pp 127-140
A Statistic on 𝑛-Color Compositions and Related Sequences
A composition of a positive integer in which a part of size 𝑛 may be assigned one of 𝑛 colors is called an 𝑛-color composition. Let $a_m$ denote the number of 𝑛-color compositions of the integer 𝑚. It is known that $a_m = F_{2m}$ for all 𝑚 ≥ 1, where $F_m$ denotes the Fibonacci number defined by $F_m = F_{m-1}+F_{m-2}$ if 𝑚 ≥ 2, with $F_0=0$ and $F_1=1$. A statistic is studied on the set of 𝑛-color compositions of 𝑚 thus providing a polynomial generalization of the sequence $F_{2m}$. The statistic may be described, equivalently, in terms of statistics on linear tilings and lattice paths. The restriction to the set of 𝑛-color compositions having a prescribed number of parts is considered and an explicit formula for the distribution is derived. We also provide 𝑞-generalizations of relations between $a_m$ and the number of self-inverse 𝑛-compositions of 2𝑚+1 or 2𝑚. Finally, we consider a more general recurrence than that satisfied by the numbers $a_m$ and note some particular cases.
pp 141-154
Repdigits in 𝑘-Lucas Sequences
For an integer 𝑘 ≥ 2, let $(L_n^{(k)})_n$ be the 𝑘-Lucas sequence which starts with $0,\ldots,0,2,1$ (𝑘 terms) and each term afterwards is the sum of the 𝑘 preceding terms. In 2000, Luca (Port. Math. 57(2) 2000 243-254) proved that 11 is the largest number with only one distinct digit (the so-called repdigit) in the sequence $(L_n^{(2)})_n$. In this paper, we address a similar problem in the family of 𝑘-Lucas sequences. We also show that the 𝑘-Lucas sequences have similar properties to those of 𝑘-Fibonacci sequences and occur in formulae simultaneously with the latter.
pp 155-167
Gaussian Curvature on Hyperelliptic Riemann Surfaces
Let 𝐶 be a compact Riemann surface of genus $g ≥ 1, 𝜔_1,\ldots,𝜔_g$ be a basis of holomorphic 1-forms on 𝐶 and let $H=(h_{ij})^g_{i,j=1}$ be a positive definite Hermitian matrix. It is well known that the metric defined as $ds_H^2=\sum^g_{i,j=1}h_{ij}𝜔_i\otimes \overline{𝜔_j}$ is a K\"a hler metric on 𝐶 of non-positive curvature. Let $K_H:C→ \mathbb{R}$ be the Gaussian curvature of this metric. When 𝐶 is hyperelliptic we show that the hyperelliptic Weierstrass points are non-degenerated critical points of $K_H$ of Morse index +2. In the particular case when 𝐻 is the 𝑔 × 𝑔 identity matrix, we give a criteria to find local minima for $K_H$ and we give examples of hyperelliptic curves where the curvature function $K_H$ is a Morse function.
pp 169-173
On 𝐼𝐴-Automorphisms that Fix the Centre Element-Wise
Let 𝐺 be a group. An automorphism of 𝐺 is called an $IA$-automorphism if it induces the identity mapping on $G/𝛾 2(G)$, where $𝛾 2(G)$ is the commutator sub-group of 𝐺. Let $IA_z(G)$ be the group of those $IA$-automorphisms, which fix the centre element-wise and let Autcent $(G)$ be the group of central automorphisms, the automorphisms that induce the identity mapping on the central quotient. It can be observed that Autcent $(G)=C_{\mathrm{Aut}(G)}(IA_z(G))$. We prove that $IA_z(G)$ and $IA_z(H)$ are isomorphic for any two finite isoclinic groups 𝐺 and 𝐻. Also, for a finite 𝑝-group 𝐺, we give a necessary and sufficient condition to ensure that $IA_z(G)=\mathrm{Autcent}(G)$.
pp 175-178
Dirichlet Problem on the Upper Half Space
In this paper, a solution of the Dirichlet problem on the upper half space for a fast growing continuous boundary function is constructed by the generalized Dirichlet integral with this boundary function.
pp 179-191
Nonexistence and Existence of Solutions for a Fourth-Order Discrete Mixed Boundary Value Problem
pp 193-203
Boundedness for Marcinkiewicz Integrals Associated with Schr\"odinger Operators
Let $L=-𝛥 +V$ be a Schr\"odinger operator, where 𝛥 is the Laplacian on $\mathbb{R}^n$, while nonnegative potential 𝑉 belongs to the reverse H\"older class. In this paper, we will show that Marcinkiewicz integral associated with Schr\"odinger operator is bounded on $BMO_L$, and from $H^1_L(\mathbb{R}^n)$ to $L^1(\mathbb{R}^n)$.
pp 205-224
Perturbation of Operators and Approximation of Spectrum
K Kumar M N N Namboodiri S Serra-Capizzano
Let $A(x)$ be a norm continuous family of bounded self-adjoint operators on a separable Hilbert space $\mathbb{H}$ and let $A(x)_n$ be the orthogonal compressions of $A(x)$ to the span of first 𝑛 elements of an orthonormal basis of $\mathbb{H}$. The problem considered here is to approximate the spectrum of $A(x)$ using the sequence of eigenvalues of $A(x)_n$. We show that the bounds of the essential spectrum and the discrete spectral values outside the bounds of essential spectrum of $A(x)$ can be approximated uniformly on all compact subsets by the sequence of eigenvalue functions of $A(x)_n$. The known results, for a bounded self-adjoint operator, are translated into the case of a norm continuous family of operators. Also an attempt is made to predict the existence of spectral gaps that may occur between the bounds of essential spectrum of $A(0)=A$ and study the effect of norm continuous perturbation of operators in the prediction of spectral gaps. As an example, gap issues of some block Toeplitz–Laurent operators are discussed. The pure linear algebraic approach is the main advantage of the results here.
pp 225-233
Homogeneous Bilateral Block Shifts
A new 3-parameter family of homogeneous 2-by-2 block shifts is described. These are the first examples of irreducible homogeneous bilateral block shifts of block size larger than 1.
pp 235-242
Using a generalized spherical mean operator, we define generalized modulus of smoothness in the space $L^2_k(\mathbb{R}^d)$. Based on the Dunkl operator we define Sobolev-type space and 𝐾-functionals. The main result of the paper is the proof of the equivalence theorem for a 𝐾-functional and a modulus of smoothness for the Dunkl transform on $\mathbb{R}^d$.
pp 243-253
On a Generalization of 𝐵_{1}(𝛺) on 𝐶*-Algebras
We discuss the unitary classification problem of a class of holomorphic curves on 𝐶*-algebras. It can been regarded as a generalization of Cowen–Doulgas operators with index one.
pp 255-265
A Note on Automorphisms of the Sphere Complex
In this note, we shall give another proof of a theorem of Aramayona and Souto, namely the group of simplicial automorphisms of the sphere complex $\mathbb{S}(M)$ associated to the manifold $M=\sharp_nS^2× S^1$ is isomorphic to the group Out $(F_n)$ of outer automorphisms of the free group $F_n$ of rank 𝑛 ≥ 3.
pp 267-279
For weighted sums of sequences of asymptotically almost negatively associated (AANA) random variables, we study the complete moment convergence by using the Rosenthal type moment in equalities. Our results extend the corresponding ones for sequences of independently identically distributed random variables of Chow [4].
Current Issue
Volume 127 | Issue 5
November 2017
© 2017 Indian Academy of Sciences, Bengaluru.