Absolute upper semicontinuity

It is proved that the following conditions are equivalent: the function ϕ [a, b]→R is absolutely upper semicontinuous (see [1]); ϕ is a function of bounded variation with decreasing singular part; there exists a summable function g: [a, b] → R such that for anyt′∈[a, b] andt″∈[t′, b], we have ϕ(t″)−ϕ(t′)⩽∫ _t′ ^t″ g (s) ds. Translated from Matematicheskie Zametki, Vol. 22, No. 3, pp. 395–399, September, 1977.     

On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima

We study the problem of existence of periodic and almost periodic solutions of the scalar equation x′ (t) = − δx(t) + pmax _{u∈[t − h, t]} x(u) + f(t) where δ, p ∈ R, with a periodic (almost periodic) perturbation f(t). For these solutions, we establish conditions of global exponential stability and prove uniqueness theorems. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 6, pp. 747–754, June, 1998.     

Normal Quadratics in Ore Extensions,Quantum Planes,and Quantized Weyl Algebras

Let R be a Noetherian domain and let (σ,δ) be a quasi-derivation of R such that σ is an automorphism. There is an induced quasi-derivation on the classical quotient ring Q of R. Suppose F=t ²−v is normal in the Ore extension R[t;σ,δ] where v∈R. We show F is prime in R[t;σ,δ] if and only if F is irreducible in Q[t;σ,δ] if and only if there does not exist w∈Q such that v=σ(w)w−δ(w). We apply this result to classify prime quadratic forms in quantum planes and quantized Weyl algebras.     

Rings with nil and power centralK-th commutators

We define thek-th commutator forx, y in a ringR inductively as follows: [x,y]₁=[x,y]=xy−yx and [x,y] _k =[[x,y] _{k−1, y} ]. Assume thatR is a ring without nonzero nil onesided ideals. The following are shown: (1) If [x,y] _k is nilpotent for allx,y∈R, thenR must be commutative. (2) If [x,y] _k is power central for allx,y∈R, thenR must satisfy the standard polynomial of degree 4. 1980 Mathematics Subject Classification (1985 Revision). Primary 16A70, Secondary 16A12.     

On the asymptotic stability for impulsive functional differential equations

We consider the asymptotic stability problems by Lyapunov functionals V for a class of functional differential equations with impulses of the form x′(t)=f(t,x _t ), x∈R ⁿ , t≧t ₀, t≠t _k ; △x=I _k (t,x(t ⁻)), t=t _k , k∈Z ⁺ . Some new asymptotic stability results are presented by using an idea originated by Burton and Makay [6] and developed by Zhang [1]. We generalize some known results about impulsive functional differential equations in the literature in which we only require the derivative of V to be negative definite on a sequence of intervals I _n =[s _n ,ξ _n ] which may or may not be contained in the sequence of impulsive time intervals [t _n ,t _n+1).     

The <Emphasis Type="Italic">f</Emphasis>-Vector of the Descent Polytope

For a positive integer n and a subset S⊆[n−1], the descent polytope DP  _S is the set of points (x ₁,…,x _n ) in the n-dimensional unit cube [0,1] ⁿ such that x _i ≥x _i+1 if i∈S and x _i ≤x _i+1 otherwise. First, we express the f-vector as a sum over all subsets of [n−1]. Second, we use certain factorizations of the associated word over a two-letter alphabet to describe the f-vector. We show that the f-vector is maximized when the set S is the alternating set {1,3,5,…}∩[n−1]. We derive a generating function for F _S (t), written as a formal power series in two non-commuting variables with coefficients in ℤ[t]. We also obtain the generating function for the Ehrhart polynomials of the descent polytopes.     

Nonuniqueness ing-functions

We give an example of two distinct stationary processes {X _n} and {X′ _n} on {0, 1} for whichP[X₀=1|X₋₁=a₋₁,X₋₂=a₋₂, …]=P[X′₀=1|X′₋₁=a₋₁,X′₋₂=a₋₂, …] for all {a _i},i=−1, −2, …, even though these probabilities are bounded away from 0 and 1, and are continuous in {a _i}. Supported in part by NSF Grant DMS 89-01545. Supported in part by the US Army Research Office.     

A generalized beta integral on two intervals

We give an explicit formula for the expansion coefficients of a generalized beta integral on the set [−1,−b]∪[b,1] b∈(0,1), in a power series in the parameter b, thus defining a generalized beta function of two complex variables.     

