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Articles 631 - 655 of 655
Full-Text Articles in Mathematics
Locality And Stability Of The Cascades Of Two-Dimensional Turbulence, Eleftherios Gkioulekas
Locality And Stability Of The Cascades Of Two-Dimensional Turbulence, Eleftherios Gkioulekas
School of Mathematical & Statistical Sciences Faculty Publications
We investigate and clarify the notion of locality as it pertains to the cascades of two-dimensional turbulence. The mathematical framework underlying our analysis is the infinite system of balance equations that govern the generalized unfused structure functions, first introduced by L’vov and Procaccia. As a point of departure we use a revised version of the system of hypotheses that was proposed by Frisch for three-dimensional turbulence. We show that both the enstrophy cascade and the inverse energy cascade are local in the sense of nonperturbative statistical locality. We also investigate the stability conditions for both cascades. We have shown that …
Some Applications Of Dirac's Delta Function In Statistics For More Than One Random Variable, Santanu Chakraborty
Some Applications Of Dirac's Delta Function In Statistics For More Than One Random Variable, Santanu Chakraborty
School of Mathematical & Statistical Sciences Faculty Publications
In this paper, we discuss some interesting applications of Dirac's delta function in Statistics. We have tried to extend some of the existing results to the more than one variable case. While doing that, we particularly concentrate on the bivariate case.
Winterberg’S Conjectured Breaking Of The Superluminal Quantum Correlations Over Large Distances, Eleftherios Gkioulekas
Winterberg’S Conjectured Breaking Of The Superluminal Quantum Correlations Over Large Distances, Eleftherios Gkioulekas
School of Mathematical & Statistical Sciences Faculty Publications
We elaborate further on a hypothesis by Winterberg that turbulent fluctuations of the zero point field may lead to a breakdown of the superluminal quantum correlations over very large distances. A phenomenological model that was proposed by Winterberg to estimate the transition scale of the conjectured breakdown, does not lead to a distance that is large enough to be agreeable with recent experiments. We consider, but rule out, the possibility of a steeper slope in the energy spectrum of the turbulent fluctuations, due to compressibility, as a possible mechanism that may lead to an increased lower-bound for the transition scale. …
A Fragment On Euler's Constant In Ramanujan's Lost Notebook, Bruce C. Berndt, Timothy Huber
A Fragment On Euler's Constant In Ramanujan's Lost Notebook, Bruce C. Berndt, Timothy Huber
School of Mathematical & Statistical Sciences Faculty Publications
A formula for Euler’s constant found in Ramanujan’s lost notebook and also in a problem he submitted to the Journal of the Indian Mathematical Society is proved and discussed.
An Action For A Classical String, The Equation Of Motion And Group Invariant Classical Solutions, Paul Bracken
An Action For A Classical String, The Equation Of Motion And Group Invariant Classical Solutions, Paul Bracken
School of Mathematical & Statistical Sciences Faculty Publications
A string action which is essentially a Willmore functional is presented and studied. This action determines the physics of a surface in Euclidean three space which can be used to model classical string configurations. By varying this action an equation of motion for the mean curvature of the surface is obtained which is shown to govern certain classical string configurations. Several classes of classical solutions for this equation are discussed from the symmetry group point of view and an application is presented.
Quantum Phases For A Generalized Harmonic Oscillator, Paul Bracken
Quantum Phases For A Generalized Harmonic Oscillator, Paul Bracken
School of Mathematical & Statistical Sciences Faculty Publications
An effective Hamiltonian for the generalized harmonic oscillator is determined by using squeezed state wavefunctions. The equations of motion over an extended phase space are determined and then solved perturbatively for a specific choice of the oscillator parameters. These results are used to calculate the dynamic and geometric phases for the generalized oscillator with this choice of parameters.
