FEM-Simulation of “Die-Less Hydroforming”

The following contribution focuses on the FEM-simulation of “Die-Less-Hydroforming” using LS-DYNA. That specific forming technology is used to create structures by inflating initial seal-welded flat 2D blanks as well as 3D hollow bodies without using any die, mould or punch in contrast to conventional hydroforming (e.g. tube hydroforming or hydromechanical deep drawing). (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Macros and modules in geometry

The paper highlights the importance of “macros” or modules for teaching and learning Geometry using Dynamical Geometry Software (DGS). The role of modules is analyzed in terms of “writing” and “reading” Geometry. At first, modules are taken as tools for geometrical construction tasks and as tools to describe and analyze these constructions. For proofs, decomposing a given geometrical statement may be supported by using prototypical pictures representing theorems of geometry (“modules”). Reading theorems into geometry and constructing proofs is still a major achievement of the student—which may be reached by using macros and modules as a major heuristic strategy     

Internal Diagrams and Archetypal Reasoning in Category Theory

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like “Set is the archetypal Cartesian Closed Category”, which means that proofs about CCCs can be done in the “archetypal language” and then lifted to proofs in the general setting. The method works even when our archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry.     

Understanding kindergarten teachers’ perspectives of teaching basic geometric shapes: a phenomenographic research

Geometry is one of the disciplines children involve within early years of their lives. However, there is not much information about geometry education in Turkish kindergarten classes. The current study aims to examine teachers’ perspectives on teaching geometry in kindergarten classes. The researchers inquired about teachers’ in-class experiences in geometry and asked a series of questions such as “what are the benchmarks in your kindergarten class?”; “what kind of tools and materials you use to teach geometry in your class?”; “what shape do you teach first in your kindergarten class?”; “what do you expect to hear when you asked your students ‘what is square’?”; “how do you teach rectangular?”. The study utilized one of the qualitative research methods, namely phenomenography, to collect the data and analyze the data. The study involved with eight kindergarten teachers who work in different schools in central Kutahya, Turkey. The researchers collected data by conducting face-to-face half-structured interviews. The findings of this phenomenographic research showed that kindergarten teachers have some difficulties in teaching geometry and have lack of knowledge and skills in teaching geometry in kindergarten classes.     

On the existence of certain error formulas for a special class of ideal projectors

In this paper, we focus on a special class of ideal projectors. With the aid of algebraic geometry, we prove that for this special class of ideal projectors, there exist “good” error formulas as defined by C. de Boor. Furthermore, we completely analyze the properties of the interpolation conditions matched by this special class of ideal projectors, and show that the ranges of this special class of ideal projectors are the minimal degree interpolation spaces with regard to their associated interpolation conditions.     

The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students’ empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students’ reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.     

Modeling of local particle accumulation in disperse flows with kinematic singularities

The aim of the study is to model the formation of local particle accumulation zones near several typical kinematic singularities. The flows considered are: (i) a steady two–dimensional flow with localized vorticity of the Kelvin cat's eye type (vortex in a mixing layer), (ii) a steady axisymmetric flow formed by a vortex filament normal to a plane in viscous fluid (simple model of tornado), (iii) a neighbourhood of a zero acceleration point in two–dimensional unsteady (harmonic) flow. From parametric numerical calculations, we investigated the inertial mechanisms of forming local particle accumulation zones and found the threshold values of governing parameters separating qualitatively different particle velocity and density patterns. In particular, it is shown that the zero–acceleration point can either “attract” or “scatter” the particles. Zones of concentrated vorticity are typically devoid of particles. In the tornado–like flow, an axisymmetric “cup-shaped” particle accumulation region is formed. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Shifting the foundations: Descartes's transformation of ancient geometry

The aim of this paper is to analyse how the bases of Descartes's geometry differed from those of ancient geometry. Particular attention is paid to modes of specifying curves of which two types are distinguished — “Specification by genesis” and “Specification by property”. For both Descartes and most of Greek geometry the former was fundamental, but Descartes diverged from ancient pure geometry by according an essential place to the imagination of mechanical instruments. As regards specification by property, Descartes's interpretation of the multiplication of (segments of) straight lines as giving rise to a straight line (segment), together with newer methods of articifical symbolism, led to more concise and suggestive modes of representation. Descartes's account of ancient procedures is historically very misleading, but it allowed him to introduce his own ideas more naturally.     

