• Volume 115, Issue 2

May 2005,   pages  117-240

• Irreducibility Results for Compositions of Polynomials in Several Variables

We obtain explicit upper bounds for the number of irreducible factors for a class of compositions of polynomials in several variables over a given field. In particular, some irreducibility criteria are given for this class of compositions of polynomials.

• On Harder–Narasimhan Reductions for Higgs Principal Bundles

The existence and uniqueness of 𝐻–𝑁 reduction for the Higgs principal bundles over nonsingular projective variety is shown. We also extend the notion of 𝐻–𝑁 reduction for (𝛤, 𝐺)-bundles and ramified 𝐺-bundles over a smooth curve.

• Twisted Vector Bundles on Pointed Nodal Curves

Motivated by the quest for a good compactification of the moduli space of 𝐺-bundles on a nodal curve we establish a striking relationship between Abramovich’s and Vistoli’s twisted bundles and Gieseker vector bundles.

• Some BMO Estimates for Vector-Valued Multilinear Singular Integral Operators

In this paper, we prove some BMO end-point estimates for some vector-valued multilinear operators related to certain singular integral operators.

• $L^p$-Continuity for Calderón–Zygmund Operator

Given a Calderón–Zygmund (𝐶-𝑍 for short) operator 𝑇, which satisfies Hörmander condition, we prove that: if 𝑇 maps all the characteristic atoms to $W L^1$, then 𝑇 is continuous from $L^p$ to $L^p(1 < p < ∞)$. So the study of strong continuity on arbitrary function in $L^p$ has been changed into the study of weak continuity on characteristic functions.

• A Note on Convex Renorming and Fragmentability

Using the game approach to fragmentability, we give new and simpler proofs of the following known results: (a) If the Banach space admits an equivalent Kadec norm, then its weak topology is fragmented by a metric which is stronger than the norm topology. (b) If the Banach space admits an equivalent rotund norm, then its weak topology is fragmented by a metric. (c) If the Banach space is weakly locally uniformly rotund, then its weak topology is fragmented by a metric which is stronger than the norm topology.

• Some Necessary and Sufficient Conditions for Hypercyclicity Criterion

We give necessary and sufficient conditions for an operator on a separable Hilbert space to satisfy the hypercyclicity criterion.

• New Criteria to Identify Spectrum

In this paper we give some new criteria for identifying the components of a probability measure, in its Lebesgue decomposition. This enables us to give new criteria to identify spectral types of self-adjoint operators on Hilbert spaces, especially those of interest.

• The Contact Angle in Inviscid Fluid Mechanics

We show that in general, the specification of a contact angle condition at the contact line in inviscid fluid motions is incompatible with the classical field equations and boundary conditions generally applicable to them. The limited conditions under which such a specification is permissible are derived;however, these include cases where the static meniscus is not flat. In view of this situation, the status of the many solutions’ in the literature which prescribe a contact angle in potential flows comes into question. We suggest that these solutions which attempt to incorporate a phenomenological, but incompatible, condition are in some, imprecise sense weak-type solutions’;they satisfy or are likely to satisfy, at least in the limit, the governing equations and boundary conditions everywhere except in the neighbourhood of the contact line. We discuss the implications of the result for the analysis of inviscid flows with free surfaces.

• # Proceedings – Mathematical Sciences

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September 2017