arXiv subjects

Abhishek Sarkar

Publications and source records attributed to Abhishek Sarkar.

At least 37 records · Page 2Linked to original sources

arXiv2025

Global branching for semilinear fractional Laplace with sublinear nonlinearity

This article investigates the existence, nonexistence, and multiplicity of positive solutions to the sublinear fractional elliptic problem $(P_λ^s)$. We begin by establishing several a priori estimates that provide regularity results and describe the qualitative behavior of solutions. A critical threshold level for the parameter $λ$ is identified, which plays a crucial role in determining the existence or nonexistence of solutions. The sub and supersolution method is employed to obtain a weak solution. Furthermore, we establish a relation between the local minimizers of $\mathcal{D}^{s,2}(\mathbb{R}^N)$ versus $C(\mathbb{R}^N; 1+|x|^{N-2s})$. Combining these results with the Classical Linking Theorem, we demonstrate the existence of at least two distinct positive weak solutions to $(P_λ^s)$. This work extends the results of Yang, Abrantes, Ubilla, and Zhou (J. Differential Equations, 416:159-189, 2025) to the nonlocal setting, i.e., when $s \in (0,1)$. Several technical challenges arise in this framework, such as the lack of a standard comparison principle in $\mathbb{R}^N$ in the fractional setting.

math.AP↗

arXiv2025

Higher topological complexity of planar polygon spaces having small genetic codes

We study the higher (sequential) topological complexity, a numerical homotopy invariant for the planar polygon spaces. For these spaces with a small genetic codes and dimension $m$, Davis showed that their topological complexity is either $2m$ or $2m+1$. We extend these bounds to the setting of higher topological complexity. In particular, when $m$ is power of $2$, we show that the $k$-th higher topological complexity of these spaces is either $km$ or $km+1.$

math.AT↗

arXiv2025

Identification Over Noisy Permutation Channels

We study message identification over the noisy permutation channel. For discrete memoryless channels (DMCs), the number of identifiable messages grows doubly exponentially, and the maximum second-order exponent is same as the Shannon capacity of the DMC. We consider a $q$-ary noisy permutation channel where the transmitted vector is first permuted by a permutation chosen uniformly at random, and then passed through a DMC with strictly positive entries in its transition probability matrix $U$. In an earlier work, we showed that over $q$-ary noiseless permutation channel, $2^{c_n n^{q-1}}$ messages can be identified if $c_n\rightarrow 0$, and a strong converse holds for $2^{c_n n^{q-1}}$ messages if $c_n\rightarrow \infty$. For the $q$-ary noisy permutation channel, we show that message sizes growing as $2^{R_n \left( \frac{n}{\log n}\right)^{(r-1)/2}}$, where $r$ be the rank of $U$, are identifiable for any $R_n\rightarrow 0$. We also prove a strong converse result showing that for any sequence of identification codes with $$2^{\left(R_n n^{(q-1)/2}(\log n)^{1+\frac{(q-1)(q-2)}{2}}\right)},$$ messages, where $R_n \rightarrow \infty$, the sum of Type-I and Type-II error probabilities approaches at least $1$ as $n\rightarrow \infty$. Our converse proof uses the idea of channel resolvability. We propose a novel deterministic quantization scheme for quantization of a distribution over the set of all compositions/types by an $M$-type input distribution when the distortion is measured on the output distribution in total variation distance. This plays a key role in the converse proof. We have also studied identification with deterministic encoder and decoder, and proved tight achievability, weak converse, and strong converse.

cs.IT↗

arXiv2025

Principal eigenvalues for the weighted p-Laplacian and antimaximum principle in $\mathbb{R}^N$

We study the existence of principal eigenvalues and principal eigenfunctions for weighted eigenvalue problems of the form: \begin{equation*} - \mbox{div} ( L (x) |\nabla u|^{p-2} \nabla u ) = λK(x) |u|^{p-2} u \hspace{.1cm} \mbox { in } \hspace{.1cm} \mathbb{R}^N , \end{equation*} where $λ\in \mathbb{R}$, $p>1$, $K : \mathbb{R}^N \rightarrow \mathbb{R}$, $L : \mathbb{R}^N \rightarrow \mathbb{R}^+$ are locally integrable functions. The weight function $K$ is allowed to change sign, provided it remains positive on a set of nonzero measure. We establish the existence, regularity, and asymptotic behavior of the principal eigenfunctions. We also prove local and global antimaximum principles for a perturbed version of the problem.

