pp 1-18
Divisibility of Class Numbers of Imaginary Quadratic Function Fields by a Fixed Odd Number
Pradipto Banerjee Srinivas Kotyada
In this paper we find a new lower bound on the number of imaginary quadratic extensions of the function field $\mathbb{F}_q(x)$ whose class groups have elements of a fixed odd order. More precisely, for 𝑞, a power of an odd prime, and 𝑔 a fixed odd positive integer ≥ 3, we show that for every $\epsilon > 0$, there are $\gg q^{L\left(\frac{1}{2}+\frac{3}{2(g+1)}-\epsilon\right)}$ polynomials $f\in \mathbb{F}_q[x]$ with $\deg f=L$, for which the class group of the quadratic extension $\mathbb{F}_q(x,\sqrt{f})$ has an element of order 𝑔. This sharpens the previous lower bound $q^{L\left(\frac{1}{2}+\frac{1}{g}\right)}$ of Ram Murty. Our result is a function field analogue which is similar to a result of Soundararajan for number fields.
pp 19-25
Density of Primes in 𝑙-th Power Residues
R Balasubramanian Prem Prakash Pandey
Given a prime number 𝑙, a finite set of integers $S=\{a_1,\ldots,a_m\}$ and 𝑚 many 𝑙-th roots of unity $𝜁^{r_i}_l,i=1,\ldots,m$ we study the distribution of primes 𝑝 in $\mathbb{Q}(𝜁_l)$ such that the 𝑙-th residue symbol of $a_i$ with respect to 𝑝 is $𝜁^{r_i}_l$, for all 𝑖. We find out that this is related to the degree of the extension $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$. We give an algorithm to compute this degree. Also we relate this degree to rank of a matrix obtained from $S=\{a_1,\ldots,a_m\}$. This latter argument enables one to describe the degree $\mathbb{Q}\left(a^{\frac{1}{l}}_1,\ldots,a^{\frac{1}{l}}_m\right)/\mathbb{Q}$ in much simpler terms.
pp 27-31
On Commuting Operator Exponentials, II
We prove that, under sufficient conditions on the spectra,
$$e^M e^N\subseteq e^N e^M\Rightarrow MN\subseteq NM,$$
where 𝑁 is an unbounded normal operator and 𝑀 is a bounded normal operator in the Hilbert space.
pp 33-46
The Multiplication Operators on some Analytic Function Spaces of the Unit Ball
We give equivalent definitions of the multipliers of the space of functions of bounded mean oscillation, the Bloch space and their logarithmic counterparts.
pp 47-54
A Note on Stable Teichmüller Quasigeodesics
In this note, we prove that for a cobounded, Lipschitz path $𝛾:I→ \mathcal{T}$ in the Teichmüller space $\mathcal{T}$ of a hyperbolic surface, if the pull back bundle $\mathcal{H}_𝛾→ I$ of the cannonical $\mathbb{H}^2$-bundle $\mathcal{H}→ T$ is a strongly relatively hyperbolic metric space then there exists a geodesic 𝜉 of 𝑇 such that 𝛾(𝐼) and 𝜉 are close to each other.
pp 55-65
One-Parameter Family of Solitons from Minimal Surfaces
In this paper, we discuss a one parameter family of complex Born–Infeld solitons arising from a one parameter family of minimal surfaces. The process enables us to generate a new solution of the B–I equation from a given complex solution of a special type (which are abundant). We illustrate this with many examples. We find that the action or the energy of this family of solitons remains invariant in this family and find that the well-known Lorentz symmetry of the B–I equations is responsible for it.
pp 67-73
Multiplier Convergent Series and Uniform Convergence of Mapping Series
In this paper, we introduce the frame property of complex sequence sets and study the uniform convergence of nonlinear mapping series in 𝛽-dual of spaces consisting of multiplier convergent series.
