Auswahl der wissenschaftlichen Literatur zum Thema „Local Tate duality“

In this paper, we formulate and prove a duality for cohomology of curves over perfect fields of positive characteristic with coefficients in Néron models of abelian varieties. This is a global function field version of the author’s previous work on local duality and Grothendieck’s duality conjecture. It generalizes the perfectness of the Cassels–Tate pairing in the finite base field case. The proof uses the local duality mentioned above, Artin–Milne’s global finite flat duality, the nondegeneracy of the height pairing and finiteness of crystalline cohomology. All these ingredients are organized under the formalism of the rational étale site developed earlier.

We use the theory of Condensed Mathematics to build a condensed cohomology theory for the Weil group of a p -adic field. The cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups.

The single most influential and widely accepted objection against any form of dualism, the belief that human beings are both body and soul, is the objection that dualism violates conservation laws in physics. The conservation laws objection against dualism posits that body and soul interaction is at best mysterious, and at worst impossible. While this objection has been both influential from the time of its initial formulation until present, this paper occupies itself with arguing that this objection is a fleeting one, and has successful answers from both scientific and philosophical perspectives. It is to this end that I provide three groups of responses to the conservation laws objection. First, I outline responses which take the ‘laws of nature’ as the proper entry point into the discussion. Secondly, I provide an analysis of those who argue that contemporary quantum physical data requires that the objection itself involves scientifically unjustified premises. Finally, I layout a philosophically oriented answer which argues that the objection is linguistically problematic since its demands on the dualist are categorically fallacious. From these groups of answers, I conclude that while the conservation laws objection has been arguably the most widely accepted objection against dualism, the objection is without philosophical justification.

For a smooth, simple closed curve α in the plane, the perpendicular bisector map P associates to each pair of distinct points (p, q) on α the perpendicular bisector of the chord joining p and q. To a pair (p, p), the map P associates the normal to α at p. The set of critical values of this map is the union of the dual of the symmetry set of α and the dual of the evolute. (The symmetry set is the locus of the centres of circles bitangent to α.) We study the mapP and use it to give a complete list of the transitions which take place on the dual of the symmetry set and the dual of the evolute, as α varies in a generic one-parameter family of plane curves.

We take up a nonsmooth multiobjective optimization problem with tangentially convex objective and constraint functions. In employing a suitable constraint qualification, we formulate both necessary and sufficient optimality conditions for (local) quasi efficient solutions in terms of tangential subdifferentials. Furthermore, under generalized convexity assumptions, we state strong, weak and converse duality relations of Wolfe and Mond–Weir types. We give a number of examples to illustrate the new concepts and main results of this paper.

Following a suggestion by Vafa, we present a quantum-mechanical model for S duality symmetries observed in the quantum theories of fields, strings and branes. Our formalism may be understood as the topological limit of Berezin's metric quantization of the upper half-plane H, in that the metric dependence of Berezin's method has been removed. Being metric-free, our prescription makes no use of global quantum numbers. Quantum numbers arise only locally, after the choice of a local vacuum to expand around. Our approach may be regarded as a manifestly nonperturbative formulation of quantum mechanics, in that we take no classical phase space and no Poisson brackets as a starting point. Position and momentum operators satisfying the Heisenberg algebra are defined and their spectra are analysed. We provide an explicit construction of the Hilbert space of states. The latter carries no representation of SL (2,R), due to the lifting of the metric dependence. Instead, the reparametrization invariance of H under SL (2,R) induces a natural SL (2,R) action on the quantum-mechanical operators that implements S duality. We also link our approach with the equivalence principle of quantum mechanics recently formulated by Faraggi–Matone.

