Geometry Deals Research Articles

Not Quite Yet a Hazy Limbo of Mystery: Intuition in Russell’s An Essay on the Foundations of Geometry

Abstract I argue that in Bertrand Russell’s An Essay on the Foundations of Geometry (1897), his forms of externality serve the same fundamental role in grounding the possibility of geometry that Immanuel Kant’s forms of intuition serve in grounding geometry in his critical philosophy. Specifically, both provide knowledge of bare numerical difference, where we have no concept of this difference. Because geometry deals with such conceptually homogeneous magnitudes and their composition on both accounts, forms of intuition or externality (respectively) are at its foundation.

Flexible polyurethane foam as personal protective mask material: a numerical and experimental study

This work outlines a method for characterizing flexible polyurethane foam (FPF) for use as a filter material, particularly for personal protective equipment. Two different FPF samples were tested experimentally to establish their geometric and flow resistance properties, which were then used to calibrate a geometric idealization of the foam created using a discrete element modeling software. The idealized model was then used to conduct pore-level numerical simulations to estimate the filtration efficiency of the foam for particles with average of diameters in the range 0.2–200 µm for various airflow velocities. A parametric study was then done to estimate the particle penetration rate of a facile mask made of FPF under different breathing conditions. The results indicate that during inhalation, pressure and velocity variations on the inner surface of the mask are not significant and that estimates of the filtration efficiency can be made by considering the time-dependent filter velocity combined with prior results for filter effectiveness as a function of velocity and particle size. The results also show that a non-medical FPF mask can drastically reduce the risk of particle penetration for particles larger than 0.2 µm. The process demonstrated in this work can be used to estimate the overall effectiveness of an FPF mask when exposed to clouds of different sized particles and to study the impacts of mask shape and fit on filtration effectiveness.

Об изображениях

In this paper, a completely different statement about the ownership of images is proposed from the generally accepted opinion. First there are definitions of well-known concepts, such as the purpose of descriptive geometry, its method, its subject. Then, gradually approaching the opinion that descriptive geometry is the theory of images, it is said about those areas of knowledge and human activity that are impossible without descriptive geometry. It is shown that images are characteristic of absolutely all areas of human activity. Mention is made of painters, graphic artists and sculptors who study descriptive geometry during training. The first drawings made on the walls of caves are mentioned, which indicates the use of descriptive geometry even before the conscious division of science into different branches of research. It is shown that any image is a mental activity of the brain, which includes the works of abstractionists to any works of painters, as well as to the halo of belonging to descriptive geometry. And since descriptive geometry deals with abstract points, lines and compartments of surfaces, then all directions of images consisting of points, lines and compartments of surfaces simply have to be attached to the field of activity of descriptive geometry.

The place of theoretical wisdom in the application of architectural geometry by Islamic mathematicians. The period under study of the fourth to eleventh centuries AH

The place of theoretical wisdom in the application of architectural geometry by Islamic mathematicians. The period under study of the fourth to eleventh centuries AH

Povezanost slikovnih i pojmovnih svojstava ugla i kocke u geometrijskom rezonovanju budućih učitelja

Drawing upon Fischbein's theory of figural concepts, the starting point of the paper is the use and value of the history of geometry in mathematics education - first point that we make historical reference are theorems of Eudemus of Rhodes and Thales of Miletus and the second one is elaboration of these theorems in work of Serbian mathematicians Mihailo Petrovic Alas. Fischbein's theory is mainly based on a hypothesis that geometry deals with mental entities, the so-called geometric figures, which simultaneously possess conceptual and figural properties. Fischbein called the geometric figures figural concepts because of their nature. We have analyzed the internal tensions of the concepts of angle and cube, which may appear in figural concepts because of their double nature, developmental aspects and didactical implications. The goal of the research was to examine the pre-service primary school teachers' geometric reasoning regarding the correlation between figural (pictoral) and conceptual properties of geometric objects (angle and cube) in order to obtain a framework for creating didactic situations that reduce the gap between figure and concept. The results of this study show that the figural (pictoral) structure of an angle (and cube) dominates in the geometric reasoning of the pre-service primary school teachers over its formal conceptual constraints. There were some differences in situations when an image is helpful in solving tasks involving the concepts of the net of the cube and the cube itself.

