On The Existence of A Unique Solution for Systems of Ordinary Differential Equations of First Order

In this paper, we state and prove a theorem for local existence of a unique solution for a system of nonlinear ordinary differential equations ( NODE ) of first order by proving that the nonlinear operator of this system is contractive in a metric space subset of the Banach space consisting of the bounded differentiable functions on the real numbers and equipped with a Bielescki#39s type norm.nbsp Finally, we give examples to illustrate our result

Towards justifying computer algebra algorithms in Isabelle/HOL

As verification efforts using interactive theorem proving grow, we are in need of certified algorithms in computer algebra to tackle problems over the real numbers. This is important because uncertified procedures can drastically increase the size of the trust base and under- mine the overall confidence established by interactive theorem provers, which usually rely on a small kernel to ensure the soundness of derived results. This thesis describes an ongoing effort using the Isabelle theorem prover to certify the cylindrical algebraic decomposition (CAD) algorithm, which has been widely implemented to solve non-linear problems in various engineering and mathematical fields. Because of the sophistication of this algorithm, people are in doubt of the correctness of its implementation when deploying it to safety-critical verification projects, and such doubts motivate this thesis. In particular, this thesis proposes a library of real algebraic numbers, whose distinguishing features include a modular architecture and a sign determination algorithm requiring only rational arithmetic. With this library, an Isabelle tactic based on univariate CAD has been built in a certificate-based way: external, untrusted code delivers solutions in the form of certificates that are checked within Isabelle. To lay the foundation for the multivariate case, I have formalised various analytical results including Cauchy’s residue theorem and the bivariate case of the projection theorem of CAD. During this process, I have also built a tactic to evaluate winding numbers through Cauchy indices and verified procedures to count complex roots in some domains. The formalisation effort in this thesis can be considered as the first step towards a certified computer algebra system inside a theorem prover, so that various engineering projections and mathematical calculations can be carried out in a high-confidence framework

KAJIAN TEOREMA BOLZANO-WEIERSTRASS UNTUK MENGKONSTRUKSI BARISAN YANG KONVERGEN DI R^n DAN APLIKASINYA DALAM PEMBUKTIAN TEOREMA EKSISTENSI MAX-MIN

A sequence is a function from the set of natural numbers to the set of real numbers . In sequences there is the concept of sequence convergence. Testing the convergence of a sequence can be done using the Bolzano-Weierstrass Theorem. This theorem states that every finite sequence has a convergent sequence. The relationship between convergent sequences and finite sequences is also important to study further. Apart from being used to prove the convergence of sequences, the Bolzano-Weirstrass Theorem can also be applied to prove the Max-Min Existence Theorem. This research was conducted to examine the relationship between convergent sequences and finite sequences, the relationship between convergence and continuous functions and the relationship between continuous functions and max-min values ​​with the aim of constructing a convergent sequence in R^n and its application in proving the Max-Min Existence Theorem. This research is a literature study. This research was conducted through a literature review of books and other literature. From the literature review, the materials are then discussed in depth. The results of the literature study show that a convergent sequence is a finite sequence, but a finite sequence is not necessarily convergent. In determining the convergence of a sequence using the Bolzano-Weierstrass Theorem, it is necessary to first show the limitations of the sequence. Furthermore, to prove the Max-Min Existence Theorem it is necessary to require that the sequence is finite and then this theorem can be proven using the Bolzano-Weierstrass Theorem and Apit Theorem. Keywords: Monotonous Sequence, Finite Sequence, Continuity, Bolzano-Weierstrass Theorem, Max-Min Existence Theorem

A Non-Trivial Minoration for the Set of Salem Numbers

The set of Salem numbers is proved to be bounded from below by $\theta_{31}^{-1}= 1.08544\ldots$ where $\theta_{n}$, $n \geq 2$, is the unique root in $(0,1)$ of the trinomial $-1+x+x^n$. Lehmer's number $1.176280\ldots$ belongs to the interval $(\theta_{12}^{-1}, \theta_{11}^{-1})$. We conjecture that there is no Salem number in $(\theta_{31}^{-1}, \theta_{12}^{-1}) = (1.08544\ldots, 1.17295\ldots)$. For proving the Main Theorem, the algebraic and analytic properties of the dynamical zeta function of the R\'enyi-Parry numeration system are used, with real bases running over the set of real reciprocal algebraic integers, and variable tending to 1.Comment: text dedicated to the 75th birthday of Christiane Frougny - NUMERATION2023 -. arXiv admin note: text overlap with arXiv:1911.1059

A New Cryptosystem Based on Decimal Numbers and Nonlinear Function

We introduce a new cryptosystem, with new digital signature procedure. This cryptosystem is based on decimal numbers and nonlinear function. Unlike other cryptosystems, this system does not depend on prime or integer numbers and it does not depend on group structure, ¯nite ¯eld or discrete logarithmic equation. The idea is about using decimal numbers and thus the only integer number will be the plaintext. A secret key is shared between Alice and Bob to construct their decimal public keys, so that, we can say this cryptosystem is symmetric and asym- metric. We choose to distribute the secret key through the classic Di±e-Hellman protocol after introducing some modi¯cation on it; this modi¯cation depends on the extension of the theory of the modulus to be applicable on the real numbers. We prove there is no loss of any bit of information when we use the decimal num- bers during the encryption and the decryption by introducing the rounding o® concept as a function. This function plays the primary role in proving a new the- orem called \Rounding theorem". A new theory for the security is established, where basically, it depends on the properties of the decimal numbers and the cu- mulative and truncation errors that will occur during the attack, because every attack requires solving a system of non-linear equations. The decimal cryptosys- tem does not depend on large numbers because it deals with numbers between zero and one. Therefore, this cryptosystem can be considered faster than the known cryptosystems

On the analysis needs when verifying state-based software requirements: an experience report

AbstractIn a previous investigation we formally defined procedures for analyzing hierarchical state-based requirements specifications for two properties: (1) completeness with respect to a set of criteria related to robustness (a response is specified for every possible input and input sequence) and (2) consistency (the specification is free from conflicting requirements and undesired nondeterminism). Informally, the analysis involves determining if large Boolean expressions are tautologies. We implemented the analysis procedures in a prototype tool and evaluated their effectiveness and efficiency on a large real world requirements specification expressed in an hierarchical state-based language called Requirements State Machine Language. Although our initial approach was largely successful, there were some drawbacks with the original tools. In our initial implementation we abstracted all formulas to propositional logic. Unfortunately, since we are manipulating the formulas without interpreting any of the functions in the individual predicates, the abstraction can lead to large numbers of spurious (or false) error reports. To increase the accuracy of our analysis we have continually refined our tool with decision procedures and, finally, come to the conclusion that theorem proving is often needed to avoid large numbers of spurious error reports. This paper discusses the problems with spurious error reports and describes our experiences analyzing a large commercial avionics system for completeness and consistency