A contribution to shape theory for the ising model at low temperature

Summary We consider low temperature limits of Gibbs states of the ferromagnetic nearest-neighbour Ising Hamiltonian in the positive orthant of the lattice ^d ,d=1, 2,..., under a negative boundary condition and a small positive external fieldh that decreases linearly with the temperatureT. It is shown that positive and negative spins are separated by a staircase-shaped random boundary. Its explicit distribution is computed in the case that the ratio =h/T exceeds some positive ₀. For < ₀, our results do not rule out infinite negative areas.     

Galois theory for the family of partial differential equationsΔ α ψ=0

Summary The partial differential fields most suited for the purpose of construction of Galois theory for the family (1) are endowed with the symmetric bilinear form (2iv) and are called -differential fields. In Section 1 are defined certain algebraic notions related to the symmetric bilinear form (2iv) and which are necessary for the construction of any Galois theory. Necessary and sufficient condition for the extension of the domain of the operator (this operator is not a derivation although it commutes with the partial derivations of the -differential field) from an -differential fieldK to a finitely generated -differential extension field is given in Theorem 1.Section 2 defines the notion of -differential mapping as linear mappings which preserve the symmetric bilinear form and commute with the partial derivations. The group properties of the set of -differential mappings are discussed and the Galois correspondence theorems set up for -differential fields.Section 3 sets up the notion of -Liouvillian extensions of -differential fields and briefly discusses the Galois groups associated with these -Liouvillian extension fields.Section 4 points to the procedure for the algebraic characterization of -simple--differential field extensions by elementary solutions of the partial differential equation _m =0.     

K-theory for finite covariant systems

LetA be a real or complex Banach algebra and assume that is an action of a finite groupG onA by means of continuous automorphisms. To such a finite covariant system (A, G, ), we associate an Abelian groupK(A, G, ). We obtain some classical exact sequences for an algebraA and a closed invariant idealI. We also compute the group in a few important special cases. Doing so, we relate our new invariant to the classicalK ₀ andK ₁ of a Banach algebra and to theK-theory of ₂-graded Banach algebras. Finally, we obtain a result that gives a close relationship of our groupK(A, G, ) with theK-theory of the crossed productA G. In particular, we prove a six-term exact sequence involving our groupK(A, G, ) and theK-groups ofA G. In this way, we hope to contribute to the well-known problem of finding theK-theory of the crossed productA G in the case of an action of a finite group.     

Numerical application of Euler's series transformation and its generalizations

Summary A nonlinear generalizationÊ _z of Euler's series transformation is compared with the (linear) Euler-Knopp transformationE _z and a twoparametric methodE . It is shown how to applyE orE _, to compute the valuef(z_o) of a functionf from the power series at 0 iff is holomorphic in a half plane or in the cut plane. BothE andE _, are superior toÊ _z . A compact recursive algorithm is given for computingE andE _,.     

Sum of Divisors in a Ring of Gaussian Integers

We construct an asymptotic formula for a sum function for _a (), where _a () is the sum of the ath powers of the norms of divisors of the Gaussian integer on an arithmetic progression ₀ (mod ) and in a narrow sector ₁ arg < ₂. For this purpose, we use a representation of _a (n) in the form of a series in the Ramanujan sums.     

Inner functions and related spaces of pseudocontinuable functions

Let be an inner function, let C, ¦¦=1. Then the harmonic function [(+)]/(–)] is the Poisson integral of a singular measure D. N. Clark's known theorem enables us to identify in a natural manner the space H² H² with the space L² ( ).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova Akademii Nauk SSSR, Vol. 170, pp. 7–33, 1989.     

A Note on a Boundary Value Problem

In this note, we prove that, for Robins boundary value problem, a unique solution exists if f_x(t, x, x), f_x(t, x, x), (t), and (t) are continuous, and f_x -(t), f_x -(t), 4(t) ² + ²(t) ++ 2(t), and 4(t) ² + ²(t) + 2(t).AMS Subject Classification (2000) 34B15     

On cyclic biquadratic fields related to a problem of Hasse

In this note we shall prove that there exist infinitely many cyclic biquadratic fieldsK whose integral bases are neither {1, , ², } nor {1, , , ³) for any numbers , inK. Next, we shall construct infinitely many cyclic biquadratic fieldsK which have the index 1, but still have not the integral basis {1, , ², ³) for every inK. Finally we shall give a class of biquadratic fields for a problem of Hasse concerning an integral basis.     

The Product of Independent Random Variables with Regularly Varying Tails

A distribution is said to have regularly varying tail with index – (0) if lim _x(kx,)/(x,)=k ^– for each k>0. Let X and Y be independent positive random variables with distributions and , respecitvely. The distribution of product XY is called Mellin–Stieltjes convolution (MS convolution) of and . It is known that D() (the class of distributions on (0,) that have regularly varying tails with index –) is closed under MS convolution. This paper deals with decomposition problem of distributions in D() related to MS convolution. A representation of a regularly varying function F of the following form is investigated: F(x)= _k=0 ^n–1 b _k f(a ^k x), where f is a measurable function and a and b _k (k=1,...,n–1) are real constants. A criterion is given for these constants in order that f be regularly varying. This criterion is applicable to show that there exist two distributions and such that neither nor belongs to D() (>0) and their MS convolution belongs to D().     