L’objectif de cette thèse est double. Premièrement, on construit une théorie de cohomologie topologique pour le groupe de Weil d’un corps p-adique. En second lieu, on utilise cette théorie pour prouver des théorèmes de dualité, qui se manifestent sous la forme de la dualité de Pontryagin entre groupes abéliens localement compacts. Ces résultats améliorent des théorèmes de dualité existants et leur confèrent une perspective topologique. De tels objectifs peuvent être atteints grâce aux Mathématiques Condensées, qui fournissent un cadre dans lequel il est possible de faire de l’algèbre avec des objets topologiques. On définit une théorie cohomologique pour les groupes condensés et pro-condensés et on étudie ses propriétés. Ensuite, on applique cela au groupe de Weil d’un corps p-adique, considéré comme un groupe pro-condensé. On démontre que, dans certains cas particuliers, les groupes de cohomologie correspondants sont des groupes abéliens localement compacts de rangs finis. Ceci nous permet d’étendre la dualité locale de Tate à une catégorie plus générale de coefficients non nécessairement discrets, o`u elle prend la forme d’une dualité de Pontryagin entre groupes abéliens localement compacts. Dans la dernière partie de la thèse, on utilise le même cadre pour retrouver une version “à la Weil” de la dualité de Tate avec coefficients dans les variétés abéliennes, et plus généralement dans les 1- motifs, en exprimant ces dualités comme des accouplements parfaits entre groupes abéliens condensés. Pour ce faire, on associe à chaque groupe algébrique, resp. 1-motif, un groupe abélien condensé, resp. un complexe de groupes abéliens condensés, avec une action du groupe de Weil (pro-condensé). On appelle cette association la réalisation de Weil-étale condensée. On montre l’existence d’un accouplement de Poincaré condensé pour les variétés abéliennes, et on prouve une version condensée et “à la Weil” de la dualité de Tate à coefficients dans les variétés abéliennes, qui améliore le résultat correspondant de Karpuk. Enfin, on montre l’existence d’un accouplement de Poincaré condensé pour les 1-motifs. On prouve que cet accouplement est compatible à la filtration par les poids et on démontre un théorème de dualité à coefficients dans les 1- motifs, qui améliore un résultat de Harari-Szamuely
The goal of this thesis is twofold. First, we build a topological cohomology theory for the Weil group of p-adic fields. Secondly, we use this theory to prove duality theorems for such fields, which manifest as Pontryagin duality between locally compact abelian groups. These results improve existing duality theorems and give them a topological flavour. Condensed Mathematics allow us to reach these objectives, providing a framework where it is possible to do algebra with topological objects. We define and study a cohomology theory for condensed groups and pro-condensed groups, and we apply it to the Weil group of a p-adic field, considered as a pro-condensed group. The resulting cohomology groups are proved to be locally compact abelian groups of finite ranks in some special cases. This allows us to enlarge the local Tate duality to a more general category of non-necessarily discrete coefficients, where it takes the form of a Pontryagin duality between locally compact abelian groups. In the last part of the thesis, we use the same framework to recover a Weil-version of the Tate duality with coefficients in abelian varieties and more generally in 1-motives, expressing those dualities as perfect pairings between condensed abelian groups. To do this, we associate to every algebraic group, resp. 1-motive, a condensed abelian group, resp. a complex of condensed abelian groups, with an action of the (pro-condensed) Weil group. We call this association the condensed Weil-´etale realisation. We show the existence of a condensed Poincar´e pairing for abelian varieties and we prove a condensed-Weil version of the Tate duality with coefficients in abelian varieties, which improves the correspondent result of Karpuk. Lastly, we exhibit a condensed Poincar´e pairing for 1-motives. We show that this pairing is compatible with the weight filtration and we prove a duality theorem with coefficients in 1-motives, which improves a result of Harari-Szamuely

