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2021

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Articles 121 - 150 of 1314

Full-Text Articles in Mathematics

Why Rectified Linear Neurons: A Possible Interval-Based Explanation, Jonathan Contreras, Martine Ceberio, Vladik Kreinovich Nov 2021

Why Rectified Linear Neurons: A Possible Interval-Based Explanation, Jonathan Contreras, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

At present, the most efficient machine learning techniques are deep neural networks. In these networks, a signal repeatedly undergoes two types of transformations: linear combination of inputs, and a non-linear transformation of each value v -> s(v). Empirically, the function s(v) = max(v,0) -- known as the rectified linear function -- works the best. There are some partial explanations for this empirical success; however, none of these explanations is fully convincing. In this paper, we analyze this why-question from the viewpoint of uncertainty propagation. We show that reasonable uncertainty-related arguments lead to another possible explanation of why rectified linear functions …

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How Probabilistic Methods For Data Fitting Deal With Interval Uncertainty: A More Realistic Analysis, Vladik Kreinovich, Sergey P. Shary Nov 2021

How Probabilistic Methods For Data Fitting Deal With Interval Uncertainty: A More Realistic Analysis, Vladik Kreinovich, Sergey P. Shary

Departmental Technical Reports (CS)

In our previous paper, we showed that a simplified probabilistic approach to interval uncertainty leads to the known notion of a united solution set. In this paper, we show that a more realistic probabilistic analysis of data fitting under interval uncertainty leads to another known notion -- the notion of a tolerable solution set. Thus, the notion of a tolerance solution set also has a clear probabilistic interpretation. Good news is that, in contrast to the united solution set whose computation is, in general, NP-hard, the tolerable solution set can be computed by a feasible algorithm.

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Homflypt Skein Theory, String Topology And 2-Categories, Uwe Kaiser Nov 2021

Homflypt Skein Theory, String Topology And 2-Categories, Uwe Kaiser

Mathematics Faculty Publications and Presentations

We show that relations in Homflypt type skein theory of an oriented 3-manifold M are induced from a 2-groupoid defined from the fundamental 2-groupoid of a space of singular links M. The module relations are defined by homomorphisms related to string topology. They appear from a representation of the groupoid into free modules on a set of model objects. The construction on the fundamental 2-groupoid is defined by the singularity stratification and relates Vassiliev and skein theory. Several explicit properties are discussed, and some implications for skein modules are derived.

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On The Relationship Between Pain Variability And Relief In Randomized Clinical Trials, Siddharth Tiwari '22 Nov 2021

On The Relationship Between Pain Variability And Relief In Randomized Clinical Trials, Siddharth Tiwari '22

Student Publications & Research

Previous research suggests greater baseline variability is associated with greater pain relief in those who receive a placebo. However, studies that evidence this association do not control for confounding effects (natural history and regression-to-the-mean); for this reason, we analyzed data from two randomized clinical trials (Placebo I and Placebo II, N = 134) while adjusting for confounding effects via a no-treatment group. Results agree between the two placebo groups: both placebo groups showed a negligible correlation between baseline variability and adjusted response (r sp (CI 95% ) = 0.13 (−0.09, 0.37) and 0.01 (−0.15, 0.20) for Placebo I and II, …

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System Identification Through Lipschitz Regularized Deep Neural Networks, Elisa Negrini, Giovanna Citti, Luca Capogna Nov 2021

System Identification Through Lipschitz Regularized Deep Neural Networks, Elisa Negrini, Giovanna Citti, Luca Capogna

Mathematics Sciences: Faculty Publications

In this paper we use neural networks to learn governing equations from data. Specifically we reconstruct the right-hand side of a system of ODEs x˙(t)=f(t,x(t)) directly from observed uniformly time-sampled data using a neural network. In contrast with other neural network-based approaches to this problem, we add a Lipschitz regularization term to our loss function. In the synthetic examples we observed empirically that this regularization results in a smoother approximating function and better generalization properties when compared with non-regularized models, both on trajectory and non-trajectory data, especially in presence of noise. In contrast with sparse regression approaches, since neural networks …

