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.

     

On Composition series relative to a module

ABSTRACT

A commutative algebra with the identity (a * b) * (c * d) ? (a * d) * (c * b) = (a, b, c) * d ? (a, d, c) * b is called Novikov–Jordan. Example: K[x] under multiplication a * b = ?(ab) is Novikov–Jordan. A special identity for Novikov–Jordan algebras of degree 5 is constructed. Free Novikov–Jordan algebras with q generators are exceptional for any q ≥ 1.

     

Real Zeros of Algebraic Polynomials with Dependent Random Coefficients

The expected number of real zeros of polynomials a ₀ + a ₁ x + a ₂ x ² +…+a _n?1 x ^n?1 with random coefficients is well studied. For n large and for the normal zero mean independent coefficients, irrespective of the distribution of coefficients, this expected number is known to be asymptotic to (2/π)log n. For the dependent cases studied so far it is shown that this asymptotic value remains O(log n). In this article, we show that when cov(a _i , a _j ) = 1 ? |i ? j|/n, for i = 0,…, n ? 1 and j = 0,…, n ? 1, the above expected number of real zeros reduces significantly to O(log n)^1/2.     

On the Simplicity of Lie Algebras Associated to Leavitt Algebras

For any field 𝕂 and integer n ≥ 2, we consider the Leavitt algebra L _𝕂(n); for any integer d ≥ 1, we form the matrix ring S = M _d (L _𝕂(n)). S is an associative algebra, but we view S as a Lie algebra using the bracket [a, b] = ab ? ba for a, b ∈ S. We denote this Lie algebra as S ^?, and consider its Lie subalgebra [S ^?, S ^?]. In our main result, we show that [S ^?, S ^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1 and char(𝕂) does not divide d. In particular, when d = 1, we get that [L _𝕂(n)^?, L _𝕂(n)^?] is a simple Lie algebra if and only if char(𝕂) divides n ? 1.     

Generalizations of Prime Ideals

Let R be a commutative ring with identity. Various generalizations of prime ideals have been studied. For example, a proper ideal I of R is weakly prime (resp., almost prime) if a, b ∈ R with ab ∈ I ? {0} (resp., ab ∈ I ? I ²) implies a ∈ I or b ∈ I. Let φ:?(R) → ?(R) ∪ {?} be a function where ?(R) is the set of ideals of R. We call a proper ideal I of R a φ-prime ideal if a, b ∈ R with ab ∈ I ? φ(I) implies a ∈ I or b ∈ I. So taking φ_?(J) = ? (resp., φ₀(J) = 0, φ₂(J) = J ²), a φ_?-prime ideal (resp., φ₀-prime ideal, φ₂-prime ideal) is a prime ideal (resp., weakly prime ideal, almost prime ideal). We show that φ-prime ideals enjoy analogs of many of the properties of prime ideals.     

On Certain Weak Engel-Type Conditions in Groups

Let w(x, y) be a word in two variables and 𝔚 the variety determined by w. In this paper we raise the following question: if for every pair of elements a, b in a group G there exists g ∈ G such that w(a ^g , b) = 1, under what conditions does the group G belong to 𝔚? In particular, we consider the n-Engel word w(x, y) = [x, _n y]. We show that in this case the property is satisfied when the group G is metabelian. If n = 2, then we extend this result to the class of all solvable groups.     

Rado Numbers fora(x+y)=bz

In the case of existence the smallest numberN=Ra_kis called a Rado number if it is guaranteed that anyk-coloring of the numbers 1, 2, …, Ncontains a monochromatic solution of a given system of linear equations. We will determine Ra_k(a, b) for the equationa(x+y)=bzifb=2 andb=a+1. Also, the case of monochromatic sequences {x_n} generated bya(x_n+x_n+1)=bx_n+2 is discussed.     

Zero-Semidistributive Inverse Semigroups

An inverse semigroup S is said to be 0-semidistributive if its lattice ?F (S) of full inverse subsemigroups is 0-semidistributive. We show that it is sufficient to study simple inverse semigroups which are not groups. Our main theorem states that such a simple inverse semigroup S is 0-semidistributive if and only if (1) S is E-unitary, (2) S is aperiodic, (3) for any a,b ∈ S/σ with ab ≠ 1, there exist nonzero integers n and m such that (ab) ^m  = a ⁿ or (ab) ^m  = b ⁿ , where σ is the minimum group congruence on S.     

Pseudodifferential Boundary Value Problems with Non-Smooth Coefficients

ABSTRACT

In this contribution, we establish a calculus of pseudodifferential boundary value problems with Hölder continuous coefficients. It is a generalization of the calculus of pseudodifferential boundary value problems introduced by Boutet de Monvel. We discuss their mapping properties in Bessel potential and certain Besov spaces. Although having non-smooth coefficients and the operator classes being not closed under composition, we will prove that the composition of Green operators a ₁(x, D _x )a ₂(x, D _x ) coincides with a Green operator a(x, D _x ) up to order m ₁ + m ₂ ? Θ, where Θ ∈ (0, τ₂) is arbitrary, a _j (x, ξ) is in C ^{τ _j} (? ⁿ ) w.r.t. x, and m _j is the order of a _j (x, D _x ), j = 1, 2. Moreover, a(x, D _x ) is obtained by the asymptotic expansion formula of the smooth coefficient case leaving out all terms of order less than m ₁ + m ₂ ? Θ. This result is used to construct a parametrix of a uniformly elliptic Green operator a(x, D _x ).     