Reliability assessment of drag embedment anchors in sand and the effect of idealized anchor geometry

In this study, the reliability of drag embedment anchors in the sand was assessed and the effect of anchor geometrical idealization on reliability indices was investigated as an inherited characteristic of analytical approaches. The anchor holding capacity was obtained by performing a series of iterative limit state analyses and a probabilistic model was developed for the selected anchor families. The tensions of the mooring lines connected to a semisubmersible platform were obtained by performing a series of time-domain dynamic mooring analyses using the OrcaFlex software. The uncertainties in environmental loads, metocean variables, and stress distribution along the catenary mooring lines were incorporated into the line tensions through the response surfaces. An iterative procedure was performed by adopting the first-order reliability method (FORM) to calculate the comparative failure probabilities in sand and clay. The study showed significant dependence of the anchoring system reliability on geometrical configuration of anchors, the seabed soil properties, and the environmental loads. It was observed that the implementation of the reliability-based design into the existing in-filed trial procedures could significantly improve the efficiency and cost-effectiveness of the design practice.

Spatial information processing of seventh grader in solving Geometry problems

Geometry deals with spatial objects and requires skills to cultivate them. Information processing requires cognitive processes such as recognizing, transferring, rotating and communicating with others. Apparently, there are students who have difficulty in processing spatial information that affects the results of geometric achievement. This study aims to explore information about how students process spatial information. The method used is a descriptive exploratory approach with the subject of three seventh graders in junior high school students. Subjects are asked to complete tasks about the triangle and three-dimensional objects. The results of the study indicate that there are obstacles in processing spatial information, namely in the encoding process. The first and second subjects have difficulty in recognizing objects and represents rotated objects. The third subject has difficulty imagining spatial objects without any media modelling them.

The Method of Analytical Preconditioning in the Analysis of the Propagation in Dielectric Waveguides With Wedges

The recent interest in dielectric waveguides with complicated structures and the improvement in the etching techniques frequently lead to a geometrical idealization in which the wedges cannot be ignored when accurate solutions are required. The aim of this paper is the introduction of a guaranteed-convergence method for the analysis of the propagation in dielectric waveguides with a polygonal cross section belonging to the class of methods of analytical preconditioning. A surface integral equation formulation in the spectral domain is discretized by means of Galerkin's method with analytical Fourier transformable expansion bases reconstructing the physical behavior of the fields even on the wedges. The obtained matrix equation has many symmetries due to reciprocity and the matrix coefficients are improper integrals of the oscillating functions efficiently computed by means of an analytical asymptotic acceleration technique and a vectorized C++ code implementation. Comparisons with the literature and the commercial software CST Microwave Studio are produced to show the effectiveness of the presented method.

Indoor scene modeling from a single image using normal inference and edge features

We present in this paper an interactive approach for semantically modeling the indoor environment given only a single indoor image as input, without requiring access to the scene or using any additional measurements like RGBD cameras. Our key insight is that, although depth estimation from a single image is notoriously difficult, we can conveniently obtain a relatively accurate normal map, which essentially conveys a great deal of scene geometry. This enables us to model each object in a data-driven manner by representing the object as a normal-based graph and retrieving a similar model from the database by graph matching. Moreover, edge information is integrated to further improve the searching result. We hypothesize a set of sparse surface orientations for the image and further refine them in an intuitive and straightforward manner. With a small amount of simple user interaction, our approach is able to generate a plausible model of the scene. To verify the effectiveness of our proposed method, we show the modeling results on a variety of indoor images.

Stability assessment of enzyme stabilized road embankments

Road infrastructure is one of the most important factors for a country’s development. The vital role that they play in the present economy is reflected in many kilometres of roads worldwide. The external loads due to recurring traffic loads and geohazards can result in road failures with consequences in terms of economic loss to asset managers, public safety and vehicle damage. Due to such adverse effects of road failures, soil stabilization is being widely used as a preventive measure of road failures since recently. However, the assessment (or design) of stabilized road embankments using current standards assume linear material response and two-dimensional geometric idealization. Such assumptions can lead to non-conservative results, leading the road embankments to fail in long-term mainly due to improper assessments in the designs. The current research investigates the assessment of enzyme based stabilized road embankments under external traffic loads using three-dimensional (3-D) finite element (FE) analysis. Firstly, laboratory tests were conducted to characterize the soil/stabilized soil and to calibrate the numerical models. Then, a series of FE analyses were conducted to investigate the performance of stabilized road embankments under external traffic loads. Results showed that the prediction of required pavement thicknesses using elasto-plastic 3-D modelling could be substantially different in comparison to the results obtained from available standard pavement design tools and 2-D idealization of pavement modelling. The study revealed the importance of considering realistic material performance and load applications during the assessment of pavement design in order to maximize the benefits of enzyme stabilization that can be effective in prevention of the road embankment failures.

