Special Linear Systems Research Articles

Companion varieties for root systems and Fermat arrangements

Unexpected hypersurfaces are a brand name for some special linear systems. They were introduced around 2017 and are a field of intensive study since then. They attracted a lot of attention because of their close connections to various other areas of mathematics including vector bundles , arrangements of hyperplanes , geometry of projective varieties. Our research is motivated by what is now known as the BMSS duality, which is a new way of deriving projective varieties out of already constructed ones. The last author coined the concept of companion surfaces in the setting of unexpected curves admitted by the B 3 root system. Here we extend this construction in various directions. We revisit the configurations of points associated to either root systems or to Fermat arrangements and we study the geometry of the associated varieties. In the case of configurations of points associated to root systems, the geometry of their companions is also described.

Bounded controls for discrete-time linear systems subject to input time delay

This paper investigates the global stabilization of discrete-time linear systems with input time delay by bounded controls. Based on some special canonical forms containing time delays both in its input and state, two special discrete-time linear systems---multiple integrators and oscillators are first considered. The global stabilizing controllers are respectively established, and moreover, explicit conditions are established to guarantee the stability of the closed-loop systems. Subsequently, a concise design method is proposed for globally stabilizing general discrete-time linear system by combining the design methods for multiple integrators and oscillators. The designed controller is in the explicit form with explicit stability conditions being given, and thus is easier to use than the existing results. Finally, numerical simulations illustrate the effectiveness of the proposed approaches.

Limit cycles bifurcated from a focus-fold singularity in general piecewise smooth planar systems

We consider codimension-two focus-fold bifurcation for general piecewise smooth (PWS) planar systems. Some changes of variables and parameters are employed to derive a normal form, which is topologically equivalent to the original system. Making use of the normal form, we verify that observations of focus-fold bifurcation in some special piecewise linear systems also hold for general PWS systems, including the existence of two types of sliding limit cycles and coexistence of crossing and sliding limit cycles. It is shown that two nested crossing limit cycles can be bifurcated from the focus-fold singularity. This result is the improvement of the work [Han and Zhang (2010) [29]], in which only one bifurcated limit cycle is obtained. In addition, the sliding limit cycles are proved to be convergent in finite time.

Global geometry on moduli of local systems for surfaces with boundary

AbstractWe show that every coarse moduli space, parametrizing complex special linear rank-2 local systems with fixed boundary traces on a surface with nonempty boundary, is log Calabi–Yau in that it has a normal projective compactification with trivial log canonical divisor. We connect this to a novel symmetry of generating series for counts of essential multicurves on the surface.

Explicit estimates and limit formulae for the solutions of linear delay functional differential systems with nonnegative Volterra type operators

Abstract In this paper we discuss the asymptotic properties of solutions of special inhomogeneous linear delay functional differential systems generated by nonnegative Volterra type operators. In a recent paper of us a variation of constants formula has been given for such systems, and this is particularly suited to handling the studied problems. First, we investigate the boundedness of the solutions. Then some estimates are given for the upper and lower limits of the solutions. By using this, we study the existence of the limit of the solutions, and obtain a formula for the limit. The applicability of our results is illustrated by studying the dependence of limits of the solutions on the choice of inhomogeneities, synchronisation, and Lotka-Volterra type delay functional differential systems.

Analytical Solutions to Aeroassisted Orbital Transfer Problem

In this article, the atmospheric flight of aeroassisted orbit transfer vehicle is investigated analytically. As the flight speed is greater than the first cosmic velocity, the centrifugal force is also larger than gravity. Due to this, the vehicle has to fly with a downward lift and the flight altitude is extremely unstable. To cope with this challenge, the high-precision formulae of altitude and flight-path angle for steady glide are deduced and used to develop the steady glide control technique to stabilize the flight altitude. Upon steady glide, the analytical solutions for 3-D aeroassisted maneuvering trajectory are derived. In the derivation, a reduced-order system with negative lift-to-drag ratio as control variable is obtained by subtly simplifying the nonlinear flight dynamics. To address the complex influence of Coriolis force, the regular perturbation method is used such that the system is further decomposed into two special linear time-varying systems, which can be solved analytically by the spectral-decomposition-based method. Due to the full consideration of the influence of the Earth's curvature and rotation, the analytical solutions can predict the downrange, crossrange, and heading angle of the 3-D trajectory accurately.

