Large localizations of finite groups

We construct examples of localizations in the category of groups which take the Mathieu group M₁₁ to groups of arbitrarily large cardinality which are “abelian up to finitely many generators.” The paper is part of a broader study on the group theoretic properties which are or are not preserved by localizations.     

Resolutions of monomial ideals of projective dimension 1

We show that a monomial ideal I in a polynomial ring S has projective dimension ≤ 1 if and only if the minimal free resolution of S∕I is supported on a graph that is a tree. This is done by constructing specific graphs which support the resolution of the S∕I. We also provide a new characterization of quasi-trees, which we use to give a new proof to a result by Herzog, Hibi, and Zheng which characterizes monomial ideals of projective dimension 1 in terms of quasi-trees.     

Exactness of homotopy functors of spaces

We will provide an analysis of the generalized Atiyah-Hirzebruch spectral sequence (GAHSS), which was introduced by Hakim-Hashemi and Kahn. To do so, we introduce a new class of functors, called n-exact functors, which are analogous to Goodwillie’s n-excisive functors. In the study of these functors, we introduce a new spectral sequence, the homological Barratt-Goerss spectral sequence (HBGSS), which has properties similar to those of the classical Barratt-Goerss Spectral Sequence on homotopy. We close by giving an identification of the E² term of the GAHSS in the case of 2-exact functors on Moore spaces.     

Epimorphisms in Certain Varieties of Algebras

We prove a lemma which, under restrictive conditions, shows that epimorphisms in V are surjective if this is true for epimorphisms from irreducible members of V. This lemma is applied to varieties of orthomodular lattices which are generated by orthomodular lattices of bounded height, and to varieties of orthomodular lattices which are generated by orthomodular lattices which are the horizontal sum of their blocks. The lemma can also be applied to obtain known results for discriminator varieties.     

The behavior of the principal eigenvalue of a mixed elliptic problem with respect to a parameter

In this paper we determine the exact asymptotic behavior of the principal eigenvalue of a mixed elliptic eigenvalue problem which depends on a positive parameter λ when λ→∞. We analyze the case in which the problem is considered in a smooth bounded domain Ω of RN, and also the case of planar domains which are smooth except for a finite number of corner points.     

Pathwise uniqueness for solutions of systems of stochastic differential equations

A general theorem which obtains pathwise uniqueness for solutions of systems of Ito stochastic differential equations is given. It is shown that this theorem contains as special cases basic criteria which generalize Ito's result in which the coefficients satisfy Lipschitz conditions in the second variable. Also some new results which assume t-dependent modulus of continuity conditions on the coefficients are given as corollaries. The main result is established by means of Lyapunov type functions and comparison principle techniques.     

Probabilistic solution of the American options

The existence and uniqueness of probabilistic solutions of variational inequalities for the general American options are proved under the hypothesis of hypoellipticity of the infinitesimal generator of the underlying diffusion process which represents the risky assets of the stock market with which the option is created. The main tool is an extension of the Itô formula which is valid for the tempered distributions on Rd and for nondegenerate Itô processes in the sense of the Malliavin calculus.     

Some classes of integral matrices

Integral matrices A for which the system Ax=b will have an integral basic solution are considered. Results for these matrices are presented which parallel results concerning other matrices for which the system has integral solutions.     

Composition operators on a Banach space of analytic functions

In this paper, we study composition operators on a Banach space of analytic functions, denoted byX, which includes the Bloch space. This space arises naturally as the dual space of analytic functions in the Bergman spaceL _α ¹ (D) which admit an atomic decomposition. We characterize the functions which induce compact composition operators and those which induce Fredholm operatorson this space. We also investigate when a composition operator has a closed range. Supported by NNSFC No.19671036     

Perturbations of Dynamic Equations

We give conditions on the coefficient matrix for certain perturbed linear dynamic equations on time scales ensuring that there exists a bounded solution (which is explicitly given) to which all other solutions converge, and similarly conditions ensuring a bounded solution from which all other solutions diverge. We also consider periodic time scales and corresponding linear dynamic equations with periodic coefficients and prove similar statements about periodic solutions to which all other solutions converge or from which all other solutions diverge.     

