m-Systems and Partial m-Systemsof Polar Spaces

LetP be a finite classical polar space of rankr, withr 2. A partialm-systemM ofP, with 0 m r - 1, is any set (₁), ₂,..., _k ofk ( 0) totally singularm-spaces ofP such that no maximal totally singular space containing _i has a point in common with (_{1 2} ... _k) — _i,i = 1, 2,...,k. In a previous paper an upper bound for ¦M¦ was obtained (Theorem 1). If ¦M¦ = , thenM is called anm-system ofP. Form = 0 them-systems are the ovoids ofP; form =r - 1 them-systems are the spreads ofP. In this paper we improve in many cases the upper bound for the number of elements of a partialm-system, thus proving the nonexistence of several classes ofm-systems.Dedicated to Hanfried Lenz on the occasion of his 80th birthday     

On the properties of ε(≥0) optimal policies in discounted unbounded return model

This paper investigates the properties of (0) optimal policies in the model of [2]. It is shown that, if * = ( ₀ ^* , ₁ ^* ,..., _n ^* , _n ^+1/* , ...) is a-discounted optimal policy, then ( ₀ ^* , ₁ ^* , ..., _n ^* ) for alln0 is also a-discounted optimal policy. Under some condition we prove that stochastic stationary policy _n ^* corresponding to the decision rule _n ^* is also optimal for the same discounting factor. We have also shown that for each-optimal stochastic stationary policy ₀ ^* , ₀ ^* can be decomposed into several decision rules to which the corresponding stationary policies are also-optimal separately; and conversely, a proper convex combination of these decision rules is identified with the former ₀ ^* . We have further proved that for any (,)-optimal policy, say *=( ₀ ^* , ₁ ^* , ..., _n ^* , _n ^+1/* , ...), _n–1 ^* ) is ((1– ⁿ )^–1 e, ) optimal forn>0. At the end of this paper we mention that the results about convex combinations and decompositions of optimal policies of § 4 in [1] can be extended to our case.Project supported by the Science Fund of the Chinese Academy of Sciences.     

Lifting-Probleme für Vektorfunktionen und ⊗-Sequenzen

In this article the topologically exact sequences of locally convex spaces are characterized for which for every locally convex space F the map id : FE F Q is a homomorphism, or equivalently, the map id L : FK F E is a topological injection. This is motivated by the problem of lifting Q-valued functions with certain given properties to E-valued functions with the same or slightly weaker properties, which may also be considered as the investigation of parameter dependences of solutions of linear (differential) equations. Applications to partial differential equations and to Fredholm functions are given.     

Divergent Fourier series with respect to orthonormalized systems of smooth functions

In this paper it is proved that for any functionf L ² [–; ], f>0, there exists a complete orthonormalized system of uniformly bounded trigonometric polynomials with respect to which the Fourier series of this function is divergent almost everywhere in the interval [–; ].Translated from Matematicheskie Zametki, Vol. 20, No. 1, pp. 69–78, July, 1976.The authoress expresses her gratitude to A. M.     

Limites dégénérées de séries discrètes, formes automorphes et variétés de Griffiths–Schmid: le cas du groupe U (2, 1)

Let G be an inner anisotropic form of an unitary group of 3 variables over Q, such that G_RU(2,1), and be an automorphic representation of G(A) whose archimedean component is a degenerate limit of discrete series; such a never occurs in the cohomology (coherent or étale) of a Shimura variety. We show that however it does appear in the coherent cohomology of some line bundle over an associated Griffiths-Schmid variety. Moreover we study cup products between such cohomology classes and some other automorphic cohomology classes and we prove some non-vanishing results.     

Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the -contamination class ={=(1–)₀+q:qQ} is derived. Here, ₀ is the elicited prior which is assumed unimodal, is the amount of uncertainty in ₀, andQ is the set of all probability densitiesq for which =(1–)₀+q is unimodal with the same mode as that of ₀. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1–)₀ except in one interval (resp. two disjoint intervals) where it is constant.     

Hyperaffine planes over hyperfields

Summary In this paper we solve the problem of finding suitable conditions for an hyperaffine plane (,) under which a planar hyperfield K exists such that (,) is isomorphic to the hyperaffine plane (K²,').Research partially supported by C.N.R.-G.N.S.A.G.A.     

The arithmetic of the characteristic Pólya functions

In the framework of the theory of D. Kendall's delphic semigroups are considered problems of divisibility in the semigroup of convex characteristic functions on the semiaxis (0,). Letn ()={:₁¦₁1 or ₁=}, and I_o()={: ₁¦ ₁ N()}. The following results are proved: 1) The semigroup is almost delphic in the sense of R. Davidson. 2) N() is a set of the type G which is dense in (in the topology of uniform convergence on compacta). 3) The class I_o() contains only the function identically equal to one.Translated from Matematicheskie Zametki, Vol. 21, No. 5, pp. 717–725, May, 1977.The author thanks I. V. Ostrovskii for the formulation of the problem and valuable remarks.     

K-interpolation in problems of uniform approximation of functions

Given an arbitrary metric compactum Q, we compute the K-functional of a pair (C(Q), C(Q)) and derive two-sided bounds for (C_[–,], C² _[–,]). We prove interpolation theorems and study applications to problems of the theory of approximation.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 44, No. 4, pp. 523–533, April, 1992.     

