Geometric loci and homothetic transformations

In this paper, we use a property of the centroids to study two general types of geometric loci: one in the plane and one in the space. Then from this result, we examine in detail the case of quadrilaterals and tetrahedra and, in particular, we determine some properties of the nine-point circle of a triangle. The study of those geometric loci is carried out by means of the homothetic transformations that facilitate and make more immediate the proofs of the properties.     

Some curious properties and loci problems associated with cubics and other polynomials

The article begins with a well-known property regarding tangent lines to a cubic polynomial that has distinct, real zeros. We were then able to generalize this property to any polynomial with distinct, real zeros. We also considered a certain family of cubics with two fixed zeros and one variable zero, and explored the loci of centroids of triangles associated with the family. Some fascinating connections were observed between the original family of the cubics and the loci of the centroids of these triangles. For example, we were able to prove that the locus of the centroid of certain triangles associated with the family of cubics is another cubic whose zeros are in arithmetic progression. Motivated by this, in the last section of the article, we considered families of cubic polynomials whose zeros are in arithmetic progression, along with the loci of the special points of certain triangles arising from such families. Special points include the centroid, circumcentre, orthocentre, and nine-point centre of the triangles. Throughout the article, we used the computer algebra system, Mathematica®, to form conjectures and facilitate calculations. Mathematica® was also used to create various animations to explore and illustrate many of the results.     

Proof of Oppenheim's area inequalities for triangles and quadrilaterals

Leta ₁,b ₁,c ₁,A ₁ anda ₂,b ₂,c ₂,A ₂ be the sides and areas of two triangles. Ifa=(a ₁ ^p +a ₂ ^p )^1/p ,b=(b ₁ ^p +b ₂ ^p )^1/p ,c=(c ₁ ^p +c ₂ ^p )^1/p , and 1p4, thena, b, c are the sides of a triangle and its area satisfiesA ^p/2A ₁ ^p/2 +A ₂ ^p/2 . If obtuse triangles are excluded,p>4 is allowed. For convex cyclic quadrilaterals, a similar inequality holds. Also, leta, b, c, A be the sides and area of an acute or right triangle. Iff(x) satisfies certain conditions,f(a),f(b),f(c) are the sides of a triangle having areaA _f, which satisfies (4A _f/3)^1/2f((4A/3)^1/2).     

Understanding rigid geometric transformations: Jeff's learning path for translation

This article describes the development of knowledge and understanding of translations of Jeff, a prospective elementary teacher, during a teaching experiment that also included other rigid transformations. His initial conceptions of translations and other rigid transformations were characterized as undefined motions of a single object. He conceived of transformations as movement and showed no indication about what defines a transformation. The results of the study indicate that the development of his thinking about translations and other rigid transformations followed an order of (1) transformations as undefined motions of a single object, (2) transformations as defined motions of a single object, and (3) transformations as defined motions of all points on the plane. The case of Jeff is part of a bigger study that included four prospective teachers and analyzed their development in understanding of rigid transformations. The other participants also showed a similar evolution.     

Partial geometric difference sets and partial geometric difference families

The concept of a partial geometric difference set (or $1\frac{1}{2}$-difference set) was introduced by Olmez in 2014. Recently, Nowak, Olmez and Song introduced the notion of a partial geometric difference family, which generalizes both the classical difference family and the partial geometric difference set. It was shown that partial geometric difference sets and partial difference families give rise to partial geometric designs. In this paper, a number of new infinite families of partial geometric difference sets and partial geometric difference families are constructed. From these partial geometric difference sets and difference families, we generate a list of infinite families of partial geometric designs.     

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one‐dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

矩阵代数上的局部合同变换和局部相似变换

令F表示任意域,Mn（F）表示由F上所有n×n矩阵形成的结合代数.本文的目的是研究Mn（F）上具有如下性质的两类线性映射,其中一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个合同变换在该点的取值相同,另一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个相似变换在该点的取值相同,随着Mn（F）上的点不同,这些合同变换和相似变换可能也不同.利用矩阵的秩、幂等阵以及幂零阵的性质,通过矩阵计算的方法证明了第一类线性映射或者是合同变换或者是合同变换与转置变换的复合,第二类线性映射或者是相似变换或者是相似变换与转置变换的复合.由这个结果可知存在真正意义上的局部合同变换和局部相似变换,从而丰富了局部映射理论的研究。     

On properties of geometric random problems in the plane

In this paper, we present results dealing with properties of well-known geometric random problems in the plane, together with their motivations. The paper specifically concentrates on the traveling salesman and minimum spanning tree problems, even though most of the results apply to other problems such as the Steiner tree problem and the minimum weight matching problem.     

Generalized inverse of linear transformations: a geometric approach

A generalized inverse of a linear transformation A:

→

, where

and

are arbitrary finite dimensional vector spaces, is defined using only geometrical concepts of linear transformations. The inverse is uniquely defined in terms of specified subspaces $\text{L}$ ?

