Geometric Theorem Research Articles

THE PROBLEM OF CORRELATION OF INDUCTION AND DEDUCTION IN THE HISTORY OF MATHEMATICS AND MATHEMATICAL EDUCATION

The analysis of the scientists’ views on the role of induction and deduction in the development of mathematics is carried out. There are no substantial difference in the estimation of the importance of deductive substantiation of mathematical concepts, theorems and theories in these views. But we had to establish the big differences and even absolutely opposite views on the role of induction in the process of generation, extension and integration of mathematical knowledge. The most conclusive are the opinions of the thing that an induction is a match for a deduction (L. Euler, F. Klein, G. Polya and others). The importance of inductive method of research in the scientific development, especially in mathematics, was even greater emphasized by L. de Broglie. Absolutely opposite are the views of those scientists who consider the mathematics to be an absolutely deductive science (T. Huxley, J. Murray, B. Pierce and others). The most categorical point of such view was expressed by J. Murray.In the mathematical education the false opinion of the thing that this science is supposedly to be absolutely deductive inevitably lead to pedagogical mistakes. The threat of these mistakes concerns mainly the geometrical training. The deductive character of statement of proving the geometrical theorems in the manuals and at the lessons leads to the pupils’ material difficulties in the process of digestion of knowledge. Inductive method of education may lighten the process of digestion of geometry. Not of less importance this method is at the lessons of algebra.If the importance of role of the deduction in the mathematical development is old-confirmed by the perfection of deductive proofs of Euclid, the consciousness of the importance of induction in the mathematical researches is set just in the first quarter of the XVII century with the publication of “Novum Organum” by Francis Bacon. The first thinker who consciously combined deduction with induction in his researches was R. Descartes. Indissoluble correlation of induction and deduction is brilliantly shown in many researches of L. Euler.In the second half of the XVIII century the induction comes to gaining practical application in the mathematical education. For the first time the inductive method was realized in the “Universal arithmetic” manual the author of which was an outstanding educator N. G. Kurganov. This manual was notable for the simplicity of teaching materialexposition.The increasing of attention to the mathematical education by inductive method is founded in the manuals, methodical works and educational activity of F. I. Busse, P. S. Hurjev, O. M. Strannoliubskiy,S. I. Shohor-Trotskiy. The valuable contribution to this problem development is issued to K. F. Lebedintsev, the author of specific inductive method of mathematical education. In the second half of the XX century the many lines of the application of induction in the secondary and higher mathematical education became a subject of long-term fundamental investigation by G. Polya.Hence, it is established that the requisite condition of the improving of mathematical education is following the principle of indissoluble correlation of induction and deduction in the educational process. This principle claims active energies of the manuals authors and teachers of mathematics at general secondary schools and higher educational establishments which provide practical realization of the unique methodical heritage of K. F. Lebedintsev and G. Polya.

Generalized negative binomial distributions as mixed geometric laws and related limit theorems*

In this paper, we study a wide and flexible family of discrete distributions, the so-called generalized negative binomial (GNB) distributions, which are mixed Poisson distributions with the mixing laws belonging to the class of generalized gamma (GG) distributions. This family was introduced by E.W. Stacy as a particular family of lifetime distributions containing gamma, exponential power, and Weibull distributions. These distributions seem to be very promising in the statistical description of many real phenomena and very convenient and almost universal models for the description of statistical regularities in discrete data. We study analytic properties of GNB distributions. We prove that a GG distribution is a mixed exponential distribution if and only if the shape and exponent power parameters are no greater than one. We write the mixing distribution explicitly as a scale mixture of strictly stable laws concentrated on the nonnegative half-line. As a corollary, we obtain a representation for the GNB distribution as a mixed geometric distribution. We consider the corresponding scheme of Bernoulli trials with random probability of success. Within this scheme, we prove a random analog of the Poisson theorem establishing the convergence of mixed binomial distributions to mixed Poisson laws. We prove limit theorems for random sums of independent random variables in which the number of summands has the GNB distribution and the summands have both light- and heavy-tailed distributions. We obtain that the class of limit laws is wide enough and includes the so-called generalized variance gamma distributions. We obtain various representations for the limit laws in terms of mixtures of Mittag-Leffler, Linnik, or Laplace distributions. We prove limit theorems establishing the convergence of the distributions of statistics constructed from samples with random sizes obeying the GNB distribution to generalized variance gamma distributions. We also discuss some applications of GNB distributions in meteorology.

