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University of Texas Rio Grande Valley

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Series

2013

Articles 1 - 12 of 12

Full-Text Articles in Physical Sciences and Mathematics

Peakon, Pseudo-Peakon, And Cuspon Solutions For Two Generalized Camassa- Holm Equations, Jibin Li, Zhijun Qiao Dec 2013

Peakon, Pseudo-Peakon, And Cuspon Solutions For Two Generalized Camassa- Holm Equations, Jibin Li, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

In this paper, we study peakon, cuspon, and pseudo-peakon solutions for two generalized Camassa-Holm equations. Based on the method of dynamical systems, the two generalized Camassa-Holm equations are shown to have the parametric representations of the solitary wave solutions such as peakon, cuspon, pseudo-peakons, and periodic cusp solutions. In particular, the pseudo-peakon solution is for the first time proposed in our paper. Moreover, when a traveling system has a singular straight line and a heteroclinic loop, under some parameter conditions, there must be peaked solitary wave solutions appearing.

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Optimal Control In The Treatment Of Retinitis Pigmentosa, Erika T. Camacho, Luis A. Melara, Cristina Villalobos, Stephen Wirkus Nov 2013

Optimal Control In The Treatment Of Retinitis Pigmentosa, Erika T. Camacho, Luis A. Melara, Cristina Villalobos, Stephen Wirkus

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Numerous therapies have been implemented in an effort to minimize the debilitating effects of the degenerative eye disease Retinitis Pigmentosa (RP), yet none have provided satisfactory long-term solution. To date there is no treatment that can halt the degeneration of photoreceptors. The recent discovery of the RdCVF protein has provided researchers with a potential therapy that could slow the secondary wave of cone death. In this work, we build on an existing mathematical model of photoreceptor interactions in the presence of RP and incorporate various treatment regiments via RdCVF. Our results show that an optimal control exists for the administration …

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Hydro-Thermal Convective Solutions For An Aquifer System Heated From Below, Dambaru Bhatta Jun 2013

Hydro-Thermal Convective Solutions For An Aquifer System Heated From Below, Dambaru Bhatta

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We investigate the effect of hydro-thermal convection in an aquifer system. It is assumed that the aquifer is bounded below and above by impermeable boundaries and it is heated from below. The solution of the governing system is expressed in terms of the basic steady state solution and perturbed solution. We obtain the critical Rayleigh number and critical wavenumber using Runge-Kutta method in combination of shooting method and present the marginal stability curve. The amplitude equation is derived by introducing the adjoint system. After amplitude is obtained, we compute the linear solutions for super-critical and sub-critical cases. Numerical results for …

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Generalized Local Test For Local Extrema In Single-Variable Functions, Eleftherios Gkioulekas May 2013

Generalized Local Test For Local Extrema In Single-Variable Functions, Eleftherios Gkioulekas

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We give a detailed derivation of a generalization of the second derivative test of single-variable calculus which can classify critical points as local minima or local maxima (or neither), whenever the traditional second derivative test fails, by considering the values of higher-order derivatives evaluated at the critical points. The enhanced test is local, in the sense that it is only necessary to evaluate all relevant derivatives at the critical point itself, and it is reasonably robust. We illustrate an application of the generalized test on a trigonometric function where the second derivative test fails to classify some of the critical …

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Algebro-Geometric Solutions For The Degasperis--Procesi Hierarchy, Yu Hou, Peng Zhao, Engui Fan, Zhijun Qiao May 2013

Algebro-Geometric Solutions For The Degasperis--Procesi Hierarchy, Yu Hou, Peng Zhao, Engui Fan, Zhijun Qiao

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Though the completely integrable Camassa--Holm (CH) equation and Degasperis--Procesi (DP) equation are cast in the same peakon family, they possess the second- and third-order Lax operators, respectively. From the viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great difficulty in the construction of algebro-geometric solutions of the DP equation. In this paper, we derive the DP hierarchy with the help of Lenard recursion operators. Based on the characteristic polynomial of a Lax matrix for the DP hierarchy, we introduce a third order algebraic curve $\mathcal{K}_{r-2}$ with genus $r-2$, from which the …

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The Schwinger Action Principle And Its Applications To Quantum Mechanics, Paul Bracken Apr 2013

The Schwinger Action Principle And Its Applications To Quantum Mechanics, Paul Bracken

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

No abstract provided.

