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We study two types of problems for polynomial Farey fractions. For a positive integer $Q$, and polynomial $P(x)\in\mathbb{Z}[X]$ with $P(0)=0$, we define polynomial Farey fractions as \[\mathcal{F}_{Q,P}:=\left\{\frac{a}{q}: 1\leq a\leq q\leq Q,\ \gcd (P(a),q)=1\right\}.\] The classical Farey fractions are obtained by considering $P(x)=x$. In this article, we determine the global and local distribution of the sequence of polynomial Farey fractions via discrepancy and pair correlation measure, respectively. In particular, we establish that the sequence of polynomial Farey fractions is uniformly distributed modulo one and that the pair correlation measure of $\mathcal{F}_{Q,P}$ exists and provide an upper bound. Further, restricting the polynomial Farey denominators to certain subsets of primes, we explicitly find the pair correlation measure and show it to be Poissonian. Finally, we study Chebyshev's bias type of questions for the classical and polynomial Farey denominators along arithmetic progressions and obtain an $\Omega$-result for the error term of its counting function., Comment: 31 pages

Cardiovascular disease refers to any critical condition that impacts the heart. Because heart diseases can be life-threatening. Researchers are focusing on designing smart systems to accurately diagnose them based on electronic health data, with the aid of machine learning algorithms. Heart disease classification using machine learning (ML) algorithms such as Support Vector Machine(SVM), Na\"ive Bayes(NB), Decision Trees (DTs) and Random Forests (RFs) are often hindered by overfitting. These ML algorithms need extensive hyperparameter tuning. Random Search offers a faster, and, more efficient exploration of hyperparameter space, but, it may overlook optimal regions. Grid Search, though exhaustive, but, it is computationally expensive and inefficient, particularly with high-dimensional data. To address these limitations, Randomized-Grid Search, a novel hybrid optimization method is proposed that combines the global exploration strengths of Random Search with the focused, and, exhaustive search of Grid Search in the most promising regions. This hybrid approach efficiently balances exploration and exploitation. The proposed model optimizes the hyperparameter for Decision Tree model. The proposed model is applied to UCI heart disease dataset for classification. It enhances model performance, provides improved accuracy, generalization, and computational efficiency. Experimental results demonstrate that Randomized-Grid Search outperforms traditional methods by significant margins. The proposed model provides a more effective solution for machine learning applications in healthcare diagnosis.

The recovery of damaged or resected ribcage structures requires precise, custom-designed implants to restore the integrity and functionality of the thoracic cavity. Traditional implant design methods rely mainly on manual processes, making them time-consuming and susceptible to variability. In this work, we explore the feasibility of automated ribcage implant generation using deep learning. We present a framework based on 3D U-Net architecture that processes CT scans to generate patient-specific implant designs. To the best of our knowledge, this is the first investigation into automated thoracic implant generation using deep learning approaches. Our preliminary results, while moderate, highlight both the potential and the significant challenges in this complex domain. These findings establish a foundation for future research in automated ribcage reconstruction and identify key technical challenges that need to be addressed for practical implementation.

This paper develops a control strategy for pursuit-evasion problems in environments with occlusions. We address the challenge of a mobile pursuer keeping a mobile evader within its field of view (FoV) despite line-of-sight obstructions. The signed distance function (SDF) of the FoV is used to formulate visibility as a control barrier function (CBF) constraint on the pursuer's control inputs. Similarly, obstacle avoidance is formulated as a CBF constraint based on the SDF of the obstacle set. While the visibility and safety CBFs are Lipschitz continuous, they are not differentiable everywhere, necessitating the use of generalized gradients. To achieve non-myopic pursuit, we generate reference control trajectories leading to evader visibility using a sampling-based kinodynamic planner. The pursuer then tracks this reference via convex optimization under the CBF constraints. We validate our approach in CARLA simulations and real-world robot experiments, demonstrating successful visibility maintenance using only onboard sensing, even under severe occlusions and dynamic evader movements., Comment: 7 pages, 7 figures

