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We define finite-time hyperbolic coordinates, describe their geometry, and prove various results on both their convergence as the time scale increases, and on their variation in the state space. Hyperbolic coordinates reframe the classical paradigm of hyperbolicity: rather than define a hyperbolic dynamical system in terms of a splitting of the tangent space into stable and unstable subspaces, we define hyperbolicity in terms of the co-eccentricity of the map. The co-eccentricity describes the distortion of unit circles in the tangent space under the differential of the map. Finite-time hyperbolic coordinates have been used to demonstrate the existence of SRB measures for the Henon map; our eventual goal is to both elucidate these techniques and to extend them to a broad class of nonuniformly and singular hyperbolic systems., Comment: 31 pages

This manuscript is concerned with the approximate controllability of fractional nonlinear differential equations with nonlocal conditions of order $1

Let Zn denote the additive group of residue classes modulo n. Let c(l,m,n) denote the number of cyclic subgroups of Zl *Zm *Zn. For any x > 1, we consider the asymptotic behavior of D3c(x):= \sum_{lmn\leq x} c(l,m,n), obtain an asymptotic formula by complex method, and get an upper bound for the integral mean-square of the error term in that asymptotic formula., Comment: no

This article develops a novel data assimilation methodology, addressing challenges that are common in real-world settings, such as severe sparsity of observations, lack of reliable models, and non-stationarity of the system dynamics. These challenges often cause identifiability issues and can confound model parameter initialization, both of which can lead to estimated models with unrealistic qualitative dynamics and induce deeper parameter estimation errors. The proposed methodology's objective function is constructed as a sum of components, each serving a different purpose: enforcing point-wise and distribution-wise agreement between data and model output, enforcing agreement of variables and parameters with a model provided, and penalizing unrealistic rapid parameter changes, unless they are due to external drivers or interventions. This methodology was motivated by, developed and evaluated in the context of estimating blood glucose levels in different medical settings. Both simulated and real data are used to evaluate the methodology from different perspectives, such as its ability to estimate unmeasured variables, its ability to reproduce the correct qualitative blood glucose dynamics, how it manages known non-stationarity, and how it performs when given a range of dense and severely sparse data. The results show that a multicomponent cost function can balance the minimization of point-wise errors with global properties, robustly preserving correct qualitative dynamics and managing data sparsity.

Wavelet-based segmentation approaches are widely used for texture segmentation purposes because of their ability to characterize different textures. In this paper, we assess the influence of the chosen wavelet and propose to use the recently introduced empirical wavelets. We show that the adaptability of the empirical wavelet permits to reach better results than classic wavelets. In order to focus only on the textural information, we also propose to perform a cartoon + texture decomposition step before applying the segmentation algorithm. The proposed method is tested on six classic benchmarks, based on several popular texture images.

The empirical wavelet transform is an adaptive multiresolution analysis tool based on the idea of building filters on a data-driven partition of the Fourier domain. However, existing 2D extensions are constrained by the shape of the detected partitioning. In this paper, we provide theoretical results that permits us to build 2D empirical wavelet filters based on an arbitrary partitioning of the frequency domain. We also propose an algorithm to detect such partitioning from an image spectrum by combining a scale-space representation to estimate the position of dominant harmonic modes and a watershed transform to find the boundaries of the different supports making the expected partition. This whole process allows us to define the empirical watershed wavelet transform. We illustrate the effectiveness and the advantages of such adaptive transform, first visually on toy images, and next on both unsupervised texture segmentation and image deconvolution applications.

The recently proposed empirical wavelet transform was based on a particular type of filter. In this paper, we aim to propose a general framework for the construction of empirical wavelet systems in the continuous case. We define a well-suited formalism and then investigate some general properties of empirical wavelet systems. In particular, we provide some sufficient conditions to the existence of a reconstruction formula. In the second part of the paper, we propose the construction of empirical wavelet systems based on some classic mother wavelets.

When we merge information in Dempster-Shafer Theory (DST), we are faced with anomalous behavior: agents with equal expertise and credibility can have their opinion disregarded after resorting to the belief combination rule of this theory. This problem is interesting because belief fusion is an inherent part of dealing with situations where available information is imprecise, as often occurs in Artificial Intelligence. We managed to identify an isomorphism betwin the DST formal apparatus into that of a Probabilistic Logic. Thus, we solved the problematic inputs affair by replacing the DST combination rule with a new fusion process aiming at eliminating anomalies proposed by that rule. We apply the new fusion method to the DST paradox Problem., Comment: 12 pages

Let $Q$ be a quiver and $R$ an associative ring. A representation by $R$-modules of $Q$ is called strongly fp-injective if it admits a pure acyclic injective resolution in the category of representations. It is shown that such representations possess many nice properties. We characterize strongly fp-injective representations under some mild assumptions, which is closely related to strongly fp-injective $R$-modules. Subsequently, we use such representations to define relative Gorenstein injective representations, called Gorenstein strongly fp-injective representations, and give an explicit characterization of the Gorenstein strongly fp-injective representations of right rooted quivers. As an application, a model structure in the category of representations is given., Comment: 26 pages

Due to their adaptive nature, empirical wavelets had several successes in many fields from engineering, science, medical signal/image processing. Recently, a general theoretical framework has been developed in the one-dimensional case, showing the possibility to build empirical wavelets from any classic mother wavelets. Given extensive literature both in theory and applications of classic wavelet frames, it is legitimate to ask about the feasibility of building empirical wavelet frames. We address this question in this paper. We prove several results which provide conditions on the existence of empirical wavelet frames taking into account the above mentioned adaptability.

