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Our purpose of this paper is to focus on fixed point property in fuzzy metric space. To achieve our objective, we will introduce a new contraction condition to examine the fixed point for multi-valued mapping, then we will be investigating the obtained result to ensure the existence and uniqueness of this property for single-valued mapping. To show the use of our main result, we will give the relative result in the ordinary metric space.

In this paper, we lay the groundwork on the comparison of phylogenetic networks based on edge contractions and expansions as edit operations, as originally proposed by Robinson and Foulds to compare trees. We prove that these operations connect the space of all phylogenetic networks on the same set of leaves, even if we forbid contractions that create cycles. This allows to define an operational distance on this space, as the minimum number of contractions and expansions required to transform one network into another. We highlight the difference between this distance and the computation of the maximum common contraction between two networks. Given its ability to outline a common structure between them, which can provide valuable biological insights, we study the algorithmic aspects of the latter. We first prove that computing a maximum common contraction between two networks is NP-hard, even when the maximum degree, the size of the common contraction, or the number of leaves is bounded. We also provide lower bounds to the problem based on the Exponential-Time Hypothesis. Nonetheless, we do provide a polynomial-time algorithm for weakly-galled networks, a generalization of galled trees., Comment: full version (with complete set of proofs) of WABI 2024 submission

In this chapter, we consider a reaction-diffusion SVIR infection model with dis-tributed delay and nonlinear incidence rate. The wellposedness of the proposed model is proved. By means of Lyapunov functionals, we show that the disease-free equilibrium state is globally asymptotically stable when the basic reproduction number is less or equal than one, and that the disease endemic equilibrium is globally asymptotically stable when the basic reproduction number is greater than one. Numerical simulations are provided to illustrate the obtained theoretical results.

Large Language Models (LLMs) such as ChatGPT and LlaMA are advancing rapidly in generative Artificial Intelligence (AI), but their immense size poses significant challenges, such as huge training and inference costs, substantial energy demands, and limitations for on-site deployment. Traditional compression methods such as pruning, distillation, and low-rank approximation focus on reducing the effective number of neurons in the network, while quantization focuses on reducing the numerical precision of individual weights to reduce the model size while keeping the number of neurons fixed. While these compression methods have been relatively successful in practice, there is no compelling reason to believe that truncating the number of neurons is an optimal strategy. In this context, this paper introduces CompactifAI, an innovative LLM compression approach using quantum-inspired Tensor Networks that focuses on the model's correlation space instead, allowing for a more controlled, refined and interpretable model compression. Our method is versatile and can be implemented with - or on top of - other compression techniques. As a benchmark, we demonstrate that a combination of CompactifAI with quantization allows to reduce a 93% the memory size of LlaMA 7B, reducing also 70% the number of parameters, accelerating 50% the training and 25% the inference times of the model, and just with a small accuracy drop of 2% - 3%, going much beyond of what is achievable today by other compression techniques. Our methods also allow to perform a refined layer sensitivity profiling, showing that deeper layers tend to be more suitable for tensor network compression, which is compatible with recent observations on the ineffectiveness of those layers for LLM performance. Our results imply that standard LLMs are, in fact, heavily overparametrized, and do not need to be large at all., Comment: 5 pages, 4 figures, 2 tables, and supplementary information of 2 pages and 1 figure. Revised version with new benchmarks for LlaMA2-7B

The proximal gradient method is a generic technique introduced to tackle the non-smoothness in optimization problems, wherein the objective function is expressed as the sum of a differentiable convex part and a non-differentiable regularization term. Such problems with tensor format are of interest in many fields of applied mathematics such as image and video processing. Our goal in this paper is to address the solution of such problems with a more general form of the regularization term. An adapted iterative proximal gradient method is introduced for this purpose. Due to the slowness of the proposed algorithm, we use new tensor extrapolation methods to enhance its convergence. Numerical experiments on color image deblurring are conducted to illustrate the efficiency of our approach., Comment: 54 figures

In this paper, we present a new framework for the recent multidimensional extrapolation methods: Tensor Global Minimal Polynomial (TG-MPE) and Tensor Global Reduced Rank Extrapolation (TG-RRE) methods. We develop a new approach to the one presented in \cite{17}. The proposed framework highlights, in addition their polynomial feature, the connection of TG-MPE and TG-RRE with nonlinear Krylov subspace methods. A unified algorithm is proposed for their implemention. Theoretical results are given and some numerical experiments on linear and nonlinear problems are considered to confirm the performance of the proposed algorithms.

