On non-linearity and affine equivalence of permutations over an arbitrary finite commutative ring with unity

Cryptographic properties of permutations viz non-linearity, affine equivalence, and mode transform have been studied in the literature, treating them as bijections on ?_n. In this article, the authors consider them as bijection on an arbitrary finite commutative ring with unity. Treating them this way, they achieve a generalization of the above mentioned properties. The authors also propose an algorithm for computing non-linearity in their generalized scenario, which is faster compared to the direct approach in many cases.     

Roundoff-Free Number Fields for Interval Computations

Roundoff errors are inevitable if the exact result of some operation is not representable in a computer, and has, therefore, to be approximated. To avoid roundoff errors, it is hence necessary to choose a set of computer-representable numbers in such a way that the results of all basic operations will be still in this set. In this paper, we prove that if we include arithmetic operations and computing the interval range into this operations list, then the set F of numbers will be roundoff-free iff F is a real closed field; therefore, the smallest such set is the set of all real algebraic numbers (i.e., solutions of polynomial equations with rational coefficients).     

Semi-synchronous transductions

Semi-synchronously rational relations generalise synchronised rational relations in a natural way. We discuss here some of their basic properties, among them a “Cobham–Semenov-like” dichotomy theorem. Our main result is a characterisation of bijective semi-synchronously rational transductions as those bijections mapping regular relations to regular ones and non-regular relations to non-regular ones.     

Semantical considerations on intention dynamics for BDI agents

The BDI paradigm is a powerful means for constructing intelligent agents in terms of their beliefs, desires, and intentions. For this paradigm to bear its full potential, it must incorporate considerations from rationality. This paper develops a set of postulates for intelligent agents who deliberate about their intentions and actions. However, even simple postulates can lead to paradoxical results when formalized naively. We propose an approach based on temporal possibility and action that avoids those problems. This approach incorporates a formal model based on branching time in which a probabilistic analysis of choice can be captured. In this manner, the intuitions of the BDI paradigm can be reconciled with those of rational agency.     

Robust stability of a family of disc polynomials

In his well-known theorem, V. L. Kharitonov established that Hurwitz stability of a set f1 of interval polynomials with complex coefficients (polynomials where each coefficient varies in an arbitrary but prescribed rectangle of the complex plane) is equivalent to the Hurwitz stability of only eight polynomials in this set. In this paper we consider an alternative but equally meaningful model of uncertainty by introducing a set fD of disc polynomials, characterized by the fact that each coefficient of a typical element P(s) in fD can be any complex number in an arbitrary but fixed disc of the complex plane. Our result shows that the entire set is Hurwitz stable if and only if the ‘center’ polynomial is stable, and the H _∞-norms of two specific stable rational functions are less than one. Our result can be readily extended to deal with the Schur stability problem and the resulting condition is equally simple.     

On the boundedness of the set of stabilizing controllers

This paper shows that the set of rational, strictly proper, robustly stabilizing controllers for single‐input single‐output linear‐time invariant plants will form a bounded (can even be empty) set in the controller parameter space if and only if the order of the stabilizing controller cannot be reduced any further; if the set of proper stabilizing controllers of order r is not empty and the set of strictly proper controllers of order r is bounded, then r is the minimal order of stabilization. The paper also extends this result to characterize the set of controllers that guarantee some pre‐specified performance specifications. In particular, it is shown here that the minimal order of a controller that guarantees specified performance is l iff (1) there is a controller of order l guaranteeing the specified performance and (2) the set of strictly proper, robustly stabilizing controllers of order l and guaranteeing the performance is bounded. Moreover, if the order of the controller is increased, the set of higher‐order controllers which satisfies the specified performance will necessarily be unbounded. This characterization is provided for performance specifications, such as gain margin and robust stability, which require a one‐parameter family of real polynomials to be Hurwitz, where the parameter is in a closed interval. Other performance specifications, such as phase margin and ℋ︁_∞ norm, can be reduced to the problem of determining a set of stabilizing controllers that renders a family of complex polynomials Hurwitz. The characterization of the set of controllers for the stabilization of complex polynomials is provided and is used to show the boundedness properties for the set of controllers that guarantee a given phase margin or an upper bound on the ℋ︁_∞ norm. Copyright © 2007 John Wiley & Sons, Ltd.     