On Mathematical Modeling, Nonlinear Properties And Stability Of Secondary Flow In A Dendrite Layer, Daniel N. Riahi
On Mathematical Modeling, Nonlinear Properties And Stability Of Secondary Flow In A Dendrite Layer, Daniel N. Riahi
School of Mathematical & Statistical Sciences Faculty Publications
This paper studies instabilities in the flow of melt within a horizontal dendrite layer with deformed upper boundary and in the presence or absence of rotation during the solidification of a binary alloy. In the presence of rotation, it is assumed that the layer is rotating about a vertical axis at a constant angular velocity. Linear and weakly nonlinear stability analyses provide results about various flow features such as the critical mode of convection, neutral stability curve, preferred flow pattern and the solid fraction distribution within the dendrite layer. The preferred shape of the deformed upper boundary of the layer, …
Is The Subdominant Part Of The Energy Spectrum Due To Downscale Energy Cascade Hidden In Quasi-Geostrophic Turbulence?, Eleftherios Gkioulekas, Ka Kit Tung
Is The Subdominant Part Of The Energy Spectrum Due To Downscale Energy Cascade Hidden In Quasi-Geostrophic Turbulence?, Eleftherios Gkioulekas, Ka Kit Tung
School of Mathematical & Statistical Sciences Faculty Publications
In systems governing two-dimensional turbulence, surface quasi-geostrophic turbulence, (more generally $\alpha$-turbulence), two-layer quasi-geostrophic turbulence, etc., there often exist two conservative quadratic quantities, one "energy''-like and one "enstrophy''-like. In a finite inertial range there are in general two spectral fluxes, one associated with each conserved quantity. We derive here an inequality comparing the relative magnitudes of the "energy'' and "enstrophy'' fluxes for finite or infinitesimal dissipations, and for hyper or hypo viscosities. When this inequality is satisfied, as is the case of 2D turbulence,where the energy flux contribution to the energy spectrum is small, the subdominant part will be effectively hidden. …
New Integrable Hierarchy, Its Parametric Solutions, Cuspons, One-Peak Solitons, And M/W-Shape Peak Solitons, Zhijun Qiao
New Integrable Hierarchy, Its Parametric Solutions, Cuspons, One-Peak Solitons, And M/W-Shape Peak Solitons, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
In this paper, we propose a new completely integrable hierarchy. Particularly in the hierarchy we draw two new soliton equations: 1 mt= 1 2 1/m2xxx− 1 2 1/m2 x; 2 mt+mx u2−ux 2+2m2ux=0, m=u−uxx. The first one is the second positive member in the hierarchy while the second one is the second negative member in the hierarchy. Both equations can be derived from the two-dimensional Euler equation by using the approximation procedure. All equations in the hierarchy are proven to have bi-Hamiltonian operators and Lax pairs through solving a crucial matrix equation. Moreover, we develop parametric solutions of the entire …
On The Elimination Of The Sweeping Interactions From Theories Of Hydrodynamic Turbulence, Eleftherios Gkioulekas
On The Elimination Of The Sweeping Interactions From Theories Of Hydrodynamic Turbulence, Eleftherios Gkioulekas
School of Mathematical & Statistical Sciences Faculty Publications
In this paper, we revisit the claim that the Eulerian and quasi-Lagrangian same time correlation tensors are equal. This statement allows us to transform the results of an MSR quasi-Lagrangian statistical theory of hydrodynamic turbulence back to the Eulerian representation. We define a hierarchy of homogeneity symmetries between incremental homogeneity and global homogeneity. It is shown that both the elimination of the sweeping interactions and the derivation of the 4/5-law require a homogeneity assumption stronger than incremental homogeneity but weaker than global homogeneity. The quasi-Lagrangian transformation, on the other hand, requires an even stronger homogeneity assumption which is many-time rather …
Solving Ramanujan's Differential Equations For Eisenstein Series Via A First Order Riccati Equation, James M. Hill, Bruce C. Berndt, Timothy Huber
Solving Ramanujan's Differential Equations For Eisenstein Series Via A First Order Riccati Equation, James M. Hill, Bruce C. Berndt, Timothy Huber
School of Mathematical & Statistical Sciences Faculty Publications
In this paper we prove that Ramanujan's differential equations for the Eisenstein series P, Q, and R are invariant under a simple one-parameter stretching group of transformations. Using this, we show that the three differential equations may be reduced to a first order Riccati differential equation, the solution of which may be represented in terms of hypergeometric functions. The resulting formulas allow for the derivation of parametric representations of P, Q, and R, analogous to representations in Ramanujan's second notebook. In contrast, in the classical approach, one first needs to derive the fundamental formula connecting theta functions with elliptic integrals. …
A New Integrable Equation With Cuspons And W/M-Shape-Peaks Solitons, Zhijun Qiao
A New Integrable Equation With Cuspons And W/M-Shape-Peaks Solitons, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
In this paper, we propose a new completely integrable wave equation: mt+mx u2 −ux 2 +2m2ux=0, m=u−uxx. The equation is derived from the two dimensional Euler equation and is proven to have Lax pair and bi-Hamiltonian structures. This equation possesses new cusp solitons—cuspons, instead of regular peakons ce− −ct with speed c. Through investigating the equation, we develop a new kind of soliton solutions—“W/M”-shape-peaks solitons. There exist no smooth solitons for this integrable water wave equation.
