Let R be a ring of nonzero characteristic and let G be a finite group with subgroup H. It is shown that H is a normal subgroup of the group of units of the group ring RG if and only if H is contained in the centre of G or R is the field with 2 elements, G is the symmetric group on 3 letters and H is normal in G.

We establish an error term in the Sato–Tate theorem of Birch. That is, for $p$ prime, $q=p^{r}$ and an elliptic curve $E:y^{2}=x^{3}+ax+b$, we show that

$$\begin{eqnarray}\#\{(a,b)\in \mathbb{F}_{q}^{2}:\unicode[STIX]{x1D703}_{a,b}\in I\}=\unicode[STIX]{x1D707}_{ST}(I)q^{2}+O_{r}(q^{7/4})\end{eqnarray}$$

$I\subseteq [0,\unicode[STIX]{x1D70B}]$

$\unicode[STIX]{x1D703}_{a,b}$

$2\sqrt{q}\cos \unicode[STIX]{x1D703}_{a,b}=q+1-E(\mathbb{F}_{q})$

$\unicode[STIX]{x1D707}_{ST}(I)$

$I$

From the viewpoint of $C^{\ast }$-dynamical systems, we define a weak version of the Haagerup property for the group action on a $C^{\ast }$-algebra. We prove that this group action preserves the Haagerup property of $C^{\ast }$-algebras in the sense of Dong [‘Haagerup property for $C^{\ast }$-algebras’, J. Math. Anal. Appl.377 (2011), 631–644], that is, the reduced crossed product $C^{\ast }$-algebra $A\rtimes _{{\it\alpha},\text{r}}{\rm\Gamma}$ has the Haagerup property with respect to the induced faithful tracial state $\widetilde{{\it\tau}}$ if $A$ has the Haagerup property with respect to ${\it\tau}$.

Let a prime $p$ divide the order of a finite real reflection group. We classify the reflection subgroups up to conjugacy that are minimal with respect to inclusion, subject to containing a $p$-Sylow subgroup. For Weyl groups, this is achieved by an algorithm inspired by the Borel–de Siebenthal algorithm. The cases where there is not a unique conjugacy class of reflection subgroups minimally containing the $p$-Sylow subgroups are the groups of type $F_{4}$ when $p=2$ and $I_{2}(m)$ when $m\geq 6$ is even but not a power of $2$ for each odd prime divisor $p$ of $m$. The classification significantly reduces the cases required to describe the $p$-Sylow subgroups of finite real reflection groups.

Let $ (G_n)_{n=0}^{\infty } $ be a nondegenerate linear recurrence sequence whose power sum representation is given by $ G_n = a_1(n) \alpha _1^n + \cdots + a_t(n) \alpha _t^n $ . We prove a function field analogue of the well-known result in the number field case that, under some nonrestrictive conditions, $ |{G_n}| \geq ( \max _{j=1,\ldots ,t} |{\alpha _j}| )^{n(1-\varepsilon )} $ for $ n $ large enough.

We present an example showing that a classical result due to Glaeser about the closedness of composition subalgebras of infinitely differentiable functions cannot be extended to the case of weakiy uniformly differentiable functions on Banach spaces.

A new set of polynomials in powers of xh (h > 0) which are related to the Laguerre polynomials , have been defined by a Rodrigues type formula involving the forward difference operator Δh of increment h. Besides obtaining shift operator formulae for , a definition of fractional power of Δh has been given. The definition has been illustrated by establishing known relations of the confluent hypergeometric function.

Let X be a non-empty set and let H(X) denote a Hibert space of complex-valued functions on X. Let T be a mapping from X to X and θ a mapping from X to C such that for all f in H(X), f ° T is in H(x) and the mappings CT taking f to f ° T and M taking f to θ.f are bounded linear operators on H(X). Then the operator CTMθ is called a weighted composition operator on H(X). This note is a report on the characterization of weighted composition operators on functional Hilbert spaces and the computation of the adjoint of such operators on L2 of an atomic measure space. Also the Fredholm criteria are discussed for such classes of operators.

We present a characterisation of all quartic polynomials with exactly three distinct roots and the property that it and all its derivatives have rational roots. It turns out that there are an infinite number of distinct such quartics, each of which corresponds to a point on a related elliptic curve. Furthermore the collection of these points forms a proper subgroup of the group of rational points on the curve.

In this paper, it is shown that the theory of perturbed generalised Hamiltonian systems provides an effective method for understanding the description of flow patterns of some three-dimensional flows. Firstly, theorems for the persistence of periodic solutions of three-dimensional generalised Hamiltonian systems under perturbation are given by developing Melnikov's method. Then, three different systems of three-dimensional steady fluid flows are discussed and the existence or non-existence of periodic solutions of these systems is proved.

