Planes and ovals

In the present paper two results are essentially given. The first one concerns the isomorphism between a projective plane , endowed of sufficiently many elations, and a projective plane whose blocks are ovals of . The second one shows that every oval of a desarguesian plane can be interpreted as a conic of a suitable plane isomorphic to .Lavoro eseguito nell'ambito del Gruppo Nazionale per le Strutture Algebriche e Geometriche e loro Applicazioni del C.N.R.     

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.     

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.     

On the Absolute Convergence of Fourier Series

The necessary and sufficient conditions of the absolute convergence of a trigonometric Fourier series are established for continuous 2-periodic functions which in [0, 2] have a finite number of intervals of convexity, and whose nth Fourier coefficients are O((l/n; f)/n), where ( f) is the continuity modulus of the function f.     

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.     

Classes caractéristiques d''une π-algèbre et suite spectrale en K-théorie bivariante

Résumé Pour tout groupe discret et pour toute -algèbre D, la C ^*-algèbre D(E) (dont la définition exacte est donnée dans la section 4) est la version équivariante de la C ^*-algèbre C(B, D) des fonctions continues sur B, le classifiant du groupe, à valeurs dans D et qui s'annulent à l'infini. Si D désigne une autre -algèbre, nous définissons une suite spectrale en K-théorie bivariante dont les premiers termes sont donnés par les groupes H ^p (B, KK(D, D)) et qui converge (lorsque B est de dimension finie) vers KK(B; D(E), D(E)). Cette suite spectrale généralise celle de Kasparov mais est obtenue de manière différente: en étendant la définition des quasihomomorphismes aux C(X)-algèbres (X est une espace topologique localement compact), on a recours à des méthodes homotopiques telles les décompositions de Postnikov et le calcul des groupes d'homotopie des espaces d'équivalences d'homotopie. Sous certaines hypothèses, ces mÊmes constructions nous permettent de définir, pour toute -algèbre D, une obstruction, appelée classe secondaire de la -algèbre D, qui détermine la différentielle d ₂ de la suite spectrale de Kasparov.

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For all discrete group and all -algebra D, the C ⁺-algebra D(E) (whose exact definition is given in Section 4) is the equivariant version of the C ^*-algebra C(B, D) of continuous functions from B (the classifiant of the group) to D, vanishing at infinity. If D is another -algebra, we define a spectral sequence in bivariant K-theory whose first terms are given by the groups H ^p (B, KK(D, D)) and which converges (if B of finite dimension) to KK(B; D(E), D(E)). This spectral sequence generalises the spectral sequence given by Kasparov but it is obtained in a quite different way: by extending the definition of quasihomomorphisms to the C(X)-algebras (where X is a locally compact topological space), we use homotopical methods, like Postnikov decompositions and the calculus of homotopy groups of spaces of homotopy equivalences. Furthermore, under certain hypotheses, with these constructions, we define an obstruction, called the secondary class of the -algebra D, which determines the differential d ₂ of the Kasparov spectral sequence.

     

227-01227-01227-01-property and normal subgroups

Summary D-property (=set of primes) in finite groups is not in general inherited by subgroups. In this paper, as evidence in favor of the following conjecture (F. Gross): (o) If a finite group G satisfies D then its normal subgroups satisfy D-property as well. the Author shows that if the D and the D-properties (=set of the primes not in ) hold together in a finite group G, then both are inherited by the normal subgroups of G. As a corollary, the characterization of the groups satisfying both the properties D and D is given in terms of the composition factors.     

1-formes fermées singulières et groupe fondamental

Summary We study the influence of the ₁ of a closed manifoldM ⁿ (n3) on the foliations ofM defined by closed differential 1-forms with Morse singularities (of index 0,n). Every nonexact form is cohomologous to a weakly complete one, that is one whose leaf space is of the same type as that of a nonsingular form. Generically, a form has compact leaves or is weakly complete. If ₁ M has no quotient isomorphic to *, then every nonexact form onM is weakly complete. We also say a form is complete if every path inM is homotopic to either a path transverse to or a path contained in a leaf of . Completeness of depends only on its de R ham cohomology class. The set of complete cohomology classes depends only on ₁ M and is related to finitely generated normal subgroups of ₁ M with quotient . If ₁ M is nilpotent (or even polycyclic), every nonexact form onM is complete. On irreducible 3-manifolds, a form is complete iff it is cohomologous to a nonsingular one.     