H2-optimal control of linear systems with tracking/disturbance rejection constraints

This paper considers the design of two-degree-of-freedom, linear multivariable controllers for four-block matrix transfer functions with objectives pertaining to both random and deterministic signals. The former are taken into account in the cost functional whereas the latter entail that the controller is constrained to solve a servomechanism problem involving a class of reference signals and two classes of disturbances. The cost functional is set up by adding quadratic terms to the usual LQG/Wiener-Hopf cost in such a way as to ensure the existence of (unique) solutions even when the signals to be tracked/rejected, as well as models, have poles on the imaginary axis. The solutions of the servomechanism problem are explicitly parametrized by free, stable, rational and proper parameters under two sets of assumptions: one which is necessary and sufficient for the existence of solutions and another, stronger set which avoids Kronecker products and the associated higher dimensional factorizations involved in the first case. On the basis of this parametrization the servomechanism constraint is incorporated into an unconstrained problem which is then quite explicitly solved.     

The zero divisor problem of multivariable stochastic adaptive control

Stochastic adaptive minimum variance control algorithms require a division by a function of a recursively computed parameter estimate at each instant of time. In order that the analysis of these algorithms is valid, zero divisions must be events of probability zero. This property is established for the stochastic gradient adaptive control algorithm under the condition that the initial state of the system and all finite segments of its random disturbance process have a joint distribution which is absolutely continuous with respect to Lebesgue measure. This result is deduced from the following general result established in this paper: a non-constant rational function of a finite set of random variables {x₁},x_n} is absolutely continuous with respect to Lebesgue measure if the joint distribution function of {x₁,…,x_n} has this property.     

An equivalence between rational H2 and Hankel-norm approximations

Let H(z) be a given function in H₂ A classical problem in engineering analysis is to find a rational function G (z) ε H₂ degree M say, which is closest to H(z) in 2-norm. This problem is typically approached using the cost function |H(z) − G(z)|₂, in which G(z) is allowed to vary over the set of Mth-order rational functions in H₂ and for which stationary points are sought. We show that each stationary point of degree M of this functional coincides with a weighted Hankel-norm approximant to H(z). The weighting function derives from the outer factor of the error function H(z) − G(z) stationary point of the rational H₂ approximation problem.     

Möbius变换下四次有理抛物-PH曲线的C2 Hermite插值

目的 曲线插值问题在机器人设计、机械工业、航天工业等诸多现代工业领域都有广泛的应用，而已知端点数据的Hermite插值是计算机辅助几何设计中一种常用的曲线构造方法，本文讨论了一种偶数次有理等距曲线，即四次抛物-PH曲线的C² Hermite插值问题。方法 基于M bius变换引入参数，利用复分析的方法构造了四次有理抛物-PH曲线的C² Hermite插值，给出了具体插值算法及相应的Bézier曲线表示和控制顶点的表达式。结果 通过给出"合理"的端点插值数据，以数值实例表明了该算法的有效性，所得12条插值曲线中，结合最小绝对旋转数和弹性弯曲能量最小化两种准则给出了判定满足插值条件最优曲线的选择方法，并以具体实例说明了与其他插值方法的对比分析结果。结论 本文构造了M bius变换下的四次有理抛物-PH曲线的C² Hermite插值，在保证曲线次数较低的情况下，达到了连续性更高的插值条件，计算更为简单，插值效果明显，较之传统奇数次PH曲线具有更加自然的几何形状，对偶数次PH曲线的相关研究具有一定意义。     

Constructive difference and disagreement: A suprA-cooperation among agents

Differences among agents may be constructive in so far as they can bring solution enhancements or conflicts, the second case leading to solution modifications. What is dealt with in this paper is a cooperation involving different rational agents resulting in more than a mere addition of the agents' individual skills, thanks to a process of approval and refutation of the current solution. Lakatos' work is taken as a basis and adapted to a set of cooperating agents, so as to define the concept of supfA-cooperation and the corresponding interaction model. Three case studies are given, involving SuprA-cooperating human or artificial agents.     