Dans cette thèse, nous étudions de divers problèmes arithmétiques, notamment l'existence de points rationnels et l'approximation faible sur certaines variétés sur des corps de nombres et leurs analogues géométriques.Après le premier chapitre d'introduction, nous présentons dans le deuxième chapitre quelques résultats obtenus par des calculs cocycliques en cohomologie galoisienne (abélienne ou non abélienne). Nous considérons d'abord une formule de Borovoi-Demarche-Harari pour le groupe de Brauer algébrique non ramifié d'espaces homogènes. Nous établissons ensuite le principe de Hasse et l'approximation faible pour une classe d'espaces homogènes de SLn sur un corps de nombres, à stabilisateurs géométriques finis nilpotents de classe 2, construits par Borovoi et Kunyavskii. Il s'agit d'un petit pas de plus vers une conjecture de Colliot-Thélène sur l'obstruction de Brauer-Manin pour les variétés rationnellement connexes.Le troisième chapitre est consacré à une conjecture récemment formulée par Wittenberg, qui concerne la théorie de la descente (une méthode qui a été initialement développée par Colliot-Thélène et Sansuc). Nous démontrons cette conjecture pour les torseurs sous un groupe algébrique linéaire connexe, généralisant un résultat antérieur de descente d'Harpaz et Wittenberg pour les torseurs sous un tore. Nous le faisons en adaptant leur technique avec la machinerie d'abélianisation de cohomologie galoisienne non abélienne à la Borovoi. Nous allons également prouver une version de cette « conjecture de descente » dans le contexte des zéro-cycles.Dans le dernier chapitre, nous suivons les travaux de Harari-Scheiderer-Szamuely, Izquierdo et Tian, en étudiant le principe local-global et l'approximation faible sur les corps de fonctions p-adiques. Sur ces corps, qui sont des corps de dimension cohomologique 3, il existe un analogue de dimension supérieure de l'obstruction de Brauer-Manin, qui repose sur la loi de réciprocité de Weil généralisée. Nous appliquons ici les théorèmes de dualité de type Poitou-Tate pour obtenir des résultats arithmétiques pour certains espaces homogènes. Nous y considérons également quelques corps de fonctions de dimension cohomologique supérieure à 3
In this thesis, we study various arithmetic problems, notably the existence of rational points and weak approximation on certain varieties over number fields and their geometric analogues.After the first introductory chapter, we present in the second one some results obtained by computations with (abelian or nonabelian) Galois cocycles. First, we consider a formula of Borovoi-Demarche-Harari for the unramified algebraic Brauer group of homogeneous spaces. Then, we establish the Hasse principle and weak approximation for a class of homogeneous spaces of SLn over number fields, whose geometric stabilizers are finite of nilpotency class 2, constructed by Borovoi and Kunyavskii. This is a small step towards a conjecture of Colliot-Thélène on the Brauer-Manin obstruction for rationally connected varieties.The third chapter is devoted to a recently formulated conjecture by Wittenberg, which concerns descent theory (a method orginially developped by Colliot-Thélène and Sansuc). We prove this conjecture for torsors under a connected linear algebraic group, generalizing a previous result of Harpaz and Wittenberg for torsors under a torus. We do this by adapting their technique with Borovoi's machinery of abelianization of non-abelian Galois cohomology. We shall also prove a version of this “descent conjecture” in the context of zero-cycles.In this last chapter, we follow the works of Harari-Scheiderer-Szamuely, Izquierdo and Tian, by studying the local-global principle and weak approximation over p-adic function fields. Over these fields, which are fields of cohomological dimension 3, there exists a higher-dimensional analogue of the Brauer-Manin obstruction that relies on the generalized Weil reciprocity law. Here, we apply Poitou-Tate style duality theorems to obtain some results for certain homogeneous spaces. We also consider some function fields of cohomological dimension greater than 3

Climate Change is a global phenomenon that has geographically varying impacts. To fulfill Hungary’s climate obligations and implement effective adaptation practices, we need to understand the working mechanism of climate change in smaller territorial units. Regional differentiating is of paramount importance in regional strategy making. As part of an on-going research that aims to identify the local impacts of climate change and the local answers against it, this paper is analyzing the local properties and opportunities of the case study of Sarkad LAU 1 region. Sarkad region is one of the most underdeveloped yet one of the richest areas in natural resources like biodiversity, landscape, and cultural heritage. This duality highlights the need to act against the negative outcomes of climate change. The local main climate effects of climate change are indicated by using the cartograms of the National Adaptation and Geoinformation System database. It is crucial to identify the local vulnerability in order to take effective measurements promoting adaptivity and mitigation. As a result of the research, the unique properties of the LAU 1 region the ways of adaptation in connection with climate change are indicated.