Phylogenetic trees (i.e. evolutionary trees, additive trees or X-trees) play a key role in the processes of modeling and representing species evolution. Genome evolution of a given group of species is usually modeled by a species phylogenetic tree that represents the main patterns of vertical descent. However, the evolution of each gene is unique. It can be represented by its own gene tree which can differ substantially from a general species tree representation. Consensus trees and supertrees have been widely used in evolutionary studies to combine phylogenetic information contained in individual gene trees. Nevertheless, if the available gene trees are quite different from each other, then the resulting consensus tree or supertree can either include many unresolved subtrees corresponding to internal nodes of high degree or can simply be a star tree. This may happen if the available gene trees have been affected by different reticulate evolutionary events, such as horizontal gene transfer, hybridization or genetic recombination. Thus, the problem of inferring multiple alternative consensus trees or supertrees, using clustering, becomes relevant since it allows one to regroup in different clusters gene trees having similar evolutionary patterns (e.g. gene trees representing genes that have undergone the same horizontal gene transfer or recombination events). We critically review recent advances and methods in the field of phylogenetic tree clustering, discuss the methods' mathematical properties, and describe the main advantages and limitations of multiple consensus tree and supertree approaches. In the application section, we show how the multiple supertree clustering approach can be used to cluster aaRS gene trees according to their evolutionary patterns.

Point defects are responsible for a wide range of optoelectronic properties in materials, making it crucial to engineer their concentrations for novel materials design. However, considering the plethora of defects in co-doped semiconducting and dielectric materials and the dependence of defect formation energies on heat treatment parameters, process design based on an experimental trial and error approach is not an efficient strategy. This makes it necessary to explore computational pathways for predicting defect equilibria during heat treatments. The accumulated experimental knowledge on defect transformations in diamond is unparalleled. Therefore, diamond is an excellent material for benchmarking computational approaches. By considering nitrogen, hydrogen, and silicon doped diamond as a model system, we have investigated the pressure dependence of defect formation energies and calculated the defect equilibria during heat treatment of diamond through ab-initio calculations. We have plotted monolithic-Kr\"oger-Vink diagrams for various defects, representing defect concentrations based on process parameters, such as temperature and partial pressure of gases used during heat treatments of diamond. The method demonstrated predicts the majority of experimental data, such as nitrogen aggregation path leading towards the formation of the B center, annealing of the B, H3, N3, and NVHx centers at ultra high temperatures, the thermal stability of the SiV center, and temperature dependence of NV concentration. We demonstrate the possibility of designing heat treatments for a wide range of semiconducting and dielectric materials by using a relatively inexpensive yet robust first principles approach, significantly accelerating defect engineering and high-throughput novel materials design., Comment: This preprint includes 17 pages, 1 cover photo, 7 figures and 2 appendices

In this paper, we study an epidemic model with Atangana-Baleanu-Caputo (ABC) fractional derivatives. We obtain a special solution using an iterative scheme via Laplace transformation. Uniqueness and existence of solution using the Banach fixed point theorem are studied. A detailed analysis of the stability of the special solution is presented. Finally, our generalized model in the ABC derivative sense is solved numerically by the Adams-Bashforth-Moulton method., Comment: This is a preprint of a Research Article published at [https://doi.org/10.31559/glm2021.10.2.7]. Cite this paper as: M. R. Sidi Ammi, M. Tahiri and D. F. M. Torres, Local existence and uniqueness for a fractional SIRS model with Mittag-Leffler law, Gen. Lett. Math. 10 (2021), no. 2, 61--71

The main aim of the present work is to study and analyze a reaction-diffusion fractional version of the SIR epidemic mathematical model by means of the non-local and non-singular ABC fractional derivative operator with complete memory effects. Existence and uniqueness of solution for the proposed fractional model is proved. Existence of an optimal control is also established. Then, necessary optimality conditions are derived. As a consequence, a characterization of the optimal control is given. Lastly, numerical results are given with the aim to show the effectiveness of the proposed control strategy, which provides significant results using the AB fractional derivative operator in the Caputo sense, comparing it with the classical integer one. The results show the importance of choosing very well the fractional characterization of the order of the operators., Comment: This is a preprint of a paper whose final and definite form is with 'Discrete and Continuous Dynamical Systems -- Series S' (DCDS-S), ISSN 1937-1632 (print), ISSN 1937-1179 (online), available at [http://aimsciences.org/journal/1937-1632]

