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We push a little further the study of two recent characterizations of almost perfect nonlinear (APN) functions. We state open problems about them, and we revisit in their perspective a well-known result from Dobbertin on APN exponents. This leads us to a new result about APN power functions and more general APN polynomials with coefficients in a subfield \begin{document}$ \mathbb{F}_{2^k} $\end{document} , which eases the research of such functions. It also allows to construct automatically many differentially uniform functions from them (this avoids calculations for proving their differential uniformity as done in a recent paper, which are tedious and specific to each APN function). In a second part, we give simple proofs of two important results on Boolean functions, one of which deserves to be better known but needed clarification, while the other needed correction.

We consider relationships between Vandermonde sets and hyperovals. Hyperovals are Vandermonde sets, but in general, Vandermonde sets are not hyperovals. We give necessary and sufficient conditions for a Vandermonde set to be a hyperoval in terms of power sums. Therefore, we provide purely algebraic criteria for the existence of hyperovals. Furthermore, we give necessary and sufficient conditions for the existence of hyperovals in terms of \begin{document}$ g $\end{document} -functions, which can be considered as an analog of Glynn's Theorem for \begin{document}$ o $\end{document} -polynomials. We also get some important applications to Niho bent functions.

Almost all event-triggered control (ETC) strategies were designed for discrete-time or continuous-time systems. In order to unify these existing theoretical results of ETC and develop ETC strategies for nonlinear systems, whose state variables evolve steadily at one time and change intermittently at another time, this article investigates quasisynchronization of delayed neural networks (NNs) on time scales with discontinuous activation functions via ETC approaches. First, the existence of the Filippov solutions is proved for discontinuous NNs with finite discontinuities. Second, two static event-triggered conditions and two dynamic event-triggered conditions are established to avoid continuous communication between the master-slave systems under algebraic/matrix inequality criteria. Third, under static/dynamic event-triggered conditions, a positive lower bound of event-triggered intervals is demonstrated to be greater than a positive number for each event-based controller, which shows that the Zeno behavior will not occur. Finally, two numerical simulations are carried out to show the effectiveness of the presented theoretical results in this article.

In this paper, we first introduce a class of quartic Boolean functions. And then, the construction of weightwise perfectly balanced Boolean functions on \begin{document}$ 2^m $\end{document} variables are given by modifying the support of the quartic functions, where \begin{document}$ m $\end{document} is a positive integer. The algebraic degree, the weightwise nonlinearity, and the algebraic immunity of the newly constructed weightwise perfectly balanced functions are discussed at the end of this paper.

Let $r, \,m$ be positive integers. Let $x$ be a rational number with $0 \le x, Comment: Corrected typos

We develop a theory of modular forms on the groups SO ( 3 , n + 1 ) , n ≥ 3 . This is very similar to, but simpler, than the notion of modular forms on quaternionic exceptional groups, which was initiated by Gross-Wallach and Gan-Gross-Savin. We prove the results analogous to those of earlier papers of the author on modular forms on exceptional groups, except now in the familiar setting of classical groups. Moreover, in the setting of SO ( 3 , n + 1 ) , there is a family of absolutely convergent Eisenstein series, which are modular forms. We prove that these Eisenstein series have algebraic Fourier coefficients, like the classical holomorphic Eisenstein series on SO ( 2 , n ) . As an application, using a local result of Savin, we prove that the so-called “next-to-minimal” modular form on quaternionic E 8 has rational Fourier expansion.

This article investigates the fixed-time synchronization problem of a class of delayed complex-valued inertial neural networks (CVINNs). The analysis does not rely on the traditional reduced-order transformation, but constructing Lyapunov functions directly focused on the original system. Based on the direct method and the separation method, different control strategies are proposed, under which the addressed CVINNs can achieve synchronization perfectly in a fixed time. The corresponding synchronization criteria in terms of matrix inequalities are derived, which are more concise and easier to verify than algebraic inequalities conditions, and the estimation of the settling times. The direct method makes full use of some innovative inequalities in the complex field; the exponential parameters in the designed controllers are independent. Finally, in order to fully support the theoretical results, based on two typical activation functions, the proposed theoretical results are numerically validated and compared.

Let f be an algebraic valued periodic arithmetical function and L ( s , f ) , defined as L ( s , f ) : = ∑ n = 1 ∞ f ( n ) / n s for ℜ ( s ) > 1 , be the associated Dirichlet series. In this paper, we study the vanishing and arithmetic nature of the special values L ( k , f ) when k > 1 is a positive integer. We prove a generalization of the Baker-Birch-Wirsing theorem conditional on the Polylog conjecture. Adopting a new approach, we define an induction operator on the space of periodic arithmetic functions, which makes precise the notion of an “imprimitive” arithmetic function. This enables us to obtain an analog of Okada's criterion for L ( 1 , f ) = 0 and derive a natural decomposition of the vector space O k ( N ) = { f : Z → Q | f ( n + N ) = f ( n ) for all n ∈ Z , L ( k , f ) = 0 } .