Обобщенные критерии устойчивости особых линейных систем автоматического управления с запаздыванием

Matters concerned with stability of special linear automatic control systems with a time delay are considered. The determining feature of such systems is that the control loop contains a link of pure (or transport) delay by time T (or for the path S) of the signal at its output with respect to the signal at its input. Apart from time delay links, these systems also contain other linear elements. Such systems include both longstudied systems for transporting various materials and relatively new selflearning repetitive control systems for accurately reproducing cyclically repeated motions or other signals with a period T (or S). The stability of special linear systems can be studied with fully resting on the Nyquist frequency criterion. According to the Nyquist frequency criterion, the stability of a closedloop system is judged by analyzing the frequency transfer function (FTF) loci of the openloop system that includes the controller and the controlled object with respect to the critical point (1, j0) on the complex plane. However, in the majority of cases, the presence of transcendental links results in that the FTF loci of the openloop system have the shape of a star with infinitely long rays, which makes it difficult to interpret the stability of systems with a time delay according to the Nyquist criterion. In such cases, it is more convenient to use a group of graphic criteria, the application of which makes it possible to establish sufficient stability conditions without examining the entire openloop system (the controller plus the object) by analyzing the relative position of the controlled object frequency response loci (obtained without taking into account the controller properties) and a certain stability region boundary, which is determined by the controller properties (obtained without taking into account the controlled object properties). To date, stability region boundaries have already been found for many types of specific structures of systems with a time delay. The problem is that an individually shaped stability region boundary has to be found for each specific type of controller. It is shown, based on a structural transposition of systems with a time delay, that their stability can be studied by using almost any stability boundaries that were previously proposed for some specific types of transcendental controllers. By generalizing the criteria it will be possible not only to estimate for the first time the stability of systems with a time delay in combination with a linear part of an arbitrary kind, but also to expand the variety of applied types of transcendental controllers.

The PEk Algorithm of Solving the Quasi-Block Tridiagonal Systems

In this paper, the PEk method of solving special linear algebraic systems-block quasi-tridiagonal systems is presented. On the condition that the coefficient matrix is Hermitian positive definite, the solvability and convergence of the new method are proved, and the selection range for the parameters k is also given.

On a notion of toric special linear systems

In this note, we study linear systems on complete toric varieties X with an invariant point whose orbit under the action of Aut(X) contains the dense torus T of X. We give a characterization of such varieties in terms of its defining fan and introduce a new definition of expected dimension of linear systems which consider the contribution given by certain toric subvarieties. Finally, we study degenerations of linear systems on these toric varieties induced by toric degenerations.

Linear systems on rational elliptic surfaces and elliptic fibrations on K3 surfaces

We consider K3 surfaces which are double covers of rational elliptic surfaces. The former are endowed with a natural elliptic fibration, which is induced by the latter. There are also other elliptic fibrations on such K3 surfaces, which are necessarily induced by special linear systems on the rational elliptic surfaces. We describe these linear systems. In particular, we observe that every conic bundle on the rational surface induces a genus 1 fibration on the K3 surface and we classify the singular fibers of the genus 1 fibration on the K3 surface it terms of singular fibers and special curves on the conic bundle on the rational surface.

Some Simple Criteria for the Solvability of Block $$2 \times 2$$ 2 × 2 Linear Systems

In many areas, there arise linear systems of the form $$\begin{aligned} \left[ \begin{array}{ll} A &{}\quad B^\top \\ B &{}\quad -\,D \\ \end{array} \right] \left[ \begin{array}{c} x \\ y \\ \end{array} \right] =\left[ \begin{array}{c} f \\ g \\ \end{array} \right] , \end{aligned}$$ where $$A \in {\mathbf {R}}^{n\times n}, D \in {\mathbf {R}}^{p\times p}$$ are symmetric and positive semi-definite and $$B \in {\mathbf {R}}^{p \times n}.$$ In this paper, some simple criteria for this special linear systems to have solutions and the unique solution are provided, and the solvability conditions are expressed by $$A, B, D, f$$ and $$g$$ .

