Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

We consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time.Comment: 48 pages, 11 figure

Invariant types in model theory

We study how the product of global invariant types interacts with the preorder of domination, i.e. semi-isolation by a small type, and the induced equivalence relation, domination-equivalence. We provide sufficient conditions for the latter to be a congruence with respect to the product, and show that this holds in various classes of theories. In this case, we develop a general theory of the quotient semigroup, the domination monoid, and carry out its computation in several cases of interest. Notably, we reduce its study in o-minimal theories to proving generation by 1-types, and completely characterise it in the case of Real Closed Fields. We also provide a full characterisation for the theory of dense meet-trees, and moreover show that the domination monoid is well-defined in certain expansions of it by binary relations. We give an example of a theory where the domination monoid is not commutative, and of one where it is not well-defined, correcting some overly general claims in the literature. We show that definability, finite satisfiability, generic stability, and weak orthogonality to a fixed type are all preserved downwards by domination, hence are domination-equivalence invariants. We study the dependence on the choice of monster model of the quotient of the space of global invariant types by domination-equivalence, and show that if the latter does not depend on the former then the theory under examination is NIP

The domination monoid in henselian valued fields

We study the domination monoid in various classes of structures arising from the model theory of henselian valuations, including RV-expansions of henselian valued fields of residue characteristic 0 (and, more generally, of benign valued fields), p-adically closed fields, monotone D-henselian differential valued fields with many constants, regular ordered abelian groups, and pure short exact sequences of abelian structures. We obtain Ax-Kochen-Ershov type reductions to suitable fully embedded families of sorts in quite general settings, and full computations in concrete ones.Comment: 35 pages. Minor revisio

Some definable types that cannot be amalgamated

We exhibit a theory where definable types lack the amalgamation property.Comment: 4 page

Traveling waves for some nonlocal 1D Gross-Pitaevskii equations with nonzero conditions at infinity

International audienceWe consider a nonlocal family of Gross-Pitaevskii equations with nonzero conditions at infinity in dimension one. We provide conditions on the nonlocal interaction such that there is a branch of traveling waves solutions with nonvanishing conditions at infinity. Moreover, we show that the branch is orbitally stable. In this manner, this result generalizes known properties for the contact interaction given by a Dirac delta function. Our proof relies on the minimization of the energy at fixed momentum. As a by-product of our analysis, we provide a simple condition to ensure that the solution to the Cauchy problem is global in time

Efficacy of spa-therapy, mud-pack therapy, balneotherapy and mud-bath therapy in the management of knee osteoarthritis. A systematic review

Background: Osteoarthritis (OA) is the most common musculoskeletal disease in the world. OA is the result of an inflammatory and degenerative process affecting the entire joint. Osteoarthritis, especially involving the knee, has a relevant socio-economic impact in terms of drugs, hospital admissions, work absences and temporary or permanent invalidity. Therapy of knee osteoarthritis is based on pharmacological and non-pharmacological measures. Methods: We conducted a systematic review of the studies published between 2002 and 2017 on spa-therapy, mud-pack therapy, balneotherapy and mud-bath therapy in the treatment of knee osteoarthritis in order to investigate the evidence of the efficacy of such treatment on pain, functional limitation, drug use and quality of life. Overall, 35 studies were examined among which 12 were selected and included in the review if trial comparative. Each report was reviewed to identify the criteria used for study enrolment and for assignment to experimental vs control groups, sample size, type and characteristics of treatment, features of mineral water, control intervention, assessment point, endpoints, outcome measures, tests used for statistical analysis of the results. We have been able to illustrate the main results obtained in the individual studies and to elaborate these results in order to allow as much a unitary presentation as possible, and hence an overall judgment. Results: Because the studies we reviewed differed markedly from one another in terms of the methods used, we were unable to conduct a quantitative analysis (meta-analysis) of pooled data from the 12 studies. For the purposes of the present review, we re-evaluated the results of the different studies using the same statistical method, the Student’s t test, which is used to compare the means of two frequency distributions. Among all the studies, the most relevant indexes used to measure effectiveness of spa therapy were improved including VAS, Lequesne’s and WOMAC Score. Conclusions: The mud-pack therapy, balneotherapy, mud-bath therapy and spa therapy has proved to be effective in the treatment and in the secondary prevention of knee osteoarthritis, by reducing pain, non-steroidal antiinflammatory drug consumption, functional limitation and improving quality of life of affected patients. Is a noninvasive, complication-free, and cost-effective alternative modality for the conservative treatment of knee osteoarthritis