Jointly Hyponormal Pairs Of Commuting Subnormal Operators Need Not Be Jointly Subnormal, Raúl E. Curto, Jasang Yoon
Jointly Hyponormal Pairs Of Commuting Subnormal Operators Need Not Be Jointly Subnormal, Raúl E. Curto, Jasang Yoon
School of Mathematical & Statistical Sciences Faculty Publications
We construct three different families of commuting pairs of subnormal operators, jointly hyponormal but not admitting commuting normal extensions. Each such family can be used to answer in the negative a 1988 conjecture of R. Curto, P. Muhly and J. Xia. We also obtain a sufficient condition under which joint hyponormality does imply joint subnormality.
Recent Developments In Understanding Two-Dimensional Turbulence And The Nastrom-Gage Spectrum, Eleftherios Gkioulekas, Ka Kit Tung
Recent Developments In Understanding Two-Dimensional Turbulence And The Nastrom-Gage Spectrum, Eleftherios Gkioulekas, Ka Kit Tung
School of Mathematical & Statistical Sciences Faculty Publications
Two-dimensional turbulence appears to be a more formidable problem than three-dimensional turbulence despite the numerical advantage of working with one less dimension. In the present paper we review recent numerical investigations of the phenomenology of two-dimensional turbulence as well as recent theoretical breakthroughs by various leading researchers. We also review efforts to reconcile the observed energy spectrum of the atmosphere (the spectrum) with the predictions of two-dimensional turbulence and quasigeostrophic turbulence.
Disintegration-Of-Measure Techniques For Commuting Multivariable Weighted Shifts, Raúl E. Curto, Jasang Yoon
Disintegration-Of-Measure Techniques For Commuting Multivariable Weighted Shifts, Raúl E. Curto, Jasang Yoon
School of Mathematical & Statistical Sciences Faculty Publications
We employ techniques from the theory of disintegration of measures to study the Lifting Problem for commuting $n$-tuples of subnormal weighted shifts. We obtain a new necessary condition for the existence of a lifting, and generate new pathology associated with bringing together the Berger measures associated to each individual weighted shift. For subnormal $2$-variable weighted shifts, we then find the precise relation between the Berger measure of the pair and the Berger measures of the shifts associated to horizontal rows and vertical columns of weights.
K-Hyponormality Of Multivariable Weighted Shifts, Raúl E. Curto, Sang Hoon Lee, Jasang Yoon
K-Hyponormality Of Multivariable Weighted Shifts, Raúl E. Curto, Sang Hoon Lee, Jasang Yoon
School of Mathematical & Statistical Sciences Faculty Publications
We characterize joint k-hyponormality for 2-variable weighted shifts. Using this characterization we construct a family of examples which establishes and illustrates the gap between k-hyponormality and (k+1) -hyponormality for each k ≥ 1. As a consequence, we obtain an abstract solution to the Lifting Problem for Commuting Subnormals.
