• Volume 109, Issue 1

February 1999,   pages  1-115

• A combinatorial Lefschetz fixed point formula

We define a class of simplicial maps — those which are “expanding directions preserving” — from a barycentric subdivision to the original simplicial complex. These maps naturally induce a self map on the links of their fixed points. The local index at a fixed point of such a map turns out to be the Lefschetz number of the induced map on the link of the fixed point in relative homology. We also show that a weakly hyperbolic [4] simplicial map sdnK →K is expanding directions preserving.

• Dolbeault cohomology of compact complex homogeneous manifolds

We show that ifM is the total space of a holomorphic bundle with base space a simply connected homogeneous projective variety and fibre and structure group a compact complex torus, then the identity component of the automorphism group ofM acts trivially on the Dolbeault cohomology ofM. We consider a class of compact complex homogeneous spacesW, which we call generalized Hopf manifolds, which are diffeomorphic to S1 ×K/L whereK is a compact connected simple Lie group andL is the semisimple part of the centralizer of a one dimensional torus inK. We compute the Dolbeault cohomology ofW. We compute the Picard group of any generalized Hopf manifold and show that every line bundle over a generalized Hopf manifold arises from a representation of its fundamental group.

• Torus quotients of homogeneous spaces — II

We classify the homogeneous spacesX for which there is aT linearised ample line bundleL onX such thatXTss(L)=XTs(L).

• Parabolic ample bundles III: Numerically effective vector bundles

In this continuation of [Bi2] and [BN], we define numerically effective vector bundles in the parabolic category. Some properties of the usual numerically effective vector bundles are shown to be valid in the more general context of numerically effective parabolic vector bundles.

• On the existence of automorphisms with simple Lebesgue spectrum

It is shown that ifT is a measure preserving automorphism on a probability space (Ω,B, m) which admits a random variable X0 with mean zero such that the stochastic sequence X0 o Tn,n ε ℤ is orthonormal and spans L02(Ω,B,m), then for any integerk ≠ 0, the random variablesX o Tnk,n ε ℤ generateB modulom.

• When isf(x,y) = u(x) + v(y)?

LetX andY be arbitrary non-empty sets and letS a non-empty subset ofX ×Y. We give necessary and sufficient conditions onS which ensure that every real valued function onS is the sum of a function onX and a function onY.

• New integral mean estimates for polynomials

In this paper we prove someLP inequalities for polynomials, wherep is any positive number. They are related to earlier inequalities due to A Zygmund, N G De Bruijn, V V Arestov, etc. A generalization of a polynomial inequality concerning self-inversive polynomials, is also obtained.

• Denting and strongly extreme points in the unit ball of spaces of operators

For 1 ≤p ≤ ∞ we show that there are no denting points in the unit ball of ℓ(lp). This extends a result recently proved by Grząślewicz and Scherwentke whenp = 2 [GS1]. We also show that for any Banach spaceX and for any measure space (Ω, A, μ), the unit ball of ℓ(L1 (μ), X) has denting points iffL1(μ) is finite dimensional and the unit ball ofX has a denting point. We also exhibit other classes of Banach spacesX andY for which the unit ball of ℓ(X, Y) has no denting points. When X* has the extreme point intersection property, we show that all ‘nice’ operators in the unit ball of ℓ(X, Y) are strongly extreme points.

• A note on the non-commutative neutrix product of distributions

The distributionF(x+, −r) Inx+ andF(x, −s) corresponding to the functionsx+−r lnx+ andx−s respectively are defined by the equations$$\left\langle {F(x_ + , - r)\ln x_ + ,\phi (x)} \right\rangle = \int_0^\infty {x^{ - r} \ln x\left[ {\phi (x) - \sum\limits_{i = 0}^{r - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}x^i \frac{{\phi ^{(i)} (0)}}{{(r - 1)!}}H(1 - x)x^{r - 1} } } \right]dx}$$ (1) and$$\left\langle {F(x_ + , - s),\phi (x)} \right\rangle = \int_0^\infty {x^{ - s} \left[ {\phi (x) - \sum\limits_{i = 0}^{s - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}( - x^i )\frac{{\phi ^{(s - 1)} (0)}}{{(s - 1)!}}H(1 - x)x^{s - 1} } } \right]dx}$$ (2) whereH(x) denotes the Heaviside function. In this paper, using the concept of the neutrix limit due to J G van der Corput [1], we evaluate the non-commutative neutrix product of distributionsF(x+, −r) lnx+ andF(x, −s). The formulae for the neutrix productsF(x+, −r) lnx+ ox−s, x+−r lnx+ ox−s andx−s o F(x+, −r) lnx+ are also given forr, s = 1, 2, ...

• Thermoelasticity with thermal relaxation: An alternative formulation

The theory of thermoelasticity with thermal relaxation for homogeneous materials is formulated upon the basis of the law of balance of energy and the law of balance of entropy, proposed by Green and Naghdi [5]. The non-linear theory is formulated first; then the linearized theory is deduced. The uniqueness of solution of a typical initial, mixed boundary value problem is established.

• Statistical stationary states for a two-layer quasi-geostrophic system

Existence of a family of locally invariant probability measures for large scale flows in enclosed temperate sea is proved. This model is extremely important for understanding the meso-scale phenomena in oceans. The techniques used are those developed by Albeverio and his collaborators.

• # Proceedings – Mathematical Sciences

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