Directed packings with block size 5 and even ν

Let andk be positive integers. A transitively orderedk-tuple (a ₁,a ₂,...,a _k) is defined to be the set {(a _i, a_j) 1i<jk} consisting ofk(k–1)/2 ordered pairs. A directed packing with parameters ,k and index =1, denoted byDP(k, 1; ), is a pair (X, A) whereX is a -set (of points) andA is a collection of transitively orderedk-tuples ofX (called blocks) such that every ordered pair of distinct points ofX occurs in at most one block ofA. The greatest number of blocks required in aDP(k, 1; ) is called packing number and denoted byDD(k, 1; ). It is shown in this paper that for all even integers , where [x] is the floor ofx.     

Limits of balayage measures

LetA be a subset of a balayage space (X,W) and a measure onX. It is shown that for every sequence _n of measures such that lim_nn and lim_{n n} ^A = the limit measure is of the formf+[(1-f)]^A for some (unique) Borel function 0f1_Cb(A). Furthermore, conditions are given such that any such functionf occurs.     

Compatible Almost Complex Structures on Quaternion Kähler Manifolds

Let (M⁴ⁿ,g,Q) be a quaternion Kähler manifold with reduced scalar curvature = K/4n(n + 2). Suppose J is an almost complex structure which is compatible with the quaternionic structure Q and let = – F J be the Lee form of J. We prove the following local results: (1) if J is conformally symplectic, then it is parallel and = 0; (2) if J is cosymplectic, then 0 with equality if and only if J is parallel; (3) if J is integrable, then d is Q-Hermitian and harmonic; and (4) any closed self-dual 2-form = f(g J) ² ₊ = g Q ² is parallel. In Section 5, extending previous results of Salamon [24], we describe a correspondence among conformally balanced J, Killing vector fields X and self-dual 2-forms satisfying the twistor equation.When M⁴ⁿ is compact our main global results are the following: (1) if > 0, then there exists no compatible almost complex structure J; (2) if the first Chern class c₁(T^(1,0) _J M) = 0, then = 0; (3) if = 0 a compatible complex structure J is parallel; and (4) if 0, then no compatible complex structure J exists. The last two results have been proved in [23] by twistor methods.     

Fourier series of additive vector measures and their term-by-term differentiation

On a measurable space (T, , ) we choose an additive measure: Z (Z is a Banach space) with the following property: for alle , we have ; this measure defines an indefinite integral over the measure onL ² (T, ,). We prove that if { _n (t)} _n ^=1/ is an orthonormal basis inL ² and _n (e)=e _n (t) d, then any additive measure: Z whose Radon-Nikodým derivatived/d belongs toL ² is uniquely expandable in a series(e)= _n ^=1/ _n n(e) that converges to(e) uniformly with respect toe can be differentiated term-by-term, and satisfies _n ^=1/ _n ^/2 <. In the caseL ²[0,2],Z=, the Fourier series of a 2-periodic absolutely continuous functionF(t) such thatF'(t) L ²[0, 2] is superuniformly convergent toF(t).Translated fromMatematicheskie Zametki, Vol. 64, No. 2, pp. 180–184, August, 1998.     

Threshold theorems for a stochastic model of an epidemic with natural immunization

A stochastic model of an epidemic is investigated, taking account of the removal of ill members of the population (by death, by recovery with immunization, by isolation) and natural immunization. Limiting distributions are found for the size of the epidemic, the number immunized ₁, and their sum, under the assumption that the original number of susceptible individuals n and the number of ill individuals m , while n 1,n ₀< , where and are the coefficients for the contraction of the disease and of immunization respectively.Translated from Matematicheskie Zametki, Vol. 8, No. 3, pp. 385–392, September, 1970.     

Decktransformationen transnormaler Mannigfaltigkeiten

For every transnormal m-manifold V (see [3] or [7]) in ⁿ :VW, mapping pV into its normal plane (p) is a covering map onto a submanifold W of the open Grassmannian H_n,n–m of all (n–m)-dimensional planes in ⁿ. The transnormal frame T:=^–1((p)) admits a transitive operation by a group J of isometries. The group action of the covering transformations of (V,,W) on T commutes with the action of J. The elements of J, which are restrictions of covering transformations to T, are exactly the elements of the centre of J. This property is applied to show the existence of nontrivial covering transformations of (V,,W) for n–m3.

