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The book provides a self-contained mathematical introduction to linear optimization for  undergraduate students of Mathematics. This book is equally suitable for Science,  Engineering, and Economics students who are interested in gaining a deeper understanding of the mathematical aspects of the subject. The linear optimization problem is analyzed from different perspectives: topological, algebraic, geometrical, logical, and algorithmic. Nevertheless, no previous knowledge of these subjects is required. The essential details are always provided in a special section at the end of each chapter. The technical material is illustrated with multiple examples, problems with fully-worked solutions, and a range of proposed exercises. In Chapter 1, several formulations of the linear optimization problem are presented and related concerning admissible vectors and optimizers. Then, sufficient conditions for the existence of optimizers based on topological techniques are discussed in Chapter 2. The main objective of Chapter 3 is to provide a way for deciding whether or not an admissible vector is an optimizer, relying on Farkas' Lemma. In Chapter 4, linear algebra is used for computing optimizers via basic admissible vectors. A geometrical characterization of these vectors is the goal of Chapter 5. Duality is discussed in Chapter 6, giving yet a new technique for finding optimizers. An introduction to computational complexity is presented in Chapter 7 with the aim to analyze the efficiency of linear optimization algorithms. The complexity of a brute-force algorithm is shown not to be polynomial. Chapter 8 is targeted at the Simplex Algorithm. It includes the proof of its soundness and completeness and an explanation on its non-polynomial complexity. Finally, Chapter 9 concentrates on the integer optimization problem with an emphasis on totally unimodularity. An algorithm based on the Branch and Bound Technique is analyzed.

The Journal of Applied Logics - IfCoLog Journal of Logics and their Applications (FLAP) covers all areas of pure and applied logic, broadly construed. All papers published are open access, and available via the College Publications website. This Journal is open access, and available in both printed and electronic formats. It is published by College Publications, on behalf of IfCoLog (www.ifcolog.net).

Logic is-arguably-all about proving, but proofs can be "costly," often impossibly so, and today most are delegated to (partly) automatic provers, namely by so-called SAT solvers, software based on the (Boolean) satisfiability problem, or SAT. This is the dual of the (Boolean) validity problem, or VAL, at the core of the conception of the digital computer via Hilbert's Entscheidungsproblem and the Universal Turing Machine. While these problems-VAL significantly less so than SAT-feature in introductory logic textbooks aimed at computer science students, they are largely or wholly absent from textbooks targeting a mathematical or philosophical studentship. Formal logic: Classic problems and proofs corrects this-in our view-misguided state of affairs by providing the basics of formal classical logic from the central viewpoint of a formal, or computer, language that distinguishes itself from the other formal or computer languages by its ability to preserve truth, thus potentially providing solutions to decision problems formulated in terms of VAL and/or SAT. This fundamental aspect of classical logic, truth-preservation, is elaborated on from three main formal semantics, to wit, Tarskian, Herbrand, and algebraic (Boolean) semantics, which, in turn, via the adequateness results for the standard first-order logic, underlie the main proof systems of direct and indirect, or refutation, proofs, associated to VAL and SAT, respectively.Not focusing on the history of classical logic, this book nevertheless provides discussions and quotes central passages on its origins and development, namely from a philosophical perspective. Not being a book in mathematical logic, it takes formal logic from an essentially mathematical perspective. Biased towards a computational approach, with SAT and VAL as its backbone, this is thus an introduction to logic that covers essential aspects of the three branches of logic, to wit, philosophical, mathematical, and computational.