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High Precision Numerical Computation Of Principal Points For Univariate Distributions, Santanu Chakraborty, Mrinal Kanti Roychowdhury, Josef A. Sifuentes Nov 2021

High Precision Numerical Computation Of Principal Points For Univariate Distributions, Santanu Chakraborty, Mrinal Kanti Roychowdhury, Josef A. Sifuentes

School of Mathematical & Statistical Sciences Faculty Publications

Principal points were first introduced in 1990: for a positive integer n, n principal points of a random variable are the n points that minimize the mean squared distance between the random variable and the nearest of the n points. In this paper, we give a high precision numerical method for calculating the n principal points and the n th quantization errors for all positive integers n. For some absolutely continuous univariate distributions, we calculate the n principal points for different n using Newton’s method. Additionally, we also provide the corresponding values of mean squared distances.

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Exploratory Data Mining Techniques (Decision Tree Models) For Examining The Impact Of Internet-Based Cognitive Behavioral Therapy For Tinnitus: Machine Learning Approach, Hansapani Rodrigo, Eldré W. Beukes, Gerhard Andersson, Vinaya Manchaiah Nov 2021

Exploratory Data Mining Techniques (Decision Tree Models) For Examining The Impact Of Internet-Based Cognitive Behavioral Therapy For Tinnitus: Machine Learning Approach, Hansapani Rodrigo, Eldré W. Beukes, Gerhard Andersson, Vinaya Manchaiah

School of Mathematical & Statistical Sciences Faculty Publications

Background: There is huge variability in the way that individuals with tinnitus respond to interventions. These experiential variations, together with a range of associated etiologies, contribute to tinnitus being a highly heterogeneous condition. Despite this heterogeneity, a “one size fits all” approach is taken when making management recommendations. Although there are various management approaches, not all are equally effective. Psychological approaches such as cognitive behavioral therapy have the most evidence base. Managing tinnitus is challenging due to the significant variations in tinnitus experiences and treatment successes. Tailored interventions based on individual tinnitus profiles may improve outcomes. Predictive models of treatment …

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Fundamental Solutions For The Dirac Equation In Curved Spacetime And Generalized Euler-Poisson-Darboux Equation, Karen Yagdjian, Anahit Galstian Nov 2021

Fundamental Solutions For The Dirac Equation In Curved Spacetime And Generalized Euler-Poisson-Darboux Equation, Karen Yagdjian, Anahit Galstian

School of Mathematical & Statistical Sciences Faculty Publications

We present the fundamental solutions for the spin-1/2 fields propagating in spacetimes with power type expansion/contraction and the fundamental solution of the Cauchy problem for the Dirac equation. The derivation of these fundamental solutions is based on formulas for the solutions to the generalized Euler-Poisson-Darboux equation, which are obtained by the integral transform approach.

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Formal Oscillatory Distributions, Alexander Karabegov Oct 2021

Formal Oscillatory Distributions, Alexander Karabegov

Mathematics

The formal asymptotic expansion of an oscillatory in- tegral whose phase function has one nondegenerate critical point is a formal distribution supported at the critical point which is applied to the amplitude. This formal distribution is called a formal oscillatory integral (FOI). We introduce the notion of a formal oscillatory distribution supported at a point. We prove that a formal distribution is given by some FOI if and only if it is an oscillatory distribution that has a certain nondegeneracy property. We also prove that a star product ? on a Poisson manifold M is natural in the sense of …

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Effects Of Refuge Prey On Stability Of The Prey-Predator Model Subject To Immigrants: A Mathematical Modelling Approach, Mussa Amos Stephano, Il Hyo Jung Oct 2021

Effects Of Refuge Prey On Stability Of The Prey-Predator Model Subject To Immigrants: A Mathematical Modelling Approach, Mussa Amos Stephano, Il Hyo Jung

Tanzania Journal of Science

Prey-predator system is enormously complex and nonlinear interaction between species. Such complexity regularly requires development of new approaches which involves more factors in analysis of its population dynamics. In this paper, we formulate a modified Lotka-Volterra model that incorporates factors such as refuge prey and immigrants. We investigate the effects of refuge prey and immigrants by varying the refuge factor, with and without immigrants. The results show that with Holling’s type I functional response, the proposed model is asymptotically convergent when a refuge prey factor is introduced. Moreover, with Holling’s type II functional response, the proposed mathematical model is unstable …