Practical optimization problems may contain different kinds of difficulties that are often not tractable if one relies on a particular optimization method. Different optimization approaches offer different strengths that are good at tackling one or more difficulty in an optimization problem. For instance, evolutionary algorithms have a niche in handling complexities like discontinuity, non-differentiability, discreteness and non-convexity. However, evolutionary algorithms may get computationally expensive for mathematically well behaved problems with large number of variables for which classical mathematical programming approaches are better suited. In this paper, we demonstrate a decomposition strategy that allows us to synergistically apply two complementary approaches at the same time on a complex optimization problem. Evolutionary algorithms are useful in this context as their flexibility makes pairing with other solution approaches easy. The decomposition idea is a special case of bilevel optimization that separates the difficulties into two levels and assigns different approaches at each level that is better equipped at handling them. We demonstrate the benefits of the proposed decomposition idea on a wide range of test problems., Comment: 9 pages

We provide effective methods to construct and manipulate trilinear birational maps $\phi:(\mathbb{P}^1)^3\dashrightarrow \mathbb{P}^3$ by establishing a novel connection between birationality and tensor rank. These yield four families of nonlinear birational transformations between 3D spaces that can be operated with enough flexibility for applications in computer-aided geometric design. More precisely, we describe the geometric constraints on the defining control points of the map that are necessary for birationality, and present constructions for such configurations. For adequately constrained control points, we prove that birationality is achieved if and only if a certain $2\times 2\times 2$ tensor has rank one. As a corollary, we prove that the locus of weights that ensure birationality is $\mathbb{P}^1\times\mathbb{P}^1\times\mathbb{P}^1$. Additionally, we provide formulas for the inverse $\phi^{-1}$ as well as the explicit defining equations of the irreducible components of the base loci. Finally, we introduce a notion of "distance to birationality" for trilinear rational maps, and explain how to continuously deform birational maps., Comment: 30 pages, 12 figures

We leverage the framework of hyperplane arrangements to analyze potential regions of (stable) fixed points. We provide an upper bound on the number of fixed points for multi-layer neural networks equipped with piecewise linear (PWL) activation functions with arbitrary many linear pieces. The theoretical optimality of the exponential growth in the number of layers of the latter bound is shown. Specifically, we also derive a sharper upper bound on the number of stable fixed points for one-hidden-layer networks with hard tanh activation.

Modeling non-stationary data is a challenging problem in the field of continual learning, and data distribution shifts may result in negative consequences on the performance of a machine learning model. Classic learning tools are often vulnerable to perturbations of the input covariates, and are sensitive to outliers and noise, and some tools are based on rigid algebraic assumptions. Distribution shifts are frequently occurring due to changes in raw materials for production, seasonality, a different user base, or even adversarial attacks. Therefore, there is a need for more effective distribution shift detection techniques. In this work, we propose a continual learning framework for monitoring and detecting distribution changes. We explore the problem in a latent space generated by a bio-inspired self-organizing clustering and statistical aspects of the latent space. In particular, we investigate the projections made by two topology-preserving maps: the Self-Organizing Map and the Scale Invariant Map. Our method can be applied in both a supervised and an unsupervised context. We construct the assessment of changes in the data distribution as a comparison of Gaussian signals, making the proposed method fast and robust. We compare it to other unsupervised techniques, specifically Principal Component Analysis (PCA) and Kernel-PCA. Our comparison involves conducting experiments using sequences of images (based on MNIST and injected shifts with adversarial samples), chemical sensor measurements, and the environmental variable related to ozone levels. The empirical study reveals the potential of the proposed approach., Comment: Revised version of the accepted manuscript to IJCNN'2024. Main corrections were in Section 2.2 and Section 3.3. In Section 2.2 was corrected expression (3), and in Section 3.3 in the definition of the elements of the matrix $D$ it was a typo where $\phi(x)$ was written instead of $x$

We review the application of BRST and BRST-BV approaches to construct the generic off-shell local Lorentz covariant cubic interaction vertices for irreducible massless and massive higher integer spin fields (as the candidates for massive particles in the Dark Matter problem) on $d$-dimensional Minkowski spaces. It is shown that equivalence among two Lagrangian dynamics for the same cubically interacting fields with given masses and spins obtained by means of the approach with complete BRST, $Q$, operator and of one with incomplete BRST, $Q_c$, operator in presence of consistent off-shell holonomic (traceless) constraints can be uplifted from the equivalence of the Lagrangians for free higher spin fields. We found that to get non-contradictory Lagrangians for irreducible interacting higher-spin fields within approach with $Q_c$ operator, together with off-shell algebraic constraints in addition to necessary condition of superconmmuting of $Q_c$ with appropriate holonomic constraints on the field and gauge parameter vectors, these constraints should form Abelian superalgebra both with BRST operator above and with operators of cubic vertices., Comment: 4 pages, 1 figure, revtex, Contribution in Proceedings of 21th Lomonosov Conference on Elementary Particle Physics, Moscow, 24 August- 30 August, 2023, published version

