An existence theorem is given for differential equations which fail the simple Lipschitz condition on some surfaces but whose solution curve is not tangential to any of these surfaces.

This paper examines a class of common numerical methods for computing eigenvectors of a compact linear operator. A necessary and sufficient condition is established for every element of an arbitrary given eigenspace to be the limit of a sequence of the approximate eigenvectors obtained by any given method in this class.

The term “rigid set” appears often in the literature in different contexts and with different meanings. In many cases the notions are unrelated, while in other they refer to related facts. Here we study infinite sets which are rigid according to [2] and analyse the relationship with the notion of rigidity used in [1]. We make several remarks, summarise different results in the area and prove a few new results.

The paper considers problems connected with the asymptotic equivalence of the system of ordinary differential equations

and the system of operator differential equations

The generality of the operator At guarantees a number of its important implementations. By a specific choice of the operator At the system (2) can be one of the concentrated delay, a system of distributed delay, a system with maxima, etcetera.

According to a result of Lee, varieties of pseudocomplemented distributive lattices form an ω+1 chain in which is the trivial variety and is the variety of Boolean algebras. In the present paper it is shown that the variety contains an almost universal subcategory B in which the members of Hom(B,B') associated with minimal prime ideals of B form a countably infinite set for any B,B' ∈ B. In particular, B3contains arbitrarily large algebras whose nontrivial endomorphisms form the countably infinite right zero semigroup. Our earlier results concerning categorical properties of varieties of pseudocomplemented distributive lattices show that no further reduction of the right zero count is possible.

If the characteristic of a field K is not zero then the Schur group S(K) = 0. In this paper we ask a similar question for the projective Schur group PS(K) and prove that the subgroup of PS(K) consisting of radical algebras is trivial. This disproves the conjecture that every projective Schur algebra is similar to a radical algebra.

We show that the normal version (R, R.) of the Banach simplicial cohomology of operator algebras can be expressed in terms of the functor Ext on the category of Banach R-bimodules. As an application, we prove that (R, R*) = Hn(R, R*) and thus the latter space vanishes for some von Heumann algebras for positive τ.

A ring R is called right co-FPF if every finitely generated cofaithful right R-module is a generator in mod-R. This definition can be carried over from rings to modules. We say that a finitely generated projective distinguished right R-module P is a co-FPF module (quasi-co-FPF module) if every P-finitely generated module, which finitely cogenerates P, generates σ[P] (P, respectively). We shall prove a result about the relationship between a co-FPF module and its endomorphism ring, and apply it to study some co-FPF rings.

This article is a sequel to a paper in which a quasilinear partial differential equation with nonlinear boundary condition was approximated using mollifiers, and the existence of solutions to the approximating problem shown under quite general conditions. In this paper we show that standard a priori Hölder estimates ensure the convergence of these solutions to a classical solution of the original problem. Some partial results giving such estimates for special cases are described.

We establish some weighted integral inequalities of Ostrowski, Čebyšev and Lupaş type and give applications for continuous probability density functions supported on infinite intervals.

We investigate the composition operators Cφ acting on the Bergman space of the unit disc D, where φ is a holomorphic self-map of D. Some new conditions for Cφ to belong to the Schatten class 𝒮p are obtained. We also construct a compact composition operator which does not belong to any Schatten class.

The object of the present paper is to study the delay differential equation of arbitrary order namely y(n)(t) + a(t)yτ(t) = f(t), n≥2 (an integer) and prove a nonoscillation theorem under the general situation in which a(t) and f(t) are allowed to oscillate arbitrarily often on some positive half real line. This is accomplished by way of two differential inequalities of nth order.

It is known that a finite non-abelian group G has a proper centralizer of order if, for example, |G| is even and |Z(G)| is odd, or whenever G is solvable. Often the exponent can be improved to , for example when G is supersolvable, or metabelian, or |G = pαqβ. Here we show more generally that this improvement is possible in many situations where G is factorizable into the product of two subgroups. In particular, much more evidence is presented to support the conjecture that some proper centralizer has order whenever G is a finite non-abelian solvable group.

For a finite group $G$, denote by $\unicode[STIX]{x1D707}(G)$ the degree of a minimal permutation representation of $G$. We call $G$ exceptional if there is a normal subgroup $N\unlhd G$ with $\unicode[STIX]{x1D707}(G/N)>\unicode[STIX]{x1D707}(G)$. To complete the work of Easdown and Praeger [‘On minimal faithful permutation representations of finite groups’, Bull. Aust. Math. Soc.38(2) (1988), 207–220], for all primes $p\geq 3$, we describe an exceptional group of order $p^{5}$ and prove that no exceptional group of order $p^{4}$ exists.

Let Cr[0,t] be the function space of the vector-valued continuous paths x:[0,t]→ℝr and define Xt:Cr[0,t]→ℝ(n+1)r by Xt(x)=(x(0),x(t1),…,x(tn)), where 0<t1<⋯<tn=t. In this paper, using a simple formula for the conditional expectations of the functions on Cr[0,t] given Xt, we evaluate the conditional analytic Feynman integral Eanfq[Ft∣Xt] of Ft given by where θ(s,⋅) are the Fourier–Stieltjes transforms of the complex Borel measures on ℝr, and provide an inversion formula for Eanfq[Ft∣Xt]. Then we present an existence theorem for the solution of an integral equation including the integral equation which is formally equivalent to the Schrödinger differential equation. We show that the solution can be expressed by Eanfq[Ft∣Xt] and a probability distribution on ℝr when Xt(x)=(x(0),x(t)).