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A generalized grover/zeta correspondence

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arXiv 2022年

We introduce a generalized Grover matrix of a graph and present an explicit formula for its characteristic polynomial. As a corollary, we give the spectra for the generalized Grover matrix of a regular graph. Next, we define a zeta function and a generalized zeta function of a graph G with respect to its generalized Grover matrix as an analog of the Ihara zeta function and present explicit formulas for their zeta functions for a vertex-transitive graph. As applications, we express the limit on the generalized zeta functions of a family of finite vertex-transitive regular graphs by an integral. Furthermore, we give the limit on the generalized zeta functions of a family of finite tori as an integral *** Codes 60F05, 05C50, 15A15, 05C25 © 2022, CC0.

关键词： Functions

Graph Fourier MMD for Signals on Graphs

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arXiv 2023年

作者： Leone, Samuel Venkat, Aarthi Huguet, Guillaume Tong, Alexander Wolf, Guy Krishnaswamy, Smita Applied Math Program New HavenCT United States Computational Biology & Bioinformatics Program Yale University New HavenCT United States Dept. of Math. & Stat Université de Montréal Mila - Quebec AI Institute MontrealQC Canada Dept. of Comp. Sci. & Oper. Res. Université de Montréal Mila - Quebec AI Institute MontrealQC Canada Department of Computer Science Department of Genetics Yale University New HavenCT United States

While numerous methods have been proposed for computing distances between probability distributions in Euclidean space, relatively little attention has been given to computing such distances for distributions on graphs. However, there has been a marked increase in data that either lies on graph (such as protein interaction networks) or can be modeled as a graph (single cell data), particularly in the biomedical sciences. Thus, it becomes important to find ways to compare signals defined on such graphs. Here, we propose Graph Fourier MMD (GFMMD), a novel distance between distributions and signals on graphs. GFMMD is defined via an optimal witness function that is both smooth on the graph and maximizes difference in expectation between the pair of distributions on the graph. We find an analytical solution to this optimization problem as well as an embedding of distributions that results from this method. We also prove several properties of this method including scale invariance and applicability to disconnected graphs. We showcase it on graph benchmark datasets as well on single cell RNA-sequencing data analysis. In the latter, we use the GFMMD-based gene embeddings to find meaningful gene clusters. We also propose a novel type of score for gene selection called gene localization score which helps select genes for cellular state space characterization. Copyright © 2023, The Authors. All rights reserved.

关键词： Genes

Intelligent model for measuring learners' satisfaction towards new contents 15

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Intelligent model for measuring learners' satisfaction towar...

15th International Conference on Information Technology Based Higher Education and Training, ITHET 2016

作者： Elghibari, Fatiha Elouahbi, Rachid El Khoukhi, Fatima Ezzahri, Layla Macs, Labo Amali, Said Department of Mathematics and Computer Science Moulay Ismail University Faculty of Sciences Meknes Morocco Team Modeling Applied Informatics in Humanities MIASH Moulay Ismail University Faculty of Humanities Meknes Morocco Team Théorie des Systèmes et Informatique TSI Moulay Ismail University Faculty of Sciences Meknes Morocco Team Outils Mathématiques en Economie Gestion et Actuariat OMEGA Moulay Ismail University Faculty of Juridical Economic and Social Sciences Meknes Morocco

ISBN: (纸本)9781509007783

Learner satisfaction is an important factor for a successful e-Learning. It allows learners to continue learning and provide their conveniences to facilitate the knowledge. That is why, the main purpose of this study is to enhance learning services in online environments, in order to keep learners' motivation and track their cognition' evolution. Namely, identify their satisfaction toward new inserted contents. In this paper, we present a learning model that detects satisfaction' degree of learners at two levels;during and after completing the course. During learning by using satisfaction agent (SA) that collects data and calculates satisfaction level towards each content' unit. This process allows to integrate improvement if we have a low levels of satisfaction. And after completing course in order to evaluate the evolution of learners' knowledge, and test the effectiveness of the methods and tools used in. © 2016 IEEE.

关键词： Motivation

Tractability of multivariate problems for standard and linear information in the worst case setting: Part II

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arXiv 2017年

作者： Novak, Erich Wozniakowski, Henryk Jena University Math Institute Ernst Abbe Platz 2 Jena Germany Department of Computer Science Columbia University New YorkNY10027 United States Institute of Applied Mathematics University of Warsaw ul. Banacha 2 Warszawa02-097 Poland

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: Copyright © 2017, The Authors. All rights reserved.

关键词： Vector spaces

Advances on Links Between math.matics and Industry 1

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丛书名： SxI - Springer for Innovation / SxI - Springer per l'Innovazione

1000年

作者： Peregrina Quintela Estévez Bartomeu Coll Rosa M. Crujeiras José Durany Laureano Escudero

Global Software Engineering Experience through International Capstone Project Exchanges

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Global Software Engineering Experience through International...

