Peano continua which are images of the unit interval [0,1] or the circle S under a continuous and irreducible map are investigated. Necessary conditions for a space to be the irreducible image of [0,1] are given, and it is conjectured that these conditions are sufficient as well. Also, various results on irreducible images of [0,1] and S are given within some classes of regular curves. Some of them involve inverse limits of inverse sequences of Euler graphs with monotone bonding maps.

We give a normal form for the elements in the free field, following the lines of the minimization theory of noncommutative rational series.

It is shown that the space hp(D,X) has the Kadec-Klee property with respect to pointwise norm convergence in the Banach space X if and only if X has the Radon-Nikodym property and every point of the unit sphere of X is a denting point of the unit ball of X. In addition, it is shown that hp(D,X) is locally uniformly rotund if and only if X is locally uniformly rotund and has the Radon-Nikodym property.

Let A be a finite set with |A| > 2. We describe all clones on A containing the set SA of all permutations of A among its unary operations. (A clone on A is a composition closed set of finitary operations on A containing all projections). With a few exceptions such a clone C is either essentially unary or cellular i.e. there exists a monoid M of self-maps of A containing SA such that either C = (= all essentially unary operations agreeing with some f ∊ M) or C = ∪ Гh where 1 < h ≤ |A| and Гh consists of all finitary operations on A taking at most h values. The exceptions are subclones of Burle's clone or of its variant (provided |A| is even).

A module M is said to be weakly N-projective if it has a projective cover π: P(M) ↠M and for each homomorphism : P(M) → N there exists an epimorphism σ:P(M) ↠M such that (kerσ) = 0, equivalently there exists a homomorphism :M ↠N such that σ= . A module M is said to be weakly projective if it is weakly N-projective for all finitely generated modules N. Weakly N-injective and weakly injective modules are defined dually. In this paper we study rings over which every weakly injective right R-module is weakly projective. We also study those rings over which every weakly projective right module is weakly injective. Among other results, we show that for a ring R the following conditions are equivalent:

(1) R is a left perfect and every weakly projective right R-module is weakly injective.

(2) R is a direct sum of matrix rings over local QF-rings.

(3) R is a QF-ring such that for any indecomposable projective right module eR and for any right ideal I, soc(eR/eI) = (eR/eJ)n for some positive integer n.

(4) R is right artinian ring and every weakly injective right R-module is weakly projective.

(5) Every weakly projective right R-module is weakly injective and every weakly injective right R-module is weakly projective.

The purpose of this paper is to examine which classes of functions from can be topologized in a sense that there exist topologies τ1 and τ2 on and respectively, such that is equal to the class C(τ1 , τ2) of all continuous functions . We will show that the Generalized Continuum Hypothesis GCH implies the positive answer for this question for a large number of classes of functions for which the sets {x : f(x) = g(x)} are small in some sense for all f, g ∈ f ≠ g. The topologies will be Hausdorff and connected. It will be also shown that in some model of set theory ZFC with GCH these topologies could be completely regular and Baire. One of the corollaries of this theorem is that GCH implies the existence of a connected Hausdorff topology T on such that the class L of all linear functions g(x) = ax + b coincides with . This gives an affirmative answer to a question of Sam Nadler. The above corollary remains true for the class of all polynomials, the class of all analytic functions and the class of all harmonic functions.

We will also prove that several other classes of real functions cannot be topologized. This includes the classes of C∞ functions, differentiable functions, Darboux functions and derivatives.

This is about results on certain regular graphs that yield information about the structure of the ideal class group of quadratic number fields associated with these graphs. Some of the results can be formulated in terms of the quadratic forms x2 + 27y2, x2 + 32y2, x2 + 64y2.

Let A be an algebra of continuous real functions on a topological space X. We study when every nonzero algebra homomorphism φ: A → R is given by evaluation at some point of X. In the case that A is the algebra of rational functions (or real-analytic functions, or Cm-functions) on a Banach space, we provide a positive answer for a wide class of spaces, including separable spaces and super-reflexive spaces (with nonmeasurable cardinal).

