Volume 108, Issue 1
February 1998, pages 1-94
pp 1-12 February 1998
Torus quotients of homogeneous spaces
We study torus quotients of principal homogeneous spaces. We classify the Grassmannians for which semi-stable=stable and as an application we construct smooth projective varieties as torus quotients of certain homogeneous spaces. We prove the finiteness of the ring ofT invariants of the homogeneous co-ordinate ring of the GrassmannianG_{2,n} (n odd) over the ring generated byR_{1}, the first graded part of the ring ofT invariants.
pp 13-21 February 1998
Finite dimensional imbeddings of harmonic spaces
For a noncompact harmonic manifoldM we establish finite dimensionality of the eigensubspacesV_{γ} generated by radial eigenfunctions of the form coshr+c. As a consequence, for such harmonic manifolds, we give an isometric imbedding ofM into (V_{γ},B), whereB is a nondegenerate symmetric bilinear indefinite form onV_{γ} (analogous to the imbedding of the real hyperbolic spaceH^{n} into ℝ^{n+1} with the indefinite formQ(x,x)=−x_{0}^{2}+Σx_{i}^{2}). This imbedding is minimal in a ‘sphere’ in (V_{γ},B). Finally we give certain conditions under whichM is symmetric.
pp 23-30 February 1998
Remarks on Banaschewski-Fomin-Shanin extensions
Vishvajit V S Gautam Vinod Kumar
The notion ofB^{*}-continuous andB_{c}^{*}-continuous maps is introduced. The problem of epireflection of Banaschewski-Fomin-Shanin extension for a general Hausdorff space is investigated with the help of_{p}B^{*} and_{P}B_{c}^{*}-continuous maps.
pp 31-40 February 1998
Weyl multipliers for invariant Sobelev spaces
Ramakrishnan Radha Sundaram Thangavelu
A concrete characterization for theL^{P}-multipliers (1<p<∞) for the Weyl transform is obtained. This is used to study the Weyl multipliers for Laguerre Sobolev spacesW^{m,p}(ℂ^{n}). A dual space characterization is obtained for the Weyl multiplier classM_{W}(W_{L}^{m,1}(ℂ^{n})).
pp 41-49 February 1998
On the neutrix convolution product ofx_{−}^{r} Inx_{−} andx_{+}^{−3}
The existence of the neutrix convolution product of distributionx_{−}^{r} Inx_{−} andx_{+}^{−3} is proved and some convolution products are evaluated.
pp 51-53 February 1998
A seminorm with square property is automatically submultiplicative
The result stated in the title is proved in a linear associative algebra, answering a problem posed in [3].
pp 55-62 February 1998
A class of convolution integral equations involving a generalized polynomial set
The aim of this paper is to derive a solution of a certain class of convolution integral equation of Fredholm type whose kernel involves a generalized polynomial set. Our main result is believed to be general and unified in nature. A number of (known or new) results follow as special cases, simply by specializing the coefficients and parameters involved in the generalized polynomial set. For the sake of illustration, some special cases are mentioned briefly.
pp 63-68 February 1998
L^{p} inequalities for polynomials with restricted zeros
LetP(z) be a polynomial of degreen which does not vanish in the disk |z|<k. It has been proved that for eachp>0 andk≥1,$$\begin{gathered} \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P^{(s)} (e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} \leqslant n(n - 1) \cdots (n - s + 1) B_p \hfill \\ \times \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {P(e^{i\theta } )} \right|^p d\theta } } \right\}^{1/p} , \hfill \\ \end{gathered} $$ where$$B_p = \left\{ {\frac{1}{{2\pi }}\int_0^{2\pi } {\left| {k^s + e^{i\alpha } } \right|^p d\alpha } } \right\}^{ - 1/p} $$ andP^{(s)}(z) is thesth derivative ofP(z). This result generalizes well-known inequality due to De Bruijn. Asp→∞, it gives an inequality due to Govil and Rahman which as a special case gives a result conjectured by Erdös and first proved by Lax.
pp 69-79 February 1998
Dispersion equation is derived for the propagation of Rayleigh type surface waves in a liquid saturated porous solid layer lying over an inhomogeneous elastic solid half-space. Effect of heterogeneity on the phase velocity is studied by taking different numerical values of heterogeneity factor for particular models. Dispersion curves have been drawn showing the effect of heterogeneity on the phase velocity.
pp 81-94 February 1998
The problem of scattering of Love waves due to the presence of a rigid barrier of finite depth in the crusfal layer of the earth is studied in the present paper. The barrier is in the slightly dissipative surface layer and the surface of the layer is a free surface. The Wiener-Hopf technique is the method of solution. Evaluation of the integrals along appropriate contours in the complex plane yields the reflected, transmitted and the scattered waves. The scattered waves behave as a decaying cylindrical wave at distant points. Numrical computations for the amplitude of the scattered waves have been made versus the wave number. The amplitude falls off rapidly as the wave number increases very slowly.
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