Volume 105, Issue 2
May 1995, pages 123-257
pp 123-134 May 1995
Badly approximablep-adic integers
It is known that thep-adic integers that are badly approximable by rationals form a null set with respect to Haar measure. We define a [0,1]-valued dimension function on thep-adic integers analogous to Hausdorff dimension inR and show that with respect to this function the dimension of the set of badly approximablep-adic integers is 1.
pp 135-151 May 1995
Uncertainty principles on certain Lie groups
A Sitaram M Sundari S Thangavelu
There are several ways of formulating the uncertainty principle for the Fourier transform on ℝ^{n}. Roughly speaking, the uncertainty principle says that if a functionf is ‘concentrated’ then its Fourier transform$$\tilde f$$ cannot be ‘concentrated’ unlessf is identically zero. Of course, in the above, we should be precise about what we mean by ‘concentration’. There are several ways of measuring ‘concentration’ and depending on the definition we get a host of uncertainty principles. As several authors have shown, some of these uncertainty principles seem to be a general feature of harmonic analysis on connected locally compact groups. In this paper, we show how various uncertainty principles take form in the case of some locally compact groups including ℝ^{n}, the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.
pp 153-156 May 1995
On subsemigroups of semisimple Lie groups
In this paper we classify the subsemigroups of any connected semisimple Lie groupG which areK-bi-invariant, whereG=KAN is an Iwasawa decomposition ofG.
pp 157-167 May 1995
Induced representation and Frobenius reciprocity for compact quantum groups
Unitary representations of compact quantum groups have been described as isometric comodules. The notion of an induced representation for compact quantum groups has been introduced and an analogue of the Frobenius reciprocity theorem is established.
pp 169-186 May 1995
Differential sobordination and Bazilevič functions
LetM(z)=z^{n}+…,N(z)=z^{n}+… be analytic in the unit disc Δ and let λ(z)=N(z)/zN′(z). The classical result of Sakaguchi-Libera shows that Re(M′(z)/N′(z))<0 implies Re(M(z)/N(z))>0 in Δ whenever Re(λ(z))>0 in Δ. This can be expressed in terms of differential subordination as follows: for anyp analytic in Δ, withp(0)=1,p(z)+λ(z)zp′(z)<1+z/1−z impliesp(z)<1+z/1−z, for Reλ(z)>0,z∈Δ.
In this paper we determine different type of general conditions on λ(z),h(z) and ϕ(z) for which one hasp(z)+λ(z)zp′(z)<h(z) impliesp(z)<ϕ(z)<h(z) z∈Δ. Then we apply the above implication to obtain new theorems for some classes of normalized analytic funotions. In particular we give a sufficient condition for an analytic function to be starlike in Δ.
pp 187-192 May 1995
K C Gupta Rashmi Jain Pawan Agrawal
In this paper we first solve a convolution integral equation involving product of the general class of polynomials and theH-function of several variables. Due to general nature of the general class of polynomials and theH-function of several variables which occur as kernels in our main convolution integral equation, we can obtain from it solutions of a large number of convolution integral equations involving products of several useful polynomials and special functions as its special cases. We record here only one such special case which involves the product of general class of polynomials and Appell's functionF_{3}. We also give exact references of two results recently obtained by Srivastavaet al [10] and Rashmi Jain [3] which follow as special cases of our main result.
pp 193-199 May 1995
OnL^{1}-convergence of modified complex trigonometric sums
Satvinder Singh Bhatia Babu Ram
We study hereL^{1}-convergence of a complex trigonometric sum and obtain a new necessary and sufficient condition for theL^{1}-convergence of Fourier series.
pp 201-205 May 1995
Absolute summability of infinite series
It is shown in [4] that if a normal matrix,A satisfies some conditions then |C,1|_{k} summability implies |A|_{k} summability wherek≥1. In the present paper, we consider the converse implication.
pp 207-218 May 1995
Solution of a system of nonstrictly hyperbolic conservation laws
In this paper we study a special case of the initial value problem for a 2×2 system of nonstrictly hyperbolic conservation laws studied by Lefloch, whose solution does not belong to the class ofL^{∞} functions always but may contain δ-measures as well: Lefloch's theory leaves open the possibility of nonuniqueness for some initial data. We give here a uniqueness criteria to select the entropy solution for the Riemann problem. We write the system in a matrix form and use a finite difference scheme of Lax to the initial value problem and obtain an explicit formula for the approximate solution. Then the solution of initial value problem is obtained as the limit of this approximate solution.
pp 219-225 May 1995
Oscillation in odd-order neutral delay differential equations
Consider the odd-order functional differential equation$$\left( {x\left( t \right) - ax\left( {t - \tau } \right)} \right)^n + p\left( t \right)f\left( {x\left( {t - \sigma } \right)} \right) = 0$$ where 0≤α<1, τ, σ∈(0, ∞),p∈C([0, ∞), (0, ∞)),f∈C^{1}(R,R) such thatf is increasing,xf(x)>0 forx≠0 andf satisfies a generalized linear condition$$\mathop {\lim \inf }\limits_{x \to 0} \left| {\frac{{df}}{{dx}}} \right| = 1$$ Here we prove that every solution of (*) oscillates if$$\mathop {\lim \inf }\limits_{x \to 0} \int_{t - \sigma /n}^t {\sigma ^{n - 1} p\left( s \right)ds > \frac{1}{e}\left( {1 - a} \right)\left( {n - 1} \right)!\left( {\frac{n}{{n - 1}}} \right)^{n - 1} } $$ This result generalizes a recent result of Gopalsamyet al. [6].
pp 227-239 May 1995
Surface waves due to blasts on and above inviscid liquids of finite depth
For the problem of waves due to an explosion above the surface of a homogeneous ocean of finite depth, asymptotic expressions of the velocity potential and the surface displacement are determined for large times and distances from the pressure area produced by the incident shock. It is shown that the first item in Sakurai's approximation scheme for the pressure field inside the, blast wave as well as the results of Taylor's point blast theory can be used to yield realistic expressions of surface displacement. Some interesting features of the wave motion in general are described. Finally some numerical calculations for the surface elevation were performed and included as a particular case.
pp 241-249 May 1995
In this paper the generation and propagation ofSH-type waves due to stress discontinuíty in a linear viscoelastic layered medium is studied. Using Fourier transforms and complex contour integration technique, the displacement is evaluated at the free surface in closed form for two special types of stress discontinuity created at the interface. The numerical result for displacement component is evaluated for different values of nondimensional station (distance) and is shown graphically. Graphs are compared with the corresponding graph of classical elastic case.
pp 251-257 May 1995
Mihir B Banerjee R G Shandil Vinay Kanwar
The present paper on the linear instability of nonviscous homogeneous parallel shear flows mathematically demonstrates the correctness of Howard's [4] prediction, for a class of velocity distributions specified by a monotone functionU of the altitudey and a single point of inflexion in the domain of flow, by showing not only the existence of a critical wave numberk_{c}>0 but also deriving an explicit expression for it, beyond which for all wave numbers the manifesting perturbations attain stability. An exciting conclusion to which the above result leads to is that the necessary instability criterion of Fjortoft has the seeds of its own destruction in the entire range of wave numbersk>k_{c}—a result which is not at all evident either from the criterion itself or from its derivation and has thus remained undiscovered ever since Fjortoft enunciated [3].
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