Acyclic models is a powerful technique in algebraic topology and homological algebra in which facts about homology theories are verified by first verifying them on "models" (on which the homology theory is trivial) and then showing that there are enough models to present arbitrary objects. One version of the theorem allows one to conclude that two chain complex functors are naturally homotopic and another that two such functors are object-wise homologous. Neither is entirely satisfactory. The purpose of this paper is to provide a uniform account of these two, fixing what is unsatisfactory and also finding intermediate forms of the theorem.

Let G be a finite π-separable group, where π is a set of primes. The π-partial characters of G are the restrictions of the ordinary characters to the set of π-elements of G. Such an object is said to be irreducible if it is not the sum of two nonzero partial characters and the set of irreducible π- partial characters of G is denoted Iπ(G). (If p is a prime and π = p′, then Iπ(G) is exactly the set of irreducible Brauer characters at p.)

From their definition, it is obvious that each partial character φ ∊ Iπ(G) can be “lifted” to an ordinary character χ ∊ Irr(G). (This means that φ is the restriction of χ to the π-elements of G.) In fact, there is a known set of canonical lifts Bπ(G) ⊆ Irr(G) for the irreducible π-partial characters. In this paper, it is proved that if 2 ∉ π, then there is an alternative set of canonical lifts (denoted Dπ(G)) that behaves better with respect to character induction.

An application of this theory to M-groups is presented. If G is an M-group and S ⊆ G is a subnormal subgroup, consider a primitive character θ ⊆ Irr(S). It was known previously that if |G : S| is odd, then θ must be linear. It is proved here without restriction on the index of S that θ(1) is a power of 2.

We consider the surface obtained from the projective plane by blowing up the points of intersection of two plane curves meeting transversely. We find minimal generating sets of the defining ideals of these surfaces embedded in projective space by the sections of a very ample divisor class. All of the results are proven over an algebraically closed field of arbitrary characteristic.

Analogues of a theorem of Greenberg about finitely generated subgroups of free groups are proved for quasiconvex subgroups of word hyperbolic groups. It is shown that a quasiconvex subgroup of a word hyperbolic group is a finite index subgroup of only finitely many other subgroups.

Let F be a local non-Archimedean field and let Ꮪ(GL2(F)) be the set of equivalence classes of irreducible admissible representations of GL(F). When K/F be a Galois field extension, there is a map, called lifting, from Ꮪ(GL2(F)) to Ꮪ(GL2(K)). We give an explicit form of lifting when K/F is a quadratic wildly ramified extension and the given representations are Weil supercuspidal. We also provide a comparison between Weil representations and induced representations of GL2(F).

This paper is devoted to extending formulas for the geometric approximate subdifferential and the Clarke subdifferential of extended-real-valued functions on Banach spaces. The results are strong enough to include completely the finite dimensional setting.

Let k be an algebraically closed field and A = kQ/I be a basic finite dimensional k-algebra such that Q is a connected quiver without oriented cycles. Assume that A is strongly simply connected, that is, for every convex subcategory B of A the first Hochschild cohomology H1(B, B) vanishes. The algebra A is sincere if it admits an indecomposable module having all simples as composition factors. We study the structure of strongly simply connected sincere algebras of tame representation type. We show that a sincere, tame, strongly connected algebra A which contains a convex subcategory which is either representation-infinite tilted of type Ẽp, p = 6,7,8, or a tubular algebra, is of polynomial growth.

We consider the C*-algebras constructed from certain hyperbolic dynamical systems. The construction, due to Ruelle, generalizes the C*-algebras of Cuntz and Krieger. We discuss relations between the C*-algebras, show the existence of natural asymptotically abelian systems and investigate the K-theory and E-theory of these C*-algebras.

