The aim of this paper is to develop the theory of groups definable in the p-adic field ${{\mathbb {Q}}_p}$, with “definable f-generics” in the sense of an ambient saturated elementary extension of ${{\mathbb {Q}}_p}$. We call such groups definable f-generic groups.So, by a “definable f-generic” or $dfg$ group we mean a definable group in a saturated model with a global f-generic type which is definable over a small model. In the present context the group is definable over ${{\mathbb {Q}}_p}$, and (...) the small model will be ${{\mathbb {Q}}_p}$ itself. The notion of a $\mathrm {dfg}$ group is dual, or rather opposite to that of an $\operatorname {\mathrm {fsg}}$ group (group with “finitely satisfiable generics”) and is a useful tool to describe the analogue of torsion-free o-minimal groups in the p-adic context.In the current paper our group will be definable over ${{\mathbb {Q}}_p}$ in an ambient saturated elementary extension $\mathbb {K}$ of ${{\mathbb {Q}}_p}$, so as to make sense of the notions of f-generic type, etc. In this paper we will show that every definable f-generic group definable in ${{\mathbb {Q}}_p}$ is virtually isomorphic to a finite index subgroup of a trigonalizable algebraic group over ${{\mathbb {Q}}_p}$. This is analogous to the o-minimal context, where every connected torsion-free group definable in $\mathbb {R}$ is isomorphic to a trigonalizable algebraic group [5, Lemma 3.4]. We will also show that every open definable f-generic subgroup of a definable f-generic group has finite index, and every f-generic type of a definable f-generic group is almost periodic, which gives a positive answer to the problem raised in [28] of whether f-generic types coincide with almost periodic types in the p-adic case. (shrink)

Given a lovely pair P ≺ M of models of a simple theory T, we study the structure whose universe is P and whose relations are the traces on P of definable (in ℒ with parameters from M) sets in M. We give a necessary and sufficient condition on T (which we call weak lowness) for this structure to have quantifier-elimination. We give an example of a non-weakly-low simple theory.

Let T be a complete O-minimal theory in a language L. We first give an elementary proof of the result (due to Marker and Steinhorn) that all types over Dedekind complete models of T are definable. Let L * be L together with a unary predicate P. Let T * be the L * -theory of all pairs (N, M), where M is a Dedekind complete model of T and N is an |M| + -saturated elementary extension of N (and (...) M is the interpretation of P). Using the definability of types result, we show that T * is complete and we give a simple set of axioms for T * . We also show that for every L * -formula φ(x) there is an L-formula ψ(x) such that $T^\ast \models (\forall \mathbf{x})(P(\mathbf{x}) \rightarrow (\phi(\mathbf{x}) \mapsto \psi (\mathbf{x}))$ . This yields the following result: Let M be a Dedekind complete model of T. Let φ(x, y) be an L-formula where l(y) = k. Let $\mathbf{X} = \{X \subset M^k$ : for some a in an elementary extension N of M, X = φ (a,y N ∩ M k }. Then there is a formula ψ(y, z) of L such that X = {ψ (y, b) M : b in M}. (shrink)

We prove (Proposition 2.1) that if $\mu$ is a generically stable measure in an NIP (no independence property) theory, and $\mu(\phi(x,b))=0$ for all $b$ , then for some $n$ , $\mu^{(n)}(\exists y(\phi(x_{1},y)\wedge \cdots \wedge\phi(x_{n},y)))=0$ . As a consequence we show (Proposition 3.2) that if $G$ is a definable group with fsg (finitely satisfiable generics) in an NIP theory, and $X$ is a definable subset of $G$ , then $X$ is generic if and only if every translate of $X$ does not (...) fork over $\emptyset$ , precisely as in stable groups, answering positively an earlier problem posed by the first two authors. (shrink)

In this paper, we consider reducts of p-adically closed fields. We introduce a notion of shadows: sets Mf={∈K2∣|y|=|f|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${M_f = \{ \in K^2 \mid |y| = |f|\}}$$\end{document}, where f is a semi-algebraic function. Adding symbols for such sets to a reduct of the ring language, we obtain expansions of the semi-affine language where multiplication is nowhere definable, thus giving a negative answer to a question posed by Marker, Peterzil and Pillay. The (...) second main result of this paper is the fact that in p-adic fields, full multiplication becomes definable if we add a rational function to the semi-affine language. (shrink)

A physical basis for the minimal time-energy uncertainty relation is formulated from basic high-energy hadronic properties such as the resonance mass spectrum, the form factor behavior, and the peculiarities of Feynman's parton picture. It is shown that the covariant oscillator formalism combines covariantly this time-energy uncertainty relation with Heisenberg's space-momentum uncertainty relation. A pictorial method is developed to describe the spacetime distribution of the localized probability density.

