The structure of groups which have at most two isomorphism classes of derived subgroups ($\mathfrak{D}$2-groups) is investigated. A complete description of $\mathfrak{D}$2-groups is obtained in the case where the derived subgroup is finite: the solution leads an interesting number theoretic problem. In addition, detailed information is obtained about soluble $\mathfrak{D}$2-groups, especially those with finite rank, where algebraic number fields play an important role. Also, detailed structural information about insoluble $\mathfrak{D}$2-groups is found, and the locally free $\mathfrak{D}$2-groups are characterized.

We give the necessary and sufficient conditions for Lagrangian submanifolds in Kähler manifolds to be biharmonic. We classify biharmonic PNMC Lagrangian H-umbilical submanifolds in the complex space forms. Furthermore, we classify biharmonic PNMC Lagrangian surfaces in the two-dimensional complex space forms.

In the present paper we prove that every 2-local derivation on a semi-finite von Neumann algebra is a derivation.

Let R be a commutative Noetherian ring, is an ideal of R and M is an R-module. We intend to establish the dual of Grothendieck's Vanishing Theorem for local homology modules. We conjecture that =0 for all i>magRM. We prove this in several cases.

We investigate the spectral theory of integrable actions of a locally compact abelian group on a locally convex vector space. We start with an analysis of various spectral subspaces induced by the action of the group. This is applied to analyse the spectral theory of operators on the space, generated by measures on the group. We apply these results to derive general Tauberian theorems that apply to arbitrary locally compact abelian groups acting on a large class of locally convex vector spaces, which includes Fréchet spaces. We show how these theorems simplify the derivation of Mean Ergodic theorems.

The mean value inequality is characteristic for upper semi-continuous functions to be subharmonic. Quasinearly subharmonic functions generalise subharmonic functions. We find the necessary and sufficient conditions under which subsets of balls are big enough for the characterisation of non-negative, quasinearly subharmonic functions by mean value inequalities. Similar result is obtained also for generalised mean value inequalities where, instead of balls, we consider arbitrary bounded sets, which have non-void interiors and instead of the volume of ball some functions depending on the radius of this ball.

A real algebraic curve of genus g is a pair (S,〈 τ 〉), where S is a closed Riemann surface of genus g and τ: S → S is a symmetry, that is, an anti-conformal involution. A Schottky uniformization of (S,〈 τ 〉) is a tuple (Ω,Γ,P:Ω → S), where Γ is a Schottky group with region of discontinuity Ω and P:Ω → S is a regular holomorphic cover map with Γ as its deck group, so that there exists an extended Möbius transformation $\widehat{\tau}$ keeping Ω invariant with P o $\widehat{\tau}$=τ o P. The extended Kleinian group K=〈 Γ, $\widehat{\tau}$〉 is called an extended Schottky groups of rank g. The interest on Schottky uniformizations rely on the fact that they provide the lowest uniformizations of closed Riemann surfaces. In this paper we obtain a structural picture of extended Schottky groups in terms of Klein–Maskit's combination theorems and some basic extended Schottky groups. We also provide some insight of the structural picture in terms of the group of automorphisms of S which are reflected by the Schottky uniformization. As a consequence of our structural description of extended Schottky groups, we get alternative proofs to results due to Kalliongis and McCullough (J. Kalliongis and D. McCullough, Orientation-reversing involutions on handlebodies, Trans. Math. Soc. 348(5) (1996), 1739–1755) on orientation-reversing involutions on handlebodies.

Considered here is the pullback attractor of the process associated with the first initial boundary value problem for the non-autonomous semilinear degenerate parabolic equation

\begin{linenomath}u_t-\text{div}(\sigma(x)\nabla u)+f(u)=g(x,t)\end{linenomath}

$D_0^2(\Omega,\sigma)$

About 15 years ago, Lutwak and Zhang (E. Lutwak and G. Zhang, Blaschke–Santalo inequalities, J. Differ. Geom. 47 (1997), 1–16) introduced the notion of Lp moment bodies and established important volume inequalities for them, which were recently generalized by Haberl and Schuster (C. Haberl and E. Schuster, General Lp affine isoperimetric inequalities, J. Differ. Geom. 83 (2009), 1–26). In this paper, we establish new Brunn–Minkowski-type inequalities with respect to Blaschke Lp harmonic addition for the quermassintegrals and dual quermassintegrals of Lp moment bodies.