The Importance of Hemorheology and Patient Anatomy on the Hemodynamics in the Inferior Vena Cava.

Inferior vena cava (IVC) filters have been used for nearly half a century to prevent pulmonary embolism in at-risk patients. However, complications with IVC filters remain common. In this study, we investigate the importance of considering the hemorheological and morphological effects on IVC hemodynamics by simulating Newtonian and non-Newtonian blood flow in three IVC models with varying levels of geometric idealization. Partial occlusion by an IVC filter and a thrombus is also considered. More than 99% of the infrarenal IVC volume is found to contain flow in the nonlinear region of the shear rate-viscosity curve for blood (less than 100s-1) in the unoccluded IVCs. Newtonian simulations performed using the asymptotic viscosity for blood over-predict the non-Newtonian Reynolds numbers by more than a factor of two and under-predict the mean wall shear stress (WSS) by 28-54%. Agreement with the non-Newtonian simulations is better using a characteristic viscosity, but local WSS errors are still large (up to 50%) in the partially occluded cases. Secondary flow patterns in the IVC also depend on the viscosity model and IVC morphological complexity. Non-Newtonian simulations required only a marginal increase in computational expense compared with the Newtonian simulations. We recommend that future studies of IVC hemodynamics consider the effects of hemorheology and IVC morphology when accurate predictions of WSS and secondary flow features are desired.

The “Carey Number” continued—Mechanisms governing the horizontal array coherence length in shallow water acoustics

The use of simplified “feature models” (geometric idealizations of specific, isolated ocean features) for coastal oceanographic features can allow one to calculate acoustically useful quantities approximately and even generate analytic forms for them. Feature models for coastal fronts, eddies, internal tides, linear and nonlinear internal waves, and spice are presented, and the scattering of sound from these objects is calculated. This allows one to estimate the useful quantity Lcoh, the horizontal coherence length that represents a physical limit for array signal processing. Continuing work on calculations of Lcoh, including source/receiver motion, will be presented. [Work sponsored by the Office of Naval Research.]

Simple expressions for the horizontal array coherence length in shallow water acoustics

The use of simplified “feature models” (geometric idealizations of specific, isolated ocean features) for coastal oceanographic features can allow one to calculate acoustically useful quantities approximately and even generate analytic forms for them. Feature models for coastal fronts, eddies, internal tides, linear and nonlinear internal waves, and spice are presented and the scattering of sound from these objects is calculated. This allows one to estimate the useful quantity Lcoh, the horizontal coherence length that represents a physical limit for array signal processing. Calculations of Lcoh and their comparisons with data will be presented. The effects of acoustic multipath propagation, which can compete with the medium scattering in the estimates of Lcoh, are also estimated using basic models. [Work sponsored by the Office of Naval Research.]

Fractal Pattern of Different Alteration Zones in Porphyry Copper Deposits of RekoDik, Chagai Belt, Pakistan

Fractal is a mathematical set that has a fractal dimension that usually exceeds its topological dimension and may fall between integers. They represent the idea of detailed self-similar repetition at same scale or they may nearly be the same at different scales. Fractals are geometrical objects with the fractal dimension (D). Their fractal geometry deals with the objects and spaces. They occupy space (x, y, z) of any dimension greater than or equal to the dimension of the objects. In this study, hydrothermally altered zones of porphyry copper deposits of Reko-Dik, Chagai Belt, are taken into account to see whether alternation patterns follow any fractal law. Rowan et al. (2006) carried out detailed spectral analyses of different alteration zones nicely exposed around the Reko-Dik. We have used the same images as base map in our calculations for spatial analysis. A refined box-counting method is used, where the number of boxes N(r) containing the alteration versus the box size (r) repeatedly tested. To start with, the scaling properties of box size (r) were taken at 0.5 km interval, following 0.5 km addition in each analysis ending at 5 km. The spatial distributions of hydrothermally altered zones show following fractal dimension values: (D) for argillic (D=1.0 ± 0.05), phyllic (muscovite + jarosite) (D=1.2 ± 0.04), phyllic (muscovite) (D=0.8 ± 0.05), propylitic (D=1.2 ± 0.04), and siliceous (D=0.8 ± 0.07). Further work is in progress to refine the D values using improved box-counting method.

