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On taxicab distance mean functions and their geometric applications: methods, implementations and examples
No full textA distance mean function measures the average distance of points from theelements of a given set of points (focal set) in the space. The level sets of adistance mean function are called generalized conics. In case of infinite focalpoints the average distance is typically given by integration over the focalset. The paper contains a survey on the applications of taxicab distance meanfunctions and generalized conics' theory in geometric tomography: bisection ofthe focal set and reconstruction problems by coordinate X-rays. The theoreticalresults are illustrated by implementations in Maple, methods and examples aswell.Comment: 26 pages, 6 figures, the paper is based on the plenary lecture presented at Meeting on Tomography and Applications (Discrete Tomography, Neuroscience and Image Reconstruction) 16th Edition, IN MEMORIAM OF CARLA PERI, 2 - 4 May 2022, Mathematics Department, Politecnico di Milano, Milano, Ital
Large-scale weighted sequence alignment for the study of intertextuality in Finnic oral folk poetry
No full textThe digitization of large archival collections of oral folk poetry in Finland and Estonia has opened possibilities for large-scale quantitative studies of intertextuality. As an initial methodological step in this direction, I present a method for pairwise line-by-line comparison of poems using the weighted sequence alignment algorithm (a.k.a. ‘weighted edit distance’). The main contribution of the paper is a novel description of the algorithm in terms of matrix operations, which allows for much faster alignment of a poem against the entire corpus by utilizing modern numeric libraries and GPU capabilities. This way we are able to compute pairwise alignment scores between all pairs from among a corpus of over 280,000 poems. The resulting table of over 40 million pairwise poem similarities can be used in variousways to study the oral tradition. Some starting points for such research are sketched in the latter part of the article
On Minimization and Learning of Deterministic -Automata in the Presence of Don't Care Words
No full textWe study minimization problems for deterministic -automata in thepresence of don't care words. We prove that the number of priorities indeterministic parity automata can be efficiently minimized under an arbitraryset of don't care words. We derive that from a more general result from whichone also obtains an efficient minimization algorithm for deterministic parityautomata with informative right-congruence (without don't care words). We then analyze languages of don't care words with a trivialright-congruence. For such sets of don't care words it is known that weakdeterministic B\"uchi automata (WDBA) have a unique minimal automaton that canbe efficiently computed from a given WDBA (Eisinger, Klaedtke 2006). We give acongruence-based characterization of the corresponding minimal WDBA, and showthat the don't care minimization results for WDBA do not extend todeterministic -automata with informative right-congruence: for thisclass there is no unique minimal automaton for a given don't care set withtrivial right congruence, and the minimization problem is NP-hard. Finally, weextend an active learning algorithm for WDBA (Maler, Pnueli 1995) to thesetting with an additional set of don't care words with trivialright-congruence.Comment: Version 2 is a minor revision with a few references added, some additional explanations, and a few typos corrected Version 3: Added "On" to title, and added a reference for Corollary 4.5 Version 4: Small change in intro Version 5: final version typeset by journa
Gabor frames from contact geometry in models of the primary visual cortex
No full textWe analyze the interplay between contact geometry and Gabor filters signalanalysis in geometric models of the primary visual cortex. We show inparticular that a specific framed lattice and an associated Gabor system isdetermined by the Legendrian circle bundle structure of the -manifold ofcontact elements on a surface (which models the V1-cortex), together with thepresence of an almost-complex structure on the tangent bundle of the surface(which models the retinal surface). We identify a scaling of the lattice, alsodictated by the manifold geometry, that ensures the frame condition issatisfied. We then consider a -dimensional model where receptor profilesalso involve a dependence on frequency and scale variables, in addition to thedependence on position and orientation. In this case we show that a proposedprofile window function does not give rise to frames (even in a distributionalsense), while a natural modification of the same generates Gabor frames withrespect to the appropriate lattice determined by the contact geometry.Comment: LaTeX, 33 pages (v2: expanded introduction, v3: typos correction and short explanatory comments added; v4: journal overlay published version
On equations of fake projective planes with automorphism group of order
No full textWe study Dolgachev elliptic surfaces with a double and a triple fiber andfind explicit equations of two new pairs of fake projective plane with automorphisms, thus finishing the task of finding explicit equations of fakeprojective planes with this automorphism group. This includes, in particular,the fake projective plane discovered by J. Keum.Comment: 19 pages + supplemental file
