On the integrability of hyperbolic systems of riccati-type equations

We consider equations of the form U_xy = U * U_x, where U(x, y) is a function taking values in an arbitrary finite-dimensional algebra T over the field ℂ. We show that every such equation can be naturally associated with two characteristic Lie algebras, L_x and L_y. We define the notion of a ℤ-graded Lie algebraB corresponding to a given equation. We prove that for every equation under consideration, the corresponding algebraB can be taken as a direct sum of the vector spaces L_x and L_y if we define the commutators of the elements from L_x and L_y by means of the zero-curvature relations. Assuming that the algebra T has no left ideals, we classify the equations of the specified type associated with the finite-dimensional characteristic Lie algebras L_x and L_y. All of these equations are Darboux-integrable. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 261–275, November, 1997.     

On the proximinality of compact operators over spaces of measurable functions

The main results in this paper are, the characterization of all the σfinite positive measures μ and v, for which K(L₁(μ, L₁(v)) proximinal in K(L₁(μ, L₁(v)) all the a-finite positive measures fi and v, for which K(L_∞(μ), L_∞(v)) is proximinal in L(L_∞μ)), L_v(v)) and all the compact Hausdorff spaces Q, for which K(C(Q), L_∞(μ)) is proximinal in L(C(Q), :_∞(μ))     

On the bounded approximation property in Banach spaces

We prove that the kernel of a quotient operator from an L ₁-space onto a Banach space X with the Bounded Approximation Property (BAP) has the BAP. This completes earlier results of Lusky-case ? ₁-and Figiel, Johnson and Pe?czyński-case X* separable. Given a Banach space X, we show that if the kernel of a quotient map from some L ₁-space onto X has the BAP, then every kernel of every quotient map from any L ₁-space onto X has the BAP. The dual result for L _∞-spaces also holds: if for some L _∞-space E some quotient E/X has the BAP, then for every L _∞-space E every quotient E/X has the BAP.     

Homomorphisms between the chevalley groups over any fields of characteristic zero

Given an irreducible root system ∑, let G(F,L) denote the Cheval- ley group over a field F corresponding to a lattice L between the root lattice and the weight lattice of ∑,. We will determine all nontnvial homomorphisms from G(k,L ₁) to G(K,L ₂when k and K are any fields of characteristic zero, and we will verify that any nontrivial homomorphism from G(k,L ₁) to G(K,L ₂are induced by a field homomorphism from k to K by multiplying an automorphism of G(K,L ₂.     

Recognition by Spectrum of Some Linear Groups over the Binary Field

We focus our attention on the linear groups L _n (2) and obtain some general properties of these groups. We will show then that the linear groups L _p (2), where 2 is a primitive root mod p (p odd prime), are recognizable by spectrum. For example, the linear groups L ₃(2), L ₅(2), L ₁₁(2), L ₁₃(2), L ₁₉(2), L ₂₉(2), L ₃₇(2), L ₅₃(2), etc. are recognizable by spectrum.     

The failure of the Hardy inequality and interpolation of intersections

The main idea of this paper is to clarify why it is sometimes incorrect to interpolate inequalities in a “formal” way. For this we consider two Hardy type inequalities, which are true for each parameter α≠0 but which fail for the “critical” point α=0. This means that we cannot interpolate these inequalities between the noncritical points α=1 and α=?1 and conclude that it is also true at the critical point α=0. Why? An accurate analysis shows that this problem is connected with the investigation of the interpolation of intersections (N∩L _p(w₀), N∩L_p(w₁)), whereN is the linear space which consists of all functions with the integral equal to 0. We calculate theK-functional for the couple (N∩L _p(w₀),N∩L _p (w₁)), which turns out to be essentially different from theK-functional for (L _p(w₀), L_p(w₁)), even for the case whenN∩L _p(w_i) is dense inL _p(w_i) (i=0,1). This essential difference is the reason why the “naive” interpolation above gives an incorrect result.     