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Inversion–Free Iterative Method For Finding Symmetric Solution Of The Nonlinear Matrix Equation X-A^* X^Q A=I (Q≥2), Chacha Stephen Chacha Oct 2021

Inversion–Free Iterative Method For Finding Symmetric Solution Of The Nonlinear Matrix Equation X-A^* X^Q A=I (Q≥2), Chacha Stephen Chacha

Tanzania Journal of Science

In this paper, we propose the inversion free iterative method to find symmetric solution of the nonlinear matrix equation  where  is an unknown symmetric solution,  is a given Hermitian matrix and  is a positive integer. The convergence of the proposed method is derived. Numerical examples demonstrate that the proposed iterative method is quite efficient and converges well when the initial guess is sufficiently close to the approximate solution. Keywords: Symmetric solution, nonlinear matrix equation, inversion free, iterative method.

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A Mathematical Analysis Of An In-Vivo Ebola Virus Transmission Dynamics Model, Seleman Ismail, Adeline Peter Mtunya Oct 2021

A Mathematical Analysis Of An In-Vivo Ebola Virus Transmission Dynamics Model, Seleman Ismail, Adeline Peter Mtunya

Tanzania Journal of Science

Ebola virus (EBOV) infection is a hemorrhagic and hazardous disease, which is among the most shocking threats to human health causing a large number of deaths. Currently, there are no approved curative therapies for the disease. In this study, a mathematical model for in-vivo Ebola virus transmission dynamics was analyzed. The analysis of the model mainly focused on the sensitivity of basic reproductive number,  pertaining to the model parameters. Particularly, the sensitivity indices of all parameters of  were computed by using the forward normalized sensitivity index method. The results showed that a slight change in the infection rate immensely influences …

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Participatory Action Research: Undergraduate Researchers Engaging Secondary Students In Social Justice Mathematics, Isabelle Miller, Alexis Grimes, Camryn Adkison Oct 2021

Participatory Action Research: Undergraduate Researchers Engaging Secondary Students In Social Justice Mathematics, Isabelle Miller, Alexis Grimes, Camryn Adkison

The Journal of Purdue Undergraduate Research

No abstract provided.

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A Clark-Ocone Type Formula Via Itô Calculus And Its Application To Finance, Takuji Arai, Ryoichi Suzuki Oct 2021

A Clark-Ocone Type Formula Via Itô Calculus And Its Application To Finance, Takuji Arai, Ryoichi Suzuki

Journal of Stochastic Analysis

No abstract provided.

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The Phase Transition Of Discrepancy In Random Hypergraphs, Calum Macrury, Tomáš Masarík, Leilani Pai, Xavier Perez Gimenez Oct 2021

The Phase Transition Of Discrepancy In Random Hypergraphs, Calum Macrury, Tomáš Masarík, Leilani Pai, Xavier Perez Gimenez

Department of Mathematics: Faculty Publications

Motivated by the Beck-Fiala conjecture, we study the discrepancy problem in two related models of random hypergraphs on n vertices and m edges. In the first (edge-independent) model, a random hypergraph H1 is constructed by fixing a parameter p and allowing each of the n vertices to join each of the m edges independently with probability p. In the parameter range in which pn ⟶ ∞ and pm ⟶ ∞, we show that with high probability (w.h.p.) H1 has discrepancy at least Ω(2-n/m √pn) when m = O(n …

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Three Essays On The Political Economics Of Conflict., Dripto Bakshi Dr. Oct 2021

Three Essays On The Political Economics Of Conflict., Dripto Bakshi Dr.

Doctoral Theses

A standard practice in the sociological and comparative-politics literature is to interpret ethnicity broadly as identity cleavages deriving from non-class categories such as language, religion, race and caste. In recent decades, conflicts between ‘ethnic’ communities, i.e., groups divided along non-class identity dimensions such as race, language, and, in particular, religion, have attained increasing salience in many countries. In response to this, a large analytical and empirical literature has developed both in political science and in political economics that seeks to explicate various aspects of this phenomenon. This thesis aims to throw light on two major aspects of ethnic conflicts. (1) …

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Some Contributions To Free Probability And Random Matrices., Sukrit Chakraborty Dr. Oct 2021

Some Contributions To Free Probability And Random Matrices., Sukrit Chakraborty Dr.

