On the well-definedness of the order of an ordinary differential equation

It is proved that if the differential equations y ^{( n )}=f(x, y, y′, …, y ^{( n ?1 )}) and y ^{( m )}=g(x, y, y′, …, y ^{( n ?1 )}) have the same particular solutions in a suitable region where f and g are continuous real-valued functions with continuous partial derivatives (alternatively, continuous functions satisfying the classical Lipschitz condition), then n?=?m and the functions f and g are equal. This note could find classroom use in a course on differential equations as enrichment material for the unit on the existence and uniqueness theorems for solutions of initial value problems.     

On the Cardinalities of the Row spaces of Non-full Rank Boolean Matrices

n -2 integers 2 ^{n -2}+2 ^{n -3}+2 ^s, where s=0,1,2,..., n-3, in the interval (2 ^{n -2}+2 ^{n -3},2 ^{n -1}] such that these integers are the cardinalities of row spaces R(A) of non-full rank Boolean matrices A of order n. We also show that for each s, where s=0,1,2,..., n-3, there exists A epsilon B _n such that A is non-full rank and the cardinality of R(A) equals 2 ^{n -2}+2 ^{n -3}+2 ^s.     

Commutativity for a certain class of rings

We discuss the commutativity of certain rings with unity 1 and one-sideds-unital rings under each of the following conditions:x ^r [x ^s ,y]=±[x,y ^t ]x ⁿ x ^r [x ^s ,y]=±x ⁿ [x,y ^t ]x ^r [x ^s ,y]=±[x,y ^t ]y ^m , andx ^r [x ^s ,y]=±y ^m [x,y ^t ], wherer, n, andm are non-negative integers andt>1,s are positive integers such that eithers, t are relatively prime ors[x,y]=0 implies [x,y]=0. Further, we improve the result of [6, Theorem 3] and reprove several recent results.     

Weyl quantization of Lebesgue spaces

We study boundedness and compactness properties for the Weyl quantization with symbols in L^q (?^2d ) acting on L^p (?^d ). This is shown to be equivalent, in suitable Banach space setting, to that of the Wigner transform. We give a short proof by interpolation of Lieb's sufficient conditions for the boundedness of the Wigner transform, proving furthermore that these conditions are also necessary. This yields a complete characterization of boundedness for Weyl operators in L^p setting; compactness follows by approximation. We extend these results defining two scales of spaces, namely L_*^q (?^2d ) and L_?^q (R^2d ), respectively smaller and larger than the L^q (?^2d ),and showing that the Weyl correspondence is bounded on L_*^q (R^2d ) (and yields compact operators), whereas it is not on L_?^q (R^2d ). We conclude with a remark on weak‐type L^p boundedness (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

The pinched Veronese is Koszul

In this paper we prove that the coordinate ring of the pinched Veronese, k[X ³,X ² Y,XY ²,Y ³,X ² Z,Y ² Z,XZ ²,YZ ²,Z ³], is Koszul. The result is obtained by combining the use of a flat deformation induced by a distinguished weight together with a generalization of the notion of Koszul filtrations.     

Cylindricity of tangent bundles of strongly parabolic metrics and strongly parabolic surfaces

It is proved that if the intrinsic zero-index of the Sasaki metric of a tangent bundleTM ⁿ isk, thenk is even andM ⁿ is the metric product of a Riemannian manifoldM ^n–k/2 by a Euclidean spaceE ^k/2, whileTM ⁿ is the metric product ofTM ^n–k/2 byE ^k . An expression is obtained for the second fundamental forms of the imbeddingTF ^l TM ⁿ in terms of the second fundamental forms of the imbeddingF ^l M ⁿ and the curvature tensor ofM ⁿ . It is proved thatTF ^l is totally geodesic inTM ⁿ if and only ifF ^l is totally geodesic inM ⁿ .Translated from Ukrainskií Geometricheskií Sbornik, Issue 28, 1985, pp. 12–32.     