In this paper we present a study of the applicability and feasibility of quantum-inspired algorithms and techniques in tensor networks for industrial environments and contexts, with a compilation of the available literature and an analysis of the use cases that may be affected by such methods. In addition, we explore the limitations of such techniques in order to determine their potential scalability., Comment: 10 pages, in Spanish language, 5 figures

A novel addition to the family of integral transforms, the quadratic phase Fourier transform (QPFT) embodies a variety of signal processing tools, including the Fourier transform (FT), fractional Fourier transform (FRFT), linear canonical transform (LCT), and special affine Fourier transforms. Due to its additional degrees of freedom, QPFT performs better in applications than other time-frequency analysis methods. Recently, quaternion quadratic phase Fourier (QQPFT), an extension of the QPFT in quaternion algebra, has been derived and since received noticeable attention because of its expressiveness and grace in the analysis of multidimensional quaternion-valued signals and visuals. To the best of our knowledge, the discrete form of the QQPFT is undefined, making it impossible to compute the QQPFT using digital techniques at this time. It initiated us to introduce the two-dimensional (2D) discrete quaternion quadratic phase Fourier (DQQPFT) that is analogous to the 2D discrete quaternion Fourier transform (DQFT). Some fundamental properties including Modulation, the reconstruction formula and the Plancherel theorem of the 2D DQQPFT are obtained. Crucially, the fast computation algorithm and convolution theorem of 2D DQQPFT which are essential for engineering applications are also taken into account. Finally, we present an application of the DQQPFT to study the two-dimensional discrete linear time-varying systems., Comment: 19 pages

Partitioned methods for coupled problems rely on data transfers between subdomains to synchronize the subdomain equations and enable their independent solution. By treating each subproblem as a separate entity, these methods enable code reuse, increase concurrency and provide a convenient framework for plug-and-play multiphysics simulations. However, accuracy and stability of partitioned methods depends critically on the type of information exchanged between the subproblems. The exchange mechanisms can vary from minimally intrusive remap across interfaces to more accurate but also more intrusive and expensive estimates of the necessary information based on monolithic formulations of the coupled system. These transfer mechanisms are separated by accuracy, performance and intrusiveness gaps that tend to limit the scope of the resulting partitioned methods to specific simulation scenarios. Data-driven system identification techniques provide an opportunity to close these gaps by enabling the construction of accurate, computationally efficient and minimally intrusive data transfer surrogates. This approach shifts the principal computational burden to an offline phase, leaving the application of the surrogate as the sole additional cost during the online simulation phase. In this paper we formulate and demonstrate such a \emph{dynamic flux surrogate-based} partitioned method for a model advection-diffusion transmission problem by using Dynamic Mode Decomposition (DMD) to learn the dynamics of the interface flux from data. The accuracy of the resulting DMD flux surrogate is comparable to that of a dual Schur complement reconstruction, yet its application cost is significantly lower. Numerical results confirm the attractive properties of the new partitioned approach.

This paper presents a novel approach at the intersection of machine learning and number theory, focusing on the classification of prime and non-prime numbers. At the core of our research is the development of a highly sparse encoding method, integrated with conventional neural network architectures. This combination has shown promising results, achieving a recall of over 99\% in identifying prime numbers and 79\% for non-prime numbers from an inherently imbalanced sequential series of integers, while exhibiting rapid model convergence before the completion of a single training epoch. We performed training using $10^6$ integers starting from a specified integer and tested on a different range of $2 \times 10^6$ integers extending from $10^6$ to $3 \times 10^6$, offset by the same starting integer. While constrained by the memory capacity of our resources, which limited our analysis to a span of $3\times10^6$, we believe that our study contribute to the application of machine learning in prime number analysis. This work aims to demonstrate the potential of such applications and hopes to inspire further exploration and possibilities in diverse fields.

Response solutions are quasi-periodic ones with the same frequency as the forcing term. The present work is devoted to the construction of response solutions for $d$-dimensional beam equations with nonlocal nonlinear damping, which model frictional mechanisms affecting the bodies based on the average. By considering $\epsilon$ in a domain that does not include the origin and imposing a small quasi-periodic forcing with Diophantine frequency vector, we can show the existence of the response solution for such a model. We present an alternative approach to the contraction mapping principle (cf. [5, 33]) through a combination of reduction together with the Nash--Moser iteration technique. The reason behind this approach lies in the derivative losses caused by the nonlocal nonlinearity., Comment: 22 pages

The Additive Transform of an arithmetic function represents a novel approach to examining the interplay between multiplicative arithmetic function and additive functions. This transform concept introduces a method to systematically generate new arithmetic functions by combining the values of an existing function under an additive operation. The resulting framework not only extends our understanding of classical arithmetic functions but also provides a versatile tool for exploring additive relationships within the realm of number theory. In this article, we present the fundamental principles of the Additive Transform and illustrate its application through various examples, shedding light on its potential implications for diverse mathematical domains. For all positive integer $n$. a motivation for the present study is to give a new concept named the Additive transform of an arithmetic function $f$ when $f$ equals some special arithmetic functions, that new concept can help us to prove many results like : \begin{equation} \big(\mu*f.Id\big)(n) =\varphi(n)f(n)+\varphi(n) \sum \limits_{p^{\alpha}||n} \frac{f(p^{\alpha})-f(p^{\alpha-1})}{p-1} \end{equation} where $f$ is an additive function., Comment: 7 pages