Varieties of the form $G\times S_{\!\text{reg}}$, where $G$ is a complex semisimple group and $S_{\!\text{reg}}$ is a regular Slodowy slice in the Lie algebra of $G$, arise naturally in hyperkähler geometry, theoretical physics and the theory of abstract integrable systems. Crooks and Rayan [‘Abstract integrable systems on hyperkähler manifolds arising from Slodowy slices’, Math. Res. Let., to appear] use a Hamiltonian $G$-action to endow $G\times S_{\!\text{reg}}$ with a canonical abstract integrable system. To understand examples of abstract integrable systems arising from Hamiltonian $G$-actions, we consider a holomorphic symplectic variety $X$ carrying an abstract integrable system induced by a Hamiltonian $G$-action. Under certain hypotheses, we show that there must exist a $G$-equivariant variety isomorphism $X\cong G\times S_{\!\text{reg}}$.

Kennison's concept of an ordered sheaf is used to show that any member of the variety of subresiduated lattices is canonically isomorphic to the algebra of all ordered sections in a certain ordered sheaf, whose base is the Priestley space of the residuating sublattice.

We give a simple proof of the strong law of large numbers with rates, assuming only finite variance. This note also serves as an elementary introduction to the theory of large deviations, assuming only finite variance, even when the random variables are not necessarily independent.

A question asked by G. Kowol is answered by the construction, to an arbitrarily given natural number n, of groups G with automorphisms that agree with inner automorphisms on each set of fewer than n elements of G, but fail to agree with any inner automorphism on at least one set of n elements.

In this paper we consider the discreteness of spectrum for higher-order differential operators in weighted function spaces. Using the method of embedding theorems of weighted Sobolev spaces Hnp in weighted spaces Ls,r, we obtain a new sufficient and necessary condition to ensure that the spectrum is discrete, which can be easily used to judge the discreteness of some differential operators.

A killer of a group $G$ is an element that normally generates $G$. We show that the group of a cable knot contains infinitely many killers such that no two lie in the same automorphic orbit.

Explicit formulas for Cotes numbers of the Gaussian Hermite quadrature formula based on the zeros of the nth Chebyshev polynomial of the second kind and their asymptotic behaviour as n → ∞ are given. This provides an answer to an analogue of Problem 26 of Tur´n.

It would seem from results of Foltz, Lair, and Kelly that symmetric monoidal closed structures, and even monoidal biclosed ones, are quite rare on one-sorted algebraic or essentially-algebraic categories. They showed many such categories to admit no such structures at all, and others to admit only one or two; no such category is known to admit an infinite set of such structures.

Among concrete categories, topological ones are in some sense at the other extreme from essentially-algebraic ones; and one is led to ask whether a topological category may admit many such structures. On the category of topological spaces itself, only one such structure - in fact symmetric - is known; although Greve has shown it to admit a proper class of monoidal closed structures. One of our main results is a proof that none of these structures described by Greve, except the classical one, is biclosed.

Our other main result is that, nevertheless, there exist topological categories (of quasi-topological spaces) which admit a proper class of symmetric monoidal closed structures. Even if we insist (like most authors) that topological categories must be wellpowered, we can still exhibit ones with more such structures than any small cardinal.

A quotient algebra of a given algebra is said to be cogenerated by a quotient object if it is contained in the quotient object and is the biggest with this property. Triples T over a regular category are characterized which have the property that every quotient object of a T-algebra cogenerates some quotient algebra: these are precisely the right exact triples, preserving colimits of quotient chains. This improves a result of Michael Barr (J. Pure Appl. Algebra 4 (1974), 1–8) that every right exact, finitary triple has the investigated property. This result is related to categorical automata, since a triple has the above property iff triple machines admit a minimal realization of every behavior.

Let (X, S) be a measurable space and S be a σ-algebra of subsets of X. A nonempty class M is said to be a class of null sets if M ⊂ S, M is closed under countable unions of sets and E ∩ F ∈ M whenever E ∈ M and F ∈ S. It is possible to show that such concepts as absolute continuity, singularity and independence of measures can be studied simply by classes of null sets and that similar results can be obtained under the condition that each disjoint subclass of S – M is countable, denoted (S – M)C. If (X, S,μ) is a measure space then M = {E ∈ S: μ(E) = O} is a class of null sets of S and S – M the class of all sets of positive measure. We say that a measure μ has the property σ if there exists a sequence of totally finite measures on S such that σ(E) = for all E ∈ S. This property of measures is weaker than total σ–finiteness of measures. The main result of the present paper is as follows: Let (X, S,μ) be a measure space and M = {E ∈ S: μ(E) = O}. Then (S – M)C if and only if μ has the property σ.