pp 75-84
Two Remarks on Normality Preserving Borel Automorphisms of $\mathbb{R}^n$
Let 𝑇 be a bijective map on $\mathbb{R}^n$ such that both 𝑇 and 𝑇^{-1} are Borel measurable. For any $𝜃\in\mathbb{R}^n$ and any real 𝑛 × 𝑛 positive definite matrix 𝛴 , let $N(𝜃,𝛴)$ denote the 𝑛-variate normal (Gaussian) probability measure on $\mathbb{R}^n$ with mean vector 𝜃 and covariance matrix 𝛴 . Here we prove the following two results:
Suppose $N(0,I)T^{-1}$ and $N(0,𝛴_j)T^{-1}, 1≤ j≤ n$ are gaussian. Then $T(x)=𝛴_s 1_{E_s}(x)VsUx a.e.x$, where 𝑠 runs over the set of $2^n$ diagonal matrices of order 𝑛 with diagonal entries $± 1,U,V$ are 𝑛 × 𝑛 orthogonal matrices and $\{E_s\}$ is a collection of 2^{𝑛} Borel subsets of $\mathbb{R}^n$ such that $\{E_s\}$ and $\{VsU(E_s)\}$ are partitions of $\mathbb{R}^n$ modulo Lebesgue-null sets and for every $j,VsU𝛴_j(VsU)^{-1}$ is independent of all 𝑠 for which the Lebesgue measure of $E_s$ is positive. The converse of this result also holds.
Our results constitute a sharpening of the results of Nabeya and Kariya (J. Multivariate Anal. 20 (1986) 251–264) and part of Khatri (Sankhyā Ser. A 49 (1987) 395–404).
pp 85-100
Gopal K Basak Arunangshu Biswas
In this paper we show that the continuous version of the self-normalized process $Y_{n,p}(t)=S_n(t)/V_{n,p}+(nt-[nt])X_{[nt]+1}/V_{n,p},0 < t ≤ 1;p>0$ where $S_n(t)=\sum^{[nt]}_{i=1}X_i$ and $V_{(n,p)}=\left(\sum^n_{i=1}|X_i|^p\right)^{1/p}$ and $X_i i.i.d.$ random variables belong to $DA(𝛼)$, has a non-trivial distribution $\mathrm{iff}$ 𝑝=𝛼=2. The case for 2>𝑝>𝛼 and 𝑝 ≤𝛼 < 2 is systematically eliminated by showing that either of tightness or finite dimensional convergence to a non-degenerate limiting distribution does not hold. This work is an extension of the work by Csörgő et al. who showed Donsker’s theorem for $Y_{n,2}(\cdot p)$, i.e., for $p=2$, holds $\mathrm{iff}$ 𝛼=2 and identified the limiting process as a standard Brownian motion in sup norm.
pp 101-143
Multiplicity of Summands in the Random Partitions of an Integer
In this paper, we prove a conjecture of Yakubovich regarding limit shapes of `slices’ of two-dimensional (2D) integer partitions and compositions of 𝑛 when the number of summands $m\sim An^𝛼$ for some 𝐴>0 and $𝛼 < \frac{1}{2}$. We prove that the probability that there is a summand of multiplicity 𝑗 in any randomly chosen partition or composition of an integer 𝑛 goes to zero asymptotically with 𝑛 provided 𝑗 is larger than a critical value. As a corollary, we strengthen a result due to Erdös and Lehner (Duke Math. J. 8(1941) 335–345) that concerns the relation between the number of integer partitions and compositions when $𝛼=\frac{1}{3}$.
pp 145-150
Remark on an Infinite Semipositone Problem with Indefinite Weight and Falling Zeros
G A Afrouzi S Shakeri N T Chung
In this work, we consider the positive solutions to the singular problem
\begin{equation*}\begin{cases}-𝛥 u=am(x)u-f(u)-\frac{c}{u^𝛼} & \text{in}\quad𝛺,\\ u=0 & \text{on}\quad𝜕𝛺,\end{cases}\end{equation*}
where $0 < 𝛼 < 1,a>0$ and $c>0$ are constants, 𝛺 is a bounded domain with smooth boundary 𝜕𝛺,𝛥 is a Laplacian operator, and $f:[0,∞]\longrightarrow\mathbb{R}$ is a continuous function. The weight functions $m(x)$ satisfies $m(x)\in C(𝛺)$ and $m(x)>m_0>0$ for $x\in𝛺$ and also $\|m\|_∞=l < ∞$. We assume that there exist $A>0, M>0,p>1$ such that $alu-M≤ f(u)≤ Au^p$ for all $u\in[0,∞)$. We prove the existence of a positive solution via the method of sub-supersolutions when $m_0 a>\frac{2𝜆_1}{1+𝛼}$ and 𝑐 is small. Here 𝜆_{1} is the first eigenvalue of operator -𝛥 with Dirichlet boundary conditions.
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.