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Diese Arbeit faßt die Kapitel 5, 6 und 7 der von der Fakultät für Allgemeine Ingenieurwissenschaften der TU Berlin genehmigten Dissertation [6] zusammen.     

Integral points on strictly convex closed curves

A negative answer is given to Swinnerton-Dyer's question: Is it true that for any > 0 there exists a positive integer n such that for any planar closed strictly convex n-times differentiable curve , when it is blown up a sufficiently large number of times, the number of integral points on the resultant curve will be less than . An example has been constructed when this number for an infinite number is not less than ^1/2, while is infinitely differentiable.Translated from Matematicheskie Zametki, Vol. 21, No. 6, pp. 799–805, June, 1977.The author thanks S. B. Stechkin for attention to the work.     

On the approximation of discrete functions by fourier-hahn sums

— f: {0,1, ..., N–1}R. .     

Equation de Choquet-Deny sur le dual d'un groupe compact

Résumé On étudie les solutions de l'équation de convolution *= où et sont respectivement un poids et un état sur l'algèbre de von Neumann d'un groupe compactG, la convolution * étant déterminée par la structure d'algèbre de Hopf de cette algèbre de von Neumann.

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Summary We study the solutions of the convolution equation *= where and are a weight and a state on the von Neumann algebra of a compact group, and the convolution is given by the Hopf algebra structure of this von Neumann algebra.

     

Weak Sequential Completeness of β-duals of Double Sequence Spaces

We consider dual pairs E,E ⁽⁾ of double sequence spaces E and E ⁽⁾, where E ⁽⁾ is the -dual space of E with respect to the -convergence of double sequences for = p (Pringsheim convergence), bp (bounded p-convergence) and r (regular convergence). Motivated by Boos, Fleming and Leiger [3], we introduce two oscillating properties (signed P_OSCP(k), k {1,2}) for a double sequence space E such that the signed P_OSCP(1) guarantees the (E ^(p), E)-sequential completeness of E ^(p), whereas the signed P_OSCP(2) implies the equalities E ^(r) = E ^(bp) = E ^(p) and the (E ⁽⁾, E)-sequentialcompleteness of E ⁽⁾ for = bp and r.     

Über beschränkte Mengen in induktiven Limiten topologischer Vektorräume

In topological linear spaces convex hulls of bounded sets are, in general, not bounded. The question arises whether there is at least for every bounded set B a sequence {|} of strictly positive numbers such that the set { _l ⁿ _v B|n} is bounded. When this obtains, bounded sets share several of the properties known in locally convex spaces. The main result of this note is an example of a countable inductive limit of complete metrizable topological linear spaces which is neither regular nor sequentially complete and also fails to have the above bounded summability property.     

On the immersion of metrics close to immersible metrics

For a C^r,-immersion z:X E, r 2, 0 < < 1, of an n-dimensional (n 1) simply-connected C^r+2,-manifold X into Euclidean space E, the metric I(z) induced by z has a neighborhood in C^r,-topology in which every metric from a given subbundle of metrics is C^r,-immersible into E. In particular, it is proved that metric ds ₀ ² of the Riemannian product of p spheres of dimensions ₁, , _p 2 has a neighborhood in C^2,-topology from which any conformally equivalent metric to ds ₀ ² , is immersible into E with dimE = ₁ + + _p + p. The proofs are based on the investigation of a varied system of Gauss—Codazzi—Ricci equations for an infinitely small deformation of surface z(X) in E with a prescribed variation of the metric.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 49–67, 1992.     

On the E- and MV-optimality of block designs having k≥v

In this paper we consider the problem of determining and constructing E- and MV-optimal block designs to use in experimental settings where treatments are applied to experimental units occurring in b blocks of size k, k. It is shown that some of the well-known methods for constructing E- and MV-optimal unequally replicated designs having k fail to yield optimal designs in the case where . Some sufficient conditions are derived for the E- and MV-optimality of block designs having and methods for constructing designs satisfying these sufficient conditions are given.     