Recent Applications Of Fractional Calculus To Science And Engineering, Lokenath Debnath
Recent Applications Of Fractional Calculus To Science And Engineering, Lokenath Debnath
School of Mathematical & Statistical Sciences Faculty Publications
This paper deals with recent applications of fractional calculus to dynamical systems in control theory, electrical circuits with fractance, generalized voltage divider, viscoelasticity, fractional-order multipoles in electromagnetism, electrochemistry, tracer in fluid flows, and model of neurons in biology. Special attention is given to numerical computation of fractional derivatives and integrals.
Category Of Nonlinear Evolution Equations, Algebraic Structure, And R-Matrix, Zhijun Qiao, Cewen Cao, Walter Strampp
Category Of Nonlinear Evolution Equations, Algebraic Structure, And R-Matrix, Zhijun Qiao, Cewen Cao, Walter Strampp
School of Mathematical & Statistical Sciences Faculty Publications
In this paper we deal with the category of nonlinear evolution equations ~NLEEs! associated with the spectral problem and provide an approach for constructing their algebraic structure and r-matrix. First we introduce the category of NLEEs, which is composed of various positive order and negative order hierarchies of NLEEs both integrable and nonintegrable. The whole category of NLEEs possesses a generalized Lax representation. Next, we present two different Lie algebraic structures of the Lax operator: one of them is universal in the category, i.e., independent of the hierarchy, while the other one is nonuniversal in the hierarchy, i.e., dependent on …
The Weierstrass–Enneper System For Constant Mean Curvature Surfaces And The Completely Integrable Sigma Model, Paul Bracken, A. M. Grundland, L. Martina
The Weierstrass–Enneper System For Constant Mean Curvature Surfaces And The Completely Integrable Sigma Model, Paul Bracken, A. M. Grundland, L. Martina
School of Mathematical & Statistical Sciences Faculty Publications
The integrability of a system which describes constant mean curvature surfaces by means of the adapted Weierstrass–Enneper inducing formula is studied. This is carried out by using a specific transformation which reduces the initial system to the completely integrable two-dimensional Euclidean nonlinear sigma model. Through the use of the apparatus of differential forms and Cartan theory of systems in involution, it is demonstrated that the general analytic solutions of both systems possess the same degree of freedom. Furthermore, a new linear spectral problem equivalent to the initial Weierstrass–Enneper system is derived via the method of differential constraints. A new procedure …
On Different Integrable Systems Sharing The Same Nondynamical R-Matrix, Zhijun Qiao, Walter Strampp
On Different Integrable Systems Sharing The Same Nondynamical R-Matrix, Zhijun Qiao, Walter Strampp
School of Mathematical & Statistical Sciences Faculty Publications
In a recent paper @Zhijun Qiao and Ruguang Zhou, Phys. Lett. A 235, 35 ~1997!#, the amazing fact was reported that a discrete and a continuous integrable system share the same r-matrix with the interesting property of being nondynamical. Now, we present three further pairs of different continuous integrable systems sharing the same r-matrix again being nondynamical. The first pair is the finite-dimensional constrained system ~FDCS! of the famous AKNS hierarchy and the Dirac hierarchy; the second pair is the FDCS of the well-known geodesic flows on the ellipsoid and the Heisenberg spin chain hierarchy; and the third pair is …
Effective Characteristic Polynomials And Two-Point Pade Approximants As Summation Techniques For The Strongly Divergent Perturbation Expansions Of The Ground State Energies Of Anharmonic Oscillators, Jiri Cizek, Ernst Joachim Weniger, Paul Bracken, Vladimir Spirko
Effective Characteristic Polynomials And Two-Point Pade Approximants As Summation Techniques For The Strongly Divergent Perturbation Expansions Of The Ground State Energies Of Anharmonic Oscillators, Jiri Cizek, Ernst Joachim Weniger, Paul Bracken, Vladimir Spirko
School of Mathematical & Statistical Sciences Faculty Publications