An implicitly defined second order ODE is said to be singular if the second derivative cannot be smoothly written in terms of lower order variables. The standard existence and uniqueness theory cannot be applied to such ODE and the graphs of solutions may fail to be regular curves (i.e., the solutions may have isolated C0-points or may fail to be single valued). In this paper we describe a local analysis for a large class of implicit second order ODE whose singular points satisfy a regularity condition. Within this class of ODE there is a secondary notion of (contact) singularity which is analogous to rest points for regular ODE. Theorems 5, 6, 7 and 8 produce invariants for these singularities which control the existence, uniqueness and the level of regularity in solutions.

In this paper we consider questions of recapturing an analytic function in a Banach space from its values on a uniqueness set. The principal method is to use reproducing kernels to construct a sequence in the Banach space which converges in norm to the given functions. The method works for several classical Banach spaces of analytic functions including some Hardy and Bergman spaces.

We prove that if φ is an (anti-) automorphism of a ring R with finite orbits on R, or integral over the integers, and if R contains a finite maximal φ-invariant subring, then R must be finite. Special cases are when φ has finite order or is an involution. Two corollaries are that R must be finite when R contains only finitely many φ-invariant subrings or has both ascending and descending chain conditions on φ invariant subrings. These generalize results in the literature for the special case when φ = idR.

Let C be an n x n positive definite matrix. If C ≥ 0 in the sense that Cij ≥ 0 and if p > n — 2, then Cp ≥ 0. This implies the following "positive minorant property" for the norms ‖A‖p = [tr(A*A)p/2]1/P. Let 2 < p ≠ 4, 6, … . Then 0 ≤ A ≤ B => ‖A‖p ≥ ‖B‖P if and only if n < p/2 + 1.

We show that , the E-homology of the Ω-spectrum for P(n), is an E* free Hopf ring for E a complex oriented theory with In sent to 0. This covers the cases and . The generators of the Hopf ring are those necessary for the stable result. The motivation for this paper is to show that P(n) satisfies all of the conditions for the machinery of unstable cohomology operations set up in [BJW95]. This can then be used to produce splittings analogous to those for BP done in [Wil75]

We completely determine the residual spectrum of the split exceptional group of type G2, thus completing the work of Langlands and Moeglin-Waldspurger on the part of the residual spectrum attached to the trivial character of the maximal torus. We also give the Arthur parameters for the residual spectrum coming from Borel subgroups. The interpretation in terms of Arthur parameters explains the “bizarre” multiplicity condition in Moeglin-Waldspurger's work. It is related to the fact that the component group of the Arthur parameter is non-abelian.

Several characterizations of weak cotype 2 and weak Hilbert spaces are given in terms of basis constants and other structural invariants of Banach spaces. For finite-dimensional spaces, characterizations depending on subspaces of fixed proportional dimension are proved.

We construct two classes of Lie subalgebras of the infinite matrix Lie algebra 𝔤𝔩∞(ℂ) and prove that they are all simple Lie algebras.

Let G be a locally compact topological group. A number of characterizations are given of the class of compact groups in terms of the geometric properties such as Radon-Nikodym property, Dunford-Pettis property and Schur property of Ap(G), and the properties of the multiplication operator on PFp(G). We extend and improve several results of Lau and Ülger [17] to Ap(G) and Bp(G) for arbitrary p.

A compact hypergroup is called almost discrete if it is homeomorphic to the one-point-compactification of a countably infinite discrete set. If the group Up of all p-adic units acts multiplicatively on the p-adic integers, then the associated compact orbit hypergroup has this property. In this paper we start with an exact projective sequence of finite hypergroups and use successive substitution to construct a new surjective projective system of finite hypergroups whose limit is almost discrete. We prove that all compact almost discrete hypergroups appear in this way—up to isomorphism and up to a technical restriction. We also determine the duals of these hypergroups, and we present some examples coming from partitions of compact totally disconnected groups.

Some oscillation criteria are given for the second order neutral delay differential equation

where τ and σ are nonnegative constants, . These results generalize and improve some known results about both neutral and delay differential equations.

We show that p-groups of maximal class and order p5 are determined by their group algebras over the field of p elements. The most important information requisite for the proof is obtained from a detailed study of the unit group of a quotient algebra of the group algebra, larger than the small group algebra.