By the theory of pseudoinverse matrices and orthoprojectors, we establish a criterion for the solvability and find the general form of solutions of an integrodifferential equation with with impulse action and control. The general form of control for which these solutions exist is also determined. 9 pages. The publication contains the research results of project No.2020.02/0089 with the grant support of the National Research Fund of Ukraine., Comment: 9 pages

We study the period map of configurations of n points on the projective line constructed via a cyclic cover branching along these points. By considering the decomposition of its Hodge structure into eigenspaces, we establish the codimension of the locus where the eigenperiod map is still pure. Furthermore, we show that the period map extends to the divisors of a specific moduli space of weighted stable rational curves, and that this extension satisfies a local Torelli map along its fibers., Comment: 26 pages, To appear in Nagoya. Math. J

We consider the lossless compression bound of any individual data sequence. If we fit the data by a parametric model, the entropy quantity $nH({\hat \theta}_n)$ obtained by plugging in the maximum likelihood estimate is an underestimate of the bound, where $n$ is the number of words. Shtarkov showed that the normalized maximum likelihood (NML) distribution or code length is optimal in a minimax sense for any parametric family. We show by the local asymptotic normality that the NML code length for the exponential families is $nH(\hat \theta_n) +\frac{d}{2}\log \, \frac{n}{2\pi} +\log \int_{\Theta} |I(\theta)|^{1/2}\, d\theta+o(1)$, where $d$ is the model dimension or dictionary size, and $|I(\theta)|$ is the determinant of the Fisher information matrix. We also demonstrate that sequentially predicting the optimal code length for the next word via a Bayesian mechanism leads to the mixture code, whose pathwise length is given by $nH({\hat \theta}_n) +\frac{d}{2}\log \, \frac{n}{2\pi} +\log \frac{|\, I({\hat \theta}_n)|^{1/2}}{w({\hat \theta}_n)}+o(1) $, where $w(\theta)$ is a prior. The asymptotics apply to not only discrete symbols but also continuous data if the code length for the former is replaced by the description length for the latter. The analytical result is exemplified by calculating compression bounds of protein-encoding DNA sequences under different parsing models. Typically, the highest compression is achieved when the parsing is in phase of the amino acid codons. On the other hand, the compression rates of pseudo-random sequences are larger than 1 regardless parsing models. These model-based results are in consistency with that random sequences are incompressible as asserted by the Kolmogorov complexity theory. The empirical lossless compression bound is particularly more accurate when dictionary size is relatively large., Comment: 3 tables

Reversible computation is an emerging computing paradigm that allows any sequence of operations to be executed in reverse order at any point during computation. Its appeal lies in its potential for lowpower computation and its relevance to a wide array of applications such as chemical reactions, quantum computation, robotics, and distributed systems. Reversing Petri nets are a recently-proposed extension of Petri nets that implements the three main forms of reversibility, namely, backtracking, causal reversing, and out-of-causal-order reversing. Their distinguishing feature is the use of named tokens that can be combined together to form bonds. Named tokens along with a history function, constitute the means of remembering past behaviour, thus, enabling reversal. In recent work, we have proposed a structural translation from a subclass of RPNs to the model of Coloured Petri Nets (CPNs), an extension of traditional Petri nets where tokens carry data values. In this paper, we extend the translation to handle RPNs with token multiplicity under the individual-token interpretation, a model which allows multiple tokens of the same type to exist in a system. To support the three types of reversibility, tokens are associated with their causal history and, while tokens of the same type are equally eligible to fire a transition when going forward, when going backwards they are able to reverse only the transitions they have previously fired. The new translation, in addition to lifting the restriction on token uniqueness, presents a refined approach for transforming RPNs to CPNs through a unifying approach that allows instantiating each of the three types of reversibility. The paper also reports on a tool that implements this translation, paving the way for automated translations and analysis of reversible systems using CPN Tools., Comment: The paper is planned to be published in a reputable journal

We prove local connectivity of the boundaries of invariant simply connected attracting basins for a class of transcendental meromorphic maps. The maps within this class need not be geometrically finite or in class $\mathcal B$, and the boundaries of the basins (possibly unbounded) are allowed to contain an infinite number of post-singular values, as well as the essential singularity at infinity. A basic assumption is that the unbounded parts of the basins are contained in regions which we call `repelling petals at infinity', where the map exhibits a kind of `parabolic' behaviour. In particular, our results apply to a wide class of Newton's methods for transcendental entire maps. As an application, we prove local connectivity of the Julia set of Newton's method for $\sin z$, providing the first non-trivial example of a locally connected Julia set of a transcendental map outside class $\mathcal B$, with an infinite number of unbounded Fatou components., Comment: V2: We correct some misprints and explaining how to extend the construction in the case of parabolic periodic (instead of fixed) points lying in the boundary of $U$. 49 pages, 8 figures

