Finite difference forms containing derivatives of higher order

Finite difference forms containing derivatives of higher orde

Bilinearization of Discrete Soliton Equations and Singularity Confinement

Bilinear forms for some nonlinear partial difference equations(discrete soliton equations) are derived based on the results of singularity confinement. Using the bilinear forms, the N-soliton and algebraic solutions of the discrete potential mKdV equation are constructed.Comment: 14 pages, LaTe

Characterization of solutions of the discrete-time algebraic Riccati equation based on quadratic difference forms

This paper is concerned with a characterization of all symmetric solutions to the discrete-time algebraic Riccati equation (DARE). Dissipation theory and quadratic difference forms from the behavioral approach play a central role in this paper. Along the line of the continuous-time results due to Trentelman and Rapisarda [H.L. Trentelman, P. Rapisarda, Pick matrix conditions for sign-definite solutions of the algebraic Riccati equation, SIAM J. Contr. Optim. 40 (3) (2001) 969–991], we show that the solvability of the DARE is equivalent to a certain dissipativity of the associated discrete-time state space system. As a main result, we characterize all unmixed solutions of the DARE using the Pick matrix obtained from the quadratic difference forms. This characterization leads to a necessary and sufficient condition for the existence of a non-negative definite solution. It should be noted that, when we study the DARE and the dissipativity of the discrete-time system, there exist two difficulties which are not seen in the continuous-time case. One is the existence of a storage function which is not a quadratic function of state. Another is the cancellation between the zero and infinite singularities of the dipolynomial spectral matrix associated with the DARE, due to the infinite generalized eigenvalues of the associated Hamiltonian pencil. One of the main contributions of this paper is to demonstrate how to resolve these difficulties

The moss Bryum argenteum var. muticum Brid. is well adapted to cope with high light in continental Antarctica

The net photosynthetic rate (NP), chlorophyll fluorescence, carotenoid content and chlorophyll content of the cosmopolitan moss Bryum argenteum were measured in the field at Botany Bay, southern Victoria Land, continental Antarctica (77°S). Comparisons were made between sun- and shade-adapted forms, and changes were followed as the moss emerged from under the snow and during exposure of shade and sun forms to ambient light. Shade forms had lower light compensation and saturation values for NP but little difference in maximal NP rates. Shade forms exposed to ambient light changed rapidly (within five days) towards the performance of the sun forms. Surprisingly, this change was not by acclimation of shoots but by the production of new shoots. Chlorophyll and carotenoid levels measured on a molar chlorophyll basis showed no difference between sun and shade forms and also little change during emergence. The constant molar relationship between carotenoids and chlorophyll plus the high levels of the xanthophyll cycle pigments suggest that protection of the chlorophyll antenna was constitutive. This is an adaptation to the very high light levels that occur when the plants are active in continental Antarctica and contrasts to the situation in more temperate areas where high light is normally avoided by desiccation