A counterexample in copositive approximation

The present paper gives a converse result by showing that there exists a functionf ∈C _[−1,1], which satisfies that sgn(x)f(x) ≥ 0 forx ∈ [−1, 1], such that {fx75-1} whereE _n ⁽⁰⁾ (f, 1) is the best approximation of degreen tof by polynomials which are copositive with it, that is, polynomialsP withP(x(f(x) ≥ 0 for allx ∈ [−1, 1],E _n(f) is the ordinary best polynomial approximation off of degreen.     

Asymptotic representations of solutions of nonautonomous ordinary differential equations with regularly varying nonlinearities

We obtain asymptotic representations as t ↑ ω, ω ≤ + ∞, for all possible types of P _ω(Y ₀, λ ₀)-solutions (where Y ₀ is zero or ±∞ and −∞ ≤ λ₀ ≤ +∞) of nonlinear differential equations y ⁽ⁿ⁾ = α ₀ p(t)φ(y), where α ₀ ∈ {−1, 1}, p: [a, ω[→]0,+∞[ is a continuous function, and φ is a continuous regularly varying function in a one-sided neighborhood of Y ₀.     

Derivations having divergence zero onR[X,Y]

In this paper it is proved that for any ℚ-algebraR any locally nilpotentR-derivationD onR[X,Y] having divergence zero and 1 ∈ (D(X),D(Y)) (i) has a slice, and (ii)A ^D =R[P] for someP. Furthermore, it is shown that any surjectiveR-derivation onR[X,Y] having divergence zero is locally nilpotent. Connections with the Jacobian Conjecture are made.     

The minimizing of the Nielsen root classes

Given a map f: X→Y and a Nielsen root class, there is a number associated to this root class, which is the minimal number of points among all root classes which are H-related to the given one for all homotopies H of the map f. We show that for maps between closed surfaces it is possible to deform f such that all the Nielsen root classes have cardinality equal to the minimal number if and only if either N R[f]≤1, or N R[f]>1 and f satisfies the Wecken property. Here N R[f] denotes the Nielsen root number. The condition “f satisfies the Wecken property is known to be equivalent to |deg(f)|≤N R[f]/(1−χ(M ₂)−χ(M ₁0/(1−χ(M ₂)) for maps between closed orientable surfaces. In the case of nonorientable surfaces the condition is A(f)≤N R[f]/(1−χ(M ₂)−χ(M ₂)/(1−χ(M ₂)). Also we construct, for each integer n≥3, an example of a map f: K _n →N from an n-dimensionally connected complex of dimension n to an n-dimensional manifold such that we cannot deform f in a way that all the Nielsen root classes reach the minimal number of points at the same time.     

On the semigroup nilpotency and the Lie nilpotency of associative algebras

To each associative ringR we can assign the adjoint Lie ringR ⁽⁻⁾ (with the operation(a,b)=ab−ba) and two semigroups, the multiplicative semigroupM(R) and the associated semigroupA(R) (with the operationaob=ab+a+b). It is clear that a Lie ringR ⁽⁻⁾ is commutative if and only if the semigroupM(R) (orA(R)) is commutative. In the present paper we try to generalize this observation to the case in whichR ⁽⁻⁾ is a nilpotent Lie ring. It is proved that ifR is an associative algebra with identity element over an infinite fieldF, then the algebraR ⁽⁻⁾ is nilpotent of lengthc if and only if the semigroupM(R) (orA(R)) is nilpotent of lengthc (in the sense of A. I. Mal'tsev or B. Neumann and T. Taylor). For the case in whichR is an algebra without identity element overF, this assertion remains valid forA(R), but fails forM(R). Another similar results are obtained. Translated fromMatematicheskie Zametki, Vol. 62, No. 4, pp. 510–519, October, 1997. Translated by A. I. Shtern     

How many cyclic subpolytopes can a non-cyclic polytope have?

Acyclic d-polytope is ad-polytope that is combinatorially equivalent to a polytope whose vertices lie on the moment curve {(t, t ², …,t ^d):t∈R}. Every subpolytope of an even-dimensional cyclic polytope is again cyclic. We show that a polytope [or neighborly polytope] withv vertices that is not cyclic has at mostd+1 [respectivelyd]d-dimensional cyclic subpolytopes withv−1 vertices, providedd is even andv≧d+5.     

Existence of linear Pfaff systems with lower characteristic set consisting of countably many segments in the space <Emphasis Type="Italic">R</Emphasis><Superscript>3</Superscript>

We prove the existence of a completely integrable Pfaff system ∂x/∂t _i = A _i (t)x, x ∈ R _n , t = (t ₁, t ₂, t ₃) ∈ R ₊³, i = 1, 2, 3, with infinitely differentiable bounded coefficient matrices and with lower characteristic set being the union of countably many segments in the space R ³.     

