• Volume 112, Issue 2

May 2002,   pages  257-365

• Two-dimensional weak pseudomanifolds on eight vertices

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.

• Sums of two polynomials with each having real zeros symmetric with the other

Consider the polynomial equation$$\prod\limits_{i = 1}^n {(x - r_i )} + \prod\limits_{i = 1}^n {(x + r_i )} = 0,$$ where 0 &lt;r1 ⪯ {irt}2⪯... ⪯rn 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 casesn = 6, 8, 10.

• Some intersections and identifications in integral group rings

LetZG be the integral group ring of a groupG and I(G) its augmentation ideal. For a free groupF andR a normal subgroup ofF, the intersectionIn+1 (F) ∩In+1 (R) is determined for alln≥ 1. The subgroupsF ∩ (1+ZFI (R) I (F) I (S)) ANDF ∩ (1 + I (R)I3 (F)) of F are identified whenR and S are arbitrary subgroups ofF.

• 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 §2-§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 Holder metric (see §5).

• 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.

• Homomorphisms of certain Banach function algebras

In this note, we study homomorphisms with domainDn(X) orLipα(X, d) of which ranges are certain Banach function algebras and determine in which cases these homomorphisms are compact.

• 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).

• Reflected backward stochastic differential equations in an orthant

We consider RBSDE in an orthant with oblique reflection and with time and space dependent coefficients, viz.$$Z(t) = \xi + \int_t^T {b(s, Z(s))} ds + \int_t^T {R(s, Z(s))} dY(s) - \int_t^T {\left\langle {U(s), dB(s)} \right\rangle }$$ with Zi(·)≥0, Yi(·) nondecreasing and Yi(·) increasing only when Zi(·) = 0, 1 ≤i ≤d. Existence of a unique solution is established under Lipschitz continuity ofb, R and a uniform spectral radius condition onR. On the way we also prove a result concerning the variational distance between the ‘pushing parts’ of solutions of auxiliary one-dimensional problem.

• Stability of a bubble expanding and translating through an inviscid liquid

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.