A Helly-type theorem for simple polygons

Let be a family of simple polygons in the plane. If every three (not necessarily distinct) members of have a simply connected union and every two members of have a nonempty intersection, then {P:P in } . Applying the result to a finite family of orthogonally convex polygons, the set {C:C in } will be another orthogonally convex polygon, and, in certain circumstances, the dimension of this intersection can be determined.Supported in part by NSF grant DMS-9207019.     

Exponential approximations in the classes of life distributions

In this paper we discuss the approximation of life distributions by exponential ones. The main results are: (1) F NBUE, where its mean is 1, we have , 0, where = 1 - ₂/2, ₂ being the second moment ofF. The inequality is sharp. (2) In the case ofFIFR, the upper bound is . (3) For the HNBUE class, the upper bound is min . Furthermore, the improved upper bound is . In addition, we show 0} |\bar G(t) - e^{ - t} | \leqslant \sqrt {\frac{\rho }{2}} $$ " align="middle" border="0"> , where (4) For the IMRL class, the upper bound is /(1+) ([1]). Here we give a simple proof.Project supported by the National Natural Science Fund of China.     

Lower bounds for faithful,preinjective modules

Let be the path algebra for some representation-infinite quiver over some field k. There exists a bound such that ^mI is faithful for all indecomposable injective -modules I and all , and such that there exists an indecomposable injective -module J satisfying that J is not faithful, denotes the Auslander-Reiten-translation. Let m() be the maximum of the taken over all possible orientations of the underlying graph . In this article we determine the bounds m() for representation-infinite quivers for which is a tree.     

Sechzehndimensionale lokalkompakte Translationsebenen,deren Kollineationsgruppe G2 enthält

New results of Salzmann and Hubig say that a 16-dimensional (locally) compact topological projective plane in which the group of continuous collineations has dimension 40 is a translation plane. It is therefore important to determine all 16-dimensional locally compact translation planes with dim 40. From previous work of the author ([10]), it is known that such a plane is either the classical octonion plane, or dim =40 and contains a subgroup isomorphic to the compact exceptional group G₂, but no larger compact simple subgroup. In the present paper, all planes satisfying the latter property more generally with dim 38 are explicitly determined. Together with the classification of all 16-dimensional locally compact translation planes in which contains Spin(7) given by the author in [8], one thus knows all 16-dimensional locally compact translation planes with containing G₂ and dim 38. Via suitable Baer subplanes, the classification makes use of analogous results for 8-dimensional planes ([7]).

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Meinem verehrten Lehrer Helmut Salzmann zum 60. Geburtstag     

Right distributive quasigroups on algebraic varieties

In this paper we develop a structure theory of algebraic right distributive quasigroups which correspond to closed and connected conjugacy classes generating algebraic Fischer groups (in the sense of [6]) such that the mappingx x ^–1 ax, fora , is an automorphism of (as variety). We also give examples of algebraic Fischer groups where this does not happen. It becomes clear that the class of algebraic right distributive quasigroups has nice properties concerning subquasigroups, normal subquasigroups and direct product.We give a complete classification of one- and two-dimensional as well as of minimal algebraic right distributive quasigroups.     

Extremal problems of Bloch function spaces

Letf be analytic in a hyperbolic region . The Bloch constant _f off is defined by , where (z)|dz| is the Poincaré metric in . Suppose is hyperbolic and where . Then for allf withf() , we have _f 1/(). In this paper we study the extremal functions defined by _f =1/() and the existence of those functions.Supported by the National Natural Science Foundation of China.     

Four-dimensional compact projective planes with doubly transitive action on the fixed pencil

We consider a four-dimensional compact projective plane =( , ) whose collineation group is six-dimensional and solvable with a nilradical N isomorphic to Nil × R, where Nil denotes the three-dimensional, simply connected, non-Abelian, nilpotent Lie group. We assume that fixes a flag pW, acts transitively on _p \{W}, and fixes no point in the set W{p}. We study the actions of and N on and on the pencil _p \{W}, in the case that does not contain a three-dimensional elation group. In the special situation that acts doubly transitively on _p {W}, we will determine all possible planes . There are exactly two series of such planes.     

Blocking sets and partial spreads in finite projective spaces

A t-blocking set in the finite projective space PG(d, q) with dt+1 is a set of points such that any (d–t)-dimensional subspace is incident with a point of and no t-dimensional subspace is contained in . It is shown that | |q ^t +...+1+q ^t–1q and the examples of minimal cardinality are characterized. Using this result it is possible to prove upper and lower bounds for the cardinality of partial t-spreads in PG(d, q). Finally, examples of blocking sets and maximal partial spreads are given.     

Symmetric planes with non-classical tangent translation planes

We construct symmetric planes associated with an arbitrary locally compact connected nearfield . If is a proper nearfield, i.e. {;;}, then the tangent translation plane of this symmetric plane is not classical. All previously known examples of symmetric planes have classical tangent translation planes.Herrn Professor Dr. H. Salzmann zum 65. Geburtstag gewidmet     

Lattices of interpretability types of varieties

Let be the set of all primes, the field of all algebraic numbers, and Z the set of square-free natural numbers. We consider partially ordered sets of interpretability types such as , and , where AD is a variety of -divisible Abelian groups with unique taking of the pth root _p(x) for every p , is a variety of -modules over a normal field , contained in , and G_n is a variety of n-groupoids defined by a cyclic permutation (12 ...n). We prove that , and are distributive lattices, with and where ub and ub_f are lattices (w.r.t. inclusion) of all subsets of the set and of finite subsets of , respectively.Deceased.__________Translated from Algebra i Logika, Vol. 44, No. 2, pp. 198–210, March–April, 2005.     

