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Articles 271 - 300 of 440
Full-Text Articles in Mathematics
Iterated Oscillation Criteria For Delay Dynamic Equations Of First Order, B. Karpuz, O. Öcalan, Martin Bohner
Iterated Oscillation Criteria For Delay Dynamic Equations Of First Order, B. Karpuz, O. Öcalan, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.
On The Asymptotic Integration Of Nonlinear Dynamic Equations, Elvan Akin, Smail Djebali, Toufik Moussaoui, Martin Bohner
On The Asymptotic Integration Of Nonlinear Dynamic Equations, Elvan Akin, Smail Djebali, Toufik Moussaoui, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
The purpose of this paper is to study the existence and asymptotic behavior of solutions to a class of second-order nonlinear dynamic equations on unbounded time scales. Four different results are obtained by using the Banach fixed point theorem, the Boyd and Wong fixed point theorem, the Leray-Schauder nonlinear alternative, and the Schauder fixed point theorem. For each theorem, an illustrative example is presented. The results provide unification and some extensions in the time scale setup of the theory of asymptotic integration of nonlinear equations both in the continuous and discrete cases
Transition To Turbulence, Small Disturbances, And Sensitivity Analysis Ii: The Navier-Stokes Equations, John R. Singler
Transition To Turbulence, Small Disturbances, And Sensitivity Analysis Ii: The Navier-Stokes Equations, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
Recent research has shown that small disturbances in the linearized Navier-Stokes equations cause large energy growth in solutions. Although many researchers believe that this interaction triggers transition to turbulence in flow systems, the role of the nonlinearity in this process has not been thoroughly investigated. This paper is the second of a two part work in which sensitivity analysis is used to study the effects of small disturbances on the transition process. In the first part, sensitivity analysis was used to predict the effects of a small disturbance on solutions of a motivating problem, a highly sensitive one dimensional Burgers' …
Transition To Turbulence, Small Disturbances, And Sensitivity Analysis I: A Motivating Problem, John R. Singler
Transition To Turbulence, Small Disturbances, And Sensitivity Analysis I: A Motivating Problem, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
For over 100 years, researchers have attempted to predict transition to turbulence in fluid flows by analyzing the spectrum of the linearized Navier-Stokes equations. However, for many simple flows this approach fails to match experimental results. Recently, new scenarios for transition have been proposed that are based on the interaction of the linearized equations of motion with small disturbances to the flow system. These new "mostly linear" theories have increased our understanding of the transition process, but the role of nonlinearity has not been explored in detail. This paper is the first of a two part work in which sensitivity …
Asymptotic Behavior Of The Global Attractors To The Boussinesq System For Rayleigh-Bénard Convection At Large Prandtl Number, Xiaoming Wang
Asymptotic Behavior Of The Global Attractors To The Boussinesq System For Rayleigh-Bénard Convection At Large Prandtl Number, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We study asymptotic behavior of the global attractors to the Boussinesq system for Rayleigh-Bénard convection at large Prandtl number. in particular, we show that the global attractors to the Boussinesq system for Rayleigh-Bénard convection converge to that of the infinite-Prandtl- number model for convection as the Prandtl number approaches infinity. This offers partial justification of the infinite-Prandtl-number model for convection as a valid simplified model for convection at large Prandtl number even in the long-time regime. © 2006 Wiley Periodicals, Inc.
Double Integral Calculus Of Variations On Time Scales, Gusein Sh. Guseinov, Martin Bohner
Double Integral Calculus Of Variations On Time Scales, Gusein Sh. Guseinov, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
We consider a version of the double integral calculus of variations on time scales, which includes as special cases the classical two-variable calculus of variations and the discrete two-variable calculus of variations. Necessary and sufficient conditions for a local extremum are established, among them an analogue of the Euler-Lagrange equation.