The lattice of strict completions of a finite poset

For a given finite poset , we construct strict completions of P which are models of all finite lattices L such that the set of join-irreducible elements of L is isomorphic to P. This family of lattices, , turns out to be itself a lattice, which is lower bounded and lower semimodular. We determine the join-irreducible elements of this lattice. We relate properties of the lattice to properties of our given poset P, and in particular we characterize the posets P for which . Finally we study the case where is distributive. Received October 13, 2000; accepted in final form June 13, 2001.     

A theorem of Jon F. Carlson on filtrations of modules

We give an alternative proof to a theorem of Carlson [J.F. Carlson, Cohomology and induction from elementary abelian subgroups, Quart. J. Math. 51 (2000) 169-181] which states that if G is a finite group and k is a field of characteristic p, then any kG-module is a direct summand of a module which has a filtration whose sections are induced from elementary abelian p-subgroups of G. We also prove two new theorems which are closely related to Carlson’s theorem.     

Completions of Goldschmidt Amalgams of Type G4 in Dimension 3

The subgroups of GL₃(k) which are completions of the Goldschmidt G ₄-amalgam are determined. We also draw attention to five related graphs which are remarkable in that they have large girth and few vertices.     

Construction of General Linear Methods with Runge–Kutta Stability Properties

We describe the construction of explicit general linear methods of order p and stage order q=p with s=p+1 stages which achieve good balance between accuracy and stability properties. The conditions are imposed on the coefficients of these methods which ensure that the resulting stability matrix has only one nonzero eigenvalue. This eigenvalue depends on one real parameter which is related to the error constant of the method. Examples of methods are derived which illustrate the application of the approach presented in this paper.     

Examples of uniformly homeomorphic Banach spaces

We give several examples of separable Banach spaces which are nonisomorphic but uniformly homeomorphic. For example, we show that for every 1 < p ≠ 2 < ∞ there are two uniformly homeomorphic subspaces (respectively, quotients) of ? _p which are not linearly isomorphic; similarly c ₀ has two uniformly homeomorphic subspaces which are not isomorphic. We also give an example of two non-isomorphic separable L _∞-spaces which are coarsely homeomorphic (i.e. have Lipschitz equivalent nets).     

Extensions of rooted trees and their applications

Informally, an almost-tree is a rooted graph which has the form of a tree in which some leaves are bent back to earlier nodes of the branch on which they sit. The paper provides a construction for unfurling any finite rooted graph to obtain a strong homomorphic image with desirable properties. As an application, it is shown that for each flow-diagram G there can be constructed an unfurling G¹ which is an almost-tree with semantic properties equivalent to those of G in a very strong sense.     

Convexity of acceptance regions of multivariate tests

We show that if ? is an increasing convex function of the eigenvalues of a symmetric matrix argument, then $\text{Y→?((YY′)}\text{1}\text{2}\text{)}$ is a convex function. This has the consequence for the multivariate analysis of variance that statistical tests for μ=0 which are defined by increasing convex functions of the square roots of the eigenvalues of (ZZ′)^?1YY′ have acceptance regions which are convex in Y for fixed Z. Tests of this form have some good properties, including unbiasedness and power functions which are increasing in the noncentrality parameters.     

Generalized inverses of matrices over fields of characteristic two

For F a field of characteristic two, the problem of determining which m×n matrices of rank r have normalized generalized inverses and which have pseudoinverses is solved. For F_q a finite field of characteristic two, both the number of m×n matrices of rank r over F which have normalized generalized inverses and the number of m×n matrices of rank r over F_q which have pseudoinverses are determined.     

A note on approximation of optimal solutions of free problems of the calculus of variations

M. Lavrentiev, and later B. Manià, have given examples of free problems of the Calculus of Variations in which it is impossible to find a piecewiseC ¹ function, satisfying the same boundary conditions as the absolutely continuous optimal solution, and which simultaneously approximates both the optimal cost and, uniformly, the optimal trajectory. We investigate this question of approximation for free problems inE ⁿ. Dispensing with differentiability assumptions on the integrand of the cost functional, we present sufficient conditions under which this approximation problem may be solved thus extending earlier work of Manià, Tonelli, and others for free problems in the plane.     

Two-generator Frattini subgroups of finitep-groups

This paper deals with nonabelianp-groupsT (p a prime andp>2) which are either metacyclic or Redei. These groups are classified into those which are Frattini subgroups of a finitep-groupG and those which are not. Finally, it is shown that a nonabelian two-generator group of orderp ⁿ (n>4) which is the Frattini subgroup of ap-group must be metacyclic. This work is contained in the author’s dissertation.