Von Neumann Theory Research Articles

A Novel Numerical Approach for Solving the Newell-Whitehead Equation

Numerical solutions of Newell-Whitehead equation are investigated by collocation method in this study. Since higher order functions produce better approximations, septic B-spline basis functions is used for analysis and approximation. Error norms are calculated for the adequacy and effectiveness of the current method. Unconditional stability is proved using Von-Neumann theory. The numerical results are obtained and the comparisons are presented in the tables. Additionally, simulations of all numerical results are plotted to show the numerical behavior of the solution. Numerical results make the method more convenient and systematically handle the nonlinear solution process. The numerical solutions found make the method attractive and reliable for the solution of Fitzhugh-Nagumo type equations.

Lévy-Leblond equation and Eisenhart–Duval lift in Koopman–von Neumann mechanics

The Koopman–von Neumann (KvN) mechanics is an approach that was formulated long ago to answer the question regarding the existence of a Hilbert space representation of classical mechanics. KvN mechanics is a non-relativistic theory, and it is not clear how spin can be included in it, since spin is widely regarded as a relativistic property. Cabrera et al., in Eur. Phys. J. Spec. Top. 227, 2195 (2019) argued that the Spohn equation [Spohn, Ann. Phys. 282, 420 (2000)] is the correct classical framework for the Koopman–von Neumann theory corresponding to the Dirac equation. However, after Lévy-Leblond’s seminal work on this topic, it became clear that spin naturally arises also from the Galilean invariant wave equations, without any need of relativistic considerations. Inspired by this, we propose another possibility of including spin in the KvN formalism: the Lévy-Leblond equation in the Koopman–von Neumann formalism can be obtained as a null reduction of the massless Dirac equation in the Eisenhart–Duval lift metric. To illustrate the idea, we implement it for a one-dimensional classical system without magnetic interactions.

Analytical model for curved-shock Mach reflection

Mach reflection (MR) is an essential component in the development of the shock theory, as the incident shock curvature is found to have a significant effect on the MR patterns. Curved-shock Mach reflection (CMR) is not yet adequately understood due to the rotational complexity behind curved shocks. Here, CMR in steady, planar/axisymmetric flows is analyzed to supplement the well-studied phenomena caused by oblique-shock Mach reflection (OMR). The solution from the von Neumann's three-shock theory does not fully describe the CMR case. A CMR structure is presented and characterized by an incident shock, reflected shock, Mach stem, and expansion/compression waves over the slipline or occasionally an absence of waves due to pressure equilibrium. On the basis of this CMR structure, an analytical model for predicting the Mach stem in the CMR case is established. The model reduces to the OMR case if the shock curvature is not applicable. Predictions of the Mach stem geometry and shock structure based on the model exhibit better agreement with the numerical results than predictions using previous models. It is found that the circumferential shock curvature plays a key role in the axisymmetric doubly curved CMR case, which results in a different outcome from the planar case.

Operational classical mechanics: holonomic systems

We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.

A collocation algorithm based on septic B-splines and bifurcation of traveling waves for Sawada–Kotera equation

In present work, a mathematical model standing for the numerical solutions of the nonlinear Sawada–Kotera (S–K) equation has been examined. For this purpose, we have used collocation finite element method for the model problem. The solution for each unknown is written as a linear combination of time parameters and combination of the septic B-splines. Next, applying Von-Neumann theory, we demonstrate that the scheme is marginally stable. Error norms L2 and L∞ are computed for single soliton solutions to display the practicality and robustness the suggested scheme. The obtained numerical results have been illustrated with tables. Furthermore, to show the numerical behavior of the single soliton, some of the simulations are depicted in 2D and 3D. Present results show that the method procures high accurate solutions so present scheme will be useful for the other nonlinear scientific problems. After that, bifurcation behavior of traveling waves for the S-K equation has been examined depending on the velocity (v) of the traveling wave. It has been seen that velocity (v) of the traveling wave plays a key role for the existence of different kind of quasiperiodic motions of the nonlinear S-K equation. Coexisting quasiperiodic motions for traveling wave solutions of the equation have been presented.