, $\text{M}$ ?

and a linear transformation N satisfying some conditions. Such an inverse is called the $\text{LM}$N-inverse. A Moore-Penrose type inverse is obtained by choosing N = 0. Some optimization problems are considered by choosing

and

as inner product spaces. Our results extend without any major modification of proofs to bounded linear operators with closed range on Hilbert spaces.     

Coxeter transformations: Construction of deltoids

The Coxeter transformations associated with deltoids (i.e., with graphs in which all simple cycles have length 3) are considered. The structure of the set of all connected deltoids whose spectra do not contain ?1 is described.     

Sensitivity analysis in geometric programming: Theory and computations

This paper surveys the main developments in the area of sensitivity analysis for geometric programming problems, including both the theoretical and computational aspects. It presents results which characterize solution existence, continuity, and differentiability properties for primal and dual geometric programs as well as the optimal value function differentiability properties for primal and dual programs. It also provides an overview of main computational approaches to sensitivity analysis in geometric programming which attempt to estimate new optimal solutions resulting from perturbations in some problem parameters.     

On Darboux and singular Darboux transformations of G-unitons

We present a unified method for constructing generalized flag transformations of unitons intoa Lie group G via singular Darboux transformations. These flag transformations provide the possibility toestablish some factorizations for G-unitons.     

On some methods of deriving geometric expectation

Formulas presented for the calculation of ∑ _j=1 ⁿ j^k (n, k ∈ N do not have a closed form; they are in the form of recursive or complex formulas. Here an attempt is made to present a simple formula in which it is only necessary to compute the numerical coefficients in a recursive form, and the coefficients in turn follow a simple pattern (almost similar to Pascal's Triangle). Although the pattern for calculating numerical coefficients based on forming a table is easy, non-recursive formulas are presented to determine the numerical coefficients.     

Some properties for the largest component of random geometric graphs with applications in sensor networks

In this paper we consider the standard Poisson Boolean model of random geometric graphs G（Hλ,s; 1） in Rd and study the properties of the order of the largest component L1 （G（Hλ,s; 1）） . We prove that ElL1 （G（Hλ,s; 1））] is smooth with respect to A, and is derivable with respect to s. Also, we give the expression of these derivatives. These studies provide some new methods for the theory of the largest component of finite random geometric graphs （not asymptotic graphs as s - co） in the high dimensional space （d 〉 2）. Moreover, we investigate the convergence rate of E[L1（G（Hλ,s; 1））]. These results have significance for theory development of random geometric graphs and its practical application. Using our theories, we construct and solve a new optimal energy-efficient topology control model of wireless sensor networks, which has the significance of theoretical foundation and guidance for the design of network layout.     

On Hardy–Steklov and geometric Steklov operators

A new (non‐Muckenhoupt type) weight characterization for the boundedness of the general Hardy–Steklov operator is obtained in the case 1 < p ≤ q < ∞. The estimates obtained for the norm of the Hardy–Steklov operator allow the limiting procedure and as a result the boundedness of the corresponding geometric Steklov operator is investigated. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Lie symmetries and form-preserving transformations of reaction-diffusion-convection equations

New Lie symmetry classification of the known class of reaction-diffusion-convection equations is presented. The classification method is based on combining the standard group classification method and the form-preserving transformation approach.     

Optimum metallic-bond scheme: Theoretical study of geometric structures for ground-state sodium clusters

We present an optimum metallic-bond scheme to study the geometric structures of sodium clusters Na_n(n⩽15) systematically by combining the characteristics of metallic bonds and the first principle molecular dynamics simulation. The scheme provides an optimum way to examine almost all stable structures of sodium clusters and to determine their ground state structures. It is interesting to note that for the larger sodium clusters (13⩽n⩽15), there are some plane-like substructures on their surfaces, which resemble the fragments of the (110) plane with the highest atomic area density in the bulk bcc sodium crystal. We also propose a possible way to understand the formation of large icosahedral sodium clusters (1500<n<22000).     

Matrix multiplication and transformations: an APOS approach

This study presents a contribution to research in undergraduate teaching and learning of linear algebra, in particular, the learning of matrix multiplication. A didactical experience consisting on a modeling situation and a didactical sequence to guide students’ work on the situation were designed and tested using APOS theory. We show results of research on students’ activity and learning while using the sequence and through analysis of student’s work and assessment questions. The didactic sequence proved to have potential to foster students’ learning of function, matrix transformations and matrix multiplication. A detailed analysis of those constructions that seem to be essential for students understanding of this topic including linear transformations is presented. These results are contributions of this study to the literature.     

Characterizations of the exponential distribution by stochastic ordering properties of the geometric compound

Under the reliability NBU/NWU conditions, the exponential distribution is characterized by stochastic ordering properties which link the geometric compound with minimum order statistics or spacings of order statistics. This somewhat answers a question posed by Kakosyan, Klebanov and Melamed (1984,Characterization of Distributions by the Method of intensively Monotone Operators, Springer, New York). We also show the related results based on the residual life in a renewal process and on record values. Finally, some fundamental properties of the NBUC/NWUC classes of life distributions are investigated.