Physical Sciences and Mathematics Commons^™

Nabla Fractional Calculus And Its Application In Analyzing Tumor Growth Of Cancer, Fang Wu

Masters Theses & Specialist Projects

This thesis consists of six chapters. In the first chapter, we review some basic definitions and concepts of fractional calculus. Then we introduce fractional difference equations involving the Riemann-Liouville operator of real number order between zero and one. In the second chapter, we apply the Brouwer fixed point and Contraction Mapping Theorems to prove that there exists a solution for up to the first order nabla fractional difference equation with an initial condition. In chapter three, we define a lower and an upper solution for up to the first order nabla fractional difference equation with an initial condition. Under certain …

Discrete-State Stochastic Models Of Calcium-Regulated Calcium Influx And Subspace Dynamics Are Not Well-Approximated By Odes That Neglect Concentration Fluctuations, Seth H. Weinberg, Gregory D. Smith

Arts & Sciences Articles

Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted “domains” associated with calcium influx are small enough (e.g., 10−17 liters) that local signaling may involve 1–100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result …

Incomplete Market Models Of Carbon Emissions Markets, Walid Mnif

Electronic Thesis and Dissertation Repository

New regulatory frameworks have been developed with the aim of decreasing global greenhouse gas emissions over both short and long time periods. Incentives must be established to encourage the transition to a clean energy economy. Emissions taxes represent a "price" incentive for this transition, but economists agree this approach is suboptimal. Instead, the "quantity" instrument provided by cap-and-trade markets are superior from an economic point of view. This thesis focuses on the cap-and-trade instrument. Carbon emissions markets have recently been implemented in different countries. We summarize the state of world cap-and-trade schemes. We also provide a literature review of existing …

Phase Transitions And Change Of Type In Low-Temperature Heat, Ralph A. Saxton, Katarzyna Saxton

Ralph Saxton

Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, …

Global Existence Of Some Infinite Energy Solutions For A Perfect Incompressible Fluid, Ralph Saxton, Feride Tiğlay

Ralph Saxton

This paper provides results on local and global existence for a class of solutions to the Euler equations for an incompressible, inviscid fluid. By considering a class of solutions which exhibits a characteristic growth at infinity we obtain an initial value problem for a nonlocal equation. We establish local well-posedness in all dimensions and persistence in time of these solutions for three and higher dimensions. We also examine a weaker class of global solutions.

Dark Solitons Of The Qiao's Hierarchy, Rossen Ivanov, Tony Lyons

Articles

We obtain a class of soliton solutions of the integrable hierarchy which has been put forward in a series of works by Z. Qiao. The soliton solutions are in the class of real functions approaching constant value fast enough at infinity, the so-called 'dark solitons'.

Convex Combinations Of Quadrant Dependent Copulas, Martin Egozcue, Luis Fuentes García, Wing Wong, Ricardas Zitikis

Martin Egozcue

It is well known that quadrant dependent (QD) random variables are also quadrant dependent in expectation (QDE). Recent literature has offered examples rigorously establishing the fact that there are QDE random variables which are not QD. The examples are based on convex combinations of specially chosen QD copulas: one negatively QD and another positively QD. In this paper we establish general results that determine when convex combinations of arbitrary QD copulas give rise to negatively or positively QD/QDE copulas. In addition to being an interesting mathematical exercise, the established results are helpful when modeling insurance and financial portfolios.

Local Fractional Fourier Series With Application To Wave Equation In Fractal Vibrating String, Yang Xiaojun

Xiao-Jun Yang

We introduce the wave equation in fractal vibrating string in the framework of the local fractional calculus. Our particular attention is devoted to the technique of the local fractional Fourier series for processing these local fractional differential operators in a way accessible to applied scientists. By applying this technique we derive the local fractional Fourier series solution of the local fractional wave equation in fractal vibrating string and show the fundamental role of the Mittag- Leffler function.