math.AP↗

arXiv2024

Identification over Permutation Channels

We study message identification over a q-ary uniform permutation channel, where the transmitted vector is permuted by a permutation chosen uniformly at random. For discrete memoryless channels(DMCs), the number of identifiable messages grows doubly exponentially. Identification capacity, the maximum second-order exponent, is known to be the same as the Shannon capacity of the DMC. Permutation channels support reliable communication of only polynomially many messages. A simple achievability result shows that message sizes growing as 2^{ε_nn^{q-1}} are identifiable for any ε_n\rightarrow0. We prove two converse results. A ``soft'' converse shows that for any R>0, there is no sequence of identification codes with message size growing as 2^{Rn^{q-1}} with a power-law decay (n^{-μ}) of the error probability. We also prove a ``strong" converse showing that for any sequence of identification codes with message size 2^{R_n n^{q-1}}, where R_n\rightarrow\infty, the sum of type I and type II error probabilities approaches at least 1 as n\rightarrow\infty. To prove the soft converse, we use a sequence of steps to construct a new identification code with a simpler structure which relates to a set system, and then use a lower bound on the normalized maximum pairwise intersection of a set system. To prove the strong converse, we use results on approximation of distributions. The achievability and converse results are generalized to the case of coding over multiple blocks. We finally study message identification over a q-ary uniform permutation channel in the presence of causal block-wise feedback from the receiver, where the encoder receives an entire n-length received block after the transmission of the block is complete. We show that in the presence of feedback, the maximum number of identifiable messages grows doubly exponentially, and we present a two-phase achievability scheme.

cs.IT↗

arXiv2024

On the study of $(p, Q)$-Laplace Choquard equations with critical Trudinger-Moser nonlinearity in $\mathbb{H}^N$

This paper deals with the existence and multiplicity of nontrivial solutions for $(p, Q)$-Laplace equations with the Stein-Weiss reaction under critical exponential nonlinearity in the Heisenberg group $\mathbb{H}^N$. In addition, a weight function and two positive parameters have also been included in the nonlinearity. The developed analysis is significantly influenced by these two parameters. Further, the mountain pass theorem, the Ekeland variational principle, the Trudinger-Moser inequality, the doubly weighted Hardy-Littlewood-Sobolev inequality and a completely new Brézis-Lieb type lemma for Choquard nonlinearity play key roles in our proofs.

math.AP↗

arXiv2024

Existence of Positive Solutions for Generalized Fractional Brézis-Nirenberg Problem

In this article, we study the fractional Brézis-Nirenberg type problem on whole domain $\mathbb{R}^N$ associated with the fractional $p$-Laplace operator. To be precise, we want to study the following problem: \begin{equation*} (-Δ)_{p}^{s}u - λw |u|^{p-2}u= |u|^{p_{s}^{*}-2}u \quad \text{in} ~\mathcal{D}^{s,p}(\mathbb{R}^{N}), \end{equation*} where $s\in (0,1),~p \in (1,\frac{N}{s}), ~p_{s}^{*}= \frac{Np}{N-sp}$ and the operator $(-Δ)_{p}^{s}$ is the fractional $p$-Laplace operator. The space $\mathcal{D}^{s,p}(\mathbb{R}^{N})$ is the completion of $C_c^\infty(\mathbb{R}^N)$ with respect to the Gaglairdo semi-norm. In this article, we prove the existence of a positive solution to this problem by allowing the Hardy weight $w$ to change its sign.

math.AP↗

arXiv2023

Machine Learning (ML)-assisted Beam Management in millimeter (mm)Wave Distributed Multiple Input Multiple Output (D-MIMO) systems

Beam management (BM) protocols are critical for establishing and maintaining connectivity between network radio nodes and User Equipments (UEs). In Distributed Multiple Input Multiple Output systems (D-MIMO), a number of access points (APs), coordinated by a central processing unit (CPU), serves a number of UEs. At mmWave frequencies, the problem of finding the best AP and beam to serve the UEs is challenging due to a large number of beams that need to be sounded with Downlink (DL) reference signals. The objective of this paper is to investigate whether the best AP/beam can be reliably inferred from sounding only a small subset of beams and leveraging AI/ML for inference of best beam/AP. We use Random Forest (RF), MissForest (MF) and conditional Generative Adversarial Networks (c-GAN) for demonstrating the performance benefits of inference.

eess.SP↗

arXiv2023

Degenerate Schr{ö}dinger-Kirchhoff $(p, N)$-Laplacian problem With singular Trudinger-Moser nonlinearity in $\mathbb{R}^N$

In this paper, we deal with the existence of nontrivial nonnegative solutions for a $(p, N)$-Laplacian Schr{ö}dinger-Kirchhoff problem in $\mathbb{R}^N$ with singular exponential nonlinearity. The main features of the paper are the $(p, N)$ growth of the elliptic operators, the double lack of compactness, and the fact that the Kirchhoff function is of degenerate type. To establish the existence results, we use the mountain pass theorem, the Ekeland variational principle, the singular Trudinger-Moser inequality, and a completely new Brézis-Lieb type lemma for singular exponential nonlinearity.