On A Theorem Of Shchepin And Repovš Concerning The Smoothness Of Compacta, Ethan Akin, William Ott
On A Theorem Of Shchepin And Repovš Concerning The Smoothness Of Compacta, Ethan Akin, William Ott
Mathematics and Statistics Faculty Research & Creative Works
We show that a theorem of Shchepin and Repovš concerning the smoothness of compacta follows from the theory of semicontinuous relations. © 2007 Elsevier B.V. All rights reserved.
Discrete Kato-Type Theorem On Inviscid Limit Of Navier-Stokes Flows, Wenfang Cheng, Xiaoming Wang
Discrete Kato-Type Theorem On Inviscid Limit Of Navier-Stokes Flows, Wenfang Cheng, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
The inviscid limit of wall bounded viscous flows is one of the unanswered central questions in theoretical fluid dynamics. Here we present a somewhat surprising result related to numerical approximation of the problem. More precisely, we show that numerical solutions of the incompressible Navier-Stokes equations converge to the exact solution of the Euler equations at vanishing viscosity and vanishing mesh size provided that small scales of the order of U in the directions tangential to the boundary are not resolved in the scheme. Here is the kinematic viscosity of the fluid and U is the typical velocity taken to be …
Note On The Emergence Of Large Scale Coherent Structure Under Small Scale Random Bombardments: The Discrete Case, Andrew Majda, Xiaoming Wang
Note On The Emergence Of Large Scale Coherent Structure Under Small Scale Random Bombardments: The Discrete Case, Andrew Majda, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We continue our study on mathematical justification of the emergence of large-scale coherent structure in a two-dimensional fluid system under small scale random bombardments. We treat the case of small-scale random bombardments at discrete times which is different from our earlier work [Commun. Pure Appl. Math. 59, 467 (2006)], where we approximated the small-scale random kicks by a continuous in time random process. in the absence of geophysical effects, the large-scale structure emerging out of the small-scale random forcing is the same as the case of continuous in time forcing that we studied before. © 2007 American Institute of Physics.
Balanced Proper Orthogonal Decomposition For Model Reduction Of Infinite Dimensional Linear Systems, John R. Singler, Belinda A. Batten
Balanced Proper Orthogonal Decomposition For Model Reduction Of Infinite Dimensional Linear Systems, John R. Singler, Belinda A. Batten
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we extend a method for reduced order model derivation for finite dimensional systems developed by Rowley to infinite dimensional systems. The method is related to standard balanced truncation, but includes aspects of the proper orthogonal decomposition in its computational approach. The method is also applicable to nonlinear systems. The method is applied to a convection diffusion equation.
Periodic Solutions Of Functional Dynamic Equations With Infinite Delay, Li Bi, Meng Fan, Martin Bohner
Periodic Solutions Of Functional Dynamic Equations With Infinite Delay, Li Bi, Meng Fan, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
In this paper, sufficient criteria are established for the existence of periodic solutions of some functional dynamic equations with infinite delays on time scales, which generalize and incorporate as special cases many known results for differential equations and for difference equations when the time scale is the set of the real numbers or the integers, respectively. The approach is mainly based on the Krasnosel'skilatin small letter i with breve fixed point theorem, which has been extensively applied in studying existence problems in differential equations and difference equations but rarely applied in studying dynamic equations on time scales. This study shows …
The Convolution On Time Scales, Gusein Sh. Guseinov, Martin Bohner
The Convolution On Time Scales, Gusein Sh. Guseinov, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
The main theme in this paper is an initial value problem containing a dynamic version of the transport equation. via this problem, the delay (or shift) of a function defined on a time scale is introduced, and the delay in turn is used to introduce the convolution of two functions defined on the time scale. In this paper, we give some elementary properties of the delay and of the convolution and we also prove the convolution theorem. Our investigation contains a study of the initial value problem under consideration as well as some results about power series on time scales. …
Trench's Perturbation Theorem For Dynamic Equations, Stevo Stevic, Martin Bohner
Trench's Perturbation Theorem For Dynamic Equations, Stevo Stevic, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
We consider a nonoscillatory second-order linear dynamic equation on a time scale together with a linear perturbation of this equation and give conditions on the perturbation that guarantee that the perturbed equation is also nonoscillatory and has solutions that behave asymptotically like a recessive and dominant solutions of the unperturbed equation. As the theory of time scales unifies continuous and discrete analysis, our results contain as special cases results for corresponding differential and difference equations by William F. Trench.