The modeling of high-frequency data that qualify financial asset transactions has been an area of relevant interest among statisticians and econometricians -- above all, the analysis of time series of financial durations. Autoregressive conditional duration (ACD) models have been the main tool for modeling financial transaction data, where duration is usually defined as the time interval between two successive events. These models are usually specified in terms of a time-varying mean (or median) conditional duration. In this paper, a new extension of ACD models is proposed which is built on the basis of log-symmetric distributions reparametrized by their quantile. The proposed quantile log-symmetric conditional duration autoregressive model allows us to model different percentiles instead of the traditionally used conditional mean (or median) duration. We carry out an in-depth study of theoretical properties and practical issues, such as parameter estimation using maximum likelihood method and diagnostic analysis based on residuals. A detailed Monte Carlo simulation study is also carried out to evaluate the performance of the proposed models and estimation method in retrieving the true parameter values as well as to evaluate a form of residuals. Finally, the proposed class of models is applied to a price duration data set and then used to derive a semi-parametric intraday value-at-risk (IVaR) model., Comment: 19 pages, 9 figures

The effect of composition on microstructure both at the length scale of the secondary dendrite arm spacing and nano-sized dispersoids during homogenization of Al-Zn-Cu-Mg-Zr alloys has been studied. A comprehensive model that can predict the microstructure at both the length scales has been used for the study. The microstructure predicted has been compared to that for two homogenized samples from a directionally solidified AA7050 sample and a reasonable match has been found. The initial as-cast microstructure for different compositions is calculated using Scheil type solidification from Thermo-Calc. The initial microstructure has a considerable influence on microstructural evolution during homogenization. To take advantage of the decreased solid solubility of Zr in {\alpha}-fcc, cooling rates during solidification must be high enough to prevent precipitation of primary Al3Zr. Under solidification with industrial cooling conditions solute rich alloys leads to fewer dispersoids. Based on the study, an improved composition range of 6-8%Zn, 1-2%Cu, 1-2%Mg and 0.1-0.15%Zr for 7XXX alloys has been proposed., Comment: 23 pages, 8 figures, Part of PhD Dissertation work by Pikee Priya at Purdue University (2016)

A "through-process" solidification homogenization numerical model to study the microstructural evolution has been developed for AA7050 alloy. A continuum scale Direct Chill Casting (DCC) solidification model has been coupled with a meso-scale homogenization precipitation model to evaluate the radial variation of microstructure in a cylindrical DC cast billet after homogenization. The Local Solidification Times predicted by the DCC numerical model is used to predict the Secondary Dendritic Arm Spacing (SDAS) across the radius from an empirical relationship. It also predicts the macrosegregation used to estimate the radial as-cast microstructures (phases and phase fractions) using Thermo-Calc. Macrosegregation affects the initial and homogenized microstructures across the radius, making the surface prone to recrystallization, due to lesser dispersoids and larger precipitated particles during cooling, caused by lesser extent of Zenner pinning and Particle Stimulated Nucleation (PSN) of recrystallized grains respectively., Comment: 20 Pages, 8 Figures, part of the PhD Dissertation work by Pikee Priya at Purdue University (2016). Cited for reference

Using strategic timetables as input for railway network design has become increasingly popular among western European railway infrastructure operators. Although both railway timetabling and railway network design on their own are well covered by academic research, there is still a gap in the literature concerning timetable-based network design. Therefore, we propose a mixed-integer linear program to design railway infrastructure so that the demand derived from a strategic timetable can be satisfied with minimal infrastructure costs. The demand is given by a list of trains, each featuring start and destination nodes as well as time bounds and a set of frequency and transfer constraints that capture the strategic timetable's main characteristics. During the optimization, the solver decides which railway lines need to be built or expanded and whether travel or headway times must be shortened to meet the demand. Since strategic timetables are subject to uncertainty, we expand the optimization model to a robust version. Uncertain timetables are modelled as discrete scenarios, while uncertain freight train demand is modelled using optional trains, which can be inserted into the resulting timetable if they do not require additional infrastructure. We present computational results for both the deterministic and the robust case and give an outlook on further research., Comment: 23 pages, 5 figures, 6 tables

In this paper, we firstly introduce nonlinear truncated Baskakov operators on compact intervals and obtain some direct theorems. Also, we give the approximation of fuzzy numbers by truncated nonlinear Baskakov operators., Comment: 14 pages

In this paper, we propose an exact general algorithm for solving non-convex optimization problems, where the non-convexity arises due to the presence of an inverse S-shaped function. The proposed method involves iteratively approximating the inverse S-shaped function through piece-wise linear inner and outer approximations. In particular, the concave part of the inverse S-shaped function is inner-approximated through an auxiliary linear program, resulting in a bilevel program, which is reduced to a single level using KKT conditions before solving it using the cutting plane technique. To test the computational efficiency of the algorithm, we solve a facility location problem involving economies and dis-economies of scale for each of the facilities. The computational experiments indicate that our proposed algorithm significantly outperforms the previously reported methods. We solve non-convex facility location problems with sizes up to 30 potential facilities and 150 customers. Our proposed algorithm converges to the global optimum within a maximum computational time of 3 hours for 95 percent of the datasets. For almost 60 percent of the test cases, the proposed algorithm outperforms the benchmark methods by an order of magnitude. The paper ends with managerial insights on facility network design involving economies and dis-economies of scale. One of the important insights points out that it may be optimal to increase the number of production facilities operating under dis-economies of scale with an overall decrease in transportation costs., Comment: 5 Figures, 8 Tables