Contextual advertising serves ads that are aligned to the content that the user is viewing. The rapid growth of video content on social platforms and streaming services, along with privacy concerns, has increased the need for contextual advertising. Placing the right ad in the right context creates a seamless and pleasant ad viewing experience, resulting in higher audience engagement and, ultimately, better ad monetization. From a technology standpoint, effective contextual advertising requires a video retrieval system capable of understanding complex video content at a very granular level. Current text-to-video retrieval models based on joint multimodal training demand large datasets and computational resources, limiting their practicality and lacking the key functionalities required for ad ecosystem integration. We introduce ContextIQ, a multimodal expert-based video retrieval system designed specifically for contextual advertising. ContextIQ utilizes modality-specific experts-video, audio, transcript (captions), and metadata such as objects, actions, emotion, etc.-to create semantically rich video representations. We show that our system, without joint training, achieves better or comparable results to state-of-the-art models and commercial solutions on multiple text-to-video retrieval benchmarks. Our ablation studies highlight the benefits of leveraging multiple modalities for enhanced video retrieval accuracy instead of using a vision-language model alone. Furthermore, we show how video retrieval systems such as ContextIQ can be used for contextual advertising in an ad ecosystem while also addressing concerns related to brand safety and filtering inappropriate content., Comment: Accepted at WACV 2025

In this article, we examine the dynamical system of $f(R,\mathcal{G})$ gravity model. This $f(R,G)$ model framework is composed of the interactions between dark matter and scalar field through the linear coupling term. The key objective of present study is to describe the cosmological viability of modified gravity theory formulated with the $ f(R, \mathcal{G}) $ gravity. We transform the cosmological equations into an autonomous system of ordinary differential equations by suitable transformation of variables. The $ f(R,\mathcal{G})$ model governed by $ f(R, \mathcal{G}) = \alpha R^m + \beta \mathcal{G}^n $ has been investigated in detail to characterize the stability properties of the critical points of the autonomous system. The model may explain the late-time accelerating universe expansion corresponding to the attractor in model. Depending on the effective equation of state parameter values corresponding to the critical points, we study the observational viability of model using low-redshift observational data such as the observational Hubble data and Pantheon data. Furthermore, we investigate the effects of parameters by using the effective equation of state parameter and statefinder diagnostics.

We study a set of many-body wave-functions of Fermions that are naturally written using momentum space basis and allow for quantum superposition of Fermion occupancy, $\{n_{\bf k}\}$. This {enables} us to capture the fluctuations of the Fermi-surface {(FS)} -- the singularly most important signature of a metal. We bench-mark our results in one spatial dimensions (1D) to show that these wave-functions allow for quantitative understanding of the Tomonaga-Luttinger liquid (TLL); computations of certain correlators using them can in fact be extended to larger systems sizes compared to conventional exact diagonalization (ED) allowing for a more systematic comparison with bosonization techniques. Finally we show that this basis may be useful for obtaining fixed-point wave-function for strongly correlated metals {in dimensions greater that one}. In particular, we study the case of coherent (equal) superposition of elliptical FS {in continuum (2D) and on a} square lattice{. In case of the former, our variational wave-function systematically interpolates between the phenomenology of the Fermi liquid ground state, i.e., finite single-Fermion residue at a sharp FS, to a non-Fermi liquid (NFL) with zero residue. In the NFL the jump in $\langle n_{\bf k}\rangle$ at the FS is replaced by a point of inflection (similar to a 1D TLL) whose contour is consistent with the Luttinger Theorem. In case of the square lattice, we} find highly anisotropic distribution of the quasi-particle residue, which, at finite resolution has an uncanny resemblance to the Fermi-arcs{, albeit at zero temperature,} seen in the pseudo-gap state of the cuprates.