Oscillation And Nonoscillation Of Forced Second Order Dynamic Equations, Christopher C. Tisdell, Martin Bohner
Oscillation And Nonoscillation Of Forced Second Order Dynamic Equations, Christopher C. Tisdell, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
Oscillation and nonoscillation properties of second order Sturm-Liouville dynamic equations on time scales — for example, second order self-adjoint differential equations and second order Sturm-Liouville difference equations — have attracted much interest. Here we consider a given homogeneous equation and a corresponding equation with forcing term. We give new conditions implying that the latter equation inherits the oscillatory behavior of the homogeneous equation. We also give new conditions that introduce oscillation of the inhomogeneous equation while the homogeneous equation is nonoscillatory. Finally, we explain a gap in a result given in the literature for the continuous and the discrete case. …
The Dynamics And Interaction Of Quantized Vortices In The Ginzburg-Landau-Schrödinger Equation, Yanzhi Zhang, Weizhu Bao, Qiang Du
The Dynamics And Interaction Of Quantized Vortices In The Ginzburg-Landau-Schrödinger Equation, Yanzhi Zhang, Weizhu Bao, Qiang Du
Mathematics and Statistics Faculty Research & Creative Works
The dynamic laws of quantized vortex interactions in the Ginzburg-Landau-Schrödinger equation (GLSE) are analytically and numerically studied. A review of the reduced dynamic laws governing the motion of vortex centers in the GLSE is provided. The reduced dynamic laws are solved analytically for some special initial data. By directly simulating the GLSE with an efficient and accurate numerical method proposed recently in [Y. Zhang, W. Bao, and Q. Du, Numerical simulation of vortex dynamics in Ginzburg-Landau-Schrödinger equation, European J. Appl. Math., to appear], we can qualitatively and quantitatively compare quantized vortex interaction patterns of the GLSE with those from the …
On A Sub-Supersolution Method For The Prescribed Mean Curvature Problem, Vy Khoi Le
On A Sub-Supersolution Method For The Prescribed Mean Curvature Problem, Vy Khoi Le
Mathematics and Statistics Faculty Research & Creative Works
The paper is about a sub-supersolution method for the prescribed mean curvature problem. We formulate the problem as a variational inequality and propose appropriate concepts of sub- and supersolutions for such inequality. Existence and enclosure results for solutions and extremal solutions between sub- and supersolutions are established.
Differentiability With Respect To Parameters Of Weak Solutions Of Linear Parabolic Equations, John R. Singler
Differentiability With Respect To Parameters Of Weak Solutions Of Linear Parabolic Equations, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
We consider the differentiability of weak solutions of linear parabolic equations with respect to parameters and initial data. under natural assumptions, it is shown that solutions possess as much differentiability with respect to the data as do the terms appearing in the equation. The derivatives are shown to satisfy the appropriate sensitivity equations. The theoretical results are illustrated with an example.
Oscillation Criteria For A Certain Class Of Second Order Emden-Fowler Dynamic Equations, Elvan Akin, S. H. Saker, Martin Bohner
Oscillation Criteria For A Certain Class Of Second Order Emden-Fowler Dynamic Equations, Elvan Akin, S. H. Saker, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
By means of Riccati transformation techniques we establish some oscillation criteria for the second order Emden-Fowler dynamic equation on a time scale. Such equations contain the classical Emden-Fowler equation as well as their discrete counterparts. The classical oscillation results of Atkinson (in the superlinear case) and Belohorec (in the sublinear case) are extended in this paper to Emden-Fowler dynamic equations on any time scale.
