On upper bounds of Fourier-Walsh coefficients

An upper bound is established for the upper bounds of the Fourier-Walsh coefficients a_n(f) whose modulus of continuity (,f) does not exceed a given modulus of continuity (). In the case of convex majorants of (), these bounds are attained for individual ordinal numbers n.Translated from Matematicheskie Zametki, Vol. 6, No. 6, pp. 725–736, December, 1969.     

Generalization of Gδ*-Diagonals and ωΔ-Spaces

In this paper we introduce the concepts of a quasi-G*-diagonal and quasi--space as generalizations of the concepts of G*-diagonal and -space respectively. It is shown that a quasi-Moore space may be characterized in terms of these concepts. As a consequence we obtain the following metrization theorems: every paracompact -space with quasi-G-diagonal is metrizable and every collectionwise normal quasi--space is metrizable.     

Problem of correctness of the best approximation in the space of continuous functions

Let W^rH _w be the subclass of those functions of C^r[a, b], for which (f ^(r),)(), where () is a given modulus of continuity, and P_n be the space of algebraic polynomials of degree at most n and _n(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for and moduli of continuity of the operators of best approximation on W^rH _w are established. For example, if ()=, then Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.     

О метрических свойст вах особых множеств г оломорфных функций нескольких п еременных

LetG be a domain inC ⁿ ,EG, mes E=0 for (r)=r ^2n–1(r), where (r) is a nondecreasing non-negative function (r>0). Iff(z) is holomorphic inGE and (,f, GE)(), C=const, thenf(z) is holomorphic inG.The impossibility of the relaxation of the stipulations on () and(r) is also established.The statement above is a corollary to a more general result about the representation of a holomorphic function from a certain class in the form of an integral with respect to -measure, extended over the set of singular points of the function.     

Finite-dimensional attainable sets for stochastic control systems

Letx _t ^u () be a stochastic control system on the probability space (, ,P) intoR ⁿ. We say that the pointxR ⁿ is (, ) attainable at timet if there exists an admissible controlu such thatP _xo{x _t ^u ()S (x)}, wherex ₀()=x ₀, 0, 10, andS (x) is the closed Euclidean -ball inR ⁿ centered atx. We define the attainable setA (t) to be the set of all pointsxR ⁿ which are (, ) attainable at timet. For a large class of stochastic control systems, it is shown thatA (t) is compact for eacht and continuous as a function oft in an appropriate metric. From this, the existence of stochastic time-optional controls is established for a large class of nonlinear stochastic differential equations.This research was supported by the National Research Council of Canada, Grant No. A-9072.     

Thermal stability of a reactive slab with time-periodic heating of one surface

Summary The stability of time-periodic thermal oscillations in a slab with chemical reaction is examined. One surface is at a fixed temperature and the temperature of the other varies with amplitude and frequency. The reaction, which is of zero-order, is characterised by the Frank-Kamenetskii parameter(,,), where(=RT ₀/E) is the activation energy parameter. Perturbation series in and are used to calculate the critical values _crit; for _crit the slab can support time-periodic temperature oscillations.

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Zusammenfassung Die Stabilität von zeitperiodischen thermischen Schwingungen in einem Stab mit chemischer Reaktion wird untersucht. Eine Oberfläche hat eine feste Temperatur und auf der anderen verändert sich die Temperatur mit der Amplitude und der Frequenz. Die Reaktion der Ordnung null, wird vom Frank-Kamenetskii Parameter(,,) bestimmt, wobei(=RT ₀/E) der Aktivierungs-Energie Parameter ist. Reihenentwicklungen in und werden für die Berechnung des kritischen Wertes _crit benützt; für _crit können im Stab zeitperiodische Temperaturschwingungen bestehen.

