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Atanu Manna

Publications and source records attributed to Atanu Manna.

16 recordsLinked to original sources
arXiv2026

Existence and non-existence of nonradial solutions to an elliptic equation on hyperbolic space with critical and subcritical nonlinearities

In this article, we study the semi-linear elliptic problem: \begin{equation*} -\Delta_{\mathbb{B}^N} u \, - \, \lambda u = |u|^{p-1}u, \quad u \in H^{1}(\mathbb{B}^N), \end{equation*} where $N \geq 3$, $\lambda > \frac{N(N-2)}{4}$, and $1 < p \leq 2^*-1$. Here, $\mathbb{B}^N$ represents the Poincar\'e ball model of the hyperbolic space, and $H^{1}(\mathbb{B}^N)$ denotes the Sobolev space on $\mathbb{B}^N$. In the critical case $p = 2^*-1$, with $\lambda < \frac{(N-1)^2}{4}$ and $N \geq 4$, we establish the existence and multiplicity of nonradial sign-changing solutions. Moreover, our result answers an open question: ``the existence of sign-changing solutions in the lower dimensions $4 \leq N \leq 6$". We also prove a partial non-existence result for a large class of symmetric solutions when $\lambda > \frac{(N-1)^2}{4}$, based on a quantitative orbit packing condition.

math.AP↗
arXiv2026

A scalar field equation on hyperbolic space with indefinite sign nonlinearity

In this article, we study threshold phenomena for the semilinear double-power elliptic equation $$-\Delta_{\mathbb{B}^N} u - \lambda u = |u|^{p-1}u - |u|^{q-1}u, \quad u \in H^1(\mathbb{B}^N),$$ on the hyperbolic space $\mathbb{B}^N$ for $N \ge 3$. For parameters $1 < p \le 2^*-1$ (though we occasionally allow for supercritical exponents) and $q > 0$, we seek to identify the optimal spectral regimes for $\lambda \in \mathbb{R}$ that delineate the existence and non-existence of positive-energy solutions. We achieve a complete resolution of these thresholds across all exponent configurations: $p < q$, $0 < q < 1 < p$, and $1 < q < p$. Our results demonstrate that the boundary separating these regimes is governed by an explicit critical spectral parameter, which depends on $p$, $q$, and $N$ in the regime where $p < q$, but depends solely on $N$ in the remaining cases.

math.AP↗
arXiv2025

Extending Hridaya Kolam to Multiple Loops: A Study of Non-Coprime Dot--Arm Structures

This paper extends Hridaya Kolam patterns to cases where the number of dots ($m$) and arms ($n$) are not coprime, i.e., $\gcd(m, n) \ne 1$. Such configurations give rise to multiple disjoint closed loops. We propose a modular-arithmetic-based algorithm to systematically generate such patterns, and illustrative patterns for various non-coprime $(m, n)$ pairs are provided to demonstrate the resulting multi-loop structures. These multi-loop Kolam designs can inspire architectural motifs and ornamental patterns in floor plans, facades, and decorative elements.

math.GM↗
arXiv2025

An improved Copson inequality

In this paper, we prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 49-51) of one-dimension in general cases admits an improvement. In fact we study the improvement of the following Copson's inequality \begin{align*} &\displaystyle\sum_{n=1}^{\infty}\frac{Q_{n}^{\alpha}|A_n-A_{n-1}|^{2}}{q_{n}}\geq\frac{(\alpha-1)^2}{4}\displaystyle\sum_{n=1}^{\infty} \frac{q_{n}}{Q_{n}^{2-\alpha}}|A_{n}|^{2}, \end{align*}where $\alpha\in[0,1)$, $A_{n}=q_{1}a_{1}+ q_{2}a_{2}+ \ldots +q_{n}a_{n}$, $Q_{n}=q_1+q_2+\ldots+q_{n}$ for $n\in \mathbb{N}$, $\{q_n\}$ is a positive real sequence and $\{a_n\}$ is a sequence of complex numbers. We show that if $\{q_n\}$ is decreasing then the above inequality has an improvement for $\alpha\in [1/3, 1)$. We also prove that for some increasing sequences $\{q_n\}$ the above inequality can also be improved. Indeed, we prove that for $q_{n}=n$ and $q_n=n^3$, $n\in \mathbb{N}$ the corresponding Copson inequalities admit an improvement for $\alpha\in[\frac{17}{50}, 1)$ and $\alpha\in[0, \frac{1}{2}]$, respectively. Further, we show that in case of $q_{n}=1$, $n\in \mathbb{N}$ the reduced Copson inequality (known as Hardy's inequality with power weights) has achieved an improvement for $\alpha\in[0, 1)$.