A Survey on Algorithms for Computing Comprehensive Gröbner Systems and Comprehensive Gröbner Bases

Weispfenning in 1992 introduced the concepts of comprehensive Grobner system/basis of a parametric polynomial system, and he also presented an algorithm to compute them. Since then, this research field has attracted much attention over the past several decades, and many efficient algorithms have been proposed. Moreover, these algorithms have been applied to many different fields, such as parametric polynomial equations solving, geometric theorem proving and discovering, quantifier elimination, and so on. This survey brings together the works published between 1992 and 2018, and we hope that this survey is valuable for this research area.

Self-evident Automated Proving Based on Point Geometry from the Perspective of Wu’s Method Identity

The algebraic methods represented by Wu’s method have made significant breakthroughs in the field of geometric theorem proving. Algebraic proofs usually involve large amounts of calculations, thus making it difficult to understand intuitively. However, if the authors look at Wu’s method from the perspective of identity,Wu’s method can be understood easily and can be used to generate new geometric propositions. To make geometric reasoning simpler, more expressive, and richer in geometric meaning, the authors establish a geometric algebraic system (point geometry built on nearly 20 basic properties/formulas about operations on points) while maintaining the advantages of the coordinate method, vector method, and particle geometry method and avoiding their disadvantages. Geometric relations in the propositions and conclusions of a geometric problem are expressed as identical equations of vector polynomials according to point geometry. Thereafter, a proof method that maintains the essence of Wu’s method is introduced to find the relationships between these equations. A test on more than 400 geometry statements shows that the proposed proof method, which is based on identical equations of vector polynomials, is simple and effective. Furthermore, when solving the original problem, this proof method can also help the authors recognize the relationship between the propositions of the problem and help the authors generate new geometric propositions.

Introduction to Discrete Mathematics and Fourier Analysis. A Practical Guide for third Year Undergraduate Students.

This book covers a quick introduction to high school mathematics, arithmetic and geometric series, infinite geometric series, mathematical induction, binomial theorem, factorials, permutations and combinations, determinants, systems of linear equations, boolean algebra, orthogonal functions, Legendre, Hermite, and Laguerre polynomials and orthogonal properties. For example, mathematical induction is a proof based on logical assumptions of equations that involve integers of one number that shows the proof for the next number. It is based on the base step and the inductive step. The base step is to prove the integers based on a formula. The inductive step involves statement of one integer number that shows the proof of the statement for the next integer. The determinant is a scalar and it is found by multiplying the two elements of the principal diagonal and subtracting them from the multiplication of the elements of the other diagonal.

Geometrical representation theorems for cylindric-type algebras

In this paper, we give new proofs of the celebrated Andréka-Resek-Thompson representability results of certain axiomatized cylindric-like algebras. Such representability results provide completeness theorems for variants of first order logic, that can also be viewed as multi-modal logics. The proofs herein are combinatorial and we also use some techniques from game theory.

Hamstring Autograft Too Small: How Much Allograft Do You Need to Supplement to a Desired Hybrid Graft Size?