Each gene has its own evolutionary history which can substantially differ from the evolutionary histories of other genes. For example, some individual genes or operons can be affected by specific horizontal gene transfer and recombination events. Thus, the evolutionary history of each gene should be represented by its own phylogenetic tree which may display different evolutionary patterns from the species tree that accounts for the main patterns of vertical descent. The output of traditional consensus tree or supertree inference methods is a unique consensus tree or supertree. We describe a new efficient method for inferring multiple alternative consensus trees and supertrees to best represent the most important evolutionary patterns of a given set of gene phylogenies. We show how an adapted version of the popular k-means clustering algorithm, based on some interesting properties of the Robinson and Foulds distance, can be used to partition a given set of trees into one (for homogeneous data) or multiple (for heterogeneous data) cluster(s) of trees. Moreover, we adapt the popular Cali\'nski-Harabasz, Silhouette, Ball and Hall, and Gap cluster validity indices to tree clustering with k-means. A special attention is given to the relevant but very challenging problem of inferring alternative supertrees. The use of the Euclidean property of the objective function of the method makes it faster than the existing tree clustering techniques, and thus perfectly suitable for analyzing large evolutionary datasets. We apply the new method to discover alternative supertrees characterizing the main patterns of evolution of SARS-CoV-2 and genetically related betacoronaviruses., Comment: Bioinformatics (in press)

It is well-known that Microsoft Word/Excel compatible documents or PDF files can contain malicious content. LaTeX files are unfortunately no exception either. LaTeX users often include third-party codes through sources or packages (.sty or .cls files). But those packages can execute malicious commands on the users' system, in order to capture sensitive information or to perform denial of service attacks. Checkoway et al. [3] were the first to warn LaTeX users of these threats. Collaborative cloud-based LaTeX editors and services compiling LaTeX sources are particularly concerned. In this paper, we have created a LaTeX package that collects system data and hides them inside the PDF file produced by the target. Then, we have measured what can be recovered by hackers using malicious LaTeX file on online services, and which measures those services have enforced to thwart the threats. Services defend themselves using sandbox or commands restrictions. Commands restrictions are more difficult to setup and we found one service (PMLatex) which is too permissive.

We introduce a fractional order SIRS model with non-linear incidence rate. Existence of a unique positive solution to the model is proved. Stability analysis of the disease free equilibrium and positive fixed points are investigated. Finally, a numerical example is presented., Comment: This is a preprint of a paper whose final and definite form is with 'Math. Comput. Sci.', ISSN 1661-8270 (print), ISSN 1661-8289 (electronic), available at [https://www.springer.com/journal/11786]. Submitted 1-Jun-2019; Revised 5-Feb-2020; Accepted 6-Feb-2020

In the past years, the IceCube Collaboration has reported in several analyses the observation of astrophysical high-energy neutrino events. Despite a compelling evidence for the first identification of a neutrino source, TXS 0506+056, the origin of the majority of these events is still unknown. In this paper, a possible transient origin of the IceCube astrophysical events is searched for using neutrino events detected by the ANTARES telescope. The arrival time and direction of 6894 track-like and 160 shower-like events detected over 2346 days of livetime are examined to search for coincidences with 54 IceCube high-energy track-like neutrino events, by means of a maximum likelihood method. No significant correlation is observed and upper limits on the one-flavour neutrino fluence from the direction of the IceCube candidates are derived. The non-observation of time and space correlation within the time window of 0.1 days with the two most energetic IceCube events constrains the spectral index of a possible point-like transient neutrino source, to be harder than $-2.3$ and $-2.4$ for each event, respectively., Comment: 14 pages, 5 figures

We show that the transition laws for a 2-connection can be recovered by discretizing the base 2-space of a 2-bundle into an Euclidean hypercubic lattice. The aim of this work is to serve as an example of how important results in higher gauge theory, which have been derived in a continuous setting, can also be derived in the lattice scheme.

