Generalizations of von Neumann regular rings, <Emphasis Type="Italic">PP</Emphasis> rings,and Baer rings

We introduce the notions of IDS modules, IP modules, and Baer* modules, which are new generalizations of von Neumann regular rings, PP rings, and Baer rings, respectively, in a general module theoretic setting. We obtain some characterizations and properties of IDS modules, IP modules and Baer* modules. Some important classes of rings are characterized in terms of IDS modules, IP modules, and Baer* modules.     

Universal inequalities in Ehrhart theory

In this paper, we show the existence of universal inequalities for the h*-vector of a lattice polytope P, that is, we show that there are relations among the coefficients of the h*-polynomial that are independent of both the dimension and the degree of P. More precisely, we prove that the coefficients h* ₁ and h* ₂ of the h*-vector (h* ₀, h* ₁,..., h* _d) of a lattice polytope of any degree satisfy Scott’s inequality if h* ₃ = 0.     

Symmetry and Short Interval Mean-Squares

The weighted Selberg integral is a discrete mean-square that generalizes the classical Selberg integral of primes to an arithmetic function f, whose values in a short interval are suitably attached to a weight function. We give conditions on f and select a particular class of weights in order to investigate non-trivial bounds of weighted Selberg integrals of both f and f * μ. In particular, we discuss the cases of the symmetry integral and the modified Selberg integral, the latter involving the Cesaro weight. We also prove some side results when f is a divisor function.     

Global asymptotic stability for half-linear differential systems with generalized almost periodic coefficients

The following system considered in this paper:

$$x' = -\,e(t)x + f(t)\phi_{p^*}(y), \qquad y'= -\,(p-1)g(t)\phi_p(x) - (p-1)h(t)y,$$

where \({p > 1, p^* > 1 (1/p + 1/p^* = 1)}\) and \({\phi_q(z) = |z|^{q-2}z}\) for q = p or q = p ^*. This system is referred to as a half-linear system. The coefficient f(t) is assumed to be bounded, but the coefficients e(t), g(t) and h(t) are not necessarily bounded. Sufficient conditions are obtained for global asymptotic stability of the zero solution. Our results can be applied to not only the case that the signs of f(t) and g(t) change like the periodic function but also the case that f(t) and g(t) irregularly have zeros. Some suitable examples are included to illustrate our results.

     

A characterization of *-minimal algebras with involution

Let (A, *) be a *-PI algebra with involution over a field of characteristic zero and let c _m (A, *) denote its m-th *-codimension. Giambruno and Zaicev, in [10], proved that, if A is finite dimensional, there exists the \({\lim _{m \to + \infty }}\root m \of {{c_m}(A,*)} \), and it is an integer, which is called the *-exponent of A. As a consequence of the presence of this invariant, in a natural manner in [2] the definition of *-minimal algebra was introduced. Our goal in this paper is to characterize, up to *-PI equivalence, *-minimal algebras.     

Stratified modules over an extension algebra

Let A be a standard Koszul standardly stratified algebra and X an A-module. The paper investigates conditions which imply that the module Ext* _A (X) over the Yoneda extension algebra A* is filtered by standard modules. In particular, we prove that the Yoneda extension algebra of A is also standardly stratified. This is a generalization of similar results on quasi-hereditary and on graded standardly stratified algebras.     

Boundedness,univalence and quasiconformal extension of Robertson functions

This article contains several results for λ-Robertson functions, i.e., analytic functions f defined on the unit disk ? satisfying f(0) = f′(0) ? 1 = 0 and Re e ^?iλ {1 + zf″(z)/f′(z)} > 0 in ? where λ ∈ (?π/2, π/2). We will discuss about conditions for boundedness and quasiconformal extension of Robertson functions. In the last section we provide another proof of univalence for Robertson functions by using the theory of Löwner chains.     

Representations of the general linear group over symmetry classes of polynomials

Let V be the complex vector space of homogeneous linear polynomials in the variables x₁,..., x _m . Suppose G is a subgroup of S _m , and χ is an irreducible character of G. Let H _d (G, χ) be the symmetry class of polynomials of degree d with respect to G and χ.

For any linear operator T acting on V, there is a (unique) induced operator K _χ (T) ∈ End(H _d (G, χ)) acting on symmetrized decomposable polynomials by

$${K_\chi }\left( T \right)\left( {{f_1} * {f_2} * \cdots * {f_d}} \right) = T{f_1} * T{f_2} * \cdots * T{f_d}.$$

In this paper, we show that the representation T ? K _χ (T) of the general linear group GL(V) is equivalent to the direct sum of χ(1) copies of a representation (not necessarily irreducible) T ? B _χ ^G (T).