A Modified Resource Distribution Fairness Measure, Zhenmin Chen

Department of Mathematics and Statistics

An important issue of resource distribution is the fairness of the distribution. For example, computer network management wishes to distribute network resource fairly to its users. To describe the fairness of the resource distribution, a quantitative fairness score function was proposed in 1984 by Jain et al. The purpose of this paper is to propose a modified network sharing fairness function so that the users can be treated differently according to their priority levels. The mathematical properties are discussed. The proposed fairness score function keeps all the nice properties of and provides better performance when the network users have different …

Should Voting Be Mandatory? Democratic Decision Making From The Economic Viewpoint, Olga Kosheleva, Vladik Kreinovich, Boakun Li

Departmental Technical Reports (CS)

Many decisions are made by voting. At first glance, the more people participate in the voting process, the more democratic -- and hence, better -- the decision. In this spirit, to encourage everyone's participation, several countries make voting mandatory. But does mandatory voting really make decisions better for the society? In this paper, we show that from the viewpoint of decision making theory, it is better to allow undecided voters not to participate in the voting process. We also show that the voting process would be even better -- for the society as a whole -- if we allow partial …

Ubiquity Of Data And Model Fusion: From Geophysics And Environmental Sciences To Estimating Individual Risk During An Epidemic, Omar Ochoa, Aline Jaimes, Christian Servin, Craig Tweedie, Aaron Velasco, Martine Ceberio, Vladik Kreinovich

Departmental Technical Reports (CS)

In many practical situations, we need to combine the results of measuring a local value of a certain quantity with results of measuring average values of this same quantity. For example, in geosciences, we need to combine the seismic models (which describe density at different locations and depths) with gravity models which describe density averaged over certain regions. Similarly, in estimating the risk of an epidemic to an individual, we need to combine probabilities describe the risk to people of the corresponding age group, to people of the corresponding geographical region, etc. In this paper, we provide general techniques for …

G-Strands, Darryl Holm, Rossen Ivanov, James Percival

Articles

A G-strand is a map g(t,s): RxR --> G for a Lie group G that follows from Hamilton's principle for a certain class of G-invariant Lagrangians. The SO(3)-strand is the G-strand version of the rigid body equation and it may be regarded physically as a continuous spin chain. Here, SO(3)_K-strand dynamics for ellipsoidal rotations is derived as an Euler-Poincar'e system for a certain class of variations and recast as a Lie-Poisson system for coadjoint flow with the same Hamiltonian structure as for a perfect complex fluid. For a special Hamiltonian, the SO(3) …

Effects Of Electrostatic Correlations On Electrokinetic Phenomena, Brian Storey, Martin Bazant

Brian Storey

The classical theory of electrokinetic phenomena is based on the mean-field approximation that the electric field acting on an individual ion is self-consistently determined by the local mean charge density. This paper considers situations, such as concentrated electrolytes, multivalent electrolytes, or solvent-free ionic liquids, where the mean-field approximation breaks down. A fourth-order modified Poisson equation is developed that captures the essential features in a simple continuum framework. The model is derived as a gradient approximation for nonlocal electrostatics of interacting effective charges, where the permittivity becomes a differential operator, scaled by a correlation length. The theory is able to capture …

Revisiting The Newsboy Problem-Optimization With A Little Help From The Airline Industry, Tamas Lengyel

Tamas Lengyel

In a typical inventory planning problem with a life cycle of only one planning period, we incur the cost of production per unit produced, profit per unit sold, loss per unit not sold, and lost revenue per unit ordered but not matched due to the lack of availability. The goal is to find the inventory level that maximizes the expected net profit. Textbooks often use the newsboy problem to illustrate the inventory management paradigm. The derivation of the formulas for the optimal level is usually done on an ad hoc basis, by dull and rote mathematical manipulations, for each modification …

Thermal Detection Of Inaccessible Corrosion, Matthew Charnley, Andrew Rzeznik

Mathematical Sciences Technical Reports (MSTR)

In this paper, we explore the mathematical inverse problem of detecting corroded material on the reverse side of a partially accessible metal plate. We will show how a linearization can be used to simplify the initial problem and explain a regularization method used to obtain acceptable results for the corrosion profile. We will also state and perform some calculations for the full three-dimensional problem for possible future work.