Maximum of the fourth diameter in the family of continua with prescribed capacity

We obtain a complete solution of the problem of the maximum of the fourth diameter in the family of continua with capacity 1. Let E(o, eⁱ, e^–i). 0<i, e^–i; H(=cap E(o, eⁱ, e^–i). Let C() be the common point of three analytic arcs which form E(o, eⁱ, e^–i). One shows that the indicated maximum is realized by the continuum ={z:H(₀)z ²E(o, eⁱ, e^–i)} where ₀, o<₀z eⁱ z+C ( is a real and C is a complex constant). One finds the value of the required maximum. The paper contains a brief exposition of the proof of this result.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 59, pp. 60–79, 1976.     

Approximate Inverse Preconditioning for Shifted Linear Systems

In this paper we consider the problem of preconditioning symmetric positive definite matrices of the form A =A+I where >0. We discuss how to cheaply modify an existing sparse approximate inverse preconditioner for A in order to obtain a preconditioner for A . Numerical experiments illustrating the performance of the proposed approaches are presented.     

Latteninterpolation bei einer Funktion von zwei Veränderlichen

Summary In each lattice point , of a rectangular net a numerical valueu is given. A bicubical and twice continuously differentiable function is constructed interpolating the valuesu . The method is known as «spline interpolation».     

Properties of certain integral operators

Two integral operatorsP andQ for analytic functions in the open unit disk are introduced. The object of the present paper is to derive some properties of integral operatorsP andQ .     

On the value function in constrained control of Markov chains

It is known that the value function in an unconstrained Markov decision process with finitely many states and actions is a piecewise rational function in the discount factor a, and that the value function can be expressed as a Laurent series expansion about = 1 for close enough to 1. We show in this paper that this property also holds for the value function of Markov decision processes with additional constraints. More precisely, we show by a constructive proof that there are numbers O = _o <₁ <... < _m–1 < _m = 1 such that for everyj = 1, 2, ...,m – 1 either the problem is not feasible for all discount factors in the open interval (_j–1, _j) or the value function is a rational function in a in the closed interval [_j–1, _j]. As a consequence, if the constrained problem is feasible in the neighborhood of = 1, then the value function has a Laurent series expansion about = 1. Our proof technique for the constrained case provides also a new proof for the unconstrained case.     

On translations of double rays in graphs

Let be an infinite graph, let be a double ray in , and letd andd denote the distance functions in and in , respectively. One calls anaxis ifd(x,y)=d (x,y) and aquasi-axis if lim infd(x,y)/d (x,y)>0 asx, y range over the vertex set of andd (x,y). The present paper brings together in greater generality results of R. Halin concerning invariance of double rays under the action of translations (i.e., graph automorphisms all of whose vertex-orbits are infinite) and results of M. E. Watkins concerning existence of axes in locally finite graphs. It is shown that if is a translation whose directionD() is a thin end, then there exists an axis inD() andD(^–1) invariant under ^r for somer not exceeding the maximum number of disjoint rays inD().The thinness ofD() is necessary. Further results give necessary conditions and sufficient conditions for a translation to leave invariant a quasi-axis.     

Lipschitz and darbo conditions for the superposition operator in ideal spaces

Summary In this paper we give necessary and sufficient conditions for the superposition operator Fx(s)=f(s, x(s)) to satisfy a Lipschitz condition Fx₁ - Fx₂kx₁ - x₂ or a Darbo condition (FN)k(N) in ideal spaces of measurable functions, where is the Hausdorff measure of noncompactness. Moreover, we characterize a large class of spaces in which the above mentioned two conditions are equivalent.

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Sunto In questo lavoro diamo delle condizioni necessarie e sufficienti perchè l'operatore di sovrapposizione Fx(s)=f (s, x(s)) soddisfi alla condizione di Lipschitz Fx₁–Fx₂ kx₁–x₂ o quella di Darbo (FN)k(N) in spazi ideali di funzioni misurabili, ove è la misura di non compattezza di Hausdorff. Inoltre, caratterizziamo un'ampia classe di spazi in cui le suddette due condizioni sono equivalenti.

     

The Parametric Buffer Phenomenon for a Singularly Perturbed Telegraph Equation with a Pendulum Nonlinearity

We consider the boundary-value problem u _tt + u _t + (1 + cos2)sin u =² u _xx, u _x|_x=0=u_x|_x==0, where 0<1, =(1+)t, ,> 0, and the sign of is arbitrary. It is proved that for an appropriate choice of the external parameters and and for sufficiently small the number of exponentially stable solutions 2-periodic in can be made equal to an arbitrary predefined number.     

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.     

Almost sure behaviour ofF-valued random fields

Summary LetX _n, n ^d be a field of independent random variables taking values in a semi-normed measurable vector spaceF. For a broad class of fields _n, ^d of positive numbers, the almost sure behaviour of _knX_k/_n, n ^d is studied. The main result allows us to deduce some new and well-known theorems for fields of independentF random variables from related results for fields of independent real random variables.Supported in part by the Youth Science Foundation of China, No. 19001018Supported by the National Natural Science Foundation of China