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.     

Generating alternating permutations lexicographically

A permutation _{1 2} ... _n is alternating if ₁<₂>₃<₄ .... We present a constant average-time algorithm for generating all alternating permutations in lexicographic order. Ranking and unranking algorithms are also derived.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

Finite presentability of Steinberg groups over group rings

LetA be a finitely generated commutative -algebra with Krull dimensiond, and let be an arbitrary finite group. It is proved that the Steinberg groupSt _n (A) is finitely presented whenevern4. If, in addition,nd+3, andK ₁ (A) andK ₂ (A) are finitely generated, thenE _n (A) andGL _n (A) are finitely presented.The Project supported by National Natural Science Foundation of China.     

Best approximation of periodic functions by trigonometric polynomials in L2

Estimates are gotten for the best approximations in L₂ (0, 2) of a periodic function by trigonometric polynomials in terms of its m-th continuity modulus or in terms of the continuity modulus of its r-th derivative. The inequality E_n–1(f)L₂<(C _2m ^m _m(2/n;f)L₂(1 const) is proved, where the constant (C _2m ^m )^–1/2 is unimprovable for the whole space L₂ (0, 2). Two titles are cited in the bibliography.Translated from Matematicheskie Zametki, Vol. 2, No. 5, pp. 513–522, November, 1967.     

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.     

Series ∑F(m)qm,where F(m) is the number of odd classes of binary quadratic forms of determinant −m

Consideration of the analytic continuation of the Eisenstein series of weight 3/2 for the group ₀(4) leads to a new proof of Mordell's formula connecting the values X()= _m=1 F(m)e^im, Im > 0, and (–1/). The behavior of the function () for ₀(4) is examined by the same method.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 64, pp. 69–79, 1976.     

Transposition generation of alternating permutations

A permutation _{1 2}... _n is alternating if ₁< ₂> ₃< ₄.... Alternating permutations are counted by the Euler numbers. Here we show that alternating permutations can be listed so that successive permutations differ by a transposition, ifn is odd. Extensions and open problems are mentioned.Research supported by the Natural Sciences and Engineering Research Council of Canada under grant A3379.     

Isolated points in duals of certain locally compact groups

Summary LetG be a separable locally compact group with dual space. consists of all equivalence classes of irreducible unitary representations ofG, and is endowed with the Fell-topology. We study the topological properties in of the square-integrable representations ofG. [ is square-integrable provided there is a coordinate functiong((g)v, v),gG, for which is inL ²(G) w.r.t. left Haar measure onG.]SupposeG contains an open normal subgroupN of the formeKN ⁿ e whereK is compact. (All groups with a compact invariant neighborhood of the identity, [IN] groups, satisfy this condition.) In this case we show that if is square-integrable then {} is an open point of.Finally, our techniques are used to prove this result for arbitrary (non connected) nilpotent Lie groups.     

Relative komplexe quotienten

Let XS be a holomorphic map, and let RX×SX be an equivalence relation. The restriction of R to the fibre ^–1(S) is denoted by R_s. The quotient X/R is called a relative complex quotient, if the quotient map XX/R is holomorphic over S. Two cases are studied: (C) All fibres of are locally R_s-separable (relative Cartan quotient); (R) All fibres of are holomorphically convex, and R_s is given by tke holomorphic functions on ^–1 (s) (relative Remmert quotient).     

The Number of Isomorphism Classes of Finite Groups with Given Element Orders

Let G be a finite group and _e(G) the set of element orders of G. Denote by h( _e(G)) the number of isomorphism classes of finite groups H satisfying _e(H) = _e(G). We prove that if G has at least three prime graph components, then h( _e (G)){1, }.     

Optimal Control of the Periodic String Equation with Internal Control

This paper is concerned with the existence and the maximum principle for optimal control problems governed by the periodic vibrating string equation on (0, )×(0, T) with Dirichlet boundary conditions. The case of internal controllers supported on (0, ) is examined.