To determine a simple rule for choosing supplemental allograft size for hybrid anterior cruciate ligament reconstruction using mathematical and cadaveric models. Mathematical and cadaveric models were used to determine the rule. The mathematical model required application of the geometric Pythagorean theorem to add areas of circles. Cadaveric semitendinosus and gracilis tendons were combined in multiple quadrupled hamstring size combinations and then sized using standard surgical techniques to confirm the mathematical model. Geometric measurement, not simple addition, of graft diameters was required to determine the final graft size. Direct comparison of cadaveric and mathematical models showed close relations. If a final graft size of 7mm is desired, an added diameter of all grafts of approximately 9.5mm is needed. If a final graft size of 8mm is desired, an added diameter of all grafts of approximately 11mm is needed. If a final graft size of 9mm is desired, an added diameter of all grafts of approximately 12.5mm is needed. If a final graft size of 10mm is desired, an added graft diameter of approximately 14mm is needed. Cadaveric hamstring measurements were similar to the mathematical model. By use of mathematical and cadaveric models, simple rules for determining the additional size of allograft diameter needed to supplement undersized hamstring autograft were created. With the increasing availability of allograft types and sizes, surgeons currently have no guidelines on the size of allograft that is required to supplement an undersized hamstring autograft. Simple rules were created for determining the amount of allograft supplementation required for undersized hamstrings and are easily applied to clinical situations.

Taxonomies of geometric problems

In the current Information Society the organisation of the information is key to ensure the information safekeeping and retrieval. It is of utmost importance that each and every user can find the information he/she is looking for, presented in such a way that best fit his/her needs. Geometry is no exception, the servers of geometric information should be easily and successfully searchable. By classifying the information contained in the servers of geometric information accordingly to several taxonomies, it will be possible to begin applying filters to the users' queries, adjusting them to the perceived user's needs. Having that in mind, the introduction of an adaptive filtering mechanisms into servers of geometric information is considered.Different taxonomies for different goals are presented. For educational purposes, a classification like Common Core Standards should be considered, but other considerations like the complexity of the construction, the provability, by a geometry automatic theorem prover, of a given conjecture and the readability of the resulting proof, should be taken into account. For research in automated deduction purposes, other issues must be considered, e.g. efficiency and applicability of the available automated provers.To validate the usefulness of these taxonomies it will be used, as a case study, their application to a server of geometric information. In particular, Thousands of Geometric problems for geometric Theorem Provers will be considered. TGTP is a Web-based repository of geometric problems being developed to support the testing and evaluation of geometric automated theorem proving systems. Using this system it will be analysed how the taxonomies could help to tailor the search for information adapted to each and every geometer.

Schrödinger operators on graphs and geometry. III. General vertex conditions and counterexamples

Schrödinger operators on metric graphs with general vertex conditions are studied. Explicit spectral asymptotics is derived in terms of the spectrum of reference Laplacians. A geometric version of the Ambartsumian theorem is proven under the assumption that the vertex conditions are asymptotically properly connecting and asymptotically standard. By constructing explicit counterexamples, it is shown that the geometric Ambartsumian theorem does not hold in general without additional assumptions on the vertex conditions.

Scale Mixtures of Frechet Distributions as Asymptotic Approximations of Extreme Precipitation

Based on the negative binomial model for the duration of wet periods measured in days, an asymptotic approximation is proposed for the distribution of the maximum daily precipitation volume within a wet period. This approximation has the form of a scale mixture of the Frechet distribution with the gamma mixing distribution and coincides with the distribution of a positive power of a random variable having the Snedecor-Fisher distribution. The proof of this result is based on the representation of the negative binomial distribution as a mixed geometric (and hence, mixed Poisson) distribution and limit theorems for extreme order statistics in samples with random sizes having mixed Poisson distributions. Some analytic properties of the obtained limit distribution are described. In particular, it is demonstrated that under certain conditions the limit distribution is mixed exponential and hence, is infinitely divisible. It is shown that under the same conditions the limit distribution can be represented as a scale mixture of stable or Weibull or Pareto or folded normal laws. The corresponding product representations for the limit random variable can be used for its computer simulation. Several methods are proposed for the estimation of the parameters of the distribution of the maximum daily precipitation volume. The results of fitting this distribution to real data are presented illustrating high adequacy of the proposed model. The obtained mixture representations for the limit laws and the corresponding asymptotic approximations provide better insight into the nature of mixed probability ("Bayesian") models.