math.AP↗

arXiv2023

Magnetic functionalization and Catalytic behavior of magnetic nanoparticles during laser photochemical graphitization of polyimide

We report laser-assisted photochemical graphitization of polyimides (PI) into functional magnetic nanocomposites using laser irradiation of PI in the presence of magnetite nanoparticles (MNP). PI Kapton sheets covered with MNP were photochemically treated under ambient conditions using a picosecond pulsed laser (1064nm) to obtain an electrically conductive material. Scanning electron microscopy of the treated material revealed layered magnetic nanoparticles/graphite nanocomposite structure (MNP/graphite). Four probe conductivity measurements indicated that nanocomposite has an electrical conductivity of 1550 S/m. Superconducting quantum interference device (SQUID) magnetometer-based magnetic characterization of the treated material revealed an anisotropic ferromagnetic response in the MNP/graphite nanocomposite compared to the isotropic response of MNP. Raman spectroscopy of MNP/graphite nanocomposite revealed a four-fold improvement in graphitization, suppression in disorder, and decreased nitrogenous impurities compared to the graphitic material obtained from laser treatment of just PI sheets. X-ray photoelectron spectroscopy, x-ray diffraction, and energy-dispersive x-ray spectroscopy were used to delineate the phase transformations of MNP during the formation of MNP/graphite nanocomposite. Post-mortem characterization indicates a possible photocatalytic effect of MNP during MNP/graphite nanocomposite formation. Under laser irradiation, MNP transformed from the initial Fe3O4 phase to γ-Fe2O3 and Fe5C2 phases and acted as nucleation spots to catalyze the graphitization process of PI.

cond-mat.mtrl-sci↗

arXiv2023

Lithium plating induced degradation during fast charging of batteries subjected to compressive loading

We report the lithium plating associated capacity loss during fast charging of compressively loaded lithium-ion batteries (LIBs). The charging and discharging of LIB under compressive loading during service may affect the cell performance or initiate localized defects in the electrodes. Pouch cells of capacity 20mAh were compressively loaded to nominal pressures of 0-440 kPa and subjected to 10 cycles of fast charging at 1C and 4C. Experimental results show that cells charged at 4C-rate experienced significant capacity fade, and applying compressive loads exacerbated the capacity loss. The coulombic efficiency study shows that active lithium loss was higher for the initial cycles before gradually reducing to a minimal capacity loss for the tenth charging cycle. The cell voltage relaxation immediately after charging was monitored to identify the stripping of plated lithium after fast charging cycles and showed that the duration of lithium stripping was higher for cells under mechanical compressive loading. Scanning electron microscopy (SEM) and Electron paramagnetic resonance spectroscopy (EPR) characterization of the anode showed significantly higher lithium deposits on the anodes charged at a 4C rate under compressive loads. These results indicate that applied mechanical compression causes increased lithium plating during fast charging of batteries.

cond-mat.mtrl-sci↗

arXiv2023

On a fractional system of NLS-KDV equations with Hardy potentials

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of nonlinear Schrödinger-Korteweg-De Vries (NLS-KdV, in short) equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1} &= 2νh(x) u^{}v^{} & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} &= νh(x) u^{2} & \quad \mbox{in} ~ \mathbb{R}^{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1)~\text{and}~λ_{i}\in (0, Λ_{N,s_{i}})$ with $Λ_{N,s_{i}} = 2 π^{N/2} \frac{Γ^{2}(\frac{N+2s_i}{4}) Γ(\frac{N+2s_i}{2})}{Γ^{2}(\frac{N-2s_i}{4}) ~|Γ(-s_{i})|}, (i=1,2)$. By imposing certain assumptions on the parameter $ν$ and on the function $h$, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

math.AP↗

arXiv2023

On the extreme value of Nehari manifold for nonlocal singular Schr{ö}dinger-Kirchhoff equations in $\mathbb{R}^N$

This article investigates the existence, non-existence, and multiplicity of weak solutions for a parameter-dependent nonlocal Schrödinger-Kirchhoff type problem on $\mathbb R^N$ involving singular non-linearity. By performing fine analysis based on Nehari submanifolds and fibre maps, our goal is to show the problem has at least two positive solutions even if $λ$ lies beyond the extremal parameter $λ_\ast$.

math.AP↗

arXiv2023

On critically coupled (s_1, s_2)-fractional system of Schrödinger equations with Hardy potential

