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.     

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 the homogeneity property for certain quantifier logics

Summary The local homogeneity property is defined as in [Mak]. We show thatL (Q₁) and some related logics do not have the local homogeneity property, whereas cofinality logicL (Q ^cf) has the homogeneity property. Both proofs use forcing and absoluteness arguments.     

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.     

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.     

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} .     

Classification de certains feuilletages associés à un cusp

We study singularities of foliations given by R. Moussu closely related to a conjecture of R. Thom. We give their analytic classification and prove their topological rigidity.

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Notations , ,F 0 - H 0, III - H ₁,H _p I - H-cusp,H _r -cusp I-3 - H _p I-3, III-2 - -cusp II-1 - _r -cusp III-1     

Cross-cuts in the power set of an infinite set

In the power setP(E) of a setE, the sets of a fixed finite cardinalityk form across-cut, that is, a maximal unordered setC such that ifX, Y E satisfyXY, X someX inC, andY someY inC, thenXZY for someZ inC. ForE=, ₁, and ₂, it is shown with the aid of the continuum hypothesis thatP(E) has cross-cuts consisting of infinite sets with infinite complements, and somewhat stronger results are proved for and ₁.The work reported here has been partially supported by NSERC Grant No. A8054.     

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     

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.     

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.     

ω-Hyponormal operators

The class of -hyponormal operators is introduced. This class contains allp-hyponormal operators. Certain properties of this class of operators are obtained. Among other things, it is shown that ifT is -hyponormal, then its spectral radius and norm are identical, and the nonzero points of its joint point spectrum and point spectrum are identical. Conditions under which a -hyponormal operator becomes normal, self-adjoint and unitary are given.     

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.     

On the width of ordered sets and Boolean algebras

Thewidth (chain number) of a partial order P, < is the smallest cardinal such that ¦A¦< 1 + whenever A is an antichain (chain) in P. We prove that, if a partial order (P, <) has width and cf()=, then P contains antichains A_n (n<) such that ¦A ₀¦<¦A₁¦ <...<={¦A_n¦: n < < } and either A₀1 A₂< ... or A₀>A₁ >A₂> ... A similar structure result is obtained for partial orders with chain number if cf()=. As an application we solve a problem of van Douwen, Monk and Rubin [1] by showing that if a Boolean algebra has width , thencf() .This work has been partially supported by NATO grant No. 339/84.Presented by Bjarni Jonsson.     

Differentiability of the minimal average action as a function of the rotation number

Let be a finite composition of exact twist diffeomorphisms. For any real number , letA() denote the minimal average action of -invariant measures with angular rotation number . We prove thatA() is differentiable at every irrational number and that for generic it is not differentiable at rational , thus verifying conjectures of S. Aubry. Moreover, we show that these results are valid for a variational principleh which satisfies the condition which we have called elsewhere (H). As a consequence, we generalize a result due to Bangert concerning geodesics on a two dimensional torus with an arbitrary, but sufficiently smooth metric.supported by NSF grant no. DMS-8806067.01 and a Guggenheim Fellowship.     

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.     

Stability of a spline collocation method for strongly elliptic multidimensional singular integral equations

Summary We consider a spline collocation method for strongly elliptic zero order pseudodifferential equationsp _gw Au=f on a cube =(0, 1) ^m . Utilizing multilinear spline functions which are zero at the boundary we collocate at the meshpoints inside . For classical strongly elliptic translation invariant pseudodifferential operators, we verify the stability of the considered collocation method inL ²(). Afterwards, form2 and a right hand sidefH ⁸(),s>m/2, we prove an asymptotic convergence estimate.The author has been supported by a grant of Deutsche Forschungsgemeinschaft under grant number Ko 634/32-1     

Effective bounds for the number of solutions of certain diophantine equations

It is proved that the number of solutions of the diophantine equation Norm (z ₁, ₁),+...+z _mm) =f (z, ...,z _m), is finite, where ₁, ..., _m are algebraic numbers of a special type, the left side of the equation is the norm with respect to a quadratic field, andf is a low-degree polynomial.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 361–371, September, 1970.     

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.     

Isomorphisms of Some Complex Poisson Brackets

Given a complex manifold M_i, structures of Poisson algebras on (M_i)=C(M_i,C) which are associated with a nondegenerate -closed (2,0)-form _i on M_i are considered. It is shown that every isomorphism of Poisson structures (M₁) (M₂) is generated by a biholomorphic map :M₂ M₁ such that ₂ = *₁