We present a new category of physics-informed neural networks called physics informed variational embedding generative adversarial network (PI-VEGAN), that effectively tackles the forward, inverse, and mixed problems of stochastic differential equations. In these scenarios, the governing equations are known, but only a limited number of sensor measurements of the system parameters are available. We integrate the governing physical laws into PI-VEGAN with automatic differentiation, while introducing a variational encoder for approximating the latent variables of the actual distribution of the measurements. These latent variables are integrated into the generator to facilitate accurate learning of the characteristics of the stochastic partial equations. Our model consists of three components, namely the encoder, generator, and discriminator, each of which is updated alternatively employing the stochastic gradient descent algorithm. We evaluate the effectiveness of PI-VEGAN in addressing forward, inverse, and mixed problems that require the concurrent calculation of system parameters and solutions. Numerical results demonstrate that the proposed method achieves satisfactory stability and accuracy in comparison with the previous physics-informed generative adversarial network (PI-WGAN)., Comment: 23 pages

As developers of libraries implementing interval arithmetic, we faced the same difficulties when it comes to testing our libraries. What must be tested? How can we devise relevant test cases for unit testing? How can we ensure a high (and possibly 100%) test coverage? Before considering these questions, we briefly recall the main features of interval arithmetic and of the IEEE 1788-2015 standard for interval arithmetic. After listing the different aspects that, in our opinion, must be tested, we contribute a first step towards offering a test suite for an interval arithmetic library. First we define a format that enables the exchange of test cases, so that they can be read and tried easily. Then we offer a first set of test cases, for a selected set of mathematical functions. Next, we examine how the Julia interval arithmetic library, IntervalArithmetic.jl, actually performs to these tests. As this is an ongoing work, we list extra tests that we deem important to perform., Comment: 2 figures

The Euclidean operator radius of two bounded linear operators in the Hilbert $C^*$-module over $\A$ is given some precise bounds. Their relationship to recent findings in the literature that offer precise upper and lower bounds on the numerical radius of linear operators is also established., Comment: 12 pages

In this paper, a new approach to bivariate modeling of autoregressive conditional duration (ACD) models is proposed. Specifically, we consider the joint modeling of durations and the number of transactions made during the spell. The proposed bivariate ACD model is based on log-symmetric distributions, which are useful for modeling strictly positive, asymmetric and light- and heavy-tailed data, such as transaction-level high-frequency financial data. A Monte Carlo simulation is performed for the assessment of the estimation method and the evaluation of a form of residuals. A real financial transactions data set is analyzed in order to illustrate the proposed method., Comment: 21 pages, 4 figures

Disruptions in the operational flow of rail traffic can lead to conflicts between train movements, such that a scheduled timetable can no longer be realised. This is where dispatching is applied, existing conflicts are resolved and a dispatching timetable is provided. In the process, train paths are varied in their spatio-temporal course. This is called the train dispatching problem (TDP), which consists of selecting conflict-free train paths with minimum delay. Starting from a path-oriented formulation of the TDP, a binary linear decision model is introduced. For each possible train path, a binary decision variable indicates whether the train path is used by the request or not. Such a train path is constructed from a set of predefined path parts (speed-profiles) within a time-space network. Instead of modelling pairwise conflicts, stronger MIP formulation are achieved by a cliques formulated over the complete train path. The combinatorics of speed-profiles and different departure times results in a large number of possible train paths, so that the column generation method is used here. Within the subproblem, the shadow prices of conflict cliques must be taken into account. When constructing a new train path, it must be determined whether this train path belongs to a clique or not. This problem is tackled by a MIP. The methodology is tested on instances from a dispatching area in Germany. Numerical results show that the presented method achieves acceptable computation times with good solution quality while meeting the requirements for real-time dispatching., Comment: 19 pages, 10 figures, 5 tables

This article deals with three types of mutually inverse series relating Ferrers and associated Legendre functions of arbitrary complex indexes and orders established on the base of integral representations by using a number of generating functions (some of them are novel) for polynomials expressed in terms of hypergeometric or generalized hypergeometric polynomials. High-order asymptotics of above-mentioned polynomials are indicated. In particular, a uniform asymptotics for Gegenbauer polynomials of degree n and index a-n is obtained for large n. In special cases, the inverse series turn into novel mutually inverse finite sums relating Gegenbauer or associated Legendre polynomials of different arguments as well as into novel connection formulas for Gegenbauer and associated Legendre polynomials., Comment: 27 pages

This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions., Comment: 24 pages, 2 figures. Published in journal of Groups, Complexity, Cryptology. arXiv admin note: text overlap with arXiv:2112.09812

In this paper, we propose a penalty dual-primal augmented lagrangian method for solving convex minimization problems under linear equality or inequality constraints. The proposed method combines a novel penalty technique with updates the new iterates in a dual-primal order, and then be extended to solve multiple-block separable convex programming problems with splitting version and partial splitting version. We establish the convergence analysis for all the introduced algorithm in the lens of variational analysis. Numerical results on the basic pursuit problem and the lasso model are presented to illustrate the efficiency of the proposed method.

