We present a systematic and self-contained exposition of the generalized Riemann integral in a locally compact Hausdorff space, and we show that it is equivalent to the Perron and variational integrals. We also give a necessary and sufficient condition for its equivalence to the Lebesgue integral with respect to a suitably chosen measure.

We discuss generalizations of the Lie-Kolchin-Mal'cev theorem. For example we show that if G is a soluble linear group of degree n, then G contains a triangularizable subgroup T whose index in G is bounded by function of n only and such that T is normalized by every automorphism of G normalizing G0, the Zariski connected component of G containing the identity. We also prove that in certain situations at least the index of G0 in G can be bounded in terms of the degree and the ground field.

Let name be a class of finite soluble groups with the properties: (1) is a Fitting class (i.e. normal subgroup closed and normal product closed) and (2) if N ≦ H ≦ G ∈, N ⊲ G and H/N is a p-group for some prime p, then H ∈. Then is called a Fischer class. In any finite soluble group G, there exists a unique conjugacy class of maximal -subgroups V called the -injectors which have the property that for every N◃◃G, N ∩ V is a maximal -subgroup of N [3]. 3. By Lemma 1 (4) [7] an -injector V of G covers or avoids a chief factor of G. As in [7] we will call a chief factor -covered or -avoided according as V covers or avoids it and -complemented if it is complemented and each of its complements contains some -injector. Furthermore we will call a chief factor partially-complemented if it is complemented and at least one of its complements contains some -injector of G.

We study the embeddings of a finite p-group U into Sylow p-subgroups of Sym (U) induced by the right regular representation p: U→ Sym(U). It turns out that there is a one-to-one correspondence between the chief series in U and the Sylow p-subgroups of Sym (U) containing Up. Here, the Sylow p-subgroup Pσ of Sym (U) correspoding to the chief series σ in U is characterized by the property that the intersections of Up with the terms of any chief series in Pσ form σp. Moreover, we see that p: U→ Pσ are precisely the kinds of embeddings used in a previous paper to construct the non-trivial countable algebraically closed locally finite p-groups as direct limits of finite p-groups.

The concept of simultaneous or collective continuity of a family of single valued functions was introduced by Gale [3] for regular spaces to replace equicontinuiry in metric spaces. Smithson [6] extended the standard point-open and compact-open function space topologies to include multi-valued functions. The aim of this paper is to use these topologies and extend the notion of collective continuity in order to obtain an Ascoli type theorem for multi-valued functions analogous to Theorem 1 in [3, p. 304]. We have the following theorem in mind:

We study the harmonicity of maps to or from cosymplectic manifolds by relating them to maps to or from Kähler spaces.

By a theorem of G. Birkhoff, every algebra in an equationally defined class of algebras K is a subdirect product of subdirectly irreducible algebras of K. In this paper we show that this result is true for any class of structures. not necessarily algebraic, closed under isomorphisms and direct limits. Quasivarieties in the sense of Malcev are examples of such classes of structures. This includes Birkhoffs result as a particular case.

We give an account of the minimal volume of the plane, as defined by Gromov, and first computed by Bavard and Pansu. We also describe some related geometric inequalities.

Throughout this paper F is an algebraically closed field of characteristic p (≠ 0) and g is a finite group whose order is divisible by p. We define in the usual way an F-representation of g (or F G-representation) and its corresponding module. The isomorphism class of the, F G-representation module M is written {M} or, where no confusion arises, M. A (G) denotes the F-representation algebra of G over the complex field G (as defined on pages 73 and 82 of [6]).

The higher Lie characters of the symmetric group Sn arise from the Poincaré-Birkhoff-Witt basis of the free associative algebra. They are indexed by the partitions of n and sum up to the regular character of Sn. A combinatorial description of the multiplicities of their irreducible components is given. As a special case the Kraśkiewicz-Weyman result on the multiplicities of the classical Lie character is obtained.

We study the Lw-norm (2 ≤ W < ∞) of the discrepancy of a sequence of points in the unit cube relative to similar copies of a given convex polygon. In particular, we prove the rather surprising result that the estimates obtained have the same order of magnitude as the analogous question when the sequence of points is replaced by a set of points.

In the paper we study the asymptotic behaviour of the number of trees with n vertices and diameter k = k(n), where k/√n → ∞ as n → ∞ but k = o(n).

We give a complete description of maximal compatible partial orders on the mono-unary algebra (A, f), where f: A → A is an arbitrary unary operation.

In this paper we are going to prove the following results:

Theorem 1. There exist two bounded hopfian lattices such that their {0,1}- free product is not hopfian.

Theorem 2. There exist two hopfian lattices such that their free product is not hopfian.

In Theorem 2 free product (coproduct, sum) has its usual meaning (see, for instance, [4]); in Theorem 1 we use the usual definition but all lattices are assumed to be bounded (that is, having a least element 0 and largest element 1) and all homomorphisms are assumed to be {0, l}-homomorphisms (that is, homomorphisms preserving 0 and 1).

Let $T_{1}$, $T_{2}$ be two Calderón–Zygmund operators and $T_{1,b}$ be the commutator of $T_{1}$ with symbol $b\in \text{BMO}(\mathbb{R}^{n})$. In this paper, by establishing new bilinear sparse dominations and a new weighted estimate for bilinear sparse operators, we prove that the composite operator $T_{1}T_{2}$ satisfies the following estimate: for $\unicode[STIX]{x1D706}>0$ and weight $w\in A_{1}(\mathbb{R}^{n})$,

$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log \bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx,\nonumber\end{eqnarray}$$

$T_{1,b}T_{2}$

$$\begin{eqnarray}\displaystyle & & \displaystyle w(\{x\in \mathbb{R}^{n}:\,|T_{1,b}T_{2}f(x)|>\unicode[STIX]{x1D706}\})\nonumber\\ \displaystyle & & \displaystyle \qquad \lesssim [w]_{A_{1}}[w]_{A_{\infty }}^{2}\log (\text{e}+[w]_{A_{\infty }})\int _{\mathbb{R}^{n}}\frac{|f(x)|}{\unicode[STIX]{x1D706}}\log ^{2}\bigg(\text{e}+\frac{|f(x)|}{\unicode[STIX]{x1D706}}\bigg)w(x)\,dx.\nonumber\end{eqnarray}$$

We construct the Weil representation of the Kantor-Koecher-Tits Lie algebra g associated to a simple real Jordan algebra V. Later we introduce a family of integral operators intertwining the Weil representation with the infinitesimal representations of the degenerate principal series of the conformal group G of the Jordan algebra V. The decomposition of L2(V) in the case of Jordan algebra of real square matrices is given using this construction.

A semigroup is a nonvoid Hausdorff space together with a continuous associative multiplication. (The latter phrase will generally be abbreviated to CAM and the multiplication in a semigroup will be denoted by juxta position unless the contrary is made explicit.)

In this paper, we give certain analytic functional relations for the double harmonic series related to the double Euler numbers. These can be regarded as continuous generalizations of the known discrete relations obtained by the author recently.

Following up on a paper of Balamohan et al. [‘On the behavior of a variant of Hofstadter’s $q$-sequence’, J. Integer Seq. 10 (2007)], we analyze a variant of Hofstadter’s $Q$-sequence and show that its frequency sequence is 2-automatic. An automaton computing the sequence is explicitly given.

A semigroup is totally commutative if each of its essentially binary polynomials is commutative, or equivalently, if in every polynomial (word) every two essential variables commute. In the present paper we describe all varieties (equational classes) of totally commutative semigroups, lattices of subvarieties for any variety, and their free spectra.