A note on limits of sequences of binary trees
No full textWe discuss a notion of convergence for binary trees that is based on subtreesizes. In analogy to recent developments in the theory of graphs, posets andpermutations we investigate some general aspects of the topology, such as acharacterization of the set of possible limits and its structure as a metricspace. For random trees the subtree size topology arises in the context ofalgorithms for searching and sorting when applied to random input, resulting ina sequence of nested trees. For these we obtain a structural result based on alocal version of exchangeability. This in turn leads to a central limittheorem, with possibly mixed asymptotic normality.Comment: Journal versio
Several Roman domination graph invariants on Kneser graphs
No full textThis paper considers the following three Roman domination graph invariants onKneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph , we present exact values for Roman dominationnumber and total Roman domination number proving that for ,. For signed Romandomination number , the new lower and upper bounds for are provided: we prove that for , the lower bound isequal to 2, while the upper bound depends on the parity of and is equal to3 if is odd, and equal to if is even. For graphs of smallerdimensions, exact values are found by applying exact methods from literature
Enumerating Independent Linear Inferences
No full textA linear inference is a valid inequality of Boolean algebra in which eachvariable occurs at most once on each side. In this work we leverage recently developed graphical representations oflinear formulae to build an implementation that is capable of more efficientlysearching for switch-medial-independent inferences. We use it to find four`minimal' 8-variable independent inferences and also prove that no smaller onesexist; in contrast, a previous approach based directly on formulae reachedcomputational limits already at 7 variables. Two of these new inferences derivesome previously found independent linear inferences. The other two (which aredual) exhibit structure seemingly beyond the scope of previous approaches weare aware of; in particular, their existence contradicts a conjecture of Dasand Strassburger. We were also able to identify 10 minimal 9-variable linear inferencesindependent of all the aforementioned inferences, comprising 5 dual pairs, andpresent applications of our implementation to recent `graph logics'
Comment penser l'interdisciplinarité en pratique ? Une question de disposition, d’indisciplinarité et de complexité
No full textCe texte constitue le prologue du numéro 11 du Journal on Interdisciplinary Methodologies and Issues in Science : Penser l'interdisciplinarité en pratique. L'ensemble des 10 contributions est brièvement présenté à partir des fondements qui font l'interdisciplinarité en pratique : les dispositions des chercheurs, leur posture indisciplinaire et le contexte complexe dans lequel s'opère l'interdisciplinarité
Quelle est la place des exemples dans l'enseignement de l'algèbre abstraite ?
No full textTeaching abstract algebra, seen as the study of structures and properties of structures, at a university level appears to be a challenge for both students and faculty. The professors in our study describe this passage through abstraction for the student body as a "killing game", an "impassable wall" or a "leap through abstraction". In this paper, based on a case study (Candy, 2020) we will investigate the choices of professors teaching abstract algebra at university. These professors were chosen because they teach abstract algebra at university in France, and they gave us access to their course corpus and agreed to be interviewed. In this article, we will choose to study in particular the teaching of the concept of ideal. An epistemological analysis will allow us to highlight its central role in the construction of abstract algebra. Then, using the anthropological theory of didactics (Chevallard, 1998), we will try to specify the place of examples and exercises in the praxis of the student body.Enseigner l'algèbre abstraite, vue comme l'étude des structures et des propriétés des structures, à un niveau universitaire apparaît être un challenge pour les corps étudiant et professoral. Les professeurs de notre étude décrivent ce passage à travers l'abstraction pour le corps étudiant comme un « jeu de massacre », un « mur infranchissable » ou encore « un saut à travers l'abstraction ». Dans cet article, en appui sur une étude de cas (Candy, 2020) nous allons étudier les choix de professeurs enseignant l'algèbre abstraite à l'université. Ces professeurs ont été choisis car ils enseignent l'algèbre abstraite à l'université en France, de plus ils nous ont donné accès à leur corpus de cours et ont accepté d'être interviewés. Dans cet article, nous choisirons d'étudier en particulier l'enseignement du concept d'idéal. Une analyse épistémologique nous permettra de mettre en évidence son rôle central dans la construction de l'algèbre abstraite. Puis, en utilisant la théorie anthropologique du didactique (Chevallard, 1998), nous chercherons à préciser la place des exemples et exercices dans la praxis du corps étudiant