Volume 112, Issue 2
May 2002, pages 257-365
pp 257-281
Two-Dimensional Weak Pseudomanifolds on Eight Vertices
Basudeb Datta Nandini Nilakantan
We explicitly determine all the two-dimensional weak pseudomanifolds on 8 vertices. We prove that there are (up to isomorphism) exactly 95 such weak pseudomanifolds, 44 of which are combinatorial 2-manifolds. These 95 weak pseudomanifolds triangulate 16 topological spaces. As a consequence, we prove that there are exactly three 8-vertex two-dimensional orientable pseudomanifolds which allow degree three maps to the 4-vertex 2-sphere.
pp 283-288
Sums of Two Polynomials with Each having Real Zeros Symmetric with the Other
Consider the polynomial equation
$$\prod_{i=1}^n(x-r_i)+\prod_{i=1}^n(x+r_i)=0,$$
where $0 < r_1 ≤ r_2 ≤ \cdots ≤ r_n$. All zeros of this equation lie on the imaginary axis. In this paper, we show that no two of the zeros can be equal and the gaps between the zeros in the upper half-plane strictly increase as one proceeds upward. Also we give some examples of geometric progressions of the zeros in the upper half-plane in cases 𝑛 = 6, 8, 10.
pp 289-297
Some Intersections and Identifications in Integral Group Rings
Let 𝑍𝐺 be the integral group ring of a group 𝐺 and 𝐼(𝐺) its augmentation ideal. For a free group 𝐹 and 𝑅 a normal subgroup of 𝐹, the intersection $I^{n+1}(F) \cap I^n(R)$ is determined for all 𝑛 ≥ 1. The subgroups $F \cap (1+ZFI(R)I(F)I(S))$ and $F \cap (1+I(R)I^3(F))$ of 𝐹 are identified when 𝑅 and 𝑆 are arbitrary subgroups of 𝐹.
pp 299-319
A New Trigonometric Method of Summation and its Application to the Degree of Approximation
The object of the present investigation is to introduce a new trigonometric method of summation which is both regular and Fourier effective and determine its status with reference to other methods of summation (see $\mathcal{x}$2–$\mathcal{x}$4) and also give an application of this method to determine the degree of approximation in a new Banach space of functions conceived as a generalized Hölder metric (see $\mathcal{x}$5).
pp 321-330
The Heat Kernel and Hardy's Theorem on Symmetric Spaces of Noncompact Type
For symmetric spaces of noncompact type we prove an analogue of Hardy's theorem which characterizes the heat kernel in terms of its order of magnitude and that of its Fourier transform.
pp 331-336
Homomorphisms of Certain Banach Function Algebras
In this note, we study homomorphisms with domain $D^n(X)$ or $Lip_𝛼(X, d)$ of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.
pp 337-346
On the Limit Matrix Obtained in the Homogenization of an Optimal Control Problem
A new formulation for the limit matrix occurring in the cost functional of an optimal control problem on homogenization is obtained. It is used to obtain an upper bound for this matrix (in the sense of positive definite matrices).
pp 347-360
Reflected Backward Stochastic Differential Equations in an Orthant
We consider RBSDE in an orthan with oblique reflection and with time and space dependent coefficients, viz.
$$Z(t)=𝜉 +\int_t^T b(s, Z(s))ds + \int_t^T R(s, Z(s))dY(s)-\int_t^T\langle U(s), dB(s)\rangle$$
with $Z_i(\cdot p)≥ 0, Y_i(\cdot p)$ nondecreasing and $Y_i(\cdot p)$ increasing only when $Z_i(\cdot p) = 0, 1 ≤ i ≤ d$. Existence of a unique solution is established under Lipschitz continuity of 𝑏, 𝑅 and a uniform spectral radius condition on 𝑅. On the way we also prove a result concerning the variational distance between the `pushing parts' of solutions of auxiliary one-dimensional problem.
pp 361-365
Stability of a Bubble Expanding and Translating Through an Inviscid Liquid
Dinesh Khattar B B Chakraborty
A bubble expands adiabatically and translates in an incompressible and inviscid liquid. We investigate the number of equilibrium points of the bubble and the nature of stability of the bubble at these points. We find that there is only one equilibrium point and the bubble is stable there.
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