We consider ideals of minors of a matrix, ideals of minors of a symmetric matrix, and ideals of Pfaffians of an alternating matrix. Assuming these ideals are of generic height, we characterize the condition $G_{s}$ for these ideals in terms of the heights of smaller ideals of minors or Pfaffians of the same matrix. We additionally obtain bounds on the generation and concentration degrees of the defining equations of Rees rings for a subclass of such ideals via specialization of the Rees rings in the generic case. We do this by proving that, given sufficient height conditions on ideals of minors or Pfaffians of the matrix, the specialization of a resolution of a graded component of the Rees ring in the generic case is an approximate resolution of the same component of the Rees ring in question. We end the paper by giving some examples of explicit generation and concentration degree bounds.

No description available.

The mathematical modeling and performance analysis of cellular networks have seen a major paradigm shift with the application of stochastic geometry. The main purpose of stochastic geometry is to endow probability distributions on the locations of the base stations (BSs) and users in a network, which, in turn, provides an analytical handle on the performance evaluation of cellular networks. To preserve the tractability of analysis, the common practice is to assume complete spatial randomness} of the network topology. In other words, the locations of users and BSs are modeled as independent homogeneous Poisson point processes (PPPs). Despite its usefulness, the PPP-based network models fail to capture any spatial coupling between the users and BSs which is dominant in a multi-tier cellular network (also known as the heterogeneous cellular networks (HetNets)) consisting of macro and small cells. For instance, the users tend to form hotspots or clusters at certain locations and the small cell BSs (SBSs) are deployed at higher densities at these locations of the hotspots in order to cater to the high data demand. Such user-centric deployments naturally couple the locations of the users and SBSs. On the other hand, these spatial couplings are at the heart of the spatial models used in industry for the system-level simulations and standardization purposes. This dissertation proposes fundamentally new spatial models based on stochastic geometry which closely emulate these spatial couplings and are conductive for a more realistic and fine-tuned performance analysis, optimization, and design of cellular networks. First, this dissertation proposes a new class of spatial models for HetNets where the locations of the BSs and users are assumed to be distributed as Poisson cluster process (PCP). From the modeling perspective, the proposed models can capture different spatial couplings in a network topology such as the user hotspots and user BS coupling occurring due to the user-centric deployment of the SBSs. The PCP-based model is a generalization of the state-of-the-art PPP-based HetNet model. This is because the model reduces to the PPP-based model once all spatial couplings in the network are ignored. From the stochastic geometry perspective, we have made contributions in deriving the fundamental distribution properties of PCP, such as the distance distributions and sum-product functionals, which are instrumental for the performance characterization of the HetNets, such as coverage and rate. The focus on more refined spatial models for small cells and users brings to the second direction of the dissertation, which is modeling and analysis of HetNets with millimeter wave (mm-wave) integrated access and backhaul (IAB), an emerging design concept of the fifth generation (5G) cellular networks. While the concepts of network densification with small cells have emerged in the fourth generation (4G) era, the small cells can be realistically deployed with IAB since it solves the problem of high capacity wired backhaul of SBSs by replacing the last-mile fibers with mm-wave links. We have proposed new stochastic geometry-based models for the performance analysis of IAB-enabled HetNets. Our analysis reveals some interesting system-design insights: (1) the IAB HetNets can support a maximum number of users beyond which the data rate drops below the rate of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no rate gain is observed with the addition of more SBSs. The third and final direction of this dissertation is the combination of machine learning and stochastic geometry to construct a new class of data driven network models which can be used in the performance optimization and design of a network. As a concrete example, we investigate the classical problem of wireless link scheduling where the objective is to choose an optimal subset of simultaneously active transmitters (Tx-s) from a ground set of Tx-s which will maximize the network-wide sum-rate. Since the optimization problem is NP-hard, we replace the computationally expensive heuristic by inferring the point patterns of the active Tx-s in the optimal subset after training a determinantal point process (DPP). Our investigations demonstrate that the DPP is able to learn the spatial interactions of the Tx-s in the optimal subset and gives a reasonably accurate estimate of the optimal subset for any new ground set of Tx-s. / Doctor of Philosophy / The high speed global cellular communication network is one of the most important technologies, and it continues to evolve rapidly with every new generation. This evolution greatly depends on observing performance-trends of the emerging technologies on the network models through extensive system-level simulations. Since these simulation models are extremely time-consuming and error prone, the complementary analytical models of cellular networks have been an area of active research for a long time. These analytical models are intended to provide crisp insights on the network behavior