Joint quantum–classical Hamilton variational principle in the phase space

We show that the dynamics of a closed quantum system obeys the Hamilton variational principle. Even though quantum particles lack well-defined trajectories, their evolution in the Husimi representation can be treated as a flow of multidimensional probability fluid in the phase space. By introducing the classical counterpart of the Husimi representation in a close analogy to the Koopman–von Neumann theory, one can largely unify the formulations of classical and quantum dynamics. We prove that the motions of elementary parcels of both classical and quantum Husimi fluid obey the Hamilton variational principle, and the differences between associated action functionals stem from the differences between classical and quantum pure states. The Husimi action functionals are not unique and defined up to the Skodje flux gauge fixing (Skodje et al 1989 Phys. Rev. A 40 2894). We demonstrate that the gauge choice can dramatically alter flux trajectories. Applications of the presented theory for constructing semiclassical approximations and hybrid classical–quantum theories are discussed.

A NOVEL SCHEME BASED ON COLLOCATION FINITE ELEMENT METHOD TO GENERALISED OSKOLKOV EQUATION

This article is concerned with designing numerical schemes for the generalised Oskolkov equation using the quintic B-spline collocation finite element method. Applying the von-Neumann theory, it is shown that the proposed method is marginally unconditionally stable. It was obtained the theoretical bound of the error in the full discrete scheme for the first time in the literature. The accuracy and effectiveness of the method checked with three model problems, consisting of a single solitary wave, Gaussian initial condition and growth of an undular bore. The performance of the new method is demonstrated by calculating invariant I and error norms L2 and L∞. Results are displayed both numerically and graphically. Numerical experiments support the correctness and robustness of the method which can be further used for solving such problems.

Theoretical and computational structures on solitary wave solutions of Benjamin Bona Mahony-Burgers equation

This paper aims to obtain exact and numerical solutions of the nonlinear Benjamin Bona Mahony-Burgers (BBM-Burgers) equation. Here, we propose the modified Kudryashov method for getting the exact traveling wave solutions of BBM-Burgers equation and a septic B-spline collocation finite element method for numerical investigations. The numerical method is validated by studying solitary wave motion. Linear stability analysis of the numerical scheme is done with Fourier method based on von-Neumann theory. To show suitability and robustness of the new numerical algorithm, error norms $L_{2}$, $L_{\infty }$ and three invariants $I_{1},I_{2}$ and $I_{3}$ are calculated and obtained results are given both numerically and graphically. The obtained results state that our exact and numerical schemes ensure evident and they are penetrative mathematical instruments for solving nonlinear evolution equation.

Self-adjoint extensions of bipartite Hamiltonians

AbstractWe compute the deficiency spaces of operators of the form $H_A{\hat {\otimes }} I + I{\hat {\otimes }} H_B$, for symmetric $H_A$ and self-adjoint $H_B$. This enables us to construct self-adjoint extensions (if they exist) by means of von Neumann's theory. The structure of the deficiency spaces for this case was asserted already in Ibort et al. [Boundary dynamics driven entanglement, J. Phys. A: Math. Theor.47(38) (2014) 385301], but only proven under the restriction of $H_B$ having discrete, non-degenerate spectrum.

“Optimize user experience”: optimization techniques and the simulation of life, from the model to the algorithm

ABSTRACT This article takes up the issue of optimization to consider the relationship between predictive algorithms and platform user experience. Corporate data analytic practices increasingly rely on machine learning algorithms that apply models to user behaviors, producing “knowledge” about users that can be bought and sold. This article considers the opacity of algorithms today in relation to optimization. Using a conceptual apparatus that draws from the study of cultural techniques, the following argues that optimization—the task of finding a sufficient solution to a well-defined problem—makes use of models to simulate possible answers to problems around the incomputablity of behavior. Tracing a set of examples that deal with the problem of predicting behavior—the “minimum point” problem, John von Neumann's automata theory, and the Facebook pixel—optimization is characterized by a shift from statistical model making towards predictive and algorithmic techniques. This shift is seen within the context of the decline of Cold War rationality towards the embeddedness of “intelligent” algorithms across technoculture.