Bounds for the lattice point enumerator

For the lattice point enumerator of a lattice and a convex body K we give bounds in terms of the intrinsic volumes of K and of minimal determinants of . The intrinsic volumes are the normalized Minkowski quermassintegrals and the minimal determinants are analogous functionals of .     

Théorème de division avec des conditions au bord

Let be an open subset of ⁿ and be a subalgebra of the algebra of analytic functions on . We suppose that satisfies some weak conditions of noetherianity such that we can construct a finite stratification for each ideal of . We also suppose that satifies global £ojasiewicz's inequalities. We prove the following: Let andf C on flat on ; if for eacha the Taylor's serie off ata, T _a f, is in the ideal generated byT _a f ₁,...,T _a f _p in the ring of formal power series, then there exist ₁,..., _p ,C on flat on such that . This result extends the classic Hormander's theorem of division (for a polynomial) or the £ojasiewicz-Malgrange theorem in the local analytic case.Reherches menées dans le cadre du Programme d'Appui à la Recherche Scientifique (PARS MI 33)     

Commutator-finite orthomodular lattices

The class of orthomodular lattices which have only finitely many commutators is investigated. The following theorems are proved: contains the block-finite orthomodular lattices. Every irreducible element of is simple. Every element of is a direct product of a Boolean algebra and finitely many simple orthomodular lattices. The irreducible elements of which are modular, or are M-symmetric with at least one atom, have height two or less.     

Infimal generators and dualities between complete lattices

Summary We give some applications of infimal generators of complete lattices, in the sense of Kutateladze and Rubinov [12], to the study of dualities between two complete lattices E and F (i.e., mappings : E F satisfying for all and all index sets I, including the empty set I=Ø). We give some additional results for E=(2^X, ), F=(2^W, ) and E=(¯R^X, ), F=(¯R^W, ) (where X and W are arbitrary sets), with suitable families of infimal generators. We obtain some lattice- theoretic properties of the relations of [22] between dualities : (2^X, ) (2^W, ), binary relations X×W, polarities (): (2^X, ) (2^W, ), coupling functionals : X×W ¯R and Fenchel-Moreau conjugations c():(¯R^X, ) (¯R^W,).     

Packing convex bodies in the plane with density greater than 3/4

Conclusion By an argument of a topological nature, the Theorem whose proof has just been completed can be somewhat strengthened. Since the collection of affine equivalence classes of all convex plane bodies of area 1 is a compact set, and since the function assigning to each such equivalence class the minimum area of a p-hexagon containing a representative of that class is continuou s,there exists a minimum value for that function, taken on a specific element of that compact set. Let us denote that minimum value by . We have proved in this , thus we can conclude that there exists a number \frac{3}{4}$$ " align="middle" border="0"> (namely ) such that every convex body can be packed in the plane with density at least d. The value of A remains unknown.     

Groupwise density and related cardinals

We prove several theorems about the cardinal associated with groupwise density. With respect to a natural ordering of families of nond-ecreasing maps from to, all families of size are below all unbounded families. With respect to a natural ordering of filters on, all filters generated by sets are below all non-feeble filters. If then and . (The definitions of these cardinals are recalled in the introduction.) Finally, some consequences deduced from by Laflamme are shown to be equivalent to .     

On a condition on the radical of a Banach algebra ensuring strong decomposability

The main result is the following theorem. Let be a commutative Banach algebra with radical R, where the factor algebra is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If rⁿ¹ /n 0 as n uniformly for r R, rl, then the algebra is strongly decomposable, i.e., there exists a closed subalgebra B isomorphic to such that =BR.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed . There are 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 589–592, December, 1967.     

On the convergence of powers of interval matrices (2)

Summary Let be a real irreduciblen×n interval matrix. Then a necessary and sufficient condition is given for the sequence of the powers of an interval matrix to converge to a matrix which is not the null matrix. In addition a criterion for is proved to decide whether the limit matrix satisfies the condition of symmetry .     

Tissus linéaires et théorèmes d'algébrisation de type Abdel-inverse et Reiss-inverse

A d-web in ( ,0) is given by d complex analytic foliations of codimension one in ( ,0) which are in general position. A d-web in ( ,0) is linear if all the leaves are (pieces of) hyperplanes in and is algebraic if it is associated, by duality, to a nondegenerate algebraic curve in of degree d. We characterize linear webs in ( ,0). We give explicit conditions under which a linear d-web in ( ,0) is algebraic and we obtain equations for in this case. Some related problems are discussed and some questions are posed.     

The Martin boundary and completion of Markov chains

Summary In this paper we treat a time-symmetrical Martin boundary theory for continuous parameter Markov chains. This is done by reversing the time sense of a Markov chainX _t in such a way as to obtain a dual Markov chain , and considering the two chains together. Various relations between the Martin exit boundaries and of these processes are studied. The exit boundary of , is in a sense an entrance boundary forX _t and vice versa. After a natural identification of certain points in and one can topologizeI in such a way thatboth X _t and have standard modifications in this space which are right continuous, have left limits, and are strongly Markov.Research supported in part at Stanford University, Stanford, California under AFOSR 0049.