Connecting Mathematics and Science in Undergraduate Teacher Education Programs: Faculty Voices from the Maryland Collaborative for Teacher Preparation

The study reported in this paper investigated perceptions concerning connections between mathematics and science held by university/college instructors who participated in the Maryland Collaborative for Teacher Preparation (MCTP), an NSF-funded program aimed at developing special middle-level mathematics and science teachers. Specifically, we asked (a) “What are the perceptions of MCTP instructors about the ‘other’ discipline?” (b) “What are the perceptions of MCTP instructors about the connections between mathematics and science?” and (c) “What are some barriers perceived by MCTP instructors in implementing mathematics and science courses that emphasize connections?” The findings suggest that the benefits of emphasizing mathematics and science connections perceived by MCTP instructors were similar to the benefits reported by school teachers. The barriers reported were also similar. The participation in the project appeared to have encouraged MCTP instructors to grapple with some fundamental questions, like “What should be the nature of mathematics and science connections?” and “What is the nature of mathematics/science in relationship to the other discipline?”     

The mathematics of war

During the Desert Storm, the Gulf war, it was possible to read in the newspapers words such as: “Inmathematical terms, was is becoming more and more electronically controlled and, as a result, it is moving away from the battlefield. Then, when war comes down to earth, it becomes bloody, it loses its mathematical asceticism” Reading the newspapers in those days, one had the impression that modern warfare is based on mathematics, as if it were not men but computers that decided where to carry out “surgical operations”. By contrast, the volume published a few years before the Gulf war conceived as a didactic unit to be used in schools with a guide for the teacher with the titleLa matematica della guerra (The Mathematics of War) published by Gruppo Abele in Turin begins with the words “Mathematics, like any other discipline, lends itself to building several paths towards education for peace”. The volume, written by a group of teachers belonging to an anti-violence organisation forming part of the “education for peace” project, highlights the power or ambiguitiy of mathematical models used to simulate war or conflict situations and demonstrates that in some cases the use of mathematics leads to a better understanding of the situation, but in other cases, the mathematical model itself can lead to conclusions which are either wrong or morally unacceptable.     

To the memory of A. A. Logunov: General relativity theory and the relativistic theory of gravity

We briefly review the contributions of A. A. Logunov to understanding the problems of general relativity and gravity with special attention to the issue of the possibility of a catastrophic stellar collapse forming a “black hole.”     

On a new class of hyperbolic functions

This article presents the results of some new research on a new class of hyperbolic functions that unite the characteristics of the classical hyperbolic functions and the recurring Fibonacci and Lucas series. The hyperbolic Fibonacci and Lucas functions, which are the being extension of Binet's formulas for the Fibonacci and Lucas numbers in continuous domain, transform the Fibonacci numbers theory into “continuous” theory because every identity for the hyperbolic Fibonacci and Lucas functions has its discrete analogy in the framework of the Fibonacci and Lucas numbers. Taking into consideration a great role played by the hyperbolic functions in geometry and physics, (“Lobatchevski's hyperbolic geometry”, “Four-dimensional Minkowski's world”, etc.), it is possible to expect that the new theory of the hyperbolic functions will bring to new results and interpretations on mathematics, biology, physics, and cosmology. In particular, the result is vital for understanding the relation between transfinitness i.e. fractal geometry and the hyperbolic symmetrical character of the disintegration of the neural vacuum, as pointed out by El Naschie [Chaos Solitons & Fractals 17 (2003) 631].     

Semialgebraic complexity of functions

In this paper we study the rate of the best approximation of a given function by semialgebraic functions of a prescribed “combinatorial complexity”. We call this rate a “Semialgebraic Complexity” of the approximated function. By the classical Approximation Theory, the rate of a polynomial approximation is determined by the regularity of the approximated function (the number of its continuous derivatives, the domain of analyticity, etc.). In contrast, semialgebraic complexity (being always bounded from above in terms of regularity) may be small for functions not regular in the usual sense. We give various natural examples of functions of low semialgebraic complexity, including maxima of smooth families, compositions, series of a special form, etc. We show that certain important characteristics of the functions, in particular, the geometry of their critical values (Morse–Sard Theorem) are determined by their semialgebraic complexity, and not by their regularity.     