     

*-regularity of certain generalized group algebras

Let B be a *-semisimple Banach algebra with a bounded approximate identity and \({\alpha: G \longrightarrow {\rm Aut}_{*}(B)}\) (isometric *-automorphisms group of B) an action of a locally group G on B. Let (D, G, γ) be the associated dynamical system, where D = C ₀(G, B) is the Banach *-algebra of all continuous B-valued functions on G vanishing at infinity and the action γ : G → Aut D is given by γ _s (y)(t) = α _s (y(s ^?1 t)) for \({y \in D}\) and \({s, t \in G}\) . Recall that B is said to be *-regular if the natural mapping \({I\in {\rm Prim} \, C^{*}(B) \mapsto I\cap B\in {\rm Prim}_{*}(B)}\) is a homeomorphism under the hull-kernel topology. When G is amenable, we show that if B is *-regular, then the generalized group algebra L ¹(G, D; γ) is *-regular. The converse is also true if we further assume that G is countable discrete. Finally the case of compact groups is studied.     

On Ranks of Polynomials

Let V be a vector space over a field k, P : V → k, d ≥?3. We show the existence of a function C(r, d) such that rank(P) ≤ C(r, d) for any field k, char(k) > d, a finite-dimensional k-vector space V and a polynomial P : V → k of degree d such that rank(?P/?t) ≤ r for all t ∈ V ??0. Our proof of this theorem is based on the application of results on Gowers norms for finite fields k. We don’t know a direct proof even in the case when k = ?.     

On Generalized Derivations and Centralizers of Operator Algebras with Involution

Let B(H) be the algebra of all bounded linear operators on a complex Hilbert space H and A(H) ? B(H) be a standard operator algebra which is closed under the adjoint operation. Let F: A(H)→ B(H) be a linear mapping satisfying F(AA*A) = F(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H), where the associated linear mapping d: A(H) → B(H) satisfies the relation d(AA*A) = d(A)A*A + Ad(A*)A + AA*d(A) for all A ∈ A(H). Then F is of the form F(A) = SA ? AT for all A ∈ A(H) and some S, T ∈ B(H), that is, F is a generalized derivation. We also prove some results concerning centralizers on A(H) and semisimple H^*-algebras.     

Compositions,derivations and polynomials

Let R be a prime ring with extended centroid C. In this paper, we discuss the case when the composition of a generalized derivation δ and a polynomial map f(Y) ∈ C[Y] of R is commutative on a non-zero right ideal ρ and a non-commutative Lie ideal L of R respectively, i.e., when the identity δ ○ f(x) = f ○ δ(x) holds on ρ or L. As applications of our main theorems, we clarify the generalized derivations which act as n-Jordan homomorphisms (S _n -homomorphisms) on ρ or L.     

Sieve Functions in Arithmetic Bands,II

An arithmetic function f is called a sieve function of range Q if its Eratosthenes transform g = f * μ is supported in [1,Q] ∩ N, where g(q) ? _ε q ^ε (?ε > 0). We continue our study of the distribution of f(n) over short arithmetic bands, n ≡ ar + b (mod q), with n ∈ (N,2N] ∩ N, 1 ≤ a ≤ H = o(N) and r,b ∈ Z such that g:c:d:(r,q) = 1. In particular, the optimality of some results is discussed.     

On the commutation graph of cyclic <Emphasis Type="Italic">TI</Emphasis>-subgroups in linear groups

We study the commutation graph Γ(A) of a cyclic TI-subgroup A of order 4 in a finite group G with quasisimple generalized Fitting subgroup F*(G). It is proved that, if F*(G) is a linear group, then the graph Γ(A) is either a coclique or an edge-regular graph but not a coedge-regular graph.     

On the existence of closed magnetic geodesics via symplectic reduction

Let (M, g) be a closed Riemannian manifold and \(\sigma \) be a closed 2-form on M representing an integer cohomology class. In this paper, using symplectic reduction, we show how the problem of existence of closed magnetic geodesics for the magnetic flow of the pair \((g,\sigma )\) can be interpreted as a critical point problem for a Rabinowitz-type action functional defined on the cotangent bundle \(T^*E\) of a suitable \(S^1\)-bundle E over M or, equivalently, as a critical point problem for a Lagrangian-type action functional defined on the free loopspace of E. We thenstudy the relation between the stability property of energy hypersurfacesin \((T^*M,dp\wedge dq+\pi ^*\sigma )\) and of the corresponding codimension2 coisotropic submanifolds in \((T^*E,dp\wedge dq)\) arising via symplecticreduction. Finally, we reprove the main result of Asselle and Benedetti (J Topol Anal 8(3):545–570, 2016) in this setting.     