THE GLOBAL APPROXIMATION BY LEFT-BERNSTEIN-DURRMEYER QUASI-INTERPOLANTS IN Lp[0, 1]

In this paper,we will use the 2r-th Ditzian-Totik modulus of smoothness wp^2r(f,t)p to discuss the direct and inverse theorem of approximation by Left-Bernstein-Durrmeyer quasi-interpolants Mn^[2r-1]f for functions of the space Lp[0,1](1≤p≤ ∞)。     

Derivations with engel conditions in prime and semiprime rings

Let R be a prime ring, I a nonzero ideal of R, d a derivation of R and m, n fixed positive integers. (i) If (d[x, y]) ^m = [x, y] _n for all x, y ∈ I, then R is commutative. (ii) If Char R ≠ 2 and [d(x), d(y)] _m = [x, y] ⁿ for all x, y ∈ I, then R is commutative. Moreover, we also examine the case when R is a semiprime ring.     

On the distribution of reducible polynomials

Let ℛ _n (t) denote the set of all reducible polynomials p(X) over ℤ with degree n ≥ 2 and height ≤ t. We determine the true order of magnitude of the cardinality |ℛ _n (t)| of the set ℛ _n (t) by showing that, as t → ∞, t ² log t ≪ |ℛ₂(t)| ≪ t ² log t and t ⁿ ≪ |ℛ _n (t)| ≪ t _n for every fixed n ≥ 3. Further, for 1 < n/2 < k < n fixed let ℛ _k,n (t) ⊂ ℛ _n (t) such that p(X) ∈ ℛ _k,n (t) if and only if p(X) has an irreducible factor in ℤ[X] of degree k. Then, as t → ∞, we always have t ^k+1 ≪ |ℛ _k,n (t)| ≪ t ^k+1 and hence |ℛ _n−1,n (t)| ≫ |ℛ _n (t)| so that ℛ _n−1,n (t) is the dominating subclass of ℛ _n (t) since we can show that |ℛ _n (t)∖ℛ _n−1,n (t)| ≪ t ⁿ⁻¹(log t)².On the contrary, if R _n ^s (t) is the total number of all polynomials in ℛ _n (t) which split completely into linear factors over ℤ, then t ²(log t) ⁿ⁻¹ ≪ R _n ^s (t) ≪ t ² (log t) ⁿ⁻¹ (t → ∞) for every fixed n ≥ 2.      

Non-Strebel points and variability sets

This paper studies the subset of the non-Strebel points in the universal Teichmuller space T. Let Z0 ∈ △be a fixed point. Then we prove that for every non-Strebel point h, there is a holomorphic curve γ : [0, 1]→ T with h as its initial point satisfying the following conditions.(1) The curve γ is on a sphere centered at the base-point of T, i.e. dT(id, γ(t)) = dT(id, h), (t ∈ [0, 1]).(2) For every t ∈ (0,1], the variability set Vγ(t)[Z0] of γ(t) has non-empty interior, i.e. Vγ(t) [Z0] ≠ .     

On classification of immersions ofn-manifolds in (2n−1)-manifolds

Under the assumption of (f, M ⁿ ,N ²ⁿ⁻¹) being trivial, the classification of immersions homotopic tof: M ⁿ →N ²ⁿ⁻¹ is obtained in many cases. The triviality of (f, M ⁿ ,P ²ⁿ⁻¹) is proved for anyM ⁿ andf. LetM, N be differentiable manifolds of dimensionn and2n−1 respectively. For a mapf: M → N, denote byI[M, N] _f the set of regular homotopy classes of immersions homotopic tof. It has been proved in [1] that, whenn>1,I[M, N] _f is nonempty for anyf. In this paper we will determine the setI[M, N] _f in some cases. For example, ifN=P ²ⁿ⁻¹ or more generally, the lens spacesS _m ²ⁿ⁻¹ =Z _m /S ²ⁿ⁻¹,M is any orientablen-manifold or nonorientable butn≡0, 1, 3 mod 4, then, for anyf, theI[M, N] _f is determined completely. WhenN=R ²ⁿ⁻¹, the setI[M, N] of regular homotopy classes of all immersions has been enumerated by James and Thomas in [2] and McClendon in [3] forn>3. Applying our results toN=R ²ⁿ⁻¹ we obtain their results again, except for the casen≡2 mod 4 andM nonorientable. Whenn=3, McClendon's results cannot be used. Our results include the casesn=3,M orientable or not (for orientableM, I[M, R⁵] is known by Wu [4]).