We deduce a non-linear commutator higher-spin (HS) symmetry algebra which encodes unitary irreducible representations of the AdS group -- subject to a Young tableaux $Y(s_1,\ldots ,s_k)$ with $k\geq 2$ rows -- in a $d$-dimensional anti-de-Sitter space. Auxiliary representations for a specially deformed non-linear HS symmetry algebra in terms of a generalized Verma module, applied to additively convert a subsystem of second-class constraints in the HS symmetry algebra into one with first-class constraints, are found explicitly in the case of a $k=2$ Young tableaux. An oscillator realization over the Heisenberg algebra for the Verma module is constructed. The results generalize the method of constructing auxiliary representations for the symplectic $sp(2k)$ algebra used for mixed-symmetry HS fields in flat spaces [arXiv:1110.5044[hep-th]], and can be extended to the case of arbitrary HS fields in AdS spaces. Polynomial deformations of the $su(1,1)$ algebra related to the conversion are studied as a by-product. A nilpotent BRST operator for a non-linear HS symmetry algebra of the converted constraints is found, with non-vanishing terms (resolving the Jacobi identities) of third order in powers of ghost coordinates. A gauge-invariant unconstrained reducible Lagrangian formulation for a free bosonic HS field of generalized spin $(s_1,s_2)$ is deduced. Following the results of [arXiv:2105.12030[hep-th]], [arXiv:2212.07097[hep-th]], we develop a BRST approach to constructing general off-shell local cubic interaction vertices for irreducible massive higher-spin fields. A new reducible gauge-invariant Lagrangian formulation for an antisymmetric massive tensor field of spin $(1,1)$ is obtained., Comment: 69 pages, presentation clarified and improved, 11 references added; elaborated and developed conference paper [arXiv:1111.5516[hep-th]]

We study experimental convergence rates of three shock-capturing schemes for hyperbolic systems of conservation laws: the second-order central-upwind (CU) scheme, the third-order Rusanov-Burstein-Mirin (RBM), and the fifth-order alternative weighted essentially non-oscillatory (A-WENO) scheme. We use three imbedded grids to define the experimental pointwise, integral, and $W^{-1,1}$ convergence rates. We apply the studied schemes to the shallow water equations and conduct their comprehensive numerical convergence study. We verify that while the studied schemes achieve their formal orders of accuracy on smooth solutions, after the shock formation, a part of the computed solutions is affected by shock propagation and both the pointwise and integral convergence rates reduce there. Moreover, while the $W^{-1,1}$ convergence rates for the CU and A-WENO schemes, which rely on nonlinear stabilization mechanisms, reduce to the first order, the RBM scheme, which utilizes a linear stabilization, is clearly second-order accurate. Finally, relying on the conducted experimental convergence rate study, we develop two new combined schemes based on the RBM and either the CU or A-WENO scheme. The obtained combined schemes can achieve the same high-order of accuracy as the RBM scheme in the smooth areas while being non-oscillatory near the shocks., Comment: 33 pages

We study the aspects of constructing the interactions for the higher spin fields in the framework of BRST approach. The main object of such an approach is BRST operator acting in the appropriate Fock space and building on the base of constraints that define the irreducible higher spin representations. In its turn, the constrains are divided into differential and purely algebraic or holonomic. The necessary and sufficient conditions to derive the consistent Lagrangian formulations for irreducible interacting higher-spin fields within approach with incomplete BRST operator, where the algebraic constraints are not included into definition of the BRST operator but imposed "by hands" on the field and gauge parameter vectors, are considered. It is shown that in addition to that such constraints should (anti)commute with the BRST operator and annihilate the fields and gauge parameters Fock space vectors, they must form the Abelian (super)algebra both with the BRST operator above and with operators of cubic, quartic and etc. vertices. Only under the above conditions, the formulations with complete and incomplete BRST charges turn out to be equivalent and yield to the same interaction vertices in terms of irreducible fields., Comment: 11 pages, minor changes, presentation improved, Contribution to Proceedings of International Workshop "Supersymmetries and Quantum Symmetries 2022"

In this paper we study of the structure of non-commutative Poisson algebras with an arbitrary set $\ss.$ We show that any of such an algebra $\pp$ decomposes as $$\pp=\uu\oplus\sum_{[\lambda]\in(\Lambda_\ss\setminus\{0\})/\sim}\pp_{[\lambda]},$$ where $\uu$ is a linear subspace complement of $\span_{\bbbf}\{ [\pp_{\mu}, \pp_{\eta}]+\pp_{\mu}\pp_{\eta} : \mu, \eta\in[\lam]\}\cap\pp_0$ in $\pp_0$ and any $\pp_{[\lambda]}$ a well-described graded ideals of $\pp,$ satisfying $[\pp_{[\lambda]}, \pp_{[\mu]}]+\pp_{[\lambda]} \pp_{[\mu]}=0$ if $[\lambda]\neq[\mu].$ Under certain conditions, the simplicity of $\pp$ is characterized and it is shown that $\pp$ is the direct sum of the family of its graded simple ideals., Comment: 19 pages. arXiv admin note: text overlap with arXiv:2303.13832

Parrondo's paradox was introduced by Juan Parrondo in 1996. In game theory, this paradox is described as: A combination of losing strategies becomes a winning strategy. At first glance, this paradox is quite surprising, but we can easily explain it by using simulations and mathematical arguments. Indeed, we first consider some examples with the Parrondo's paradox and, using the software R, we simulate one of them, the coin tossing. Actually, we see that specific combinations of losing games become a winning game. Moreover, even a random combination of these two losing games leads to a winning game. Later, we introduce the major definitions and theorems over Markov chains to study our Parrondo's paradox applied to the coin tossing problem. In particular, we represent our Parrondo's game as a Markov chain and we find its stationary distribution. In that way, we exhibit that our combination of two losing games is truly a winning combination. We also deliberate possible applications of the paradox in some fields such as ecology, biology, finance or reliability theory.