Ganea comonads

We construct for all topological space X and all n∈N a natural section e ⁿ _X :G ⁿ X→G ⁿ G ⁿ X of the Ganea projection :G ⁿ G ⁿ X→G ⁿ X and show that the triple (G ⁿ ,g ⁿ ,e ⁿ ) is a comonad on Top ^*. Received: 6 March 2000     

SOME EULER SPACES OF DIFFERENCE SEQUENCES OF ORDER m

Kizmaz [13] studied the difference sequence spaces e∞(△), c(△), and c0(△).Several article dealt with the sets of sequences of m-th order difference of which are bounded, convergent, or convergent to zero. Altay and Basar [5] and Altay, Basar, and Mursaleen [7] introduced the Euler sequence spaces eτ0, eτ0, andeτ∞, respectively. The main purpose of this article is to introduce the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)) consisting of all sequences whose mth order differences are in the Euler spaces eτ0, eτc, and eτ∞, respectively. Moreover, the authors give some topological properties and inclusion relations, and determine the α-, β-, and γ-duals of the spaces eτ0(△(m)), eτc(△(m)), and eτ∞(△(m)), and the Schauder basis of the spaces eτ0(△(m)), eτc(△(m)). The last section of the article is devoted to the characterization of some matrix mappings on the sequence space eτc(△(m)).     

NONLINEAR HOMOTOPY AND PRE-POINCARÉ LEMMAS

Abstract

Most homotopies considered in the literature are linear homotopies of the form h ⁱ (λ) = λx ⁱ + (1—λ)y ⁱ , 0 ≤ λ ≤ 1. Although these prove to be adequate in most instances, they lack direct geometric significance because {h ⁱ (λ) | 0 ≤ λ ≤ 1} are not orbits of a vector field. On the other hand, the nonlinear homotopy g ⁱ (s) = e ^s x ⁱ + (1—e ^s )y ⁱ ,—∞ ≤ s ≤ 0, are orbits of a vector field (i.e., dg ⁱ /ds = g ⁱ —y ⁱ , g ⁱ (0) = x ⁱ ), and thus have direct geometric significance. This suggests that useful results can be obtained by replacing linear homotopy by transport along flows of smooth vector fields. The purpose of this paper is to elaborate on this simple idea. We define prehomotopy operators induced by vector fields on a manifold. These allow us to obtain finite transport relations and pre-Poincaré lemmas that generalize the classical results. They are shown to reproduce the classical results as asymptotic limits and to obtain representations of all solutions of complete systems of exterior differential equations on a star shaped region of a manifold.     

A class of regular semigroups with regular *- transversals

Let S be a regular semigroup. If there is a subsemigroup S ^* of S and a unary operation * in S satisfying: (1) x ^* ∈ S ^* \cap V_ S ^* (x) for all x∈ S; (2) (x ^* ) ^* =x for all x∈ S ^* ; (3) (x ^* y) ^* =y ^* x ^** and (xy ^* ) ^* =y ^** x ^* for all x,y∈ S, then S ^* is called a regular *- transversal of S ; if (3) is replaced with (xy) ^* =y ^* x ^* for all x,y∈ S, then S ^* is called a strongly regular *- transversal of S. In this paper we consider the class of regular semigroups with a strongly regular *- transversal. It is proved that these semigroups are P - regular semigroups. We characterize the structure of the regular semigroups with a strongly regular *- transversal.     

Recursive constructions of -polynomials over

This paper presents procedures for constructing irreducible polynomials over GF(2^s) with linearly independent roots (or normal polynomials or N-polynomials). For a suitably chosen initial N-polynomial F₀(x)GF(2^s) of degree n, polynomials F_k(x)GF(2^s) of degrees n2^k are constructed by iteratively applying the transformation x→x+x^-1, and their roots are shown to form a normal basis of GF(2^sn2k) over GF(2^s). In addition, the sequences are shown to be trace compatible, i.e., the trace map T_{GF(2^sn2k+1)/GF(2^sn2k)} fromGF(2^sn2k+1) onto GF(2^sn2k) maps the roots of F_k+1(x) onto those of F_k(x).     

Codimension Two PL Embeddings of Spheres with Nonstandard Regular Neighborhoods

For a given polyhedron K(?)M,the notation RM(K)denotes a regular neigh- borhood of K in M.The authors study the following problem:find all pairs(m,k) such that if K is a compact k-polyhedron and M a PL m-manifold,then R_M(f(K))≌R_M(g(K))for each two homotopic PL embeddings f,g:K→M.It is proved that R_S~(k 2)(S~k)(?)S~k×D~2 for each k(?)2 and some PL sphere S~k(?)S~(k 2)(even for any PL sphere S~k(?)S~(K 2)having an isolated non-locally flat point with the singularity S~(k-1)(?) S~(k 1)such thatπ_1(S~(k 1)-S~(k-1))(?)Z).     

