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arXiv 2020年

作者： Le, Huu Phuoc El Din, Mohab Safey Sorbonne Université Cnrs Laboratoire d'Informatique de Paris 6 LIP6 Équipe Polsys Paris Cedex 05F-75252 France

We design a new algorithm for solving parametric systems of equations having finitely many complex solutions for generic values of the parameters. More precisely, let f = (f1, ⋯, fm) ⊂ [y][x] with y = (y1, ⋯, yt) and x = (x1, ⋯, xn), V ⊂ ℂt× ℂnbe the algebraic set defined by the simultaneous vanishing of the fi's and π be the projection (y, x) → y. Under the assumptions that f admits finitely many complex solutions when specializing y to generic values and that the ideal generated by f is radical, we solve the following algorithmic problem. On input f, we compute semi-algebraic formulas defining open semi-algebraic sets S1, ⋯, Sin the parameters' space tsuch that ∪ i=1Siis dense in tand, for 1 ≤ i ≤ , the number of real points in V ∩ π-1(η) is invariant when η ranges over Si. This algorithm exploits special properties of some well chosen monomial bases in the quotient algebra (y)[x]/I where I ⊂ (y)[x] is the ideal generated by f in (y)[x] as well as the specialization property of the so-called Hermite matrices which represent Hermite's quadratic forms. This allows us to obtain "compact" representations of the semi-algebraic sets Siby means of semialgebraic formulas encoding the signature of a given symmetric matrix. When f satisfies extra genericity assumptions (such as regularity), we use the theory of Gröbner bases to derive complexity bounds both on the number of arithmetic operations in and the degree of the output polynomials. More precisely, letting d be the maximal degrees of the fi's and D = n(d - 1)dn, we prove that, on a generic input f = (f1, ⋯, fn), one can compute those semialgebraic formulas using Õ((t+D/t)23tn2t+1d3nt+2(n+t)+1) arithmetic operations in and that the polynomials involved in these formulas have degree bounded by D. We report on practical experiments which illustrate the efficiency of this algorithm, both on generic parametric systems and parametric systems coming from applications since it allows us to solve systems which were out of rea

关键词： polynomials

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THE RELATIVE MANIN–MUMFORD CONJECTURE

arXiv

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arXiv 2023年

作者： Gao, Ziyang Habegger, Philipp Institute of Algebra Number Theory and Discrete Mathematics Leibniz University Hannover Welfengarten 1 Hannover30167 Germany Department of Mathematics and Computer Science University of Basel Spiegelgasse 1 Basel4051 Switzerland

MSC Codes 11G30, 11G50, 14G05, 14G25We prove the Relative Manin–Mumford Conjecture for families of abelian varieties in characteristic 0. We follow the Pila–Zannier method to study special point problems, and we use the Betti map which goes back to work of Masser and Zannier in the case of curves. The key new ingredients compared to previous applications of this approach are a height inequality proved by both authors of the current paper and Dimitrov, and the first-named author’s study of certain degeneracy loci in subvarieties of abelian schemes. We also strengthen this result and prove a criterion for torsion points to be dense in a subvariety of an abelian scheme over C. The Uniform Manin–Mumford Conjecture for curves embedded in their Jacobians was first proved by Kühne. We give a new proof, as a corollary to our main theorem, that does not use equidistribution. © 2023, CC BY.

