Projective Metric Geometry and Clifford Algebras

Each vector space that is endowed with a quadratic form determines its Clifford algebra. This algebra, in turn, contains a distinguished group, known as the Lipschitz group. We show that only a quotient of this group remains meaningful in the context of projective metric geometry. This quotient of the Lipschitz group can be viewed as a point set in the projective space on the Clifford algebra and, under certain restrictions, leads to an algebraic description of so-called kinematic mappings.

Measurements and geometry

The paper is aimed at demonstrating the points of contact between measurements and geometry, which is done by modelling the main elements of the measurement process by the elements of geometry. It is shown that the basic equation for measurements can be established from the expression of projective metric and represents its particular case. Commonly occurring groups of functional transformations of the measured value are listed. Nearly all of them are projective transformations, which have invariants and are useful if greater accuracy of measurements is desired. Some examples are given to demonstrate that real measurement transformations can be dealt with via fractional-linear approximations. It is shown that basic metrological and geometrical categories are related, and a concept of seeing a multitude of physical values as elements of an abstract geometric space is introduced. A system of units can be reasonably used as the basis of this space. Two tensors are introduced in the space. One of them (the affinor) describes the interactions within the physical object, the other (the metric tensor) establishes the summation rule on account of the random nature of components.

Interpretation of Geometry on Manifolds as a Geometry in a Space with Projective Metric

In this paper, we give essential concepts of geometry of three-dimensional spaces in vector formulation in an affine-vector space An.

Effects of primary school’s physical environment on children’s spatial perception and behavior

Purpose The relationship between the child and his/her physical environment is an area of interaction that includes social, psychological and cultural factors along with the spatial experience, perception and behavior of the child. This study is based on the effects of spatial perception and behavior of the child within the physical environment of primary schools. In this direction, the purpose of this paper is to investigate how spatial and physical characteristics of primary school typologies affect the spatial perception and behavior of the child. Also, the parameters affecting spatial perception and behavior are examined. Design/methodology/approach The question to be investigated is how the spatial and physical characteristics of the school’s physical environment affect the child’s spatial perception and behavior in primary schools with different typologies. Within this scope, Istanbul’s Kagithane region is selected as a case study. Schools are chosen for their similar spatial and dimensional features and similar socio-economic environment. The methodology of the study consists of a literature review, an observational study carried out to discover the interaction between the child and his/her school building and the analysis of the student’s cognitive maps. These maps were evaluated according to topological, projective, metric and imaginative parameters. Findings The results show spatial organization and physical characteristics of primary school buildings with a structure that allows for change and transformation, and contributes to the physical and cognitive development of children. Originality/value This study will provide an opportunity to develop the design of future primary school buildings that can support the spatial perception and spatial experiences of the children.

Positive μ-pseudo almost periodic solutions for some nonlinear infinite delay integral equations arising in epidemiology using Hilbert’s projective metric

The purpose of this work is to give sufficient conditions which guarantee the existence and the uniqueness of positive μ-pseudo almost periodic solutions for the nonlinear infinite delay integral equation . We improve the original work of [H. S. Ding, Y. Y. Chen and G. M. N’Guérékata, Existence of positive pseudo almost periodic solutions to a class of neutral integral equations, Nonlinear Analysis. Theorey, Methods and Applications 74 (2011) 7356-7364] by dropping the hypotheses of monotonicity on the functions f and h. The main results are proved by using the Hilbert’s projective metric combined with the contraction mapping principle. Our results can deal with some cases to which many results are not applicable. An example is provided to illustrate the main results of this work.

POSITIVE ALMOST PERIODIC SOLUTIONS FOR THE HEMATOPOIESIS MODEL VIA THE HILBERT PROJECTIVE METRIC

The aim of this work is to prove the existence of a positive almost periodic solution to a multifinite time delayed nonlinear differential equation that describes the so-called hematopoiesis model. The approach uses the Hilbert projective metric in a cone. With some additional assumptions, we construct a fixed point theorem to prove the desired existence and uniqueness of the solution.