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This article is focused on the development of organizational and pedagogical conditions of college students’ vocational training in terms of social partnership and experimental study of their effectiveness. The leading methods in the study of this problem are the observations, conversations, monitoring, questioning, psychological testing and pedagogical experiment, allowing checking of the effectiveness of the proposed organizational and pedagogical conditions of students’ vocational training in terms of social partnership. The article deals with the role of social partnership of colleges and enterprises; identifies organizational and pedagogical conditions of students’ vocational training in terms of social partnership between the college and the company; justifies forms of cooperation between the educational institutions and its social partners; identifies criteria for production and pedagogical management of students’ vocational training in the conditions of Social Partnership between the college and the company; proves the effectiveness of college students’ vocational training in the conditions of social partnership. The paper submissions are of a definite value in the organization of social partnership in the system of secondary and higher vocational education.

A longitudinal failure criterion for UD composites based on a kinking model was re-constructed guided by the principles of analytic geometry. The proposed method for determining the fiber initial m...

Geometry is a very interesting, applicable and beautiful part of mathematics. However, geometry is often difficult for students to understand and demanding for teachers to teach [1]. Constructing proofs in geometric problems turns out to be particularly difficult, even for high attaining students [2]. Sometimes, students do not even know where to start when trying to solve these [3].

A new determinantal representation for the implicit equation of offsets to conics and quadrics is derived. It is simple, free of extraneous components and provides a very compact expanded form, these representations being very useful when dealing with geometric queries about offsets such as point positioning or solving intersection purposes. It is based on several classical results in ?A Treatise on the Analytic Geometry of Three Dimensions? by G. Salmon for offsets to non-degenerate conics and central quadrics. This research was funded by the Spanish Ministerio de Economía y Competitividad and by the European Regional Development Fund (ERDF), under the project MTM2017-88796-P.

We study a couple of geometrical elements of a monumental synagogue (Dohany Synagogue in Budapest). Then we describe how undergraduate students from the Jewish orthodox population have performed a ...

The theory of $F$-manifolds, and more generally, manifolds endowed with commutative and associative multiplication of their tangent fields, was discovered and formalised in various models of quantum field theory involving algebraic and analytic geometry, at least since 1990's. The focus of this paper consists in the demonstration that various spaces of probability distributions defined and studied at least since 1960's also carry natural structures of $F$-manifolds. This fact remained somewhat hidden in various domains of the vast territory of models of information storing and transmission that are briefly surveyed here.

This paper argues that recent topological expressions of surface in architecture have an intellectual lineage to late-modernism. Several key developments of that era challenged the limitations of a simplified Cartesian understanding of form. This paper commences with a discussion of Cartesian geometry in Modernism, relative to innovations in the building industry that promoted constancy, repetition and standardisation. Despite the same logic being transferred to digital working platforms where the architectural elements are normally designed as geometric prisms and their immediate derivatives, an interest in computation has enabled architecture to bypass orthogonal regularity. Concepts investigated during late-modernism—such as adaptability, disequilibrium, and smooth transitioning—are now central to performance and parametric-based design, examined through the interaction with data inputs, further pointing at critical updates of common software production tools.

We construct a derived enhancement of Hom spaces between rigid analytic spaces. It encodes the hidden deformation-theoretic informations of the underlying classical moduli space. The main tool in our construction is the representability theorem in derived analytic geometry, which has been established in our previous work. The representability theorem provides us sufficient and necessary conditions for an analytic moduli functor to possess the structure of a derived analytic stack. In order to verify the conditions of the representability theorem, we prove several general results in the context of derived non-archimedean analytic geometry: derived Tate acyclicity, projection formula, and proper base change. These results also deserve independent interest themselves. Our main motivation comes from non-archimedean enumerative geometry. In our subsequent works, we will apply the derived mapping stacks to obtain non-archimedean analytic Gromov-Witten invariants.

