• Volume 121, Issue 2

May 2011,   pages  111-244

• Group Graded Associated Ideals with Flat Base Change of Rings and Short Exact Sequences

This paper deals with the study of behaviour of 𝐺-associated ideals and strong Krull 𝐺-associated ideals with flat base change of rings and behaviour of 𝐺-associated ideals with short exact sequences over rings graded by finitely generated abelian group 𝐺.

• Reduced Multiplication Modules

An 𝑅-module 𝑀 is called a multiplication module if for each submodule 𝑁 of $M, N=IM$ for some ideal 𝐼 of 𝑅. As defined for a commutative ring 𝑅, an 𝑅-module 𝑀 is said to be reduced if the intersection of prime submodules of 𝑀 is zero. The prime spectrum and minimal prime submodules of the reduced module 𝑀 are studied. Essential submodules of 𝑀 are characterized via a topological property. It is shown that the Goldie dimension of 𝑀 is equal to the Souslin number of Spec $(M)$. Also a finitely generated module 𝑀 is a Baer module if and only if Spec $(M)$ is an extremally disconnected space; if and only if it is a $CS$-module. It is proved that a prime submodule 𝑁 is minimal in 𝑀 if and only if for each $x\in N,\mathrm{Ann}(x)\nsubseteq(N:M)$. When 𝑀 is finitely generated; it is shown that every prime submodule of 𝑀 is maximal if and only if 𝑀 is a von Neumann regular module ($VNM$); i.e., every principal submodule of 𝑀 is a summand submodule. Also if 𝑀 is an injective 𝑅-module, then 𝑀 is a $VNM$

• Irreducible Multivariate Polynomials Obtained from Polynomials in Fewer Variables, II

We provide several irreducibility criteria for multivariate polynomials and methods to construct irreducible polynomials starting from irreducible polynomials in fewer variables.

• Approximation of the Inverse 𝐺-Frame Operator

In this paper, we introduce the concept of (strong) projection method for 𝑔-frames which works for all conditional 𝑔-Riesz frames. We also derive a method for approximation of the inverse 𝑔-frame operator which is efficient for all 𝑔-frames. We show how the inverse of 𝑔-frame operator can be approximated as close as we like using finite-dimensional linear algebra.

• Fusion Frames and 𝐺-Frames in Banach Spaces

Fusion frames and 𝑔-frames in Hilbert spaces are generalizations of frames, and frames were extended to Banach spaces. In this article we introduce fusion frames, 𝑔-frames, Banach 𝑔-frames in Banach spaces and we show that they share many useful properties with their corresponding notions in Hilbert spaces. We also show that 𝑔-frames, fusion frames and Banach 𝑔-frames are stable under small perturbations and invertible operators.

• Principal Bundles whose Restrictions to a Curve are Isomorphic

Let 𝑋 be a normal projective variety defined over an algebraically closed field 𝑘. Let $|O_X(1)|$ be a very ample invertible sheaf on 𝑋. Let 𝐺 be an affine algebraic group defined over 𝑘. Let $E_G$ and $F_G$ be two principal 𝐺-bundles on 𝑋. Then there exists an integer $n \gg 0$ (depending on $E_G$ and $F_G$) such that if the restrictions of $E_G$ and $F_G$ to a curve $C\in |O_X(n)|$ are isomorphic, then they are isomorphic on all of 𝑋.

• Real Hypersurfaces of a Complex Projective Space

In this paper, we classify real hypersurfaces in the complex projective space $CP\frac{n+1}{2}$ whose structure vector field is a 𝜑-analytic vector field (a notion similar to analytic vector fields on complex manifolds). We also define Jacobi-type vector fields on a Riemannian manifold and classify real hypersurfaces whose structure vector field is a Jacobi-type vector field.

• On Conformal Minimal 2-Spheres in Complex Grassmann Manifold $G(2,n)$

For a harmonic map 𝑓 from a Riemann surface into a complex Grassmann manifold, Chern and Wolfson [4] constructed new harmonic maps $\partial f$ and $\overline{\partial} f$ through the fundamental collineations 𝜕 and $\overline{\partial}$ respectively. In this paper, we study the linearly full conformal minimal immersions from $S^2$ into complex Grassmannians $G(2,n)$, according to the relationships between the images of $\partial f$ and $\overline{\partial}f$. We obtain various pinching theorems and existence theorems about the Gaussian curvature, Kähler angle associated to the given minimal immersions, and characterize some immersions under special conditions. Some examples are given to show that the hypotheses in our theorems are reasonable.

• A Note on Existence and Stability of Solutions for Semilinear Dirichlet Problems

We provide existence and stability results for a fourth-order semilinear Dirichlet problem in the case when both the coefficients of the differential operator and the nonlinear term depend on the numerical parameter. We use a dual variational method.

• The almost Sure Local Central Limit Theorem for the Product of Partial Sums

We derive under some regular conditions an almost sure local central limit theorem for the product of partial sums of a sequence of independent identically distributed positive random variables.

• Bias Expansion of Spatial Statistics and Approximation of Differenced Lattice Point Counts

Investigations of spatial statistics, computed from lattice data in the plane, can lead to a special lattice point counting problem. The statistical goal is to expand the asymptotic expectation or large-sample bias of certain spatial covariance estimators, where this bias typically depends on the shape of a spatial sampling region. In particular, such bias expansions often require approximating a difference between two lattice point counts, where the counts correspond to a set of increasing domain (i.e., the sampling region) and an intersection of this set with a vector translate of itself. Non-trivially, the approximation error needs to be of smaller order than the spatial region’s perimeter length. For all convex regions in 2-dimensional Euclidean space and certain unions of convex sets, we show that a difference in areas can approximate a difference in lattice point counts to this required accuracy, even though area can poorly measure the lattice point count of any single set involved in the difference. When investigating large-sample properties of spatial estimators, this approximation result facilitates direct calculation of limiting bias, because, unlike counts, differences in areas are often tractable to compute even with non-rectangular regions. We illustrate the counting approximations with two statistical examples.

• # Proceedings – Mathematical Sciences

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