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`The Very Beautiful Principles Of Natural Philosophy': Michael Faraday, Paper Marbling And The Physics Of Natural Forms, Robert Pepperell

LASER Journal

In 1854, Michael Faraday wrote to thank the author who had sent him a book on the art of paper marbling. In the letter, Faraday referred to `the very beautiful principles of natural philosophy' involved in the process of dropping ink on thickened water. What are the `beautiful principles' that Faraday referred to, and how are they involved in the art of paper marbling? Here I consider some of the physical processes that occur in paper marbling and how the patterns that emerge represent `dissipative structures' that are governed by fundamental principles of nature, in particular the tendency for physical …

Advances And Applications Of Dsmt For Information Fusion. Collected Works, Volume 5, Florentin Smarandache, Jean Dezert, Albena Tchamova

Branch Mathematics and Statistics Faculty and Staff Publications

This fifth volume on Advances and Applications of DSmT for Information Fusion collects theoretical and applied contributions of researchers working in different fields of applications and in mathematics, and is available in open-access. The collected contributions of this volume have either been published or presented after disseminating the fourth volume in 2015 (available at fs.unm.edu/DSmT-book4.pdf or www.onera.fr/sites/default/files/297/2015-DSmT-Book4.pdf) in international conferences, seminars, workshops and journals, or they are new. The contributions of each part of this volume are chronologically ordered.

First Part of this book presents some theoretical advances on DSmT, dealing mainly with modified Proportional Conflict Redistribution Rules (PCR) of …

Spline Modeling And Localized Mutual Information Monitoring Of Pairwise Associations In Animal Movement, Andrew Benjamin Whetten

Theses and Dissertations

to a new era of remote sensing and geospatial analysis. In environmental science and conservation ecology, biotelemetric data recorded is often high-dimensional, spatially and/or temporally, and functional in nature, meaning that there is an underlying continuity to the biological process of interest. GPS-tracking of animal movement is commonly characterized by irregular time-recording of animal position, and the movement relationships between animals are prone to sudden change. In this dissertation, I propose a spline modeling approach for exploring interactions and time-dependent correlation between the movement of apex predators exhibiting territorial and territory-sharing behavior. A measure of localized mutual information (LMI) is …

A Super Fast Algorithm For Estimating Sample Entropy, Weifeng Liu, Ying Jiang, Yuesheng Xu

Mathematics & Statistics Faculty Publications

: Sample entropy, an approximation of the Kolmogorov entropy, was proposed to characterize complexity of a time series, which is essentially defined as − log(B/A), where B denotes the number of matched template pairs with length m and A denotes the number of matched template pairs with m + 1, for a predetermined positive integer m. It has been widely used to analyze physiological signals. As computing sample entropy is time consuming, the box-assisted, bucket-assisted, x-sort, assisted sliding box, and kd-tree-based algorithms were proposed to accelerate its computation. These algorithms require O(N²) or …

The Existence And Quantum Approximation Of Optimal Pure State Ensembles, Ryan Thomas Mcgaha

Theses and Dissertations

In this manuscript we study entanglement measures defined via the convex roof construction. In the first chapter we build the notion of an entanglement measure from the ground up and discuss various issues that arise when trying to measure the amount of entanglement present in an arbitrary density operator. Through this introduction we will motivate the use of the convex roof construction. In the second chapter we will show that the infimum in the convex roof construction is achieved for a specific set of entanglement measures and provide canonical examples of such measures. We also describe LOCC operations via a …

Extractable Entanglement From A Euclidean Hourglass, Takanori Anegawa, Norihiro Iizuka, Daniel Kabat

Publications and Research

We previously proposed that entanglement across a planar surface can be obtained from the partition function on a Euclidean hourglass geometry. Here we extend the prescription to spherical entangling surfaces in conformal field theory. We use the prescription to evaluate log terms in the entropy of a conformal field theory in two dimensions, a conformally coupled scalar in four dimensions, and a Maxwell field in four dimensions. For Maxwell we reproduce the extractable entropy obtained by Soni and Trivedi. We take this as evidence that the hourglass prescription provides a Euclidean technique for evaluating extractable entropy in quantum field theory.

Existence And Transportation Inequalities For Fractional Stochastic Differential Equations, Abdelghani Ouahab, Mustapha Belabbas, Johnny Henderson, Fethi Souna

Turkish Journal of Mathematics

In this work, we establish the existence and uniqueness of solutions for a fractional stochastic differential equation driven by countably many Brownian motions on bounded and unbounded intervals. Also, we study the continuous dependence of solutions on initial data. Finally, we establish the transportation quadratic cost inequality for some classes of fractional stochastic equations and continuous dependence of solutions with respect Wasserstein distance.

