Von Neumann Type Research Articles

Asymptotic integration of a certain second-order linear delay differential equation

We construct the asymptotic formulas for solutions of a certain linear second-order delay differential equation as independent variable tends to infinity. When the delay equals zero this equation turns into the so-called one-dimensional Schrodinger equation at energy zero with Wigner–von Neumann type potential. The question of interest is how the behaviour of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients.

Distribution of zeros in the rough geometry of fluctuating interfaces.

We study numerically the correlations and the distribution of intervals between successive zeros in the fluctuating geometry of stochastic interfaces, described by the Edwards-Wilkinson equation. For equilibrium states we find that the distribution of interval lengths satisfies a truncated Sparre-Andersen theorem. We show that boundary-dependent finite-size effects induce nontrivial correlations, implying that the independent interval property is not exactly satisfied in finite systems. For out-of-equilibrium nonstationary states we derive the scaling law describing the temporal evolution of the density of zeros starting from an uncorrelated initial condition. As a by-product we derive a general criterion of the von Neumann's type to understand how discretization affects the stability of the numerical integration of stochastic interfaces. We consider both diffusive and spatially fractional dynamics. Our results provide an alternative experimental method for extracting universal information of fluctuating interfaces such as domain walls in thin ferromagnets or ferroelectrics, based exclusively on the detection of crossing points.

Asymptotic Integration of a Certain Second-Order Linear Delay Differential Equation

We construct some asymptotic formulas for solutions of a certain linear second-order delay differential equation when the independent variable tends to infinity. Two features concerning the considered equation should be emphasized. First, the coefficient of this equation has an oscillatory decreasing form. Second, when the delay equals zero, this equation turns into the so-called one-dimensional Schr¨odinger equation at energy zero with Wigner–von Neumann type potential. Dynamics of the latter is well-known. The question of interest is how the behavior of solutions changes qualitatively and quantitatively when the delay is introduced in this dynamical model. This equation also attracts interest from the standpoint of the theory of oscillations of solutions of functional differential equations. We apply the method of asymptotic integration that is based on the ideas of the centre manifold theory in its presentation with respect to the systems of functional differential equations with oscillatory decreasing coefficients. The essence of the method is to construct a so-called critical manifold in the phase space of the considered dynamical system. This manifold is attractive and positively invariant, and, therefore, the dynamics of all solutions of the initial equation is determined by the dynamics of the solutions lying on the critical manifold. The system that describes the dynamics of the solutions lying on the critical manifold is a linear system of two ordinary differential equations. To construct the asymptotics for solutions of this system, we use the averaging changes of variables and transformations that diagonalize variable matrices. We reduce the system on the critical manifold to what is called the L-diagonal form. The asymptotics of the fundamental matrix of L-diagonal system may be constructed by the use of the classical Levinson’s theorem.

Beyond Gisin's Theorem and its Applications: Violation of Local Realism by Two-Party Einstein-Podolsky-Rosen Steering.

We demonstrate here that for a given mixed multi-qubit state if there are at least two observers for whom mutual Einstein-Podolsky-Rosen steering is possible, i.e. each observer is able to steer the other qubits into two different pure states by spontaneous collapses due to von Neumann type measurements on his/her qubit, then nonexistence of local realistic models is fully equivalent to quantum entanglement (this is not so without this condition). This result leads to an enhanced version of Gisin’s theorem (originally: all pure entangled states violate local realism). Local realism is violated by all mixed states with the above steering property. The new class of states allows one e.g. to perform three party secret sharing with just pairs of entangled qubits, instead of three qubit entanglements (which are currently available with low fidelity). This significantly increases the feasibility of having high performance versions of such protocols. Finally, we discuss some possible applications.

