Two new supersymmetric equations of Harry Dym type and their supersymmetric reciprocal transformations

A new N=1$N=1$ supersymmetric Harry Dym equation is constructed by applying supersymmetric reciprocal transformation to a trivial supersymmetric Harry Dym equation, and its recursion operator and Lax formulation are also obtained. Within the framework of symmetry approach, a class of 3rd order supersymmetric equations of Harry Dym type are considered. In addition to five known integrable equations, a new supersymmetric equation, admitting 5th order generalized symmetry, is shown to be linearizable through supersymmetric reciprocal transformation. Furthermore, its Lax representation and recursion operator are given so that the integrability of this new equation is confirmed.     

On an integrable deformed affinsphären equation. A reciprocal gasdynamic connection

The integrable affinsphären equation originally arose in a geometric context but has an interesting gasdynamic connection. Here, an integrable deformed version of the affinsphären equation is derived in a novel manner via the action of reciprocal transformations on a related anisentropic gasdynamics system. A linear representation for the deformed affinsphären equation is constructed by means of the reciprocal transformations. The latter are then employed to derive a class of exact solutions in parametric form.     

Hamiltonian structures and their reciprocal transformations for the -KdV–CH hierarchy

The r-KdV–CH hierarchy is a generalization of the Korteweg–de Vries and Camassa–Holm hierarchies parameterized by r+1 constants. In this paper we clarify some properties of its multi-Hamiltonian structures including the explicit expressions of the Hamiltonians, the formulae of the central invariants of the associated bihamiltonian structures and the relationship of these bihamiltonian structures with Frobenius manifolds. By introducing a class of generalized Hamiltonian structures, we present in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain reciprocal transformations, and prove the validity of the formulae at the level of dispersionless limit.     

Time-evolution of quantum systems via a complex nonlinear Riccati equation. I. Conservative systems with time-independent Hamiltonian

The sensitivity of the evolution of quantum uncertainties to the choice of the initial conditions is shown via a complex nonlinear Riccati equation leading to a reformulation of quantum dynamics. This sensitivity is demonstrated for systems with exact analytic solutions with the form of Gaussian wave packets. In particular, one-dimensional conservative systems with at most quadratic Hamiltonians are studied.     

On the Onsager-Casimir reciprocal relations in a tilted Weyl semimetal

The Onsager-Casimir reciprocal relations are a fundamental symmetry of nonequilibrium statistical systems. Here we study an unusual chirality-dependent Hall effect in a tilted Weyl semimetal Co₃Sn₂S₂ with broken time-reversal symmetry. It is confirmed that the reciprocal relations are satisfied. Since two Berry curvature effects, an anomalous velocity and a chiral chemical potential, contribute to the observed Hall effect, the reciprocal relations suggest their intriguing connection.     

Synthesis of magnetite nanoparticles via a solvent-free thermal decomposition route

Superparamagnetic nanoparticles have been widely applied in various bio-medical applications. To date, it is still a challenge to synthesize nanosized Fe₃O₄ particles with controlled size and distribution. In this paper, a novel solvent-free thermal decomposition method is reported for synthesizing Fe₃O₄ nanoparticles. Size and morphology of the nanoparticles are determined by TEM while the structure of the nanoparticles is identified by FTIR, XPS and TGA measurements. Magnetic properties of the obtained particles are determined using VSM and SQUID measurement. The particle size of the Fe₃O₄ can be tailored by adjusting either reaction temperature or time. When the reaction temperature is increased to 330 °C and the reaction time is extended to 4 h, the average particle size of the obtained nanoparticles is ∼9 nm, while Ms value reaches ∼76 emu/g. The as synthesized Fe₃O₄ nanoparticles show well-established superparamagnetic properties with the blocking temperature at around 100 K.     

On the problem of finding a suitable distribution of students to universities in Germany

For many years, the problem of how to distribute students to the various universities in Germany according to the preferences of the students has remained unsolved. Various approaches, like the centralized method to let a central agency organize the distribution to the various universities or the decentralized method to let the students apply directly at their preferred universities, turned out to lead to a significant fraction of frustrated students ending up at universities not being on their preference list or even not having a place to study at all. With our centralized approach, we are able to decrease the fraction of frustrated students as well as the bureaucratic expenses for applicants and universities drastically.     

