We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.

Let G be a finite group and p a prime number. About five years ago I. M. Isaacs and S. D. Smith [5] gave several character-theoretic characterizations of finite p-solvable groups with p-length 1. Indeed, they proved that if P is a Sylow p-subgroup of G then the next four conditions (l)–(4) are equivalent:

(1) G is p-solvable of p-length 1.

(2) Every irreducible complex representation in the principal p-block of G restricts irreducibly to NG(P).

(3) Every irreducible complex representation of degree prime to p in the principal p-block of G restricts irreducibly to NG(P).

(4) Every irreducible modular representation in the principal p-block of G restricts irreducibly to NG(P).

Square grid circle patterns with prescribed intersection angles, mimicking holomorphic maps $z^{\gamma }$ and ${\rm log}(z)$ are studied. It is shown that the corresponding circle patterns are embedded and described by special separatrix solutions of discrete Painlevé and Riccati equations. The general solution of this Riccati equation is expressed in terms of the hypergeometric function. Global properties of these solutions, as well as of the discrete $z^{\gamma }$ and ${\rm log}(z)$, are established.

We consider the Jack–Laurent symmetric functions for special values of parameters p0=n+k−1m, where k is not rational and m and n are natural numbers. In general, the coefficients of such functions may have poles at these values of p0. The action of the corresponding algebra of quantum Calogero–Moser integrals $\mathcal{D}$(k, p0) on the space of Laurent symmetric functions defines the decomposition into generalised eigenspaces. We construct a basis in each generalised eigenspace as certain linear combinations of the Jack–Laurent symmetric functions, which are regular at p0=n+k−1m, and describe the action of $\mathcal{D}$(k, p0) in these eigenspaces.

A tight Riesz group is a partially ordered group which satisfies a strengthened form of the Riesz interpolation property. The term “tight” was introduced by Miller in [8], and the tight interpolation property has been considered in papers by Fuchs [3], Miller [8, 9], Loy and Miller [7] and Wirth [12]. If the closure of the cone P, in the interval topology, of such a partially ordered group G contains no pseudozeros, then is itself the cone of a partial order on G. Loy and Miller found of particular interest the case in which this associated partial orderis a lattice order. This situation was then considered in reverse by A. Wirth [12] who investigated under what circumstances a lattice ordered group would permit the existence of a tight Riesz order (called a compatible tight Riesz order) for which the initial lattice order is the order defined by the closure of the cone of the tight Riesz order.Wirth gave two fundamental anduseful characterizations of those subsets of the cone of a lattice ordered group that canbe the strict cone of a compatible tight Riesz order; one is in terms of archimedean classes and the other is an elementwise characterization. Although Loy, Miller and Wirth restricted their attention to abelian groups, much of what they do carries over verbatim to nonabelian groups. In the main result of this paper (Theorem 2.6) a description of the strict cone of a compatible tight Riesz order on a lattice ordered group Gis given in terms of the prime subgroups of G.This is particularly useful when one is attempting to identify the compatible tight Riesz orders on some particular lattice ordered group or class of lattice ordered groups, since it narrows down to a convenient family of subsets the possible candidates for strict cones of compatible tight Riesz orders. These can then be tested under Wirth's criteria. This technique is illustrated in § 5, where the compatible tight Riesz orders are determined o a lattice ordered group of the type V(Γ, Gγ), where Γ is of finite width, and in § 6, where two examples are considered.

A compact bordered Klein surface of genus g ≥ 2 has maximal symmetry [4] if its automorphism group is of order 12(g − 1), the largest possible. An M*-group [8] acts on a bordered surface with maximal symmetry. The first important result about these groups was that they must have a certain partial presentation [8, p. 5]. However, research has tended to focus more on the surfaces with maximal symmetry than on the M*-groups, and results about these groups typically deal with existence.

In this paper, we study the existence of positive solutions to a semilinear nonlocal elliptic problem with the fractional α-Laplacian on Rn, 0 < α < n. We show that the problem has infinitely many positive solutions in $ {C^\tau}({R^n})\bigcap H_{loc}^{\alpha /2}({R^n}) $. Moreover, each of these solutions tends to some positive constant limit at infinity. We can extend our previous result about sub-elliptic problem to the nonlocal problem on Rn. We also show for α ∊ (0, 2) that in some cases, by the use of Hardy’s inequality, there is a nontrivial non-negative $ H_{loc}^{\alpha /2}({R^n}) $ weak solution to the problem

$$ {( - \Delta )^{\alpha /2}}u(x) = K(x){u^p} \quad {\rm{ in}} \ {R^n}, $$

In this paper, we give a characterization of Clifford tori and in a unit sphere S n+1 (1). Our results extend the results due to Cheng and Yau [4], and Wang and Xia [11].

It is well-known (see e.g. [1, p. 5]) that a class ℳ of (not necessarily associative) rings is the semisimple class for some radical class, relative to some universal class if and only if it has the following properties:

(a)if ℳ, then every non-zero ideal I of Rhas a non-zero homomorphic image I/J∈ℳ.

(b) If R∈ but R∉ℳ, then R has a non-zero ideal I∈, where ℳ = {K ∈ | every non-zero K/H∉ℳ}. In fact ℳ is the radical class whose semisimple class is ℳ. On the other hand, if ℘ is a radical class, then ℐ℘ = {K∈/ if I is a non-zero ideal of K, then I∉℘} is its semisimple class. If a class ℳ is hereditary (that is, when R∈ℳ, then all its ideals are in ℳ), it clearly satisfies (a), but there do exist non-hereditary semisimple classes (see [2]). The condition (satisfied in all associative or alternative classes) is that ℘ is hereditary for a radical class ℘ if and only if ℘(I) ⊆ ℘(R) for all ideals I of all rings R∈ [3, Lemma 2, p. 595].