Differential structures in lie superalgebras

We develop a formal construction of an U-system as a fundamental concept of noncommutative differential geometry. By using the notion of “conditional differential” (an analog of the Hamiltonian mapping), we construct a series of brackets that generalize the classical Poisson brackets.     

Phenomena in Inverse Stackelberg Games,Part 1: Static Problems

Games are considered in which the role of the players is a hierarchical one. Some players behave as leaders, others as followers. Such games are named after Stackelberg. In the current paper, a special type of these games is considered, known in the literature as inverse Stackelberg games. In such games, the leader (or: leaders) announces his strategy as a mapping from the follower (or: followers) decision space into his own decision space. Arguments for studying such problems are given. The routine way of analysis, leading to a study of composed functions, is not very fruitful. Other approaches are given, mainly by studying specific examples. Phenomena in problems with more than one leader and/or follower are studied within the context of the inverse Stackelberg concept. As a side issue, expressions like “two captains on a ship” and “divide and conquer” are given a mathematical foundation.     

Teachers' Understandings of Realistic Contexts to Capitalize on Students' Prior Knowledge

The theory of realistic mathematics education establishes that framing mathematics problems in realistic contexts can provide opportunities for guided reinvention. Using data from a study group, I examine geometry teachers' perspectives regarding realistic contexts during a lesson study cycle. I ask the following. (a) What are the participants' perspectives regarding realistic contexts that elicit students' prior knowledge? (b) How are the participants' perspectives of realistic contexts related to teachers' instructional obligations? (c) How do the participants draw upon these perspectives when designing a lesson? The participants identified five characteristics that are needed for realistic contexts: providing entry points to mathematics, using “catchy” and “youthful” contexts, selecting personal contexts for the students, using contexts that are not “too fake” or “forced,” and connecting to the lesson's mathematical content. These characteristics largely relate to the institutional, interpersonal, and individual obligations with some connections with the disciplinary obligation. The participants considered these characteristics when identifying a realistic context for a problem‐based lesson. The context promoted mathematical connections. In addition, the teachers varied the context to increase the relevance for their students. The study has implications for supporting teachers' implementation of problem‐based instruction by attending to teachers' perspectives regarding the obligations shaping their work.     

Brouwer and Euclid

We explore the relationship between Brouwer’s intuitionistic mathematics and Euclidean geometry. Brouwer wrote a paper in 1949 called The contradictority of elementary geometry. In that paper, he showed that a certain classical consequence of the parallel postulate implies Markov’s principle, which he found intuitionistically unacceptable. But Euclid’s geometry, having served as a beacon of clear and correct reasoning for two millennia, is not so easily discarded.Brouwer started from a “theorem” that is not in Euclid, and requires Markov’s principle for its proof. That means that Brouwer’s paper did not address the question whether Euclid’s Elements really requires Markov’s principle. In this paper we show that there is a coherent theory of “non-Markovian Euclidean geometry”. We show in some detail that our theory is an adequate formal rendering of (at least) Euclid’s Book I, and suffices to define geometric arithmetic, thus refining the author’s previous investigations (which include Markov’s principle as an axiom).Philosophically, Brouwer’s proof that his version of the parallel postulate implies Markov’s principle could be read just as well as geometric evidence for the truth of Markov’s principle, if one thinks the geometrical “intersection theorem” with which Brouwer started is geometrically evident.     

A smooth and unified proof of cases 6, 5 and 3 of the ringel-youngs theorem

A proof of the Heawood conjecture for Cases 6, 5, and 3 is given; “unified” means that the same “geometry” and “zigzag” are used in each of the three cases; “smooth” means that the zigzag is of simplest possible type and that the related “chord problems” are trivial or nearly so.     

Loop quantum mechanics and the fractal structure of quantum spacetime

We discuss the relation between string quantization based on the Schild path integral and the Nambu-Goto path integral. The equivalence between the two approaches at the classical level is extended to the quantum level by a saddle-point evaluation of the corresponding path integrals. A possible relationship between M-Theory and the quantum mechanics of string loops is pointed out. Then, within the framework of “loop quantum mechanics”, we confront the difficult question as to what exactly gives rise to the structure of spacetime. We argue that the large scale properties of the string condensate are responsible for the effective Riemannian geometry of classical spacetime. On the other hand, near the Planck scale the condensate “evaporates”, and what is left behind is a “vacuum” characterized by an effective fractal geometry.