This book is a collection of papers related to a workshop organized in Geneva in January 2017, part of a big event celebrating the centenary of Ferdinand de Saussure's famous "Cours de Linguistique Générale" (CLG). The topic of this workshop was THE FIRST PRINCIPLE stated in the second section of the first part of the CLG entitled: THE ABITRARINESS OF THE SIGN. Discussions are developed according to the three perspectives presented in the call for papers: (1) The details of the formulation of this principle in the CLG, its proper place (cf. the following sentence of section 2: "No one disputes the principle of the arbitrariness of the sign but it is often easier to discover a truth than to assign it its proper place"). Discussions about the question of arbitrariness of the sign in works by Saussure before the CLG are also welcome. (2) How the arbitrariness of the sign has been formulated and stressed before the CLG by other people than Saussure, in particular, but not exclusively, by people of the second part of the 19th century. Three important names: Boole, Peirce, Bréal. (3) The import and value of this principle and the criticisms it received after the publication of the CLG. Special focus will be given on the opposition between arbitrary sign and symbol (as characterized in the CLG: "the symbol is never arbitrary; it is not empty, for there is the rudiment of a natural bond between the signifier and the signified") in the context of mathematical and logical languages (visual reasoning), traffic signs and pictograms (cf. Neurath's Isotype), typefaces (cf. the work of Adrian Frutiger).

Cet ouvrage propose une analyse épistémologique des modèles scientifiques et de leurs modes d'application, centrée sur la définition et l'identification des objets théoriques. La question des idéalisations en science y est traitée en examinant la portée des connaissances pouvant être tirées de modèles scientifiques idéalisés, et les conditions selon lesquelles elles peuvent concerner certains aspects du monde actuel. En discutant différentes approches courantes de cette question en philosophie des sciences, comme certaines formes de réalisme, de fictionnalisme, ou encore de structuralisme, cet ouvrage développe une épistémologie modale définissant notamment l'identité des propriétés et des relations scientifiques dans une perspective causale et nomologique

Languages, machines, and classical computation is a new undergraduate course book on the conjoined subjects of Formal Languages and Automata & Computability and Complexity. By new, we mean more than its recent publication: It is (more) clearly structured around the Chomsky hierarchy, which acts as its backbone; It has an overall algorithmic approach, with many central algorithms thoroughly and clearly described in a step-by-step manner; No programming language or software plays any role whatsoever in it, guaranteeing thus the (mathematical) generality of the diverse contents; It is to some extent a return to the original textbook approaches of the late 1970's / early 1980's, now often-wrongly-seen as too hard for an undergraduate audience; and, Last but not least, it takes into consideration the fact, largely or wholly ignored by other course books on the aforementioned subjects, that to speak of computation today turns out to be an extremely equivocal business, as many other forms of computation have developed outside what we can call the Turing-von Neumann paradigm.A vast selection of exercises is a crucial component to this course book, with exercises ranging from simple tasks to research projects and explorations of creative skills.All the mathematical topics necessary to the satisfactory grasping of the contents discussed are provided in an introductory chapter, making of this a largely self-contained course book.The present second edition corrects addenda and errata, has both improved and new figures, an additional algorithm, and redesigned exercises.

Is it possible to conceive two perfectly identical objects? Is identity even possible withoutindividuality? How would a perfectly symmetrical universe be? The current philosophical debate on identity, and in particular on the necessity of the Leibniz's principle of the identity of indiscernibles, is complex and multi-faceted. Recent works have indicated that the problem becomes increasingly complex if we apply it to mathematical objects. Is it possible to speak of 'identity' for numbers? How can we identify numbers?Drawing on philosophical accounts on identity and individuality in contemporary metaphysics (analytic and continental), this book explores a new path. The author argues that an identity without individuality is possible. By means of a critique of the idea of the identity of indiscernibles, the book formulates the concept of 'manifold identity', through the concept of 'iteration'. Iteration is a specific transgression of the identity of indiscernibles arising from the collision of two forms of identity: qualitative identity and numerical identity. Nonetheless, a pair of perfectly identical objects is still a paradox, a contradiction.The first thesis of the book is that iteration is a paraconsistent and dialethetical logical structure, which allows for true contradiction. The author applies recent works in non-standard logic and dialetheism (Priest, Routley, Berto) to illustrate how we can make sense of the idea that objects can be perfectly identical but discernible.The second thesis of the book is that iteration is the basis of enumerability and computability. A 'computable object' is an object constructed on the basis of an iterative logic. It is possible to re-interpret all the primary concepts of computability theory through the logic of iteration.