such as the dependence of network performance metrics (such as coverage or rate) on key system-level parameters (such as transmission powers, base station (BS) density) which serve as the prior knowledge for more fine-tuned simulations. Over the last decade, the analytical modeling of the cellular networks has been driven by stochastic geometry. The main purpose of stochastic geometry is to endow the locations of the base stations (BSs) and users with probability distributions and then leverage the properties of these distributions to average out the spatial randomness. This process of spatial averaging allows us to derive the analytical expressions of the system-level performance metrics despite the presence of a large number of random variables (such as BS and user locations, channel gains) under some reasonable assumptions. The simplest stochastic geometry based model of cellular networks, which is also the most tractable, is the so-called Poisson point process (PPP) based network model. In this model, users and BSs are assumed to be distributed as independent homogeneous PPPs. This is equivalent to saying that the users and BSs independently and uniformly at random over a plane. The PPP-based model turned out to be a reasonably accurate representation of the yesteryear’s cellular networks which consisted of a single tier of macro BSs (MBSs) intended to provide a uniform coverage blanket over the region. However, as the data-hungry devices like smart-phones, tablets, and application like online gaming continue to flood the consumer market, the network configuration is rapidly deviating from this baseline setup with different spatial interactions between BSs and users (also termed spatial coupling) becoming dominant. For instance, the user locations are far from being homogeneous as they are concentrated in specific areas like residential and commercial zones (also known as hotspots). Further, the network, previously consisting of a single tier of macro BSs (MBSs), is becoming increasingly heterogeneous with the deployment of small cell BSs (SBSs) with small coverage footprints and targeted to serve the user hotspots. It is not difficult to see that the network topology with these spatial couplings is quite far from complete spatial randomness which is the basis of the PPP-based models. The key contribution of this dissertation is to enrich the stochastic geometry-based mathematical models so that they can capture the fine-grained spatial couplings between the BSs and users. More specifically, this dissertation contributes in the following three research directions. Direction-I: Modeling Spatial Clustering. We model the locations of users and SBSs forming hotspots as Poisson cluster processes (PCPs). A PCP is a collection of offspring points which are located around the parent points which belong to a PPP. The coupling between the locations of users and SBSs (due to their user-centric deployment) can be introduced by assuming that the user and SBS PCPs share the same parent PPP. The key contribution in this direction is the construction of a general HetNet model with a mixture of PPP and PCP-distributed BSs and user distributions. Note that the baseline PPP-based HetNet model appears as one of the many configurations supported by this general model. For this general model, we derive the analytical expressions of the performance metrics like coverage probability, BS load, and rate as functions of the coupling parameters (e.g. BS and user cluster size). Direction-II: Modeling Coupling in Wireless Backhaul Networks. While the deployment of SBSs clearly enhances the network performance in terms of coverage, one might wonder: how long network densification with tens of thousands of SBSs can meet the everincreasing data demand? It turns out that in the current network setting, where the backhaul links (i.e. the links between the BSs and core network) are still wired, it is not feasible to densify the network beyond some limit. This backhaul bottleneck can be overcome if the backhaul links also become wireless and the backhaul and access links (link between user and BS) are jointly managed by an integrated access and backhaul (IAB) network. In this direction, we develop the analytical models of IAB-enabled HetNets where the key challenge is to tackle new types of couplings which exist between the rates on the wireless access and backhaul links. Such couplings exist due to the spatial correlation of the signal qualities of the two links and the number of users served by different BSs. Two fundamental insights obtained from this work are as follows: (1) the IAB HetNets can support a maximum number of users beyond which the network performance drops below that of a single-tier macro-only network, and (2) there exists a saturation point of SBS density beyond which no performance gain is observed with the addition of more SBSs. Direction-III: Modeling Repulsion. In this direction, we focus on modeling another aspect of spatial coupling imposed by the intra-point repulsion. Consider a device-to-device (D2D) communication scenario, where some users are transmitting some on-demand content locally cached in their devices using a common channel. Any reasonable multiple access scheme will ensure that two nearly users are never simultaneously active as they will cause severe mutual interference and thereby reducing the network-wide sum rate. Thus the active users in the network will have some spatial repulsion. The locations of these users can be modeled as determinantal point processes (DPPs). The key property of DPP is that it forms a bridge between stochastic geometry and machine learning, two otherwise non-overlapping paradigms for wireless network modeling and design. The main focus in this direction is to explore the learning framework of DPP and bring together advantages of stochastic geometry and machine learning to construct a new class of data-driven analytical network models.

