• Volume 119, Issue 1

February 2009,   pages  1-135

• On an Extension of a Combinatorial Identity

Using Frobenius partitions we extend the main results of [4]. This leads to an infinite family of 4-way combinatorial identities. In some particular cases we get even 5-way combinatorial identities which give us four new combinatorial versions of Göllnitz–Gordon identities.

• Hyperbolic Unit Groups and Quaternion Algebras

We classify the quadratic extensions $K=\mathbb{Q}[\sqrt{d}]$ and the finite groups 𝐺 for which the group ring $\mathfrak{o}_K[G]$ of 𝐺 over the ring $\mathfrak{o}_K$ of integers of 𝐾 has the property that the group $\mathcal{U}_1(\mathfrak{o}_K[G])$ of units of augmentation 1 is hyperbolic. We also construct units in the $\mathbb{Z}$-order $\mathcal{H}(\mathfrak{o}_K)$ of the quaternion algebra $\mathcal{H}(K)=\left\frac{-1,-1}{k}(\right)$, when it is a division algebra.

• Vanishing of the Top Local Cohomology Modules over Noetherian Rings

Let 𝑅 be a (not necessarily local) Noetherian ring and 𝑀 a finitely generated 𝑅-module of finite dimension 𝑑. Let $\mathfrak{a}$ be an ideal of 𝑅 and $\mathfrak{M}$ denote the intersection of all prime ideals $\mathfrak{p}\in\mathrm{Supp}_R H^d_a(M)$. It is shown that

$$H^d_a(M)\simeq H^d_{\mathfrak{M}}(M)/\sum\limits_{n\in\mathbb{N}}\langle \mathfrak{M}\rangle(0:_{H^d_{\mathfrak{M}}(M)}a^n),$$

where for an Artinian 𝑅-module 𝐴 we put $\langle\mathfrak{M}\rangle A=\cap_{n\in\mathbb{N}}\mathfrak{M}^n A$. As a consequence, it is proved that for all ideals $\mathfrak{a}$ of 𝑅, there are only finitely many non-isomorphic top local cohomology modules $H^d_a(M)$ having the same support. In addition, we establish an analogue of the Lichtenbaum–Hartshorne vanishing theorem over rings that need not be local.

• On the Zeros of a Polynomial

For a polynomial of degree 𝑛, we have obtained an upper bound involving coefficients of the polynomial, for moduli of its 𝑝 zeros of smallest moduli, and then a refinement of the well-known Eneström–Kakeya theorem (under certain conditions).

• On the Theorem of 𝑀 Golomb

Let $X_1,\ldots,X_n$ be compact spaces and $X=X_1×\cdots× X_n$. Consider the approximation of a function $f\in C(X)$ by sums $g_1(x_1)+\cdots+g_n(x_n)$, where $g_i\in C(X_i),i=1,\ldots,n$. In [8], Golomb obtained a formula for the error of this approximation in terms of measures constructed on special points of 𝑋, called projection cycles’. However, his proof had a gap, which was pointed out by Marshall and O’Farrell [15]. But the question if the formula was correct, remained open. The purpose of the paper is to prove that Golomb’s formula holds in a stronger form.

• Hölder Seminorm Preserving Linear Bijections and Isometries

Let (𝑋, 𝑑) be a compact metric and $0 < 𝛼 < 1$. The space $\mathrm{Lip}^𝛼(X)$ of Hölder functions of order 𝛼 is the Banach space of all functions 𝑓 from 𝑋 into $\mathbb{K}$ such that $\| f\|=\max \{\| f\|_∞,L(f)\} <∞$, where

$$L(f)=\sup\{|f(x)-f(y)|/d^𝛼(x,y):x,y\in X, x≠ y\}$$

is the Hölder seminorm of 𝑓. The closed subspace of functions 𝑓 such that

$$\lim\limits_{d(x,y)→ 0}|f(x)-f(y)|/d^𝛼(x,y)=0$$

is denoted by $\mathrm{lip}^𝛼(X)$. We determine the form of all bijective linear maps from $\mathrm{lip}^𝛼(X)$ onto $\mathrm{lip}^𝛼(Y)$ that preserve the Hölder seminorm.

• On Equivariant Embedding of Hilbert 𝐶* Modules

We prove that an arbitrary (not necessarily countably generated) Hilbert $G-\mathcal{A}$ module on a 𝐺-𝐶* algebra $\mathcal{A}$ admits an equivariant embedding into a trivial $G-\mathcal{A}$ module, provided 𝐺 is a compact Lie group and its action on $\mathcal{A}$ is ergodic.

• On Kähler–Norden Manifolds-Erratum

This paper is concerned with the problem of the geometry of Norden manifolds. Some properties of Riemannian curvature tensors and curvature scalars of Kähler–Norden manifolds using the theory of Tachibana operators is presented.

• Torus Quotients of Homogeneous Spaces of the General Linear Group and the Standard Representation of Certain Symmetric Groups

We give a stratification of the $GIT$ quotient of the Grassmannian $G_{2,n}$ modulo the normaliser of a maximal torus of $SL_n(k)$ with respect to the ample generator of the Picard group of $G_{2,n}$. We also prove that the flag variety $GL_n(k)/B_n$ can be obtained as a $GIT$ quotient of $GL_{n+1}(k)/B_{n+1}$ modulo a maximal torus of $SL_{n+1}(k)$ for a suitable choice of an ample line bundle on $GL_{n+1}(k)/B_{n+1}$.

• On the Torus Cobordant Cohomology Spheres

Let 𝐺 be a compact Lie group. In 1960, P A Smith asked the following question: Is it true that for any smooth action of 𝐺 on a homotopy sphere with exactly two fixed points, the tangent 𝐺-modules at these two points are isomorphic?" A result due to Atiyah and Bott proves that the answer is yes’ for $\mathbb{Z}_p$ and it is also known to be the same for connected Lie groups. In this work, we prove that two linear torus actions on $S^n$ which are 𝑐-cobordant (cobordism in which inclusion of each boundary component induces isomorphisms in $\mathbb{Z}$-cohomology) must be linearly equivalent. As a corollary, for connected case, we prove a variant of Smith’s question.

• On some Frobenius Restriction Theorems for Semistable Sheaves

We prove a version of an effective Frobenius restriction theorem for semistable bundles in characteristic 𝑝. The main novelty is in restricting the bundle to the 𝑝-fold thickening of a hypersurface section. The base variety is 𝐺/𝑃, an abelian variety or a smooth projective toric variety.

• On Upper Bounds for the Growth Rate in the Extended Taylor–Goldstein Problem of Hydrodynamic Stability

For the extended Taylor–Goldstein problem of hydrodynamic stability governing the stability of shear flows of an inviscid, incompressible but density stratified fluid in sea straits of arbitrary cross-section a new estimate for the growth rate of an arbitrary unstable normal mode is given for a class of basic flows. Furthermore the Howard’s conjecture, namely, the growth rate $kc_i→ 0$ as the wave number $k→∞$ is proved for two classes of basic flows.

• # Proceedings – Mathematical Sciences

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