We show that for N = 1 supersymmetric Yang-Mills theory it is possible to build an off-shell nilpotent BRST and anti-BRST algebra in terms of a BRST superspace formalism. This is based on the introduction of the basic fields of the quantized theory together with an auxiliary real field via the lowest components of the superfield components of a superYang-Mills connection. Here, the associated supercurvature is constrained by horizontality conditions as in ordinary Yang-Mills theory. We also show how the off-shell BRST-invariant quantum action can be constructed starting from a gauge-fixed superaction., Comment: accepted for publication in Acta Polonica B and will appear in the October issue of the journal. arXiv admin note: substantial text overlap with arXiv:hep-th/0510206

Fractal analysis has been shown to be useful in image processing for characterizing shape and gray-scale complexity. The fractal feature is a compact descriptor used to give a numerical measure of the degree of irregularity of the medical images. This descriptor property does not give ownership of the local image structure. In this paper, we present a combination of this parameter based on Box Counting with GLCM Features. This powerful combination has proved good results especially in classification of medical texture from MRI and CT Scan images of trabecular bone. This method has the potential to improve clinical diagnostics tests for osteoporosis pathologies., Comment: 13 pages, 6 figures, Signal & Image Processing : An International Journal(SIPIJ)Vol.5, No.4, August 2014

We show that for open gauge theories, it is possible to build an off-shell Becchi-Rouet-Stora-Tyutin (BRST) algebra together with an invariant extension of the classical action. This is based on the introduction of auxiliary fields, after having defined an on-shell invariant quantum action, where the gauge-fixing action is written as in Yang-Mills type theories up to a modified BRST operator. An application to simple supergravity is performed., Comment: 12 pages, no figures

The Autonomous System (AS)-level topology of the Internet that currently comprises 40k ASs, is growing at a rate of about 10% per year. In these conditions, Border Gateway Protocol (BGP), the inter-domain routing protocol of the Internet starts to show its limits, among others in terms of the number of routing table entries it can dynamically process and control. To overcome this challenging situation, the design but also the evaluation of alternative dynamic routing models and their comparison with BGP shall be performed by means of simulation. For this purpose, DRMSim, a Dynamic Routing Model Simulator, was developed that provides the means for large-scale simulations of various routing models including BGP. By means of this discrete-event simulator, execution of path-vector routing, e.g. BGP, and other compact routing models have been successfully performed on network topologies comprising more than ten thousand (abstract) nodes. However, to simulate dynamic routing schemes like BGP, DRMSim needs enhancements to support current Internet size (40k ASs) and even more by considering its evolution (up to 100k ASs). This paper proposes a feasibility study of the extension of DRMSim so as to support the Distributed Parallel Discrete Event paradigm. We first detail the possible distribution models and their associated communication overhead. Then, we analyze the communication overhead of such a distributed simulator by executing BGP on a partitioned topology according to different scenarios. Finally, we conclude on the feasibility of such a simulator by computing the expected additional time required by a distributed simulation of BGP compared to its sequential simulation.

The Autonomous System (AS) topology of the Internet (up to 61k ASs) is growing at a rate of about 10% per year. The Border Gateway Protocol (BGP) starts to show its limits in terms of the number of routing table entries it can dynamically process and control. Due to the increasing routing information processing and storage, the same trend is observed for routing model simulators such as DRMSim specialized in large-scale simulations of routing models. Therefore, DRMSim needs enhancements to support the current size of the Internet topology and its evolution (up to 100k ASs). To this end, this paper proposes a feasibility study of the extension of DRMSim so as to support the Distributed Parallel Discrete Event paradigm. We first detail the possible distribution models and their associated communication overhead. Then, we analyze this overhead by executing BGP on a partitioned topology according to different scenarios. Finally, we conclude on the feasibility of such a simulator by computing the expected additional time required by a distributed simulation of BGP compared to its sequential simulation., Comment: 26th ACM/IEEE/SCS Workshop on Principles of Advanced and Distributed Simulation (2012)

A modification of the usual extended N = 2 supersymmetry algebra implementing the two dimensional permutation group is performed. It is shown that one can found a multiplet that forms an off-shell realization of this alternative extension of standard supersymmetry.