A novel implementation of Petrov-Galerkin method to shallow water solitary wave pattern and superperiodic traveling wave and its multistability: Generalized Korteweg-de Vries equation

This work deals with the constitute of numerical solutions of the generalized Korteweg-de Vries (GKdV) equation with Petrov-Galerkin finite element approach utilising a cubic B-spline function as the trial function and a quadratic function as the test function. Accurateness and effectiveness of the submitted methods are shown by employing propagation of single solitary wave. The L2,L∞error norms and I1,I2and I3invariants are used to validate the applicability and durability of our numerical algorithm. Implementing the Von-Neumann theory, it is manifested that the suggested method is marginally stable. Furthermore, supernonlinear traveling wave solution of the GKdV equation is presented using phase plots. It is seen that the GKdV equation supports superperiodic traveling wave solution only and it is significantly affected by velocity and nonlinear parameters. Also, considering a superficial periodic forcing multistability of traveling waves of perturbed GKdV equation is presented. It is found that the perturbed GKdV equation supports coexisting chaotic and various quasiperiodic features with same parametric values at different initial conditions.

POLYNOMIAL AND RATIONAL WAVE SOLUTIONS OF KUDRYASHOV-SINELSHCHIKOV EQUATION AND NUMERICAL SIMULATIONS FOR ITS DYNAMIC MOTIONS

Polynomial and rational wave solutions of Kudryashov-Sinelshchikov equation and numerical simulations for its dynamic motions are investigated. Conservation flows of the dynamic motion are obtained utilizing multiplier approach. Using the unified method, a collection of exact solitary and soliton solutions of Kudryashov-Sinelshchikov equation is presented. Collocation finite element method based on quintic B-spline functions is implemented to the equation to evidence the accuracy of the proposed method by test problems. Stability analysis of the numerical scheme is studied by employing von Neumann theory. The obtained analytical and numerical results are in good agreement.

Hydrogenoid Spectra with Central Perturbations

Through the Kre\u{\i}n-Vi\v{s}ik-Birman extension scheme, unlike the classical analysis based on von Neumann's theory, we reproduce the construction and classification of all self-adjoint realisations of three-dimensional hydrogenoid-like Hamiltonians with singular perturbation supported at the Coulomb centre (the nucleus), as well as of Schr\"{o}dinger operators with Coulomb potentials on the half-line. These two problems are technically equivalent, albeit sometimes treated by their own in the the literature. Based on such scheme, we then recover the formula to determine the eigenvalues of each self-adjoint extension, as corrections of the non-relativistic hydrogenoid energy levels. We discuss in which respect the Kre\u{\i}n-Vi\v{s}ik-Birman scheme is somewhat more natural in yielding the typical boundary condition of self-adjointness at the centre of the perturbation.

Positive Self-adjoint Operator Extensions with Applications to Differential Operators

In this paper we consider extensions of positive operators. We study the connections between the von Neumann theory of extensions and characterisations of positive extensions via decompositions of the domain of the associated form. We apply the results to elliptic second order differential operators and look in particular at examples of the Laplacian on a disc and the Aharonov–Bohm operator.

Numerical experiments for long nonlinear internal waves via Gardner equation with dual-power law nonlinearity

This paper studies Gardner equation, which represents long nonlinear internal waves. The collocation method based on B-splines is applied to the equation. The stability of the proposed numerical scheme is analyzed by using von Neumann theory. To observe some physical properties of long nonlinear internal waves, three test problems which contain the propagation of solitary waves, the interaction of solitary waves and evolution of solitons are considered. Also, the effect of nonlinearity on physical problems is investigated. In order to see this effect clearly, the same parameters are used during the computation for different degrees of nonlinearity in each problem.

Koopman wavefunctions and classical-quantum correlation dynamics.

Upon revisiting the Hamiltonian structure of classical wavefunctions in Koopman–von Neumann theory, this paper addresses the long-standing problem of formulating a dynamical theory of classical–quantum coupling. The proposed model not only describes the influence of a classical system onto a quantum one, but also the reverse effect—the quantum backreaction. These interactions are described by a new Hamiltonian wave equation overcoming shortcomings of currently employed models. For example, the density matrix of the quantum subsystem is always positive definite. While the Liouville density of the classical subsystem is generally allowed to be unsigned, its sign is shown to be preserved in time for a specific infinite family of hybrid classical–quantum systems. The proposed description is illustrated and compared with previous theories using the exactly solvable model of a degenerate two-level quantum system coupled to a classical harmonic oscillator.

