Algebraic Sufficient Conditions Research Articles

Non-affine n-valued maps on tori

In this paper we construct n-valued maps on k-dimensional tori, where n,k≥2, that are not homotopic to affine n-valued maps. This is in high contrast with the single valued case, where any such map is homotopic to an affine (even linear) map. We do this by investigating necessary and sufficient algebraic conditions on certain induced morphisms.

From area metric backgrounds to the cosmological constant and corrections to the Polyakov action

Area metrics and area metric backgrounds provide a unified framework for quantum gravity. They encode physical degrees of freedom beyond those of a metric. These nonmetric degrees of freedom must be suppressed by a potential at sufficiently high energy scales to ensure that in the infrared regime classical gravity is recovered. On this basis, we first study necessary and sufficient algebraic conditions for an area metric to be induced by a metric. Second, we consider candidate potentials for the area metric and point out a possible connection between the reduction of area metric geometry to metric geometry on the one hand, and the smallness of the cosmological constant on the other. Finally, we consider modifications of the Nambu-Goto action for a string from a metric background to an area metric background. We demonstrate that area metric perturbations introduce an interaction corresponding to a singular vertex operator in the classically equivalent Polyakov action. The implications of these types of vertex operators for the quantum theory remain to be understood. Published by the American Physical Society 2024

On a Geometry Problem in Plato’s Meno

In this paper, we determine necessary and sufficient algebraic conditions for the solution of the triangle inscription problem in Plato’s Meno, in the form of an isosceles triangle, to be constructed with straightedge and compass. We apply these conditions to find infinitely many solutions, in the form of an isosceles triangle, that cannot be constructed with straightedge and compass. We prove that only countably many solutions in the form of an isosceles triangle can be constructed with straightedge and compass.

Finite-time synchronization of discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances

In this paper, the finite-time synchronization issue is addressed for discontinuous fuzzy neural networks with mixed time-varying delays and impulsive disturbances. The dynamic impulsive gain about the impulsive moment is proposed as an alternative to the previous static impulsive gain. The time-varying impulsive gain can combine the synchronizing and desynchronizing impulses to directly observe their effects on the system, avoiding the classification discussion. In order to study the finite-time synchronization of the drive–response system, we obtain an equivalent system from the original fuzzy neural networks based on the concepts of Filippov solutions and differential inclusion theory. Furthermore, by constructing new Lyapunov functions that combine the 1-norm and the comparative system method, some algebraic sufficient conditions for the synchronization of the drive system and the response system in finite time are established, which are based on two types of novel non-chattering controllers. In addition, the parameters of the system itself can simply demonstrate the sufficient conditions that have been obtained her e, avoiding the time-consuming calculation of matrix inequalities. Moreover, the settling time estimation scheme have been established. Eventually, numerical examples are provided to demonstrate the effectiveness of the control scheme.

Strong Left Inversion of Linear Systems and Input Reconstruction

Left inversion of linear time-invariant systems aims at identifying the input acting on a system, for the zero initial state, from partial information on the input and the state. In the present contribution, we propose to generalize such left inversion for linear systems in various directions, that take arbitrary initial state into consideration to address both exact and asymptotic input reconstruction. Necessary and sufficient algebraic conditions are given to achieve such strong left inversion properties. Complete characterizations of introduced concepts in terms of system zeros are provided. Relationship between inversion and input reconstruction is therefore investigated, with emphasis on causal realization. Conditions for input observer existence are proposed, and a constructive causal design for an asymptotically convergent input observer is presented.