International Conference on Global Software Engineering (ICGSE)

作者： Dean Knudson Stavros Kalafatis Carsten Kleiner Stephen Zahos Barbara Seegebarth Jonas Deeterfelt Iman Avazpour Kristian Sandahl Peter Gorder Jeewani Ginige Alex Radermacher Hugo Caballero Humberto Gomez Mikael Roos Computer Science Dept. North Dakota State University Fargo ND USA ECE Department Texas A& M University College Station TX USA Computer Science Dept. Hannover University of Applied Sciences & Arts Hannover Germany Agricultural and Biological Eng. University of Illinois at Urbana-Champaign Urbana IL USA Institute of Marketing Technische Universitat Braunschweig Braunschweig Germany Linkopings universitet Linkoping SE School of IT Deakin University Burwood Australia Department of Computer and Information Science Linkoping University Linkoping SE Sweden Department of Mechanical and Aerospace Engineering Univ. of Colorado Colorado Springs CO USA Sch. of Comput. Eng. & Math. Western Sydney Univ. Sydney NSW Australia Departamento de Ingenieria Mecanica Universidad del Norte Barranquilla Colombia Department of Computer Science and Engineering Blekinge Institute of Technology (BTH) Karlskrona Sweden

Today it is very common for software systems to be built by teams located in more than one country. For example, a project team may be located in the US while the team lead resides in Sweden. How then should students be trained for this kind of work? Senior design or capstone projects offer students real-world hands-on experience but rarely while working internationally. One reason is that most instructors do not have international business contacts that allow them to find project sponsors in other countries. Another reason is the fear of having to invest a huge amount of time managing an international project. In this paper we present the general concepts related to "International Capstone Project Exchanges", the basic model behind the exchanges (student teams are led by an industry sponsor residing in a different country) and several alternate models that have been used in practice. We will give examples from projects in the US, Germany, Sweden, Australia, and Colombia. We have extended the model beyond software projects to include engineering projects as well as marketing, and journalism. We conclude with a description of an International Capstone Project Exchange website that we have developed to aid any university in establishing their own international project exchange.

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Topological Entropy and Sequence Entropy for Hom Tree-Shifts on Unexpandable Trees

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Qualitative Theory of Dynamical Systems 2024年第3期23卷 108-108页

作者： Ban, Jung-Chao Chang, Chih-Hung Hu, Wen-Guei Lai, Guan-Yu Wu, Yu-Liang Department of Mathematical Sciences National Chengchi University Taipei 11605 Taiwan Math. Division National Center for Theoretical Science National Taiwan University Taipei 10617 Taiwan Department of Applied Mathematics National University of Kaohsiung Kaohsiung 81148 Taiwan College of Mathematics Sichuan University Chengdu 610064 China Department of Mathematical Sciences University of Oulu Oulu 90014 Finland

This article explores the topological entropy and topological sequence entropy of hom tree-shifts on unexpandable trees. Regarding topological entropy, we establish that the entropy, denoted as h(T_X) on an unexpandable tree, equals the entropy h(X) of the base shift X when X is a subshift satisfying the almost specification property. Additionally, we derive some fundamental properties such as entropy approximation and the denseness of entropy for subsystems. Concerning topological sequence entropy, we show that the set of sequence entropies of hom tree-shifts with a base shift is generated by an irreducible matrix A, forming a subset of logN. Precisely, these entropies correspond to the logarithms of the largest cardinalities of the periodic classes of A. © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024.

关键词： Entropy Sequence entropy Tree-shift

Neural FIM for learning Fisher information metrics from point cloud data

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arXiv 2023年

作者： Fasina, Oluwadamilola Huguet, Guillaume Tong, Alexander Zhang, Yanlei Wolf, Guy Nickel, Maximilian Adelstein, Ian Krishnaswamy, Smita Applied Math Program Yale University New HavenCT United States Mila - Quebec AI Institute MontrealQC Canada Department of Mathematics and Statistics Université de Montréal MontrealQC Canada Department of Computer Science and Operations Research Université de Montréal MontrealQC Canada FAIR Meta AI United States Department of Math Yale University New HavenCT United States Department of Computer Science Yale University New HavenCT United States Department of Genetics Yale University New HavenCT United States

Although data diffusion embeddings are ubiquitous in unsupervised learning and have proven to be a viable technique for uncovering the underlying intrinsic geometry of data, diffusion embeddings are inherently limited due to their discrete nature. To this end, we propose neural FIM, a method for computing the Fisher information metric (FIM) from point cloud data - allowing for a continuous manifold model for the data. Neural FIM creates an extensible metric space from discrete point cloud data such that information from the metric can inform us of manifold characteristics such as volume and geodesics. We demonstrate Neural FIM's utility in selecting parameters for the PHATE visualization method as well as its ability to obtain information pertaining to local volume illuminating branching points and cluster centers embeddings of a toy dataset and two single-cell datasets of IPSC reprogramming and PBMCs (immune cells). Copyright © 2023, The Authors. All rights reserved.

关键词： Embeddings

Schur multipliers and mixed unitary maps

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arXiv 2018年

作者： Harris, Samuel J. Levene, Rupert H. Paulsen, Vern I. Plosker, Sarah Rahaman, Mizanur Institute for Quantum Computing Dept. of Pure Math. University of Waterloo WaterlooONN2L 3G1 Canada School of Mathematics and Statistics University College Dublin Belfield Dublin 4 Ireland Department of Mathematics and Computer Science Brandon University BrandonMBR7A 6A9 Canada

We consider the tensor product of the completely depolar- ising channel on d×d matrices with the map of Schur multiplication by a k ×k correlation matrix and characterise, via matrix theory methods, when such a map is a mixed (random) unitary channel. When d = 1, this recovers a result of O'Meara and Pereira, and for larger d is equiva- lent to a result of Haagerup and Musat that was originally obtained via the theory of factorisation through von Neumann algebras. We obtain a bound on the distance between a given correlation matrix for which this tensor product is nearly mixed unitary and a correlation matrix for which such a map is exactly mixed unitary. This bound allows us to give an elementary proof of another result of Haagerup and Musat about the closure of such correlation matrices without appealing to the theory of von Neumann algebras. Copyright © 2018, The Authors. All rights reserved.

关键词： Tensors