Continuous and Discrete Age-Replacement Policies

This paper considers the combined continuous and discrete age-replacement policies when units deteriorate with age and use: a unit is replaced preventively before failure at time T of age or at number N of uses, whichever occurs first. The expected cost rate C (T,N) is derived, and both optimum time T* and number N* to minimize C (T,N) are discussed. There exist finite and unique T* and N* when the use occurs in a Poisson process under suitable conditions.     

On curves on K3 surfaces,II

We study the Clifford index c of a smooth irreducible curve X in the linear series |2H| on a special K3 surface S of degree 2n in \({{\mathbb P}}^{n+1}\), with hyperplane section H, and we look for the complete and base point free linear series of S whose restrictions to X compute c. In a more general context, we discuss the features of such series, for an assigned curve on a K3 surface; this discussion is of some independent interest.     

Interassociates of the bicyclic semigroup

An interassociate of a semigroup \((S,\cdot )\) is a semigroup \((S, *)\) such that for all \(a, b, c \in S\), \(a\cdot (b*c)=(a\cdot b) *c\) and \(a*(b\cdot c)=(a*b) \cdot c\). We investigate the bicyclic semigroup C and its interassociates. In particular, if p and q are the generators of the bicyclic semigroup and m and n are fixed nonnegative integers, the operation \(a*_{m,n} b= aq^mp^n b\) is known to be an interassociate. We show that for distinct pairs (m, n) and (s, t), the interassociates \((C, *_{m,n})\) and \((C, *_{s,t})\) are not isomorphic. We also generalize a result regarding homomorphisms on C to homomorphisms on its interassociates.     

Factorization of the reaction-diffusion equation,the wave equation,and other equations

We investigate equations of the form D _t u = Δu + ξ? _u for an unknown function u(t, x), t ∈ ?, x ∈ X, where D _t u = a ₀(u, t) + Σ _k=1 ^r a _k (t, u)? _t ^k u, Δ is the Laplace-Beltrami operator on a Riemannian manifold X, and ξ is a smooth vector field on X. More exactly, we study morphisms from this equation within the category PDE of partial differential equations, which was introduced by the author earlier. We restrict ourselves to morphisms of a special form—the so-called geometric morphisms, which are given by maps of X to other smooth manifolds (of the same or smaller dimension). It is shown that a map f: X → Y defines a morphism from the equation D _t u = Δu + ξ? _u if and only if, for some vector field Ξ and a metric on Y, the equality (Δ + ξ?)f*v = f*(Δ + Ξ?)v holds for any smooth function v: Y → ?. In this case, the quotient equation is D _t v = Δv + Ξ?v for an unknown function v(t, y), y ∈ Y. It is also shown that, if a map f: X → Y is a locally trivial bundle, then f defines a morphism from the equation D _t u = Δu if and only if fibers of f are parallel and, for any path γ on Y, the expansion factor of a fiber translated along the horizontal lift γ to X depends on γ only.     

Multiplicity and uniqueness of positive solutions for nonhomogeneous semilinear elliptic equation with critical exponent

In this paper, we consider the following problem

$$\left\{ {\begin{array}{*{20}{c}}{ - \Delta u\left( x \right) + u\left( x \right) = \lambda \left( {{u^p}\left( x \right) + h\left( x \right)} \right),\;x \in {\mathbb{R}^N},} \\ {u\left( x \right) \in {H^1}\left( {{\mathbb{R}^N}} \right),\;u\left( x \right) \succ 0,\;x \in {\mathbb{R}^N},\;} \end{array}} \right.\;\left( * \right)$$

, where λ > 0 is a parameter, p = (N+2)/(N?2). We will prove that there exists a positive constant 0 < λ* < +∞ such that (*) has a minimal positive solution for λ ∈ (0, λ*), no solution for λ > λ*, a unique solution for λ = λ*. Furthermore, (*) possesses at least two positive solutions when λ ∈ (0, λ*) and 3 ≤ N ≤ 5. For N ≥ 6, under some monotonicity conditions of h we show that there exists a constant 0 < λ** < λ* such that problem (*) possesses a unique solution for λ ∈ (0, λ**).