Learning from Demonstration allows robots to mimic human actions. However, these methods do not model constraints crucial to ensure safety of the learned skill. Moreover, even when explicitly modelling constraints, they rely on the assumption of a known cost function, which limits their practical usability for task with unknown cost. In this work we propose a two-step optimization process that allow to estimate cost and constraints by decoupling the learning of cost functions from the identification of unknown constraints within the demonstrated trajectories. Initially, we identify the cost function by isolating the effect of constraints on parts of the demonstrations. Subsequently, a constraint leaning method is used to identify the unknown constraints. Our approach is validated both on simulated trajectories and a real robotic manipulation task. Our experiments show the impact that incorrect cost estimation has on the learned constraints and illustrate how the proposed method is able to infer unknown constraints, such as obstacles, from demonstrated trajectories without any initial knowledge of the cost., Comment: (Accepted/In press) 2024 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS)

We study a set of two-loop non-planar master integrals needed for the NNLO QCD corrections to diphoton and dijet production at hadron colliders. The top-sector topology contains an internal massive fermion loop and is known to contain elliptic curves. Leveraging the method of differential equations, we provide a comprehensive discussion for deriving an $\epsilon$-factorized differential equation related to the most intricate sector within the Feynman integral family. Despite the dependence on multiple scales and the presence of two elliptic sectors, we demonstrate how to leverage the properties of their maximal cuts and the factorization of the Picard-Fuchs operator to deal with the complexity of the analytic computation. In particular, we construct a transformation matrix that brings the differential equations into a format enabling the convenient expression of analytic results in terms of Chen's iterated integrals., Comment: 19 pages, 2 figures

Assuming the generalized Lindel\"of hypothesis, we provide asymptotic formulas for the mean values of the first and second moments of Ramanujan sums over any number field. Additionally, unconditionally, we estimate the second moment of Ramanujan sums over cyclotomic number fields., Comment: 15 pages. arXiv admin note: text overlap with arXiv:2109.09398

We present the analytic and compact two-loop helicity amplitudes for QCD and QED corrections to the light-by-light scattering process with massive internal fermions. We express the master integrals either in terms of multiple polylogarithms or in terms of iterated integrals with dlog one-forms. We also elaborate on optimizing the analytic results for each phase-space region. This makes the numerical evaluation of the scattering amplitudes fast, stable and suitable for phenomenological applications., Comment: 35 pages, 1 figure, v2: journal version (two long equations move into appendix)

The recent experimental observation of Light-by-Light (LbL) scattering at the Large Hadron Collider has revived interest in this fundamental process, and especially of the accurate prediction of its cross-section, which we present here for the first time at Next-to-Leading Order (NLO) in both QCD and QED. We compare two radically different computational approaches, both exact in the fermion mass dependence, thus offering a strong cross-check of our results. The first approach is a fully analytic method to calculate compact and well-organized two-loop helicity amplitudes. The second one is entirely numerical and leverages the Local Unitarity construction. Our two calculations agree with each other and conclude that including the exact fermion mass contribution typically increases the size of the NLO corrections. Moreover, we find that the exact result converges slowly to the massless limit of the high-energy regime, thus emphasizing the importance of including the full mass dependence at NLO. We also compare our results with the ATLAS measurement of LbL in ultra-peripheral lead-lead collisions, and find that the inclusion of exact NLO corrections reduces, but does not eliminate, the existing tension with theoretical predictions., Comment: 11 pages, 6 figures (including appendix) v2: minor corrections and journal version

The Animal-AI Environment is a unique game-based research platform designed to facilitate collaboration between the artificial intelligence and comparative cognition research communities. In this paper, we present the latest version of the Animal-AI Environment, outlining several major new features that make the game more engaging for humans and more complex for AI systems. New features include interactive buttons, reward dispensers, and player notifications, as well as an overhaul of the environment's graphics and processing for significant improvements in agent training time and quality of the human player experience. We provide detailed guidance on how to build computational and behavioural experiments with the Animal-AI Environment. We present results from a series of agents, including the state-of-the-art Deep Reinforcement Learning agent, Dreamer-v3, on newly designed tests and the Animal-AI Testbed of 900 tasks inspired by research in the field of comparative cognition. The Animal-AI Environment offers a new approach for modelling cognition in humans and non-human animals, and for building biologically-inspired artificial intelligence., Comment: 37 pages, 16 figures, 3 tables