Behavior Of Entanglement Entropy Near Periodic Orbits In A Hamiltonian Dynamical System, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Entanglement entropy growth is studied under a form of dynamics that is based on iteration. This approach allows the investigation of the role of decoherence in producing increases of entropy. This has important consequences as far as the study of decoherence is concerned. It is indicated that results are generally independent of Hilbert space partitioning. It is seen that a deep relationship between classical dynamical entropy and the growth of entanglement entropy exists in this kind of model. The former acts to bound the latter and in the asymptotic region, they tend to a common limit.

An Introduction To Generalized Entropy And Some Quantum Applications, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

The concept of generalized entropy is introduced and some of its properties are studied. Irreversible time evolution can be generated by a non-Hermitian superoperator on the states of the system. The case when irreversibility comes about from embedding the system in a thermal reservoir is looked at. The time evolution is found compatible both with equilibrium thermodynamics and entropy production near the final state. Some examples are presented as well as a longer introduction as to how this might play a role in the black hole information loss paradox.

The Degeneration Of The Hilbert Metric On Ideal Pants And Its Application To Entropy, Marianne Debrito, Andrew Nguyen, Marisa O'Gara

Rose-Hulman Undergraduate Mathematics Journal

Entropy is a single value that captures the complexity of a group action on a metric space. We are interested in the entropies of a family of ideal pants groups $\Gamma_T$, represented by projective reflection matrices depending on a real parameter $T > 0$. These groups act on convex sets $\Omega_{\Gamma_T}$ which form a metric space with the Hilbert metric. It is known that entropy of $\Gamma_T$ takes values in the interval $\left(\frac{1}{2},1\right]$; however, it has not been proven whether $\frac{1}{2}$ is the sharp lower bound. Using Python programming, we generate approximations of tilings of the convex set in the projective …

Analyzing The Von Neumann Entropy Of Contact Networks, Thomas J. Brower

All Graduate Theses and Dissertations, Spring 1920 to Summer 2023

When modeling the spread of disease, ecologists use ecological or contact networks to model how species interact with their environment and one another. The structure of these networks can vary widely depending on the study, where the nodes of a network can be defined as individuals, groups, or locations among other things. With this wide range of definition and with the difficulty of collecting samples, it is difficult to capture every factor of every population. Thus ecologists are limited to creating smaller networks that both fit their budget as well as what is reasonable within the population of interest. With …

Entropy In Quantum Mechanics And Applications To Nonequilibrium Thermodynamics, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Classical formulations of the entropy concept and its interpretation are introduced. This is to motivate the definition of the quantum von Neumann entropy. Some general properties of quantum entropy are developed, such as the quantum entropy which always increases. The current state of the area that includes thermodynamics and quantum mechanics is reviewed. This interaction shall be critical for the development of nonequilibrium thermodynamics. The Jarzynski inequality is developed in two separate but related ways. The nature of irreversibility and its role in physics are considered as well. Finally, a specific quantum spin model is defined and is studied in …

Ricci Curvature Of Noncommutative Three Tori, Entropy, And Second Quantization, Rui Dong

Electronic Thesis and Dissertation Repository

In noncommutative geometry, the metric information of a noncommutative space is encoded in the data of a spectral triple $(\mathcal{A}, \mathcal{H},D)$, where $D$ plays the role of the Dirac operator acting on the Hilbert space of spinors. Ideas of spectral geometry can then be used to define suitable notions such as volume, scalar curvature, and Ricci curvature. In particular, one can construct the Ricci curvature from the asymptotic expansion of the heat trace $\textrm{Tr}(e^{-tD^2})$. In Chapter 2, we will compute the Ricci curvature of a curved noncommutative three torus. The computation is done for both conformal and a non-conformal perturbation …

Functional Dimension Of Solution Space Of Differential Operators Of Constant Strength, Morteza Shafii-Mousavi

Applications and Applied Mathematics: An International Journal (AAM)

A differential operator with constant coefficients is hypoelliptic if and only if its solution space is of finite functional dimension. We extend this property to operators with variable coefficient. We prove that an equally strong differential operator with variable coefficients has the same property. In addition, we extend the Zielezny’s result to operators with variable coefficients; prove that an operator with analytic coefficients on ℝⁿ is elliptic if and only if locally the functional dimension of its solution space is the same as the Euclidean dimension n.