Asymptotic Characteristics of Compact Sets and Stepanets Classes

We obtain estimates for the -entropy and -capacity of sets of periodic functions with mean value zero that have a (, )-derivative belonging to the space L ²(0, 2).     

Asymptotic estimates of approximation of continuous periodic functions by Fourier sums

Asymptotic estimates, expressed in terms of the value of the modulus of continuity of r-th order (r2) at the point t=/n of a functionf C ₂ or of the (, )-derivative of a functionf C _B C, are established for the deviations of continuous periodic functions from their Fourier sums.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 6, pp. 747–755, June, 1990.     

The dissection of a polygon into nearly equilateral triangles

Every polygon can be dissected into acute triangles. In this paper we prove that every polygon, such that the interior angles are at least /5, can be dissected into triangles with interior angles all less than or equal to 2/5. We find necessary conditions on the interior angles of the polygon in order to obtain a dissection into triangles with interior angles all (where /3<<2/5). The conjecture can be stated that these conditions are also sufficient.     

Criterion for π-supersolvability for finite groups

It is proved that the class of finite -supersolvable groups is precisely the class of all finite -solvable groups with the following property: For each maximal subgroup M of a -solvable group G with index p for some p , there exists a cyclic subgroup S of order p ( ) such that G = MS and S commutes with each element of the Sylow system _M of the subgroup M.Translated from Matematicheskie Zametki, vol. 52, No. 1, pp. 57–61, July, 1992.     

A conjecture on semi-field planes. II

Conclusion The results in the previous sections lend strong support to the conjecture made in the Introduction. Furthermore, if the long-standing conjecture concerning the solvability of autotopism groups for semi-field planes is correct then the probability of our conjecture being true is greatly increased. In any case the existence of a semi-field plane for which u() = 2, 3, or 4 would provide a counterexample to the earlier conjecture.There are examples of semi-field planes with u() = %. As mentioned in Example 2 of Section 3, one of the semi-field planes of order 16 has u()-5. For that plane, the five orbits of the autotopism group G in (G) have lengths 27, 36, 54, 54, 54. The union of the orbit having length 36 and one of those having 54 is the union of the points in (G) on 6 lines through a vertex U and the union of the remaining three orbits consists of the 135 points on the remaining 9 lines through U. There are also non-Desarguesian A-planes in which u() = 5; the semi-field plane of order 3⁴ coordinatized by the twisted field of Albert has u() = 5.Supported in part by NSF Grants No. MPS 75-05260 and MPS 76-06661     

Representation of real discrete Fourier transform in terms of a new set of functions based upon Möbius inversion

The trigonometric functions sin(2n/N) and cos(2n/N) are transformed into a new set of basis functions using Möbius inversion of certain types of series. The new basis functions are number theoretic series. They are used to represent the real discrete Fourier transform (RDFT) in terms of 2 matrices of factorization. The first matrix, with elements 1, -1 and 0 is obtained by replacing cos(2k/N) and sin(2k/N) by (k/N + 1/4) and (k/N), where (x) is the bipolar rectangular wave function. The second matrix is block-diagonal where each block is a circular correlation and consists of the new basis functions. Some applications of the new representation are discussed.     

Generalization of istratescu's theorem on contractive mappings in metric spaces

In this paper we consider mappings T:XX of metric spaces, satisfying the condition: , where is some right semicontinuous function. We prove that if is a nondecreasing function, ()< for >0, –() as ,, then the map T has a fixed point and for any pointxX. Interesting examples are given.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol, 83, pp. 73–82, 1979.     

High-Energy Relations Between Elastic Scattering Amplitudes of Particles and Antiparticles

We find conditions under which the ratio of particle and antiparticle total scattering cross sections as s. If the forward elastic scattering amplitudes become purely real asymptotically, then their ratio tends to –1. We prove that /_–(s)1 for ₊(s)/m ², where m is the -meson mass. We show that the asymptotic relations obtained have finite-energy analogues for some processes.     

Une application de Corps des Normes

Let k be a perfect field of characteristic p > 0, K₀ = Frac(W(k)), a uniformizer in K₀ and _n K ₀ (n N) such that ₀ = and _n+1 ^p = _n. We write K = _nN K₀ (_n), H = Gal (K₀/ K and G = Gal(K₀/ K₀). The main result of this paper is that the functor restriction of the Galois action from the category of crystalline representations of G with Hodge–Tate weights in an interval of length p - 2 to the category of p-adic representations of H is fully faithful and its essential image is stable by sub-object and quotient. The proof uses the comparison between two ways of building mod. p representations of H: one thanks to the norm field of K, the other thanks to some categories of filtered modules with divided powers previously introduced by the author.     

Mappings that preserve cones in Lobachevskii space

Let ⁿ be n-dimensional Lobachevskii space, and {l_x:x ⁿ} be a family of lines, parallel to a linel ₀, 0ⁿ (in a given direction). Let {c_x:Xⁿ} be a family of circular cones in ⁿ of opening with axes l_X and vertex X. Then, iff:ⁿⁿ(n>2) is a bijective mapping andf(C_x)=C _f(x), it follows thatf is a motion in the space ⁿ.Translated from Matematicheskie Zametki, Vol. 13, No. 5, pp. 687–694, May, 1973.