Doctoral Theses

No abstract provided.

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Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto Oct 2021

Unary-Determined Distributive ℓ -Magmas And Bunched Implication Algebras, Natanael Alpay, Peter Jipsen, Melissa Sugimoto

Mathematics, Physics, and Computer Science Faculty Articles and Research

A distributive lattice-ordered magma (dℓ-magma) (A,∧,∨,⋅) is a distributive lattice with a binary operation ⋅ that preserves joins in both arguments, and when ⋅ is associative then (A,∨,⋅) is an idempotent semiring. A dℓ-magma with a top ⊤ is unary-determined if x⋅y=(x⋅⊤∧y)∨(x∧⊤⋅y). These algebras are term-equivalent to a subvariety of distributive lattices with ⊤ and two join-preserving unary operations p, q. We obtain simple conditions on p, q such that x⋅y=(px∧y)∨(x∧qy) is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the …

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Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen Oct 2021

Algorithmic Correspondence For Relevance Logics, Bunched Implication Logics, And Relation Algebras Via An Implementation Of The Algorithm Pearl, Willem Conradie, Valentin Goranko, Peter Jipsen

Mathematics, Physics, and Computer Science Faculty Articles and Research

The non-deterministic algorithmic procedure PEARL (acronym for ‘Propositional variables Elimination Algorithm for Relevance Logic’) has been recently developed for computing first-order equivalents of formulas of the language of relevance logics LR in terms of the standard Routley-Meyer relational semantics. It succeeds on a large class of axioms of relevance logics, including all so called inductive formulas. In the present work we re-interpret PEARL from an algebraic perspective, with its rewrite rules seen as manipulating quasi-inequalities interpreted over Urquhart’s relevant algebras, and report on its recent Python implementation. We also show that all formulae on which PEARL succeeds are canonical, i.e., …

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Calculating Infinitesimal Generators, Majnu John, Yihren Wu Oct 2021

Calculating Infinitesimal Generators, Majnu John, Yihren Wu

Journal of Stochastic Analysis

No abstract provided.

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Isometric Flows Of G 2-Structures, Sergey Grigorian Oct 2021

Isometric Flows Of G 2-Structures, Sergey Grigorian

School of Mathematical & Statistical Sciences Faculty Publications

We survey recent progress in the study of flows of isometric G 2-structures on seven-dimensional manifolds, that is, flows that preserve the metric, while modifying the G 2-structure. In particular, heat flows of isometric G 2-structures have been recently studied from several different perspectives, in particular in terms of 3-forms, octonions, vector fields, and geometric structures. We will give an overview of each approach, the results obtained, and compare the different perspectives.

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Recursive And Viterbi Estimation For Semi-Markov Chains, Robert J. Elliott, W. P. Malcolm Oct 2021

Recursive And Viterbi Estimation For Semi-Markov Chains, Robert J. Elliott, W. P. Malcolm

Journal of Stochastic Analysis

No abstract provided.

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Mitigating The Externality Of Diseases Of Poverty Through Health Aid, Kamal Jnawali, Michael G. Tyshenko, Tamer Oraby Oct 2021

Mitigating The Externality Of Diseases Of Poverty Through Health Aid, Kamal Jnawali, Michael G. Tyshenko, Tamer Oraby

School of Mathematical & Statistical Sciences Faculty Publications

Externality exists in healthcare when an individual benefits from others being healthy as it reduces the probability of getting sick from illness. Healthy workers are considered to be the more productive labourers leading to a country’s positive economic growth over time. Several research studies have modelled disease transmission and its economic impact on a single country in isolation. We developed a two-country diseaseeconomy model that explores disease transmission and crossborder infection of disease for its impacts. The model includes aspects of a worsening and rapid transmission of disease juxtaposed by positive impacts to the economy from tourism. We found that …

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Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil Oct 2021