A Development Of Transfer Entropy In Continuous-Time, Christopher David Edgar

Theses and Dissertations

The quantification of causal relationships between time series data is a fundamen- tal problem in fields including neuroscience, social networking, finance, and machine learning. Amongst the various means of measuring such relationships, information- theoretic approaches are a rapidly developing area in concert with other methods. One such approach is to make use of the notion of transfer entropy (TE). Broadly speaking, TE is an information-theoretic measure of information transfer between two stochastic processes. Schreiber’s 2001 definition of TE characterizes information transfer as an informational divergence between conditional probability mass func- tions. The original definition is native to discrete-time stochastic processes …

Dynamical Entropy Of Quantum Random Walks, Duncan Wright

Theses and Dissertations

In this manuscript, we study discrete-time dynamics of systems that arise in physics and information theory, and the measure of disorder in these systems known as dy- namical entropy. The study of dynamics in classical systems is done from two distinct viewpoints: random walks and dynamical systems. Random walks are probabilistic in nature and are described by stochastic processes. On the other hand, dynami- cal systems are described algebraically and deterministic in nature. The measure of disorder from either viewpoint is known as dynamical entropy.

Entropy is an essential notion in physics and information theory. Motivated by the study of …

Surface Entropy Of Shifts Of Finite Type, Dennis Pace

Electronic Theses and Dissertations

Let χ be the class of 1-D and 2-D subshifts. This thesis defines a new function, H_S : χ x R → [0,∞] which we call the surface entropy of a shift. This definition is inspired by the topological entropy of a subshift and we compare and contrast several structural properties of surface entropy to entropy. We demonstrate that much like entropy, the finiteness of surface entropy is a conjugacy invariant and is a tool in the classification of subshifts. We develop a tiling algorithm related to continued fractions which allows us to prove a continuity result about surface …

Simplicity And Sustainability: Pointers From Ethics And Science, Mehrdad Massoudi, Ashuwin Vaidya

Department of Mathematics Facuty Scholarship and Creative Works

In this paper, we explore the notion of simplicity. We use definitions of simplicity proposed by philosophers, scientists, and economists. In an age when the rapidly growing human population faces an equally rapidly declining energy/material resources, there is an urgent need to consider various notions of simplicity, collective and individual, which we believe to be a sensible path to restore our planet to a reasonable state of health. Following the logic of mathematicians and physicists, we suggest that simplicity can be related to sustainability. Our efforts must therefore not be spent so much in pursuit of growth but in achieving …

Using The Entropy Rate Balance To Determine The Heat Transfer And Work In An Internally Reversible, Polytropic, Steady State Flow Process, Savannah Griffin

Undergraduate Journal of Mathematical Modeling: One + Two

The entropy rate equation for internally reversible steady state flow process has been used to calculate the heat transfer and work in an internally reversible, polytropic, steady state flow process.

Neuronal Correlation Parameter In The Idea Of Thermodynamic Entropy Of An N-Body Gravitationally Bounded System, Ioannis Haranas, Ioannis Gkigkitzis, Ilias S. Kotsireas, Carlos Austerlitz

Physics and Computer Science Faculty Publications

Understanding how the brain encodes information and performs computation requires statistical and functional analysis. Given the complexity of the human brain, simple methods that facilitate the interpretation of statistical correlations among different brain regions can be very useful. In this report we introduce a numerical correlation measure that may serve the interpretation of correlational neuronal data, and may assist in the evaluation of different brain states. The description of the dynamical brain system, through a global numerical measure may indicate the presence of an action principle which may facilitate a application of physics principles in the study of the human …

Multicriteria Decision Making Using Double Refined Indeterminacy Neutrosophic Cross Entropy And Indeterminacy Based Cross Entropy, Ilanthenral Kandasamy, Florentin Smarandache

Branch Mathematics and Statistics Faculty and Staff Publications

Double Refined Indeterminacy Neutrosophic Set (DRINS) is an inclusive case of the refined neutrosophic set, defined by Smarandache [1], which provides the additional possibility to represent with sensitivity and accuracy the uncertain, imprecise, incomplete, and inconsistent information which are available in real world.

More precision is provided in handling indeterminacy; by classifying indeterminacy (I) into two, based on membership; as indeterminacy leaning towards truth membership (IT ) and indeterminacy leaning towards false membership (IF ). This kind of classification of indeterminacy is not feasible with the existing Single Valued Neutrosophic Set (SVNS), but it is a particular case of the …

Mapped Tent Pitching Schemes For Hyperbolic Systems, Jay Gopalakrishnan, Joachim Schöberl, C. Wintersteiger

Portland Institute for Computational Science Publications

A spacetime domain can be progressively meshed by tent shaped objects. Numerical methods for solving hyperbolic systems using such tent meshes to advance in time have been proposed previously. Such schemes have the ability to advance in time by different amounts at different spatial locations. This paper explores a technique by which standard discretizations, including explicit time stepping, can be used within tent-shaped spacetime domains. The technique transforms the equations within a spacetime tent to a domain where space and time are separable. After detailing techniques based on this mapping, several examples including the acoustic wave equation and the Euler …