Recently A. Gutek, D. Hart, J. Jamison and M. Rajagopalan have obtained many significiant results concerning shift operators on Banach spaces. Using a result of Holsztynski they classify isometric shift operators on C(X) for any compact Hausdorff space X into two (not necessarily disjoint) classes. If there exists an isometric shift operator T: C(X) → C(X) of type II, they show that X is necessarily separable. In case T is of type I, they exhibit a paticular infinite countable set of isolated points in X. Under the additional assumption that the linear functional Γ carrying f ∊ C(X) to Tf(p) ∊ is identically zero, they show that D is dense in X. They raise the question whether D will still be dense in X even when Γ ≠ 0. In this paper we give a negative answer to this question. In fact, given any integer l ≥ 1, we construct an example of an isometric shift operator T: C(X) —> C(X) of type I with X \ having exactly / elements, where is the closure of D in X.

We obtain a number of new conditions for oscillation of the first order neutral delay equation with nonconstant coefficients of the form

Comparison results are also given as well as conditions for the existence of nonoscillatory solutions.

We consider a class of real flag manifolds which occur as Fürstenberg boundaries of ordered symmetric spaces and study the image of associated momentum maps. The presence of the order structure is responsible for much stronger convexity properties than in the general case.

Three main results are obtained in this paper: one generalizes the Atiyah-Bott-Shapiro periodicity equivalence on the category of real Clifford modules, (Theorem 2.2); another establishes the existence of two algebras for real projective representations of the symmetric group Sn and the alternating group An, (Theorem 3.2) and determines their structure, (Theorem 6.1); the third describes all the real projective representations of Sn and An except for some small numbers n, (Theorem 7.2).

The Lp-norm convergence of weighted averages μnf is ergodic theory is equivalent to the pointwise convergence of the Fourier transforms . If h(γ) = , then the behavior of h determines when the Lp-norm limit of μnf is ∫f dm. The nature of such limit functions h is the focus of this article.

For any variety of groups, the relative inner rank of a given groupG is defined to be the maximal rank of the -free homomorphic images of G. In this paper we explore metabelian inner ranks of certain one-relator groups. Using the well-known Quillen-Suslin Theorem, in conjunction with an elegant result of Artamonov, we prove that if r is any "Δ-modular" element of the free metabelian group Mn of rank n > 2 then the metabelian inner rank of the quotient group Mn/(r) is at most [n/2]. As a corollary we deduce that the metabelian inner rank of the (orientable) surface group of genus k is precisely k. This extends the corresponding result of Zieschang about the absolute inner ranks of these surface groups. In continuation of some further applications of the Quillen-Suslin Theorem we give necessary and sufficient conditions for a system g = (g1,..., gk) of k elements of a free metabelian group Mn, k ≤ n, to be a part of a basis of Mn. This extends results of Bachmuth and Timoshenko who considered the cases k = n and k < n — 3 respectively.

The summability fields of generalized Nörlund means (N,p*α,p), α ∈ Ν, are increasing with a and are contained in that of the corresponding power series method (P,p). Particular cases are the Cesàro- and Euler-means with corresponding power series methods of Abel and Borel. In this paper we generalize a convexity theorem, which is well-known for the Cesàro means and which was recently shown for the Euler means to a large class of generalized Nörlund means.

We determine abstract characterizations of the fixed point subalgebras of the rotation algebra under the automorphisms and .

We characterize Banach spaces that admit Markuševič bases with various properties connected with weak countable determining or weak Lindelöf determining of spaces or with various norming properties.

In singularity theory, J. Damon gave elegant versions of the unfolding and determinacy theorems for geometric subgroups of . and . In this work, we propose a unified treatment of the smooth stability of germs and the structural stability of versai unfoldings for a large class of such subgroups.

The paper establishes some characterizations of spaces in terms of p-summing or p-nuclear norms of the identity operator on the given space E.

In particular, for an n-dimensional Banach space E and I ≤ p < 2, E is isometric to if and only if πp(E*) ≥ n1/p and E* has cotype p' with the constant one.

Furthermore, spaces are characterized by inequalities for p-summing norms of operators related to the John's ellipsoid of maximal volume contained in the unit ball of E.