Reports on Mathematical Physics最新文献

In this work, we introduce an adopted local frame on the tangent bundle of a Finsler manifold with respect to the natural foliations of the tangent bundle. We show the prominence of using this local frame by studying some geometric properties of the foliations and distributions on the tangent bundle of a Finsler manifold. Moreover, we find the necessary and sufficient conditions on the Finsler manifold (M, F) such that the unit tangent bundle admits a Sasakian structure.

Formulas are derived for solutions of many-body wave scattering problem by small impedance particles embedded in a homogeneous medium. The limiting case is considered, when the size a of small particles tends to zero while their number tends to infinity at a suitable rate. The basic physical assumption is a << d << λ, where d is the minimal distance between neighboring particles, λ is the wavelength, and the particles can be impedance balls B(x_m, a) with centers x_m located on a grid. Equations for the limiting effective (self-consistent) field in the medium are derived. It is proved that one can create material with a desired refraction coefficient by embedding in a free space many small balls of radius a with prescribed boundary impedances. The small balls can be centered at the points located on a grid. A recipe for creating materials with a desired refraction coefficient is formulated. It is proved that materials with a desired radiation pattern, for example, wave-focusing materials, can be created.

Let R_r =$F$_q + v₁F$F$_q + ··· + v_r$F$_q, where q is the power of prime, v_i² = v_i, v_iv_j = v_jv_i = 0 for 1 ≤ i, j ≤ r and r ≥ 1. In this paper, the structure of λ-constacyclic codes over the ring R_r is studied and a Gray map ϕ from$R_r^n$ to$F$_q^(r+1)n is given. The necessary and sufficient conditions for these codes to contain their Euclidean duals are determined. As an application, we obtain many new better quantum codes from dual-containing λ-constacyclic codes over R_r, for r = 1, 2, 3, that improve the known existing quantum codes.

This paper is devoted to identifying a time-dependent source term for a multi-term time-fractional diffusion equation with a nonlocal dynamic boundary and integral type over-determination condition. The time-fractional derivatives are considered in Caputo's sense. By applying Fourier's method we obtained multi-term ordinary fractional order differential equation which has been reduced to an algebraic equation by using Laplace transform. Inverse Laplace transform is used to obtain the solution of multi-term ordinary fractional order differential equation which involves multinomial Mittag-Leffler functions. Under some regularity and consistency conditions on the data, unique existence and stability of the regular solution of the inverse problem is proved.

In 1991 I. E. Segal stated (on the basis of a certain induced representation of the conformal group, but without proof) that there are four chronometric elementary particles of spin 1/2. In terms of the same representation (but using a different parallelization), we discuss a model where there are just three of those particles. They are interpreted as the proton, the (electronic) neutrino, and the electron, respectively. Mathematically, the three particles correspond to factors of the composition series. Each of the three representation spaces (F_p, F_v, and F_e, respectively) is provided with an invariant unitary structure and, in each of the three cases, energy positivity holds. These findings may shed some light on why there are lepton generations at all. The conclusions are mathematically based on the specific formula for the action of our representation in the full representation space F. This formula is presented in terms of the so-called reproducing kernel functions which are related to coherent states.

We exploit the dynamics of entangled system between a harmonic oscillator and a single mode electromagnetic field as two harmonic oscillators, to examine quantum entanglement in coherent states. Through the Schrödinger cat state, we compute analytically its distribution function. Then, we check graphically the evolution of entanglement and the distribution function for two particular cases: the first is defined by two harmonic oscillators in quantum motion in a Paul trap; here whatever the quantum state, the system is at most entangled. The second case is visualised by two harmonic oscillators with a perturbative force, in that case, by varying the displacement numbers of the coherent states, the quantum entanglement and the distribution function present a similar evolution and they reach very large values. Comparing the two models: in the first, the dynamics of quantum entanglement exhibits a multi-frequency lines behaviour while in the second model, it shows an exponential behaviour.

In this paper, we use the double Wronskian reduction technique to consider nonlocal reductions of a nonisospectral (2+1)-dimensional breaking Ablowitz--Kaup--Newell--Segur equation. Various types of solutions, including soliton-like solutions and Jordan-block solutions, for the resulting nonlocal equations are derived. Dynamics of these obtained solutions are analyzed and illustrated.

For the SOS (solid-on-solid) model (with an external field) with spin values from the set of all integers on a Cayley tree we give gradient Gibbs measures (GGMs). Such a measure corresponds to a boundary law (a function defined on vertices of Cayley tree) satisfying an infinite system of functional equations. We give several concrete GGMs which correspond to periodic boundary laws.

We present a direct test to verify separability of symmetric pure states of n quantum bits, that is, n-qubit states that are invariant under any element of the permutation group.

Lord Kelvin believed that the electromagnetic aether must also generate gravity. Presently, we have no methods to determine the density of the electromagnetic aether, or we say the Ω(1) substratum. Thus, we also suppose that vacuum is filled with another kind of continuously distributed substance, which may be called the Ω(2) substratum. Based on a theorem of V. Fock on the mass tensor of a fluid, the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata are established. Quasi-static solutions of the gravitational field equations in vacuum are obtained. Based on an assumption, relationships between the contravariant energy-momentum tensors of the Ω(1) and Ω(2) substrata and the contravariant metric tensor are obtained. Thus, the cosmological constant is calculated theoretically. The Ω(1) and Ω(2) substrata may be a possible candidate of the dark energy. According to the theory of vacuum mechanics, only those energy-momentum tensors of discrete or continuously distributed sinks in the Ω(0) substratum are permitted to act as the source terms in the generalized Einstein's equations. Thus, the zero-point energy of electromagnetic fields is not qualified for a source term in the generalized Einstein's equations. Some people believed that all kinds of energies should act as source terms in the Einstein's equations. It may be this unwarranted belief that leads to the cosmological constant problem. The mass density of theΩ(1) and Ω(2) substrata is equivalent to that of around 3 protons contained in a box with a volume of 1 cubic metre.