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A Survey on Time Series Data Classification: Blockchain Technologies and Security Concerns

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Procedia computer Science 2024年 246卷 961-970页

作者： Ahmed Faris Alsayyad Alaa Abid Muslam Abid Ali Mohamed Mabrouk Ahmed Al-Shammari Mounir Zrigui Research Laboratory in Algebra Numbers Theory and Intelligent Systems RLANTIS University of Monastir Monastir Tunisia Al-Qadisiyah University Al-Qadisiyah Iraq Military Academy Fondok Jedid Tunisia Department of Computer Science College of Computer Science and Information Technology University of Al-Qadisiyah Al Diwaniyah 58002 Iraq

The difficulties about user security and privacy have appeared as significant concerns in recent years. The number of cyber-attacks grows at a concerning velocity, hence rendering internet users susceptible to malicious activities perpetrated by hackers. Data mining approaches are employed to extract accurate results from massive and complex databases. Furthermore, the utilization of Blockchain (BC) approaches is increasingly popular in current Internet of Things (IoT) applications as an opportunity to address issues related to privacy and security. Lots of studies have been performed on algorithms for data mining and techniques concerning blockchain. Time series data is a commonly used form of data. Time Series Classification (TSC) refers to the creation of predictive models that generate a target variable or label based on linear or sequential data inputs across a considerable duration. The possible results may be presented in either ordinal or numerical form. Even so, previous studies have shown major limitations when it comes to handling privacy and security issues that can’t be applicable in dynamic instances, as well as the substantial computational cost necessary. Moreover, correctly determining the amount of sensitive parameters required to complete the classification process remains a challenge. We have put forth a comprehensive survey on the classification of blockchain data. In the first phase of our study, we conducted an analysis and categorization of both conventional data classification approaches and contemporary time series data classification techniques. We further discussed limitations and strengths of existing techniques. Finally, we highlight future research problems and directions.

关键词： Blockchain Data Mining Time Series Classification Data Classification Security

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Faster Randomized Interior Point Methods for Tall/Wide Linear Programs

arXiv

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arXiv 2022年

作者： Chowdhury, Agniva Dexter, Gregory London, Palma Avron, Haim Drineas, Petros Computer Science and Mathematics Division Oak Ridge National Laboratory TN United States Department of Computer Science Purdue University West LafayetteIN United States Operations Research and Information Engineering Cornell University IthacaNY United States School of Mathematical Sciences Tel Aviv University Tel Aviv Israel

Linear programming (LP) is an extremely useful tool which has been successfully applied to solve various problems in a wide range of areas, including operations research, engineering, economics, or even more abstract mathematical areas such as combinatorics. It is also used in many machine learning applications, such as 1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider both feasible and infeasible IPMs for the special case where the number of variables is much larger than the number of constraints. Using tools from Randomized Linear algebra, we present a preconditioning technique that, when combined with the iterative solvers such as Conjugate Gradient or Chebyshev Iteration, provably guarantees that IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real-world and synthetic data. © 2022, CC BY.

关键词： Operations research

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Quartic scaling second-order approximate coupled cluster singles and doubles via tensor hypercontraction: THC-CC2

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JOURNAL OF CHEMICAL PHYSICS 2013年第12期138卷 124111-124111页

作者： Hohenstein, Edward G. Kokkila, Sara I. L. Parrish, Robert M. Martinez, Todd J. Stanford Univ Dept Chem Stanford CA 94305 USA Stanford Univ PULSE Inst Stanford CA 94305 USA SLAC Natl Accelerator Lab Menlo Pk CA 94025 USA Georgia Inst Technol Sch Chem & Biochem Ctr Computat Mol Sci & Technol Atlanta GA 30332 USA

The second-order approximate coupled cluster singles and doubles method (CC2) is a valuable tool in electronic structure theory. Although the density fitting approximation has been successful in extending CC2 to larger molecules, it cannot address the steep O(N-5) scaling with the number of basis functions, N. Here, we introduce the tensor hypercontraction (THC) approximation to CC2 (THC-CC2), which reduces the scaling to O(N-4) and the storage requirements to O(N-2). We present an algorithm to efficiently evaluate the THC-CC2 correlation energy and demonstrate its quartic scaling. This implementation of THC-CC2 uses a grid-based least-squares THC (LS-THC) approximation to the density-fitted electron repulsion integrals. The accuracy of the CC2 correlation energy under these approximations is shown to be suitable for most practical applications. (C) 2013 American Institute of Physics. [http://***/10.1063/1.4795514]

关键词： ELECTRONIC structure LEAST squares APPROXIMATION theory TENSOR algebra ALGORITHMS ELECTRONS

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Dynamic Clustering Recommendation Algorithm For Two-Layer Graph Attention Network

Dynamic Clustering Recommendation Algorithm For Two-Layer Gr...