The concept of a metaplectic form was introduced about 40 years ago by T. Kubota. He showed how Jacobi-Legendre symbols of arbitrary order give rise to characters of arithmetic groups. Metaplectic forms are the automorphic forms with these characters. Kubota also showed how higher analogues of the classical theta functions could be constructed using Selberg's theory of Eisenstein series. Unfortunately many aspects of these generalized theta series are still unknown, for example, their Fourier coefficients. The analogues in the case of function fields over finite fields can in principle be calculated explicitly and this was done first by J. Hoffstein in the case of a rational function field. Here we shall return to his calculations and clarify a number of aspects of them, some of which are important for recent developments.

All sets considered will be finite, and |x| will denote the cardinal number of the set X.

Let = (Ai:i∈I) be a family of subsets of a set E. A subset E′ ⊆ E is called a transversal of if there exists a bijection σ:E′→ I such that e ∈ Aσ(e) (e ∈ E′). According to a well-known theorem of P. Hall [2], the familyhas a transversal if and only iffor every subset I′ of I. Ford and Fulkerson [1] obtained (as a special case of a more general theorem) an analogous criterion for the existence of a common transversal (CT) of two families. We may state their result in the following terms.

Using the Hilbert Uniqueness Method, we study the problem of exact controllability in Neumann boundary conditions for problems of transmission of the wave equation. We prove that this system is exactly controllable for all initial states in L2(Ω)×(H1(Ω))′.

We give an asymptotic bound for the size of the n-generated relatively free semigroup in the variety generated by all combinatorial strictly 0-simple semigroups.

Let G be a finitely generated (f.g.) torsion-free nilpotent group. Then the group algebra k[G] of G over a field k is a Noetherian domain and hence has a classical division ring of fractions, denoted by k(G). Recently, the division algebras k(G) and, somewhat more generally, division algebras generated by f.g. nilpotent groups have been studied in [3] and [5]. These papers are concerned with the question to what extent the division algebra determines the group under consideration. Here we continue the study of the division algebras k(G) and investigate their Gelfand–Kirillov (GK–) transcendence degree.

We consider families of quasiplatonic Riemann surfaces characterised by the fact that – as in the case of Fermat curves of exponent n – their underlying regular (Walsh) hypermap is an embedding of the complete bipartite graph Kn,n, where n is an odd prime power. We show that these surfaces, regarded as algebraic curves, are all defined over abelian number fields. We determine their orbits under the action of the absolute Galois group, their minimal fields of definition and in some easier cases their defining equations. The paper relies on group – and graph – theoretic results by G. A. Jones, R. Nedela and M. Škoviera about regular embeddings of the graphs Kn,n [7] and generalises the analogous results for maps obtained in [9], partly using different methods.

The problem of Josephus and the forty Jewsis well known [1, 3]. In its most general form, this problem is equivalent to the problem of m-enumeration of a set, as described below.

Define the ordered set

We choose and remove cyclically, from left to right, each with element of Zn until the set is exhausted. The chosen elements are ordered into a new ordered set

which is therefore a permutation of Zn, obtained by what we call m-enumeration of theset Zn.

A number of theorems are established about positive definite functions and representations of certain topological semigroups. In particular we establish theorems which show that measurable positive definite functions and measurable representations can each be decomposed into the sum of two parts one of which is continuous and the other of which is “small”.

It is shown that the creation operator is the only (up to a multiplicative constant) injective weighted shift all of whose translations (or at least one) are still injective weighted shifts regardless of what the weight sequences and the bases are. A similar result is true for the annihilation operator as well as for the Heisenberg and Schrödinger couples.

Many special cases of the equation x2+C= yn where x and y are positive integers and n≥3 have been considered over the years, but most results for general n are of fairly recent origin. The earliest reference seems to be an assertion by Fermat that he had shown that when C=2, n=3, the only solutions are given by x = 5, y = 3; a proof was published by Euler [1]. The first result for general n is due to Lebesgue [2] who proved that when C = 1 there are no solutions. Nagell [4] generalised Fermat's result and proved that for C = 2 the equation has no solution other than x = 5, y = 3, n = 3. He also showed [5] that for C = 4 the equation has no solution except x = 2, y = 2, n = 3 and x = 11, y = 5, n = 3, and claims in [6] to have dealt with the case C = 5. The case C = -1 was solved by Chao Ko, and an account appears in [3], pp. 302–304.

In recent years a new approach to the study of compact symmetric spaces has been taken by Nagano and Chen [10]. This approach assigned to each pair of antipodal points on a closed geodesic a pair of totally geodesic submanifolds. In this paper we will show how these totally geodesic submanifolds can be used in conjunction with a theorem of Bott to compute homotopy in compact symmetric spaces. Some of the results are already known (see [1], [5], [11] for example) but we include them here for completeness and to illustrate this unified approach. We also exhibit a connection between the second homotopy group of a compact symmetric space and the multiplicity of the highest root. Using this in conjunction with a theorem of J. H. Cheng [6] we obtain a topological characterization of quaternionic symmetric spaces with antiquaternionic involutive isometry. The author would like to thank Prof T. Nagano for all his help and his detailed descriptions of the totally geodesic submanifolds mentioned above.