Using Lie Sphere Geometry To Study Dupin Hypersurfaces In R^N, Thomas E. Cecil

Mathematics and Computer Science Department Faculty Scholarship

A hypersurface M in Rn or Sn is said to be Dupin if along each curvature surface, the corresponding principal curvature is constant. A Dupin hypersurface is said to be proper Dupin if each principal curvature has constant multiplicity on M, i.e., the number of distinct principal curvatures is constant on M. The notions of Dupin and proper Dupin hypersurfaces in Rn or Sn can be generalized to the setting of Lie sphere geometry, and these properties are easily seen to be invariant under Lie sphere transformations. This makes Lie sphere geometry an effective …

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The N-Dimensional Quadratic Heisenberg Algebra As A “Non–Commutative” Sl(2,C), Luigi Accardi, Andreas Boukas, Yun-Gang Lu Oct 2021

The N-Dimensional Quadratic Heisenberg Algebra As A “Non–Commutative” Sl(2,C), Luigi Accardi, Andreas Boukas, Yun-Gang Lu

Journal of Stochastic Analysis

No abstract provided.

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The Chromatic Quasisymmetric Class Function Of A Digraph, Jacob A. White Oct 2021

The Chromatic Quasisymmetric Class Function Of A Digraph, Jacob A. White

School of Mathematical & Statistical Sciences Faculty Publications

We introduce a quasisymmetric class function associated with a group acting on a double poset or on a directed graph. The latter is a generalization of the chromatic quasisymmetric function of a digraph introduced by Ellzey, while the former is a generalization of a quasisymmetric function introduced by Grinberg. We prove representation-theoretic analogues of classical and recent results, including F-positivity, and combinatorial reciprocity theorems. We deduce results for orbital quasisymmetric functions, and study a generalization of the notion of strongly flawless sequences.

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Rough Paths And Regularization, André O. Gomes, Alberto Ohashi, Francesco Russo, Alan Teixeira Oct 2021

Rough Paths And Regularization, André O. Gomes, Alberto Ohashi, Francesco Russo, Alan Teixeira

Journal of Stochastic Analysis

No abstract provided.

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Three Questions From Cctm Teachers About Mathematical Modeling, Robyn Stankiewicz-Van Der Zanden, Rachel Levy Oct 2021

Three Questions From Cctm Teachers About Mathematical Modeling, Robyn Stankiewicz-Van Der Zanden, Rachel Levy

Colorado Mathematics Teacher

This article shares three questions and answers about mathematical modeling in the classroom from an April 2020 online conversation with participants of a CCTM webinar. We hope that the answers to these questions will motivate teachers to embrace the value of implementing math modeling tasks, help students see the math all around them in the world, and empower future professionals to reach for the mathematical tools in their pockets to make data-driven decisions.

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Why Neural Networks In The First Place: A Theoretical Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich Oct 2021

Why Neural Networks In The First Place: A Theoretical Explanation, Jonatan Contreras, Martine Ceberio, Olga Kosheleva, Vladik Kreinovich

Departmental Technical Reports (CS)

Neural networks -- specifically, deep neural networks -- are, at present, the most effective machine learning techniques. There are reasonable explanations of why deep neural networks work better than traditional "shallow" ones, but the question remains: why neural networks in the first place? why not networks consisting of non-linear functions from some other family of functions? In this paper, we provide a possible theoretical answer to this question: namely, we show that of all families with the smallest possible number of parameters, families corresponding to neurons are indeed optimal -- for all optimality criteria that satisfy some reasonable requirements: : …

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Why Daubechies Wavelets Are So Successful, Solymar Ayala Cortez, Laxman Bokati, Aaron Velasco, Vladik Kreinovich Oct 2021

Why Daubechies Wavelets Are So Successful, Solymar Ayala Cortez, Laxman Bokati, Aaron Velasco, Vladik Kreinovich

Departmental Technical Reports (CS)

In many applications, including analysis of seismic signals, Daubechies wavelets perform much better than other families of wavelets. In this paper, we provide a possible theoretical explanation for the empirical success of Daubechies wavelets. Specifically, we show that these wavelets are optimal with respect to any optimality criterion that satisfies the natural properties of scale- and shift-invariance.

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