Sharpness of the Calderón-Lozanovskii interpolation construction

We present a short proof of the sharpness of the Calderón-Lozanovskii interpolation construction in couples of weighted L _p spaces in the “lower triangle,” i.e., for operators from a couple { L _p0 (V ₀), L _p1 (V ₁)} to a couple {L _q0 (U ₀), L _q1 (U ₁)} with p ₀ ? q ₀ and p ₁ ? q ₁. This generalizes the well-known result due to Dmitriev and Semenov on the sharpness of the Riesz-Thorin interpolation theorem in the “lower triangle” for L _p spaces on intervals.     

A theorem on the factorization of matrix polynomials

A similarity condition is developed for the factorization of monic matrix polynomials L(λ) into the forms L(λ) = L_k(λ) … L₁(λ), wihtout any restriction on the spectrum of factors L_j(λ).     

Extension of isometries on the unit sphere of L p spaces

In this paper we study the isometric extension problem and show that every surjective isometry between the unit spheres of Lp (μ) (1 p ∞, p≠2) and a Banach space E can be extended to a linear isometry from Lp(μ) onto E, which means that if the unit sphere of E is (metrically) isometric to the unit sphere of Lp(μ), then E is linearly isometric to Lp(μ). We also prove that every surjective 1-Lipschitz or anti-1-Lipschitz map between the unit spheres of Lp (μ1, H1) and Lp(μ2,H2) must be an isometry and can be extended to a linear isometry from Lp (μ1,H1) to Lp (μ2,H2), where H1 and H2 are Hilbert spaces.     

First non-zero terms for the Taylor expansion at 1 of the Conway potential function

The Conway potential function ∇ _L (t ₁,...,t _l ) of an ordered oriented link L = L ₁ ∪ L ₂ ∪ ... ∪ L _l ⊂ S ³ is considered. In general, this function is not determined by the linking numbers and the Conway potential functions of the components. However, the first two nonzero terms of the Taylor expansion at the point 1 of the function ∇ _L are determined by the linking numbers only. We give the explicit formulas for these terms using summation over trees with l vertices.     

Pure quantitative characterization of linear groups over the binary field

The purpose of this paper is to discuss the pure scalar characterization of the automorphism group Aut (L ₅(2)) and the linear group L ₆(2). It is proved that Aut(L ₅(2)) and L ₆(2) can be characterized quantitatively by the set of element orders. The main results are obtained by using William’s work on prime graph components of finite groups and Brauer characters in trivializing the possible 2-subgroups. __________ Translated from Chinese Annals of Mathematics, 2003, 24A(6): 675–682. This work was supported by the National Natural Science Foundation of China under grant number 10171074     

A new formula for the volume of lattice polyhedra

LetL _n be the lattice consisting of all pointsx inR ^N such thatnx belongs to the fundamental latticeL ₁ of points with integer coordinates. When the vertices of a polyhedronP inR ^N are restricted to lie inL ₁ there is a formula which relates the volume ofP to the numbers of points ofL ₁,...,L _N in the interior and on the boundary ofP. The aim of this note is to show that the volume ofP can be determined only by means of the numbers of points ofL ₁,...,L _N lying in the interior ofP and cannot be expressed by the numbers of points ofL ₁,...,L _N lying on the boundary ofP. The latter numbers in turn can be used to compute to comopute the Euler characteristic of the boundary ofP.     

The order of the best one-sided approximation by polynomials and splines in the Lp-metric

In this paper we determine the order of the best one-sided approximation by polynomials and splines of minimal defect of the classes W^rL_p in the L_p-metric.Translated from Matematicheskie Zametki, Vol. 19, No. 3, pp. 323–329, March, 1976.     

Multi-window dilation-and-modulation frames on the half real line

Wavelet and Gabor systems are based on translation-and-dilation and translation-and-modulation operators, respectively, and have been studied extensively. However, dilation-and-modulation systems cannot be derived from wavelet or Gabor systems. This study aims to investigate a class of dilation-and-modulation systems in the causal signal space L²(ℝ₊). L²(ℝ₊) can be identified as a subspace of L²(ℝ), which consists of all L²(ℝ)-functions supported on ℝ₊ but not closed under the Fourier transform. Therefore, the Fourier transform method does not work in L²(ℝ₊). Herein, we introduce the notion of Θ_a-transform in L²(ℝ₊) and characterize the dilation-and-modulation frames and dual frames in L²(ℝ₊) using the Θ_a-transform; and present an explicit expression of all duals with the same structure for a general dilation-and-modulation frame for L²(ℝ₊). Furthermore, it has been proven that an arbitrary frame of this form is always nonredundant whenever the number of the generators is 1 and is always redundant whenever the number is greater than 1. Finally, some examples are provided to illustrate the generality of our results.

