• Volume 115, Issue 3

August 2005,   pages  241-369

• Some Properties of Complex Matrix-Variate Generalized Dirichlet Integrals

Dirichlet integrals and the associated Dirichlet statistical densities are widely used in various areas. Generalizations of Dirichlet integrals and Dirichlet models to matrix-variate cases, when the matrices are real symmetric positive definite or hermitian positive definite, are available [4]. Real scalar variables case of the Dirichlet models are generalized in various directions. One such generalization of the type-2 or inverted Dirichlet is looked into in this article. Matrix-variate analogue, when the matrices are hermitian positive definite, are worked out along with some properties which are mathematically and statistically interesting.

• Homeomorphisms and the Homology of Non-Orientable Surfaces

We show that, for a closed non-orientable surface 𝐹, an automorphism of $H_1(F,\mathbb{Z})$ is induced by a homeomorphism of 𝐹 if and only if it preserves the $(\mathrm{mod} 2)$ intersection pairing. We shall also prove the corresponding result for punctured surfaces.

• Higher Order Hessian Structures on Manifolds

In this paper we define 𝑛th order Hessian structures on manifolds and study them. In particular, when 𝑛=3, we make a detailed study and establish a one-to-one correspondence between third-order Hessian structures and a certain class of connections on the second-order tangent bundle of a manifold. Further, we show that a connection on the tangent bundle of a manifold induces a connection on the second-order tangent bundle. Also we define second-order geodesics of special second-order connection which gives a geometric characterization of symmetric third-order Hessian structures.

• Degree-Regular Triangulations of Torus and Klein Bottle-Erratum

A triangulation of a connected closed surface is called weakly regular if the action of its automorphism group on its vertices is transitive. A triangulation of a connected closed surface is called degree-regular if each of its vertices have the same degree. Clearly, a weakly regular triangulation is degree-regular. In [8], Lutz has classified all the weakly regular triangulations on at most 15 vertices. In [5], Datta and Nilakantan have classified all the degree-regular triangulations of closed surfaces on at most 11 vertices.

In this article, we have proved that any degree-regular triangulation of the torus is weakly regular. We have shown that there exists an 𝑛-vertex degree-regular triangulation of the Klein bottle if and only if 𝑛 is a composite number ≥ 9. We have constructed two distinct 𝑛-vertex weakly regular triangulations of the torus for each 𝑛 ≥ 12 and a (4𝑚+2)-vertex weakly regular triangulation of the Klein bottle for each 𝑚 ≥ 2. For 12 ≤ 𝑛 ≤ 15, we have classified all the 𝑛-vertex degree-regular triangulations of the torus and the Klein bottle. There are exactly 19 such triangulations, 12 of which are triangulations of the torus and remaining 7 are triangulations of the Klein bottle. Among the last 7, only one is weakly regular.

• Estimates and Nonexistence of Solutions of the Scalar Curvature Equation on Noncompact Manifolds

This paper is to study the conformal scalar curvature equation on complete noncompact Riemannian manifold of nonpositive curvature. We derive some estimates and properties of supersolutions of the scalar curvature equation, and obtain some nonexistence results for complete solutions of scalar curvature equation.

• Ideal Amenability of Banach Algebras on Locally Compact Groups

In this paper we study the ideal amenability of Banach algebras. Let $\mathcal{A}$ be a Banach algebra and let 𝐼 be a closed two-sided ideal in $\mathcal{A}, \mathcal{A}$ is 𝐼-weakly amenable if $H^1(\mathcal{A},I^∗)=\{0\}$. Further, $\mathcal{A}$ is ideally amenable if $\mathcal{A}$ is 𝐼-weakly amenable for every closed two-sided ideal 𝐼 in $\mathcal{A}$. We know that a continuous homomorphic image of an amenable Banach algebra is again amenable. We show that for ideal amenability the homomorphism property for suitable direct summands is true similar to weak amenability and we apply this result for ideal amenability of Banach algebras on locally compact groups.

• The Socle and Finite Dimensionality of some Banach Algebras

The purpose of this note is to describe some algebraic conditions on a Banach algebra which force it to be finite dimensional. One of the main results in Theorem 2 which states that for a locally compact group 𝐺, 𝐺 is compact if there exists a measure 𝜇 in $\mathrm{Soc} (L^1(G))$ such that 𝜇(𝐺) ≠ 0. We also prove that 𝐺 is finite if $\mathrm{Soc}(M(G))$ is closed and every nonzero left ideal in 𝑀(𝐺) contains a minimal left ideal.

• Multipliers of $A_p((0, ∞))$ with Order Convolution

The aim of this paper is to study the multipliers from $A_r(I)$ to $A_p(I), r≠ p$, where 𝐼=(0, ∞) is the locally compact topological semigroup with multiplication max and usual topology and $A_r(I)=\{f\in L_1(I):\hat{f}\in L_r(\hat{I})\}$ with norm $|||f|||_r=||f||_1+||hat{f}||_r$.

• Fixed Point Theory for Composite Maps on almost Dominating Extension Spaces

New fixed point results are presented for $\mathcal{U}^k_c(X, X)$ maps in extension type spaces.

• Wavelet Characterization of Hörmander Symbol Class $S^m_{ρ,𝛿}$ and Applications

In this paper, we characterize the symbol in Hörmander symbol class $S^m_{ρ𝛿}(m\in R,ρ,𝛿≥ 0)$ by its wavelet coefficients. Consequently, we analyse the kernel-distribution property for the symbol in the symbol class $S^m_{ρ,𝛿}(m\in R,ρ > 0,𝛿≥ 0)$ which is more general than known results; for non-regular symbol operators, we establish sharp 𝐿2-continuity which is better than Calderón and Vaillancourt's result, and establish $L^p(1≤ p≤∞)$ continuity which is new and sharp. Our new idea is to analyse the symbol operators in phase space with relative wavelets, and to establish the kernel distribution property and the operator's continuity on the basis of the wavelets coefficients in phase space.

• The Congruence Subgroup Problem

• # Proceedings – Mathematical Sciences

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