Let d be a positive integer, q=(qij)d×d be a d×d matrix, ℂq be the quantum torus algebra associated with q. We have the semidirect product Lie algebra $\mathfrak{g}$=Der(ℂq)⋉Z(ℂq), where Z(ℂq) is the centre of the rational quantum torus algebra ℂq. In this paper, we construct a class of irreducible weight $\mathfrak{g}$-modules $\mathcal{V}$α (V,W) with three parameters: a vector α∈ℂd, an irreducible $\mathfrak{gl}$d-module V and a graded-irreducible $\mathfrak{gl}$N-module W. Then, we show that an irreducible Harish Chandra (uniformaly bounded) $\mathfrak{g}$-module M is isomorphic to $\mathcal{V}$α(V,W) for suitable α, V, W, if the action of Z(ℂq) on M is associative (respectively nonzero).

We present several conjectures which would describe the Nielsen equivalence classes of generating pairs for the groups SL(2,q) and PSL(2,q). The Higman invariant, which is the union of the conjugacy classes of the commutator of a generating pair and its inverse, and the trace of the commutator play key roles. Combining known results with additional work, we clarify the relationships between the conjectures, and obtain various partial results concerning them. Motivated by the work of Macbeath (A. M. Macbeath, Generators of the linear fractional groups, in Number theory (Proc. Sympos. Pure Math., vol. XII, Houston, TX, 1967) (American Mathematical Society, Providence, RI, 1969), 14–32), we use another invariant defined using traces to develop algorithms that enable us to verify the conjectures computationally for all q up to 101, and to prove the conjectures for a highly restricted but possibly infinite set of q.

We prove uniqueness of positive solutions for the boundary value problems

\[\{\begin{array}{ll} -\Delta u=\lambda f(u)\ \ &\text{in}\Omega, \ \ \ \ \ u=0 &\text{on \partial \Omega,\]

The present paper gives an example of a rigid spherical cone-manifold and that of a flexible one, which are both Seifert fibred.

In this paper a four-dimensional macroeconomic model of a small open economy, describing the development of income, capital stock, interest rate and money stock, which was constructed in [5] (Makovínyiová, K. & Zimka, R. (2009) On stability in generalized Schinasi's macroeconomic model under fixed exchange rates. Tatra Mt. Math. Publ. 43, 115–122), is analysed. Sufficient conditions for the existence of one pair of purely imaginary eigenvalues and two eigenvalues with negative real parts in the linear approximation matrix of the model are found. Formulae for the calculation of the bifurcation coefficients of the model are derived. A statement about the existence of limit cycles is made. A numerical example is given illustrating the results.

In this paper, a non-linear mathematical model is proposed and analysed to study the role of technology in combating social crimes in a dynamic population by considering immigration and emigration rates of susceptible population and criminals. The problem is modelled by considering five interacting variables, namely the density of susceptible population, the density of criminals, the density of removed (isolated) criminals, the density of crime burden and the level of technology used to control crime. The proposed model is analysed by using the stability theory of differential equation and simulation. The model analysis shows that the crime burden decreases considerably as the level of technology increases. It is noted that the crime in a society can be controlled almost completely if criminals from the general population are removed by intensive use of technology.

We consider partitions of the positive integer n whose parts satisfy the following condition. For a given sequence of non-negative numbers {bk}k≥1, a part of size k appears in exactly bk possible types. Assuming that a weighted partition is selected uniformly at random from the set of all such partitions, we study the asymptotic behaviour of the largest part Xn. Let D(s)=∑k=1∞bkk−s, s=σ+iy, be the Dirichlet generating series of the weights bk. Under certain fairly general assumptions, Meinardus (1954) obtained the asymptotic of the total number of such partitions as n→∞. Using the Meinardus scheme of conditions, we prove that Xn, appropriately normalized, converges weakly to a random variable having Gumbel distribution (i.e., its distribution function equals e−e−t, −∞<t<∞). This limit theorem extends some known results on particular types of partitions and on the Bose–Einstein model of ideal gas.

In this paper we investigate the population dynamics of a species with age structure in the case where the diffusion and death rates of the matured population are both age-dependent. We develop a new application of the age-structure technique in terms of an integral equation. For unbounded spatial domains, we study the existence of travelling waves, whilst in bounded domains, we investigate the existence of positive steady-state solutions and their stability.