Dans ce travail nous étudions les variétés determinantales essentiellement isolées (EIDS). Ce type de singularité est une généralization de la notion de singularité isolée. La variété determinantale générique $M_{m,n}^t$ est un sous-ensemble des matrices, mxn, tels que le rang est inférieur que t, où t≤m≤n. Une variété X est determinantal si X est définie comme la pré-image d'une fonction holomorphe, $F:\mathbb{C}^N \to M$, sur la variété determinantale générique avec la condition $codim X=codim M_{m,n}^t$.Certains travaux précédents ont étudié les variétés determinantales avec singularité isolée et ils ont défini le nombre de Milnor d'une surface determinantale et la caractéristique évanescente d'Euler.Nous étudions l'ensemble des hyperplans limites d'hyperplans tangents à une surface determinantale en $\mathbb{C}^4$ et 3-variété en $\mathbb{C}^5$ pour donner une caractérisation de ces hyperplans, par le fait que le nombre de Milnor de leur section avec la surface dans le premier cas ou la 3- variété dans le deuxième cas n'est pas minimum.Nous montrons également que, si X est une EIDS, de dimension d et H et H' sont des hyperplans fortement généraux, si $P \subset H$ et $P'\subset H'$ sont des plans de codimension d-2, les nombres de Milnor des surfaces genériques sont égaux.Nous étudions aussi la modification de Nash d'une EIDS et donnons des conditions suffisantes pour que cette transformation soit lisse.Un autre objectif de notre travail est l'étude de l'obstruction d'Euler d Nous obtenons des formules inductives qui relient l'obstruction d'Euler de X à la caractéristique d'Euler évanescente du lissage essentiel de leurs sections génériques. / In this work, we study the essentially isolated determinantal singularities (EIDS). This type of singularities is a natural generalization of isolated ones. A generic determinantal variety $M_{m,n}^t$ is a subset of the space of mxn matrices, given by matrices of rank less than t, where t≤m≤n. A variety X is determinantal if X is defined as the pre-image of $M_{m,n}^t$ by a holomorphic function $F:\mathbb{C}^N \to M$ with the condition $codim X=codim M_{m,n}^t$.Several recent works investigate determinantal variety with isolated singularities and they are difened the Milnor number and the vanishing Euler characteristic.In this work we study the set of limits of tangent hyperplanes to surface in $\mathbb{C}^4$ and 3-variety in $\mathbb{C}^5$ to give a characterization of this set by the fact that the Milnor number of its section with the surface in the first case or the 3-dimensional determinantal variety in the second case is not minimum. We also prove that if X is a d- dimensional EIDS and H and H' are strongly general hyperplans, if $P \subset H$ and $P'\subset H'$ are d-2 linear plans, the Milnor number of the generic surfaces are equal.We study the Nash transformation of an EIDS and give sufficient conditions for this transformation to be smooth.Another aim of our study is the Euler obstruction of essentially isolated determinantal singularities. We obtain inductive formulas associating the Euler obstruction with the vanishing Euler characteristic of the essencial smoothing of their generic sections.

