Branching structure for the transient (1;R)-random walk in random environment and its applications

An intrinsic multitype branching structure within the transient (1;R)-RWRE is revealed. The branching structure enables us to specify the density of the absolutely continuous invariant measure for the environments seen from the particle and reprove the LLN with an drift explicitly in terms of the environment, comparing with the results in Br\'emont (2002).Comment: 25 page

A second-order scheme with nonuniform time steps for a linear reaction-sudiffusion problem

Stability and convergence of a time-weighted discrete scheme with nonuniform time steps are established for linear reaction-subdiffusion equations. The Caupto derivative is approximated at an offset point by using linear and quadratic polynomial interpolation. Our analysis relies on two tools: a discrete fractional Gr\"{o}nwall inequality and the global consistency analysis. The new consistency analysis makes use of an interpolation error formula for quadratic polynomials, which leads to a convolution-type bound for the local truncation error. To exploit these two tools, some theoretical properties of the discrete kernels in the numerical Caputo formula are crucial and we investigate them intensively in the nonuniform setting. Taking the initial singularity of the solution into account, we obtain a sharp error estimate on nonuniform time meshes. The fully discrete scheme generates a second-order accurate solution on the graded mesh provided a proper grading parameter is employed. An example is presented to show the sharpness of our analysis.Comment: 23 pages, 4 table

Solving the Jaynes-Cummings Model with Shift Operators Constructed by Means of the Matrix-Diagonalizing Technique

The Jaynes-Cummings model is solved with the raising and lowering (shift) operators by using the matrix-diagonalizing technique. Bell nonlocality is also found present ubiquitously in the excitations states of the model.Comment: 5 page

Sharp H1H^1-norm error estimates of two time-stepping schemes for reaction-subdiffusion problems

Due to the intrinsically initial singularity of solution and the discrete convolution form in numerical Caputo derivatives, the traditional $H^1$-norm analysis (corresponding to the case for a classical diffusion equation) to the time approximations of a fractional subdiffusion problem always leads to suboptimal error estimates (a loss of time accuracy). To recover the theoretical accuracy in time, we propose an improved discrete Gr\"{o}nwall inequality and apply it to the well-known L1 formula and a fractional Crank-Nicolson scheme. With the help of a time-space error-splitting technique and the global consistency analysis, sharp $H^1$-norm error estimates of the two nonuniform approaches are established for a reaction-subdiffusion problems. Numerical experiments are included to confirm the sharpness of our analysis.Comment: 22 pages, 8 table

A discrete Gr\"{o}nwall inequality with application to numerical schemes for subdiffusion problems

We consider a class of numerical approximations to the Caputo fractional derivative. Our assumptions permit the use of nonuniform time steps, such as is appropriate for accurately resolving the behavior of a solution whose derivatives are singular at~$t=0$. The main result is a type of fractional Gr\"{o}nwall inequality and we illustrate its use by outlining some stability and convergence estimates of schemes for fractional reaction-subdiffusion problems. This approach extends earlier work that used the familiar L1 approximation to the Caputo fractional derivative, and will facilitate the analysis of higher order and linearized fast schemes.Comment: 15 pages, 2 figure

Multivariate Lagrange Interpolation and an Application of Cayley-Bacharach Theorem For it