Algebraic and geometric properties of [formula omitted]-semipositive matrices and [formula omitted]-semipositive cones

Given a proper cone K in the Euclidean space Rn, a square matrix A is said to be K-semipositive if there exists an x∈K such that Ax∈int(K), the topological interior of K. The paper aims to study algebraic and geometrical properties of K-semipositive matrices with special emphasis on the self-dual proper Lorentz cone L+n={x∈Rn:xn≥0,∑i=1n−1xi2≤xn2}. More specifically, we discuss a few necessary and other sufficient algebraic conditions for L+n-semipositive matrices. Also, we provide algebraic characterizations for diagonal and orthogonal L+n-semipositive matrices. Furthermore, given a square matrix A and a proper cone K, geometric properties of the semipositive cone KA,K={x∈K:Ax∈K} and the cone of SA,K={x:Ax∈K} are discussed in terms of their extremals. As L+n is an ellipsoidal cone, at last we find results for the cones KA,L+n and SA,L+n to be ellipsoidal.

The moment exponential stability of infinite-dimensional linear stochastic switched systems

<abstract><p>This paper studies the 2nd-moment exponential stability of a class of infinite-dimensional linear stochastic switched systems comprising two unstable subsystems. We first construct an algebraic sufficient condition on the existence of multiple Lyapunov functions. Then, two switching strategies are designed to stabilize infinite-dimensional linear stochastic switched systems in terms of the multiple Lyapunov function method. Moreover, the system possesses good robust stability of the switching time with our switching strategies.</p></abstract>

Criterion of Absolute Stability of Solutions of Stochastic Diffusion Dynamic Information Systems of Automatic Regulation With External Disturbances

The conditions of asymptotic stability of the trivial solution of stochastic diffusion dynamic systems of automatic regulation with external disturbances are obtained. In work is to find criteria for the absolute stability of solutions of stochastic diffusion dynamic systems of automatic regulation (SAR) with external disturbances. The problem of determining sufficient algebraic conditions for the stability of stochastic diffusion dynamic SARs with aftereffect and taking into account external disturbances is relevant, since its solution makes it possible to study complex stochastic systems using matrix calculations.

Runge-Kutta Methods for Rough Differential Equations

We study Runge-Kutta methods for rough differential equations which can be used to calculate solutions to stochastic differential equations driven by processes that are rougher than a Brownian motion. We use a Taylor series representation (B-series) for both the numerical scheme and the solution of the rough differential equation in order to determine conditions that guarantee the desired order of the local error for the underlying Runge-Kutta method. Subsequently, we prove the order of the global error given the local rate. In addition, we simplify the numerical approximation by introducing a Runge-Kutta scheme that is based on the increments of the driver of the rough differential equation. This simplified method can be easily implemented and is computational cheap since it is derivative-free. We provide a full characterization of this implementable Runge-Kutta method meaning that we provide necessary and sufficient algebraic conditions for an optimal order of convergence in case that the driver, e.g., is a fractional Brownian motion with Hurst index $\frac{1}{4} < H \leq \frac{1}{2}$. We conclude this paper by conducting numerical experiments verifying the theoretical rate of convergence.

On the algebraic properties of exponentially stable integrable hamiltonian systems

Steepness is a geometric property which, together with complex-analyticity, is needed in order to insure stability of a near-integrable hamiltonian system over exponentially long times. Following a strategy developed by Nekhoro-shev, we construct sufficient algebraic conditions for steepness for a given function that involve algebraic equations on its derivatives up to order five. The underlying analysis suggests some interesting considerations on the gener-icity of steepness and represents a first step towards the construction of sufficient conditions for steepness involving the derivatives of the studied function up to an arbitrary order.

Nonchaotic Behavior and Transition to Chaos in Lorenz-like Systems Having Invariant Algebraic Surfaces

The famous and well-studied Lorenz system is considered a paradigm for chaotic behavior in three-dimensional continuous differential systems. After the appearance of such a system in 1963, several Lorenz-like chaotic systems have been proposed and studied in the related literature, as Rössler system, Chen-Ueta system, Rabinovich system, Rikitake system, among others. However, these systems are parameter dependent and are chaotic only for suitable combinations of parameter values. This raises the question of when such systems are not chaotic, which can be seen as a dual problem regarding chaotic systems. In this paper, we give sufficient algebraic conditions for a generalized class of Lorenz-like systems to be nonchaotic. Using the general results obtained, we give some examples of nonchaotic behavior of some classical ``chaotic'' Lorenz-like systems, including the Lorenz system itself. The nonchaotic differential systems presented here have invariant algebraic surfaces, which contain the stable (or unstable) invariant manifolds of their equilibrium points. We show that, in some cases, the deformation of these invariant manifolds through the destruction of the invariant algebraic surfaces, by perturbing the parameter values, can reorganize the global structure of the phase space, leading to a transition from nonchaotic to chaotic behavior of such differential systems.