Proclus

In the history of a specific idea there are moments of reception that radically alter a concept’s functioning, moments where its meaning is transplanted in such a way that an internal rupture emerges. In the case of imagination the Neoplatonist Proclus (410/412-485) seems responsible for such a disruptive reception. Proclus is generally held to be the first to have systematically cut imagination’s exclusive ties with sense-data in his projectionist philosophy of mathematics, presented in his Commentary of the First Book of Euclid’s Elements (Bouriau 2002, 49). Proclus’ theory of geometry consists in a frontal attack on the traditional, Aristotelian viewpoint that geometry deals with sensible objects qua planes, qua lines, etc. From Aristotle’s point of view (Metaph. 1077b24-30), geometry ‘makes abstraction from’ or ‘leaves unconsidered’ the material composition of a specific object, by filtering out the properties of sensible objects that are not mathematically relevant. According to Proclus, in geometry discursive thinking uses imagined representations of innate concepts. Due to and within imagination these unextended ideas are projected as extended and divisible. Therefore the Neoplatonist is called, along with Descartes, a philosopher of the productive imagination (Bouriau 2002, 47; Rabouin 2009, 178), a term hardly compatible with the traditional, Aristotelian vocabulary. The compatibility of this productive geometrical imagination with imagination’s functioning in other contexts of the Neoplatonist’s oeuvre, where imagination is generally understood as the deceitful receptacle of the imprints from sensible objects, is a much debated topic. Nonetheless, imagination’s ambiguous and shifting meaning in the Proclean’ corpus is generally believed to be an interpretative problem. In the present article I will argue that by introducing imagination’s productive side and by simultaneously holding on to its traditional features, Proclus wittingly or unwittingly reveals a shift in meaning and functioning that was always already proper to imagination as a mediating place between sensibility and intelligibility, between productivity and reproductivity. For that reason, imagination’s ambiguous meaning throughout the Proclian corpus should not so much be considered an interpretative problem as a symptomatic manifestation of an underlying structure – a system

Design and implementation of efficient transit networks: Procedure, case study and validity test

This paper presents and tests a method to design high-performance transit networks. The method produces conceptual plans for geometric idealizations of a particular city that are later adapted to the real conditions. These conceptual plans are generalizations of the hybrid network concept proposed in Daganzo (2010). The best plan for a specific application is chosen via optimization. The objective function is composed of analytic formulae for a concept’s agency cost and user level of service. These formulae include as parameters key demand-side attributes of the city, assumed to be rectangular, and supply-side attributes of the transit technology. They also include as decision variables the system’s line and stop spacings, the degree to which it focuses passenger trips on the city center, and the service headway. These decision variables are sufficient to define an idealized geometric layout of the system and an operating plan. This layout-operating plan is then used as a design target when developing the real, detailed master plan. Ultimately, the latter is simulated to obtain more accurate cost and level of service estimates. This process has been applied to design a high performance bus (HPB) network for Barcelona (Spain). The idealized solution for Barcelona includes 182 km of one-way infrastructure, uses 250 vehicles and costs 42,489 €/h to build and run. These figures only amount to about one third of the agency resources and cost currently used to provide bus service. A detailed design that resembles this target and conforms to the peculiarities of the city is also presented and simulated. The agency cost and user level of service metrics of the simulated system differ from those of the idealized model by less than 10%. Although the designed and simulated HPB systems provide sub-optimal spatial coverage because Barcelona lacks suitable streets, the level of service is good. Simulations suggest that if the proposed system was implemented side-by-side with the current one, it would capture most of the demand.

Design and Implementation of Efficient Transit Networks: Procedure, Case Study and Validity Test