Combination of WENO and Explicit Runge–Kutta Methods for Wind Transport in the Meso-NH Model

AbstractThis paper investigates the use of the weighted essentially nonoscillatory (WENO) space discretization methods of third and fifth order for momentum transport in the Meso-NH meteorological model, and their association with explicit Runge–Kutta (ERK) methods, with the specific purpose of finding an optimal combination in terms of wall-clock time to solution. A linear stability analysis using von Neumann theory is first conducted that considers six different ERK time integration methods. A new graphical representation of linear stability is proposed, which allows a first discrimination between the ERK methods. The theoretical analysis is then completed by tests on numerical problems of increasing complexity (linear advection of high wind gradient, orographic waves, density current, large eddy simulation of fog, and windstorm simulation), using a fourth-order-centered scheme as a reference basis. The five-stage third-order and fourth-order ERK combinations appear as the time integration methods of choice for coupling with WENO schemes in terms of stability. An explicit time-splitting method added to the ERK temporal scheme for WENO improves the stability properties slightly more. When the spatial discretizations are compared, WENO schemes present the main advantage of maintaining stable, nonoscillatory transitions with sharp discontinuities, but WENO third order is excessively damping, while WENO fifth order provides better accuracy. Finally, WENO fifth order combined with the ERK method makes the whole physics of the model 3 times faster compared to the classical fourth-order centered scheme associated with the leapfrog temporal scheme.

Scattering and Bound States of a Spin-1/2 Neutral Particle in the Cosmic String Spacetime

In this paper the relativistic quantum dynamics of a spin-1/2 neutral particle with a magnetic moment μ in the cosmic string spacetime is reexamined by applying the von Neumann theory of self-adjoint extensions. Contrary to previous studies where the interaction between the spin and the line of charge is neglected, here we consider its effects. This interaction gives rise to a point interaction: ∇·E=(2λ/α)δ(r)/r. Due to the presence of the Dirac delta function, by applying an appropriated boundary condition provided by the theory of self-adjoint extensions, irregular solutions for the Hamiltonian are allowed. We address the scattering problem obtaining the phase shift, S-matrix, and the scattering amplitude. The scattering amplitude obtained shows a dependency with energy which stems from the fact that the helicity is not conserved in this system. Examining the poles of the S-matrix we obtain an expression for the bound states. The presence of bound states for this system has not been discussed before in the literature.

Semantic processing of English sentences using statistical computation based on neurophysiological models.

Computer programs that can accurately interpret natural human language and carry out instructions would improve the lives of people with language processing deficits and greatly benefit society in general. von Neumann in theorized that the human brain utilizes its own unique statistical neuronal computation to decode language and that this produces specific patterns of neuronal activity. This paper extends von Neumann's theory to the processing of partial semantics of declarative sentences. I developed semantic neuronal network models that emulate key features of cortical language processing and accurately compute partial semantics of English sentences. The method of computation implements the MAYA Semantic Technique, a mathematical technique I previously developed to determine partial semantics of sentences within a natural language processing program. Here I further simplified the technique by grouping repeating patterns into fewer categories. Unlike other natural language programs, my approach computes three partial semantics. The results of this research show that the computation of partial semantics of a sentence uses both feedforward and feedback projection which suggest that the partial semantic presented in this research might be a conscious activity within the human brain.

One world, one beable

Is the quantum state part of the furniture of the world? Einstein found such a position indigestible, but here I present a different understanding of the wavefunction that is easy to stomach. First, I develop the idea that the wavefunction is nomological in nature, showing how the quantum It or Bit debate gets subsumed by the corresponding It or Bit debate about laws of nature. Second, I motivate the nomological view by casting quantum mechanics in a “classical” formalism (Hamilton–Jacobi theory) and classical mechanics in a “quantum” formalism (Koopman–von Neumann theory) and then comparing and contrasting classical and quantum wave functions. I argue that Humeans about laws can treat classical and quantum wave functions on a par and that doing so yields many benefits.