Approximation of Convex Bodies by ConvexBodies

For the affine distance d(C,D) between two convex bodies C, D(?) Rn, which reduces to the Banach-Mazur distance for symmetric convex bodies, the bounds of d(C, D) have been studied for many years. Some well known estimates for the upper-bounds are as follows: F. John proved d(C, D) < n1/2 if one is an ellipsoid and another is symmetric, d(C, D) < n if both are symmetric, and from F. John's result and d(C1,C2) < d(C1,C3)d(C2,C3) one has d(C,D) < n2 for general convex bodies; M. Lassak proved d(C, D) < (2n - 1) if one of them is symmetric. In this paper we get an estimate which includes all the results above as special cases and refines some of them in terms of measures of asymmetry for convex bodies.     

Characterizations of F-V-rings by Quasi-continuous Modules

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On the Impossibility of Stabilization of Solutions of a System of Linear Deterministic Difference Equations by Perturbations of Its Coefficients by Stochastic Processes of “White-Noise” Type

We consider the problem of mean-square stabilization of solutions of a system of linear deterministic difference equations with discrete time by perturbations of its coefficients by a stochastic white-noise process. The answer is negative and is based on the analysis of the corresponding matrix algebraic Sylvester equation introduced earlier by the author in the theory of stability of stochastic systems. At the same time, we answer the same question for a vector matrix system of linear difference equations with continuous time and for a vector matrix system of differential equations.     

Non-periodic Tilings of ? n by Crosses

An n-dimensional cross consists of 2n+1 unit cubes: the “central” cube and reflections in all its faces. A tiling by crosses is called a Z-tiling if each cross is centered at a point with integer coordinates. Periodic tilings of ℝ ⁿ by crosses have been constructed by several authors for all n∈N. No non-periodic tiling of ℝ ⁿ by crosses has been found so far. We prove that if 2n+1 is not a prime, then the total number of non-periodic Z-tilings of ℝ ⁿ by crosses is 2^à₀2^{\aleph _{0}} while the total number of periodic Z-tilings is only ℵ₀. In a sharp contrast to this result we show that any two tilings of ℝ ⁿ ,n=2,3, by crosses are congruent. We conjecture that this is the case not only for n=2,3, but for all n where 2n+1 is a prime.     

Homogenization of Reynolds Equation by Two-Scale Convergence

To increase the hydrodynamic performance in different machine elements, as e.g. journal bearings and thrust bearings, during lubrication it is important to understand the influence of surface roughness. In this connection one encounters homogenization of the incompressible Reynolds equation, where the roughness of the lubricated surface is assumed to be periodic. This problem has recently been studied in more engineering-oriented papers by using the formal method of multiple scale expansion. In this paper, we rigorously prove both homogenization and corrector results by using two-scale convergence, which may be regarded as a justification of the formal multiple scale expansion method described above. Moreover, some numerical illustrations and results are presented.     

Improvement of the deformative characteristics of poly-β-hydroxybutyrate by plasticization

Plasticized systems of poly--hydroxybutyrate (PHB) are studied, where low-molecular compounds traditionally used for this purpose for various systems, i.e., dioctyl sebacate, dibutyl sebacate, polyethylene glycol, Laprol 503, and Laprol 5003, are used as plasticizers. All of them are nontoxic and biodegradable compounds with a similar molecular weight and comparable polarity of molecules. The main purpose of this study is to improve the deformability of PHB taking into account the structural changes in plasticized PHBs. The plasticizers chosen are completely compatible with the polymer and form a monophase system up to a plasticizer content of 15–20%. An increase in the plasticizer content allows us to increase efficiently the deformability of the polymer (the relative breaking elongation of PHB at room temperature grows up to 250–300%). At the same time, the system becomes considerably weaker and therefore there is no point in increasing the concentration of plasticizers by more than 20 wt.%. According to the data obtained from DSC measurements, the ratio between the amorphous and crystalline regions of PHB in the presence of plasticizers mentioned remains practically constant. The changes in the elastic properties of PHB/low-molecular plasticizer systems are mainly due to efficient weakening of intermolecular interactions in the amorphous regions of the polymer. A slight decrease in the crystalline order of PHB is of secondary importance.Institute of Polymer Materials, Riga Technical University, Riga, LV-1048, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 4, pp. 527–538, July–August, 1999.     

Approximation of functions of Dirichlet class by Fejér means

For the Dirichlet classes D _p of holomorphic functions in the disk, we obtain the exact orders of best polynomial approximations and of upper bounds for deviations of Fejér means of Taylor series in the Hardy spaces H _p.     