Some 2-Categorical Aspects In Physics, Arthur Parzygnat

Dissertations, Theses, and Capstone Projects

2-categories provide a useful transition point between ordinary category theory and infinity-category theory where one can perform concrete computations for applications in physics and at the same time provide rigorous formalism for mathematical structures appearing in physics. We survey three such broad instances. First, we describe two-dimensional algebra as a means of constructing non-abelian parallel transport along surfaces which can be used to describe strings charged under non-abelian gauge groups in string theory. Second, we formalize the notion of convex and cone categories, provide a preliminary categorical definition of entropy, and exhibit several examples. Thirdly, we provide a universal description …

Quantum Dynamics, Entropy And Quantum Versions Of Maxwell’S Demon, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Several subjects which reside in the overlap area of quantum mechanics, statistical physics and thermodynamics are investigated in depth. This collection of subjects shares a common domain which is referred to as Maxwell’s demon. The classical version of this idea is introduced, and then, the contribution made by Szilard to the subject is presented. Several demons are considered, and it is shown that to best understand this area, quantum mechanics and the role information plays in it must be appreciated deeply.

Quantum Random Number Generation Using A Quanta Image Sensor, Emna Amri, Yacine Felk, Damien Stucki, Jiaju Ma, Eric Fossum

Dartmouth Scholarship

A new quantum random number generation method is proposed. The method is based on the randomness of the photon emission process and the single photon counting capability of the Quanta Image Sensor (QIS). It has the potential to generate high-quality random numbers with remarkable data output rate. In this paper, the principle of photon statistics and theory of entropy are discussed. Sample data were collected with QIS jot device, and its randomness quality was analyzed. The randomness assessment method and results are discussed.

Takens Theorem With Singular Spectrum Analysis Applied To Noisy Time Series, Thomas K. Torku

Electronic Theses and Dissertations

The evolution of big data has led to financial time series becoming increasingly complex, noisy, non-stationary and nonlinear. Takens theorem can be used to analyze and forecast nonlinear time series, but even small amounts of noise can hopelessly corrupt a Takens approach. In contrast, Singular Spectrum Analysis is an excellent tool for both forecasting and noise reduction. Fortunately, it is possible to combine the Takens approach with Singular Spectrum analysis (SSA), and in fact, estimation of key parameters in Takens theorem is performed with Singular Spectrum Analysis. In this thesis, we combine the denoising abilities of SSA with the Takens …

An Order Model For Infinite Classical States, Joe Mashburn

Joe D. Mashburn

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory,2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and …

Supplementary Balance Laws For Cattaneo Heat Propagation, Serge Preston

Mathematics and Statistics Faculty Publications and Presentations

In this work we determine for the Cattaneo heat propagation system all the supplementary balance laws (conservation laws ) of the same order (zero) as the system itself and extract the constitutive relations (expression for the internal energy) dictated by the Entropy Principle. The space of all supplementary balance laws (having the functional dimension 8) contains four original balance laws and their deformations depending on 4 functions of temperature (λ⁰(ϑ),K^A (ϑ), A = 1, 2, 3). The requirements of the II law of thermodynamics leads to the exclusion of three functional degrees (K^A= 0, A …

Bekenstein Bound Of Information Number N And Its Relation To Cosmological Parameters In A Universe With And Without Cosmological Constant, Ioannis Haranas, Ioannis Gkigkitzis

Physics and Computer Science Faculty Publications

Bekenstein has obtained is an upper limit on the entropy S, and from that, an information number bound N is deduced. In other words, this is the information contained within a given finite region of space that includes a finite amount of energy. Similarly, this can be thought as the maximum amount of information required to perfectly describe a given physical system down to its quantum level. If the energy and the region of space are finite then the number of information N required in describing the physical system is also finite. In this short letter two information number …

H-Coloring Tori, John Engbers, David Galvin

Mathematics, Statistics and Computer Science Faculty Research and Publications

For graphs G and H, an H-coloring of G is a function from the vertices of G to the vertices of H that preserves adjacency. H-colorings encode graph theory notions such as independent sets and proper colorings, and are a natural setting for the study of hard-constraint models in statistical physics. We study the set of H-colorings of the even discrete torus View the MathML source, the graph on vertex set {0,…,m−1}d (m even) with two strings adjacent if they differ by 1 (mod m) on one coordinate and agree on all others. This is a bipartite graph, with bipartition …