Portfolio theorem proving and prover runtime prediction for geometry

In recent years, portfolio problem solving found many applications in automated reasoning, primarily in SAT solving and in automated and interactive theorem proving. Portfolio problem solving is an approach in which for an individual instance of a specific problem, one particular, hopefully most appropriate, solving technique is automatically selected among several available ones and used. The selection usually employs machine learning methods. To our knowledge, this approach has not been used in automated theorem proving in geometry so far and it poses a number of new challenges. In this paper we propose a set of features which characterize a specific geometric theorem, so that machine learning techniques can be used in geometry. Relying on these features and using different machine learning techniques, we constructed several portfolios for theorem proving in geometry and also runtime prediction models for provers involved. The evaluation was performed on two corpora of geometric theorems: one coming from geometric construction problems and one from a benchmark set of the GeoGebra tool. The obtained results show that machine learning techniques can be useful in automated theorem proving in geometry, while there is still room for further progress.

Abner of Burgos: The Missing Link between Nasir al-Din al-Tusi and Nicolaus Copernicus?

Abstract The geometrical theorem known as the ‘Tusi couple’ was first discovered by Persian astronomer Nasir al-Din al-Tusi (1201–1274). The Tusi couple was believed to be discovered for Europeans by Nicolaus Copernicus (1473–1543) and it played an important role in the development of his planetary system. It has been suggested by Willy Hartner, that Copernicus borrowed it from al-Tusi, however, a particular way of transmission is not known. In this article I show that Spanish-Jewish author Abner of Burgos (1270–1340) was familiar with the Tusi couple and followed Tusi’s notation in his diagrams. This may provide a missing link in the transmission of the Muslim astronomic knowledge to Europe and advance our understanding of the European Renaissance as a multicultural phenomenon.

The areas of the trajectory surface under the Galilean motions in the Galilean space

In the literature, Holditch theorem was obtained under periodic rotation and translation motions in [H. Holditch, Geometrical theorem, Q. J. Pure Appl. Math. 2 (1858) 38] or periodic shear and translation motions in [O. Röschel, Der satz von Holditch in der isotropen ebene. Abh. Braunschweig. Wiss. Ges. 36 (1984) 27–32]. In this paper, by introducing the projection of a vector onto a plane, scalar area, area vector of a surface, we investigate Holditch theorem under periodic rotation, translation and shear motions. We give two interpretations for the Holditch type theorem in Galilean space.

How Gauss and Markov Met Pythagoras: Geometry in Econometrics

Pursuing abstractness and generality, a number of books and articles in econometrics rely heavily on standard algebraic proofs. However, most of the theorems proved in such a way have a strong geometric appeal. This paper demonstrates how shorter, less technical and even more beautiful the proofs can be when they are based on geometric theorems. We show that this technique can be extended to explain concepts in probability and statistics. Although most of the theorems and ideas are not completely new and there are even a few works on the geometry in econometrics and statistics, this paper introduces alternative explanations and more general results in some cases. Further, there are algebraic proofs provided parallel to geometric ones and illustrations which are our own work.

App-based scaffolds for writing two-column proofs

ABSTRACTThis paper presents apps designed to assist students in understanding and developing proofs in geometric theorems. These technologies focus on triangle congruence, triangle similarity and properties of parallelograms. Focus group discussions and initial testing of the apps revealed that the apps offered a more engaging medium for learning proving and were capable of facilitating proof-writing skills in geometry.

The recognition and the constitution of the theorems of closure

This papers analyzes how several geometric theorems, that were considered to be disconnected from each other in the beginning of the nineteenth century, have been progressively recognized as elements of a bigger whole called “the theorems of closure.” In particular, we show that the constitution of this set of theorems was grounded on the use of encompassing words, as well as observations of analogies, and searches for unifying points of view. In the concluding remarks, we discuss the relevancy of the notion of “family resemblance” to describe the categorization process of the theorems of closure during the nineteenth century.

