Bezout identity

This research paper aims to attach a signature scheme that enables signature generation and verification to a well-defined cryptosystem. So, the new system combines key generation, encryption, signature generation, signature verification, and decryption algorithms. The Michael O. Rabin signature scheme uses random padding to validate signatures. Analogously, the proposed cryptographic technique utilizes the same padding system and additional components, such as quadratic residue plus the floor value of the quadratic quotient. The only difference is that Michael O. Rabin's signature may double or triple in some cases, whereas the proposed model uses a four-tuple signature system. The Michael O. Rabin cryptosystem's ability to produce multiple plaintexts from a single ciphertext and identical ciphertexts from distinct plaintexts is one of its potential ambiguities. To address this issue, I constructed a cryptosystem proposing a mathematical solution. Specifically, I created a mathematical model to distinguish identical encrypted text generated from different plaintexts in the Michael O. Rabin cryptosystem. However, that model lacked an authentication mechanism to verify the authenticity of the sender and the message, as it had no signature scheme. The proposed crypto-intensive technique uses a two-time security key by slightly altering the Diffie-Hellman key exchange protocol. Through a signature verification mechanism of the suggested method, the intended recipient can recover the original plaintext, and the sender has the advantage of using the encrypted text to create a signature. The above-mentioned cryptographic technique provides secure protection against man-in-the-middle attacks. It is unforgeable, whereas Rabin’s signature is forgeable in a forgery attack. By the way, the initial research initiative was to review different cryptosystem construction techniques and signature schemes and then apply those mathematical concepts to construct an effective cryptographic technique. This research begins with an exploratory research approach and ends with a computational research method. I utilized data collection methods such as conducting a literature review, employing critical thinking strategies, solving various computational math problems, and facilitating focus group discussions. The research population includes academic professors and graduate students.

In this paper, the decentralized control of generalized systems is considered via a frequency domain approach. Firstly, based on the design of generalized decentralized observer-based controllers, a state-space representation of the decentralized Bezout identity is developed for generalized decentralized control systems with two control channels. Secondly, all the stabilizing decentralized controllers are parameterized. Thirdly, a design procedure is proposed, and an example is given to illustrate the design procedure.

In this paper, the Bezout's identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type. Differential equations, relation with the starting family as well as recurrence relations and explicit representations are given for the Bezout's pair. Extensions to classical orthogonal polynomials of a discrete variable and their q-analogues are also presented. Applications of these results for the representation of the second kind functions are given.

The objective of this paper is to demonstrate the utilization of algebraic controller design in an unconventional ring while control integrating processes with time delay. In contrast to many other methods, the proposed method is not based on the time delay approximation. A control structure combining a simple feedback loop and a two-degrees-of-freedom control structures is considered. This structure can be conceived as a simple feedback loop with inner stabilizing loop. The control design is performed in the ring of retarded quasipolynomial (RQ) meromorphic functions (RMS) - an algebraic method based on the solution of the Bézout equation with Youla-Kučera parameterization is presented. Final controllers may be of so-called anisochronic type and ensure feedback loop stability, tracking of the step reference and load disturbance attenuation. Among many possible tuning methods, the dominant pole assignment method is adopted.

In this paper, the Bezout's identity is analyzed in the context of classical orthogonal polynomials solution of a second order differential equation of hypergeometric type. Differential equations, relation with the starting family as well as recurrence relations and explicit representations are given for the Bezout's pair. Extensions to classical orthogonal polynomials of a discrete variable and their q-analogues are also presented. Applications of these results for the representation of the second kind functions are given.