     

Pinning effect in peierls impurity systems with deviation from half-filling of the energy band

A study is made of the effect of deviation from half-filling of the energy band (0) on the Fröhlich collective mode in onedimensional impurity systems. A low impurity concentration is considered, and the infinite series of impurity scattering is taken into account self-consistently in the determination of the collective mode Green's function. The conductivity () is found in terms of this Green's function, and an analytic expression is obtained for () at _T ( _T is the pinning frequency). It is shown that for the ratio Re(()/_max) a universal formula arises. It differs from the results of Kurihara in the expression for _T , which contains an essential dependence on in the incommensurate state of the charge density wave. It is also shown that the width of the peak in the dependence () and its position increase with increasing .Institute of Applied Physics, Moldovan Academy of Sciences. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 101, No. 1, pp. 110–122, October, 1994.     

Applications of the degree theorem to absolute continuity on Wiener space

Summary Let (,H, P) be an abstract Wiener space and define a shift on byT()=+F() whereF is anH-valued random variable. We study the absolute continuity of the measuresPºT ^–1and ( _F P)ºT ¹ with respect toP using the techniques of the degree theory of Wiener maps, where _F =det₂(1+F) × Exp{–F–1/2|F|²}.The work of the second author was supported by the fund for promotion of research at the Technion     

ε-Invarianten bei verallgemeinerten Carlesonmengen

Generalized Carleson-sets with modulus of continuity () and sets of uniqueness for the classA ={f(z): f regular inD={z: |z|<1} and continuous with modulus of continuity () in } are described by the behavior of their -entropy and -capacity (in the sense of Kolmogorov).     

Quasimonotonicity,regularity and duality for nonlinear systems of partial differential equations

Summary We prove partial regularity for the vector-valued differential forms solving the system (A(x, ))=0, d=0, and for the gradient of the vector-valued functions solving the system div A(x, Du)=B(x, u, Du). Here the mapping A, with A(x, w) (1+ + ¦¦²)^{(p – 2)/2} (p2), satisfies a quasimonotonicity condition which, when applied to the gradient A(x, )=Df(x, ) of a real-valued functionf, is analogous to but stronger than quasiconvexity for f. The case 1相似文献   

Equivalence Bimodule Between Non-Commutative Tori

The non-commutative torus C ^*(ⁿ,) is realized as the C*-algebra of sections of a locally trivial C*-algebra bundle over S with fibres isomorphic to C ^*n/S, ₁) for a totally skew multiplier ₁ on ⁿ/S. D. Poguntke [9] proved that A is stably isomorphic to C(S) C(^*( Zⁿ/S, ₁) C(S) A M_kl( C) for a simple non-commutative torus A and an integer kl. It is well-known that a stable isomorphism of two separable C*-algebras is equivalent to the existence of equivalence bimodule between them. We construct an A-C(S) A-equivalence bimodule.     

Cohomology of solvable lie algebras and solvmanifolds

The cohomology H^* (G/,) of the de Rham complex *(G/) of a compact solvmanifold G/ with deformed differential d = d + , where is a closed 1 -form, is studied. Such cohomologies naturally arise in Morse-Novikov theory. It is shown that, for any completely solvable Lie group G containing a cocompact lattice G, the cohomology H^*(G/, ) is isomorphic to the cohomology H^*( ) of the tangent Lie algebra of the group G with coefficients in the one-dimensional representation : defined by () = (). Moreover, the cohomology H ^*(G/,) is nontrivial if and only if -[] belongs to a finite subset of H ¹(G/,) defined in terms of the Lie algebra .Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 67–79.Original Russian Text Copyright © 2005 by D. V. Millionshchikov.This revised version was published online in April 2005 with a corrected issue number.     

Diameters of certain sets in function spaces

The precise values of the diameters (according to A. N. Kolmogorov) of the class H _C in the space C[a, b] and lower bounds for the diameters of the class H _p in the spaceL_p(0, 2) (1p), for any modulus of continuity(), are obtained. The latter bounds give the exact values of the odd-numbered diameters of the class and the exact order of decay of the diameters of the class H ₁ .Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 637–646, May, 1973.In conclusion, the author wishes to thank A. L. Garkavi, A. V. Efimov, and S. B. stechkin for their help.     