La Citologia del sedimento urinario: osservazioni su campioni di urine di cane ottenuti per minzione spontanea

RIASSUNTO Parole chiave: cane, urine, sedimento, cytospin, Wright. L’analisi completa delle urine è parte integrante dell’inquadramento complessivo delle funzioni metaboliche e d’organo e della visita clinica. Al pari di uno striscio di sangue nell’interpretazione di un emogramma, l’esame microscopico del sedimento è imprescindibile per una corretta lettura dell’analisi dell’urina. In questo studio è stata effettuata la valutazione microscopica di 61 campioni di sedimento di urine ottenute da pazienti canini per minzione spontanea. Ciascun sedimento è stato allestito per centrifugazione standard delle urine e tramite citocentrifugazione. Entrambi i vetrini, tal quale e cytospin, sono stati sottoposti alla colorazione di Wright e l’esame microscopico del sedimento colorato è stato confrontato con quello a fresco non colorato. La stima eritrocitaria e leucocitaria sui campioni di sedimento colorati ha dimostrato di differire in maniera significativa da quella condotta sui campioni di sedimento non colorati e la numerosità delle emazie e dei leucociti appare essere in relazione con la patologia dei soggetti in studio. ABSTRACT Keywords: dog, urine, sediment, cytospin, Wright. Urinalysis is an integral part of the overall framework of metabolic and organ functions and the clinical exam. As a blood smear in the interpretation of a CBC, microscopic examination of the urinary sediment is essential for a correct interpretation of the urine analysis. In this study the microscopic evaluation of 61 sediment samples of urine obtained from canine patients by spontaneous urination was carried out. Each sediment was set up using standard centrifugation of urine and via cytocentrifuging. Both the slides, as it is and cytospin, were subjected to Wright staining and microscopic examination of colored sediment was compared with the non-colored fresh one. The erythrocyte and leukocyte estimation on colored sediment samples has been shown to differ significantly from that carried out on non-colored ones and the number of red blood cells and leukocytes seems to be in relation with the pathology of the patients included in this study

