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A standard assumption in the study of logarithmic structures is "fineness", but this assumption is not preserved by intersections, fiber products, and more general limits. We explain how a coherent logarithmic scheme $X$ has a natural…
Linear Geometry studies geometric properties which can be expressed via the notion of a line. All information about lines is encoded in a ternary relation called a line relation. A set endowed with a line relation is called a liner. So,…
We use a probabilistic interpretation of solid angles to generalize the well-known fact that the inner angles of a triangle sum to 180 degrees. For the 3-dimensional case, we show that the sum of the solid inner vertex angles of a…
These lecture notes are a personal introduction to signed graphs, concentrating on the aspects that have been most persistently interesting to me. They are just a few corners of signed graph theory; I am leaving out a great deal. The…
We describe a geometric theory classified by Connes-Consani's epicylic topos and two related theories respectively classified by the cyclic topos and by the topos $[{\mathbb N}^{\ast}, \mathbf{Set}]$.
We provide an alternative unified approach for proving the Pythagorean theorem (in dimension $2$ and higher), the law of sines and the law of cosines, based on the concept of shape derivative. The idea behind the proofs is very simple: we…
An elementary general result is proved that allows for simple characterizations of well-known location/consensus functions (median, mean and center) on the n-cube. In addition, alternate new characterizations are given for the median and…
The convex and metric structures underlying probabilistic physical theories are generally described in terms of base normed vector spaces. According to a recent proposal, the purely geometrical features of these spaces are appropriately…
The setting of metric spaces is very natural for numerous questions concerning manifolds, norms, and fractal sets, and a few of the main ingredients are surveyed here.
By dividing hypergeometric series representations of the inverse sine by sin^-1 (x) and integrating, new double series representations of integers and constants arise. Binomial coefficients and the sine integral are thus combined in double…
In the present article, real number representations, that are generalizations of classical positive and alternating representations of numbers, are introduced and investigated. The main metric relation, properties of cylinder sets are…
The purpose of this work is to demonstrate how an arbitrarily chosen background of the Universe can be made a solution of a simple geometric sigma model. Geometric sigma models are purely geometric theories in which spacetime coordinates…
The standard model of particle physics represents the cornerstone of our understanding of the microscopic world. In these lectures we review its contents and structure, with a particular emphasis on the central role played by symmetries and…
In this paper, we study main properties of cone normed spaces, and prove some theorems of weighted means in cone normed spaces.
Basic concepts and definitions in differential geometry and topology which are important in the theory of solitons and instantons are reviewed. Many examples from soliton theory are discussed briefly, in order to highlight the application…
Some discussion of physical and geometrical interpretation of Einsteins theory of gravitation which is on basic of cosmology.
A special formula for the total mean curvature of an ovaloid is derived. This formula allows us to extend the notion of the mean curvature to the class of boundaries of strictly convex sets. Moreover, some integral formula for ovaloids is…
We describe a number of geometric contexts where categorification appears naturally: coherent sheaves, constructible sheaves and sheaves of modules over quantizations. In each case, we discuss how "index formulas" allow us to easily perform…
After a Hessian computation, we quickly prove the 3D simplex mean width conjecture using classical methods. Then, we generalize some components to $d$ dimensions.
In classical geometry, there is no such well-known and much-studied topic as the construction of conic sections (or briefly conics) from its five points. Its importance in many applications of mechanical engineering, civil engineering and…