We prove the representability theorem in derived analytic geometry. The theorem asserts that an analytic moduli functor is a derived analytic stack if and only if it is compatible with Postnikov towers, has a global analytic cotangent complex, and its truncation is an analytic stack. Our result applies to both derived complex analytic geometry and derived non-archimedean analytic geometry (rigid analytic geometry). The representability theorem is of both philosophical and practical importance in derived geometry. The conditions of representability are natural expectations for a moduli functor. So the theorem confirms that the notion of derived analytic space is natural and sufficiently general. On the other hand, the conditions are easy to verify in practice. So the theorem enables us to enhance various classical moduli spaces with derived structures, thus provides plenty of down-to-earth examples of derived analytic spaces. For the purpose of proof, we study analytification, square-zero extensions, analytic modules and cotangent complexes in the context of derived analytic geometry. We will explore applications of the representability theorem in our subsequent works. In particular, we will establish the existence of derived mapping stacks via the representability theorem., This version contains a revision to Section 6

The purpose of this paper is to use historical textbooks to explore the number line, a mathematical object common to our time, contextualizing its use during its initial development period in the seventeenth century until its current use as in the nineteenth century. Relevant analytic geometry textbooks are explored, mainly from France and from Germany of that time, to understand when and how this notion was used. Additionally, pertinent entries from encyclopedias of that time are used to contrast with what was not sufficiently explained in the aforementioned textbooks, until, finally, we can identify this object as it is currently characterized. I analyze how the negative numbers were used to constitute the number line and survey several number line diagrams, which were sometimes similar, but most often quite different from each other. The intention is to understand the historical contexts that configure the emergence and development of the notion of the number line.

The past decades have witnessed several well-known beautiful conclusions on four con-cyclic points. With highly promising research value, we profoundly studied circumscribed ellipses of convex quadrilaterals in this paper. Using tools of parallel projective transformation and analytic geometry, we derived several theorems including the proof of the existence of circumscribed ellipses of convex quadrilaterals, the properties of its minimal coverage area, and locus center, respectively. This simple approach lays a solid foundation for its application to three-dimensional situations, which is namely the circumscribed quadric surface of a solid figure and its wide-range utility in construction engineering. Meanwhile, we have a new insight into innate connection of conic sections as well as a taste of beauty and harmony of geometry.

In Part I [Bal14a], we defined the notion of a modest complete descriptive axiomatization and showed that HP5 and EG are such axiomatizations of Euclid’s polygonal geometry and Euclidean circle geometry1. In this paper we argue: 1) Tarski’s axiom set E is a modest complete descriptive axiomatization of Cartesian geometry (Section 2; 2) the theories EGπ,C,A and E π,C,A are modest complete descriptive axiomatizations of Euclidean circle geometry and Cartesian geometry, respectively when extended by formulas computing the area and circumference of a circle (Section 3); and 3) that Hilbert’s system in the Grundlagen is an immodest axiomatization of any of these geometries. As part of the last claim (Section 4), we analyze the role of the Archimdedean postulate in the Grundlagen, trace the intricate relationship between alternative formulations of ‘Dedekind completeness’, and exhibit many other categorical axiomatizations of related geometries.

A generalized complex manifold is locally gauge-equivalent to the product of a holomorphic Poisson manifold with a real symplectic manifold, but in possibly many different ways. In this paper, we show that the isomorphism class of the holomorphic Poisson structure occurring in this local model is independent of the choice of gauge equivalence, and is hence the unique local invariant of generalized complex manifolds. This completes the local classification of generalized complex structures. We use this result to prove that the complex locus of a generalized complex manifold naturally inherits the structure of a complex analytic space.

This article is on fast-consensus reaching in a class of multi-agent systems (MAS). We present an analytical approach to tune controllers for the agents based on the premise that delayed measurements in the controller can be preferable to standard controllers relying only on current measurements. Controller tuning in this setting is however challenging due to the presence of delays. To tackle this problem, we propose an analytic geometry approach. The key contribution is that the tuning can be implemented for complex eigenvalues of the arising graph Laplacian of the network, complementing the current state of the art, which is limited to real eigenvalues. Results, therefore, extend our knowledge beyond symmetric graphs and enable the study of the MAS under directed graphs. This article is part of the theme issue ‘Nonlinear dynamics of delay systems’.