On the nonequilibrium statistical mechanics of a binary mixture. I. The distribution functions

In this paper, we obtain the generalization of the BBGKY hierarchy for a binary mixture of chemically neutral particles. Using modified boundary conditions different from the ones proposed by Bogoliubov, we solve the hierarchy, and obtain explicitly the set of two-particle distribution functions for the several species of the mixture, up to first order in the density.     

On the layering transition of an SOS surface interacting with a wall. I. Equilibrium results

We consider the model of a 2D surface above a fixed wall and attracted toward it by means of a positive magnetic fieldh in the solid-on-solid (SOS) approximation when the inverse temperature is very large and the external fieldh is exponentially small in . We improve considerably previous results by Dinaburg and Mazel on the competition between the external field and the entropic repulsion with the wall, leading, in this case, to the phenomenon of layering phase transitions. In particular, we show, using the Pirogov-Sinai scheme as given by Zahradník, that there exists a unique critical valueh _k ^* () in the interval (1/4e ^–4k , 4e ^–4k ) such that, for allh(h _k+1 ^* ,h _k ^* ) and large enough, there exists a unique infinite-volume Gibbs state. The typical configurations are small perturbation of the ground state represented by a surface at heightk+1 above the wall. Moreover, for the same choice of the thermodynamic parameters, the influence of the boundary conditions of the Gibbs measure in a finite cube decays exponentially fast with the distance from the boundary. Whenh=h _k ^* () we prove instead the convergence of the cluster expansion for bothk andk+1 boundary conditions. This fact signals the presence of a phase transition. In the second paper of this series we will consider a Glauber dynamics for the above model and we will study the rate of approach to equilibrium in a large finite cube with arbitrary boundary conditions as a function of the external fieldh. Using the results proven in this paper, we will show that there is a dramatic slowing down in the approach to equilibrium when the magnetic field takes one of the critical values and the boundary conditions are free (absent).     

On a free-electron-laser in a uniform magnetic field. A solution for arbitrarily strong electromagnetic radiation fields

Exact expressions are derived for the gain of a free-electron laser based on a uniform longitudinal magnetic field configuration operating in the single-particle, low-gain regime. The gain is calculated for different parameters of the system. For strong enough a field, the gain decreases with an increase of the amplitude, becoming negative when passing a threshold value, which depends on the system parameters. Implications regarding the saturation of the lasing process are discussed.     

On the hydrodynamic limit of a one-dimensional Ginzburg-Landau lattice model. Thea priori bounds

The simplest Ginzburg-Landau model with conservation law is investigated. The initial state is specified by an inhomogeneous profile of the chemical potential associated with the conserved quantity, that is, the mean spin. It is shown that the mean spin satisfies a nonlinear diffusion equation in the hydrodynamic limit. The proof is based on the nice, parabolic structure of the model. A standard perturbation technique is used.     

On the parametrization of finite-gap solutions by frequency and wavenumber vectors and a theorem of I. krichever

We discuss the parametrization of real finite-gap solutions of an integrable equation by frequency and wavenumber vectors. This parametrization underlies perturbation and averaging theories for the finite-gap solutions. Out of the framework of integrable equations, the parametrization gives a convenient coordinate system on the corresponding manifold of Riemann curves.     

On invariant states and the commutant of a group of quasi-free automorphisms of the CAR algebra. III

It is shown that a state of the CAR algebra is an extreme invariant state under the group of quasi-free automorphisms α_U with unitaries u in a von Neumann algebra M on the one- particle Hilbert space if and only if it is a gauge-invariant quasi-free state Ø _A corresponding to A in M' with 0 ≤ A ≤ 1, under the assumption that M does not contain any finite type I factor direct summands.     

Editor's Note: On the Gravitational Field of a Mass Point According to Einstein's Theory by K. Schwarzschild

We comment on the famous paper by Karl Schwarzschild from 1916, which is reprinted in the present issue.