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 structure of the set bases of a degenerate point

Consider an extreme point (EP)x ⁰ of a convex polyhedron defined by a set of linear inequalities. If the basic solution corresponding tox ⁰ is degenerate,x ⁰ is called a degenerate EP. Corresponding tox ⁰, there are several bases. We will characterize the set of all bases associated withx ⁰, denoted byB ⁰. The setB ⁰ can be divided into two classes, (i) boundary bases and (ii) interior bases. For eachB ⁰, there is a corresponding undirected graphG ⁰, in which there exists a tree which connects all the boundary bases. Some other properties are investigated, and open questions for further research are listed, such as the connection between the structure ofG ⁰ and cycling (e.g., in linear programs).     

On the gehring link problem and the isoperimetric inequality of bombieri and simon

LetE ⁿ be a completen-dimensional Riemannian manifold and letM ^p andN ^n−p−1 be compact oriented submanifolds ofE ⁿ whcih are linked inE ⁿ . The problem (generalizing one due to Gehring whenE ⁿ is Euclidean) of finding sharp lower bounds on the volume ofM ^p in terms of a lower bound on the distance ofM ^p fromN is solved in (among other cases) the case whereE ⁿ orM ^p is simply connected andE ⁿ is a space form or has a nonpositive upper bound on its sectional curvatures. The main technical tool is a generalization of an isoperimetric inequality of Bombieri and Simon which they used to solve Gehring's problem. Research supported in part by a Grant from the University of South Carolina.     

On the diameter and radius of randon subgraphs of the cube

The n-dimensional cube Qⁿ is the graph whose vertices are the subsets of {1,…n}, with two vertices adjacent if and only if their symmetric difference is a singleton. Clearly Qⁿ has diameter and radious n. Write M = n2^n-1 = e(Qⁿ) for the size of Qⁿ. Let Q = (Q_t)_o^M be a random Qⁿ-process. Thus Q^t is a spanning subgraph of Qⁿ of size t, and Q_t is obtained from Q_t–1 by the random addition of an edge of Qⁿ not in Q_t–1, Let t^(k) = τ(Q;δ?k) be the hitting time of the property of having minimal degree at least k. We show that the diameter d_t = diam (Q_t) of Q_t in almost every Q? behaves as follows: d_t starts infinite and is first finite at time t⁽¹⁾, it equals n + 1 for t⁽¹⁾ ? t⁽²⁾ and d_t, = n for t ? t⁽²⁾. We also show that the radius of Q_t, is first finite for t = t⁽¹⁾, when it assumes the value n. These results are deduced from detailed theorems concerning the diameter and radius of the almost surely unique largest component of Q_t, for t = Ω(M). © 1994 John Wiley & Sons, Inc.     

Invertibility of the Gabor frame operator on the Wiener amalgam space

We use a generalization of Wiener's 1/f theorem to prove that for a Gabor frame with the generator in the Wiener amalgam space W(L^∞,?¹)(R^d), the corresponding frame operator is invertible on this space. Therefore, for such a Gabor frame, the canonical dual belongs also to W(L^∞,?¹)(R^d).     

On Complementary Spaces of the Lizorkin Spaces

Let S(Rⁿ){\cal S}(R^n) be the Schwartz space on R ⁿ . For a subspace V ì S(Rⁿ)V\subset {\cal S}(R^n), if a subspace W ì S(Rⁿ)W \subset {\cal S}(R^n) satisfies the condition that S(Rⁿ){\cal S}(R^n) is a direct sum of V and W, then W is called a complementary space of V in S(Rⁿ){\cal S}(R^n). In this article we give complementary spaces of two kinds of the Lizorkin spaces in S(Rⁿ){\cal S}(R^n).     

Component behavior near the critical point of the random graph process

We study the behavior of a random graph process (G(n, M))₀²ⁿ for M(n) = n/2 + s and ∣s∣³n^?;2 → ∞. Among others we find the number of components in G(n, M) and estimate the number of vertices and edges in the kth largest component of G(n, M), for any natural number k, Moreover, it is shown that, with probability 1 –o(1), when M(n) = n/2 + s, s³n^?2 →?∞, then during a random graph process in some step M¹ > M a “new” largest component will emerge, whereas when s³n^?2→∞, the largest component of G(n, M) remains largest until the very end of the process.     