Relativistic Solution Of The N-Body Problem (Ii), Jorge A. Franco

Jorge A Franco

This work is the continuation of the classical approach described in previous paper for constant masses. In here the solution of the movement of a group of N gravitationally attracting bodies around its center of mass CM, given their initial positions and velocities, is developed for variable masses under the Theory of Vectorial Relativity. The strategy of realizing special physical characteristics of forces on the the CM and properties of the reduced mass in the solution of the two-body problem, allowed extending the Newton’s Universal Gravitation Law for applying to two or more attracting bodies, and also allowed operating on …

How To Create A Two-Component Spinor, Charles G. Torre

How to... in 10 minutes or less

Let (M, g) be a spacetime, i.e., a 4-dimensional manifold M and Lorentz signature metric g. The key ingredients needed for constructing spinor fields on the spacetime are: a complex vector bundle E -> M ; an orthonormal frame on TM ; and a solder form relating sections of E to sections of TM (and tensor products thereof). We show how to create a two-component spinor field on the Schwarzschild spacetime using the DifferentialGeometry package in Maple. PDF and Maple worksheets can be downloaded from the links below.

A Doubling Technique For The Power Method Transformations, Mohan D. Pant, Todd C. Headrick

Mohan Dev Pant

Power method polynomials are used for simulating non-normal distributions with specified product moments or L-moments. The power method is capable of producing distributions with extreme values of skew (L-skew) and kurtosis (L-kurtosis). However, these distributions can be extremely peaked and thus not representative of real-world data. To obviate this problem, two families of distributions are introduced based on a doubling technique with symmetric standard normal and logistic power method distributions. The primary focus of the methodology is in the context of L-moment theory. As such, L-moment based systems of equations are derived for simulating univariate and multivariate non-normal distributions with …

Cyclic Universe With An Inflationary Phase From A Cosmological Model With Real Gas Quintessence, Rossen Ivanov, Emil Prodanov

Articles

Phase-plane stability analysis of a dynamical system describing the Universe as a two-fraction uid containing baryonic dust and real virial gas quintessence is presented. Existence of a stable periodic solution experiencing in ationary periods is shown. A van der Waals quintessence model is revisited and cyclic Universe solution again found.

A Unified Approach To Generalized Stirling Functions, Tian-Xiao He

Scholarship

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.

An Exercise With The He’S Variation Iteration Method To A Fractional Bernoulli Equation Arising In A Transient Conduction With A Non-Linear Boundary Heat Flux, Jordan Hristov

Jordan Hristov

Surface temperature evolution of a body subjected to a nonlinear heat flux involving counteracting convection heating and radiation cooling has been solved by the variations iteration method (VIM) of He. The surface temperature equations comes as a combination of the time-fractional (half-time) subdiffusion model of the heat conduction and the boundary condition relating the temperature field gradient at the surface through the Riemann-Liouville fractional integral. The result of this equation is a Bernoulli-type ordinary fractional equation with a nonlinear term of 4th order. Two approaches in the identification of the general Lagrange multiplier and a consequent application of VIM have …

How To Define Relative Approximation Error Of An Interval Estimate: A Proposal, Vladik Kreinovich

Departmental Technical Reports (CS)

The traditional definition of a relative approximation error of an estimate X as the ratio |X - x|/|x| does not work when the actual value x is 0. To avoid this problem, we propose a new definition |X - x|/|X|. We show how this definition can be naturally extended to the case when instead of a numerical estimate X, we have an interval estimate [x], i.e., an interval that is guaranteed to contain the actual (unknown) value x.

A Unified Approach To Generalized Stirling Functions, Tian-Xiao He

Tian-Xiao He

Here presented is a unified approach to generalized Stirling functions by using generalized factorial functions, $k$-Gamma functions, generalized divided difference, and the unified expression of Stirling numbers defined in \cite{He11}. Previous well-known Stirling functions introduced by Butzer and Hauss \cite{BH93}, Butzer, Kilbas, and Trujilloet \cite{BKT03} and others are included as particular cases of our generalization. Some basic properties related to our general pattern such as their recursive relations, generating functions, and asymptotic properties are discussed, which extend the corresponding results about the Stirling numbers shown in \cite{HS98} to the defined Stirling functions.