同源密码中Montgomery模型的w-坐标研究

椭圆曲线群律计算是传统椭圆曲线密码(ECC)的核心运算,同时也是基于同源的后量子密码计算中的重要组成部分。Montgomery曲线上的Montgomery ladder算法是一种高效(伪)群律计算方法,且经常用于预防侧信道攻击。Farashahi和Hosseini在ACISP 2017提出了Edwards曲线模型上的w-坐标可得到类似Montgomery ladder算法以进行群律计算,Kim等人在ASIACRYPT 2019将其用于优化奇数次同源计算。随后,不同曲线模型上的w-坐标陆续被提出用于优化同源计算。本质上,w-坐标是关于传统椭圆曲线有理点(x,y)-坐标的有理函数。与标准(x,y)-坐标相比,w-坐标不仅可以节约椭圆曲线群律和同源计算的计算量,还可以减少带宽。Hisil和Renes在ACM TOMS 2019提出可利用加2阶点得到更多的w-坐标。受此启发,本文提出利用Montgomery曲线上的2-同源构造出3类新的w-坐标,与x-坐标相同的是,均可应用于Montgomery ladder算法和奇数次同源计算的优化。同时,w-坐标在计算奇数次同源中,同源映射像曲线系数计算公式与像点公式类似,可利用SIMD指令集将两者并行化处理,从而得到相关计算的进一步加速。最后,由于Edwards,Huff,Jacobi等曲线模型在某些条件下可与Montgomery模型建立双有理等价,因此可由Montgomery曲线上新的w-坐标开发出其他曲线模型上更多的w-坐标,它们将有可能支持同源密码实现中更有效的算法。     

Conchoid surfaces of rational ruled surfaces

The conchoid surface G of a given surface F with respect to a point O is roughly speaking the surface obtained by increasing the radius function of F with respect to O by a constant d. This paper studies real rational ruled surfaces in this context and proves that their conchoid surfaces possess real rational parameterizations, independently of the position of O. Thus any rational ruled surface F admits a rational radius function r(u,v) with respect to any point in space. Besides the general skew ruled surfaces and examples of low algebraic degree we study ruled surfaces generated by rational motions.     

A Coherence Space of Rational Intervals for a Construction of IR

A constructive computational representation of the space of real intervals IR is introduced, in a way that makes it possible to capture both its information structure relevant from a computational standpoint, and its application features as a mathematical structure. The representation consists of the Coherence Space of Rational Intervals IIQ, introduced by defining the web (IQ, ) of rational intervals, which is obtained from the set IQ of rational intervals on which a suitable reflexive and symmetric relation is defined. A two-fold construction of IIQ is performed, such that the internal construction of its domain-like structure leads the transformation of a suitable external algebraic structure defined on IQ into a certain one on IIQ which, when restricted to the set tot(IIQ) of total objects, becomes a structure which is order and algebraically isomorphic to the complete ordered field of the real numbers, R, and such that the family of quasi-total objects when extended with R is order and algebraically isomorphic to the set of the real intervals R.     

