Theory of Hermite Calculus

Abstract We show that entanglement in one-dimensional spin and electron systems, with one excitation, depends only on the system size and has very simple form in both multipartite and bipartite case. We consider the exact solutions given by the Fourier transform known in literature as the basis of wavelets. Regarding the multipartite case, we present very simple expressions for global entanglement and N-concurrence, and show that they are mutually related. In the bipartite case, we give expressions for I-concurrence and negativity, and show that they are also depend on each other. Presented formulas allow one to calculate entanglement for an arbitrary size N, while the original definitions practically work only for the systems consisting of a few qubits. We expect that the size dependence of the entanglement of states with elementary excitation may help to understand entanglement in the systems with greater number of elementary excitations. ? ? ? Tadeusz KOSZTOLOWICZ Institute of Physics, Jan Kochanowski University, ul. ´Swi¸etokrzyska 15, 25-406 Kielce, Poland e-mail: tadeusz.kosztolowicz@ujk.edu.pl Normal diffusion, subdiffusion, and slow subdiffusion in a membrane system Abstract We consider various kinds of diffusion in a system with a thin membrane. The membrane is treated as a partially permeable wall. Using the random walk model with discrete time and space variables the probabilities describing a particle?s random walk are found. Probabilities are transformed to the system in which the variables are continuous. From the obtained probabilities we derive boundary conditions at the membrane. One of the condition demands the continuity of flux at the membrane, but the other one is unexpected and contains the Riemann-Liouville fractional time derivative. The additional memory effect, represented by the fractional derivative, is created by the membrane even for normal diffusion case. In the presented model a kind of diffusion is defined by a probability density of time which is needed for the particle to take its next step. ? ? ? Silvia LICCIARDI ENEA - Centro Ricerche Frascati, via E. Fermi, 45, IT 00044 Frascati (Roma), Italy e-mail: silviakant@gmail.com Theory of Hermite Calculus Abstract The use of umbral methods and of other concepts borrowed from algebraic theory of operators is a powerful tool to treat problems concerning the theory of special functions and the relevant applications in physical problems. The Hermite Calculus allows tremendous simplifications in a va- riety of problems involving Hermite polynomials and functions. It yields the possibility of providing explicit integration of families of function hardly achievable with conventional means and open new scenarios for the definition of new integral transforms of noticeable interest in application in optics.