Abelianization of Fuchsian systems on a $4$-punctured sphere and applications

In this paper we consider special linear Fuchsian systems of rank $2$ on a $4-$punctured sphere and the corresponding parabolic structures. Through an explicit abelianization procedure we obtain a $2-$to$-1$ correspondence between flat line bundle connections on a torus and these Fuchsian systems. This naturally equips the moduli space of flat $SL(2,\mathbb C)-$connections on a $4-$punctured sphere with a new set of Darboux coordinates. Furthermore, we apply our theory to give a complex analytic proof of Witten's formula for the symplectic volume of the moduli space of unitary flat connections on the $4-$punctured sphere.

A metric graph satisfying w41=1$w_4^1 = 1$ that cannot be lifted to a curve satisfying dim⁡ (W41)=1$\dim \;(W_4^1 ) = 1$

Abstract For all integers g ≥ 6 we prove the existence of a metric graph G with w 4 1 = 1 $w_4^1 = 1$ such that G has Clifford index 2 and there is no tropical modification G′ of G such that there exists a finite harmonic morphism of degree 2 from G′ to a metric graph of genus 1. Those examples show that not all dimension theorems on the space classifying special linear systems for curves have immediate translation to the theory of divisors on metric graphs.

On linear systems of $$\mathbb {P}^3$$ P 3 with nine base points

We study special linear systems of surfaces of $$\mathbb {P}^3$$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in terms of linear obstructions for a quasi-homogeneous class. By degeneration, we also prove a Nagata type result for the blown-up projective plane in points that implies a base locus lemma for the quadric. As an application, we establish Laface–Ugaglia Conjecture for linear systems with multiplicities bounded by 8 and for homogeneous linear systems with multiplicity m and degree up to $$2m+1$$ .

Very special divisors on 4-gonal real algebraic curves

Abstract Given a real curve, we study special linear systems called “very special” for which the dimension does not satisfy a Clifford type inequality.We classify all these very special linear systems when the gonality of the curve is small.

Uncertain linear systems

In this paper, an uncertain linear system is defined and some special uncertain linear systems are studied by using uncertainty distributions. An approach for solving some special uncertain linear systems is designed and conditions for the existence of a solution to an uncertain linear system are presented. And, two examples are given to show the effectiveness of the proposed approach. Finally, an application to diet is given to show the practical significance of uncertain linear systems.

Sliding Mode <i>H</i><sup>∞</sup>-Control with Stochastic Time Delay Systems

In this paper, we concerned the problem of sliding mode of-control with stochastic stabilization of uncertainty. Some sufficient conditions are derived for this class of robust feedback stabilization of time delay systems. The stochastic time delay systems may switch from one to one corresponds of linear filter, such that the dynamics of estimation error is guaranteed to be stochastically stable in mean square. Moreover, it is shown that for a class of special linear stochastic neutral systems, the H-sliding mode control design can be obtained by solving linear matrix inequalities (LMIs).

Oscillation and spectral theory for linear Hamiltonian systems with nonlinear dependence on the spectral parameter

AbstractIn this paper, we consider linear Hamiltonian differential systems which depend in general nonlinearly on the spectral parameter and with Dirichlet boundary conditions. Our results generalize the known theory of linear Hamiltonian systems in two respects. Namely, we allow nonlinear dependence of the coefficients on the spectral parameter and at the same time we do not impose any controllability and strict normality assumptions. We introduce the notion of a finite eigenvalue and prove the oscillation theorem relating the number of finite eigenvalues which are less than or equal to a given value of the spectral parameter with the number of proper focal points of the principal solution of the system in the considered interval. We also define the corresponding geometric multiplicity of finite eigenvalues in terms of finite eigenfunctions and prove that the algebraic and geometric multiplicities coincide. The results are also new for Sturm–Liouville differential equations, being special linear Hamiltonian systems.

Special systems through double points on an algebraic surface

Let S be a smooth projective algebraic surface satisfying the following property: H i (S, B) = 0 for i > 0, for any irreducible and reduced curve B of S. The aim of this paper is to provide a characterization of special linear systems on S which are singular along a set of double points in very general position. As an application, the dimension of such systems is evaluated in case S is a simple Abelian surface, a K3 surface which does not contain elliptic curves or an anticanonical rational surface.

Linear systems over P1×P1with base points of multiplicity bounded by three

We propose a combinatorial method of proving non-specialty of a linear system of curves with multiple points in general position. As an application, we obtain a classification of special linear systems on ${\mathbb P}^1\times{\mathbb P}^1$ with multiplici