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IEEE International Symposium on Parallel and Distributed Processing with applications and IEEE International Conference on Ubiquitous Computing and Communications (ISPA/IUCC)

作者： Zhihui Wang Jianrui Chen Bo Wang School of Computer Science Shaanxi Normal University Xi’an China College of Science Inner Mongolia University of Technology Hohhot China

ISBN: (纸本)9781728152103

Collaborative filtering recommendation algorithm is one of the most widely used personalized recommendation algorithms in e-commerce websites. The traditional collaborative filtering recommendation algorithm has a high recommendation complexity and low accuracy with the increasing number of users and items in recent years. The previous differential clustering evolution process only recommended a single clustering results of users or items. Besides, the node state of network was only a scalar, which ignored the integration of user layer and item layer and could not better represent the attribute characteristics of users and items. This paper proposes an effective collaborative filtering recommendation algorithm for the above three problems. We fully explore the changes of interests of users and their attention to the items over time. Firstly, a time-weighted scoring matrix is constructed by combining the forgetting function. According to the new scoring matrix, the user-item attention matrix is obtained. Then, according to the differential equations, users and items with high relevance are gathered to obtain the user communities and item communities. Stabilizing the same user status values mean that they have similar interests and then they are assigned to the same community. Finally, the real time prediction results are obtained through improved prediction method and dynamic similarity measurement in each community. The effectiveness of the proposed algorithm is verified by comparison with several better algorithms.

关键词： collaborative filtering electronic commerce evolutionary computation graph theory matrix algebra pattern clustering recommender systems Web sites collaborative filtering electronic commerce Cluster Analysis websites Recommender systems matrix algebra evolutionary algorithm Graph theory Consumers algorithms

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Anomaly-free policy composition in software-defined networks

Anomaly-free policy composition in software-defined networks

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IFIP Networking Conference

作者： Mohsen Rezvani Aleksandar Ignjatovic Maurice Pagnucco Sanjay Jha School of Computer Science and Engineering UNSW Australia

Software Defined Networking (SDN) provides considerable simplification of design and deployment of various network applications for large networks. Each application has its own view of network policy and sends its policy to a network hypervisor in which a composed policy is generated from the application policies and deployed into the data plane. A significant challenge for the hypervisor is to detect and resolve both intra and inter policy anomalies during the policy composition. However, current SDN compilers do not consider the policy anomalies well and generate large number of unnecessary rules for the data plane. This leads to a considerable inefficiency in both policy composition and policy deployment. In this paper, we propose a novel framework for policy composition in a SDN hypervisor which takes into account both inter and intra policy anomalies. Moreover, we augment the framework with an efficient insertion transformation mechanism which allows the applications to issue rule insertion and priority change updates. Our evaluation shows that our method is several orders of magnitude more efficient than the state of the art in both policy composition and compiling the rule insertion updates.

关键词： Virtual machine monitors Electronics packaging Protocols Complexity theory algebra Data structures

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Faster randomized infeasible interior point methods for tall/wide linear programs 20

Faster randomized infeasible interior point methods for tall...

