? is central;

(2) Or ? ? M ₂(𝒞), the 2 × 2 matrix ring over 𝒞, with char(𝒞) = 2.     

An Annihilator Condition with Generalized Derivations on Multilinear Polynomials

Let R be a non-commutative prime ring of characteristic different from 2, U its right Utumi quotient ring, C its extended centroid, F a generalized derivation on R, and f(x ₁,…, x _n ) a noncentral multilinear polynomial over C. If there exists a ∈ R such that, for all r ₁,…, r _n  ∈ R, a[F ²(f(r ₁,…, r _n )), f(r ₁,…, r _n )] = 0, then one of the following statements hold: 1. a = 0;

2. There exists λ ∈C such that F(x) = λx, for all x ∈ R;

3. There exists c ∈ U such that F(x) = cx, for all x ∈ R, with c ² ∈ C;

4. There exists c ∈ U such that F(x) = xc, for all x ∈ R, with c ² ∈ C.

     

Vanishing Derivations and Co-Centralizing Generalized Derivations on Multilinear Polynomials in Prime Rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, and f(x₁,…, x_n) be a multilinear polynomial over C, which is not central valued on R. Suppose that F and G are two generalized derivations of R and d is a nonzero derivation of R such that d(F(f(r))f(r) ? f(r)G(f(r))) = 0 for all r = (r₁,…, r_n) ∈ Rⁿ, then one of the following holds:

1. There exist a, p, q, c ∈ U and λ ∈C such that F(x) = ax + xp + λx, G(x) = px + xq and d(x) = [c, x] for all x ∈ R, with [c, a ? q] = 0 and f(x₁,…, x_n)² is central valued on R;

2. There exists a ∈ U such that F(x) = xa and G(x) = ax for all x ∈ R;

3. There exist a, b, c ∈ U and λ ∈C such that F(x) = λx + xa ? bx, G(x) = ax + xb and d(x) = [c, x] for all x ∈ R, with b + αc ∈ C for some α ∈C;

4. R satisfies s₄ and there exist a, b ∈ U and λ ∈C such that F(x) = λx + xa ? bx and G(x) = ax + xb for all x ∈ R;

5. There exist a′, b, c ∈ U and δ a derivation of R such that F(x) = a′x + xb ? δ(x), G(x) = bx + δ(x) and d(x) = [c, x] for all x ∈ R, with [c, a′] = 0 and f(x₁,…, x_n)² is central valued on R.

     

The Ideal Structure of Semigroups of Linear Transformations with Upper Bounds on Their Nullity or Defect

Suppose V is a vector space with dim V = p ≥ q ≥ ?₀, and let T(V) denote the semigroup (under composition) of all linear transformations of V. For α ∈ T (V), let ker α and ran α denote the “kernel” and the “range” of α, and write n(α) = dim ker α and d(α) = codim ran α. In this article, we study the semigroups AM(p, q) = {α ∈ T(V):n(α) < q} and AE(p, q) = {α ∈ T(V):d(α) < q}. First, we determine whether they belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide. Then, for each semigroup, we describe its maximal regular subsemigroup, and we characterise its Green's relations and (two-sided) ideals. As a precursor to further work in this area,, we also determine all the maximal right simple subsemigroups of AM(p, q).     

Subsets with Generalized Derivations Having Nilpotent Values on Lie Ideals

Let R be a prime ring, with no nonzero nil right ideal, Q the two-sided Martindale quotient ring of R, F a generalized derivation of R, L a noncommutative Lie ideal of R, and b ∈ Q. If, for any u, w ∈ L, there exists n = n(u, w) ≥1 such that (F(uw) ? bwu)ⁿ = 0, then one of the following statements holds:

1. F = 0 and b = 0;

2. R ? M₂(K), the ring of 2 × 2 matrices over a field K, b² = 0, and F(x) = ?bx, for all x ∈ R.

     

Finite free resolutions of length two and koszul generalized complex

Abstract

In 1956, Ehrenfeucht proved that a polynomial f ₁(x ₁) + · + f _n (x _n ) with complex coefficients in the variables x ₁, …, x _n is irreducible over the field of complex numbers provided the degrees of the polynomials f ₁(x ₁), …, f _n (x _n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f ₁(x ₁) + · + f _n (x _n ) belonging to K[x ₁, …, x _n ] is irreducible over any field K of characteristic zero provided the degree of each f _i is positive. Clearly a polynomial F = f ₁(x ₁) + · + f _n (x _n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g ₁(x ₁)) ^p  + (g ₂(x ₂)) ^p  + · + (g _n (x _n )) ^p  + c[g ₁(x ₁) + g ₂(x ₂) + · + g _n (x _n )] where c is in K and each g _i (x _i ) is in K[x _i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3.     

Multiple positive solutions of boundary value problems for second-order discrete equations on the half-line

In this paper, we study the existence of multiple positive solutions of boundary value problems for second-order discrete equations Δ² x(n ? 1) ? pΔx(n ? 1) ? qx(n ? 1)+f(n, x(n)) = 0, n ∈ {1,2,…}, αx(0) ? βΔx(0) = 0, x(∞) = 0. The proofs are based on the fixed point theorem in Fréchet space (see Agarwal and O'Regan, 2001, Cone compression and expansion and fixed point theorems in Fréchet spaces with application, Journal of Differential Equations, 171, 412–42).