In an attempt to prove Amitsur's Conjecture for cyclic subgroups of the Brauer group, we look at rational embeddings of the Severi Brauer variety of an algebra into its norm hypersurface. We enlarge the collection of such embeddings, and generalize them to embeddings of generalized Severi Brauer varieties into determinantal varieties. / text

Ce manuscrit est dédié à l'étude de l'estimation paramétrique d'une famille de processus ponctuels appelée processus déterminantaux. Ces processus sont utilisés afin de générer et modéliser des configurations de points possédant de la dépendance négative, dans le sens où les points ont tendance à se repousser entre eux. Plus précisément, nous étudions les propriétés asymptotiques de divers estimateurs classiques de processus déterminantaux paramétriques, stationnaires et non-stationnaires, dans les cas où l'on observe une unique réalisation d'un tel processus sur une fenêtre bornée. Ici, l'asymptotique se fait sur la taille de la fenêtre et donc, indirectement, sur le nombre de points observés. Dans une première partie, nous montrons un théorème limite central pour une classe générale de statistiques sur les processus déterminantaux. Dans une seconde partie, nous montrons une inégalité de béta-mélange générale pour les processus ponctuels que nous appliquons ensuite aux processus déterminantaux. Dans une troisième partie, nous appliquons le théorème limite central obtenu à la première partie à une classe générale de fonctions estimantes basées sur des méthodes de moments. Finalement, dans la dernière partie, nous étudions le comportement asymptotique du maximum de vraisemblance des processus déterminantaux. Nous donnons une approximation asymptotique de la log-vraisemblance qui est calculable numériquement et nous étudions la consistance de son maximum. / This manuscript is devoted to the study of parametric estimation of a point process family called determinantal point processes. These point processes are used to generate and model point patterns with negative dependency, meaning that the points tend to repel each other. More precisely, we study the asymptotic properties of various classical parametric estimators of determinantal point processes, stationary and non stationary, when considering that we observe a unique realization of such a point process on a bounded window. In this case, the asymptotic is done on the size of the window and therefore, indirectly, on the number of observed points. In the first chapter, we prove a central limit theorem for a wide class of statistics on determinantal point processes. In the second chapter, we show a general beta-mixing inequality for point processes and apply our result to the determinantal case. In the third chapter, we apply the central limit theorem showed in the first chapter to a wide class of moment-based estimating functions. Finally, in the last chapter, we study the asymptotic behaviour of the maximum likelihood estimator of determinantal point processes. We give an asymptotic approximation of the log-likelihood that is computationally tractable and we study the consistency of its maximum.

No description available.

Neste trabalho, estudamos a classificação de singularidades de curvas espaciais simples que não são intersecções completas. O Teorema de Hilbert-Burch nos permite usar a matriz de representação para estudar a variedade definida pelo ideal gerado por seus menores maximais. Da mesma forma, as deformações da variedade determinantal podem ser representadas por perturbações da matriz e qualquer perturbação da matriz fornece uma deformação da variedade. Assim, o estudo das singularidades de curvas determinantais pode ser formulado em termos da matriz de representação da curva / In this work, we study the classification of simple space curve singularities which are not complete intersections. The Theorem of Hilbert-Burch enables us to deal with the presentation matrices instead of the ideals defined by their maximal minors. In the same way, deformations of the determinantal variety can be represented by perturbations of the matrix and any perturbation of the matrix gives rise to a deformation of the variety. Therefore, the study of determinantal curves can be formulated in terms of the presentation matrices

A point process is a mechanism, which realizes randomly locally finite point measures. One of the main results of this thesis is an existence theorem for a new class of point processes with a so called signed Levy pseudo measure L, which is an extension of the class of infinitely divisible point processes. The construction approach is a combination of the classical point process theory, as developed by Kerstan, Matthes and Mecke, with the method of cluster expansions from statistical mechanics. Here the starting point is a family of signed Radon measures, which defines on the one hand the Levy pseudo measure L, and on the other hand locally the point process. The relation between L and the process is the following: this point process solves the integral cluster equation determined by L. We show that the results from the classical theory of infinitely divisible point processes carry