In this paper,we deeply research Lagrange interpolation of n-variables and give an application of Cayley-Bacharach theorem for it. We pose the concept of sufficient intersection about s algebraic hypersurfaces in n-dimensional complex Euclidean space and discuss the Lagrange interpolation along the algebraic manifold of sufficient intersection. By means of some theorems (such as Bezout theorem, Macaulay theorem and so on) we prove the dimension for the polynomial space P(n) m along the algebraic manifold S of sufficient intersection and give a convenient expression for dimension calculation by using the backward difference operator. According to Mysovskikh theorem, we give a proof of the existence and a characterizing condition of properly posed set of nodes of arbitrary degree for interpolation along an algebraic manifold of sufficient intersection. Further we point out that for s algebraic hypersurfaces of sufficient intersection, the set of polynomials must constitute the H-base of ideal. As a main result of this paper, we deduce a general method of constructing properly posed set of nodes for Lagrange interpolation along an algebraic manifold, namely the superposition interpolation process. At the end of the paper, we use the extended Cayley-Bacharach theorem to resolve some problems of Lagrange interpolation along the 0-dimensional and 1-dimensional algebraic manifold. Just the application of Cayley-Bacharach theorem constitutes the start point of constructing properly posed set of nodes along the high dimensional algebraic manifold by using the superposition interpolation process.Comment: 21 pages, 3 figure

Unconditional convergence of a fast two-level linearized algorithm for semilinear subdiffusion equations

A fast two-level linearized scheme with unequal time-steps is constructed and analyzed for an initial-boundary-value problem of semilinear subdiffusion equations. The two-level fast L1 formula of the Caputo derivative is derived based on the sum-of-exponentials technique. The resulting fast algorithm is computationally efficient in long-time simulations because it significantly reduces the computational cost $O(MN^2)$ and storage $O(MN)$ for the standard L1 formula to $O(MN\log N)$ and $O(M\log N)$, respectively, for $M$ grid points in space and $N$ levels in time. The nonuniform time mesh would be graded to handle the typical singularity of the solution near the time $t=0$, and Newton linearization is used to approximate the nonlinearity term. Our analysis relies on three tools: a new discrete fractional Gr\"{o}nwall inequality, a global consistency analysis and a discrete $H^2$ energy method. A sharp error estimate reflecting the regularity of solution is established without any restriction on the relative diameters of the temporal and spatial mesh sizes. Numerical examples are provided to demonstrate the effectiveness of our approach and the sharpness of error analysis.Comment: 23 pages, 5 figure

The coexistence of p-wave spin triplet superconductivity and itinerant ferromagnetism

A model for coexistence of p-wave spin-triplet superconductivity (SC) and itinerant ferromagnetism (FM) is presented. The Hamiltonian can be diagonalized by using the so(5) algebraic coherent state. We obtain the coupling equations of the magnetic exchange energy and superconducting gaps through the double-time Green function. It is found that the ferromagnetisation gives rise to the phase transitions of p-wave superconducting states or superfluid of $^{3}He$.Comment: 9 pages, no figure

A Model for the Coexistence of p-wave Superconductivity and Ferroelectricity

A model for the coexistence of p-wave superconductivity (SC) and ferroelectricity (FE) is presented. The Hamiltonian of SC sector and FE sector can be diagonalized by using the $so(5)$ and $h(4)$ algebraic coherent states respectively. We assume a minimal symmetry-allow coupling and simplify the total Hamiltonian through a double mean-field approximation (DMFA). A variational coherent-state (VCS) trial wave-function is applied for the ground state. It is found that the ferroelectricity gives rise to the magnetic field effect of p-wave superconductivity.Comment: 11pages,no figur

Optical force on toroidal nanostructures: toroidal dipole versus renormalized electric dipole

We study the optical forces acting on toroidal nanostructures. A great enhancement of optical force is unambiguously identified as originating from the toroidal dipole resonance based on the source-representation, where the distribution of the induced charges and currents is characterized by the three families of electric, magnetic, and toroidal multipoles. On the other hand, the resonant optical force can also be completely attributed to an electric dipole resonance in the alternative field-representation, where the electromagnetic fields in the source-free region are expressed by two sets of electric and magnetic multipole fields based on symmetry. The confusion is resolved by conceptually introducing the irreducible electric dipole, toroidal dipole, and renormalized electric dipole. We demonstrate that the optical force is a powerful tool to identify toroidal response even when its scattering intensity is dwarfed by the conventional electric and magnetic multipoles.Comment: 25 pages, 7 figure