Symplectic Action Around Loops In Ham(<Emphasis Type="Italic">M</Emphasis>)

Let Ham(M) be the group of Hamiltonian symplectomorphisms of a quantizable, compact, symplectic manifold (M, ω). We prove the existence of an action integral around loops in Ham(M), and determine the value of this action integral on particular loops when the manifold is a coadjoint orbit.     

Bogolyubov averaging and normalization procedures in nonlinear mechanics. IV

In this paper, we apply the theory developed in parts I-III [Ukr. Math. Zh.,46, No. 9, 1171–1188; No. 11, 1509–1526; No. 12, 1627–1646 (1994)] to some classes of problems. We consider linear systems in zero approximation and investigate the problem of invariance of integral manifolds under perturbations. Unlike nonlinear systems, linear ones have centralized systems, which are always decomposable. Moreover, restrictions connected with the impossibility of diagonalization of the coefficient matrix in zero approximation are removed. In conclusion, we apply the method of local asymptotic decomposition to some mechanical problems.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 8, pp. 1044–1068, August, 1995.This research was partially supported by the International Science Foundation, grant No. UB2000.     

The Structural Nature of the Nerve Functor for n-Groupoids

Given a left exact category B, the construction of the nerve functor _n for n-groupoids in B is related to a certain property of the category S-S i m p l _{n – 1} B of the split (n – 1)-truncated simplicial objects in B, which allows us to define the split n-truncated simplicial objects in B completely internally to S-S i m p l _{n – 1} B and thus to construct intrisincally from it the category S-S i m p l _n B.     

A characterization of discrete unimodality with applications to variance upper bounds

Bertin and Theodorescu (1984,Statist. Probab. Lett.,2, 23–30) developed a characterization of discrete unimodality based on convexity properties of a discretization of distribution functions. We offer a new characterization of discrete unimodality based on convexity properties of a piecewise linear extension of distribution functions. This reliance on functional convexity, as in Khintchine's classic definition, leads to variance dilations and upper bounds on variance for a large class of discrete unimodal distributions. These bounds are compared to existing inequalities due to Muilwijk (1966,Sankhy, Ser. B,28, p. 183), Moors and Muilwijk (1971,Sankhy, Ser. B,33, 385–388), and Rayner (1975,Sankhy, Ser. B,37, 135–138), and are found to be generally tighter, thus illustrating the power of unimodality assumptions.     

Normal quasi-solutions of the integral equations of the theory of elasticity

To find solutions of integral equations of first kind in the boundary value problems of the theory of elasticity we use a variational approach connected with the minimization of the discrepancy function on a compact set. We prove that the problem is well-posed in the sense of Hadamard, Bibliography: 4 titles.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 30, 1989, pp. 17–20.     

A Multivariate CLT for Decomposable Random Vectors with Finite Second Moments

Stein's method is used to derive a CLT for dependent random vectors possessing the dependence structure from Barbour et al. J. Combin. Theory Ser. B 47, 125–145, but under the assumption of second moments only. This allows us to derive Lindeberg–Feller type theorems for sums of random vectors with certain dependence structures. We apply the main theorem to the study of three problems: local dependence, random graph degree statistics and finite population statistics. In particular, we consider U-statistics of independent observations as well as of observations drawn without replacement.     

Eigenvalue bounds in the gaps of Schrödinger operators and Jacobi matrices

We consider C=A+B where A is selfadjoint with a gap (a,b) in its spectrum and B is (relatively) compact. We prove a general result allowing B of indefinite sign and apply it to obtain a (δV)^d/2 bound for perturbations of suitable periodic Schrödinger operators and a (not quite) Lieb–Thirring bound for perturbations of algebro-geometric almost periodic Jacobi matrices.     

The Yoneda Algebra of a Graded Ore Extension

Let A be a connected-graded algebra with trivial module 𝕜, and let B be a graded Ore extension of A. We relate the structure of the Yoneda algebra E(A): = Ext _A (𝕜, 𝕜) to E(B). Cassidy and Shelton have shown that when A satisfies their 𝒦₂ property, B will also be 𝒦₂. We prove the converse of this result.     

Weighted Kaplan-Meier Tests for Umbrella Alternatives

The problem of testing for umbrella alternatives in a one-way layout with right-censored survival data is considered. Testing procedures based on the two-sample weighted Kaplan-Meier statistics suggested by Pepe and Fleming (1989, Biometrics, 45, 497–507; 1991, J. Roy. Statist. Soc. Ser. B, 53, 341–352) are suggested for both cases when the peak of the umbrella is known or unknown. The asymptotic relative efficiency of the weighted Kaplan-Meier test and the weighted logrank test proposed by Chen and Wolfe (2000, Statist. Sinica, 10, 595–612) is computed for the umbrella peak-known setting where the piecewise exponential survival distributions have the proportional or crossing hazards, or the related hazards differ at early or late times. Moreover, the results of a Monte Carlo study are presented to investigate the level and power performances of the umbrella tests. Finally, application of the proposed procedures to an appropriated data set is illustrated.     

Zur Theorie der Schwingungen

Ohne ZusammenfassungDie Kenntnis der beiden früheren Mitteilungen (Monatsh. f. Math. u. Phys.37, 1930, S. 325–342 und39, 1932, S. 173–220) wird nicht vorausgesetzt.     

Paley–Wiener and Boas theorems for singular Sturm–Liouville integral transforms

This paper deals with a class of integral transforms arising from a singular Sturm–Liouville problem y″−q(x)y=−λy, x(a,b), in the limit-point case at one end or both ends of the interval (a,b). The paper completely solves the problem of characterization of the image of a function that has compact support (Paley–Wiener theorem) and also of a function that vanishes on some interval (Boas problem) under this class of transforms. The characterizations are obtained with no restriction on q(x) other than being locally integrable.     