In this article, our main concern is to study the existence of bound and ground state solutions for the following fractional system of Schrödinger equations with Hardy potentials: \begin{equation*} \left\{ \begin{aligned} (-Δ)^{s_{1}} u - λ_{1} \frac{u~~}{|x|^{2s_{1}}} - u^{2_{s_{1}}^{*}-1} = ναh(x) u^{α-1}v^β & \quad \mbox{in} ~ \mathbb{R}^{N}, (-Δ)^{s_{2}} v - λ_{2} \frac{v~~}{|x|^{2s_{2}}} - v^{2_{s_{2}}^{*}-1} = νβh(x) u^αv^{β-1} & \quad \mbox{in} ~ \mathbb{R}^{N}, u,v >0 \quad \mbox{in} ~ \mathbb{R}^{N} \setminus \{0\}, \end{aligned} \right. \end{equation*} where $s_{1},s_{2} \in (0,1)~\text{and}~λ_{i}\in (0, Λ_{N,s_{i}})$ with $Λ_{N,s_{i}} = 2 π^{N/2} \frac{Γ^{2}(\frac{N+2s_i}{4}) Γ(\frac{N+2s_i}{2})}{Γ^{2}(\frac{N-2s_i}{4}) ~|Γ(-s_{i})|}, (i=1,2)$. By imposing certain assumptions on the parameters and on the function h, we obtain ground-state solutions using the concentration-compactness principle and the mountain-pass theorem.

math.AP↗

arXiv2022

LogAnMeta: Log Anomaly Detection Using Meta Learning

Modern telecom systems are monitored with performance and system logs from multiple application layers and components. Detecting anomalous events from these logs is key to identify security breaches, resource over-utilization, critical/fatal errors, etc. Current supervised log anomaly detection frameworks tend to perform poorly on new types or signatures of anomalies with few or unseen samples in the training data. In this work, we propose a meta-learning-based log anomaly detection framework (LogAnMeta) for detecting anomalies from sequence of log events with few samples. LoganMeta train a hybrid few-shot classifier in an episodic manner. The experimental results demonstrate the efficacy of our proposed method

cs.LG↗

arXiv2022

On Lie algebroid over algebraic spaces

We consider Lie algebroids over algebraic spaces (in short we call it as $a$-spaces) by considering the sheaf of Lie-Rinehart algebras. We discuss about properties of universal enveloping algebroid $\mathscr{U}(\mathcal{O}_X,\mathcal{L})$ of a Lie algebroid $\mathcal{L}$ over an $a$-space $(X, \mathcal{O}_X)$. This is done by sheafification of the presheaf of universal enveloping algebras for Lie-Rinehart algebras. We review the extent to which structure of the universal enveloping algebroid of Lie algebroids (over special $a$-spaces) resembles a sheaf of bialgebras. In the sequel we present a version of Poincaré-Birkhoff-Witt theorem and Cartier-Milnor-Moore theorem for the Lie algebroid.

math.RA↗

arXiv2022

On the fourth order semipositone problem in $\mathbb{R}^N$

For $N \geq 5$ and $a>0$, we consider the following semipositone problem \begin{align*} Δ^2 u= g(x)f_a(u) \text { in } \mathbb{R}^N, \, \text{ and } \, u \in \mathcal{D}^{2,2}(\mathbb{R}^N),\ \ \ \qquad \quad \mathrm{(SP)} \end{align*} where $g \in L^1_{loc}(\mathbb{R}^N)$ is an indefinite weight function, $f_a:\mathbb{R} \to \mathbb{R}$ is a continuous function that satisfies $f_a(t)=-a$ for $t \in \mathbb{R}^-$, and $\mathcal{D}^{2,2}(\mathbb{R}^N)$ is the completion of $\mathcal{C}_c^{\infty}(\mathbb{R}^N)$ with respect to $(\int_{\mathbb{R}^N} (Δu)^2)^{1/2}$. For $f_a$ satisfying subcritical nonlinearity and a weaker Ambrosetti-Rabinowitz type growth condition, we find the existence of $a_1>0$ such that for each $a \in (0,a_1)$, (SP) admits a mountain pass solution. Further, we show that the mountain pass solution is positive if $a$ is near zero. For the positivity, we derive uniform regularity estimates of the solutions of (SP) for certain ranges in $(0,a_1)$, relying on the Riesz potential of the biharmonic operator.

math.AP↗

arXiv2022

Multiple solutions for a weighted $p$-Laplacian problem

We prove the existence of at least three solutions for a weighted $p$-Laplacian operator involving Dirichlet boundary condition in a weighted Sobolev space. The main tool we use here is a three solution theorem in reflexive Banach spaces due to G. Bonanno and B. Ricceri.

math.AP↗