Abstract:
It is well known that an irreducible algebraic curve is rational (i.e. parametric) if and only if its genus is zero. In this paper, given a tolerance $\epsilon>0$ and an $\epsilon$-irreducible algebraic affine plane curve $\mathcal C$ of proper degree $d$, we introduce the notion of $\epsilon$-rationality, and we provide an algorithm to parametrize approximately affine $\epsilon$-rational plane curves, without exact singularities at infinity, by means of linear systems of $(d-2)$-degree curves. The algorithm outputs a rational parametrization of a rational curve $\bar{\mathcal C}$ of degree at most $d$ which has the same points at infinity as $\mathcal C$. Moreover, although we do not provide a theoretical analysis, our empirical analysis shows that $\bar{\mathcal C}$ and $\mathcal C$ are close in practice.

Abstract:
This manuscript contains technical details of recent results developed by the authors on adaptive model predictive control for constrained linear systems that exhibits exploring property and uses basis function model parametrization.

Abstract:
On the basis of Luenberger state observers, a new parametrization of all function observers for linear systems is presented. The physical significance and the state space realization of this parametrization are also analyzed in detail. The order of the class of all function observers is minimal, which is lower than the known one.

Abstract:
A good state-time quantized symbolic abstraction of an already input quantized control system would satisfy three conditions: proximity, soundness and completeness. Extant approaches for symbolic abstraction of unstable systems limit to satisfying proximity and soundness but not completeness. Instability of systems is an impediment to constructing fully complete state-time quantized symbolic models for bounded and quantized input unstable systems, even using supervisory feedback. Therefore, in this paper we come up with a way of parametrization of completeness of the symbolic model through the quintessential notion of Trimmed-Input Approximate Bisimulation which is introduced in the paper. The amount of completeness is specified by a parameter called trimming of the set of input trajectories. We subsequently discuss a procedure of constructing state-time quantized symbolic models which are near-complete in addition to being sound and proximate with respect to the time quantized models.

Abstract:
The paper considers pseudo-differential boundary value control systems. The underlying operators form an algebra D with the help of which we are able to formulate typical boundary value control problems. The symbolic calculus gives tools to form e.g. compositions, formal adjoints, generalized right or left inverses and compatibility conditions. By a parametrizability we mean that for a given control system Au=0 one finds an operator S such that Au=0 if and only if u=Sf. The computation rules of D (or its appropriate subalgebra D') guarantee that in many applications S can be refinely analyzed or even explicitly calculated. We outline some methods of homological algebra for the study of parametrization S. Especially the projectivity of a certain factor module (defined by the system equations) implies the parametrizability. We give some examples to illustrate our computational methods.

Abstract:
Given a general Enriques surface T and a genus g linear system |C| on T, we consider the relative compactified Jacobian N=\Jac(|C|) \to |C|. Let f:S \to T be the universal K3 cover and set D=f^*C. We show that N is a smooth (2g-1)-dimensional variety and that it admits an \'etale double cover to a Lagrangian subvariety of the relative compactified Jacobian M=\Jac(|D|). The main results are that, under some technical assumption that can be verified for low values of g, we prove that \pi_1(N)=\Z/(2), that \omega_N={\mc O}_N$, and that h^{p,0}(N)=0 for p not equal to 0 or to 2g-1.

Abstract:
We consider multidimensional systems of PDEs of generalized evolution form with t-derivatives of arbitrary order on the left-hand side and with the right-hand side dependent on lower order t-derivatives and arbitrary space derivatives. For such systems we find an explicit necessary condition for existence of higher conservation laws in terms of the system's symbol. For systems that violate this condition we give an effective upper bound on the order of conservation laws. Using this result, we completely describe conservation laws for viscous transonic equations, for the Brusselator model, and the Belousov-Zhabotinskii system. To achieve this, we solve over an arbitrary field the matrix equations SA=A^tS and SA=-A^tS for a quadratic matrix A and its transpose A^t, which may be of independent interest.

Abstract:
The bases of the theory of integrals for multidimensional differential systems are stated. The integral equivalence of total differential systems, linear homogeneous systems of partial differential equations, and Pfaff systems of equations is established.

Abstract:
We extend the known piecewise linear parametrization of the canonical basis of the plus part of an enveloping algebra of type ADE to the nonsimplylaced case.

Abstract:
This paper is concerned with a dissipativity theory for dynamical systems governed by linear Ito stochastic differential equations driven by random noise with an uncertain drift. The deviation of the noise from a standard Wiener process in the nominal model is quantified by relative entropy. We discuss a dissipation inequality for the noise relative entropy supply. The problem of minimizing the supply required to drive the system between given Gaussian state distributions over a specified time horizon is considered. This problem, known in the literature as the Schroedinger bridge, was treated previously in the context of reciprocal processes. A closed-form smooth solution is obtained for a Hamilton-Jacobi equation for the minimum required relative entropy supply by using nonlinear algebraic techniques.