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.     

Some Results for Decomposition Numbers for the Hecke Algebra of Type A with q = −1

Let ? = ?_{?, ?1}(𝔖 _n ) be the Hecke algebra of the symmetric group 𝔖 _n . For partitions λ and ν with ν 2 ? regular, define the Specht module S(λ) and the irreducible module D(ν). Define d _λν = [S(λ): D(ν)] to be the composition multiplicity of D(ν) in S(λ). In this paper we compute the decomposition numbers d _λν for all partitions of the form λ = (a, c, 1 ^b ) and ν 2 ? regular.     

Boundedness properties of the difference equation x n+1=(α+βx n +δx n−2)/(x n−1)

Consider the third-order difference equation x _n+1 = (α+βx _n +δx _n ? 2)/(x _n ? 1) with α ∈ [0,∞) and β,δ ∈ (0,∞). It is shown that this difference equation has unbounded solutions if and only if δ>β.     

Localization on the Boundary of Blow-up for Reaction–Diffusion Equations with Nonlinear Boundary Conditions

Abstract

In this work we analyze the existence of solutions that blow-up in finite time for a reaction–diffusion equation u _t  ? Δu = f(x, u) in a smooth domain Ω with nonlinear boundary conditions ?u/?n = g(x, u). We show that, if locally around some point of the boundary, we have f(x, u) = ?βu ^p , β ≥ 0, and g(x, u) = u ^q then, blow-up in finite time occurs if 2q > p + 1 or if 2q = p + 1 and β < q. Moreover, if we denote by T _b the blow-up time, we show that a proper continuation of the blowing up solutions are pinned to the value infinity for some time interval [T, τ] with T _b  ≤ T < τ. On the other hand, for the case f(x, u) = ?βu ^p , for all x and u, with β > 0 and p > 1, we show that blow-up occurs only on the boundary.     

Rings Over which the Krull Dimension and the Noetherian Dimension of All Modules Coincide

We denote by 𝒜(R) the class of all Artinian R-modules and by 𝒩(R) the class of all Noetherian R-modules. It is shown that 𝒜(R) ? 𝒩(R) (𝒩(R) ? 𝒜(R)) if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)), for all centrally prime ideals P (i.e., ab ∈ P, a or b in the center of R, then a ∈ P or b ∈ P). Equivalently, if and only if 𝒜(R/P) ? 𝒩(R/P) (𝒩(R/P) ? 𝒜(R/P)) for all normal prime ideals P of R (i.e., ab ∈ P, a, b normalize R, then a ∈ P or b ∈ P). We observe that finitely embedded modules and Artinian modules coincide over Noetherian duo rings. Consequently, 𝒜(R) ? 𝒩(R) implies that 𝒩(R) = 𝒜(R), where R is a duo ring. For a ring R, we prove that 𝒩(R) = 𝒜(R) if and only if the coincidence in the title occurs. Finally, if Q is the quotient field of a discrete valuation domain R, it is shown that Q is the only R-module which is both α-atomic and β-critical for some ordinals α,β ≥ 1 and in fact α = β = 1.     

The Incidence Divisor of Cycles in Projective Space

For nonnegative integers a, b with a + b + 1 = n, we show the incidence locus has the structure of an effective Cartier divisor in the product of Chow varieties 𝒞 _a (? ⁿ ) × 𝒞 _b (? ⁿ ).     

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.     

A note: Every homogeneous difference equation of degree one admits a reduction in order

Every difference equation x _n+1 = f _n (x _n ,x _n ? 1,…,x _n ? k ) of order k+1 with each mapping f _n being homogeneous of degree 1 on a group G is shown to be equivalent to a system consisting of an equation of order k and a linear equation of order 1.     

Hopf-Type Algebras

In this article, we provide an alternative approach to the definition of a weak Hopf algebra (WHA). For an associative unital algebra A with a coassociative comultiplication Δ ∈Alg ^u (A, A ? A), the set of homomorphisms from A to A ? A, which do not preserve the units. If the linear maps Ξ₁, Ξ₂ ∈ End(A ? A), defined by Ξ₁(a ? b) = Δ(a)(1 ? b), Ξ₂(a ? b) = (a ? 1)Δ(b), are von Neumann regular elements in the ring End(A ? A) of endomorphisms of A ? A satisfying some appropriate assumptions, we call the A a Hopf-type algebra. We show the existence of a target, a source, a counit, and an antipode of A as in the usual WHA.     

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.