In the last decades, fractional di erential equations have become popular among scientists in order to model various stable physical phenomena with anomalous decay, s a y that are not of exponential type. Moreover in discrete-time series analysis, so-called fractional ARMA models have b e e n proposed in the literature in order to model stochastic processes, the autocorrelation of which also exhibits an anomalous decay. Both types of models stem from a common property of complex variable functions: namely, m ultivalued functions and their behaviour in the neighborhood of the branching point, and asymptotic expansions performed along the cut between branching points. This more abstract point o f v i e w p r o ves very much useful in order to extend these models by c hanging the location of the classical branching points (the origin of the complex plane, for continuous-time systems). Hence, stability properties of and modelling issues by generalized fractional di erential systems will be adressed in the present paper: systems will be considered both in the time-domain and in the frequency-domain when necessary a distinction will be made between fractional di erential systems of commensurate a n d incommensurate orders. R esum e. Ces derni eres ann ees, les equations di erentielles fractionnaires ont et e de plus en plus utilis ees par les scienti ques d esireux de mod eliser divers ph enom enes physiques stables mais pr esentant u n e d ecroissance lente, c'esta-dire qui ne soit pas de type exponentiel. D'autre part, dans le domaine de l'analyse des s eries temporelles, des mod eles ARMA fractionnaires ont et e p r opos es de fa con a m o d eliser des processus stochastiques dont l'autocorr elation est aussi a d ecroissance lente. Ces deux types de mod eles proviennent d'une propri et e c o m m une des fonctions de la variable complexe : a s a voir, les fonctions multivalu eeset leur comportement a uv oisinage du point d e b r a n c hement, ainsi que des d eveloppements asymptotiques e ectu es le long de la coupure qui relie les points de branchement. Ce point de vue plus abstrait r ev ele toute son utilit e lorsqu'on veut etendre ces mod eles en changeant la position des points de branchement classiques (l'origine du plan complexe, pour les syst emes en temps continu). Ainsi, nous etudierons les propri et es de stabilit e des syst emes di erentiels fractionnaires g en eralis es et les cons equences sur la mod elisation : nous consid ererons les syst emes tant dans le domaine temporel que dans le domaine fr equentiel et si n ecessaire nous ferons la distinction entre syst emes di erentiels fractionnaires d'ordres commensurables ou incommensurables.

A large class of viscoelastic and elastoplastic systems, frequently encountered in physics, are based on causal pseudo-di erential operators, which are hereditary: the whole past of the state is involved in the dynamic expression of the system evolution. This generally induces major technical di culties. We consider a speci c class of pseudo-di erential damping operators, associated to the so-called di usive representation which enables to built augmented state-space realizations without heredity. Dissipativity property is expressed in a straightforward and precise way. Thanks to state-space realizations, standard analysis and approximation methods as well as control-theory concepts may therefore be used.

The stability of non-linear fractional di erential equations is studied. A su cient stability condition on the non-linearity is given for the input-output stability, thanks to many di erent reformulations of the system using di usive representations of dissipative pseudo-di erential operators. The problem of asymptotic internal stability is analyzed by a more involved functional analytic method. Finally, a fractional version of the classical Hartman{Grobman theorem for hyperbolic dynamical systems of order 1 is conjectured and reformulated, based upon known necessary and su cient stability conditions for linear fractional di erential equations.

We give a frequency-domain approach to stabilization for a large class of systems with transfer functions involving fractional powers of s. A necessary and su cient criterion for BIBO stability is given, and it is shown how to construct coprime factorizations and associated BÃ ezout factors in order to parametrize all stabilizing controllers of these systems.

In this paper we establish a connection between subresultants and locally nilpotent derivations over commutative rings containing the rationals. As consequence of this connection, we prove that for any commutative ring with unit and any polynomials P and Q in A[y], the ith subresultant of P and Q is the determinant of a matrix, depending only on the degrees of P and Q, whose entries are taken from the list built with P , Q and their successive Hasse derivatives.

The robust control issue for uncertain nonlinear system is discussed by using the method of right coprime factorization. As it is difficult to obtain the inverse of the right factor due to the high nonlinearity, the proving of the Bezout identity becomes troublesome. Therefore, two sufficient conditions are derived to manage this problem with the nonlinear feedback system as well as that with the uncertain nonlinear feedback system under the definition of Lipschitz norm. A simulation of temperature control is given to demonstrate the validity of the proposed method.

In this paper we give an algorithmic characterization of rank two locally nilpotent derivations in dimension three. Together with an algorithm for computing the plinth ideal, this gives a method for computing the rank of a locally nilpotent derivation in dimension three.