Is Classical Reality Completely Deterministic?

We interpret the concept of determinism for a classical system as the requirement that the solution to the Cauchy problem for the equations of motion governing this system be unique. This requirement is generally believed to hold for all autonomous classical systems. Our analysis of classical electrodynamics in a world with one temporal and one spatial dimension provides counterexamples of this belief. Given the initial conditions of a particular type, the Cauchy problem may have an infinite set of solutions. Therefore, random behavior of closed classical systems is indeed possible. With this finding, we give a qualitative explanation of how classical strings can split. We propose a modified path integral formulation of classical mechanics to include indeterministic systems.     

Curve crossing problem with Gaussian type coupling: analytically solvable model

We give a general method for finding an exact analytical solution for the two state curve crossing problem. The solution requires the knowledge of the Green's function for the motion on the uncoupled potentials. We use the method to find the solution of the problem in the case of parabolic potentials coupled by Gaussian interaction. Our method is applied to this model system to calculate the effect of curve crossing on the electronic absorption spectrum and the resonance Raman excitation profile.     

Least Energy Solitary Waves for a System of Nonlinear Schrödinger Equations in $${\mathbb{R}^n}$$

In this paper we consider systems of coupled Schrödinger equations which appear in nonlinear optics. The problem has been considered mostly in the one-dimensional case. Here we make a rigorous study of the existence of least energy standing waves (solitons) in higher dimensions. We give: conditions on the parameters of the system under which it possesses a solution with least energy among all multi-component solutions; conditions under which the system does not have positive solutions and the associated energy functional cannot be minimized on the natural set where the solutions lie.     

New Solution of a Problem of the Dipole Radiation and Quantum Properties of the Electromagnetic Field

The Planck's formula connecting an energy and frequency of quantum is known more than half-century. However there are obscure appearances and processes defining nature of the electromagnetic field quantum and reducing to this formula. The path to a solution of these problems uncloses a new solution of a problem of the dipole radiation. Hertz obtained the first solution describing of a dipole radiator field. However this solution and all subsequent ones are incomplete and they not give a complete exposition of the dipole field. In this paper we will show a possibility of complete exposition of a dipole radiator field by using both a complete set of the electromagnetic field equations obtained in [1] and also solution for a radiator of a longitudinal electromagnetic wave. The obtained solution is of great importance in understanding of a nature of quantum properties of electromagnetic field. The work contains series of assumptions about existence of some constants characterizing electromagnetic field.     

A dynamical formulation of one-dimensional scattering theory and its applications in optics

We develop a dynamical formulation of one-dimensional scattering theory where the reflection and transmission amplitudes for a general, possibly complex and energy-dependent, scattering potential are given as solutions of a set of dynamical equations. By decoupling and partially integrating these equations, we reduce the scattering problem to a second order linear differential equation with universal initial conditions that is equivalent to an initial-value time-independent Schrödinger equation. We give explicit formulas for the reflection and transmission amplitudes in terms of the solution of either of these equations and employ them to outline an inverse-scattering method for constructing finite-range potentials with desirable scattering properties at any prescribed wavelength. In particular, we construct optical potentials displaying threshold lasing, antilasing, and unidirectional invisibility.     

Quantum entanglement,indistinguishability, and the absent-minded driver's problem

The absent-minded driver's problem illustrates that probabilistic strategies can give higher pay-offs than deterministic ones. We show that there are strategies using quantum entangled states that give even higher pay-offs, both for the original problem and for the generalized version with an arbitrary number of intersections and any possible set of pay-offs.     

Multiplicative dichotomic markov noise- some exact dynamical results and applications

We give the exact time dependent solution for the case that a dichotomic noise assuming the values 1 and 0, multiplies the possibly stochastic evolution equation of a system. The result is applied to the chromatographic problem and the problem of electrodiffusion.     

Regularization of inverse boundary design radiative heat transfer problems

Inverse boundary design heat transfer problems, including only radiation, are considered. Variational methods of regularization are used for solving these (mathematically ill-posed) problems, they give possibility to take into account various a priori information about the desired solution. For minimizing the discrepancy functional we use the adjoint problem method; however, we use it not only in iterative regularization, but in Tikhonov and parametric ones as well. We use all the available a priori information about desired solutions in all the techniques; this allows to obtain physically feasible solutions in all the cases.     

Census of planar maps: from the one-matrix model solution to a combinatorial proof

We consider the problem of enumeration of planar maps and revisit its one-matrix model solution in the light of recent combinatorial techniques involving conjugated trees. We adapt and generalize these techniques so as to give an alternative and purely combinatorial solution to the problem of counting arbitrary planar maps with prescribed vertex degrees.     

