The conformal structure of Riemann surfaces with boundary parametrizing minimal surfaces

Let F denote a surface with boundary F, being contained in a Riemann surface R, such that R\F is somedisk. If we vary the boiundary curve _o parametrizing F, we will get a manifold of real dimension 6g–3, such that any bounds some F and any local deformation of F is conformally equivalent to just one F for .This result also implies that none of the conformal invariants of R will be an invariant of this F, since its neighbors {F|} cover all possible deformations of F at all.     

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.     

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.     

Transformation of Wiener measure under anticipative flows

Summary LetT()=+F() be a transformation from the Wiener space to itself with the range ofF() assumed to be in the Cameron-Martin space. The absolute continuity and the density function associated withT is considered;T is assumed to be embedded in or defined through a parameterizationT _t =+F _t () andF _t is assumed to be differentiable int. The paper deals first with the case where the range of thet-derivative ofF _t () is also in the Cameron-Martin space and new representations for the Radon-Nikodym derivative and the Carleman-Fredholm determinant are derived. The case where thet-derivative ofF _t is not in the Cameron-Martin space is considered next and results on the absolute continuity and the density function, under conditions which are considerably weaker than previously known conditions, are presented.The work of the second author was supported by the fund for promotion of research at the Technion     

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.     

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.     

On meromorphic solutions of algebraic differential equations in angular domains

We obtain asymptotic estimates of meromorphic solutions to the differential equationP _n (z, , )=P _n–1 (z, , ,..., ^(m) ) in the angular domain P={z: arg z · }. Here P_n(z, w, w) is a polynomial in all variables, and of degree n with respect to w and w; P_n–1(z, w, w, ..., w^(m)) is a polynomial in all variables, and of degree n –1 with respect to w, w, ..., w^(m) In the particular case, when the solutions are entire functions, these estimates are more precise than the known estimates that are obtained by using the method of Wiman-Valiron, which cannot be applied to meromorphic solutions in the domain P.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 514–523, April, 1992.     

Divisor Functions in the Ring of Gaussian Integers Weighted by the Generalized Klosterman Sum

Let _a,b,c () be the number of factorizations of a Gaussian number in the form = ₁ ^a ₂ ^b ₃ ^c , where a, b, and c are natural numbers. In the ring of Gaussian numbers, we construct an asymptotic formula for a summatory function of _a,b,c () weighted by the generalized Klosterman function.     

Exact bounds for norms of differentiable periodic functions in L metric

Exact values are obtained for the upper bounds on the norms,fL, on the classes W ^(r) H(r =0, 1, 2,...) of r-fold differentiable functions f(x), of periodicity2, for which (f^(r);t)(t),where (t) is a given convex modulus of continuity.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 569–576, December, 1967.     

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.     

On Piecewise-Constant Approximation of Continuous Functions of n Variables in Integral Metrics

We consider the approximation by piecewise-constant functions for classes of functions of many variables defined by moduli of continuity of the form (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ), where _i ( _i ) are ordinary moduli of continuity that depend on one variable. In the case where _i ( _i ) are convex upward, we obtain exact error estimates in the following cases: (i) in the integral metric L ₂ for (₁, ..., _n ) = ₁(₁) + ... + _n ( _n ); (ii) in the integral metric L _p (p 1) for (₁, ..., _n ) = c ₁₁ + ... + c _{n n} ; (iii) in the integral metric L _{(2, ..., 2, 2r)} (r = 2, 3, ...) for (₁, ..., _n ) = ₁(₁) + ... + _{n – 1}( _{n – 1}) + c _{n n} .     

Measures induced on Wiener space by monotone shifts

Summary In this work we study the absolute continuity of the image of the Wiener measure under the transformations of the formT()=+u(), the shiftu is a random variable with values in the Cameron-Martin spaceH and is monotone in the sense that (T(+h-T(),h) _H 0 a.s. for allh inH.     

Singular Rank One Perturbations of Self-Adjoint Operators and Krein Theory of Self-Adjoint Extensions

Gesztesy and Simon recently have proven the existence of the strong resolvent limit A_, for A_, = A + (·), where A is a self-adjoint positive operator, being the A-scale). In the present note it is remarked that the operator A_, also appears directly as the Friedrichs extension of the symmetric operator :=A \{f (A)| f,=0\}. It is also shown that Krein's resolvents formula: (A_{_b,}-z)^-1 =(A-z)^-1+ (·, ) _z, with b=b-(1+z) (_z,_-1),_z= (A-z)^-1 defines a self-adjoint operator A_b, for each and b R¹. Moreover it is proven that for any sequence _n which goes to in there exists a sequence _n0 such that A_b, in the strong resolvent sense.     