The action for classical and consequently quantum N=1, D=4, supersymmetric Yang-Mills theories is proven to be written as a BRST exact term. In order to show this, the space of fields in the theory must be enlarged to include an extrafield as a non dynamical auxiliary field., Comment: 10 pages, Latex, no figure

We propose the superconnection formalism to construct the off-shell BRST-VSUSY superalgebra for D=4 BF theories. The method is based on the natural introduction of physical fields as well as auxiliary fields via superconnections and their associated supercurvatures defined on a superspace. We also give a prescription to build the off-shell BRST-VSUSY exact quantum action., Comment: Latex, 13 pages, no figures, comments and references added

We use a Becchi-Rouet-Stora-Tyutin (BRST) superspace approach to formulate off-shell nilpotent BRST and anti-BRST transformations in four dimensional N=1 supersymmetric Yang-Mills theory. The method is based on the possibility of introducing auxiliary fields through the supersymmetric transformations of the superpartener of the gauge potential associated to a supersymmetric Yang-Mills connection. These fields are required to achieve the off-shell nilpotency of the BRST and anti-BRST operators. We also show how this off-shell structure is used to build the BRST and anti-BRST invariant gauge-fixing quantum action., Comment: 14 pages; Latex; no figure

The quantization of gauge-affine gravity within the superfiber bundle formalism is proposed. By introducing an even pseudotensorial 1-superform over a principal superfibre bundle with superconnection, we obtain the geometrical Becchi-Rouet-Stora-Tyutin (BRST) and anti-BRST transformations of the fields occurring in such a theory. Reducing the four-dimensional general affine group double-covering to the Poincare group double-covering we also find the BRST and anti-BRST transformations of the fields present in Einstein's gravity. Furthermore, we give a prescription leading to the construction of both BRST-invariant gauge-fixing action for gauge-affine gravity and Einstein's gravity., Comment: Latex, 18 pages, no figures; minor changes made in Section 2, added references for conclusion; final published version

We discuss the structure of auxiliary fields for non-Abelian BF theories in arbitrary dimensions. By modifying the classical BRST operator, we build the on-shell invariant complete quantum action. Therefore, we introduce the auxiliary fields which close the BRST algebra and lead to the invariant extension of the classical action., Comment: 7 pages, minor changes, typos in equations corrected and acknowledgements added

We develop a method to derive the on-shell invariant quantum action of the supergravity in such a way that the quartic ghost interaction term is explicity determined. First, we reinvestigate the simple supergravity in terms of a principal superfibre bundle. This gives rise to the closed geometrical BRST algebra. Therefore we determine the open BRST algebra, which realizes the invariance of the classical action. Then, given a prescription to build the full quantum action, we obtain the quantum BRST algebra. Together with the constructed quantum action this allows us to recover the auxiliary fields and the invariant extension of the classical action., Comment: 17 pages, No figures

We present an alternative quantization for irreducible open gauge theories. The method relies on the possibility of modifying the classical BRST operator and the gauge-fixing action written as in Yang-Mills type theories, in order to obtain an on-shell invariant quantum action by using equations characterizing the full gauge algebra. From this follows then the construction of an off-shell version of the theory. We show how it is possible to build off-shell BRST algebra together with an invariant extension of the classical action. This is realized via a systematic prescription for the introduction of auxiliary fields., Comment: 24 pages, no figures, PACS added, minor changes

A model kinetic equation is solved exactly for a special stationary state describing nonlinear Couette flow in a low density system of inelastic spheres. The hydrodynamic fields, heat and momentum fluxes, and the phase space distribution function are determined explicitly. The results apply for conditions such that viscous heating dominates collisional cooling, including large gradients far from the reference homogeneous cooling state. Explicit expressions for the generalized transport coefficients (e.g., viscosity and thermal conductivity) are obtained as nonlinear functions of the coefficient of normal restitution and the shear rate.These exact results for the model kinetic equation are also shown to be good approximations to the corresponding state for the Boltzmann equation via comparison with direct Monte Carlo simulation for the latter, Comment: 22 pages (9 figures included). Minor changes with respect to v2. Fig. 1 is new. To be published in J. Stat. Phys. (June 2001 issue,Volume 103, 5/6)

By means of the Direct Simulation Monte Carlo method, the Boltzmann equation is numerically solved for a gas of hard spheres enclosed between two parallel plates kept at different temperatures and subject to the action of a gravity field normal to the plates. The profiles of pressure, density, temperature and heat flux are seen to be quite sensitive to the value of the gravity acceleration $g$. If the gravity field and the heat flux are parallel ($g>0$), the magnitudes of both the temperature gradient and the heat flux are smaller than in the opposite case ($g<0$). When considering the actual heat flux relative to the value predicted by the Fourier law, it is seen that, if $g>0$, the ratio increases as the reduced local field strength increases, while the opposite happens if $g<0$. The simulation results are compared with theoretical predictions for Maxwell molecules, Comment: 18 pages (LaTex), 7 figures (eps)