More on discrete convexity

In several recent papers some concepts of convex analysis were extended to discrete sets. This paper is one more step in this direction. It is well known that a local minimum of a convex function is always its global minimum. We study some discrete objects that share this property and provide several examples of convex families related to graphs and to two-person games in normal form

Stability and Spectral Properties in the Max Algebra with Applications in Ranking Schemes

This thesis is concerned with the correspondence between the max algebra and non-negative linear algebra. It is motivated by the Perron-Frobenius theory as a powerful tool in ranking applications. Throughout the thesis, we consider max-algebraic versions of some standard results of non-negative linear algeb- ra. We are specifically interested in the spectral and stability properties of non-negative matrices. We see that many well-known theorems in this context extend to the max algebra. We also consider how we can relate these results to ranking applications in decision making problems. In particular, we focus on deriving ranking schemes for the Analytic Hierarchy Process (AHP). We start by describing fundamental concepts that will be used throughout the thesis after a general introduction. We also state well-known results in both non-negative linear algebra and the max algebra. We are next interested in the characterisation of the spectral properties of mat- rix polynomials. We analyse their relation to multi-step difference equations. We show how results for matrix polynomials in the conventional algebra carry over naturally to the max-algebraic setting. We also consider an extension of the so-called Fundamental Theorem of Demography to the max algebra. Using the concept of a multigraph, we prove that a number of inequalities related to the spectral radius of a matrix polynomial are also true for its largest max eigenvalue. We are next concerned with the asymptotic stability of non-negative matrices in the context of dynamical systems. We are motivated by the relation of P-matrices and positive stability of non-negative matrices. We discuss how equivalent conditions connected with this relation echo similar results over the max algebra. Moreover, we consider extensions of the properties of sets of P-matrices to the max algebra. In this direction, we highlight the central role of the max version of the generalised spectral radius. We then focus on ranking applications in multi-criteria decision making prob- lems. In particular, we consider the Analytic Hierarchy Process (AHP) which is a method to deal with these types of problems. We analyse the classical Eigenvalue Method (EM) for the AHP and its max-algebraic version for the single criterion case. We discuss how to treat multiple criteria within the max-algebraic framework. We address this generalisation by considering the multi-criteria AHP as a multi-objective optimisation problem. We consider three approaches within the framework of multi-objective optimisation, and use the optimal solution to provide an overall ranking scheme in each case. We also study the problem of constructing a ranking scheme using a combi- natorial approach. We are inspired by the so-called Matrix Tree Theorem for Markov Chains. It connects the spectral theory of non-negative matrices with directed spanning trees. We prove that a similar relation can be established over the max algebra. We consider its possible applications to decision making problems. Finally, we conclude with a summary of our results and suggestions for future extensions of these

Hypohamiltonian and almost hypohamiltonian graphs

This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results

Kant in English: An Index

Kant in English: An Index / By Daniel Fidel Ferrer. ©Daniel Fidel Ferrer, 2017. Pages 1 to 2675. Includes bibliographical references. Index. 1. Ontology. 2. Metaphysics. 3. Philosophy, German. 4. Thought and thinking. 5. Kant, Immanuel, 1724-1804. 6. Practice (Philosophy). 7. Philosophy and civilization. 8). Kant, Immanuel, 1724-1804 -- Wörterbuch. 9. Kant, Immanuel, 1724-1804 -- Concordances. 10. Kant, Immanuel, 1724-1804 -- 1889-1976 – Indexes. I. Ferrer, Daniel Fidel, 1952-. MOTTO As a famous motto calls us back to Kant, Otto Liebmann’s writes (Kant and His Epigones of 1865): “Also muss auf Kant zurückgegangen werden.” “Therefore, must return to Kant.” Table of Contents 1). Preface and Introduction. 2. Background on Kant’s Philosophy (hermeneutical historical situation). 3). Main Index (pages, 25 to 2676). Preface and Introduction Total words indexed: 58,928; for the 12 volumes that are in the MAIN INDEX are indexed: pages 1 to 7321. This monograph by Daniel Fidel Ferrer is 2676 pages in total. The following is a machine index of 12 volumes written by Immanuel Kant and translated from German into English. Everything is indexed including the text, title pages, preface, notes, editorials, glossary, indexes, biographical notes, and even some typos. No stop words or words removed from this index. There are some German words in the text, bibliographies, and in the glossaries (also included in Main Index). Titles in English of Kant’s writings for this index (pages 1 to 7321). Anthropology, History, and Education [Starts on page 1 Correspondence [Starts on page 313 Critique of Pure Reason [Starts page 971 Critique of the Power of Judgment [Starts on page 1771 Lectures on Logic [Starts on page 2247 Lectures on Metaphysics [Starts on page 2991 Notes and Fragments [Starts on page 3670 Opus Postumum [Starts on page 4374 Practical Philosophy [Starts on page 4741 Religion and Rational Theology [Starts on page 5446 Theoretical Philosophy after 1781 [Starts on page 5990 Theoretical Philosophy, 1755-1770 [Starts on page 6541 Universal Natural History and Theory of the Heavens or An Essay on the Constitution and the Mechanical Origin of the Entire Structure of the Universe Based on Newtonian Principles [Starts on page 7162 The whole single file which includes all of these books ends on page 7321. 12 volumes are pages 1 to 7321. These actual texts of these books by Kant are not include here because of copyright. This is only an index of these 7321 pages by Immanuel Kant. There are some German words in the text and in the glossaries, etc. Searching this Main Index. Please note the German words that start with umlauts are at the end of the index because of machine sorting of the words. Starting with the German word “ße” on page 2674 page of this book (see in Main Index). Use the FIND FUNCTION for all examples of the words or names you are searching. Examples from the Main Index mendacium, 5171, 5329, 5389 mendation, 220 mendax, 2702, 2800 mended, 360 Mendel, 416, 925, 965 Mendelian, 2212 Mendels, 345, 363, 417, 458, 560, 572, 588, 926, 928, 929 MENDELSSOHN, 925 Mendelssohn, 8, 9, 19, 98, 99, 100, 101

