Proceedings of the National Academy of Sciences, India Section A: Physical Sciences最新文献

A b-coloring of a graph G is a proper coloring of the vertices of G such that there exists a vertex in each color class joined to at least one vertex in each other color class. The b-chromatic number of a graph G, denoted by (varphi (textit{G})), is the largest integer k such that G has a b-coloring with k colors. In this paper, the (textit{b})-chromatic number and b-chromatic sum of ((P_{n}circ G)^{p}), where ((P_{n}circ G)^p) is the power graph of corona of the path (P_n) and G with power p is discussed. Also discussed the b-continuity property of ((P_{n}circ G)^{p}.)

The initial phase of this research involves the examination of the properties of space (L_{mathbb {B}mathbb {C}}^{p}left( G,lambda right) ), for (1le p<infty ), using the (mathbb {D})-Haar measure (lambda ) on the locally compact abelian group G. It is demonstrated that (L_{mathbb {B}mathbb {C}}^{1}left( G,lambda right) ) is a (mathbb {D})-normed bicomplex Banach algebra with bounded approximate identity. Then, as an extension of the well-known Lipschitz spaces, bicomplex Lipschitz spaces are presented. The bicomplex versions of semi homogeneous and homogeneous Banach spaces are defined as semi-homogeneous bicomplex Banach-modules and homogeneous bicomplex Banach-modules, respectively. Numerous properties of these modules are investigated.

The Severe Acute Respiratory Syndrome CoronaVirus 2 (SARS-CoV-2) is the virus that causes the contagious disease and is known as coronavirus disease 2019 (COVID-19). In December 2019, the existence of the first case of COVID-19 was recorded in Wuhan city of China and the disease spread rapidly throughout the world, causing the COVID-19 pandemic. After the phase 3 trial, researchers recently reported the studies on the safety and efficacy of treatment with Molnupiravir against SARS-CoV-2 due to the urgent need to discover antiviral agents to fight COVID-19. A topological index is a mathematical tool that gives a numerical value to the molecular graph of a chemical compound and forecasts a variety of the physico-chemical, biological and structural properties of that compound. In this present research work, we are eager to examine the distance-based topological indices (namely, Wiener index, hyper-Wiener index, Tratch-Stankevitch-Zefirov index and Harary index) for an oral antiviral drug Molnupiravir via their standard distance-dependent mathematical formulas and alternatively via the Hosoya polynomial approach. Also, we geometrically comprehend the Hosoya polynomial and the associated distance-based topological indices. The findings can assist and motivate the researchers in the direction of analysing some additional properties of the drug and in developing new drugs to treat COVID-19 patients.

In this paper, we propose some fractional integral identities between the ({}_pR_q(lambda ,eta ; z)) function and Riemann–Liouville fractional integral operators. In particular, we derive explicit integral expressions of the ({}_pR_q(lambda ,eta ; z)) function in terms of Riemann–Liouville fractional integral operators associated with generalized M-series. Furthermore, an analytical solution to the free electron laser equation has also been discussed. A closed-form solution is derived in terms of the ({}_pR_q(lambda ,eta ;z)) function by using Riemann–Liouville fractional operators.

Quantitative Structure–Activity Relationship (QSAR) models rely heavily on descriptors like molecular connectivity indices to predict the physicochemical and biological properties of molecules. The connectivity index (CI) of a molecular graph is a numerical descriptor that is useful for understanding the structural features of a molecule. The use of molecular connectivity indices like the Randic index simplifies these complex molecular systems by reducing the structural information into a single descriptor. Specifically, first-order, second-order and third-order Randic CIs for cyclodextrins with 6 (α-cyclodextrin) and 7 (β-cyclodextrin) sugar rings are reported in the literature. The smallest cyclodextrins with 3 and 4 sugar units were discovered recently. In this work, we propose the Randic connectivity index of the smallest cyclodextrin with 3 and 4 sugar units. The approaches for deriving first-order, second-order and third-order CIs were developed from the molecular graph of cyclodextrin. The derived results are useful for determining the structural activity relationship between chemical structure and biological activity and in comparing different cyclodextrin-drug complexes. In specific, these analytical expressions provide a theoretical foundation for predicting the physicochemical properties of novel small-ring cyclodextrins in drug delivery applications.

In this paper, we investigate statistical convergence and statistical Cauchy sequences in quasi-metric spaces by means of the asymptotic density of subsets of natural numbers. Our primary objective is to introduce a statistical analogue of the Cauchy definition given by Doitchinov (Topol Appl 30:127–148, 1988) and to examine its relationship with statistical convergence in asymmetric settings. We establish a necessary and sufficient condition for a sequence to be statistically Cauchy in terms of its dense subsequences, thereby extending classical subsequence characterizations to the quasi-metric framework. It is shown that, unlike several existing notions of Cauchy sequences in quasi-metric spaces, the proposed definition coincides with the standard statistical Cauchy criterion in the metric case.