In this paper, the expanding nature of the universe within the context of $f(R, L_m)$ gravity theory, where $R$ is Ricci Scalar and $L_m$ is matter Lagrangian, is explored. By employing a specific form $f(R, L_m) = \Lambda + \frac{\alpha}{2} R + \beta L_m^n$, where $\Lambda$, $\alpha$, $\beta$, $n$ are model parameters, with the matter Lagrangian given by $L_m = -p$. We solve the field equations by considering the equation of state (EoS) parameter $\omega$, defined as $p = \omega \rho $, where $p$ represents the isotropic pressure and $\rho$ represents the energy density of the perfect fluid. We thoroughly investigate the model through phase plane analysis, actively studying the evolution of cosmological solutions using dynamical system techniques. To analyse the evolution equations, we have introduced suitable transformation of variables. The evolution of corresponding solutions is presented by curves in the phase-plane diagram. The nature of critical points are analysed and stable attractors are examined for $f(R, L_m)$ gravity cosmological model. We determine the classical stabilities of these cosmologies. We also examine the transition of early decelerating stage of the universe to present accelerating stage of the universe. We analyze the evolution of the effective equation of state parameters and statefinder diagnostics for different parameter values. We examine and analyze the nature of critical points from a cosmological perspective.

We present a new digital holographic microscope (DHM) with a low coherent source for the quantitative imaging of smooth and optically rough objects. The experimental design of the microscope uses an in-line experimental geometry based on the Fizeau interferometer and shows the depth sectioning capability due to the limited longitudinal coherence of the source. A polarization-phase shifting approach is implemented to extract the quantitative and speckle-free image from the experimentally recorded interference fringes. To test and experimentally demonstrate the working of the proposed DHM, we present the results of the quantitative images for different objects.

Speaker identification systems are deployed in diverse environments, often different from the lab conditions on which they are trained and tested. In this paper, first, we show the problem of generalization using fixed thresholds (computed using EER metric) for imposter identification in unseen speaker recognition and then introduce a robust speaker-specific thresholding technique for better performance. Secondly, inspired by the recent use of meta-learning techniques in speaker verification, we propose an end-to-end meta-learning framework for imposter detection which decouples the problem of imposter detection from unseen speaker identification. Thus, unlike most prior works that use some heuristics to detect imposters, the proposed network learns to detect imposters by leveraging the utterances of the enrolled speakers. Furthermore, we show the efficacy of the proposed techniques on VoxCeleb1, VCTK and the FFSVC 2022 datasets, beating the baselines by up to 10%.

In this article, we investigate the fine-scale statistics of real-valued arithmetic sequences. In particular, we focus on real-valued vector sequences and show the Poissonian behavior of the pair correlation function for certain classes of such sequences, thereby extending previous works of Boca et al. and the first author on local statistics of integer-valued and rational-valued vector sequences., Comment: 18 pages

Let $\Phi_k(n)=|\{ (x_1, x_2, \cdots, x_k)\in \left(\mathbb{Z}/n\mathbb{Z}\right)^k; \ \gcd(x_1^2+x_2^2+ \cdots+ x_k^2, n)=1\}|$ be a general totient function introduced first by Cald\'{e}ron et. al. Motivated by the classical works of Schoenberg, Erd\H{o}s, Bateman and Diamond on the distribution of $\Phi_1(n)$, we prove results on the joint distribution of $\Phi_k(n)$ for any $k\ge 1$. Additionally, we also exhibit the extremal order of $\Phi_k(n)$., Comment: 24 pages