Erratum: The Emergence Of A Large-Scale Coherent Structure Under Small-Scale Random Bombardments (Communications On Pure And Applied Mathematics (2006) 59:4 (467-500)), Andrew Majda, Xiaoming Wang
Erratum: The Emergence Of A Large-Scale Coherent Structure Under Small-Scale Random Bombardments (Communications On Pure And Applied Mathematics (2006) 59:4 (467-500)), Andrew Majda, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
No abstract provided.
On Self-Adjoint And J-Self-Adjoint Dirac-Type Operators: A Case Study, Stephen L. Clark, Fritz Gesztesy
On Self-Adjoint And J-Self-Adjoint Dirac-Type Operators: A Case Study, Stephen L. Clark, Fritz Gesztesy
Mathematics and Statistics Faculty Research & Creative Works
We provide a comparative treatment of some aspects of spectral theory for self-adjoint and non-self-adjoint (but J-self-adjoint) Dirac-type operators connected with the defocusing and focusing nonlinear Schrödinger equation, of relevance to nonlinear optics. In addition to a study of Dirac and Hamiltonian systems, we also introduce the concept of Weyl-Titchmarsh half-line m-coefficients (and 2 × 2 matrix-valued M-matrices) in the non-self-adjoint context and derive some of their basic properties. We conclude with an illustrative example showing that crossing spectral arcs in the non-self-adjoint context imply the blowup of the norm of spectral projections in the limit where the crossing point …
Multiple Lebesgue Integration On Time Scales, Gusein Sh. Guseinov, Martin Bohner
Multiple Lebesgue Integration On Time Scales, Gusein Sh. Guseinov, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
We study the process of multiple Lebesgue integration on time scales. The relationship of the Riemann and the Lebesgue multiple integrals is investigated.
Singular Second-Order Multipoint Dynamic Boundary Value Problems With Mixed Derivatives, Hua Luo, Martin Bohner
Singular Second-Order Multipoint Dynamic Boundary Value Problems With Mixed Derivatives, Hua Luo, Martin Bohner
Mathematics and Statistics Faculty Research & Creative Works
We study a certain singular second-order m-point boundary value problem on a time scale and establish the existence of a solution. The proof of our main result is based upon the Leray-Schauder continuation theorem.
Modeling Of Bioinspired Sensors For Flow Separation Detection For Micro Air Vehicles, Belinda A. Batten, John R. Singler, Benjamin T. Dickinson
Modeling Of Bioinspired Sensors For Flow Separation Detection For Micro Air Vehicles, Belinda A. Batten, John R. Singler, Benjamin T. Dickinson
Mathematics and Statistics Faculty Research & Creative Works
Autonomous micro air vehicle (MAV) flight faces inherent stability challenges. One challenge is controlling flow separation over the airfoil and an autonomous control system for MAV flight may be enhanced with closed loop separation control. In this work, we focus on modeling biologically inspired hair cell sensors for future flow control applications. We model the sensor output and present examples and numerical results.