Kant in English: An Index

Kant in English: An Index / By Daniel Fidel Ferrer. ©Daniel Fidel Ferrer, 2017. Pages 1 to 2675. Includes bibliographical references. Index. 1. Ontology. 2. Metaphysics. 3. Philosophy, German. 4. Thought and thinking. 5. Kant, Immanuel, 1724-1804. 6. Practice (Philosophy). 7. Philosophy and civilization. 8). Kant, Immanuel, 1724-1804 -- Wörterbuch. 9. Kant, Immanuel, 1724-1804 -- Concordances. 10. Kant, Immanuel, 1724-1804 -- 1889-1976 – Indexes. I. Ferrer, Daniel Fidel, 1952-. MOTTO As a famous motto calls us back to Kant, Otto Liebmann’s writes (Kant and His Epigones of 1865): “Also muss auf Kant zurückgegangen werden.” “Therefore, must return to Kant.” Table of Contents 1). Preface and Introduction. 2. Background on Kant’s Philosophy (hermeneutical historical situation). 3). Main Index (pages, 25 to 2676). Preface and Introduction Total words indexed: 58,928; for the 12 volumes that are in the MAIN INDEX are indexed: pages 1 to 7321. This monograph by Daniel Fidel Ferrer is 2676 pages in total. The following is a machine index of 12 volumes written by Immanuel Kant and translated from German into English. Everything is indexed including the text, title pages, preface, notes, editorials, glossary, indexes, biographical notes, and even some typos. No stop words or words removed from this index. There are some German words in the text, bibliographies, and in the glossaries (also included in Main Index). Titles in English of Kant’s writings for this index (pages 1 to 7321). Anthropology, History, and Education [Starts on page 1 Correspondence [Starts on page 313 Critique of Pure Reason [Starts page 971 Critique of the Power of Judgment [Starts on page 1771 Lectures on Logic [Starts on page 2247 Lectures on Metaphysics [Starts on page 2991 Notes and Fragments [Starts on page 3670 Opus Postumum [Starts on page 4374 Practical Philosophy [Starts on page 4741 Religion and Rational Theology [Starts on page 5446 Theoretical Philosophy after 1781 [Starts on page 5990 Theoretical Philosophy, 1755-1770 [Starts on page 6541 Universal Natural History and Theory of the Heavens or An Essay on the Constitution and the Mechanical Origin of the Entire Structure of the Universe Based on Newtonian Principles [Starts on page 7162 The whole single file which includes all of these books ends on page 7321. 12 volumes are pages 1 to 7321. These actual texts of these books by Kant are not include here because of copyright. This is only an index of these 7321 pages by Immanuel Kant. There are some German words in the text and in the glossaries, etc. Searching this Main Index. Please note the German words that start with umlauts are at the end of the index because of machine sorting of the words. Starting with the German word “ße” on page 2674 page of this book (see in Main Index). Use the FIND FUNCTION for all examples of the words or names you are searching. Examples from the Main Index mendacium, 5171, 5329, 5389 mendation, 220 mendax, 2702, 2800 mended, 360 Mendel, 416, 925, 965 Mendelian, 2212 Mendels, 345, 363, 417, 458, 560, 572, 588, 926, 928, 929 MENDELSSOHN, 925 Mendelssohn, 8, 9, 19, 98, 99, 100, 101