Parametric Resonance in a Time-Dependent Harmonic Oscillator

In this paper, we study the phenomenon of appearance of new resonances in a timedependent harmonic oscillator under an oscillatory decreasing force. The studied equation belongs to the class of adiabatic oscillators and arises in connection with the spectral problem for the one-dimensional Schr¨odinger equation with Wigner–von Neumann type potential. We use a specially developed method for asymptotic integration of linear systems of differential equations with oscillatory decreasing coefficients. This method uses the ideas of the averaging method to simplify the initial system. Then we apply Levinson’s fundamental theorem to get the asymptotics for its solutions. Finally, we analyze the features of a parametric resonance phenomenon. The resonant frequencies of perturbation are found and the pointwise type of the parametric resonance phenomenon is established. In conclusion, we construct an example of a time-dependent harmonic oscillator (adiabatic oscillator) in which the parametric resonances, mentioned in the paper, may occur.

Twisted Heisenberg Doubles

We introduce a twisted version of the Heisenberg double, constructed from a twisted Hopf algebra and a twisted Hopf pairing. We state a Stone--von Neumann type theorem for a natural Fock space representation of this twisted Heisenberg double and deduce the effect on the algebra of shifting the product and coproduct of the original twisted Hopf algebra. We conclude by showing that the quantum Weyl algebra, quantum Heisenberg algebras, and lattice Heisenberg algebras are all examples of the general construction.

The refraction of shock pairs

We study shock refractions at the interface between two materials with convex equations of state involving only two shocks, which we refer to as shock pairs or shock pair systems. We present experimental and computational evidence for the existence of both regular and irregular shock pair systems in gases and develop a von Neumann type theory for regular systems that one can use to predict the angle of incidence at which a regular shock pair will occur. We show regular shock pair refractions obey the Principle of Shock Reciprocity; i.e., if the two materials are exchanged there is no effect on the polar diagram of the regular shock pair system, the angle between the two shocks in the pair or the streamline deflection angle. We also study regular and irregular shock pair systems in gases numerically and compare the numerical results with experimental data and our theory.

Dilation theory in finite dimensions: The possible, the impossible and the unknown

This expository essay discusses a finite dimensional approach to dilation theory. How much of dilation theory can be worked out within the realm of linear algebra? It turns out that some interesting and simple results can be obtained. These results can be used to give very elementary proofs of sharpened versions of some von Neumann type inequalities, as well as some other striking consequences about polynomials and matrices. Exploring the limits of the finite dimensional approach sheds light on the difference between those techniques and phenomena in operator theory that are inherently infinite dimensional, and those that are not.

Trace formulae and singular values of resolvent power differences of self-adjoint elliptic operators

In this note self-adjoint realizations of second order elliptic differential expressions with non-local Robin boundary conditions on a domain $\Omega\subset\dR^n$ with smooth compact boundary are studied. A Schatten--von Neumann type estimate for the singular values of the difference of the $m$th powers of the resolvents of two Robin realizations is obtained, and for $m>\tfrac{n}{2}-1$ it is shown that the resolvent power difference is a trace class operator. The estimates are slightly stronger than the classical singular value estimates by M. Sh. Birman where one of the Robin realizations is replaced by the Dirichlet operator. In both cases trace formulae are proved, in which the trace of the resolvent power differences in $L^2(\Omega)$ is written in terms of the trace of derivatives of Neumann-to-Dirichlet and Robin-to-Neumann maps on the boundary space $L^2(\partial\Omega)$.

Saving culture from selection: Comment on “An evolutionary framework for cultural change: Selectionism versus communal exchange” by L. Gabora

Most researchers working on evolutionary approaches to culture do so within the Darwinian paradigm, and accept the role of selection in shaping variation. Liane Gabora is an exception, and has been steadfast in articulating the need for an evolutionary but “non-Darwinian” theory of cultural change. In [1] she argues that culture does not meet the definitional requirements to evolve by natural selection, and introduces an alternative process called “communal exchange.” We find her argument that culture cannot be subject to selection deeply flawed. Indeed, it seems to us that Gabora’s primary mission is to “save culture” from being subject to natural selection, by analyzing selection in three ways that allow culture to be excluded from its domain. First, Gabora’s argument is based upon a non-standard account of selection. Few if any geneticists or biologists would recognize genetic inheritance as powered by “self-replicating automata” of the von Neumann type. A “gene” (an incredibly complex concept in its own right [2]) certainly does not code both for the machinery to interpret itself, and for the protein its sequence represents. SRA’s or replicators are not needed for selection [3]. Indeed, while the “replicator” concept was useful in illuminating the “units of selection” problem, employing it as an empirical unit is problematic [3–7]. Second, Gabora adds unnecessary conditions to the requirements for selection [8]. For example, the sequestration of inherited information, as in the isolation of germ-line cells in many animal lineages, is a derived characteristic which evolved quite late in the history of life [9]. Natural selection proceeds handily in many taxa which do not sequester heritable information. Nor is it required that new variation be approximated by a random process [10]. Third, Gabora criticizes the transfer of biological concepts to culture, by confusing theoretical concepts and empirical instances. For example, she claims that “generations” are not applicable to culture. In reality, organismal

