On the homogeneous Riemannian structures of type ℐ1⊕ℐ3

We study homogeneous Riemannian structures belonging to the class ₁₃ of the classification given by Tricerri and Vanhecke. The main result is the following: a connected, simply connected Riemannian manifoldM admits a homogeneous structureT of type ₁₃,T₃, if and only ifM is isometric to a hyperbolic space ⁿ .This work was partially supported by M.P.I.     

On the approximation by rational combinations of a class of functions

() [0,1] — {(n)} — , +. , f(x) [0,1] () , x ₁ ,x ₂ [0, 1], (₁)=(₂), f(x ₁ )=f(x ₂ ).     

Fourier coefficients of functions of classC Β ψ H X ω

Sharp upper bounds are obtained from the suprema of the Fourier coefficients of functionsC H _C andC H _L of several variables defined by multipliers (·). translations in the arguments and moduli of continuity in the spaces C and L.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 11, pp. 1537–1545, November, 1990.     

A geometric property of extremal surfaces

Let the surface R³ be defined by the equation z = f(x, y), where f(x, y) is a function 3 times continuously differentiable in R². It is proved that if the total (Gaussian) curvature of the surface is nonzero almost everywhere on in the sense of Lebesgue measure in R²), then is extremal, i.e., for almost all (x,y) R² the inequality max (||qx||, ||qy, qf (x, y)) > q^–1/s–. holds for all integral q qo (f), where x is the distance from the real number x to the nearest integer and > 0 is arbitrarily small.Translated from Matematicheskie Zametki, Vol. 23, No. 2, pp. 177–181, February, 1978.In conclusion, the author thanks V. G. Sprindzhuk for suggesting the problem.     

The Orders of Related Elements of a Finite Field

All finite fields q (q 2, 3, 4, 5, 7, 9, 13, 25, 121) contain a primitive element for which + 1/ is also primitive. All fields of square order q ² (q 3, 5) contain an element of order q + 1 for which + 1/ is a primitive element of the subfield q. These are unconditional versions of general asymptotic results.     

Lévy–Khintchine Formula and Dual Convolution Semigroups Associated with Laguerre and Bessel Functions

In this work we consider a system of partial differential operators D ₁,D ₂ on K=[0,+[×R, whose eigenfunctions are the functions (x,t), (x,t)K, =((R0)×N)(0×[0,+[), which are related to the Laguerre functions for ((R 0)×N)(0,0) and which are the Bessel functions for (0×[0,+[). We provide K and with a convolution structure. We prove a Lévy–Khintchine formula on K, which permits us to characterize dual convolution semigroups on .     

Arithmetic of Rikuna’s generic cyclic polynomial and generalization of Kummer theory

In this paper we study the arithmetic of the Rikunas generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that and k where is a primitive n-th root of unity and =+^–1. In particular, this result with k implies the classical Kummer theory.Mathematics Subject Classification (2000): 11R20, 12E10, 12G05     

Symmetrische Permutationsmengen

A permutation set (M, I) consisting of a setM and a set of permutations ofM, is calledsymmetric, if for any two permutations, the existence of anx M with (x) (x) and ^–1 (x) = ^–1 (x) implies ^–1 = ^–1 , andsharply 3-transitive, if for any two triples (x ₁,x ₂,x ₃), (y ₁,y ₂,y ₃) M ³ with|{x ₁,x ₂,x ₃ }| = |{y ₁,y ₂,y ₃ }| = 3 there is exactly one permutation with(x ₁) =y ₁,(x ₂) =y ₂,(x ₃) =y ₃. The following theorem will be proved.THEOREM.Let (M, ) be a sharply 3-transitive symmetric permutation set with |M|3, such that contains the identity. Then is a group and there is a commutative field K such that and the projective linear group PGL(2, K) are isomorphic.     

Lengths of the periods of the continued fraction expansion of quadratic irrationalities and on the class numbers of real quadratic fields

It is proved that the relation h(d)=2 is valid for at least Cx^1/2 log^–2 x values of dx. Here h(d) is the number of the classes of binary quadratic forms of determinant d, while C>0 is a constant. Further, it is shown that for almost all primes p3 (mod 4), px, for (p), a fundamental unit of field and l(p), the length of the period of the continued fraction expansion of p, we have estimates (p)p² log^–c p, l(p)log p, which improve a result of Hooley (J. Reine Angew. Math., Vol. 353, pp. 98–131, 1984; MR 86d:11032). In addition, a generalization is given to composite discriminants of the Hirzebruch-Zagier formula, relating h(–p), p 3 (mod 4), with the continued fraction expansion of _p (Astérisque, no. 24–25, pp. 81–97, 1975; MR 51 #10293).Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 168, pp. 11–22, 1988.     

Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let denote the boundary of T, consisting of all infinite geodesics b=[b ₀,b ₁,b ₂,] beginning at the root, 0. For each b, 1, and a0 we define the approach region _,a (b) to be the set of all vertices t such that, for some j, t is a descendant of b _j and the geodesic distance of t to b _j is at most (–1)j+a. If >1, we view these as tangential approach regions to b with degree of tangency . We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition _T f ^p (t)q ^–|t|<, where p>1 and 0<<1, or p=1 and 0<1. For 11/, we show that Gf(s) has limit zero as s approaches a boundary point b within _,a (b) except for a subset E of of -dimensional Hausdorff measure 0, where H (E)=sup_>0inf _i q ^{–|t i|}:E a subset of the boundary points passing through t _i for some i,|t _i |>log _q (1/).     