One of the reasons to use technology in education is to visualize and explore the insight of mathematics with optimum possibilities. Geogebra gives visual meaning to symbolic ideas along with optimum dynamic possibilities. The purpose of this experimental study was to know about the potential effect of dynamic geometry software on diverse students’ mathematical thinking behavior. For which a sample of forty students (grade-12) of F.G Inter College for Boys Mardan Cantt was selected. On the of their previous grade-11 standardized exams two groups along with their nested factors higher and lower achievers students were constructed. To investigate students’ mathematical thinking basis in the subject of analytic geometry, a model behavior of thinking with its six aspects was constructed on the basis of this model a well-design criterion test was developed for data collection. The six dependent variables were combined together to form the total mathematical thinking in this research study. Further, six week experiments of 22 lessons were prepared and two teaching methods traditional vs DGS (Dynamic Geometry Software) aided instructions were tested for two groups with almost equal statistical background and with the same compatibility in the biological age. Two hypotheses were carried out i.e. Treatment does not significantly affect the higher and low achievers of the two groups in mathematical thinking mean scores. To check the significant effect of the treatment on the students’ overall mathematical thinking variable across the groups, the two-way (ANOVA) was used and analyzed through as statistical tool SPSS. The study findings showed that treatment did significantly affect the higher and lower achievers of the two groups in mathematical thinking mean scores.

This paper proposes a novel 6-DOF robotic crusher that combines the performance characteristics of the cone crusher and parallel robot, such as interparticle breakage and high flexibility. Kinematics and dynamics are derived from the no-load and crushing parts in order to clearly describe the whole crushing process. For the no-load case, the kinematic and dynamic equations are established by using analytical geometry and Lagrange equation. Analytical geometry is mainly used to solve the inverse kinematics and then establish the velocity relationship between generalized coordinates and actuators. Lagrange equation which takes into account the weight of the mantle and actuators is used to solve driving forces of actuators. For the crushing case, crushing pressure is related to the compression ratio and particle size distribution, but the selection and breakage functions should be established first. Because the trajectory model of the mantle is difficult to be established by using analytical method, it can be obtained by an eccentric simulation. The results of input velocities and driving forces of actuators are distinctive due to the eccentric angle and selection of the initial position. Finally, the proposed approach is verified by a numerical example and then the energy consumption is calculated.

In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting., 53 pages, 1 figure. v2: Section 7.2 on weakly stable fields added and other minor changes. Final version. To appear in Journal of the Institute of Mathematics of Jussieu

Gromov's compactness theorem for pseudo-holomorphic curves is a foundational result in symplectic geometry. It controls the compactness of the moduli space of pseudo-holomorphic curves with bounded area in a symplectic manifold. In this paper, we prove the analog of Gromov's compactness theorem in non-archimedean analytic geometry. We work in the framework of Berkovich spaces. First, we introduce a notion of K\"ahler structure in non-archimedean analytic geometry using metrizations of virtual line bundles. Second, we introduce formal stacks and non-archimedean analytic stacks. Then we construct the moduli stack of non-archimedean analytic stable maps using formal models, Artin's representability criterion and the geometry of stable curves. Finally, we reduce the non-archimedean problem to the known compactness results in algebraic geometry. The motivation of this paper is to provide the foundations for non-archimedean enumerative geometry., Comment: Improved presentation

We give a formulation for derived analytic geometry built from commutative differential graded algebras equipped with entire functional calculus on their degree 0 part, a theory well-suited to developing shifted Poisson structures and quantisations. In the complex setting, we show that this formulation recovers equivalent derived analytic spaces and stacks to those coming from Lurie's structured topoi. In non-Archimedean settings, there is a similar comparison, but for derived dagger analytic spaces and stacks, based on overconvergent functions., 24pp

We propose a derived version of non-archimedean analytic geometry. Intuitively, a derived non-archimedean analytic space consists of an ordinary non-archimedean analytic space equipped with a sheaf of derived rings. Such a naive definition turns out to be insufficient. In this paper, we resort to the theory of pregeometries and structured topoi introduced by Jacob Lurie. We prove the following three fundamental properties of derived non-archimedean analytic spaces: (1) The category of ordinary non-archimedean analytic spaces embeds fully faithfully into the $\infty$-category of derived non-archimedean analytic spaces. (2) The $\infty$-category of derived non-archimedean analytic spaces admits fiber products. (3) The $\infty$-category of higher non-archimedean analytic Deligne-Mumford stacks embeds fully faithfully into the $\infty$-category of derived non-archimedean analytic spaces. The essential image of this embedding is spanned by $n$-localic discrete derived non-archimedean analytic spaces. We will further develop the theory of derived non-archimedean analytic geometry in our subsequent works. Our motivations mainly come from intersection theory, enumerative geometry and mirror symmetry., Improved exposition, 60 pages