Banach Spaces of Solutions of the Helmholtz Equation in the Plane

The purpose of this article is to study the Hilbert space W²\mathcal{ W}^2 consisting of all solutions of the Helmholtz equation Du+u=0\Delta u+u=0 in \BbbR²\Bbb{R}^2 that are the image under the Fourier transform of L²L^2 densities in the unit circle. We characterize this space as a close subspace of the Hilbert space H²\mathcal{ H}^2 of all functions belonging to L²( | x | ^-3dx) L^2( | x | ^{-3}dx) jointly with their angular and radial derivatives, in the complement of the unit disk in \BbbR²\Bbb{R}^2. We calculate the reproducing kernel of W²\mathcal{ W}^2 and study its reproducing properties in the corresponding spaces H^p\mathcal{H}^p, for $p>1$p>1.     

On the ideal centre of the space of vector valued integrable functions

Let E be a Banach lattice and L¹(μ, E) be the space of E-valued Bochner integrable functions. Some order properties of L¹(μ, E) are given. It is shown that L_s^∞(μ, Z(E)) is the ideal centre of L¹(μ, E) and it is obtained a Radon-Nikodym type theorem for B -integrable functions.      

Error bounds for asymptotic expansion of the scale mixtures of the normal distribution

Summary LetX be a standard normal random variable and let σ be a positive random variable independent ofX. The distribution of η=σX is expanded around that ofN(0, 1) and its error bounds are obtained. Bounds are given in terms of E(σ ²V−σ ²−1) ^k whereσ ²Vσ ⁻² denotes the maximum of the two quantitiesσ ² andσ ⁻², andk is a positive integer, and of E(σ ²−1) ^k , ifk is even. The Institute of Statistical Mathematics     

Joint continuity of multiplication on the dual of the left uniformly continuous functions

Let G be a locally compact group and LUC(G) the C*-algebra of the bounded left uniformly continuous functions on G. The spectrum G ^LUC of LUC(G) is the universal semigroup compactification of G with respect to the joint continuity property: the multiplication on G×G ^LUC is jointly continuous. The paper studies the joint weak* continuity of multiplication on LUC(G)^* and, in particular, the question how the joint continuity property of G ^LUC can be related to a property concerning the whole algebra LUC(G)^*. The group G is naturally replaced by the measure algebra M(G), and LUC(G)^* can be identified with M(G ^LUC), the space of regular Borel measures on G ^LUC. It is shown that the joint weak* continuity can fail even on bounded sets of M(G)×M(G ^LUC), but, on the other hand, the multiplication on M(G)×M(G ^LUC) is positive continuous in the sense of Jewett.     

Characterizations of certain classes of Hankel operators on the Bergman spaces of the unit disk

Let f be an integrable function on the unit disk. The Hankel operator H_f is densely defined on the Bergman space A^p by H_fg = fg − P(fg), where g is a bounded analytic function and P is the Bergman projection (orthogonal projection from L² to A²) extended to L¹ via its integral formula. In this paper, the functions f for which H_f extends to a bounded operator from A^p to L^p are characterized, 1 < p < ∞. Also characterized are the functions f for which H_f extends to a compact or Schatten class operator on A². The proofs can be extended to handle any smoothly bounded domain in C in place of the unit disk.     

Lower bounds for the blowup rate of solutions of the Zakharov equation in dimension two

We consider the blowup solution ( u,n,v )( t ) of the Zakharov equations where u : R ² → C, n : R ² → R, v: R² → R² in the energy space H₁ = {(u,n,v) η H¹ × L² × L²}. We show that there is a constant c depending on the L²-norm of u₀ such that where T is the blowup time. We check that this estimate is optimal and give further applications. © 1996 John Wiley & Sons, Inc.     

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.     

