On separating a fixed point from zero by invariants

Assume a fixed point v∈V^G can be separated from zero by a homogeneous invariant f∈𝕜[V]^G of degree p^rd, where p>0 is the characteristic of the ground field 𝕜 and p,d are coprime. We show that then v can also be separated from zero by an invariant of degree p^r, which we obtain explicitly from f. It follows that the minimal degree of a homogeneous invariant separating v from zero is a p-power.     

Error Estimates for a Class of Elliptic Optimal Control Problems

In this article, functional type a posteriori error estimates are presented for a certain class of optimal control problems with elliptic partial differential equation constraints. It is assumed that in the cost functional the state is measured in terms of the energy norm generated by the state equation. The functional a posteriori error estimates developed by Repin in the late 1990s are applied to estimate the cost function value from both sides without requiring the exact solution of the state equation. Moreover, a lower bound for the minimal cost functional value is derived. A meaningful error quantity coinciding with the gap between the cost functional values of an arbitrary admissible control and the optimal control is introduced. This error quantity can be estimated from both sides using the estimates for the cost functional value. The theoretical results are confirmed by numerical tests.     

Clifford theory for infinite dimensional modules

Clifford theory of possibly infinite dimensional modules is studied.     

A Note on normalizations of orbit closures

We give a negative answer to a question by J. M. Landsberg on the nature of normalizations of orbit closures. A counterexample originates from the study of complex, ternary, cubic forms.     

Koszulity of some path categories

We prove Koszulity of certain linear path categories obtained from connected graphs with some infinite directed walk. These categories can be viewed as locally quadratic dual to preprojective algebras.     

On σ-commutativity in Menger algebras of n-place functions

We present an abstract characterization of various types of Menger algebras of n-place functions allowing certain permutations of variables.     

A summation‐integral type modification of Szász–Mirakjan operators

In this paper, we introduced a summation‐integral type modification of Szász–Mirakjan operators. Calculation of moments, density in some space, a direct result and a Voronvskaja‐type result, are obtained. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and multiplicity of systems of Kirchhoff‐type equations with general potentials

This paper is concerned with the following systems of Kirchhoff‐type equations: Under more relaxed assumptions on V(x) and F(x,u,v), we first prove the existence of at least two nontrivial solutions for the aforementioned system by using Morse theory in combination with local linking arguments. Then by using the Clark theorem, the existence results of at least 2k distinct pairs of solutions are obtained. Some recent results from the literature are extended. Copyright © 2016 John Wiley & Sons, Ltd.