This paper presents and tests a method to design high-performance transit networks. The method produces conceptual plans for geometric idealizations of a particular city that are later adapted to the real conditions. These conceptual plans are generalizations of the hybrid network concept proposed in Daganzo (2010). The best plan for a specific application is chosen via optimization. The objective function is composed of analytic formulae for a concept‘s agency cost and user level of service. These formulae include as parameters key demand-side attributes of the city, assumed to be rectangular, and supply-side attributes of the transit technology. They also include as decision variables the system‘s line and stop spacings, the degree to which it focuses passenger trips on the city center, and the service headway. These decision variables are sufficient to define an idealized geometric layout of the system and an operating plan. This layout-operating plan is then used as a design target when developing the real, detailed master plan. Ultimately, the latter is simulated to obtain more accurate cost and level of service estimates.This process has been applied to design a high performance bus (HPB) network for Barcelona (Spain). The idealized solution for Barcelona includes 182km of one-way infrastructure, uses 250 vehicles and costs 42,489 €/h to build and run. These figures only amount to about one third of the agency resources and cost currently used to provide bus service. A detailed design that resembles this target and conforms to the peculiarities of the city is also presented and simulated. The agency cost and user level of service metrics of the simulated system differ from those of the idealized model by less than 10%. Although the designed and simulated HPB systems provide sub-optimal spatial coverage because Barcelona lacks suitable streets, the level of service is good. Simulations suggest that if the proposed system was implemented side-by-side with the current one, it would capture most of the demand.

Discrete Geometry

Discrete Geometry deals with the structure and complexity of discrete geometric objects ranging from finite point sets in the plane to more complex structures like arrangements of n -dimensional convex bodies. Classical problems such as Kepler's conjecture and Hilbert's third problem on decomposing polyhedra, as well as classical works by mathematicians such as Minkowski, Steinitz, Hadwiger and Erdös are part of the heritage of this area. By its nature, this area is interdisciplinary and has relations to many other vital mathematical fields, such as algebraic geometry, topology, combinatorics, computational geometry, convexity, and probability theory. At the same time it is on the cutting edge of applications such as geographic information systems, mathematical programming, coding theory, solid modelling, computational structural biology and crystallography. The workshop was attended by 40 participants. There was a series of 12 survey talks giving an overview of developments in Discrete Geometry and related fields: In addition, there were 18 shorter talks and an open problem session chaired by János Pach on Wednesday evening — a collection of open problems resulting from this session can be found in this report. The program left ample time for research and discussions in the stimulating atmosphere of the Oberwolfach Institute.

Active skin for turbulent drag reduction

Drag reduction for aerial and underwater vehicles has a range of positive ramifications:reduced fuel consumption, larger operational range, greater endurance and higherachievable speeds. This work capitalizes on recent developments in direct numericalsimulation (DNS) studies on turbulent drag reduction techniques and active material basedactuation to develop an ‘active’ skin for turbulent drag reduction. The skin operationprinciple is based on computational evidence from DNS studies indicating thatspanwise traveling surface waves of the right amplitude, wavelength and frequencycan result in significant turbulent drag reduction. Such traveling waves can beinduced in the skin via active material actuation. The flow control techniquepursued is ‘micro’ in the sense that only microscale wave amplitudes (order of30 µm) and energy inputs are expected to produce significant benefits. In this work, a generalizedactuation principle that can generate a traveling surface wave is proposed and analyzed.Two actuation schemes, that are approximate implementations of this generalizedactuation principle, are considered. Treating these actuation schemes as design paradigms,several skin designs are developed. The feasibility of the different actuation possibilities,such as shape memory alloys and piezoelectric material based actuators, and the relativemerits of the different skin designs are discussed. The feasibility studies include analyticalsolutions based on geometric idealizations, that are further refined using quasistatic finiteelement analysis. The dynamic responses of the skin designs to complicated loadingsequences are investigated using dynamic finite element analysis. Rayleigh damping isused in the dynamic analysis to model the total damping present in the system.

Affine Algebraic Geometry

Affine geometry deals with algebro-geometric questions of affine varieties. In the last decades this area has developed into a systematic discipline with a sizeable international group of researchers, and with methods coming from commutative and non-commutative algebra, algebraic, complex analytic and differential geometry, singularity theory and topology. The meeting was attended by 48 participants, among them the most important senior researchers in this field and many promising young mathematicians. Especially helpful were the programs for young researchers: the NSF Oberwolfach program, the EU grant and JAMS grant. They allowed to increase the number of young participants considerably. The conference took place in a very lively atmosphere, made possible by the excellent facilities of the institute. There were 24 talks with a considerable number of lectures given by young researchers at the beginning of their careers. Moreover there were 4 invited lectures that gave an overview over some of the most vivid subfields of Affine Geometry. The program left plenty of time for cooperation and discussion among the participants. We highlight the areas in which new results were presented by the lecturers: In the first 4 areas there were overview talks given by D. Wright, M. Miyanishi, D. Daigle and Sh. Kaliman. Finally, in a problem session there were presented a number of open questions and problems, which are listed in at the end of this report.