math.CA↗
arXiv2025

Extending Hridaya Kolam to Even-Ordered Dot Patterns and Their Applications

This study extends the mathematical framework of Hridaya Kolam patterns by applying modular arithmetic to even-ordered dot arrangements with arm counts co-prime to the number of dots. We analyze the resulting cyclic sequences that correspond to Eulerian circuits, enabling continuous single-stroke kolam designs beyond the classical odd-ordered cases. Our method provides explicit algorithms for constructing these intricate patterns, unveiling new symmetries and structural properties. Elevating this traditional floor art, we translate these mathematically grounded motifs into striking designs, showcasing their beauty and complexity in contemporary dari art in the carpet and textile sectors.

math.GM↗
arXiv2025

Existence of a bi-radial sign-changing solution for Hardy-Sobolev-Mazya type equation

In this article, we study the following Hardy-Sobolev-Maz'ya type equation: \begin{equation} -\Delta u - \mu \frac{u}{|z|^2} = \frac{|u|^{q-2}u}{|z|^t}, \quad u \in D^{1,2} (\mathbb{R}^n), \end{equation} where $x = (y,z) \in \mathbb{R}^h \times \mathbb{R}^k = \mathbb{R}^n$, with $n \geq 5$, $2 < k <n$, and $t = n - \frac{(n-2)q}{2}$. We establish the existence of a bi-radial sign-changing solution under the assumptions $0 \leq \mu < \frac{(k-2)^2}{4}, \, 2 < q <2^* = \frac{2(n-k+1)}{n-k-1}$. We approach the problem by lifting it to the hyperbolic setting, leading to the equation: $-\Delta_{\mathbb{B}^N} u \, - \, \lambda u = |u|^{p-1}u, \; u \in H^1(\mathbb{B}^N)$, $\mathbb{B}^N$ is the hyperbolic ball model. We study the existence of a sign-changing solution with suitable symmetry by constructing an appropriate invariant subspace of $H^1(\mathbb{B}^N)$ and applying the concentration compactness principle, and the corresponding solution of the Hardy-Sobolev-Maz'ya type equation becomes bi-radial under the corresponding isometry.

math.AP↗
arXiv2023

On improvements of the Hardy, Copson and Rellich inequalities

Using a method of factorization and by introducing a generalized discrete Dirichlet's Laplacian matrix $(-\Delta_{\Lambda})$, we establish an extended improved discrete Hardy's inequality and Rellich inequality in one dimension. We prove that the discrete Copson inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.) in one-dimension admits an improvement. We also prove that the improved Copson's weights are optimal (in fact \emph{critical}). It is shown that improvement of the Knopp inequalities (Knopp in J. London Math. Soc. 3(1928), 205-211 and 5(1930), 13-21) lies on improvement of the Rellich inequalities. Further, an improvement of the generalized Hardy's inequality (Hardy in Messanger of Math. 54(1925), 150-156) in a special case is obtained.

math.FA↗
arXiv2022

On the improvements of Hardy and Copson inequalities

In this current work, we revisit the recent improvement of the discrete Hardy's inequality in one dimension and establish an extended improved discrete Hardy's inequality with its optimality. We also study one-dimensional discrete Copson's inequality (E.T. Copson, \emph{Notes on a series of positive terms}, J. London Math. Soc., 2 (1927), 9-12.), and achieve an improvement of the same in a particular case. Further, we study some fundamental structures such as completeness, K\"{o}the-Toeplitz duality, separability, etc. of the sequence spaces which originated from the improved discrete Hardy and Copson inequalities in one dimension.

math.FA↗
arXiv2022

Orlicz extension of Numerical radius inequalities

In this paper, we achieve new and improved numerical radius inequalities of operators defined on a Hilbert space by using Orlicz function and Hermite-Hadamard inequality. The upper bounds of various inequalities involving numerical radii have been obtained. Finally, we compute an upper bound of the numerical radius for block matrices of the form $\begin{bmatrix}O & P\\Q & O \end{bmatrix}$, where $P, Q$ are any bounded linear operators on a Hilbert space.