Definable groups, NIP theories, and the Ellis Group Conjecture

DEFINABLE GROUPS, NIP THEORIES, AND THE ELLIS GROUP CONJECTURE A \emph{definable group} $G$ is a group which is definable in a first-order structure. Despite the name, it is not a single group, but a family of groups given by interpreting the defining formulas in \emph{elementary extensions} of the structure defining the group. For instance, algebraic groups are definable in the complex field using first-order formulas. These include matrix groups and abelian varieties such as elliptic curves. Among groups which are definable with first-order formulas in the real field there are $\operatorname{GL}(n, \mathbb R)$, $\operatorname{SO}(n, \mathbb R)$, and other Lie groups. The two families of examples above are, in a sense, orthogonal. The field $\mathbb C$ falls into the class of \emph{stable} structures, which are, in a nutshell, the ones that do not define an order relation on an infinite set. Stable theories have been a central and fruitful topic in the model theory of the past decades (e.g. [1,11]), and there is a huge literature on stable groups (for instance [3,13]). Unfortunately, since stability is destroyed by the presence of a total infinite order, the field structure on $\mathbb R$ lives outside this realm, and more generally \emph{o-minimal} structures, another important class in which it is possible to provide a framework for \emph{tame geometry} (see [14]), are not stable. Model theorists have therefore tried to generalize methods from stability theory to broader contexts. One robust, simultaneous generalization of both stability and o-minimality is found in the class of \emph{dependent}, or \textsc{nip} theories. \textsc{Nip} structures can be roughly described as the ones that do not code a membership relation on an infinite set; this viewpoint is intimately connected to \textsc{vc}-dimension, a fundamental tool of statistical learning theory. This thesis explores a problem, which we are now going to outline, concerning the relation between two groups that can be attached to any group definable in a \textsc{nip} structure. In \textsc{nip} theories, to every definable group is associated a concrete compact Hausdorff topological group called $G/G^{00}$. As an example, it can be proven that if $G$ is a definably compact group definable over $\emptyset$ in a real closed field, for instance $\operatorname{SO}(3, M)$ for $M\succ \mathbb R$ an hyperreal field, then $G/G^{00}$ is exactly $G(\mathbb R)$, and the projection to $G/G^{00}$ behaves like a ``standard part'' map. If $G$ is not compact then this may not be true, as in the case of $\operatorname{SL}(n, M)$ where $G/G^{00}$ is trivial. In general (see [2]), for a group which is definable in an o-minimal structure, $G/G^{00}$ is a real Lie group. As a stable example, if $G$ is the additive group in the structure of the integers with sum (but without product), then $G/G^{00}$ is isomorphic to $\hat{\mathbb Z}=\varprojlim \mathbb Z/n\mathbb Z$. All these isomorphisms preserve the topology, i.e. are isomorphisms of topological groups. This canonical quotient is the first protagonist of the problem studied in the thesis. In order to introduce the second one, some preliminary explanations are needed. An important concept in the study of stable groups is the one of a \emph{generic type}. Trying to find a well-behaved analogue in the unstable context, Newelski noticed that a certain notion, namely that of a \emph{weak generic type}, is well understood when bringing topological dynamics into the picture\footnot{Briefly, in the dynamical context ``generic'' becomes ``syndetic'', and ``weak generic'' corresponds to ``piecewise syndetic''.}. In topological dynamics one is often interested in $G$-flows, actions of a group $G$ on compact Hausdorff spaces by homeomorphisms; soon one turns the attention to the ones that have a dense orbit ($G$-ambits) and to the ones in which all orbits are dense (minimal flows). A very special $G$-flow is the universal $G$-ambit $\beta G$ of ultrafilters on $G$: every $G$-ambit can be seen as a quotient of $\beta G$, and its minimal subflows enjoy a similar universal property. A ``tame'' counterpart of $\beta G$ is the space $S_G(M)$ of types over a model $M$ concentrating on $G$, i.e. the ultrafilters on definable subsets of $G(M)$, and one could develop a theory of \emph{tame topological dynamics} ([7,12]) and hope for $S_G(M)$ to be universal with respect to \emph{definable} $G(M)$-flows. Now, one important tool in the study of a $G$-flow $X$ is its \emph{enveloping semigroup} $E(X)$; it turns out that $\beta G\cong E(\beta G)$ and this equips the former with a semigroup structure. Once some technical obstacles are overcome, this construction can be carried out for $S_G(M)$ too, or at least for a certain bigger type space called $S_G^{\textnormal{ext}}(M)$. Applying the theory of enveloping semigroups to $E(\beta G)\cong\beta G$ produces a certain family of sub-semigroups that are indeed groups, and furthermore all in the same isomorphism class: this is the \emph{ideal group}, or \emph{Ellis group} associated to the flow. Modulo the complications mentioned above, an Ellis group can also be associated to $S_G(M)$. Even if this may depend on $M$, a comparison with $G/G^{00}$ can be made, and indeed the latter is always a quotient of the former, the projection $\pi$ being the restriction of a certain natural map $S_G(M)\to G/G^{00}$. Since in stable groups a similar situation arises replacing the Ellis group with the subspace of generic types of $S_G(M)$, and in that case the relevant map is injective, the next question is: is this $\pi$ an isomorphism? Even in tame context, this need not be the case: it was shown in [8] that the Ellis group of $\operatorname{SL}(2,\mathbb R)$ is the group with two elements, but its $G/G^{00}$ is trivial. A property that is \emph{not} satisfied by $\operatorname{SL}(2, \mathbb R)$ is amenability: there is no finitely additive, left-translation-invariant probability measure defined on $\mathscr P(\operatorname{SL}(2, \mathbb R))$. Another group lacking amenability is $\operatorname{SO}(3, \mathbb R)$; this is essentially the Banach-Tarski paradox. The reasons behind the non-amenability of these two groups are, however, different. If one searches for a left-translation-invariant (finitely additive) measure defined not on the whole power-set, but only on the Boolean algebra of \emph{definable} subsets of $\operatorname{SO}(3, \mathbb R)$, then such a measure \emph{does} exist, and we say that $\operatorname{SO}(3, \mathbb R)$ is \emph{definably amenable}. A similar thing happens with free groups on at least two generators. This is due to the fact that the non-measurable sets arising from the Banach-Tarski paradox are very complicated, and certainly not definable in the first-order structure of $\mathbb R$, and so this kind of obstructions to amenability disappear when we only want a measure on an algebra of ``simple'' sets. On the contrary, $\operatorname{SL}(2,\mathbb R)$ is not even definably amenable, thus being more inherently pathological under this point of view. In [4] Pillay then proposed the \emph{Ellis Group Conjecture}. Several special cases were proven in the same paper and, thereafter, the conjecture was proven true in [5] by Chernikov and Simon, hence we state it as a Theorem. \begin{theorem}[5, Theorem 5.6] If $G$ is a definably amenable \textsc{nip} group, the restriction of the natural map $S_G^{\textnormal{ext}}(M)\to G/G^{00}$ to any ideal group of $G$ is an isomorphism. \end{theorem} Remarkably, the model-theoretic techniques involved in stating, approaching, and proving the conjecture are anything but peculiar to this particular problem, and the main focus of this thesis is on the development and understanding of said techniques. This is reflected in the fact that we will deal with Ellis semigroups only in the first chapter and in the closing section. We start in Chapter 1 by studying enveloping semigroups, first in the classical context ([6]) and then in the definable one, without any kind of tameness assumption ([9, 10]). In Chapter 2 we introduce some techniques, still without assuming anything on the underlying theory beyond being first-order complete. In Chapter 3 we introduce dependent theories, see how the previously introduced tools behave in this context, and explore some constructions that heavily exploit the \textsc{nip} hypothesis. In Chapter 4 we bring in the last ingredient, i.e. definable amenability, see that under our hypotheses it is preserved when passing to Shelah's expansion, characterize it in terms of f-generic types, and conclude by studying the proof of the Ellis Group Conjecture. [1] J. T. Baldwin. Fundamentals of Stability Theory, volume 12. Springer-Verlag, 1988. [2] A. Berarducci, M. Otero, Y. Peterzil, and A. Pillay. A descending chain condition for groups definable in o-minimal structures. Annals of Pure and Applied Logic, 134:303–313, 2005. [3] A. Borovik and A. Nesin. Groups of Finite Morley Rank, volume 26 of Oxford Logic Guides. Oxford University Press, 1994. [4] A. Chernikov, A. Pillay, and P. Simon. External definability and groups in NIP theories. Journal of the London Mathematical Society, 2014. [5] A. Chernikov and P. Simon. Definably amenable NIP groups. http://arxiv.org/abs/1502.04365, submitted. [6] R. Ellis. Lectures on Topological Dynamics. Mathematics Lecture Note Series. W.A. Benjamin, 1969. [7] J. Gismatullin, D. Penazzi, and A. Pillay. On compactifications and the topological dynamics of definable groups. Annals of Pure and Applied Logic, 165:552–562, 2014. [8] J. Gismatullin, D. Penazzi, and A. Pillay. Some model theory of SL(2, R). Fundamenta Mathematicae, 229:117–128, 2015. [9] L. Newelski. Topological dynamics of definable group actions. Journal of Symbolic Logic, 74:50–72, 2009. [10] L. Newelski. Model theoretic aspects of the ellis semigroup. Israel Journal of Mathematics, 190:477–507, 2012. [11] A. Pillay. Geometric Stability Theory, volume 32 of Oxford Logic Guides. Oxford University Press, 1996. [12] A. Pillay. Topological dynamics and definable groups. The Journal of Symbolic Logic, 78:657–666, 2013. [13] B. Poizat. Stable Groups, volume 87 of Mathematical Surveys and Monographs. American Mathematical Society, 2001. translated from the 1987 original. [14] L. van den Dries. Tame Topology and O-minimal Structures, volume 248. Cambridge University Press, 1998