Weyl—Titchmarsh-type formula for periodic Schrödinger operator with Wigner—von Neumann potential

The Schrödinger operator on the half-line with periodic background potential perturbed by a certain potential of Wigner—von Neumann type is considered. The asymptotics of generalized eigenvectors for λ ϵ ℂ+ and on the absolutely continuous spectrum is established. The Weyl—Titchmarsh-type formula for this operator is proven.

Schrödinger operators with slowly decaying Wigner–von Neumann type potentials

We consider Schrödinger operators with potentials satisfying a generalized bounded variation condition at infinity and an L^p decay condition. This class of potentials includes slowly decaying Wigner–von Neumann type potentials \sin(ax)/x^b with b>0 . We prove absence of singular continuous spectrum and show that embedded eigenvalues in the continuous spectrum can only take values from an explicit finite set. Conversely, we construct examples where such embedded eigenvalues are present, with exact asymptotics for the corresponding eigensolutions.

Schrödinger Operators with δ and δ′-Potentials Supported on Hypersurfaces

Self-adjoint Schr\"odinger operators with $\delta$ and $\delta'$-potentials supported on a smooth compact hypersurface are defined explicitly via boundary conditions. The spectral properties of these operators are investigated, regularity results on the functions in their domains are obtained, and analogues of the Birman--Schwinger principle and a variant of Krein's formula are shown. Furthermore, Schatten--von Neumann type estimates for the differences of the powers of the resolvents of the Schr\"odinger operators with $\delta$ and $\delta'$-potentials, and the Schr\"odinger operator without a singular interaction are proved. An immediate consequence of these estimates is the existence and completeness of the wave operators of the corresponding scattering systems, as well as the unitary equivalence of the absolutely continuous parts of the singularly perturbed and unperturbed Schr\"odinger operators. In the proofs of our main theorems we make use of abstract methods from extension theory of symmetric operators, some algebraic considerations and results on elliptic regularity.

On a Weyl–von Neumann type theorem for antilinear self-adjoint operators

Antilinear operators on a complex Hilbert space arise in various contexts in mathematical physics. In this paper, an analogue of the Weyl–von Neumann theorem for antilinear self-adjoint operators is proved, i.e. that an antilinear self-adjoint opera

Orthogonal Polynomials with Recursion Coefficients of Generalized Bounded Variation

We consider probability measures on the real line or unit circle with Jacobi or Verblunsky coefficients satisfying an $\ell^p$ condition and a generalized bounded variation condition. This latter condition requires that a sequence can be expressed as a sum of sequences $\beta^{(l)}$, each of which has rotated bounded variation, i.e., $\sum_{n=0}^\infty | e^{i\phi_l} \beta_{n+1}^{(l)} - \beta_n^{(l)} |$ is finite for some $\phi_l$. This includes discrete Schr\"odinger operators on a half-line or line with finite linear combinations of Wigner--von Neumann type potentials. For the real line, we prove that in the Lebesgue decomposition $d\mu=f dm + d\mu_s$ of such measures, the intersection of (-2,2) with the support of $d\mu_s$ is contained in an explicit finite set S (thus, $d\mu$ has no singular continuous part), and f is continuous and non-vanishing on $(-2,2) \setminus S$. The results for the unit circle are analogous, with (-2,2) replaced by the unit circle.