On the Covering of the Vertices of a Graph by Cliques

In conversation I was told by Professor R.Brigham the following conjecture [1].Let G(n) be a graph of n vertices.Denote by f(G(n))=t the smallest integer for which the vertices of G(n) can be covered by t cliques. Denote further by h(G(n)) =l the largest integer for which there are l edges of our G(n) no two of Which are in the same clique.Clearly h(G(n)) can be much larger than f(G(n))e.g.if n=2m and G(n) is the complete bipartite graph of m white and m black vertices.Then l(G(n))=m and l(G(n))=m~2. It was conjectured that if G(n).has no isolated vertices then     

Construction of Inertial Manifolds by the Hadamard Method

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The Approximation of Continuous Functions by Jackson Operator

Let f(x): R~＇→R~＇be a continuous function with period 2π,i. e, f(x)∈C_2~·D. Jackson singular integral operator of f(x) is defined by: According to [1], for any f(x)∈C_2x, Then     

The Construction of Association Schemes by Parameter Amendment

Association scheme is a structure on a finite set that has some special relations among elements in the set. These relations are usually hidden in other relations, so how to derive them out is a problem. The paper gives a constructing method of mending parameters of the association schemes, and a new family of association schemes is obtained： quasiFmetric association scheme.     

On a Problem of Approximation by Linear Means

is valid, where ω(f,δ) is the modulus of continuity of f. Furthermore they asked that: (1.3) If Theorem A can be extended to matrix summability? Later, P. D. Kathal, A. S. B. Holland and B. N. Sahney partially answered this problem. They proved the following Theorem B If {Λ_(n,k)} is monotonic non-increasing for all 0≤k≤n and Λ_(n,k)≥0 (0≤k≤n),Λ_(n,k)=0 (k>n), then we have     

On the Degree of Approximation by Spherical Translations

The degree of approximation of spherical functions by the translations formed by a function defined on the unit sphere is dealt with. A kind of Jackson inequality is established under the condition that none of the L^2（S^q） norms of the orthogonal projection operators of the translated function are zeros. In the present paper we show that the spherical translations share the same degree of approximation as that of spherical harmonics.     

The Representation of Continuous Linear Operator by Integral

LetXoverthefieldKstandforacompletelocallyconvexspace,andweassumethatitstopologyisgeneratedbyafamilyPofcontinuousseminormsonX,Let(O,Z,p)beafinitepositivemeasurespace.Afunctionf:Q-Xissaidtobep-integrable(forthedefinition,see[1]),ifthereexistsasequence{ft}ofsimplefunctionssuchthat.,ThislimitexistsandisinX.Denote,p)asthespaceofcIassesofNintegrablefunctionswhereLfj=Lgjifandonlyiff=gP-a.e..Putq(f)=j.p(f)dp,foreachpeP.ThetopologyonL1(p,Z)consideredisthese,generatedbyalltheq.L.EggheLlIhaspro…     

ϕ-Strong approximation of functions by trigonometric polynomials

Mathematical Notes - The rate of ?-strong approximation of periodic functions by trigonometric polynomials constructed on the basis of interpolating polynomials with equidistant nodes is...     

On some problems of interpolation by ℒ-splines

_n (D) — ,s — _n (D), _v (v=1, 2, ...,s/2) — . _m={0x ₀<x ₁<...<x _2m–1<2,x _2m =x ₀+2} , x _j +1–x _j <(4s max _v )^–1,j=0, 1, ..., 2m –1, ( ) 2- - _n,m 2m , _m . , L _q - (1q) W ( _n )={f ₂ :f ^(n–1)AC ₂ , _n (D)f 1} 2- - (s _n f), _m . , - - _n,m .

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The author expresses his gratitude to Yu. N. Subbotin for a useful discussion on the results of this paper.     

Root determination by use of Padé approximants

In this paper we describe a new technique for generating iteration formulas — of arbitrary order — for determining a zero (assumed simple) of a functionf, assumed analytic in a region containing the zero. The 1/p Padé Approximant (p0) to the functiong(t)f(z) is formed wherez=w+t, using the Taylor series forf at the pointw, an approxination to the zero off. The value oft for which the 1/p Padé Approximant vanishes provides the basis of iteration formulas of orderp+2.Some known iteration formulas, e.g., Newton-Raphson's, Halley's and Kiss's of order of convergence two, three and four, are directly obtained by settingp=0,1 and 2, respectively.     

Contraction Flow by Certain Curvature of Convex Hypersurfaces

In the following mixed tensors on the hypersurface M Rn+1 under consideration willbe denoted by T={ Tijkl} ,the induced metric by g={ gij} and the second fundamentalform by A={ hij} .We always sum overrepeated indices from1 to n and use brackets forthe inner product on M:〈Tijk,Sijk〉 =gisgjαgkβTijk Ssαβ,　 | T| 2 =〈Tijk,Tijk〉.Setrkl =gkαhαl,Sm(λ1 ,… ,λn) = i1<… 相似文献