Weighted Polynomial Approximation for Weights with Logarithmic Singularities in the Extremal Measure

Let w : [0, ) be a weight function on a set R. We assume that the associated extremal measure has density function v(t) with finitely many singularities of logarithmic type. We show that any continuous function f on which vanishes outside the set where v is positive or has a logarithmic singularity, is the uniform limit on of a sequence of weighted polynomials of the form wⁿ P_n, where P_n is of degree n. This extends previous results for continuous densities to densities having logarithmic singularities.     

The correctness problem for best approximations by trigonometric polynomials in the class W o r H[ω]C

Suppose thatk, rz₊, W _o ^r H[]_C= {ff is a 2-periodic function,f C^r [–, ], (f^(r), ) ()}, T_k is the space of trigonometric polynomials of order k, p_k(f)T_k is the polynomial of best uniform approximation to f, and E_k(f) is the error of the best approximation. It is shown that for an arbitrary > 0 we have,where for 0<&#x2A7D;(1),k > 0.R () is the root of the equation , and for k = 0 or > (1) we have R()=.Translated from Matematicheskie Zametki, Vol. 22, No. 1, pp. 85–101, July, 1977.The author thanks S. B. Stechkin for posing the problem and for his attention to this work.     

Calculation of Some Integrals Describing Wave Fields

Two methods of calculating the scattering amplitude f(, ₀) of a wave scattered by the vertex of an arbitrarily shaped cone are justified. It is shown that the approximation f _d (, ₀,t) obtained by a method similar to the AbelPoisson method of summation converges uniformly in the regularity region for f. Also, the possibility of calculating f(, ₀) for N ₁( ₀) with the help of rapidly convergent integrals is proved. Bibliography: 7 titles.     

Asymptotic Formulae and Divisor Problems

We investigate the asymptotic behaviour of the summatory functions of _z(n, ), _k(n, ) z ⁽ⁿ⁾ and _k(n, ) z ⁽ⁿ⁾.     

On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials

Let H, H ^L be classes of functionsf(x) whose modulus of continuity (f; t) and, respectively, integral modulus of continuity(f; t)_L do not exceed a given modulus of continuity(t), while H_v is a class of functionsf(x) whose variation fdoes not exceed a given number V > 0. Bounds are obtained for the upper limit of the best approximations in the metric of L by Haar-system polynomials on the classes just introduced (on the class H ^L only when (t)=Kt). These bounds are exact for class H_V and, in case(t) is convex, also for the classes H and H ^L .Translated from Matematicheskie Zametki, Vol. 6, No. 1, pp. 47–54, July, 1969.The author wishes to thank N. P. Korneichuk for having posed the problem and for his constant attention to this work.     

Two congruence lattices of completely simple semigroups

For a Rees matrix semigroupS with normalized sandwich matrix and C(S), the congruence lattice ofS, we consider the lattice generated by {itpT_l, pK, pT_r, pt_l, pk, pt_r}. HerepT ₁ andpt _l are the upper and lower ends of the interval which makes up the _i -class of , _i being the left trace relation onC(S). The remaining symbols have the analogous meaning relative to the kernel and the right trace relations. We also consider the lattice generated by {T _l, K, T_r, t_l, _k, t_r} where and are the equality and the universal relations onS, respectively. In both cases, we find lattices freest relative to these lattices and represent them as distributive lattices with generators and relations.With 3 Figures     

Blow-up rates for parabolic systems

Let ⁿ be a bounded domain andB _R be a ball in ⁿ of radiusR. We consider two parabolic systems: ut=u +f(), i= +g(u) in × (0,T) withu=v=0 on × (0,T) andu _t =u, v _t =v inB _r × (0,T) withe/v=f (v), e/v=g(u) onB _R × (0,T). Whenf(v) andg(u) are power law or exponential functions, we establish estimates on the blow-up rates for nonnegative solutions of the systems.