Elimination of unknowns for systems of algebraic differential-difference equations
We prove effective Nullstellensatz and elimination theorems for difference equations in sequence rings. More precisely, we compute an explicit function of geometric quantities associated to a system of difference equations (and these geometric quantities may themselves be bounded by a function of the number of variables, the order of the equations, and the degrees of the equations) so that for any system of difference equations in variables x=(x_1,…,x_m) and u=(u_1,…,u_r), if these equations have any nontrivial consequences in the x variables, then such a consequence may be seen algebraically considering transforms up to the order of our bound. Specializing to the case of m=0, we obtain an effective method to test whether a given system of difference equations is consistent.
I give the explicit formula for the (set-theoretical) system of Resultants of m+1 homogeneous polynomials in n+1 variables
The paper presents an approach to the implementation of multiport linear dynamical blocks in a circuit simulator. Block characteristics are supposed to be presented in the form of rational transfer functions. The approach is based on the state space method, which allows avoiding problems of numerical convolution. The extension of the approach to the case of the exponential-rational form of transfer function is taken into account. The MNA formulation of Laplace element model is presented. The methodology providing construction of arbitrary multi-port Laplace elements on the base of developed set of simple Laplace elements is proposed. The approach was implemented in SPICE3 circuit simulator. Examples are given in the paper.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.