If killing a single person is considered as a major crime that forbidden by Sharia and law at the international level and at the level of all religions and divine legislation, so what about the concept of genocide!! Here, not just an individual with a weak influence on society is killed, but thousands of individuals, that means an entire nation, a future, energy and human and intellectual capabilities that can tip the scales, and on the other hand, broken and half-dead hearts are left behind from the horrific scenes of killing they witnessed before their eyes, moreover, the massacres of genocide continues to excrete its remnants and consequences for long years and for successive generations, and it may generate grudges of revenge among generations that did not receive the adequate awareness and psychological support which are necessary to rehabilitate these generations to benefit from the tragedies and bitter experiences of life to turn them into lessons and incentives to achieve progress and advancement. Genocide is a deadly poison whose toxic effect extends from generations to others unless it is wisely controlled. Here the role of the international community and its legal, legislative and humanitarian stance from these crimes is so important and supportive. Genocide can be occurred on two levels: external and internal. As for genocide on the external level: this is what happened at the hands of foreign powers against a certain people for colonial and expansionist goals in favor of the occupier or usurper. There are many examples throughout history, such as the Ottoman and British occupations...etc Whereas genocide at the internal level, can be defined as the repressive actions that governments practice against their own people for goals that could be extremist, racist or dictatorial, such as t ""Al-Anfal"" massacre in 1988 carried out by the previous regime against the Kurds in the Kurdistan region. The number of victims amounted at one hundred thousand martyrs, most of them were innocent and unarmed people from children, women and the elderly, and also the genocide which was practiced against of the organizers of Al-Shaibania Revolution in 1991 was another example of genocide in the internal level. It is possible to deduce a third level between the external and internal levels, which is the genocide that is done at the hands of internal elements from the people of the country, but in implementation of external agendas, for example, the scenes of organized and systematic sectarian killing that we witnessed daily during (2007) and (2008), followed by dozens of bloody explosions in various regions throughout the capital, which unfortunately was practiced by the people of the country who were misguided elements in order to destabilize the security of the country and we did not know until this moment in favor of which external party!! In the three aforementioned cases, nothing can justify the act of killing or genocide, but in my personal opinion, I see that genocide at the hands of foreign forces is less drastic effects than the genocides that done at the hands of internal forces that kill their own people to impose their control and to defense their survival, from the perspective of ""the survival for the strongest, the most criminal and the most dictatorial. The matter which actually dragged the country into the abyss and the ages of darkness and ignorance. As for the foreign occupier, he remains an occupier, and it is so natural for him to be resentful and spiteful and to keep moving with the bragging theory of that (the end justifies the means) and usurping lands illegally, but perhaps recently the occupier has begun to exploit loopholes in international laws and try to gain the support of the international community and international organizations to prove the legitimacy of what has no legitimacy, in the end to achieve goals which pour into the interest of the occupiers' country and from the principle of building the happiness and well-being of the occupiers' people at the expense of the misery and injustice of other peoples!! This remains absolutely dehumanizing societal crime, but at least it has a positive side, which is maximizing economic resources and thus achieving the welfare of a people at the expense of seizing the wealth of the occupied country. This remains the goal of the occupier since the beginning of creation to this day, but today the occupation associated with the horrific and systematic killing has begun to take a new template by framing the ugliness of the crime with humanitarian goals and the worst, to exploit religion to cover their criminal acts. A good example of this is the genocide that took place at the hands of the terrorist organization ISIS, that contradictory organization who adopted the religion which forbids killing and considers it as one of the greatest sins as a means to practice the most heinous types of killing that contemporary history has witnessed!! The ""Spiker"" and ""Sinjar"" massacres in 2014 are the best evidence of this duality in the ideology of this terrorist organization. We may note that the more we advance in time, the more justification for the crimes of murder and genocide increases. For example, we all know the first crimes of genocide represented by the fall of Baghdad at the hands of the Mongol leader ""Hulagu"" in 1258. At that time, the crimes of genocide did not need justification, as they were practiced openly and insolently for subversive, barbaric and criminal goals!! The question here imposes itself: why were the crimes of genocide in the past practiced openly and publicly without need to justify the ugliness of the act? And over time, the crimes of genocide began to be framed by pretexts to legitimize what is prohibited, and to permit what is forbidden!! Or to clothe brutality and barbarism in the patchwork quilt of humanity?? And with this question, crossed my mind the following ""Aya"" from the Glorious Quran (and do not kill the soul that God has forbidden except in the right) , this an explicit ""Aya"" that prohibits killing and permits it only in the right, through the use of the exception tool (except) that permits what coming after it . But the"" right"" that God describes in the glorious Quran has been translated by the human tongues into many forms and faces of falsehood!! Anyway, expect the answer of this controversial question within the results of this study. This study will discuss the axis of (ideologies of various types and genocide), as we will analyze excerpts from the speeches of the former regime that were announced on the local media after each act of genocide or purification, as the former regime described at that time, but the difference in this study is that the analysis will be according to a scientific and thoughtful approach which is far from the personal ideology of the researcher. The analysis will be based on a model proposed by the contemporary Dutch scientist ""Teun A. Van Dijk"". Born in 1943, ""Van Dijk"" is a distinguished scholar and teaching in major international universities. He has authored many approved books as curricula for teaching in the field of linguistics and political discourse analysis. In this study, Van Dijk's Model will be adopted to analyze political discourse ideologies according to forty-one criteria. The analysis process will be conducted in full transparency and credibility in accordance with these criteria without imposing the researcher's personal views. This study aims to shed light on the way of thinking that the dictatorial regimes adopt to impose their existence by force against the will of the people, which can be used to develop peoples' awareness to understand and analyze political statements in a scientific way away from the inherited ideologies imposed by customs, clan traditions, religion, doctrine and nationalism. With accurate scientific diagnosis, we put our hand on the wounds. So we can cure them and also remove the scars of these wounds. This is what we seek in this study, diagnosis and therefore suggesting the suitable treatment "