Contraction–Deletion Invariants for Graphs

We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(G/e)+T(G−e) for eE(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc.43, 26–40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z. Extending results of Tutte (1974, J. Combin. Theory Ser. B16, 168–174) and Brylawski (1981, J. Combin. Theory Ser. B30, 233–246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this.     

Idempotent probabilities on semigroups

Summary In this paper, idempotent probability measures have been considered on semigroups which are locally compact or metric and satisfy: (*) A ^–1 B and Ax ^–1 are compact whenever A and B are so, for every x in the semigroup. Such semigroups are more general than compact semigroups which do admit of such measures. On such semigroups we can construct such measures by the usual process if there is a compact sub-semigroup. It is shown in this paper that if such a measure exists in such semigroups, then it must be such an extension measure. Some related results concerning the conditions (*) are also discussed here.     

Self-packing of centrally symmetric convex bodies in ℝ2

LetB be a compact convex body symmetric around0 in ℝ² which has nonempty interior, i.e., the unit ball of a two-dimensional Minkowski space. The self-packing radiusρ(m,B) is the smallestt such thatt B can be packed withm translates of the interior ofB. Form≤6 we show that the self-packing radiusρ(m,B)=1+2/α(m,B) whereα(m,B) is the Minkowski length of the side of the largest equilateralm-gon inscribed inB (measured in the Minkowski metric determined byB). We showρ(6,B)=ρ(7,B)=3 for allB, and determine most of the largest and smallest values ofρ(m,B) form≤7. For allm we have

Table algebras with multiple P-polynomial structures

Using covering numbers we prove that a standard real integral table algebra (A, B) with |B| ≥ 6 has a P-polynomial structure with respect to every b ≠ 1 in B if and only if 2|B|-1 is prime and (A, B) is exactly isomorphic to the Bose-Mesner algebra of the association scheme of the ordinary (2|B|-1)-gon. Then we present an example showing that this result is not true if |B| ≤ 5.     

The vertex separator problem: a polyhedral investigation

The vertex separator (VS) problem in a graph G=(V,E) asks for a partition of V into nonempty subsets A, B, C such that there is no edge between A and B, and |C| is minimized subject to a bound on max{|A|,|B|}. We give a mixed integer programming formulation of the problem and investigate the vertex separator polytope (VSP), the convex hull of incidence vectors of vertex separators. Necessary and sufficient conditions are given for the VSP to be full dimensional. Central to our investigation is the relationship between separators and dominators. Several classes of valid inequalities are investigated, along with the conditions under which they are facet defining for the VSP. Some of our proofs combine in new ways projection with lifting.In a companion paper we develop a branch-and-cut algorithm for the (VS) problem based on the inequalities discussed here, and report on computational experience with a wide variety of (VS) problems drawn from the literature and inspired by various applications.Research supported by the National Science Foundation through grant #DMI-0098427 and by the Office of Naval Research through contract N00014-97-1-0196Research supported by the Brazilian agencies FAPESP (grant 01/14205–6), CAPES (grant BEX 04444/02–2) and CNPq (grants 302588/02–7 and Pronex 664107/97–4)     

Regularized Resolvent of Sums of Commuting Operators

For B ₁ and B ₂ commuting linear operators on a Banach space such that B ₁ generates a bounded strongly continuous semigroup and –B ₂ generates an exponentially decaying strongly continuous holomorphic semigroup, it is shown that (B ₁ – B ₂)^–1 B ₂ ^–r and (B ₁ – B ₂)^–1(–B ₁)^–r are bounded and everywhere defined, for any r > 0. Density of domains may also be removed. The results are applied to various abstract Cauchy problems.     

Score and Wald tests for the multivariate growth curve model with missing data

We present the score and Wald test analogues to Srivastava's (1985, Comm. Statist. A—Theory Methods, 14, 775–792) likelihood ratio tests for the multivariate growth curve model with missing data, and illustrate their use with data from an immunotherapy experiment (Fukushima et al. (1982, Int. J. Cancer, 29, 107–112, 113–117)).     

Borel Subalgebras Redux with Examples from Algebraic and Quantum Groups

Both building upon and revising previous literature, this paper formulates the general notion of a Borel subalgebra B of a quasi-hereditary algebra A. We present various general constructions of Borel subalgebras, establish a triangular factorization of A, and relate the concept to graded Kazhdan–Lusztig theories in the sense of Cline et al. (Tôhoku Math. J. 45 (1993), 511–534). Various interesting types of Borel subalgebras arise naturally in different contexts. For example, `excellent" Borel subalgebras come about by abstracting the theory of Schubert varieties. Numerous examples from algebraic groups, q-Schur algebras, and quantum groups are considered in detail.     

Anti-tori in Square Complex Groups

An anti-torus is a subgroup 〈a,b 〉 in the fundamental group of a compact non-positively curved space X, acting in a specific way on the universal covering space X such that a and b do not have any commuting nontrivial powers. We construct and investigate anti-tori in a class of commutative transitive fundamental groups of finite square complexes, in particular for the groups Γ_p,l originally studied by Mozes [Israel J. Math. 90(1–3) (1995), 253–294]. It turns out that anti-tori in Γ_p,l directly correspond to non commuting pairs of Hamilton quaternions. Moreover, free anti-tori in Γ_p,l are related to free groups generated by two integer quaternions, and also to free subgroups of . As an application, we prove that the multiplicative group generated by the two quaternions 1+2i and 1+4k is not free.