Interfaces and typical Gibbs configurations for one-dimensional Kac potentials

Summary We consider a one dimensional Ising spin system with a ferromagnetic Kac potential J(|r|),J having compact support. We study the system in the limit, »0, below the Lebowitz-Penrose critical temperature, where there are two distinct thermodynamic phases with different magnetizations. We prove that the empirical spin average in blocks of size ^–1 (for any positive ) converges, as »0, to one of the two thermodynamic magnetizations, uniformly in the intervals of size ^–p , for any given positivep1. We then show that the intervals where the magnetization is approximately constant have lengths of the order of exp(c ^–1),c>0, and that, when normalized, they converge to independent variables with exponential distribution. We show this by proving large deviation estimates and applying the Ventsel and Friedlin methods to Gibbs random fields. Finally, if the temperature is low enough, we characterize the interface, namely the typical magnetization pattern in the region connecting the two phases.The research has been partially supported by CNR, GNFM, GNSM and by grant SC1CT91-0695 of the Commission of European Communities     

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.     

∑II∑ threshold formulas

A II formula has the form, where eachL is either a variable or a negated variable. In this paper we study the computation of threshold functions by II formulas. By combining the proof of the Fredman-Komlós bound [5, 10] and a counting argument, we show that fork andn large andkn/2, every II formula computing the threshold functionT _k ⁿ has size at least exp . Fork andn large andkn ^2/3, we show that there exist II formulas for computingT _k ⁿ with size at most exp .     

On the equation of similar profiles

Summary Considerf+ ff+ (1–f²)+ f=0 together with the boundary conditionsf(0)=f(0)=0,f ()=1. If=–1,>0, arbitrary there is at least one solution which satisfies 0<f<1 on (0, ). By the additional conditionf>0 on (0, ) or, alternately 0<1, the uniqueness of the solution is demonstrated.If=1,<0, arbitrary the existence of solutions for which –1<f<0 in some initial interval (0,t) and satisfying generallyf>1 is established. In both problems, bounds forf (0) and qualitative behavior of the solutions are shown.

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Sommario Si consideri il problema definito dall'equazionef+ f f+ (1–f²)+ f=0 e dalle condizioni al contornof(0)=f (0)=0,f()=1. Assumendo=–1,>0, arbitrario si dimostra che esiste almeno una soluzione che soddisfa 0<f<1 nell'intervallo (0, ). Se in aggiunta si ipotizzaf>0 in (0, ), oppure 0<=1, l'unicità délia soluzione è assicurata.Successivamente si considéra il problema di valori al contorno con=1,<0, arbitrario. In questo caso esiste un'intera classe di soluzioni che soddisfano –1<f<0 in un intorno dell'origine e tali chef>1, in generale.Di detti problemi viene studiato il comportamento délle soluzioni e vengono determinate dalle maggiorazioni e minorazioni del valoref(0).

     

A note on sequence-covering mappings

Summary Let f: X &rarr; Y be a mapping. f is called a sequence-covering mapping if in case S is a convergent sequence containing its limit point in Y then there is a compact subset K of X such that f(K) = S. It is shown that each quotient and compact mapping of a metric space is sequence-covering.     

A family of multiply extended grids

We construct an infinite family{ _n}_n=5 of finite connected graphs _n that are multiple extensions of the well-known extended grid discovered in [1] (which is isomorphic to ₅). The graphs _n are locally _n–1 forn > 5, and have the following property: the automorphism groupG(n) of _n permutes transitively the maximal cliques of _n (which aren-cliques) and the stabilizer of somen-clique of _n inG(n) induces _n on the vertices of. Furthermore we show that the clique complexes of the graphs _n are simply connected.     

Conservation laws with continuous flux functions

In the present paper we investigate uniqueness problem for generalized entropic solution of stationnary and evolution conservation laws under anisotropic conditions on vector-flux functions in the classesL andL L ¹.This research was performed while visiting the Equipe de Mathématiques in Besançon.     

Fine Analysis of the Quasi-Orderings on the Power Set

We pursue the fine analysis of the quasi-orderings and on the power set of a quasi-ordering (Q,). We set X Y if every xX is majorized in by some yY, and X Y if every yY is minorized in by some xX. We show that both these quasi-orderings are -wqo if and only if the original quasi-ordering is ( )-wqo. For this holds also restricted to finite subsets, thus providing an example of a finitary operation on quasi-orderings which does not preserve wqo but preserves bqo.     

On the lower Markov spectrum

C. Hightower found two infinite sequences of gaps in the Markov spectrum, ( _n , _n ) and ( _n , _n ) with _n and _n both Markov elements, converging to . This paper exhibits Markov elements _n ^* and _n ^* such that, for alln 1, ( _n ^* , _n ) and ( _{n n} ^* ) are gaps in the Markov spectrum. Other results include showing that, for alln 1, _n is completely isolated, while the other endpoints of the gaps are limit points in the Markov spectrum.     

V-localizations andV-Kleisli algebras

We prove a pushout theorem for localizations and Kleisli categories over a symmetric monoidal closed categoryV. That is, suppose is aV-localizable subcategory of aV-categoryA and thatT=(T,,) is aV-monad onA. Then under suitable relations betweenT and we show that there is aV-monadT induced onA[-¹] such that the Kleisli category ofT is the pushout of the localization functor :AA[-¹] and the free functor F:AK(T). Consequently,K(T)K(T) [S-¹] for some S K(T). We give several examples of this situation.