• Volume 112, Issue 4

November 2002,   pages  477-663

• Addendum to A result concerning the stability of some difference equations and its applications" by Hark-Mahn Kim, Vol.112, No.3, pp.453-462

• Generalized (𝑚, 𝑛) Bi-Ideals of a Near-Ring

In this paper we generalize the notion of bi-ideals and obtain equivalent conditions for generalized near-fields in terms of generalized (𝑚, 𝑛) bi-ideals.

• Limits of Rank 4 Azumaya Algebras and Applications to Desingularization

It is shown that the schematic image of the scheme of Azumaya algebra structures on a vector bundle of rank 4 over any base scheme is separated, of finite type, smooth of relative dimension 13 and geometrically irreducible over that base and that this construction base-changes well. This fully generalizes Seshadri's theorem in [16] that the variety of specializations of (2 × 2)-matrix algebras is smooth in characteristic ≠ 2. As an application, a construction of Seshadri in [16] is shown in a characteristic-free way to desingularize the moduli space of rank 2 even degree semi-stable vector bundles on a complete curve. As another application, a construction of Nori over $\mathbb{Z}$ (Appendix, [16]) is extended to the case of a normal domain which is a universally Japanese (Nagata) ring and is shown to desingularize the Artin moduli space [1] of invariants of several matrices in rank 2. This desingularization is shown to have a good specialization property if the Artin moduli space has geometrically reduced fibers – for example this happens over $\mathbb{Z}$. Essential use is made of Kneser's concept [8] of semi-regular quadratic module'. For any free quadratic module of odd rank, a formula linking the half-discriminant and the values of the quadratic form on its radical is derived.

• Representability of $GL_E$

We prove a necessary and sufficient condition for the automorphisms of a coherent sheaf to be representable by a group scheme.

• A General Theorem Characterizing some Absolute Summability Methods

A general theorem is given which gives the necessary and sufficient conditions satisfied by a sequence $(𝜀_n)$ in order to have the series $\sum a_n𝜀_n$ summable to |𝐴| whenever $\sum a_n$ is summable to |𝐴| for some summability method 𝐴.

• Certain Fractional Derivative Formulae Involving the Product of a General Class of Polynomials and the Multivariable 𝐻-Function

In the present paper, we obtain three unified fractional derivative formulae (FDF). The first involves the product of a general class of polynomials and the multivariable 𝐻-function. The second involves the product of a general class of polynomials and two multivariable 𝐻-functions and has been obtained with the help of the generalized Leibniz rule for fractional derivatives. The last FDF also involves the product of a general class of polynomials and the multivariable 𝐻-function but it is obtained by the application of the first FDF twice and it involves two independent variables instead of one. The polynomials and the functions involved in all our fractional derivative formulae as well as their arguments which are of the type $x^ρ\prod_{i = 1}^s(x^{t_i} + 𝛼_i)^{𝜎_i}$ are quite general in nature. These formulae, besides being of very general character have been put in a compact form avoiding the occurrence of infinite series and thus making them useful in applications. Our findings provide interesting unifications and extensions of a number of (new and known) results. For the sake of illustration, we give here exact references to the results (in essence) of five research papers [2, 3, 10, 12, 13] that follow as particular cases of our findings. In the end, we record a new fractional derivative formula involving the product of the Hermite polynomials, the Laguerre polynomials and the product of 𝑟 different Whittaker functions as a simple special case of our first formula.

• On Integral Means of Star-Like Functions

We study univalent holomorphic functions in the unit disk $U = \{z:|z| < 1\}$ of the form $f(z) = z + 𝛴_{n=2}^∞ a_n z^n$ that satisfy the condition Re $zf'(z)/f(z) > 𝛼$ with $𝛼 \in [0, 1)$ in 𝑈. Some integral means of such functions are estimated.

• Approximation by Modified Szasz-Mirakjan Operators on Weighted Spaces

The theorems on weighted approximation and the order of approximation of continuous functions by modified Szasz–Mirakjan operators on all positive semi-axis are established.

• A Complete Analogue of Hardy's Theorem on $SL_2 (\mathbb{R})$ and Characterization of the Heat Kernel

A theorem of Hardy characterizes the Gauss kernel (heat kernel of the Laplacian) on $\mathbb{R}$ from estimates on the function and its Fourier transform. In this article we establish a full group version of the theorem for $SL_2(\mathbb{R})$ which can accommodate functions with arbitrary 𝐾-types. We also consider the heat equation' of the Casimir operator, which plays the role of the Laplacian for the group. We show that despite the structural difference of the Casimir with the Laplacian on $\mathbb{R}^n$ or the Laplace-Beltrami operator on the Riemannian symmetric spaces, it is possible to have a heat kernel. This heat kernel for the full group can also be characterized by Hardy-like estimates.

• Convexity of Spheres in a Manifold without Conjugate Points

For a non-compact, complete and simply connected manifold 𝑀 without conjugate points, we prove that if the determinant of the second fundamental form of the geodesic spheres in 𝑀 is a radial function, then the geodesic spheres are convex. We also show that if 𝑀 is two or three dimensional and without conjugate points, then, at every point there exists a ray with no focal points on it relative to the initial point of the ray. The proofs use a result from the theory of vector bundles combined with the index lemma.

• Stability Estimates for ℎ-𝑝 Spectral Element Methods for Elliptic Problems

In a series of papers of which this is the first we study how to solve elliptic problems on polygonal domains using spectral methods on parallel computers. To overcome the singularities that arise in a neighborhood of the corners we use a geometrical mesh. With this mesh we seek a solution which minimizes a weighted squared norm of the residuals in the partial differential equation and a fractional Sobolev norm of the residuals in the boundary conditions and enforce continuity by adding a term which measures the jump in the function and its derivatives at inter-element boundaries, in an appropriate fractional Sobolev norm, to the functional being minimized. Since the second derivatives of the actual solution are not square integrable in a neighborhood of the corners we have to multiply the residuals in the partial differential equation by an appropriate power of $r_k$, where $r_k$ measures the distance between the point 𝑃 and the vertex $A_k$ in a sectoral neighborhood of each of these vertices. In each of these sectoral neighborhoods we use a local coordinate system $(𝜏_k, 𝜃_k)$ where $𝜏_k = ln r_k$ and $(r_k, 𝜃_k)$ are polar coordinates with origin at $A_k$, as first proposed by Kondratiev. We then derive differentiability estimates with respect to these new variables and a stability estimate for the functional we minimize.

In [6] we will show that we can use the stability estimate to obtain parallel preconditioners and error estimates for the solution of the minimization problem which are nearly optimal as the condition number of the preconditioned system is polylogarithmic in 𝑁, the number of processors and the number of degrees of freedom in each variable on each element. Moreover if the data is analytic then the error is exponentially small in 𝑁.

• Slow Motion of a Sphere Away from a Wall: Effect of Surface Roughness on the Viscous Force

An asymptotic analysis is given for the effect of roughness exhibited through the slip parameter 𝛽 on the motion of the sphere, moving away from a plane surface with velocity 𝑉. The method replaces the no-slip condition at the rough surface by slip condition and employs the method of inner and outer regions on the sphere surface. For 𝛽 > 0, we have the classical slip boundary condition and the results of the paper are then of interest in the microprocessor industry.

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