• Volume 118, Issue 1

February 2008,   pages  1-157

• On Automorphisms of some Finite 𝑝-Groups

We give a sufficient condition on a finite 𝑝-group 𝐺 of nilpotency class 2 so that $\mathrm{Aut}_c(G)=\mathrm{Inn}(G)$, where $\mathrm{Aut}_c(G)$ and $\mathrm{Inn}(G)$ denote the group of all class preserving automorphisms and inner automorphisms of 𝐺 respectively. Next we prove that if 𝐺 and 𝐻 are two isoclinic finite groups (in the sense of 𝑃. Hall), then $\mathrm{Aut}_c(G)≅\mathrm{Aut}_c(H)$. Finally we study class preserving automorphisms of groups of order $p^5,p$ an odd prime and prove that $\mathrm{Aut}_c(G)=\mathrm{Inn}(G)$ for all the groups 𝐺 of order $p^5$ except two isoclinism families.

• Entire Functions Sharing One Polynomial with their Derivatives

In this paper, we study the growth of solutions of 𝑎𝑘-th order linear differential equation and that of 𝑎𝑘+1-th order linear differential equation. From this we affirmatively answer a uniqueness question concerning a conjecture given by Brück in 1996 under the restriction of the hyper order less than 1/2, and obtain some uniqueness theorems of a nonconstant entire function and its derivative sharing a finite nonzero complex number CM. The results in this paper also improve some known results. Some examples are provided to show that the results in this paper are best possible.

• Sums of the Squares of Terms of Sequence $\{u_n\}$

In this paper, we consider generalized Fibonacci type second order linear recurrence $\{u_n\}$. We derive a generating matrix for both the sums of squares, $\sum^n_{i=0}u^2_i$ and the products of the form $u_nu_{n+2}$. We also derive explicit formulas for the sums and products by using matrix methods. Then we give a matrix method to generate the sums of product of two consecutive terms $u_n u_{n+1}$ as well as the product, $u_n u_{n+2}$. Further we give generating functions and combinatorial representations of the sums of squares of terms of $\{u_n\}$ and the product, $u_n u_{n+2}$.

• Parabolic Bundles on Algebraic Surfaces I -- The Donaldson-Uhlenbeck Compactification

The aim of this paper is to construct the parabolic version of the Donaldson-Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algebraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non-emptiness of the moduli space of parabolic stable bundles of rank 2.

• Torsionfree Sheaves over a Nodal Curve of Arithmetic Genus One

We classify all isomorphism classes of stable torsionfree sheaves on an irreducible nodal curve of arithmetic genus one defined over $\mathbb{C}$. Let 𝑋 be a nodal curve of arithmetic genus one defined over $\mathbb{R}$, with exactly one node, such that 𝑋 does not have any real points apart from the node. We classify all isomorphism classes of stable real algebraic torsionfree sheaves over 𝑋 of even rank. We also classify all isomorphism classes of real algebraic torsionfree sheaves over 𝑋 of rank one.

• Vrănceanu Connections and Foliations with Bundle-Like Metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.

• Realizations of the Canonical Representation

A characterisation of the maximal abelian subalgebras of the bounded operators on Hilbert space that are normalised by the canonical representation of the Heisenberg group is given. This is used to classify the perfect realizations of the canonical representation.

• Low-Cost Control Problems on Perforated and Non-Perforated Domains

We study the homogenization of a class of optimal control problems whose state equations are given by second order elliptic boundary value problems with oscillating coefficients posed on perforated and non-perforated domains. We attempt to describe the limit problem when the cost of the control is also of the same order as that describing the oscillations of the coefficients. We study the situations where the control and the state are both defined over the entire domain or when both are defined on the boundary.

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