Random reals and the relation 171-1171-1171-1

It is consistent that ₁(₁,(:n))² holds in any random extension for n finite and countable.     

Classifying countable Boolean terms

We deal with the Borel and difference hierarchies in the space P of all subsets of endowed with the Scott topology. (The spaces P and 2 coincide set-theoretically but differ topologically.) We look at the Wadge reducibility in P. The results obtained are applied to the problem of characterizing ₁-terms t which satisfy C = t( ₁ ⁰ ) for a given Borel-Wadge class C. We give its solution for some levels of the Wadge hierarchy, in particular, all levels of the Hausdorff difference hierarchy. Finally, we come up with a discussion of some relevant facts and open questions.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 173–197, March–April, 2005.     

Invertible infinitary calculus without loop rules for restricted FTL

In the paper, a fragment of first-order linear time logic (with operators next and always) is considered. The object under investigation in this fragment is so-called t-D-sequents. For considered t-D-sequents, an invertible infinitary sequent calculus G ⁺ is constructed. This calculus has no loop rules, i.e., rules with duplications of the main formula in the premises of the rules. The calculus G ⁺ along with an -type rule for the temporal operator always contains an integrated separation rule (IS), which includes the traditional loop-type rule ( ), a special rule ( ) (without duplication of the main formula), and the traditional rule for the temporal operator next. The rule ( ) is incorporated in an axiom. The soundness and -completeness of the constructed calculus G ⁺ are proved. Bibliography: 43 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol.293, 2002, pp. 149–180.This revised version was published online in April 2005 with a corrected cover date and article title.     

On countably compact,locally countable spaces

We study the cardinalities of countably compact, locally countableT ₃ spaces. For alln(<), there exists one of cardinality _n . IfV=L, then there exists one of cardinalityx iffx= orx =x. MA implies that there exists one of cardinality>2.     

Constrained normalization of Hamiltonian systems and perturbed Keplerian motion

Summary Consider a Hamiltonian system (H, _2n ,). LetM be a symplectic submanifold of (_2n ,). The system (H, _2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on _2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as the lunar problem and the main problem of artificial satellite theory.

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Zusammenfassung Wir betrachten ein Hamiltonisches System (H, _2n ,). SeiMein symplectisches Submanifold von (_2n ,). Das System (H, _2n ,), aufM beschränkt, ist (HM,M,M). In der vorliegenden Arbeit wird ein Algorithmus vorgeschlagen, der dieses System so auf _2n normalisiert, daß das aufM beschränkte System auch normalisiert ist. Dieser Algorithmus wird dann auf gestörte Keplersysteme, wie z. B. das Hill-sche Mondproblem und das Hauptproblem der Theorie der künstlichen Satelliten, angewendet.

     

On semiregular conditional distributions

Let (,A,P) denote some probability space and some sub--algebra ofA. It is shown that there exists a semiregular versionQ (A),A, , of the conditional distributionP(A|), AA, i.e., Q (A), (AA fixed) is andAQ (A),AA ( fixed), is a probability charge satisfyingQ (N)=0, , for allP-zero setsN, if and only ifL ₁(,P|) has a lifting, which exists for any sub--algebra ofA ifL ₁(,A P) is separable. Separability ofL ₁(,A,P) implies also the existence of a strongly semiregular versionQ (A),A, , ofP(A|), A , i.e., Q (A), (AA fixed), is -measurable andAQ (A),A ( fixed), is a probability charge. Furthermore,P can be written as P ₁+(1–)P ₂, 01, whereP ₁ are probability measures onA such thatP ₁(A|),AA, has a semiregular version vanishing for anyP-zero setN andP ₂ is singular with respect to any probability measure onA of the type ofP ₁. In the case 0<<1 the probability measuresP _j ,j=1, 2, are uniquely determined. The decomposition can be carried over to the case, where the additional condition thatQ (N)=0 for all and anyP-zero setN is valid, is omitted respectively semiregularity is replaced by (i) strong semiregularity, or (ii) classical regularity. In the last mentioned case (ii) the decomposition is multiplicative.     

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.