In this paper there are several results, we prove approximation of periodic function by Fejer means and De La Vallee Poussin means in Lebesgue spaces the estimates are given in terms of function for and in terms of second continuity modulus., Comment: 9 pages

In this paper we introduce the class of graded Poisson color algebras as the natural generalization of graded Poisson algebras and graded Poisson superalgebras. For $\Lambda$ an arbitrary abelian group, we show that any of such $\Lambda$-graed Poisson color algebra $\mathcal{P}$, with a symmetric $\Lambda$-support is of the form $\mathcal{P} = \mathcal{U}\oplus\sum_j I_j$, with $\mathcal{U}$ a subspace of $\mathcal{P}_1$ and any $I_j$ a well described graded ideal of $\mathcal{P},$ satisfying $\{I_j, I_k\}+I_j I_k=0$ if $j\neq i.$ Furthermore, under certain conditions, the gr-simplicity of $\mathcal{P}$ is characterized and it is shown that $\mathcal{P}$ is the direct sum of the family of its graded simple ideals., Comment: 15 pages

We study the structure of a graded $3$-Lie-Rinehart algebra $\mathcal{L}$ over an associative and commutative graded algebra $A.$ For $G$ an abelian group, we show that if $(L, A)$ is a tight $G$-graded 3-Lie-Rinehart algebra, then $\mathcal{L}$ and $A$ decompose as $\mathcal{L} =\bigoplus_{i\in I}\mathcal{L}_i$ and $A =\bigoplus_{j\in J}A_j,$ where any $\mathcal{L}_i$ is a non-zero graded ideal of $\mathcal{L}$ satisfying $[\mathcal{L}_{i_1}, \mathcal{L}_{i_2}, \mathcal{L}_{i_3}]=0$ for any $i_1, i_2, i_3\in I$ different from each other, and any $A_j$ is a non-zero graded ideal of $A$ satisfying $A_j A_l=0$ for any $l, j\in J$ such that $j\neq l,$ and both decompositions satisfy that for any $i\in I$ there exists a unique $j\in J$ such that $A_j \mathcal{L}_i\neq 0.$ Furthermore, any $(\mathcal{L}_i, A_j)$ is a graded 3-Lie-Rinehart algebra. Also, under certain conditions, it is shown that the above decompositions of $\mathcal{L}$ and $A$ are by means of the family of their, respective, graded simple ideals., Comment: 27 pages. arXiv admin note: text overlap with arXiv:2108.03604. substantial text overlap with arXiv:2202.12982, arXiv:1706.07084, arXiv:1904.11821 by other authors

We develop the BRST-BV approach to construct the general off-shell Lorentz covariant cubic, quartic, $e$-tic interaction vertices for irreducible higher spin fields on $d$-dimensional Minkowski space. We consider two different cases for interacting integer higher spin fields both with massless and with massive fields. The deformation procedure to find minimal (determined with help of generalized Hilbert space) BRST-BV action for interacting higher spin fields is based on the preservation of master equation validity in each power of coupling constant $g$ starting from the Lagrangian formulation for free gauge theory. As examples we consider the construction of local cubic vertices for $k$ irreducible massless fields of integer helicities, and $k-1$ massless with one massive fields of spins $s_1, ,..., s_{k-1}, s_k$. For triple of two massless scalars and tensor field of integer spin the BRST-BV action with cubic interaction is explicitly found. Unlike the previous results on cubic vertices we follow our result for BRST approach [arXiv:2105.12030[hep-th]] for massless fields, but for the unique BRST-BV action instead of classical action with reducible gauge transformations. The procedure is based on the complete BRST operator, including the trace constraints that is used to formulate an irreducible representation with definite integer spin., Comment: 26 pages, 2 references, acknowledgments added, comments on single vertex added, contribution for Proceedings of the VII Conference "Models in Quantum Field Theory" (MQFT-2022), Saint Petersburg, Russia (10-14 October 2022), accepted for publication in Theoretical and Mathematical Physics. arXiv admin note: text overlap with arXiv:2212.07097

We are concerned with finding the set $I_{\succeq}(A,B)$ of real values $\mu$ such that the matrix pencil $A+\mu B$ is positive semidefinite. If $A, B$ are not simultaneously diagonalizable via congruence (SDC), $I_{\succeq}(A,B)$ either is empty or has only one value $\mu.$ When $A, B$ are SDC, $I_{\succeq}(A,B),$ if not empty, can be a singleton or an interval. Especially, if $I_{\succeq}(A,B)$ is an interval and at least one of the matrices is nonsingular then its interior is the positive definite interval $I_{\succ}(A,B).$ If $A, B$ are both singular, then even $I_{\succeq}(A,B)$ is an interval, its interior may not be $I_{\succ}(A,B),$ but $A, B$ are then decomposed to block diagonals of submatrices $A_1, B_1$ with $B_1$ nonsingular such that $I_{\succeq}(A,B)=I_{\succeq}(A_1,B_1).$ Applying $I_{\succeq}(A,B),$ the hard-case of the generalized trust-region subproblem (GTRS) can be dealt with by only solving a system of linear equations or reduced to the easy-case of a GTRS of smaller size., Comment: 23 pages, no figures