Controllability on a class of switched time-varying systems with impulses and multiple time delays

The problem of complete controllability is addressed for a class of switched time-varying impulsive systems under multiple input delays. Several controllability criteria are obtained with the aid of algebraic and geometric tools. Firstly, the explicit expression of solution on every impulsive interval for such systems is derived by mathematical induction and variation of parameters. Then, complete controllability is investigated for the underlying system based on the solution established. Specifically, the new necessary or/and sufficient algebraic controllability conditions are obtained without assuming that all impulse-dependent matrices are non-singular. In addition, the geometric criteria with regard to the constant matrices are further discussed when the considered system reduces to a time-invariant case. Finally, numerical examples are worked out to demonstrate the controllability tests derived in this paper. It is shown that the existence of the delayed term will potentially affect the controllability of the studied system. In contrast to the existing literatures, the criteria derived in this paper removing some constrained assumptions have less conservatism and can be used to verify the controllability of a more general linear hybrid system.

Stochastic Synchronization of Nonlinear Networks With Directed Graphs and Degenerate Noise

Noise is ubiquitous in network systems and has profound impacts on network dynamics. Despite substantial progress in stochastic synchronization of complex networks, most studies limit their focus to nondegenerate noise. However, the influence of degenerate noise on synchronization has not been studied properly. Here, we tackle this challenging problem by introducing a network of nonlinear units with a deterministic communication digraph and a noisy communication digraph. The communication links in the noisy communication digraph are affected by degenerate noise in a relative-dependent manner. Two types of deterministic communication digraphs are considered: 1) the case of strongly connected graphs and 2) the case of graphs containing a spanning tree. By using the algebraic graph theory and the stability theory of stochastic differential equations, algebraic sufficient conditions for both cases are derived, which depend on node dynamics, network structure, and coupling strengths. These criteria demonstrate that strong deterministic communication can compensate for the detrimental effect of degenerate noise, thus ensuring the stochastic synchronization in the almost sure sense. Numerical simulations support and illustrate the proposed theoretical results.

Structural Completeness of a Multichannel Linear System With Dependent Parameters

It is well known that the "fixed spectrum" {i.e., the set of fixed modes} of a multi-channel linear system plays a central role in the stabilization of such a system with decentralized control. A parameterized multi-channel linear system is said to be "structurally complete" if it has no fixed spectrum for almost all parameter values. Necessary and sufficient algebraic conditions are presented for a multi-channel linear system with dependent parameters to be structurally complete. An equivalent graphical condition is also given for a certain type of parameterization.

A Single-Adversary-Single-Detector Zero-Sum Game in Networked Control Systems*

This paper proposes a game-theoretic approach to address the problem of optimal sensor placement for detecting cyber-attacks in networked control systems. The problem is formulated as a zero-sum game with two players, namely a malicious adversary and a detector. Given a protected target vertex, the detector places a sensor at a single vertex to monitor the system and detect the presence of the adversary. On the other hand, the adversary selects a single vertex through which to conduct a cyber-attack that maximally disrupts the target vertex while remaining undetected by the detector. As our first contribution, for a given pair of attack and monitor vertices and a known target vertex, the game payoff function is defined as the output-to-output gain of the respective system. Then, the paper characterizes the set of feasible actions by the detector that ensures bounded values of the game payoff. Finally, an algebraic sufficient condition is proposed to examine whether a given vertex belongs to the set of feasible monitor vertices. The optimal sensor placement is then determined by computing the mixed-strategy Nash equilibrium of the zero-sum game through linear programming. The approach is illustrated via a numerical example of a 10-vertex networked control system with a given target vertex.

