Related papers: Hilbert's 16th Problem for Quadratic Systems. New …
We study questions of existence and uniqueness of quadrature domains using computational tools from real algebraic geometry. These problems are transformed into questions about the number of solutions to an associated real semi-algebraic…
We focus on the second part of Hilbert's 16th problem and provide an upper bound on the number of limit cycles that a polynomial, differential, planar system may have, depending exclusively on the degree $n$ of the system. Such a bound…
A quadratically constrained quadratic programming problem is considered in a Hilbert space setting, where neither the objective nor the constraint are convex functions. Necessary and sufficient conditions are provided to guarantee that the…
The quantum dynamical systems of identical particles admitting an additional integral quadratic in momenta are considered. It is found that an appropriate ordering procedure exists which allows to convert the classical integrals into their…
We consider the problem of solving a large-scale Quadratically Constrained Quadratic Program. Such problems occur naturally in many scientific and web applications. Although there are efficient methods which tackle this problem, they are…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
The new necessary and sufficient affine invariant conditions for the existence and for determining the number of centers for general quadratic system are pointed out. These conditions correspond to the partition of 12-dimensional…
Quadratic descent of hermitian and skew hermitian forms over division algebras with involution of the first kind in arbitrary characteristic is investigated and a criterion, in terms of systems of quadratic forms, is obtained. A refined…
Hilbert-Arnold (HA) problem, motivated by Hilbert 16-th problem, is to prove that for a generic k-parameter family of smooth vector fields {\dot x=v(x,\eps)}_{\eps\in B^k} on the 2-dimensional sphere S^2 has uniformly bounded number of…
This paper focuses on the numerical solution of initial value problems for fractional differential equations of linear type. The approach we propose grounds on expressing the solution in terms of some integral weighted by a generalized…
A novel canonical transformation is offered as the mean for studying properties of a system of strongly correlated electrons. As an example of the utility of the transformation, it is used to demonstrate the existence of a quantum phase…
For the system of second order quasilinear parabolic equations the problem of reducing them to the equations of diffusion type is considered. In non-degenerate case an effective algorithm for solving this problem is suggested.
Inspired by Quantum Mechanics, we reformulate Hilbert's tenth problem in the domain of integer arithmetics into either a problem involving a set of infinitely coupled differential equations or a problem involving a Shr\"odinger propagator…
An extension of Riewe's fractional Hamiltonian formulation is presented for fractional constrained systems. The conditions of consistency of the set of constraints with equations of motion are investigated. Three examples of fractional…
The question is raised whether the unique decomposition of the physical Hilbert space, as emerging in the refined algebraic quantization of a constrained system, may be understood in terms of the old Klein-Gordon type quantization.
A form of the Laplace transform is reviewed as a paradigm for an entire class of fractional functional transforms. Various of its properties are discussed. Such transformations should be useful in application to differential/integral…
The paper concerns the solvability by quadratures of linear differential systems, which is one of the questions of differential Galois theory. We consider systems with regular singular points as well as those with (non-resonant) irregular…
The Linear Quadratic Regulator (LQR), which is arguably the most classical problem in control theory, was recently related to kernel methods in (Aubin-Frankowski, SICON, 2021) for finite dimensional systems. We show that this result extends…
It is shown that the complex Bernoulli differential equations admitting the supplementary structure of a Lie-Hamilton system related to the book algebra $\mathfrak{b}_2$ can always be solved by quadratures, providing an explicit solution of…
We prove that to each real singularity $f: (\mathbb{R}^{n}, 0) \to (\mathbb{R}^k, 0)$ with $k\geq 2$ one can associate systems of differential equations $\mathfrak{g}^{k}_f$ which are pushforwards in the category of $\mathcal{D}$-modules…