• Volume 117, Issue 3

August 2007,   pages  287-427

• On the Cohomology of Orbit Space of Free $\mathbb{Z}_p$-Actions on Lens Spaces

Let $G=\mathbb{Z}_p, p$ an odd prime, act freely on a finite-dimensional $CW$-complex 𝑋 with $\mathrm{mod} p$ cohomology isomorphic to that of a lens space $L^{2m-1}(p;q_1,\ldots,q_m)$. In this paper, we determine the $\mathrm{mod} p$ cohomology ring of the orbit space $X/G$, when $p^2\nmid m$

• Equi-Gaussian Curvature Folding

In this paper we introduce a new type of folding called equi-Gaussian curvature folding of connected Riemannian 2-manifolds. We prove that the composition and the cartesian product of such foldings is again an equi-Gaussian curvature folding. In case of equi-Gaussian curvature foldings, $f:M\to P_n$, of an orientable surface 𝑀 onto a polygon $P_n$ we prove that

(i) $f\in\mathcal{F}_{EG}(S^2)\Leftrightarrow n=3$

(ii) $f\in\mathcal{F}_{EG}(T^2)\Rightarrow n=4$

(iii) $f\in\mathcal{F}_{EG}(\# 2T^2)\Rightarrow n=5, 6$

and we generalize (iii) for $\# nT^2$.

• Classification of Framed Links in 3-Manifolds

We present a short and complete proof of the following Pontryagin theorem, whose original proof was complicated and has never been published in detail. Let 𝑀 be a connected oriented closed smooth 3-manifold, $L_1(M)$ be the set of framed links in 𝑀 up to a framed cobordism, and $\deg: L_1(M)\to H_1(M;\mathbb{Z})$ be the map taking a framed link to its homology class. Then for each $\alpha\in H_1(M;\mathbb{Z})$ there is a one-to-one correspondence between the set $\deg^{-1}\alpha$ and the group $\mathbb{Z}_{2d(\alpha)}$, where $d(\alpha)$ is the divisibility of the projection of 𝛼 to the free part of $H_1(M;\mathbb{Z})$.

• A Sharp Upper Bound for the First Eigenvalue of the Laplacian of Compact Hypersurfaces in Rank-1 Symmetric Spaces

Let 𝑀 be a closed hypersurface in a simply connected rank-1 symmetric space $\overline{M}$. In this paper, we give an upper bound for the first eigenvalue of the Laplacian of 𝑀 in terms of the Ricci curvature of $\overline{M}$ and the square of the length of the second fundamental form of the geodesic spheres with center at the center-of-mass of 𝑀.

• Equivariant Embeddings of Hermitian Symmetric Spaces

We prove that equivariant, holomorphic embeddings of Hermitian symmetric spaces are totally geodesic (when the image is not of exceptional type).

• Uniqueness of Solutions to Schrödinger Equations on Complex Semi-Simple Lie Groups

In this note we study the time-dependent Schrödinger equation on complex semi-simple Lie groups. We show that if the initial data is a bi-invariant function that has sufficient decay and the solution has sufficient decay at another fixed value of time, then the solution has to be identically zero for all time. We also derive Strichartz and decay estimates for the Schrödinger equation. Our methods also extend to the wave equation. On the Heisenberg group we show that the failure to obtain a parametrix for our Schrödinger equation is related to the fact that geodesics project to circles on the contact plane at the identity.

• On the Schwartz Space Isomorphism Theorem for Rank One Symmetric Space

In this paper we give a simpler proof of the $L^p$-Schwartz space isomorphism $(0 &lt; p\leq 2)$ under the Fourier transform for the class of functions of left 𝛿-type on a Riemannian symmetric space of rank one. Our treatment rests on Anker’s [2] proof of the corresponding result in the case of left 𝐾-invariant functions on 𝑋. Thus we give a proof which relies only on the Paley–Wiener theorem.

• On an Inequality Concerning the Polar Derivative of a Polynomial

In this paper, we present a correct proof of an $L_p$-inequality concerning the polar derivative of a polynomial with restricted zeros. We also extend Zygmund’s inequality to the polar derivative of a polynomial.

• On Eneström–Kakeya Theorem and Related Analytic Functions

We prove some extensions of the classical results concerning the Eneström–Kakeya theorem and related analytic functions. Besides several consequences, our results considerably improve the bounds by relaxing and weakening the hypothesis in some cases.

• Weighted Composition Operators from Bergman-Type Spaces into Bloch Spaces

Let 𝜑 be an analytic self-map and 𝑢 be a fixed analytic function on the open unit disk 𝐷 in the complex plane $\mathbb{C}$. The weighted composition operator is defined by

$$u C_\varphi f=u\cdot p (f\circ\varphi), f\in H(D).$$

Weighted composition operators from Bergman-type spaces into Bloch spaces and little Bloch spaces are characterized by function theoretic properties of their inducing maps.

• Approximation of Functions of Two Variables by Certain Linear Positive Operators

We introduce certain linear positive operators and study some approximation properties of these operators in the space of functions, continuous on a compact set, of two variables. We also find the order of this approximation by using modulus of continuity. Moreover we define an 𝑟th order generalization of these operators and observe its approximation properties. Furthermore, we study the convergence of the linear positive operators in a weighted space of functions of two variables and find the rate of this convergence using weighted modulus of continuity.

• Continuity of Integrated Density of States - Independent Randomness

In this paper we discuss the continuity properties of the integrated density of states for random models based on that of the single site distribution. Our results are valid for models with independent randomness with arbitrary free parts. In particular in the case of the Anderson type models (with stationary, growing, decaying randomness) on the 𝑣 dimensional lattice, with or without periodic and almost periodic backgrounds, we show that if the single site distribution is uniformly 𝛼-Hölder continuous, $0 &lt; \alpha \leq 1$, then the density of states is also uniformly 𝛼-Hölder continuous.

• A Generalization of d'Alembert Formula

In this paper we find a closed form of the solution for the factored inhomogeneous linear equation

$$\prod\limits_{j=1}^n\left(\frac{d}{dt}-A_j\right)u(t)=f(t).$$

Under the hypothesis $A_1,A_2,\ldots,A_n$ are infinitesimal generators of mutually commuting strongly continuous semigroups of bounded linear operators on a Banach space 𝑋. Here we do not assume that $A_j s$ are distinct and we offer the computational method to get explicit solutions of certain partial differential equations.

• # Proceedings – Mathematical Sciences

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