We compute periods of perturbations of a Fermat variety. This allows us to consider a subspace of the Hodge cycles defined by "simple" arithmetic conditions. We explore some examples and give an upper bound for the dimension of this subspace. As an application, we find explicit expressions involving some Gauss' hypergeometric functions which are algebraic over the field of rational functions in one variable., 26 pages. Notations have been improved. Final version to appear in Journal of Number Theory

This study aims to develop the notion of general fuzzy automata (GFA) to a new one which is known as “ L B -valued general fuzzy automata”. Instead of the term L B -valued general fuzzy automata, for simplicity, L B -valued GFA is used where B is regarded as a set of propositions about the GFA, in which its underlying structure has been a complete infinitely distributive lattice. Further, L B -valued GFA is scrutinized via different operators and also the interrelationship among these operators is examined. Specifically, it is shown that L B -valued successor, L B -valued predecessor and L B -valued residuated operators play an important role in the algebraic study of L B -valued general fuzzy automaton and that under certain conditions these operators are interrelated. In addition, the concept of a homomorphism between L B -valued general fuzzy automata is introduced and studied. Finally, the concepts of equivalence and congruence are defined, the quotient L B -valued GFA with respect to congruence is formulated and the equivalence between L B -valued GFA and its quotient automaton is proved. To clarify the notions and the results obtained in this study, some examples are submitted as well.

The existing results of Lagrange stability for neural networks with distributed time delays are scale-free, which introduces conservativeness naturally. A class of Takagi-Sugeno fuzzy memrisive neural networks (FMNNs) on time scales with discrete time-varying and infinite distributed delays is brought in this paper. First, a new scale-limited Halanay inequality is demonstrated by timescale theory. Next, on the basis of inequality techniques on time scales, some new scale-limited algebraic criteria and linear matrix inequality criteria of Lagrange stability are obtained by comparison strategy and generalized Halanay inequality. All scale-limited sufficient criteria of Lagrange stability for FMNNs not only apply to continuous-time FMNNs and their discrete-time analogues, but also could deal with the arbitrary combination of them. Finally, two numerical simulations are given to verify the validity of the obtained theoretical results.

We give a Schinzel-Zassenhaus-type lower bound for the maximum modulus of roots of a monic integer polynomial with all roots symmetric with respect to the unit circle. Our results extend a recent work of Dimitrov, who proved the general Schinzel-Zassenhaus conjecture by using the P\'olya rationality theorem for a power series with integer coefficients, and some estimates for logarithmic capacity (transfinite diameter) of sets. We use an enhancement of P\'olya's result obtained by Robinson, which involves Laurent-type rational functions with small supremum norms, thereby replacing the logarithmic capacity with a smaller quantity. This smaller quantity is expressed via a weighted Chebyshev constant for the set associated with Dimitrov's function used in Robinson's rationality theorem. Our lower bound for the house confirms a conjecture of Boyd., Comment: 13 pages

A basic problem in transcendental number theory is to determine the arithmetic properties of values of special functions. Many special functions, such as Bessel functions and certain hypergeometric functions, are E-functions which are a natural generalization of the exponential function and satisfy certain linear differential equations. In this case, there exists an algorithm which determines if f ( α ) is transcendental or algebraic if f ( z ) is an E-function and α ∈ Q ‾ ⁎ is a non-zero algebraic number. In this paper, we consider the analogous question when f ( z ) satisfies an integral equation, in particular, a Fredholm integral equation of the first or second kind where the kernel and forcing term satisfy strong arithmetic properties. We show that in both periodic and non-periodic cases, there exists no algorithm to determine if f ( 0 ) ∈ Q is rational. Our results are an application of the undecidability of the Generalized Collatz Problem due to Conway [6] .

We consider a gradient-like system for real-time implementation of multivariable algebraic loops arising in input-constrained control problems. Using results from mathematical programming, we establish global asymptotic convergence under a less stringent condition as compared to existing techniques. We comment on the application of the gradient-like system in anti-windup control implementation where the closed loop can also be interpreted within the framework of singularly perturbed systems. The proposed system can easily be realized for practical circuit implementation and may be prototyped using fast analog processors.