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Proceedings of the 34th International Conference on Neural Information Processing Systems

作者： Agniva Chowdhury Palma London Haim Avron Petros Drineas Department of Statistics Purdue University West Lafayette IN ORIE Department Cornell University Ithaca NY School of Mathematical Sciences Tel Aviv University Tel Aviv Israel Department of Computer Science Purdue University West Lafayette IN

ISBN: (纸本)9781713829546

Linear programming (LP) is used in many machine learning applications, such as ℓ1-regularized SVMs, basis pursuit, nonnegative matrix factorization, etc. Interior Point Methods (IPMs) are one of the most popular methods to solve LPs both in theory and in practice. Their underlying complexity is dominated by the cost of solving a system of linear equations at each iteration. In this paper, we consider infeasible IPMs for the special case where the number of variables is much larger than the number of constraints (i.e., wide), or vice-versa (i.e., tall) by taking the dual. Using tools from Randomized Linear algebra, we present a preconditioning technique that, when combined with the Conjugate Gradient iterative solver, prov-ably guarantees that infeasible IPM algorithms (suitably modified to account for the error incurred by the approximate solver), converge to a feasible, approximately optimal solution, without increasing their iteration complexity. Our empirical evaluations verify our theoretical results on both real and synthetic data.

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Towards new multiwavelets: Associated filters and algorithms. Part I: Theoretical framework and investigation of biomedical signals, ECG and coronavirus cases

arXiv

引用

arXiv 2021年

作者： Jallouli, Malika Zemni, Makerem Mabrouk, Anouar Ben Mahjoub, Mohamed Ali Université de Sousse Ecole Nationale d’Ingénieurs de Sousse LATIS Laboratory of Advanced Technology and Intelligent Systems Sousse4023 Tunisia Department of Mathematics Higher Institute of Applied Mathematics and Computer Science Street of Assad Ibn Alfourat Kairouan3100 Tunisia Laboratory of Algebra Number Theory and Nonlinear Analysis LR18ES15 Department of Mathematics Faculty of Sciences Monastir5000 Tunisia Department of Mathematics Faculty of Sciences University of Tabuk Saudi Arabia

Biosignals are nowadays important subjects for scientific researches from both theory and applications especially with the appearance of new pandemics threatening the humanity such as the new Coronavirus. One aim in the present work is to prove that Wavelets may be a successful machinery to understand such phenomena by applying a step forward extension of wavelets to multiwavelets. We proposed in a first step to improve multiwavelet notion by constructing more general families using independent components for multi-scaling and multiwavelet mother functions. A special multiwavelet is then introduced, continuous and discrete multiwavelet transforms are associated, as well as new filters and algorithms of decomposition and reconstruction. The constructed multiwavelet framework is applied for some experimentations showing fast algorithms, ECG signal and a strain of Coronavirus *** Codes 42C40, 65T60, 94A08 © 2021, CC0.

关键词： Coronavirus

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EXTENSION OF ZORN'S LEMMA TO ARBITRARY BINARY RELATIONS

arXiv

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arXiv 2022年

作者： Andrikopoulos, Athanasios Dept. of Computer Engineering and Informatics University of Patras Patras26504 Greece

In 1935, Max Zorn introduced Zorn's lemma with the intention of shortening proofs in algebra that had previously used the Axiom of Choice or the Well-Ordering theorem. However, there are several applications in optimization, economics, decision analysis, and game theory where a binary relation is not transitive as required by Zorn's lemma. For example, a choice process in social choice theory is formally modeled as a maximization problem. More specifically, a choice process is presented by a choice function that associates with each feasible set of alternatives a certain subset of it, which consists of the maximal elements according to the viewpoint of a binary relation. However, the set of maximal elements is often empty. In this case, the crucial question which arises is what to count as a choice. That is, what sets of alternatives may be considered as reasonable solutions? To answer this question, a number of theories, called general solution theories, have been proposed to take over the role of maximality in the absence of maximal elements. Zorn's lemma cannot be applied to solve these problems because binary relations are not transitive. In this note, the Zorn lemma is extended to arbitrary binary relations and thus the Zorn lemma can do for optimization when the transitivity is broken. Zorn's extended lemma can be used to prove existence theorems of generalized solution concepts for binary relations that do not have maximal elements. Copyright © 2022, The Authors. All rights reserved.

关键词： Game theory

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