over in a natural way to the larger class of point processes with a signed Levy pseudo measure. In this way we obtain e.g. a criterium for simplicity and a characterization through the cluster equation, interpreted as an integration by parts formula, for such point processes. Our main result in chapter 3 is a representation theorem for the factorial moment measures of the above point processes. With its help we will identify the permanental respective determinantal point processes, which belong to the classes of Boson respective Fermion processes. As a by-product we obtain a representation of the (reduced) Palm kernels of infinitely divisible point processes. In chapter 4 we see how the existence theorem enables us to construct (infinitely extended) Gibbs, quantum-Bose and polymer processes. The so called polymer processes seem to be constructed here for the first time. In the last part of this thesis we prove that the family of cluster equations has certain stability properties with respect to the transformation of its solutions. At first this will be used to show how large the class of solutions of such equations is, and secondly to establish the cluster theorem of Kerstan, Matthes and Mecke in our setting. With its help we are able to enlarge the class of Polya processes to the so called branching Polya processes. The last sections of this work are about thinning and splitting of point processes. One main result is that the classes of Boson and Fermion processes remain closed under thinning. We use the results on thinning to identify a subclass of point processes with a signed Levy pseudo measure as doubly stochastic Poisson processes. We also pose the following question: Assume you observe a realization of a thinned point process. What is the distribution of deleted points? Surprisingly, the Papangelou kernel of the thinning, besides a constant factor, is given by the intensity measure of this conditional probability, called splitting kernel. / Ein Punktprozess ist ein Mechanismus, der zufällig ein lokalendliches Punktmaß realisiert. Ein Hauptresultat dieser Arbeit ist ein Existenzsatz für eine sehr große Klasse von Punktprozessen mit einem signierten Levy Pseudomaß L. Diese Klasse ist eine Erweiterung der Klasse der unendlich teilbaren Punktprozesse. Die verwendete Methode der Konstruktion ist eine Verbindung der klassischen Punktprozesstheorie, wie sie von Kerstan, Matthes und Mecke ursprünglich entwickelt wurde, mit der sogenannten Methode der Cluster-Entwicklungen aus der statistischen Mechanik. Ausgangspunkt ist eine Familie von signierten Radonmaßen. Diese definiert einerseits das Levysche Pseudomaß L; andererseits wird mit deren Hilfe der Prozess lokal definiert. Der Zusammenhang zwischen L und dem Prozess ist so, dass der Prozess die durch L bestimmte Integralgleichung (genannt Clustergleichung) löst. Wir zeigen, dass sich die Resultate aus der klassischen Theorie der unendlich teilbaren Punktprozesse auf natürliche Weise auf die neue Klasse der Punktprozesse mit signiertem Levy Pseudomaß erweitern lassen. So erhalten wir z.B. ein Kriterium für die Einfachheit und eine Charackterisierung durch die Clustergleichung für jene Punktprozesse. Unser erstes Hauptresultat in Kapitel 3 zur Analyse der konstruierten Prozesse ist ein Darstellungssatz der faktoriellen Momentenmaße. Mit dessen Hilfe werden wir die permanentischen respektive determinantischen Punktprozesse, die in die Klasse der Bosonen respektive Fermionen Prozesse fallen, identifizieren. Als ein Nebenresultat erhalten wir eine Darstellung der (reduzierten) Palm Kerne von unendlich teilbaren Punktprozessen. Im Kapitel 4 konstruieren wir mit Hilfe unseres Existenzsatzes unendlich ausgedehnte Gibbsche Prozesse sowie Quanten-Bose und Polymer Prozesse. Unseres Wissens sind letztere bisher nicht konstruiert worden. Im letzten Teil der Arbeit zeigen wir, dass die Familie der Clustergleichungen gewisse Stabilitätseigenschaften gegenüber gewissen Transformationen ihrer Lösungen aufweist. Dies wird erstens verwendet, um zu verdeutlichen, wie groß die Klasse der Punktprozesslösungen einer solchen Gleichung ist. Zweitens wird damit der Ausschauerungssatz von Kerstan, Matthes und Mecke in unserer allgemeineren Situation gezeigt. Mit seiner Hilfe können wir die Klasse der Polyaschen Prozesse auf die der von uns genannten Polya Verzweigungsprozesse vergrößern. Der letzte Abschnitt der Arbeit beschäftigt sich mit dem Ausdünnen und dem Splitten von Punktprozessen. Wir beweisen, dass die Klassen der Bosonen und Fermionen Prozesse abgeschlossen unter Ausdünnung ist. Die Ergebnisse über das Ausdünnen verwenden wir, um eine Teilklasse der Punktprozesse mit signiertem Levy Pseudomaß als doppelt stochastische Poissonsche Prozesse zu identifizieren. Wir stellen uns auch die Frage: Angenommen wir beobachten eine Realisierung einer Ausdünnung eines Punktprozesses. Wie sieht die Verteilung der gelöschten Punktkonfiguration aus? Diese bedingte Verteilung nennen wir splitting Kern, und ein überraschendes Resultat ist, dass der Papangelou-Kern der Ausdünnung, abgesehen von einem konstanten Faktor, gegeben ist durch das Intensitätsmaß des splitting Kernes.