math.FA↗
arXiv2022

Numerical radius and Berezin number inequality

We study various inequalities for numerical radius and Berezin number of a bounded linear operator on a Hilbert space. It is proved that the numerical radius of a pure two-isometry is 1 and the Crawford number of a pure two-isometry is 0. In particular, we show that for any scalar-valuednon-constant inner function $\theta$, the numerical radius and the Crawford number of a Toeplitz operator $T_{\theta}$ on a Hardy space is 1 and 0, respectively. It is also shown that numerical radius is multiplicative for a class of isometries and sub-multiplicative for a class of commutants of a shift. We have illustrated these results with some concrete examples. Finally, some Hardy-type inequalities for Berezin number of certain class of operators are established with the help of the classical Hardy's inequality.

math.FA↗
arXiv2016

Certain geometric structure of $Λ$-sequence spaces

The $Λ$-sequence spaces $Λ_p$ for $1< p\leq\infty$ and its generalization $Λ_{\hat{p}}$ for $1<\hat{p}<\infty$, $\hat{p}=(p_n)$ is introduced. The James constants and strong $n$-th James constants of $Λ_p$ for $1<p\leq\infty$ is determined. It is proved that generalized $Λ$-sequence space $Λ_{\hat{p}}$ is embedded isometrically in the Nakano sequence space $l_{\hat{p}}(\mathbb{R}^{n+1})$ of finite dimensional Euclidean space $\mathbb{R}^{n+1}$. Hence it follows that sequence spaces $Λ_p$ and $Λ_{\hat{p}}$ possesses the uniform Opial property, property $(β)$ of Rolewicz and weak uniform normal structure. Moreover, it is established that $Λ_{\hat{p}}$ possesses the coordinate wise uniform Kadec-Klee property. Further necessary and sufficient conditions for element $x\in S(Λ_{\hat{p}})$ to be an extreme point of $B(Λ_{\hat{p}})$ are derived. Finally, estimation of von Neumann-Jordan and James constants of two dimensional $Λ$-sequence space $Λ_2^{(2)}$ is being carried out.

math.FA↗
arXiv2013

Some $B$-Difference Sequence Spaces Derived by Using Generalized Means and Compact Operators

This paper presents new sequence spaces $X(r, s, t, p ; B)$ for $X \in \{l_\infty(p), c(p), c_0(p), l(p)\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete paranormed spaces and the spaces $X(r, s, t, p ; B)$ for $X \in \{c(p), c_0(p), l(p)\}$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t, p ;B)$ to $X$. Finally, some classes of compact operators on the space $l_p(r, s, t ;B)$ are characterized by using the Hausdorff measure of noncompactness.

math.FA↗
arXiv2013

"Some $m$th-order Difference Sequence Spaces of Generalized Means and Compact Operators"

In this paper, new sequence spaces $X(r, s, t ;Δ^{(m)})$ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator of order $m$ are introduced. It is shown that these spaces are complete normed linear spaces and the spaces $c_0(r, s, t ;Δ^{(m)})$, $c(r, s, t ;Δ^{(m)})$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t ;Δ^{(m)})$ to $X$. Finally, some classes of compact operators on the spaces $c_0(r, s, t ;Δ^{(m)})$ and $l_{\infty}(r, s, t ;Δ^{(m)})$ are characterized by using the Hausdorff measure of noncompactness.

math.FA↗
arXiv2013

Difference Sequence Spaces Derived by Using Generalized Means

This paper deals with new sequence spaces $X(r, s, t ;Δ) $ for $X\in \{l_\infty, c, c_0\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete normed linear spaces and the spaces $X(r, s, t ;Δ)$ for $X\in \{c, c_0\}$ have Schauder basis. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also established necessary and sufficient conditions for matrix transformations from $X(r, s, t ;Δ)$ to $X$.

math.FA↗
arXiv2013

Some Paranormed Difference Sequence Spaces Derived by Using Generalized Means

This paper presents new sequence spaces $X(r, s, t, p ;Δ)$ for $X \in \{l_\infty(p), c(p), c_0(p), l(p)\}$ defined by using generalized means and difference operator. It is shown that these spaces are complete under a suitable paranorm. Furthermore, the $α$-, $β$-, $γ$- duals of these sequence spaces are computed and also obtained necessary and sufficient conditions for some matrix transformations from $X(r, s, t, p ;Δ)$ to $X$. Finally, it is proved that the sequence space $l(r, s, t, p ;Δ)$ is rotund when $p_n>1$ for all $n$ and has the Kadec-Klee property.

math.FA↗