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Stochastic model for barrier crossings and fluctuations in local timescale
No full textThe problem of computing the rate of diffusion-aided activated barrier crossings between metastable states is one of broad relevance in physical sciences. The transition path formalism aims to compute the rate of these events by analysing the statistical properties of the transition path between the two metastable regions concerned. In this paper, we show that the transition path process is a unique solution to an associated stochastic differential equation, with a discontinuous and singular drift term. The singularity arises from a local time contribution, which accounts for the fluctuations at the boundaries of the metastable regions. The presence of fluctuations at the local time scale calls for an excursion theoretic consideration of barrier crossing events. We show that the rate of such events, as computed from excursion theory, factorizes into a local time term and an excursion measure term, which bears empirical similarity to the transition state theory rate expression. Since excursion theory makes no assumption about the presence of a transition state in the potential energy landscape, the mathematical structure underlying this factorization ought to be general. We hence expect excursion theory (and local times) to provide some physical and mathematical insights in generic barrier crossing problems
Subconvexity bound for GL(3) × GL(2) L-functions: Hybrid level aspect
No full textLet F be a GL(3) Hecke–Maass cusp form of prime level P1 and let f be a GL(2) Hecke–Maass cuspform of prime level P2. We will prove a subconvex bound for the GL(3)×GL(2) Rankin–Selberg L-function L(s, F × f) in the level aspect for certain ranges of the parameters P1 and P2
Synergistic impact of bioavailable PHEs and alkalinity on microbial diversity and traits in agricultural soil adjacent to chromium-asbestos mines
No full textSoil microbial communities undergo constant fluctuations, particularly in response to environmental factors. Although the deposition of toxic mine waste is recognized for introducing potentially hazardous elements (PHEs) into the soil, its specific impacts on microbial communities remain unclear. This study aims to explore the combined effects of soil alkalinity and bioavailable PHEs on microbial diversity and traits in agricultural soil adjacent to a chromium-asbestos mining area. By employing a comprehensive analysis, this study indicated that microbiological attributes were reduced in contaminated areas (zone 1), whereas both the levels of bioavailable PHEs (CrWs: 31.08 mg/kg, NiWs: 13.90 mg/kg) and alkalinity indices (CROSS, MCAR, MH) were significantly higher. The spatial distribution of soil alkalinity and bioavailable PHEs, primarily originating from chromium-asbestos mines, has been determined. This study also elucidates the negative relationship between soil stressors (Alkalinity and PHEs) and microbial activities (soil enzymatic activity, microbial respiration, and biomass carbon). The vector\u27s length exhibited a notable difference between zone 1 (0.51) and zone 2 (0.32), indicating a substantial limitation on carbon (C). Also, the investigation of soil bacterial diversity unveiled notable disparities in the prevalence of microbial populations inside zone 1. Proteobacteria constituted 57.18% of the total population indicating a noteworthy prevalence in the contaminated soils. Finally, the random forest (RF) algorithm from machine learning was selected and proven to be a robust choice in Taylor diagrams for predicting the causative stressors responsible for the deterioration of soil microbial health. Therefore, this research offers insights into the health and resilience of soil microbial communities under synergistic stress conditions, which will aid environmentalists in planning future interventions and improving sustainable farming techniques
The Higher Structure of Unstable Homotopy Groups
No full textWe construct certain unstable higher order homotopy operations indexed by the simplex categories of deltan for n ≥ 2, and prove that all elements in the homotopy groups of a wedge of spheres are generated under such operations by Whitehead products and the group structure. This provides a stronger unstable analogue of Cohen’s theorem on the decomposition of stable homotopy
Unifying adjacency, Laplacian, and signless Laplacian theories*
No full textLet G be a simple graph with associated diagonal matrix of vertex degrees D(G), adjacency matrix A(G), Laplacian matrix L(G) and signless Laplacian matrix Q(G). Recently, Nikiforov proposed the family of matrices Aα(G) defined for any real α ∈ [0, 1] as Aα(G):= α D(G) + (1 − α) A(G), and also mentioned that the matrices Aα(G) can underpin a unified theory of A(G) and Q(G). Inspired from the above definition, we introduce the Bα-matrix of G, Bα(G):= αA(G) + (1 − α)L(G) for α ∈ [0, 1]. Note that L(G) = B0(G), D(G) = 2B1 2 (G), Q(G) = 3B2 3 (G), A(G) = B1(G). In this article, we study several spectral properties of Bα-matrices to unify the theories of adjacency, Laplacian, and signless Laplacian matrices of graphs. In particular, we prove that each eigenvalue of Bα(G) is continuous on α. Using this, we characterize positive semidefinite Bα-matrices in terms of α. As a consequence, we provide an upper bound of the independence number of G. Besides, we establish some bounds for the largest and the smallest eigenvalues of Bα(G). As a result, we obtain a bound for the chromatic number of G and deduce several known results. In addition, we present a Sachs-type result for the characteristic polynomial of a Bα-matrix
Unique continuation inequalities for Schrödinger equation on Riemannian symmetric spaces of noncompact type