Pade approximants are able to sum effectively the Rayleigh-Schrodinger perturbation series for the ground state energy of the quartic anharmonic oscillator, as well as the corresponding renormalized perturbation expansion [E.J. Weniger, J. Cizek, and F. Vinette, J. Math. Phys. 34, 571 (1993)]. In the sextic case, Pade approximants are still able to sum these perturbation series, but convergence is so slow that they are computationally useless. In the octic case, Pade approximants are not powerful enough and fail. On the other hand, the inclusion of only a few additional data from the strong coupling domain [E.J. Weniger, Ann. Phys. (N.Y.) …
A Completely Integrable System And Parametric Representation Of Solutions Of The Wadati-Konno-Ichikawa Hierarchy, Zhijun Qiao
A Completely Integrable System And Parametric Representation Of Solutions Of The Wadati-Konno-Ichikawa Hierarchy, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
A finite-dimensional involutive system is presented, and the Wadati-Konno- Ichikawa (WKI) hierarchy of nonlinear evolution equations and their commutator representations are discussed in this article. By this finite-dimensional involutive system, it is proven that under the so-called Bargmann constraint between the potentials and the eigenfunctions, the eigenvalue problem (called the WKI eigenvalue problem) studied by Wadati, Konno, and Ichikawa [J. Phys. Sot. Jpn. 47, 1698 (1979)] is nonlinearized as a completely integrable Hamiltonian system in the Liouville sense. Moreover, the parametric representation of the solution of each equation in the WKI hierarchy is obtained by making use of the solution …
A Hierarchy Of Nonlinear Evolution Equations And Finite-Dimensional Involutive Systems, Zhijun Qiao
A Hierarchy Of Nonlinear Evolution Equations And Finite-Dimensional Involutive Systems, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
A spectral problem and an associated hierarchy of nonlinear evolution equations are presented in this article. In particular, the reductions of the two representative equations in this hierarchy are given: one is the nonlinear evolution equation rl= - ar,- 2icu/3] r2] r which looks like the nonlinear Schrijdinger equation, the other is the generalized derivative nonlinear Schrijdinger equation rt= $ar,,- ialr12r- a/3(lr12r),- a j3I r I 2r,-2iap21r14r which is just a combination of the nonlinear Schrijdinger equation and two different derivative nonlinear Schrodinger equations [D. J. Kaup and A. C. Newell, J. Math. Phys. 19, 789 (1978); M. J. Ablowitz, …
An Involutive System And Integrable C. Neumann System Associated With The Modified Korteweg-De Vries Hierarchy, Zhijun Qiao
An Involutive System And Integrable C. Neumann System Associated With The Modified Korteweg-De Vries Hierarchy, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
In this article, a system of finite-dimensional involutive functions is presented and proven to be integrable in the Liouville sense. By using the nonlinearization method, the C. Neumann system associated with the modified Korteweg-de Vries (mKdV) hierarchy is obtained. Thus, the C. Neumann system is shown to be completely integrable via a gauge transformation between it and an integrable Hamiltonian system. Finally, the solution of a stationary mKdV equation and the involutive solutions of the mKdV hierarchy are secured. As two examples, the involutive solutions are given for the mKdV equation: u,+ ;uXXX- $u2u,=0 and the 5th mKdV equation v,- …
A New Completely Integrable Liouville's System Produced By The Kaup-Newell Elgenvalue Problem, Zhijun Qiao
A New Completely Integrable Liouville's System Produced By The Kaup-Newell Elgenvalue Problem, Zhijun Qiao
School of Mathematical & Statistical Sciences Faculty Publications
Under the constraint between the potentials and eigenfunctions, the Kaup–Newell eigenvalue problem is nonlinearized as a new completely integrable Hamiltonian system (R2N,dpΛdq,H): H=i〈Λ2p,q〉+1/2〈Λq,q〉〈Λp,p〉. Furthermore, the involutive solution of the high‐order Kaup–Newell equation is obtained. Specifically, the involutive solution of the well‐known derivative Schrödinger equation ut=1/2iuxx+1/2(u‖u‖2)x is developed.