• Volume 106, Issue 3

August 1996,   pages  217-328

• Hardy’s theorem for zeta-functions of quadratic forms

LetQ(u1,…,u1) =Σdijuiuj (i,j = 1 tol) be a positive definite quadratic form inl(≥3) variables with integer coefficientsdij(=dji). Puts=σ+it and for σ&gt;(l/2) write$$Z_Q (s) = \Sigma '(Q(u_1 ,...,u_l ))^{ - s} ,$$ where the accent indicates that the sum is over alll-tuples of integer (u1,…,ul) with the exception of (0,…, 0). It is well-known that this series converges for σ&gt;(l/2) and that (s-(l/2))ZQ(s) can be continued to an entire function ofs. Let σ be any constant with 0&lt;σ&lt;1/100. Then it is proved thatZQ(s)has ≫δTlogT zeros in the rectangle(|σ-1/2|≤δ, T≤t≤2T).

• Degree of approximation of functions associated with Hardy-Littlewood series in the Hölder metric by Euler means

After establishing the Fourier character of the Hardy-Littlewood series the authors have studied the degree of approximation of functions associated with the same series in the Hölder metric using Euler means.

• On the first cohomology of cocompact arithmetic groups

Results of Matsushima and Raghunathan imply that the first cohomology of a cocompact irreducible lattice in a semisimple Lie groupG, with coefficients in an irreducible finite dimensional representation ofG, vanishes unless the Lie group isSO(n, 1) orSU(n, 1) and the highest weight of the representation is an integral multiple of that of the standard representation.

We show here that every cocompact arithmetic lattice inSO(n, 1) contains a subgroup of finite index whose first cohomology is non-zero when the representation is one of the exceptional types mentioned above.

• Solution of singular integral equations with logarithmic and Cauchy kernels

A direct method of solution is presented for singular integral equations of the first kind, involving the combination of a logarithmic and a Cauchy type singularity. Two typical cases are considered, in one of which the range of integration is a single finite interval and, in the other, the range of integration is a union of disjoint finite intervals. More such general equations associated with a finite number (greater than two) of finite, disjoint, intervals can also be handled by the technique employed here.

• Finite part representations of hyper singular integral equation of acoustic scattering and radiation by open smooth surfaces

The Green’s function solution of the Helmholtz's equation for acoustic scattering by hard surfaces and radiation by vibrating surfaces, lead in both the cases, to a hyper singular surface boundary integral equation. Considering a general open surface, a simple proof has been given to show that the integral is to be interpreted like the Hadmard finite part of a divergent integral in one variable. The equation is reformulated as a Cauchy principal value integral equation, but also containing the potential at the control point. It is amenable to numerical treatment by conventional methods. An alternative formulation in the better known form, containing the tangential derivative of the potential is also given. The two dimensional problem for an open arc is separately treated for its simpler feature.

• Eigenvalue bounds for Orr-Sommerfeld equation ‘No backward wave’ theorem

Theoretical estimates of the phase velocityCr of an arbitrary unstable, marginally stable or stable wave derived on the basis of the classical Orr-Sommerfeld eigenvalue problem governing the linear instability of plane Poiseuille flow (U(z)=1−z2,−1≤z≤+1), leave open the possibility of these phase velocities lying outside the rangeUmin&lt;Cr&lt;Umax, but not a single experimental or numerical investigation in this regard, which are concerned with unstable or marginally stable waves has supported such a possibility as yet,Umin andUmax being respectively the minimum and the maximum value ofU(z) forz∈[−1, +1]. This gap between the theory on one side and the experiment and computation on the other has remained unexplained ever since Joseph derived these estimates, first, in 1968, and has even led to the speculation of a negative phase velocity (or rather,Cr&lt;Umin=0) and hence the possibility of a ‘backward’ wave as in the case of the Jeffery-Hamel flow in a diverging channel with back flow ([1]). A simple mathematical proof of the non-existence of such a possibility is given herein by showing that the phase velocityCr of an arbitrary unstable or marginally stable wave must satisfy the inequalityUmin&lt;Cr&lt;Umax. It follows as a consequence stated here in this explicit form for the first time to the best of our knowledge, that ‘overstability’ and not the ‘principle of exchange of stabilities’ is valid for the problem of plane Poiseuille flow.

• Distributed computation of fixed points of ∞-nonexpansive maps

The distributed implementation of an algorithm for computing fixed points of an ∞-nonexpansive map is shown to converge to the set of fixed points under very general conditions.

• Moduli for principal bundles over algebraic curves: I

We classify principal bundles on a compact Riemann surface. A moduli space for semistable principal bundles with a reductive structure group is constructed using Mumford's geometric invarian theory.

• # Proceedings – Mathematical Sciences

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