GENERALIZED MINIMAX THEOREMS IN GENERALIZED CONVEX SPACES

In this work, we obtain intersection theorem, analytic alternative and von Neumann type minimax theorem in G-convex spaces. We also generalize Ky Fan minimax inequality to acyclic versions in G-convex spaces. The result is applied to formulate acyclic versions of other minimax results, a theorem of systems of inequalities and analytic alternative.

ELEMENTS OF THE KKM THEORY ON CONVEX SPACES

We introduce a new concept of convex spaces and a multimap class K having certain KKM property. From a basic KKM type theorem for a K-map defined on an convex space without any topology, we deduce ten equivalent formulations of the theorem. As applications of the equivalents, in the frame of convex topological spaces, we obtain Fan-Browder type fixed point theorems, almost fixed point theorems for multimaps, mutual relations between the map classes K and B, variational inequalities, the von Neumann type minimax theorems, and the Nash equilibrium theorems.

Equilibrium existence theorems in KKM spaces

An abstract convex space satisfying the KKM principle is called a KKM space. This class of spaces contains G -convex spaces properly. In this work, we show that a large number of results in KKM theory on G -convex spaces also hold on KKM spaces. Examples of such results are theorems of Sperner and Alexandroff–Pasynkoff, Fan–Browder type fixed point theorems, Horvath type fixed point theorems, Ky Fan type minimax inequalities, variational inequalities, von Neumann type minimax theorems, Nash type equilibrium theorems, and Himmelberg type fixed point theorems.

Nonlinear reflection of grazing acoustic shock waves: unsteady transition from von Neumann to Mach to Snell–Descartes reflections

We study the reflection of acoustic shock waves grazing at a small angle over a rigid surface. Depending on the incidence angle and the Mach number, the reflection patterns are mainly categorized into two types, namely regular reflection and irregular reflection. In the present work, using the nonlinear KZ equation, this reflection problem is investigated for extremely weak shocks as encountered in acoustics. A critical parameter, defined as the ratio of the sine of the incidence angle and the square root of the acoustic Mach number, is introduced in a natural way. For step shocks, we recover the self-similar (pseudo-steady) nature of the reflection, which is well known from von Neumann's work. Four types of reflection as a function of the critical parameter can be categorized. Thus, we describe the continuous but nonlinear and non-monotonic transition from linear reflection (according to the Snell–Descartes laws) to the weak von-Neumann-type reflection observed for almost perfectly grazing incidence. This last regime is a new, one-shock regime, in contrast with the other, already known, two-shock (regular reflection) or three-shock (von Neumann-type reflection) regimes. Hence, the transition also resolves another paradox on acoustic shock waves addressed by von Neumann in his classical paper. However, step shocks are quite unrealistic in acoustics. Therefore, we investigate the generalization of this transition for N-waves or periodic sawtooth waves, which are more appropriate for acoustics. Our results show an unsteady reflection effect necessarily associated with the energy decay of the incident wave. This effect is the counterpart of step-shock propagation over a concave surface. For a given value of the critical parameter, all the patterns categorized for the step shock may successively appear when the shock is propagating along the surface, starting from weak von-Neumann-type reflection, then gradually turning to von Neumann reflection and finally evolving into nonlinear regular reflection. This last one will asymptotically result in linear regular reflection (Snell–Descartes). The transition back to regular reflection is one of two types, depending on whether a secondary reflected shock is observed. The latter case, here described for the first time, appears to be related to the non-constant state behind the incident shock, which prevents secondary reflection.

ATTENUATION OF NON-PHYSICAL OSCILLATIONS IN SUPERNOVA SHOCK WAVES

Artificial viscosity is widely used in numerical calculations of stellar core collapse. The failure or success of the prompt mechanism explosion of type-II supernovae is strongly dependent on the numerical code, and the study of a suitable and efficient method of capturing the shock front is a current problem. We present a novel one-term artificial viscosity which is dependent on the velocity field along the shock front. We show that this form of artificial viscosity is able to capture the profile of a plane shock wave, removing the non-physical oscillations originated by the artificial viscosity of von Neumann and Richtmyer type.