Feedback Control Of Low Dimensional Models Of Transition To Turbulence, John A. Burns, John R. Singler
Feedback Control Of Low Dimensional Models Of Transition To Turbulence, John A. Burns, John R. Singler
Mathematics and Statistics Faculty Research & Creative Works
The problem of controlling or delaying transition to turbulence in shear flows has been the subject of numerous papers over the past twenty years. This period has seen the development of several low dimensional models for parallel shear flows in an attempt to explain the failure of classical linear hydrodynamic stability theory to correctly predict transition. In recent years, ideas from robust control theory have been employed to attack this problem. In this paper we use these models to develop a scenario for transition that employs both classical bifurcation theory and robust control theory. In addition, we present numerical results …
The Poweratlas: A Power And Sample Size Atlas For Microarray Experimental Design And Research, Grier P. Page, Jode W. Edwards, Gary L. Gadbury, Prashanth Yelisetti, Jelai Wang, Prinal Trivedi, David B. Allison
The Poweratlas: A Power And Sample Size Atlas For Microarray Experimental Design And Research, Grier P. Page, Jode W. Edwards, Gary L. Gadbury, Prashanth Yelisetti, Jelai Wang, Prinal Trivedi, David B. Allison
Mathematics and Statistics Faculty Research & Creative Works
Microarrays permit biologists to simultaneously measure the mRNA abundance of thousands of genes. An important issue facing investigators planning microarray experiments is how to estimate the sample size required for good statistical power. What is the projected sample size or number of replicate chips needed to address the multiple hypotheses with acceptable accuracy? Statistical methods exist for calculating power based upon a single hypothesis, using estimates of the variability in data from pilot studies. There is, however, a need for methods to estimate power and/or required sample sizes in situations where multiple hypotheses are being tested, such as in microarray …
The Emergence Of Large-Scale Coherent Structure Under Small-Scale Random Bombardments, Andrew Majda, Xiaoming Wang
The Emergence Of Large-Scale Coherent Structure Under Small-Scale Random Bombardments, Andrew Majda, Xiaoming Wang
Mathematics and Statistics Faculty Research & Creative Works
We provide mathematical justification of the emergence of large-scale coherent structure in a two-dimensional fluid system under small-scale random bombardments with small forcing and appropriate scaling assumptions. the analysis shows that the large-scale structure emerging out of the small-scale random forcing is not the one predicted by equilibrium statistical mechanics. But the error is very small, which explains earlier successful prediction of the large-scale structure based on equilibrium statistical mechanics. © 2005 Wiley Periodicals, Inc.
Dynamics Of Rotating Bose-Einstein Condensates And Its Efficient And Accurate Numerical Computation, Weizhu Bao, Qiang Du, Yanzhi Zhang
Dynamics Of Rotating Bose-Einstein Condensates And Its Efficient And Accurate Numerical Computation, Weizhu Bao, Qiang Du, Yanzhi Zhang
Mathematics and Statistics Faculty Research & Creative Works
In this paper, we study the dynamics of rotating Bose--Einstein condensates (BEC) based on the Gross--Pitaevskii equation (GPE) with an angular momentum rotation term and present an efficient and accurate algorithm for numerical simulations. We examine the conservation of the angular momentum expectation and the condensate width and analyze the dynamics of a stationary state with a shift in its center. By formulating the equation in either the two-dimensional polar coordinate system or the three-dimensional cylindrical coordinate system, the angular momentum rotation term becomes a term with constant coefficients. This allows us to develop an efficient time-splitting method which is …
The Hurwitz Zeta Function As A Convergent Series, Roman Dwilewicz, Jan Minac
The Hurwitz Zeta Function As A Convergent Series, Roman Dwilewicz, Jan Minac
Mathematics and Statistics Faculty Research & Creative Works
New series for the Hurwitz zeta function which converge on the whole plane, except s = 1, are developed. This is applied to obtain a remarkably simple evaluation of some special values of the function.
A Peano-Akô Type Theorem For Variational Inequalities, Vy Khoi Le
A Peano-Akô Type Theorem For Variational Inequalities, Vy Khoi Le
Mathematics and Statistics Faculty Research & Creative Works
We consider in this paper a Peano-Akô property of solution sets in some quasilinear elliptic variational inequalities. As consequences, variants of that property and a partial Hukuhara-Kneser theorem for inequalities are derived.
Boundedness In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef N. Raffoul
Boundedness In Functional Dynamic Equations On Time Scales, Elvan Akin, Youssef N. Raffoul
Mathematics and Statistics Faculty Research & Creative Works
Using nonnegative definite Lyapunov functionals, we prove general theorems for the boundedness of all solutions of a functional dynamic equation on time scales. We apply our obtained results to linear and nonlinear Volterra integro-dynamic equations on time scales by displaying suitable Lyapunov functionals.