Pour toute variété affine W munie d'une opération d'un groupe réductif G, le schéma de Hilbert invariant est un espace de modules qui classifie les sous-schémas fermés de W, stables par l'opération de G, et dont l'algèbre affine est somme directe de G-modules simples avec des multiplicités finies préalablement fixées. Dans cette thèse , on étudie d'abord le schéma de Hilbert invariant, noté H, qui paramètre les sous-schémas fermés GL(V)-stables Z de W=n1 V oplus n2 V^* tels que k[Z] est isomorphe à la représentation régulière de GL(V) comme GL(V)-module. Si dim(V)<3,on montre que H est une variété lisse, et donc que le morphisme de Hilbert-Chow gamma: H -> W//G est une résolution des singularités du quotient W//G. En revanche, si dim(V)=3, on montre que H est singulier. Lorsque dim(V)<3, on décrit H par des équations et aussi comme l'espace total d'un fibré vectoriel homogène au dessus d'un produit de deux grassmanniennes. On se place ensuite dans le cadre symplectique en prenant n1=n2 et en remplaçant W par la fibre en 0 de l'application moment mu: W -> End(V). On considère alors le schéma de Hilbert invariant H' qui paramètre les sous-schémas contenus dans mu^{-1}(0). On montre que H' est toujours réductible, mais que sa composante principale Hp' est lisse lorsque dim(V)<3. Dans ce cas, le morphisme de Hilbert-Chow est une résolution (parfois symplectique) des singularités du quotient mu^{-1}(0)//G. Lorsque dim(V)<3, on décrit Hp' comme l'espace total d'un fibré vectoriel homogène au dessus d'une variété de drapeaux. Enfin, on obtient des résultats similaires lorsque l'on remplace GL(V) par un autre groupe classique (SL(V), SO(V), O(V), Sp(V)) que l'on fait opérer d'abord dans W=nV, puis dans la fibre en 0 de l'application moment. / Let W be an affine variety equipped with an action of a reductive group G. The invariant Hilbert scheme is a moduli space which classifies the G-stable closed subschemes of W such that the affine algebra is the direct sum of simple G-modules with previously fixed finite multiplicities. In this thesis, we first study the invariant Hilbert scheme, denoted H. It parametrizes the GL(V)-stable closed subschemes Z of W=n1 V oplus n2 V^* such that k[Z] is isomorphic to the regular representation of GL(V) as GL(V)-module. If dim(V)<3, we show that H is a smooth variety, so that the Hilbert-Chow morphism gamma: H -> W//G is a resolution of singularities of the quotient W//G. However, if dim(V)=3, we show that H is singular. When dim(V)<3, we describe H by equations and also as the total space of a homogeneous vector bundle over the product of two Grassmannians. Then we consider the symplectic setting by letting n1=n2 and replacing W by the zero fiber of the moment map mu: W -> End(V). We study the invariant Hilbert scheme H' which parametrizes the subschemes included in mu^{-1}(0). We show that H' is always reducible, but that its main component Hp' is smooth if dim(V)<3. In this case, the Hilbert-Chow morphism is a resolution of singularities (sometimes a symplectic one) of the quotient mu^{-1}(0)//G. When dim(V)=3, we describe Hp' as the total space of a homogeneous vector bundle over a flag variety. Finally, we get similar results when we replace GL(V) by some other classical group (SL(V), SO(V), O(V), Sp(V)) acting first on W=nV, then on the zero fiber of the moment map.