We characterize sandwiched singularities in terms of their link in two different settings. We first prove that such singularities are precisely the normal surface singularities having self-similar non-archimedean links. We describe this self-similarity both in terms of Berkovich analytic geometry and of the combinatorics of weighted dual graphs. We then show that a complex surface singularity is sandwiched if and only if its complex link can be embedded in a Kato surface in such a way that its complement remains connected., 35 pages

In the early nineteenth century debate over geometric methodology, Jean-Victor Poncelet characterized pure geometry as reasoning in which the figure is never lost from view. Whether illustrated, described or constructed, Poncelet presented the figure as the primary form of geometrical evidence, a means of justification based in sensory perception. In Poncelet's pure geometry, the objects of geometry were emphatically representational and tangible. By contrast, though classified as analytic geometry, Julius Plucker's contemporary research treated coordinate equations as visual geometric objects—evidence—by focusing on their form and endeavouring to avoid calculations. Working from Poncelet's division between pure and analytic geometries we focus on five versions by three different geometers, of a single conic section construction written between 1817 and 1826. Despite the similarity of their results, Poncelet, Plucker, and Joseph Diaz Gergonne each addressed the problem from contrasting methodological perspectives. We examine how the figure-based distinction materialized in contemporary geometric practices, and what constituted geometric evidence when the figure was lost from view.

In this paper we study the $$p$$ -adic analytic geometry of the basic unitary group Rapoport–Zink spaces $$\mathcal {M}_K$$ with signature $$(1,n-1)$$ . Using the theory of Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal fur die reine und angewandte Mathematik 645:1–39, 2010), Fargues (Theorie de la reduction pour les groupes p-divisibles, prepublications. http://www.math.jussieu.fr/~fargues/Prepublications.html , 2010), and the Bruhat–Tits stratification of the reduced special fiber $$\mathcal {M}_{red}$$ defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011), we find some relatively compact fundamental domain $$\mathcal {D}_K$$ in $$\mathcal {M}_K$$ for the action of $$G(\mathbb {Q}_p)\times J_b(\mathbb {Q}_p)$$ , the product of the associated $$p$$ -adic reductive groups, and prove that $$\mathcal {M}_K$$ admits a locally finite cell decomposition. By considering the action of regular elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic spaces (Preprint). arXiv:1011.1720 , 2013).

This paper examines the problem of smoothly connecting two arcs of curves with two more arcs, and presents the application of this procedure to the drawing of a class of egg-shaped curves and to the generation of some open and closed polycentric curves. The paper provides a solid mathematical background as well as ‘ruler and compass’ constructions, some of them original techniques. The use of basic, although lengthy, Euclidean proof tools suggests these properties and techniques were within reach—if not known—by architects and scientists long before the invention of analytic geometry. The concepts described in this paper are further developed in its second part (Mazzotti, What Borromini Might Have Known About Ovals. Ruler and Compass Constructions, 2014).

We study the rigid analytic geometry of cyclic coverings of the projective line. We determine the defining equation of a cyclic covering of degree $p$ of the projective line by a Mumford curve over a complete discrete valuation field of positive characteristic $p$. Previously, Bradley studied that of any degree over a non-archimedean local field of characteristic zero., 20 pages, 2 figures, to appear in Kodai Mathematical Journal

This paper develops a theory of analytic geometry over the field with one element. The approach used is the analytic counter-part of the Toen-Vaquie theory of schemes over F 1 , i.e. the base category relative to which we work out our theory is the category of sets endowed with norms (or families of norms). Base change functors to analytic spaces over Banach rings are studied and the basic spaces of analytic geometry (e.g. polydisks) are recovered as a base change of analytic spaces over F 1 . We conclude by discussing some applications of our theory to the theory of the Fargues-Fontaine curve and to the ring of Witt vectors.