No full textWe study unique continuation inequalities for the free Schrödinger equation in the context of Riemannian symmetric spaces of noncompact type. The results imply that if the solution is small at two different times outside sets of finite measure, then the solution is small in the whole space. On the Euclidean spaces, these inequalities are equivalent to certain uncertainty principles in harmonic analysis
Well-posedness of stochastic heat equation with distributional drift and skew stochastic heat equation
No full textWe study stochastic reaction–diffusion equation (Formula presented.) where (Formula presented.) is a generalized function in the Besov space (Formula presented.), (Formula presented.) and (Formula presented.) is a space-time white noise on (Formula presented.). We introduce a notion of a solution to this equation and obtain existence and uniqueness of a strong solution whenever (Formula presented.), (Formula presented.) and (Formula presented.). This class includes equations with (Formula presented.) being measures, in particular, (Formula presented.) which corresponds to the skewed stochastic heat equation. For (Formula presented.), we obtain existence of a weak solution. Our results extend the work of Bass and Chen (2001) to the framework of stochastic partial differential equations and generalize the results of Gyöngy and Pardoux (1993) to distributional drifts. To establish these results, we exploit the regularization effect of the white noise through a new strategy based on the stochastic sewing lemma introduced in Lê (2020)
Black-Box Identity Testing of Noncommutative Rational Formulas in Deterministic Quasipolynomial Time
No full textRational Identity Testing (RIT) is the decision problem of determining whether or not a noncommutative rational formula computes zero in the free skew field. It admits a deterministic polynomial-time white-box algorithm [Garg, Gurvits, Oliveira, and Wigderson (2016); Ivanyos, Qiao, Subrahmanyam (2018); Hamada and Hirai (2021)], and a randomized polynomial-time algorithm [Derksen and Makam (2017)] in the black-box setting, via singularity testing of linear matrices over the free skew field. Indeed, a randomized NC algorithm for RIT in the white-box setting follows from the result of Derksen and Makam (2017). Designing an efficient deterministic black-box algorithm for RIT and understanding the parallel complexity of RIT are major open problems in this area. Despite being open since the work of Garg, Gurvits, Oliveira, and Wigderson (2016), these questions have seen limited progress. In fact, the only known result in this direction is the construction of a quasipolynomial-size hitting set for rational formulas of only inversion height two [Arvind, Chatterjee, and Mukhopadhyay (2022)]. In this paper, we significantly improve the black-box complexity of this problem and obtain the first quasipolynomial-size hitting set for all rational formulas of polynomial size. Our construction also yields the first deterministic quasi-NC upper bound for RIT in the white-box setting
Brief Announcement: Agent-Based Leader Election, MST, and Beyond
No full textLeader election is one of the fundamental and well-studied problems in distributed computing. In this paper, we initiate the study of leader election using mobile agents. Suppose n agents are positioned initially arbitrarily on the nodes of an arbitrary, anonymous, n-node, m-edge graph G. The agents relocate themselves autonomously on the nodes of G and elect an agent as a leader such that the leader agent knows it is a leader and the other agents know they are not leaders. The objective is to minimize time and memory requirements. Following the literature, we consider the synchronous setting in which each agent performs its operations synchronously with others and hence the time complexity can be measured in rounds. The quest in this paper is to provide solutions without agents knowing any graph parameter, such as n, a priori. We first establish that, without agents knowing any graph parameter a priori, there exists a deterministic algorithm to elect an agent as a leader in O(m) rounds with O(n log n) bits at each agent. Using this leader election result, we develop a deterministic algorithm for agents to construct a minimum spanning tree of G in O(m + n log n) rounds using O(n log n) bits memory at each agent, without agents knowing any graph parameter a priori. Finally, using the same leader election result, we provide improved time/memory results for other fundamental distributed graph problems, namely, gathering, maximal independent set, and minimal dominating sets, removing the assumptions on agents knowing graph parameters a priori
Diabetic Retinopathy Detection Using Amalgamated Deep Learning Algorithm
No full textThe eye disorder termed diabetic retinopathy (DR), which can cause diminished vision, can be brought on by diabetes. DR detection and routine diagnosis are complex tasks that may require multiple testing. Vision loss may be avoided or delayed with early detection of DR. However, the early diagnosis of DR is a challenging mission that necessitates the interpretation of fundus images by clinical specialists. Deep Learning Models (DLM) have become effective techniques for medical image analysis in recent years, promising to provide precise and automated DR identification. DLM automatically extracts the most discriminative features from training photos, but which characteristics are removed to produce predictions is unknown. This paper provides a blended DL method for categorizing DR in fundus pictures. To reduce over-fitting brought on by imbalanced datasets within a single DLN, our strategy employs several of the most well-liked DLN methods learned and validated in a balanced image set, merging their findings in a composite framework. The approach enhances robustness by reducing potential over-fitting patterns and produces more reliable predicted outputs than those obtained using individual DL designs. It does this using the advantages of developing the DLN in multiple resolutions. The suggested method successfully classifies glaucoma images with a sensitivity of 92% and a specificity of 93%