The Spectral Gaps of Generalized Flag Complexes and a Geometric Hall-type Theorem

Abstract Let $X$ be a simplicial complex on $n$ vertices without missing faces of dimension larger than $d$. Let $L_{k}$ denote the $k$-Laplacian acting on real $k$-cochains of $X$ and let $\mu _{k}(X)$ denote its minimal eigenvalue. We study the connection between the spectral gaps $\mu _{k}(X)$ for $k\geq d$ and $\mu _{d-1}(X)$. In particular, we establish the following vanishing result: if $\mu _{d-1}(X)>\big(1-\binom{k+1}{d}^{-1}\big)n$, then $\tilde{H}^{j}\left (X;{\mathbb{R}}\right )=0$ for all $d-1\leq j \leq k$. As an application we prove a fractional extension of a Hall-type theorem of Holmsen, Martínez-Sandoval, and Montejano for general position sets in matroids.

Riemannian and Lorentzian flow-cut theorems

We prove several geometric theorems using tools from the theory of convex optimization. In the Riemannian setting, we prove the max flow-min cut (MFMC) theorem for boundary regions, applied recently to develop a ‘bit-thread’ interpretation of holographic entanglement entropies. We also prove various properties of the max flow and min cut, including respective nesting properties. In the Lorentzian setting, we prove the analogous MFMC theorem, which states that the volume of a maximal slice equals the flux of a minimal flow, where a flow is defined as a divergenceless timelike vector field with norm at least 1. This theorem includes as a special case a continuum version of Dilworth’s theorem from the theory of partially ordered sets. We include a brief review of the necessary tools from the theory of convex optimization, in particular Lagrangian duality and convex relaxation.

A Novel Passive Tracking Scheme Exploiting Geometric and Intercept Theorems.

Passive tracking aims to track targets without assistant devices, that is, device-free targets. Passive tracking based on Radio Frequency (RF) Tomography in wireless sensor networks has recently been addressed as an emerging field. The passive tracking scheme using geometric theorems (GTs) is one of the most popular RF Tomography schemes, because the GT-based method can effectively mitigate the demand for a high density of wireless nodes. In the GT-based tracking scheme, the tracking scenario is considered as a two-dimensional geometric topology and then geometric theorems are applied to estimate crossing points (CPs) of the device-free target on line-of-sight links (LOSLs), which reveal the target’s trajectory information in a discrete form. In this paper, we review existing GT-based tracking schemes, and then propose a novel passive tracking scheme by exploiting the Intercept Theorem (IT). To create an IT-based CP estimation scheme available in the noisy non-parallel LOSL situation, we develop the equal-ratio traverse (ERT) method. Finally, we analyze properties of three GT-based tracking algorithms and the performance of these schemes is evaluated experimentally under various trajectories, node densities, and noisy topologies. Analysis of experimental results shows that tracking schemes exploiting geometric theorems can achieve remarkable positioning accuracy even under rather a low density of wireless nodes. Moreover, the proposed IT scheme can provide generally finer tracking accuracy under even lower node density and noisier topologies, in comparison to other schemes.

Exchange of Geometric Information Between Applications

The Web Geometry Laboratory (WGL) is a collaborative and adaptive e-learning Web platform integrating a well known dynamic geometry system. Thousands of Geometric problems for Geometric Theorem Provers (TGTP) is a Web-based repository of geometric problems to support the testing and evaluation of geometric automated theorem proving systems. The users of these systems should be able to profit from each other. The TGTP corpus must be made available to the WGL user, allowing, in this way, the exploration of TGTP problems and their proofs. On the other direction TGTP could gain by the possibility of a wider users base submitting new problems. Such information exchange between clients (e.g. WGL) and servers (e.g. TGTP) raises many issues: geometric search - someone, working in a geometric problem, must be able to ask for more information regarding that construction; levels of geometric knowledge and interest - the problems in the servers must be classified in such a way that, in response to a client query, only the problems in the user's level and/or interest are returned; different aims of each tool - e.g. WGL is about secondary school geometry, TGTP is about formal proofs in semi-analytic and algebraic proof methods, not a perfect match indeed; localisation issues, e.g. a Portuguese user obliged to make the query and process the answer in English; technical issues-many technical issues need to be addressed to make this exchange of geometric information possible and useful. Instead of a giant (difficult to maintain) tool, trying to cover all, the interconnection of specialised tools seems much more promising. The challenges to make that connection work are many and difficult, but, it is the authors impression, not insurmountable.