The Mortality Problem for Matrices of Low Dimensions

In this paper we discuss the existence of an algorithm to decide if a given set of 2 \times 2 matrices is mortal. A set F={A ₁ ,\ldots,A _m } of square matrices is said to be mortal if there exist an integer k \geq 1 and some integers i ₁ ,i ₂ ,\ldots,i _k ∈ {1, \ldots, m} with A _i1 A _i2 ⋅s A _ik =0 . We survey this problem and propose some new extensions. We prove the problem to be BSS-undecidable for real matrices and Turing-decidable for two rational matrices. We relate the problem for rational matrices to the entry-equivalence problem, to the zero-in-the-corner problem, and to the reachability problem for piecewise-affine functions. Finally, we state some NP-completeness results.     

The -basis and implicitization of a rational parametric surface

The concept of a μ-basis was introduced in the case of parametrized curves in 1998 and generalized to the case of rational ruled surfaces in 2001. The μ-basis can be used to recover the parametric equation as well as to derive the implicit equation of a rational curve or surface. Furthermore, it can be used for surface reparametrization and computation of singular points. In this paper, we generalize the notion of a μ-basis to an arbitrary rational parametric surface. We show that: (1) the μ-basis of a rational surface always exists, the geometric significance of which is that any rational surface can be expressed as the intersection of three moving planes without extraneous factors; (2) the μ-basis is in fact a basis of the moving plane module of the rational surface; and (3) the μ-basis is a basis of the corresponding moving surface ideal of the rational surface when the base points are local complete intersections. As a by-product, a new algorithm is presented for computing the implicit equation of a rational surface from the μ-basis. Examples provide evidence that the new algorithm is superior than the traditional algorithm based on direct computation of a Gröbner basis. Problems for further research are also discussed.     

Robust feedforward design for a two-degrees of freedom controller

In this paper we address the problem of designing the feedforward block of a two degrees of freedom controller in the case of single input single output (SISO) discrete-time linear systems with model uncertainty. In this work the design of the feedforward filter is formulated as a robust model matching problem. First, a closed form solution of the optimal noncausal filter, which minimizes the worst-case model matching error for each ω[0,2π], is obtained. Then, suitable rational stable approximations of the optimal solution are derived either by means of causal filters or, when a preview of the reference signal is available, by means of noncausal FIR filters. The efficiency of the proposed method is tested on a simulated example.     

On Bounded Rational Trace Languages

In this paper, for a finitely generated monoid M, we tackle the following three questions:

1. Is it possible to give a characterization of rational subsets of M which have polynomial growth?
2. What is the structure of the counting function of rational sets which have polynomial growth?
3. Is it true that every rational subset of M has either exponential growth or it has polynomial growth? Can one decide for a given rational set which of the two options holds?

We give a positive answer to all the previous questions in the case that M is a direct product of free monoids. Some of the proved results also extend to trace monoid.     

On Controlling the Transient Response of Linear Time Invariant Systems with Fixed Structure Controllers

The problem of controlling transient response is important in many industrial applications; for example, the speed and accuracy of motion control of robots directly relates to the productivity of the robot. The objective of transient control is to determine a feedback controller of a fixed structure that renders the closed loop response of a specified system to lie in a specified envelope. One may associate a set of errors which measures the deviation of the response from the envelope. The set of errors may be defined in such a way that all the errors are non‐negative if and only if the response does not deviate from the envelope at any time. The transient problem can be thus posed as the problem of determining a stabilizing controller that renders the set of all errors to be non‐negative at every time. One may associate a control parameter vector K with a controller of a specified structure. The main topic of investigation of this paper is to find a bound for the set of real control parameters, K, so that a rational, proper transfer function, has a decaying, non‐negative impulse response. It is assumed that the coefficients of the polynomials N(s, k) and D(s, K) are affine in K.     

Eigenschaften der von linearen Automaten erkennbaren Worte

Summary The paper investigates the set of words accepted by a linear sequential machine M and the algebraic structure of this set. If M is nonsingular a strong relationship exists between this set and the transition group of the accepting automaton. Thus the algebraic structure of this group is used to describe some properties of the set of accepted words. If M is nilpotent, in addition, an algorithm is given which in a direct way generates just the words accepted by M.