• Volume 105, Issue 2

May 1995,   pages  123-257

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.

• Uncertainty principles on certain Lie groups

There are several ways of formulating the uncertainty principle for the Fourier transform on ℝn. 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 ℝn, the Heisenberg group, the reduced Heisenberg groups and the Euclidean motion group of the plane.

• 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.

• 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.

• Differential sobordination and Bazilevič functions

LetM(z)=zn+…,N(z)=zn+… 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))&lt;0 implies Re(M(z)/N(z))&gt;0 in Δ whenever Re(λ(z))&gt;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)&lt;1+z/1−z impliesp(z)&lt;1+z/1−z, for Reλ(z)&gt;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)&lt;h(z) impliesp(z)&lt;ϕ(z)&lt;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.

• Convolution integral equations involving a general class of polynomials and the multivariableH-function

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 functionF3. 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.

• OnL1-convergence of modified complex trigonometric sums

We study hereL1-convergence of a complex trigonometric sum and obtain a new necessary and sufficient condition for theL1-convergence of Fourier series.

• 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.

• 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.

• 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&lt;1, τ, σ∈(0, ∞),pC([0, ∞), (0, ∞)),fC1(R,R) such thatf is increasing,xf(x)&gt;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$$\mathop {\lim \inf }\limits_{x \to 0} \int_{t - \sigma /n}^t {\sigma ^{n - 1} p\left( s \right)ds &gt; \frac{1}{e}\left( {1 - a} \right)\left( {n - 1} \right)!\left( {\frac{n}{{n - 1}}} \right)^{n - 1} }$$ This result generalizes a recent result of Gopalsamyet al. [6].

• 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.

• Generation and propagation ofSH-type waves due to stress discontinuity in a linear viscoelastic layered medium

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.

• A proof of Howard's conjecture in homogeneous parallel shear flows—II: Limitations of Fjortoft's necessary instability criterion

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 numberkc&gt;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&gt;kc—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].

• # Proceedings – Mathematical Sciences

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