We are concerned with rigid analytic geometry in the general setting of Henselian fields K with separated analytic structure, whose theory was developed by Cluckers&ndash, Lipshitz&ndash, Robinson. It unifies earlier work and approaches of numerous mathematicians. Separated analytic structures admit reasonable relative quantifier elimination in a suitable analytic language. However, the rings of global analytic functions with two kinds of variables seem not to have good algebraic properties such as Noetherianity or excellence. Therefore, the usual global resolution of singularities from rigid analytic geometry is no longer at our disposal. Our main purpose is to give a definable version of the canonical desingularization algorithm (the hypersurface case) due to Bierstone&ndash, Milman so that both of these powerful tools are available in the realm of non-Archimedean analytic geometry at the same time. It will be carried out within a category of definable, strong analytic manifolds and maps, which is more flexible than that of affinoid varieties and maps. Strong analytic objects are those definable ones that remain analytic over all fields elementarily equivalent to K. This condition may be regarded as a kind of symmetry imposed on ordinary analytic objects. The strong analytic category makes it possible to apply a model-theoretic compactness argument in the absence of the ordinary topological compactness. On the other hand, our closedness theorem enables application of resolution of singularities to topological problems involving the topology induced by valuation. Eventually, these three results will be applied to such issues as the existence of definable retractions or extending continuous definable functions. The established results remain valid for strictly convergent analytic structures, whose classical examples are complete, rank one valued fields with the Tate algebras of strictly convergent power series. The earlier techniques and approaches to the purely topological versions of those issues cannot be carried over to the definable settings because, among others, non-Archimedean geometry over non-locally compact fields suffers from lack of definable Skolem functions.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).

We develop the foundations of higher geometric stacks in complex analytic geometry and in non-archimedean analytic geometry. We study coherent sheaves and prove the analog of Grauert's theorem for derived direct images under proper morphisms. We define analytification functors and prove the analog of Serre's GAGA theorems for higher stacks. We use the language of infinity category to simplify the theory. In particular, it enables us to circumvent the functoriality problem of the lisse-\'etale sites for sheaves on stacks. Our constructions and theorems cover the classical 1-stacks as a special case., Comment: Major revision. Improved exposition

We give new examples of algebraic elliptic surfaces and non-algebraic rigid analytic elliptic surfaces by means of logarithmic transformations. In the complex analytic case, it is known that all multiple fibers of elliptic surfaces are obtained by logarithmic transformations. Using rigid analytic geometry, we construct similar transformations of elliptic surfaces over complete non-Archimedean valuation base fields. These operations yield rigid analytic elliptic fibrations with multiple fibers. When the resulting surface admits an ample line bundle, we may algebraize the surface. In the positive characteristic case, we obtain new types of algebraic elliptic surfaces. We also obtain a non-algebraic rigid analytic surface the combination of whose invariants appears neither in the algebraic case nor in the complex analytic case.

In this paper we use the connections between tropical algebraic geometry and rigid analytic geometry in order to prove two main results. We use tropical methods to prove a theorem about the Newton polygon for convergent power series in several variables: if f_1,...,f_n are n convergent power series in n variables with coefficients in a non-Archimedean field K, we give a formula for the valuations and multiplicities of the common zeros of f_1,...,f_n. We use rigid-analytic methods to show that stable complete intersections of tropical hypersurfaces compute algebraic multiplicities even when the intersection is not tropically proper. These results are naturally formulated and proved using the theory of tropicalizations of rigid-analytic spaces, as introduced by Einsiedler-Kapranov-Lind [EKL06] and Gubler [Gub07b]. We have written this paper to be as readable as possible both to tropical and arithmetic geometers., Comment: 46 pages, 11 figures

In the paper ‘Bruhat–Tits theory from Berkovich's point of view. I. Realizations and compactifications of buildings’, we investigated various realizations of the Bruhat–Tits building $\mathcal{B}(\mathrm{G},k)$ of a connected and reductive linear algebraic group G over a non-Archimedean field k in the framework of Berkovich's non-Archimedean analytic geometry. We studied in detail the compactifications of the building which naturally arise from this point of view. In the present paper, we give a representation theoretic flavour to these compactifications, following Satake's original constructions for Riemannian symmetric spaces.We first prove that Berkovich compactifications of a building coincide with the compactifications, previously introduced by the third named author and obtained by a gluing procedure. Then we show how to recover them from an absolutely irreducible linear representation of G by embedding $\mathcal{B}(\mathrm{G},k)$ in the building of the general linear group of the representation space, compactified in a suitable way. Existence of such an embedding is a special case of Landvogt's general results on functoriality of buildings, but we also give another natural construction of an equivariant embedding, which relies decisively on Berkovich geometry.

