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-locales in Formal Topology
No full textA -frame is a poset with countable joins and finite meets in whichbinary meets distribute over countable joins. The aim of this paper is to showthat -frames, actually -locales, can be seen as a branch ofFormal Topology, that is, intuitionistic and predicative point-free topology.Every -frame is the lattice of Lindel\"of elements (those for whicheach of their covers admits a countable subcover) of a formal topology of aspecific kind which, in its turn, is a presentation of the free frame over .We then give a constructive characterization of the smallest (strongly) dense-sublocale of a given -locale, thus providing a"-version" of a Boolean locale. Our development depends on the axiom ofcountable choice
Extremal digraphs on Meyniel-type condition for hamiltonian cycles in balanced bipartite digraphs
No full textLet be a strong balanced digraph on vertices. Adamus et al. haveproved that is hamiltonian if whenever and . The lower bound is tight. In this paper, we shallshow that the extremal digraph on this condition is two classes of digraphsthat can be clearly characterized. Moreover, we also show that if whenever and , then istraceable. The lower bound is tight.Comment: 12 page
A Logic for Monitoring Dynamic Networks of Spatially-distributed Cyber-Physical Systems
No full textCyber-Physical Systems (CPS) consist of inter-wined computational (cyber) andphysical components interacting through sensors and/or actuators. Computationalelements are networked at every scale and can communicate with each other andwith humans. Nodes can join and leave the network at any time or they can moveto different spatial locations. In this scenario, monitoring spatial andtemporal properties plays a key role in the understanding of how complexbehaviors can emerge from local and dynamic interactions. We revisit here theSpatio-Temporal Reach and Escape Logic (STREL), a logic-based formal languagedesigned to express and monitor spatio-temporal requirements over the executionof mobile and spatially distributed CPS. STREL considers the physical space inwhich CPS entities (nodes of the graph) are arranged as a weighted graphrepresenting their dynamic topological configuration. Both nodes and edgesinclude attributes modeling physical and logical quantities that can evolveover time. STREL combines the Signal Temporal Logic with two spatial modalitiesreach and escape that operate over the weighted graph. From these basicoperators, we can derive other important spatial modalities such as everywhere,somewhere and surround. We propose both qualitative and quantitative semanticsbased on constraint semiring algebraic structure. We provide an offlinemonitoring algorithm for STREL and we show the feasibility of our approach withthe application to two case studies: monitoring spatio-temporal requirementsover a simulated mobile ad-hoc sensor network and a simulated epidemicspreading model for COVID19.Comment: arXiv admin note: substantial text overlap with arXiv:1904.0884
A survey on t-core partitions
No full text-core partitions have played important roles in the theory of partitions and related areas. In this survey, we briefly summarize interesting and important results on -cores from classical results like how to obtain a generating function to recent results like simultaneous cores. Since there have been numerous studies on -cores, it is infeasible to survey all the interesting results. Thus, we mainly focus on the roles of -cores in number theoretic aspects of partition theory. This includes the modularity of -core partition generating functions, the existence of -core partitions, asymptotic formulas and arithmetic properties of -core partitions, and combinatorial and number theoretic aspects of simultaneous core partitions. We also explain some applications of -core partitions, which include relations between core partitions and self-conjugate core partitions, a -core crank explaining Ramanujan's partition congruences, and relations with class numbers
The last chapter of the Disquisitiones of Gauss
No full textThis exposition reviews what exactly Gauss asserted and what did he prove in the last chapter of {\sl Disquisitiones Arithmeticae} about dividing the circle into a given number of equal parts. In other words, what did Gauss claim and actually prove concerning the roots of unity and the construction of a regular polygon with a given number of sides. Some history of Gauss's solution is briefly recalled, and in particular many relevant classical references are provided which we believe deserve to be better known
Analyse réflexive du partenariat en santé à partir de mon expérience de patiente-partenaire: Constats, réflexions et recommandations
No full textIn Switzerland, the context is relatively favorable to patient engagement, which encouraged me to get involved in the health care system. Since 2019, I have been able to experience partnership with various actors, different institutional structures, and at several levels. Despite a positive environment for the deployment of patient partnership, thanks to political, institutional, and financial incentives, the implementation of the new paradigm is facing many obstacles. Obstacles that I faced in my journey as a patient-partner. The following analysis is conducted based on my last three years of engagement as a patient partner in the health system.En Suisse, le contexte relativement favorable à l’engagement patient m’a poussé à vouloir m’investir dans le partenariat en santé. Depuis 2019, j’ai pu expérimenter le partenariat avec divers acteurs, différentes structures institutionnelles et à plusieurs niveaux. Malgré un environnement propice au déploiement du partenariat, qui s’est traduit par des incitations politiques, institutionnelles et financières, la mise en pratique du nouveau paradigme qui touche le système de santé, se heurte à de nombreux freins. Freins auxquels je me suis confrontée dans mon propre parcours de patiente-partenaire. L’analyse qui suit est menée à la lumière de mes trois dernières années d’engagement dans le partenariat
Down-step statistics in generalized Dyck paths
No full textThe number of down-steps between pairs of up-steps in -Dyck paths, ageneralization of Dyck paths consisting of steps suchthat the path stays (weakly) above the line , is studied. Results areproved bijectively and by means of generating functions, and lead to severalinteresting identities as well as links to other combinatorial structures. Inparticular, there is a connection between -Dyck paths and perforationpatterns for punctured convolutional codes (binary matrices) used in codingtheory. Surprisingly, upon restriction to usual Dyck paths this yields a newcombinatorial interpretation of Catalan numbers
Domination in Kn\"odel Graphs
No full textGiven a graph and an integer , it is an NP-complete problem to decidewhether there is a dominating set of size at most . In this paper we studythis problem for the Kn\"odel Graph on vertices using elementary numbertheory techniques. In particular, we show an explicit upper bound for thedomination number of the Kn\"odel Graph on vertices any time that we canfind a prime number dividing for which is a primitive root.Comment: Comments welcome
On local antimagic vertex coloring for complete full -ary trees
No full textLet be a finite simple undirected graph without components. A bijection is called alocal antimagic labeling if for any two adjacent vertices and , theyhave different vertex sums, i.e., , where the vertex sum , and is the set of edges incident to . Thusany local antimagic labeling induces a proper vertex coloring of where thevertex is assigned the color (vertex sum) . The local antimagicchromatic number is the minimum number of colors taken over allcolorings induced by local antimagic labelings of . It was conjectured\cite{Aru-Wang} that for every tree the local antimagic chromatic number, where is the number of leaves of .In this article we verify the above conjecture for complete full -ary trees,for . A complete full -ary tree is a rooted tree in which allnodes have exactly children except leaves and every leaf is of the samedepth. In particular we obtain that the exact value for the local antimagicchromatic number of all complete full -ary trees is for odd .Comment: 15 pages, 6 figure
Summability characterizations of positive sequences
No full textIn this paper, we propose extensions for the classical Kummer test, which isa very far-reaching criterion that provides sufficient and necessary conditionsfor convergence and divergence of series of positive terms. Furthermore, wepresent and discuss some interesting consequences and examples such asextensions of the Olivier's theorem and Raabe, Bertrand and Gauss's test.Comment: 12 pages (Accepted for publication in Communications in Mathematics in June 17, 2020