The Montessori Method is accepted as an alternative education model today. Although it was spread out in USA at the beginning of the 20th century, it is currently used and accepted all over the world. Although its application in pre-education is common, it has also been adopted and applied at different levels. The Montessori method differs from traditional education not only in terms of approach to students, teachers, discipline, and school environment, but also in the way it uses certain materials (...) in its education. According to Maria Montessori, kids should be at the center of education, and the duty of teacher is to observe and guide children. The Montessori method, which is mostly applied in general education, is also used in the education of religion, values, and morals. In this study, besides Maria Montessori's thoughts on general education, her thoughts on religious education are also discussed. This research focuses on Montessori schools in Indiana, United States -. In this study, the qualitative research method and the phenomenological approach are adopted. Observation, interview, and document analysis are used as data collection tools. The subject of the study is the discussion of the Montessori method in terms of research results in religion, values and moral education and Montessori’s point of view on the subject. All participants, who work as instructors in Montessori schools, determined by maximum diversity and snowball sampling, Observation in Montessori schools and the program of one of them is used as an aid in the interpretation of the research findings. (shrink)

All these activities of living beings, and the functioning of their organs, and the functioning of instruments demand a teleological explanation. Why do human beings toil? Why do living beings try to maintain the activities of the organism? What are the functions of specific organs or instruments? Intelligible answers can be given in teleological terms. Of course, one could ask many questions about these situations that would not require the teleological explanation; for example, questions concerning the mechanical structure of living (...) beings or of instruments. Yet this fact would not eliminate the legitimacy nor the utility of the teleological category. (shrink)

Originally published 1961. Translation of "Meditaciones del Quijote :Meditacion preliminar. Meditacion primera." (1914). Contents: Reader...; Preliminary Meditation; First Meditation (Brief Treatise on the Novel).

The ArgumentYehuda Halevi'sKuzariwas written in response to the challenge posed to Judaism by a highly spiritual, nondenominational philosophy. Science, especially that embodied in the Hellenistic heritage, was a major component of philosophy; thus, if for no other reason than to make Judaism a serious competitor, Halevi had to show that the Jewish tradition as well possessed a body of scientific knowledge. The superiority of the Jewish teachings was demonstrated chiefly by appeal to the criteria of tradition, consensus, and authority, which, (...) in Halevi's judgement, were in practice the criteria most influential in deciding scientific opinion. Despite the rather unique setting for the book, and the wide range of stances Halevi develops, theKuzariwas rather quickly and smoothly absorbed into the mainstream of Jewish religious thought. (shrink)

In this study, complex dynamics of Briggs–Rauscher reaction system is investigated analytically and numerically. First, the Briggs–Rauscher reaction system is reduced into a new nonlinear parametric oscillator. The Melnikov method is used to derive the condition of the appearance of horseshoe chaos in the cases ω = Ω and ω ≠ Ω. The performed numerical simulations confirm the obtained analytical predictions. Second, the prediction of coexisting attractors is investigated by solving numerically the new nonlinear parametric ordinary differential equation via the (...) fourth-order Runge–Kutta algorithm. As results, it is found that the new nonlinear chemical system displays various coexisting behaviors of symmetric and asymmetric attractors. In addition, the system presents a rich variety of bifurcations phenomena such as symmetry breaking, symmetry restoring, period doubling, reverse period doubling, period-m bubbles, reverse period-m bubbles, intermittency, and antimonotonicity. On the contrary, emerging chaotic band attractors and period-1, period-3, period-9, and period-m bubbles routes to chaos occur in this system. (shrink)