This work proposes the Lie group method via a one-dimensional optimal system to construct some more exact invariant solutions of the (2+1)-dimensional KdV-mKdV equation. Using Lie symmetry, we demonstrate Lie infinitesimal generators, potential vector fields, and their commutative and adjoint relations. Additionally, Lie vectors are used to build an optimal system of the one-dimensional subalgebras. Meanwhile, the KdV-mKdV equation’s Lie symmetry reductions are derived from the optimal system. A repetitive process of Lie symmetry reductions between the considered vectors, utilizing single, double, triple, and quadruple combinations, turns the KdV-mKdV equation into non-linear ordinary differential equations with many group invariant solutions. In order to elucidate the significance of the governing equation, numerical simulation is used to trace the exact solutions. The resulting graphical structures include kink, line, single soliton, multisoliton, cusp, parabolic, lump, and stationary wave profiles. These results help numerous authors to understand the complex nature of various nonlinear phenomena throughout diverse scientific disciplines, including plasma physics, internal gravity waves, fluid mechanics, crystal lattice theory, nonlinear optics, and Bose-Einstein condensates.

In this paper, we prove several Tauberian remainder theorems on (lambda )-bounded sequences with respect to iterations of the logarithmic summability method. We establish the relationship between the general logarithmic control modulo and the logarithmic mean of a sequence. Additionally, we present an equality between the general logarithmic control modulo and the generator sequence. In the main results of this study, the (lambda )-boundedness of a sequence is obtained with the help of some conditions on the general logarithmic control modulo and the generator sequence. Furthermore, the results in this work generalize some Tauberian remainder theorems previously established for the logarithmic method of summability.

Multimodal Machine Translation (MMT) aims to improve translation quality by combining textual input with additional modalities such as visual context. In recent years, several promising MMT approaches have been proposed for various languages, including Indian languages. However, most of these methods struggle to accurately translate emotional or ambiguous expressions, especially in under-resourced languages like Hindi. While datasets like Hindi Visual Genome (HVG) support English-Hindi MMT, they often lack emotional depth and fail to capture real-world ambiguity. To address this gap, this research introduces the SentiCap-HIN Dataset, derived from SentiCap. It pairs English-Hindi image captions to better handle linguistic diversity and ambiguity. This dataset serves as a valuable benchmark for developing visually grounded and semantically sensitive translation models for Indian languages. The study also proposes a novel vision-aware MMT framework called MMT-CMAI-T (Cross-Modal Attention Injection with Structured FeatureTrie) to integrate image features with text using a structured attention mechanism. It employs attention entropy to selectively inject relevant visual information, organising missed semantic features during translation. This enables context-aware querying of visual memory based on the decoder’s evolving state. Experimental results show that MMT-CMAI-T significantly outperforms existing MMT and text-only baselines on the SentiCap-HIN corpus, using BLEU, chrF, and METEOR metrics. The research marks progress toward more adaptive, interpretable, and culturally aware translation systems that effectively use multimodal context while preserving linguistic structure. Significance Machine translation systems often struggle with ambiguous words, resulting in inaccurate translations that impede cross-cultural communication. This research presents a novel method that combines visual context with text, allowing translation models to clarify meanings in a way that resembles natural human understanding. The SentiCap-HIN dataset and MMT-CMAI-T framework specifically tackle the difficulties of English-to-Hindi translation, where lexical ambiguity poses a significant challenge. This approach provides notable benefits, including enhanced translation precision, improved management of context-sensitive expressions, and increased accessibility for millions of Hindi speakers engaging with global digital content.

Parkinson's Disease (PD) is a condition which reduces greatly the quality of life for patients. PD receives its diagnosis primarily through the observation of dopaminergic neuron damage within the substantia nigra that appears as motor conditions including tremors, postural instability and additional non motor manifestations like cognitive deterioration, depression and sleep problems. It is a major global public health issue as it generates substantial health expenses together with costs from long-term care and workforce reduction. A successful timely diagnosis of Parkinson's disease allows healthcare providers to deliver proper treatment that enhances patient results. Traditional diagnostic methods depend on clinical histories combined with clinical assessments yet this approach leads to misdiagnosis and delayed diagnosis mainly because symptoms are underdeveloped early in disease progression. The critical requirement for advanced diagnostic systems became vital for precise PD detection along with quicker diagnosis timing while providing necessary intervention guidance. This paper evaluates cutting-edge deep learning technologies developed from MRI data that enhance PD diagnosis precision, alongside their predictive role in transforming clinical patient care. The study aims to advance PD identification methods by enhancing the existing diagnostic techniques through its approach and try to overcome major challenges.