Let $(K, v)$ be a Henselian discrete valued field with a quasifinite residue field. This paper proves the existence of an algebraic extension $E/K$ satisfying the following: (i) $E$ has dimension dim$(E) \le 1$, i.e. the Brauer group Br$(E ^{\prime })$ is trivial, for every algebraic extension $E ^{\prime }/E$; (ii) finite extensions of $E$ are not $C _{1}$-fields. This, applied to the maximal algebraic extension $K$ of the field $\mathbb{Q}$ of rational numbers in the field $\mathbb{Q} _{p}$ of $p$-adic numbers, for a given prime $p$, proves the existence of an algebraic extension $E _{p}/\mathbb{Q}$, such that dim$(E _{p}) \le 1$, $E _{p}$ is not a $C _{1}$-field, and $E _{p}$ has a Henselian valuation of residual characteristic $p$., Comment: 17 pages, LaTeX: final form, incorporates Referee's suggestions, to appear in Journal of Number Theory

The published stability criteria for impulsive neural networks are scale-free on time line, which is only appropriate for discrete or continuous ones. The issue of global exponential stability for impulsive delayed neural networks on time scales is analyzed by employing the convex combination method in this article. Several algebraic and linear matrix inequality conditions are proved by constructing impulse-dependent Lyapunov functionals and using timescale inequality techniques. Unlike the published works, impulsive control strategies can be designed by utilizing our theoretical results to stabilize delayed neural networks on time scales if they are unstable before introducing impulses. Sufficient criteria for global exponential stability in this article are derived based on the timescale theory, and they are applicable to discrete-time impulsive neural networks, their continuous-time analogues, and neural networks whose states are discrete at one time and continuous at another time. Four numerical examples are offered to demonstrate the effectiveness and superiority of our new theoretical results in the end.

This article focuses on the problem of a distributed computation of continuous-time algebraic Riccati equations (CARE), where information of matrices is split and known by multiple agents. This article proposes a distributed optimization design of the iterative refinement technique (IRM), a well-established centralized method for CARE. By assuming that each agent only knows partial information of CARE, we reformulate IRM for CARE as three classes of distributed optimization subproblems with different formulations and constraints. Then, we propose distributed algorithms for obtained distributed optimization subproblems and prove convergence properties of proposed algorithms. Numerical results show the efficacy of the proposed distributed IRM.

Integral equations are extensively used in many physical models appearing in the field of plasma physics, atmosphere–ocean dynamics, fluid mechanics, mathematical physics and many other disciplines of physics and engineering. In this research work, a new numerical technique for the solution of the system of Fredholm integral equations (FIEs) of both first and second kinds is established, which is based on Bernstein basis functions. Here, the system of FIEs of both kinds has been taken, then reduces the equations to an algebraic linear system and can be solved using any standard rule. Convergence analysis of the proposed technique and some useful numerical results are presented so that the reader could understand this idea easily. Further, Hyers-Ulam stability analysis criteria is used for analyzing the stability of the proposed technique. The comparison of exact and approximate solutions of some problems is demonstrated in tables and their graphs are plotted to show the efficiency of the given technique.

We propose a new predictor-corrector (PC) interior-point algorithm (IPA) for solving linear complementarity problem (LCP) with P * ( κ ) -matrices. The introduced IPA uses a new type of algebraic equivalent transformation (AET) on the centering equations of the system defining the central path. The new technique was introduced by Darvay and Takacs (2018) for linear optimization. The search direction discussed in this paper can be derived from positive-asymptotic kernel function using the function φ ( t ) = t 2 in the new type of AET. We prove that the IPA has O ( ( 1 + 4 κ ) n log 3 n μ 0 4 ϵ ) iteration complexity, where κ is an upper bound of the handicap of the input matrix. To the best of our knowledge, this is the first PC IPA for P * ( κ ) -LCPs which is based on this search direction.

This article proposes a distributed optimization design to compute continuous-time algebraic Riccati inequalities (ARIs), where the information of matrices is distributed among agents. We propose a design procedure to tackle the nonlinearity, the inequality, and the coupled information structure of ARI; then, we design a distributed algorithm based on an optimization approach and analyze its convergence properties. The proposed algorithm is able to verify whether ARI is feasible in a distributed way and converges to a solution if ARI is feasible for any initial condition.

Boolean functions with five-valued Walsh spectra (shortened as five-valued functions) and semi-bent functions are two classes of Boolean functions with high nonlinearity. They have useful applications in cryptography and communications. In this paper, we first present the construction of resilient five-valued functions by modifying the support of Rothaus’s bent function. And then, we give the method of constructing resilient semi-bent functions. At the same time, the number of the newly constructed resilient semi-bent functions is determined. Lastly, we give the construction of resilient semi-bent functions with maximal algebraic degree.