Quantifying Performance Bias In Label Fusion, Alexander M. Venzin

Theses and Dissertations

Classification systems are employed to remotely assess whether an element of interest falls into a target class or non-target class. These systems have uses in fields as far ranging as biostatistics to search engine keyword analysis. The performance of the system is often summarized as a trade-off between the proportions of elements correctly labeled as target plotted against the number of elements incorrectly labeled as target. These are empirical estimates of the true positive and false positive rates. These rates are often plotted to create a receiver operating characteristic (ROC) curve that acts as a visual tool to assess classification …

The Octonions And The Exceptional Lie Algebra G2, Ian M. Anderson

Research Vignettes

The octonions O are an 8-dimensional non-commutative, non-associative normed real algebra. The set of all derivations of O form a real Lie algebra. It is remarkable fact, first proved by E. Cartan in 1908, that the the derivation algebra of O is the compact form of the exceptional Lie algebra G2. In this worksheet we shall verify this result of Cartan and also show that the derivation algebra of the split octonions is the split real form of G2.

PDF and Maple worksheets can be downloaded from the links below.

Approximate Methods For Dynamic Portfolio Allocation Under Transaction Costs, Nabeel Butt

Electronic Thesis and Dissertation Repository

The thesis provides robust and efficient lattice based algorithms for solving dynamic portfolio allocation problems under transaction costs. The early part of the thesis concentrates upon developing a toolbox based on multinomial trees. The multinomial trees are shown to provide a reasonable approximation for most popular transaction cost models in the academic literature. The tool, once forged, is implemented in the powerful Mathematica based parallel computing environment. In the second part of the thesis we provide applications of our framework to real world problems. We show re-balancing portfolios is more valuable in an investment environment where the growth and volatility …

270: How To Win The Presidency With Just 17.56% Of The Popular Vote, Charles D. Wessell

Math Faculty Publications

With the U.S. presidential election fast approaching we will often be reminded that the candidate who receives the most votes is not necessarily elected president. Instead, the winning candidate must receive a majority of the 538 electoral votes awarded by the 50 states and the District of Columbia. Someone with a curious mathematical mind might then wonder: What is the small fraction of the popular vote a candidate can receive and still be elected president? [excerpt]

Fuzzy And Adaptive Neuro-Fuzzy Inference System Of Washing Machine, R.W. Hndoosh

R. W. Hndoosh

Software estimation accuracy is among the greatest challenges for software developers. Fuzzy set theory, Fuzzy system and Neural Networks techniques seem very well suited for typical technical problems. In conjunction with software computing and conventional mathematical methods, hybrid methods can be developed that may prove to be a step forward in modeling geotechnical problems. This study aimed at building two different models, Fuzzy Inference Systems and Adaptive Neuro Fuzzy Inference System and a comparison between them, through an application to real data of the relationship between three inputs (time, temperature of water and the amount of washing powder) during the …

Higher Order Symmetries Of The Kdv Equation, Ian M. Anderson

Research Vignettes

In this worksheet we symbolically construct the formal inverse of the total derivative operator and use it to construct the recursion operator for the higher-order symmetries of the KdV equation. Using this recursion operator we generate the first 5 generalized symmetries of the KdV equation and verify that they all commute.

PDF and Maple worksheets can be downloaded from the links below.

Molecular Dynamics Studies Of Water Flow In Carbon Nanotubes, Alexander D. Marshall

Electronic Thesis and Dissertation Repository

We present classical molecular dynamics (MD) simulations providing insight into the behaviour of water. We focus on confined water, the properties of which are often significantly different from the properties of bulk water.

First, we performed several simulations investigating the handling of long-range interactions in GROMACS [1], a MD simulation package. Selection of simulation protocols such as handling of long-range interactions is often overlooked, sometimes to the significant detriment of the final result [2, 3, 4]. Ensuring that the chosen simulation protocols are appropriate is a critical step in computer simulation.

Second, we performed MD simulations where water flowed between …