Kolmogorov Width of Classes of Smooth Functions on the Sphere

Let ^d−1{(x₁,…,x_d) ^d:x²₁+···+x²_d=1} be the unit sphere of the d-dimensional Euclidean space ^d. For r>0, we denote by B^r_p (1p∞) the class of functions f on ^d−1 representable in the formwhere dσ(y) denotes the usual Lebesgue measure on ^d−1, and P^λ_k(t) is the ultraspherical polynomial.For 1p,q∞, the Kolmogorov N-width of B^r_p in L^q( ^d−1) is given bythe left-most infimum being taken over all N-dimensional subspaces X_N of L^q( ^d−1).The main result in this paper is that for r2(d−1)²,where A_NB_N means that there exists a positive constant C, independent of N, such that C⁻¹A_NB_NCA_N.This extends the well-known Kashin theorem on the asymptotic order of the Kolmogorov widths of the Sobolev class of the periodic functions.     

Reverse Order Law for the Group Inverse in Semigroups and Rings

In this paper, we provide equivalent conditions for the two-sided reverse order law for the group inverse (ab)^# = b ^# a ^# and (ba)^# = a ^# b ^#, in semigroups and rings. Moreover, we prove that, under finiteness conditions, these conditions are also equivalent with the one-sided reverse order law (ab)^# = b ^# a ^#.     

C n -almost automorphic solutions for partial neutral functional differential equations

In this work, we study the existence of C ⁿ -almost periodic solutions and C ⁿ -almost automorphic solutions (n?≥?1), for partial neutral functional differential equations. We prove that the existence of a bounded integral solution on ?⁺ implies the existence of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions. When the exponential dichotomy holds for the homogeneous linear equation, we show the uniqueness of C ⁿ -almost periodic and C ⁿ -almost automorphic strict solutions.     

Generalized commutators for Marcinkiewicz type integrals with variable kernels

Let A be a function with derivatives of order m and D γ A ∈■β (0 < β < 1, |γ| = m). The authors in the paper prove that if Ω(x, z) ∈ L ∞ (R n ) × L s (S n 1 ) (s ≥ n/(n β)) is homogenous of degree zero and satisfies the mean value zero condition about the variable z, then both the generalized commutator for Marcinkiewicz type integral μ A Ω and its variation μ A Ω are bounded from L p (R n ) to L q (R n ), where 1 < p < n/β and 1/q = 1/p β/n. The authors also consider the boundedness of μ A Ω and its variation μ A Ω on Hardy spaces.     

Dense Subsets of H1/2(S2, S1)

We prove that the maps from S ² intoS ¹ having a finite number of isolated singularities ofdegree ±1 are dense for the strong topology inH ^1/2(S ², S ¹). We also prove that smooth maps are densein H ^1/2(S ², S ¹)for the sequentially weak topology andthat this is no more the case in H ^s (S ², S ¹) for s> 1/2.     

Symmetrizations of Markov processes

Two methods for symmetrizing Markov processes are discussed. Letu ^a(x, y) be the potential density of a Lévy process on a compact Abelian groupG. A general condition is given that guarantees thatv(x, y)=u^a(x, y)+u^a(y, x) is the potential density of a symmetric Lévy process onG. The second method arises by considering the linear space of one-potentialsU ¹ f, withf inL ², endowed with the inner product (U ¹ f,U ¹ g)=fU ¹ g+gU ¹ f. If the semigroup ofX(t) is normal, then the completionH of this space is the Dirichlet space of a symmetric processY(t). A set that is semipolar forX(t) is polar forY(t).     

Boundedness in nonlinear oscillations

In this paper, the boundedness of all solutions of the nonlinear differential equation (φ_p(x′))′ + αφ_p(x⁺) – βφ_p(x^–) + f(x) = e(t) is studied, where φ_p(u) = |u|^p–2 u, p ≥ 2, α, β are positive constants such that = 2w^–1 with w ∈ ?⁺\?, f is a bounded C⁵ function, e(t) ∈ C⁶ is 2π_p‐periodic, x⁺ = max{x, 0}, x^– = max{–x, 0}. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Characterizations of Some Product Manifolds by Their Spectrum

Let Sⁿ(c) denote the n-dimensional Euclidean sphere of constant sectional curvature c and denote by CPⁿ(c) the complex projective space of complex dimension n and of holomorphic sectional curvature c. In this paper, we obtain some characterizations of the manifolds S²(c) × S²(c′), S⁴(c) × S⁴(c′), CP²(c) × CP²(c′) by their spectrum.