In order to explore measuring accuracy of coded-structured-light 3D measurement system onto high reflectance surface of turbine blade, measuring error analysis, system parameter design and uncertainty evaluation were investigated. First, device parameters and azimuth parameters of analytic geometry model and projective geometry model of coded structured light system were analyzed and unified. Second, based on analytic geometry model, quantization relationship between measuring error and device parameters, azimuth parameters were analyzed. Third, system parameters were designed by taking into account system range, shelter, depth of field, luminous power and system capacity. Finally, type B uncertainty evaluation based on error sources and transfer coefficients was done. Experimental results indicate that error of measured high reflectance unicolor standard plane is 0.60.8mm; reconstructed turbine blade shape has same visual effect as the real one; and type A uncertainty evaluation based on measured data on standard plane has similar result as type B. It can basically meet the requirements of turbine blade reconstruction in industrial site.

This article follows recent work of Miyajima on the complex-analytic approach to deformations of the regular part (i.e. the punctured smooth neighbourhood) of isolated singularities. Attention has previously focused on stably-embeddable infinitesimal deformations as those which correspond to standard algebraic deformations of the germ of a variety, and which also lead to convergent series solutions of the Kodaira-Spencer integrability equation. The emphasis of the present paper, however, is on the subspaces Z(0) of first cohomology classes containing infinitesimal deformations with vanishing Kodaira- Spencer bracket, and W(0), consisting more broadly of deformations for which the bracket represents the trivial second cohomology class. Deformations representing classes in Z(0) are automatically integrable, regardless of their analytic behaviour near the singular point. Classes in W(0) are those for which only the first formal obstruction to integrability is overcome. After some preliminary results on cohomology, the main theorem of this paper gives a partial description of the analytic geometry of Z(0) and W(0) for affine cones of arbitrary

The reader will find in this article a collection of problems, questions and exercises related to the singularities of algebraic and analytic varieties. Many of them are inspired by the work and mathematical conception of Hironaka: they are concrete, involve basic ideas and techniques from geometry and algebra, and they can immediately be attacked from scratch. Some problems rely on or use results proven by Hironaka. Simple and double asterisques indicate the more difficult problems; they may potentially serve as a topic of future research.

We present a unifying theory of fields with certain classes of analytic functions, called fields with analytic structure. Both real closed fields and Henselian valued fields are considered. For real closed fields with analytic structure, o-minimality is shown. For Henselian valued fields, both the model theory and the analytic theory are developed. We give a list of examples that comprises, to our knowledge, all principal, previously studied, analytic structures on Henselian valued fields, as well as new ones. The b-minimality is shown, as well as other properties useful for motivic integration on valued fields. The paper is reminiscent of [Denef, van den Dries, "p-adic and real subanalytic sets" Ann. of Math. (2) 128 (1988) 79--138], of [Cohen, Paul J. "Decision procedures for real and p-adic fields" Comm. Pure Appl. Math. 22 (1969) 131--151, and of [Fresnel, van der Put, "Rigid analytic geometry and its applications" Progress in Mathematics, 218 Birkhauser (2004)], and unifies work by van den Dries, Haskell, Macintyre, Macpherson, Marker, Robinson, and the authors., 73 pages

This paper investigates the geometry of the expansion RQ of the real field R by restricted quasianalytic functions. The main purpose is to establish quantifier elimination, description of definable functions by terms, the valuation property and preparation theorem (in the sense of Parusinski–Lion–Rolin). To this end, we study non-standard models R of the universal diagram T of RQ in the language L augmented by the names of rational powers. Our approach makes no appeal to the Weierstrass preparation theorem, upon which the majority of fundamental results in analytic geometry rely, but which is unavailable in the general quasianalytic geometry. The basic tools applied here are transformation to normal crossings and decomposition into special cubes. The latter method, developed in our article [32], combines modifications by blowing up with a suitable partitioning. Via an analysis of L-terms and infinitesimals, we prove the valuation property for functions given by L-terms, and next the exchange property for substructures of a given model R. Our proofs are based on the concepts of analytically independent as well as active and non-active infinitesimals, introduced in this article. Further, quantifier elimination for T is established through model-theoretic compactness. The universal theory T is thus complete and o-minimal, andRQ is its prime model. Under the circumstances, every definable function is given piecewise by L-terms, and therefore the previous results concerning L-terms generalize immediately to definable functions. In this fashion, we obtain the valuation property and preparation theorem for quasi-subanalytic functions. Finally, a quasi-subanalytic version of Puiseux’s theorem with parameter is demonstrated. AMS Classification: Primary 32S45, 14P15, 32B20; Secondary 03C10, 26E10, 03C64.

We prove that Euclidean geometry is interpretable in terms of orthogonality of its k-subspaces, and thus it can be formalized as a theory of such an orthogonality.