Mappings of finite distortion: the degree of regularity

This paper investigates the self-improving integrability properties of the so-called mappings of finite distortion. Let K(x)1 be a measurable function defined on a domain ΩRⁿ, n2, and such that exp(βK(x))L_loc¹(Ω), β>0. We show that there exist two universal constants c₁(n),c₂(n) with the following property: Let f be a mapping in W_loc^1,1(Ω,Rⁿ) with |Df(x)|ⁿK(x)J(x,f) for a.e. xΩ and such that the Jacobian determinant J(x,f) is locally in L¹ log^−c₁(n)βL. Then automatically J(x,f) is locally in L¹ log^c₂(n)βL(Ω). This result constitutes the appropriate analog for the self-improving regularity of quasiregular mappings and clarifies many other interesting properties of mappings of finite distortion. Namely, we obtain novel results on the size of removable singularities for bounded mappings of finite distortion, and on the area distortion under this class of mappings.     

On the Strong Law of Large Numbers and the Law of the Logarithm for Weighted Sums of Independent Random Variables with Multidimensional Indices

Let (X, X ; ^d} be a field of independent identically distributed real random variables, 0 < p < 2, and {a _, ; ( , ) ^d × ^d, ≤ } a triangular array of real numbers, where ^d is the d-dimensional lattice. Under the minimal condition that sup _, |a _, | < ∞, we show that | |^{− 1/p} ∑ _≤ a _, X → 0 a.s. as | | → ∞ if and only if E(|X|^p(L|X|)^{d − 1}) < ∞ provided d ≥ 2. In the above, if 1 ≤ p < 2, the random variables are needed to be centered at the mean. By establishing a certain law of the logarithm, we show that the Law of the Iterated Logarithm fails for the weighted sums ∑ _≤ a _, X under the conditions that EX = 0, EX² < ∞, and E(X²(L|X|)^{d − 1}/L₂|X|) < ∞ for almost all bounded families {a _, ; ( , ) ^d × ^d, ≤ of numbers.     

A note on the perturbation bound of the Drazin inverse

A perturbation bound for the Drazin inverse A^D with Ind(A+E)=1 has recently been developed. However, those upper bounds are not satisfied since it is not tight enough. In this paper, a sharper upper bounds for ||(A+E)^#−A^D|| with weaker conditions is derived. That new bound is also a generalization of a new general upper bound of the group inverse. We also derive a new expression of the Drazin inverse (A+E)^D with Ind(A+E)>1 and the corresponding upper bound of ||(A+E)^D−A^D|| in a special case. Numerical examples are given to illustrate the sharpness of the new bounds.     

Regularity of the obstacle problem for a fractional power of the laplace operator

Given a function φ and s ∈ (0, 1), we will study the solutions of the following obstacle problem:

• u ≥ φ in ?ⁿ,
• (??)^su ≥ 0 in ?ⁿ,
• (??)^su(x) = 0 for those x such that u(x) > φ(x),
• lim_{|x| → + ∞} u(x) = 0.

We show that when φ is C^{1, s} or smoother, the solution u is in the space C^{1, α} for every α < s. In the case where the contact set {u = φ} is convex, we prove the optimal regularity result u ∈ C^{1, s}. When φ is only C^{1, β} for a β < s, we prove that our solution u is C^{1, α} for every α < β. © 2006 Wiley Periodicals, Inc.     

Regularity of the solutions of the steady‐state Boussinesq equations with thermocapillarity effects on the surface of the liquid

In this paper we show that every variational solution of the steady‐state Boussinesq equations ( u , p, θ) with thermocapillarity effect on the surface of the liquid has the following regularity: u ∈ H²(Ω)², p ∈ H¹(Ω), θ ∈ H²(Ω) under appropriate hypotheses on the angles of the ‘2‐D’ container (a cross‐section of the 3‐D container in fact) and of the horizontal surface of the liquid with the inner surface of the container. The difficulty comes from the boundary condition on the surface of the liquid (e.g. water) which modelizes the thermocapillarity effect on the surface of the liquid (equation (68.10) of Levich [7]). More precisely we will show that u ∈ P²₂(Ω)² and that θ ∈ P²₂(Ω), where P²₂(Ω) denotes the usual Kondratiev space. This result will be used in a forthcoming paper to prove convergence results for finite element methods intended to compute approximations of a non‐singular solution [1] of this problem. Copyright © 1999 John Wiley & Sons, Ltd.