• Volume 120, Issue 3

June 2010,   pages  259-394

• Zero-Sum Problems with Subgroup Weights

In this note, we generalize some theorems on zero-sums with weights from [1], [4] and [5] in two directions. In particular, we consider $\mathbb{Z}^d_p$ for a general 𝑑 and subgroups of $Z^∗_p$ as weights.

• Theta Function Identities Associated with Ramanujan's Modular Equations of Degree 15

We present alternative proofs of some of Ramanujan’s theta function identities associated with the modular equations of composite degree 15. Along the way we also find some new theta-function identities. We also give simple proofs of his modular equations of degree 15.

• Power Cocentralizing Generalized Derivations on Prime Rings

Let 𝑅 be a prime ring, 𝑈 the Utumi quotient ring of $R,C=Z(U)$ the extended centroid of 𝑅,𝐿 a non-central Lie ideal of 𝑅,𝐻 and 𝐺 non-zero generalized derivations of 𝑅. Suppose that there exists an integer $n≥ 1$ such that $(H(u)u-uG(u))^n=0$, for all $u\in L$, then one of the following holds: (1) there exists $c\in U$ such that $H(x)=xc,G(x)=cx;(2)R$ satisfies the standard identity $s_4$ and char (𝑅)=2; (3) 𝑅 satisfies $s4$ and there exist $a, b, c\in U$, such that $H(x)=ax+xc,G(x)=cx+xb$ and $(a-b)^n=0$.

• The Atiyah Bundle and Connections on a Principal Bundle

Let 𝑀 be a 𝐶 manifold and 𝐺 a Lie a group. Let $E_G$ be a 𝐶 principal 𝐺-bundle over 𝑀. There is a fiber bundle $\mathcal{C}(E_G)$ over 𝑀 whose smooth sections correspond to the connections on $E_G$. The pull back of $E_G$ to $\mathcal{C}(E_G)$ has a tautological connection. We investigate the curvature of this tautological connection.

• Some Equivalent Multiresolution Conditions on Locally Compact Abelian Groups

Conditions under which a function generates a multiresolution analysis are investigated. The definition of the spectral function of a shift invariant space is generalized from $\mathbb{R}^n$ to a locally compact abelian group and the union density and intersection triviality properties of a multiresolution analysis are characterized in terms of the spectral functions. Finally, all multiresolution analysis conditions are characterized in terms of the scaling and the spectral functions.

• Eigenvalue Estimates of Positive Integral Operators with Analytic Kernels

In this paper, we exhibit canonical positive definite integral kernels associated with simply connected domains. We give lower bounds for the eigenvalues of the sums of such kernels.

• Regularity of the Interband Light Absorption Coefficient

In this paper we consider the interband light absorption coefficient (ILAC), in a symmetric form, in the case of random operators on the 𝑑-dimensional lattice. We show that the symmetrized version of ILAC is either continuous or has a component which has the same modulus of continuity as the density of states.

• Critical Age-Dependent Branching Markov Processes and their Scaling Limits

This paper studies: (i) the long-time behaviour of the empirical distribution of age and normalized position of an age-dependent critical branching Markov process conditioned on non-extinction; and (ii) the super-process limit of a sequence of age-dependent critical branching Brownian motions.

• On Short Wave Stability and Sufficient Conditions for Stability in the Extended Rayleigh Problem of Hydrodynamic Stability

We consider the extended Rayleigh problem of hydrodynamic stability dealing with the stability of inviscid homogeneous shear flows in sea straits of arbitrary cross section. We prove a short wave stability result, namely, if $k>0$ is the wave number of a normal mode then $k>k_c$ (for some critical wave number $k_c$) implies the stability of the mode for a class of basic flows. Furthermore, if $K(z)=\frac{-({U''}_0-T_0{U'}_0)}{U_0-U_{0s}}$, where $U_0$ is the basic velocity, $T_0$ (a constant) the topography and prime denotes differentiation with respect to vertical coordinate 𝑧 then we prove that a sufficient condition for the stability of basic flow is $0 < K(z)≤\left(\frac{𝜋^2}{D^2}+\frac{T^2_0}{4}\right)$, where the flow domain is $0≤ z≤ D$.

• # Proceedings – Mathematical Sciences

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