Curle's equation and acoustic scattering by a sphere

Recent papers have initiated interesting comparisons between aeroacoustic theory and the results of acoustic scattering problems. In this paper, we consider some aspects of these comparisons for acoustic scattering by a sphere. We give a derivation of Curle's equation for a specific class of linear acoustic scattering problems, and, in response to previous claims to the contrary, give an explicit confirmation of Curle's equation for plane wave scattering by a stationary rigid sphere of arbitrary size in an inviscid fluid. We construct the complete solution for scattering by a rigid sphere in a viscous fluid, and show that the neglect of viscous terms in Curie's equation yields an incomplete prediction of the far field dipole pressure. We also consider the null field solution of the sphere scattering problem, and give its extension to the vorticity modes associated with viscosity. Finally, we construct a solution for an elastic sphere in a viscous fluid, and show that the rigid sphere/null field solution is recovered from the limit of infinite longitudinal and shear wave speeds in the elastic solid.     

On some boundary value problems on a strip in the complex plane

The Sommerfeld integral inversion method for the Helmholtz equation in an angular region with different boundary values leads to boundary value problems in an infinite strip of the complex plane. We investigate a generic system for such boundary value problems and give the existence and uniqueness results with optimal growth estimates on the solution. We also give the solutions of the Dirichlet problem in a strip when the boundary functions grow exponentially.     

Multi-time fractional diffusion equation

We construct a fundamental solution of a multi-time diffusion equation with the Dzhrbashyan-Nersesyan fractional differentiation operator with respect to the time variables. We give a representation for a solution of the Cauchy problem and prove the uniqueness theorem in the class of functions of fast growth. The corresponding results for equations with Riemann-Liouville and Caputo derivatives are obtained as particular cases of the proved assertions.     

Radial distribution of temperature in a thin lens due to absorption of light and heat conduction

We consider the stationary radial distribution of temperature in a lens. For a thin lens, i.e. a two-dimensional model problem, we give an analytical approximation for the distribution of temperature, a quantity of great importance for the prediction of the performance of the lens. We compare this formula with the numerical solution of the heat equation for several examples. Finally, we discuss the incorporation of the heat-transfer to the environment in principle and give prospects for further generalizations of this theory.     

The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrödinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a nonlinear steepest descent analysis of an associated 3 × 3 matrix Riemann–Hilbert problem, we can give the precise asymptotic formulas for the solution of the coupled nonlinear Schrödinger equation on the half-line.     

A geometric characterisation of resonance in Hopf bifurcation from relative equilibria

We give a new characterisation of resonance in Hopf bifurcation from relative equilibria in systems with compact symmetry group. This characterisation provides a full geometric explanation of the resonance phenomenon. In addition, we develop techniques based on normal form theory to give a complete solution to the associated bifurcation problem.     

Superspace geometry and the minimal,non minimal,and new minimal supergravity multiplets

We analyse superspace constraints in a systematic way and define a set of natural constraints. We give a complete solution of the Bianchi identities subject to these constraints and obtain a reducible, but not fully reducible multiplet. By additional constraints it can be reduced to either the minimal, non minimal or new minimal multiplet. We discuss the superspace actions for the various multiplets.     

On the capacity of the multiple-hop relay channel with linear relaying

The relay nodes with linear relaying transmit linear combination of their past received signals. The capacity of the multiple-hop Gaussian relay channel with linear relaying is derived, assuming that each node in the channel only communicates with its nearest neighbor nodes. The capacity is formulated as an optimization problem over the relaying matrices and the covariance matrix of the signals transmitted from the source. It is proved that the solution to this optimization problem is equivalent to a “single-letter” optimization problem when some certain conditions are satisfied. We also show that the solution to the “single-letter” optimization problem has the same form as the expression of the rate achieved by time-sharing amplify-and-forward (TSAF). In order to solve this equivalent problem, we give an iterative algorithm. Simulation results show that the achievable rate with TSAF is close to the capacity, if channel gain of one certain hop is smaller than that of all the other hops relatively.     

Multi-channel scattering problems: an analytically solvable model

We have proposed a general method for finding an exact analytical solution for the multi-channel scattering problem in the presence of a delta function coupling. Our solution is quite general and is valid for any set of potentials, if the uncoupled diabatic potential has an exact solution. We have also discussed a few examples, where our method can easily be applied.