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Characterizations Of Exponential Distribution Based On Sample Of Size Three, George Yanev, Santanu Chakraborty Jan 2013

Characterizations Of Exponential Distribution Based On Sample Of Size Three, George Yanev, Santanu Chakraborty

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Two characterizations of the exponential distribution based on equalities among order statistics in a random sample of size three are proved. This proves two conjectures stated recently in Arnold and Villasenor [4].

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Analytic Matrix Elements Of The Schrödinger Equation, Muhammad I. Bhatti Jan 2013

Analytic Matrix Elements Of The Schrödinger Equation, Muhammad I. Bhatti

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A previously defined analytic technique of constructing matrix elements from the Bernstein-polynomials (B-poly) has been applied to Schr¨odinger equation. This method after solving generalized eigenvalue problem yields very accurate eigenenergies and eigenvectors. The numerical eigenvectors and eigenvalues obtained from this process agree well with exact results of the hydrogen-like systems. Furthermore, accuracy of the numerical spectrum of hydrogen equation depends on the number of B-polys being used to construct the analytical matrix elements. Validity of eigenvalues and quality of the constructed wavefunctions is verified by evaluating the Thomas-Reiche-Kuhn (TRK) sum rules. Excellent numerical agreement is seen with exact results of …

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Zero-Bounded Limits As A Special Case Of The Squeeze Theorem For Evaluating Single-Variable And Multivariable Limits, Eleftherios Gkioulekas Jan 2013

Zero-Bounded Limits As A Special Case Of The Squeeze Theorem For Evaluating Single-Variable And Multivariable Limits, Eleftherios Gkioulekas

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Many limits, typically taught as examples of applying the ‘squeeze’ theorem, can be evaluated more easily using the proposed zero-bounded limit theorem. The theorem applies to functions defined as a product of a factor going to zero and a factor that remains bounded in some neighborhood of the limit. This technique is immensely useful for both single-variable limits and multidimensional limits. A comprehensive treatment of multidimensional limits and continuity is also outlined.

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On Equivalent Characterizations Of Convexity Of Functions, Eleftherios Gkioulekas Jan 2013

On Equivalent Characterizations Of Convexity Of Functions, Eleftherios Gkioulekas

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

A detailed development of the theory of convex functions, not often found in complete form in most textbooks, is given. We adopt the strict secant line definition as the definitive definition of convexity. We then show that for differentiable functions, this definition becomes logically equivalent with the first derivative monotonicity definition and the tangent line definition. Consequently, for differentiable functions, all three characterizations are logically equivalent.

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On Characterizations Of Exponential Distribution Through Order Statistics And Record Values With Random Shifts, M. Ahsanullah, Imtiyaz A. Shah, George Yanev Jan 2013

On Characterizations Of Exponential Distribution Through Order Statistics And Record Values With Random Shifts, M. Ahsanullah, Imtiyaz A. Shah, George Yanev

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

Distributional relations of the form Y d= X +T where X, Y, and T are record values or order statistics and the random translator T is independent from X are considered. Characterizations of the exponential distribution when the ordered random variables are non-neighboring are proved. Corollaries for Pareto and power function distributions are also derived.

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The Two-Phase Arterial Blood Flow With Or Without A Catheter And In The Presence Of A Single Or Multi Stenosis, Ani E. Garcia, Daniel N. Riahi Jan 2013

The Two-Phase Arterial Blood Flow With Or Without A Catheter And In The Presence Of A Single Or Multi Stenosis, Ani E. Garcia, Daniel N. Riahi

School of Mathematical and Statistical Sciences Faculty Publications and Presentations

We consider the problem of blood flow in an artery with or without a catheter and in the presence of single or multi stenosis whose shape is based on the available experimental data for the stenosis in a human’s artery. The presence of stenosis in the artery, which locally narrows portion of the artery, can be a result of fatty materials such as cholesterol in the blood. The use of catheter is important as a standard tool for diagnosis and treatment in patience whose blood flow passage in the artery is affected adversely by the presence of the stenosis within …

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