To do this, we show that it is possible to introduce simple ^-vectors?those of the form ax a A ak where ax,... , ak are vectors in some Rn?as geometric objects. In calculus or an introductory physics class, one sometimes hears a vector described as an equivalence class of directed line segments. It turns out that one can, in a similar way, define a simple /c-vector as an equivalence class of oriented parallelepipeds. We shall show that this geometric definition leads to the standard properties of exterior algebra. To illustrate the use and convenience of this machinery, we then generalize the triangle law of vector addition and the law of cosines to ft-simplexes and rederive the ft-dimensional versions of the parallelogram law and the Pythagorean theorem.

We prove that a real analytic subset of a torus group that is contained in its image under an expanding endomorphism is a finite union of translates of closed subgroups. This confirms the hyperplane zeros conjecture of Lagarias and Wang for real analytic varieties. Our proof uses real analytic geometry, topological dynamics and Fourier analysis., Comment: 25 pages

V.G. Berkovich’s non-Archimedean analytic geometry provides a natural framework to understand the combinatorial aspects in the theory of toric varieties and toroidal embeddings. This point of view leads to a conceptual and elementary proof of the following results: if X is an algebraic scheme over a perfect field and if D is the exceptional normal crossing divisor of a resolution of the singularities of X, the homotopy type of the incidence complex of D is an invariant of X. This is a generalization of a theorem due to D. Stepanov.

V. L. Rvachev called R-functions ‘logically charged functions’ because they encode complete logical information within the standard setting of real analysis. He invented them in the 1960s as a means for unifying logic, geometry, and analysis within a common computational framework – in an effort to develop a new computationally effective language for modelling and solving boundary value problems. Over the last forty years, R-functions have been accepted as a valuable tool in computer graphics, geometric modelling, computational physics, and in many areas of engineering design, analysis, and optimization. Yet, many elements of the theory of R-functions continue to be rediscovered in different application areas and special situations. The purpose of this survey is to expose the key ideas and concepts behind the theory of R-functions, explain the utility of R-functions in a broad range of applications, and to discuss selected algorithmic issues arising in connection with their use.

The role geometry plays in real life makes it a core component of mathematics that students must understand and master. Conceptual knowledge of geometric concepts goes beyond the development of skills required to manipulate geometric shapes. This study is focused on errors students made when solving coordinate geometry problems in the final Grade 12 examination in South Africa. An analysis of 1000 scripts from the 2008 Mathematics examination was conducted. This entailed a detailed analysis of one Grade 12 geometry examination question. Van Hiele levels of geometrical thought were used as a lens to understand students’ knowledge of geometry. Studies show that Van Hiele levels are a good descriptor of current and future performance in geometry. This study revealed that whilst students in Grade 12 are expected to operate at level 3 and level 4, the majority were operating at level 2 of Van Hiele’s hierarchy. The majority of students did not understand most of the basic concepts in Euclidian transformation. Most of the errors were conceptual and suggested that students did not understand the questions and did not know what to do as a result. It is also noted that when students lack conceptual knowledge the consequences are so severe that they hardly respond to the questions in the examination.

It takes a good deal of historical imagination to picture the kinds of debates that accompanied the slow process, which ultimately led to the acceptance of non-Euclidean geometries little more than a century ago. The difficulty stems mainly from our tendency to think of geometry as a branch of pure mathematics rather than as a science with deep empirical roots, the oldest natural science so to speak. For many of us, there is a natural tendency to think of geometry in idealized, Platonic terms. So to gain a sense of how late nineteenth-century authorities debated over the true geometry of physical space, it may help to remember the etymological roots of geometry: “geo” plus “metria” literally meant to measure the earth, of course. In fact, Herodotus reported that this was originally an Egyptian science; each spring the Egyptians had to re-measure the land after the Nile River flooded its banks altering the property lines. Among those engaged in this land surveying were the legendary Egyptian rope-stretchers, the “harpedonaptai” who were occasionally depicted in artwork relating to Egyptian ceremonials.

Compact Levi flat real analytic hypersurfaces in complex tori of dimension two are completely classified. The method is a combination of Siu’s ∂-regularity theory on weakly pseudoconvex domains, an updated variant of the L estimate for the ∂operator originated from Berndtsson-Charpentier, and basic arguments in complex analytic geometry.

0. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 0.1 Where we came from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 0.2 Where we were heading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 0.3 What we hoped to find there . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 0.4 Why we were not disappointed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 0.5 Planning for the journey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 0.6 Division of labour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 0.7 Surveyor’s gear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171 0.8 Detour to visit a relative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 0.9 Dulcis in fundo . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 1. Algebraic preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 1.1 Power-multiplicative seminorms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 1.2 Normed modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 1.3 Henselian algebras and complete algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 2. Study of the discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 2.1 Discriminant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190 2.2 Finite ramified coverings of annuli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194 2.3 Convexity and piecewise linearity of the discriminant function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 2.4 The p-adic Riemann existence theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 3. Study of the conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.1 Algebraization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 3.2 Vanishing cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223 3.3 Conductors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 4. Local systems on the punctured disc . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.1 Break decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.2 Local systems with bounded ramification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257

We give an example of an affinoid curve without analytic continuation. We use this to produce an example of an affinoid morphism that cannot be flattened by a finite sequence of local blow-ups. Thus the global rigid analogue of Hironaka's complex analytic flattening theorem given by T. Gardener and H. Schoutens, in Theorem 2.3 of ?Flattening and subanalytic sets in rigid analytic geometry?, Proc. London Math. Soc (3) 83 (2001) 681?707, is not true. Since this is a key step in the proof of the affinoid elimination theorem (loc. cit. Theorem 3.12), that proof contains a serious gap. We also give an example of an affinoid subset of the plane that is not the image under a proper rigid analytic map of a set that is globally semianalytic in the domain of that map. This clarifies the relationship among several natural categories of rigid subanalytic sets.

The Gerritzen-Grauert theorem ([GG], [BGR, 7.3.5/1]) is one of the most important foundational results of rigid analytic geometry. It describes so called locally closed immersions between affinoid varieties, and this description implies the fact that any affinoid subdomain of an affinoid variety is a finite union of rational domains. In its turn, the latter fact allowed one to extend Tate’s theorem (see [Tate], [BGR, 8.2.1/1]) on acyclicity of the Cech complex associated to a finite rational covering of an affinoid variety to finite covering by arbitrary affinoid domains. The same fact also plays an important role in foundations of non-Archimedean analytic geometry developed by V. Berkovich in [Ber1] and [Ber2]. Recall that building blocks of the latter are affinoid spaces associated to a class of affinoid algebras broader than that considered in rigid analytic geometry (the latter were called in [Ber1] strictly affinoid) and, besides, the valuation on the ground field is not assumed to be nontrivial. In the recent papers by A. Ducros [Duc, 2.4] and the author [Tem, 3.5], the above fact on the structure of affinoid domains was extended to arbitrary affinoid spaces, but its proof was based on the case of strictly affinoid ones (i.e., affinoid varieties). The original proof of the Gerritzen-Grauert theorem is not easy, and since then the only different proof was found by M. Raynaud in the framework of his approach to rigid analytic geometry (see [Ray], [BL]). Although that proof is more conceptual, it is based on a complicated algebraic technics. The purpose of this paper is to give a new proof of the Gerritzen-Grauert theorem which uses basic properties of affinoid algebras in a standard way. The only novelty is in using the whole spectrum M(A) of an affinoid algebra A, introduced in [Ber1], instead of the maximal spectrum Max(A), considered in rigid analytic geometry. The use of the whole spectrum allows one to apply additional but standard compactness arguments. In §§1-2, we work in the setting of rigid analytic geometry, i.e., the valuation on the ground field is assumed to be nontrivial and only the class of strictly affinoid algebras is considered. In §1, we recall basic definitions of an affinoid algebra, an affinoid domain, all notions necessary for the formulation of the Gerritzen-Grauert theorem, and formulate it (Theorem 1.1). The only new fact is Proposition 1.2 which establishes the simple fact that a morphism of affinoid varieties is a locally closed immersion if and only if it is a

We generalize the construction of the eigencurve by Coleman-Mazur to the setting of totally real fields, and show that a finite slope Hilbert modular eigenform can be deformed into a one parameter family of finite slope eigenforms. The key point is to show the overconvergence of the canonical subgroup and the complete continuity of the U p operator. We deduce this form some general considerations in rigid analytic geometry.

Resume Nous demontrons une conjecture de Robert J. Zimmer dans le cas particulier des actions de reseaux par diffeomorphismes holomorphes sur les varietes kahleriennes compactes.