We present an alternative relatively easy way to understand and determine the zeros of a quintic whose Galois group is isomorphic to the group of rotational symmetries of a regular icosahedron. The extensive algebraic procedures of Klein in his famous \textit{Vorlesungen uber das Ikosaeder und die Auflosung der Gleichungen vom funften Grade} are here shortened via Heymann's theory of transformations. Also, we give a complete explanation of the so-called icosahedral equation and its solution in terms of Gaussian hypergeometric functions. As an innovative element, we construct this solution by using algebraic transformations of hypergeometric series. Within this framework, we develop a practical algorithm to compute the zeros of the quintic.

In the sensorless control system of an induction machine, the rotor speed value is not measured but reconstructed by an observer structure. The rotor speed value can be reconstructed by the classical adaptive law with the integrator. The second approach, which is the main contribution of this paper, is the non-adaptive structure without an integrator. The proposed method of the rotor speed reconstruction is based on an algebraic relationship – the rank of the mathematical model of the observer system is not increased. However, the problem with the stabilization of the observer structure does exist. For near to zero rotor speed or in the regenerating mode of an induction machine, the speed observer structure can be unstable. Therefore, in this paper, the new stabilization functions are proposed. The stability is provided by the Lyapunov theorem and the practical stability theorems in which the uncertainty of parameters is considered. In the proposed solution, the newly introduced stabilization functions guarantee observer stability during both the motoring and regenerating conditions at the chosen low rotor speed ranges and for different load torque values. All the theoretical considerations were confirmed by simulation and experimental tests during the chosen working modes and uncertainties of nominal parameters of the induction machine.

This paper discusses the stabilizability, weak stabilizability, exact observability and robust quadratic stabilizability of linear stochastic control systems. By means of the spectrum technique of the generalized Lyapunov operator, a necessary and sufficient condition is given for stabilizability and weak stabilizability of stochastic systems, respectively. Some new concepts called unremovable spectrums, strong solutions, and weakly feedback stabilizing solutions are introduced. An unremovable spectrum theorem is given, which generalizes the corresponding theorem of deterministic systems to stochastic systems. A stochastic Popov-Belevith-Hautus (PBH) criterion for exact observability is obtained. For applications, we give a comparison theorem for generalized algebraic Riccati equations (GAREs), and two results on Lyapunov-type equations are obtained, which improve the previous works. Finally, we also discuss robust quadratic stabilization of uncertain stochastic systems, and a necessary and sufficient condition is given for quadratic stabilization via a linear matrix inequality (LMI).

This chapter defines vectors and describes their algebraic and geometric properties. It also introduces another fundamental object—matrix—whose basic properties parallel those of the vector and discusses the examination of techniques that are useful for reading and writing proofs. The link between algebraic manipulation and geometric intuition is a recurring theme in linear algebra that is used to establish many important results. The study of linear algebra begins with vectors and matrices—two of the most practical concepts in mathematics. Linear algebra, in addition to having a multitude of practical applications in science and engineering, also can be used to introduce proof-writing skills. The concept of proof is central to higher mathematics. Mathematicians claim no statement as a “fact” until it is proven true using logical deduction. Therefore, no one can succeed in higher mathematics without mastering the techniques required to supply such a proof.

We introduce the principal representation category $\mathscr{O}({\bf G})$ of reductive algebraic groups with Frobenius maps and put forward a conjecture that this category is a highest weight category. When $\Bbbk$ is complex field $\mathbb{C}$, we provide some evidences of this conjecture. We also study certain kind of bound quiver algebras whose representations are related to the principal representation category $\mathscr{O}({\bf G})$ ., 14 pages

This paper studies the exponential stabilization on infinite dimensional system with impulse controls, where impulse instants appear periodically. The first main result shows that exponential stabilizability of the control system with a periodic feedback law is equivalent to one kind of weak observability inequalities. The second main result presents that, in the setting of a discrete LQ problem, the exponential stabilizability of control system with a periodic feedback law is equivalent to the solvability of an algebraic Riccati-type equation which was built up in [Qin, Wang and Yu, SIAM J. Control Optim., 59 (2021), pp. 1136-1160] for finite dimensional system. As an application, some sufficient and necessary condition for the exponential stabilization of an impulse controlled system governed by coupled heat equations is given., 26 Pages

We compute all signatures of [Formula: see text] and [Formula: see text] which classify all orientation preserving actions of the groups [Formula: see text] and [Formula: see text] on compact, connected, orientable surfaces with orbifold genus [Formula: see text]. This classification is well-grounded in the other branches of Mathematics like topology, smooth and conformal geometry, algebraic categories, and it is also directly related to the inverse Galois problem.

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.