On the strongly spherical Sasakian metric of a spherical tangent bundle

Let Mⁿ denote an n-dimensional Riemannian manifold. Its metric is called -strongly spherical if at every point Q Mⁿ there exists a -dimensional subspace _Q T_QMⁿ such that the curvature operator of the metric of Mⁿ satisfies R(X, Y) Z = k(< Y, Z > X < X, Z > Y), where k = const > 0, Y _Q , X, Z #x2208; T_QMⁿ. The number is called the index of sphericity and k the exponent of sphericity. The following theorems are proved in the paper.THEOREM 1. Let the Sasakian metric of T₁Mⁿ be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if M² has constant Gaussian curvature K 1 and k = K²/4; b) = 3 if and only if M² has constant curvature K = 1 and k = 1/4; c) = 0, otherwise.THEOREM 2. Let the Sasakian metric of T₁Mⁿ (n Mⁿ) be -strongly spherical with exponent of sphericity k. If k > 1/3 and k 1, then = 0. Let us denote by (Mⁿ, K) a space of constant curvatureK. THEOREM 3. Let the Sasakian metric of T₁(Mⁿ, K) (n 3) be -strongly spherical with exponent of sphericity k. The following assertions hold: a) = 1 if and only if K = 1/4; b) = 0, otherwise. In dimension n = 3 Theorem 2 is true for k {1/4, 1}.Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 150–159, 1992.     

Geometric methods in the study of irregularities of distribution

Let be a signed measure on E ^d with E ^d =0 and ¦¦E^d<. DefineD _s() as sup ¦H¦ whereH is an open halfspace. Using integral and metric geometric techniques results are proved which imply theorems such as the following.Theorem A. Let be supported by a finite pointsetp _i. ThenD _s()>c _d(₁/ ₂)^1/2{ _{i(p i})²}^1/2 where ₁ is the minimum distance between two distinctp _i, and ₂ is the maximum distance. The numberc _d is an absolute dimensional constant. (The number .05 can be chosen forc ₂ in Theorem A.)Theorem B. LetD be a disk of unit area in the planeE ², andp ₁,p ₂,...,p _n be a set of points lying inD. If m if the usual area measure restricted toD, while nP _i=1/n defines an atomic measure _n, then independently of _n,nD _s(m – _n) .0335n ^1/4. Theorem B gives an improved solution to the Roth disk segment problem as described by Beck and Chen. Recent work by Beck shows thatnD _s(m – _n)cn ^1/4(logn)^–7/2.     

Canonical and Boundary Representations on a Hyperboloid of One Sheet

For the hyperboloid of one sheet X=G/H, G=SO₀(1,2), H=SO₀(1,1), canonical representations R _,, C, =0,1, are defined as the restrictions to G of representations of the overgroup =SO₀(2,2) associated with a cone. They act on the torus containing two copies of X as open G-orbits. We study boundary representations generated by R _,. For some , they contain Jordan blocks. The decomposition of R _, into irreducible constituents includes a finite number (depending on ) of irreducible parts of the boundary representations.     

Sources in posets and comparability graphs

A vertex in a poset is a source if its indegree is zero. Further, a vertex in a comparability graph G is a source if there is a transitive orientation of G in which is a source. We characterize sources in comparability graphs in terms of forbidden subgraphs. Certain results follow, including a brief proof of a theorem by Olariu.     

Zeros of the densities of infinitely divisible measures on R n

Let be an infinitely divisible probability measure onR ⁿ without Gaussian component and let be its Lévy measure. Suppose that is absolutely continuous with respect to the Lebesgue measure . We investigate the structure of the set ⁿ of admissible translates of . This yields a unified presentation of previously known results. We also show that if(S)>0 then is equivalent to , under the assumption that supp =R ⁿ , whereS is the closure of the semigroup generated by the support of .The research of this author is supported by KBN Grant.The research of this author is supported by AFSOR Grant No. 90-0168, and the University of Tennessee Science Alliance, a State of Tennessee Center of Excellence.     

Central Limit Theorem for a Class of Nonhomogeneous Random Walks

A spatially nonhomogeneous random walk _t on the grid =^m X ⁿ is considered. Let _t ⁰ be a random walk homogeneous in time and space, and let _t be obtained from it by changing transition probabilities on the set A= X ⁿ, || < , so that the walk remains homogeneous only with respect to the subgroup ⁿ of the group . It is shown that if >m 2 or the drift is distinct from zero, then the central limit theorem holds for _t.     

Some Properties of Isologism of Groups and Baer-Invariants

Let be a variety of groups defined by the set of laws V. In this paper we study the concept of -isologism of groups in terms of -extensions and their connections with the Baer-invariant of groups are also discussed.AMS Subject Classification (2000): primary 20F14, 20F19, secondary 20E10