     

The level polynomials of the free distributive lattices

We show that there exist a set of polynomials {L_k?k = 0, 1?} such that L_k(n) is the number of elements of rank k in the free distributive lattice on n generators. L₀(n) = L₁(n) = 1 for all n and the degree of L_k is k?1 for k?1. We show that the coefficients of the L_k can be calculated using another family of polynomials, P_j. We show how to calculate L_k for k = 1,…,16 and P_j for j = 0,…,10. These calculations are enough to determine the number of elements of each rank in the free distributive lattice on 5 generators a result first obtained by Church [2]. We also calculate the asymptotic behavior of the L_k's and P_j's.     

Upper bounds for the L 1-risk of the minimum L 1-distance regression estimator

A new estimator of a regression function is introduced via minimizing the L ₁-distance between some empirical function and its theoretical counterpart plus penalty for the roughness. The L ₁-risk of the estimator is bounded from above for every sample size no matter what the dependence structure of the observed random variables is. In the case of independent errors of measurement with a common variance the estimator is shown to achieve the optimal L ₁-rate of convergence within the class of m-times differentiable functions with bounded derivatives.     

On the variation of co-ordinates in subspaces

Summary Let1≦k≦n−1, 2≦n. This paper examines the vectors δ_p(L_k), where L_k is a k-dimensional subspace of an n-dimensional space, and the co-ordinates of δ_p(L_k) are given below by (1.1). For fixed k, the set of such vectors as L_k varies is determined for p=2. For general p, information is given on upper and lower bounds for the sum of the co-ordinates of δ_p(L_k). Dedicated to the sixtieth birthday of Prof. Edgar R. Lorch This research was supported in part by the Office of Naval Research under contract number Nonr 3775(09), NR 047040.     

Remarks on the Lp − Lq estimates of solutions of the Klein-Gordon equation

We clarify some difficulties in the proofs of estimates, which were established earlier, in the paper by Marshall, Straus and Wainger devoted to the L_p —L_q estimates of solutions to the Klein-Gordon equation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 2, pp. 233–245, February, 1992.     

On the Uniqueness of Heat Flow of Harmonic Maps and Hydrodynamic Flow of Nematic Liquid Crystals

For any n-dimensional compact Riemannian manifold (M,g) without boundary and another compact Riemannian manifold (N,h), the authors establish the uniqueness of the heat flow of harmonic maps from M to N in the class C([0,T),W1,n). For the hydrodynamic flow (u,d) of nematic liquid crystals in dimensions n = 2 or 3, it is shown that the uniqueness holds for the class of weak solutions provided either (i) for n = 2, u ∈ Lt∞ L2x∩L2tHx1, ▽P∈ Lt4/3 Lx4/3 , and ▽d∈ L∞t Lx2∩Lt2Hx2; or (ii) for n = 3, u ∈ Lt∞ Lx2∩L2tHx1∩ C([0,T),Ln), P ∈ Ltn/2 Lxn/2 , and ▽d∈ L2tLx2 ∩ C([0,T),Ln). This answers affirmatively the uniqueness question posed by Lin-Lin-Wang. The proofs are very elementary.     

Bivariate Sign Tests Based on the Sup, L 1 and L 2 Norms

The bivariate location problem is considered. The sup, L ₁ and L ₂ norms are used to construct bivariate sign tests from the univariate sign statistics computed on the projected observations on all lines passing through the origin. The tests so obtained are affine-invariant and distribution-free under the null hypothesis. The sup-norm gives rise to Hodges' test. A class of tests derived from the L ₂-norm, with Blumen's test as a member, is seen to be related to a class proposed by Oja and Nyblom (1989, J. Amer. Statist. Assoc., 84, 249-259). The L ₁-norm gives rise to a new test. Its asymptotic null distribution is seen to be the same as that of the L ₁-norm of a certain normal process related to the standard Wiener process. An explicit expression of its cumulative distribution function is given. A simulation study will examine the merits of the three approaches.