The degrees of maps between -connected -dimensional manifolds or Poincaré complexes and their applications

In this paper, using homotopy theoretical methods we study the degrees of maps between -connected -dimensional Poincaré complexes. Necessary and sufficient algebraic conditions for the existence of mapping degrees between such Poincaré complexes are established. These conditions allow us, up to homotopy, to construct explicitly all maps with a given degree. As an application of mapping degrees, we consider maps between -connected -dimensional Poincaré complexes with degree , and give a sufficient condition for these to be homotopy equivalences. This resolves a homotopy theoretical analogue of Novikov’s question: when is a map of degree between manifolds a homeomorphism? For low , we classify, up to homotopy, torsion free -connected -dimensional Poincaré complexes. Bibliography: 29 titles.

Pinning multisynchronization of delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations

This paper concerns the multisynchronization issue for delayed fractional-order memristor-based neural networks with nonlinear coupling and almost-periodic perturbations. First, the coexistence of multiple equilibrium states for isolated subnetwork is analyzed. By means of state-space decomposition, fractional-order Halanay inequality and Caputo derivative properties, the novel algebraic sufficient conditions are derived to ensure that the addressed networks with arbitrary activation functions have multiple locally stable almost periodic orbits or equilibrium points. Then, based on the obtained multistability results, a pinning control strategy is designed to realize the multisynchronization of the N coupled networks. By the aid of graph theory, depth first search method and pinning control law, some sufficient conditions are formulated such that the considered neural networks can possess multiple synchronization manifolds. Finally, the multistability and multisynchronization performance of the considered neural networks with different activation functions are illustrated by numerical examples.

Equivariant basic cohomology under deformations

There is a natural way to deform a Killing foliation with non-closed leaves, due to Ghys and Haefliger--Salem, into a closed foliation, i.e., a foliation whose leaves are all closed. Certain transverse geometric and topological properties are preserved under these deformations, as previously shown by the authors. For instance, the basic Euler characteristic is invariant. In this article we show that the equivariant basic cohomology ring structure is preserved under these deformations, which in turn leads to a sufficient algebraic condition (namely, equivariant formality) for the Betti numbers of basic cohomology to be preserved as well. In particular, this is true for the deformation of the Reeb orbit foliation of a $K$-contact manifold. Another consequence is that there is a universal bound on the sum of basic Betti numbers of any equivariantly formal, positively curved Killing foliation of a given codimension. We also show that a Killing foliation with negative transverse Ricci curvature is closed. If the transverse sectional curvature is negative we show, furthermore, that its fundamental group has exponential growth. Finally, we obtain a transverse generalization of Synge's theorem to Killing foliations.

Mittag-Leffler Synchronization of Delayed Fractional Memristor Neural Networks via Adaptive Control.

This brief is devoted to exploring the global Mittag-Leffler (ML) synchronization problem of fractional-order memristor neural networks (FOMNNs) with leakage delay via a hybrid adaptive controller. By applying Fillipov's theory and the Lyapunov functional method, the novel algebraic sufficient condition for the global ML synchronization of FOMNNs is derived. Finally, a simulation example is presented to show the practicability of our findings.

Event-triggered control design for networked evolutionary games with time invariant delay in strategies

This paper investigates the dynamics and control problem for a class of networked evolutionary games with time invariant delay (DNEGs) in strategies by using a semi-tensor product based method. A number of new results are presented. First, algebraic forms are constructed for DNEGs. Second, based on the algebraic formulations, necessary and sufficient conditions for the global convergence of desired strategy profile are presented under a state feedback event-triggered controller. Furthermore, the constructive procedure and the number of all valid event-triggered state feedback controllers are derived, which can make the game converge globally. Finally, an illustrative example is given to show the effectiveness of the obtained results.