arXiv subjects

Ákos G. Horváth

Publications and source records attributed to Ákos G. Horváth.

35 records · Page 2Linked to original sources

arXiv2016

Relativity theory in time-space

The concept of time-space defined in an earlier paper of the author is a certain generalization of the so-called space-time. In this paper we introduce the concept of time-space manifolds. In the homogeneous case, a time-space manifold is a differentiable manifold with such tangent spaces which have a certain fixed time-space structure. We redefine the fundamental concepts of global relativity theory with respect to this general situation. We study the concepts of affine connection, parallel transport, curvature tensor and Einstein equation, respectively.

physics.gen-ph↗

arXiv2016

On the non-Paschian ordered planes

We give a non-Paschian plane based on the property of betweenness which cannot be derived from an ordering of the points of a line. In this model there is no possibility to define the congruence of segments but we can define angle, triangle and angle measure, respectively. With respect to our definitions the plane has an elliptic character, meaning that the sum of the angles of a triangle is greater than $π$.

math.MG↗

arXiv2016

Angle measures, general rotations, and roulettes in normed planes

In this paper a special group of bijective maps of a normed plane, called the group of general rotations, is introduced; it contains the isometry group as a subgroup. The concept of general rotations leads to the notion of flexible motions of the plane, and to the concept of Minkowskian roulettes. As a nice consequence of this new approach to motions the validity of a strong analogue to the Euler-Savary equations for Minkowskian roulettes is proved.

math.MG↗

arXiv2016

On the volume growth of the hyperbolic regular $n$-simplex

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the $n$-dimensional volume of a regular simplex and the $(n-1)$-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form is based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for $n=3$.

math.MG↗

arXiv2015

Semi-inner products and the concept of semi-polarity

The lack of an inner product structure in Banach spaces yields the motivation to introduce a semi-inner product with a more general axiom system, one missing the requirement for symmetry, unlike the one determing a Hilbert space. We use it on a finite dimensional real Banach space $(\X, \| \cdot\|)$ to define and investigate three concepts. First, we generalize that of \emph{antinorms}, already defined in Minkowski planes, for even dimensional spaces. Second, we introduce \emph{normality maps}, which in turn leads us to the study of \emph{semi-polarity}, a variant of the notion of polarity, which makes use of the underlying semi-inner product.

math.MG↗

arXiv2015

Isometries of Minkowski geometries

In this paper we review the known facts on isometries of Minkowski geometries and prove some new results on them. We give the normal forms of two special classes of operators and also characterize the isometry group of Minkowski $3$-spaces in which the unit sphere does not contain an ellipse.

math.MG↗

arXiv2014

On the icosahedron inequality of László Fejes-Tóth

In this paper we deal with the problem to find the maximal volume polyhedra with a prescribed property and inscribed in the unit sphere. We generalize those inequality (called by \emph{icosahedron inequality}) of L. Fejes-Tóth of which an interesting consequence the fact that regular icosahedron has maximal volume in the class of the polyhedra with twelve vertices and inscribed in the unit sphere. We give an upper bound for the volume of such star-shaped (with respect to the origin) triangular polyhedra which we know the maximal edge lengths of its faces, respectively. As a further consequence of the generalized inequality we prove the conjecture which states that the maximal volume polyhedron spanned by the vertices of two regular simplices with common centroid is the cube.

math.MG↗

arXiv2014

Hyperbolic plane geometry revisited

Using the method of C. Vörös, we establish results in hyperbolic plane geometry, related to triangles and circles. We present a model independent construction for Malfatti's problem and several trigonometric formulas for triangles.

math.MG↗

arXiv2014

On the hyperbolic triangle centers

Using the method of C. Vörös, we establish results on hyperbolic plane geometry, related to triangles. In this note we investigate the orthocenter, the concept of isogonal conjugate and some further center as of the symmedian of a triangle. We also investigate the role of the "Euler line" and the pseudo-centers of a triangle.

math.MG↗

arXiv2014

Maximum volume polytopes inscribed in the unit sphere

In this paper we investigate the problem of finding the maximum volume polytopes, inscribed in the unit sphere of the $d$-dimensional Euclidean space, with a given number of vertices. We solve this problem for polytopes with $d+2$ vertices in every dimension, and for polytopes with $d+3$ vertices in odd dimensions. For polytopes with $d+3$ vertices in even dimensions we give a partial solution.

math.MG↗

arXiv2014

Malfatti's problem on the hyperbolic plane

More than two centuries ago Malfatti (see \cite{malfatti}) raised and solved the following problem (the so-called Malfatti's construction problem):Construct three circles into a triangle so that each of them touches the two others from outside moreover touches two sides of the triangle too. It is an interesting fact that nobody investigated this problem on the hyperbolic plane, while the case of the sphere was solved simultaneously with the Euclidean case. In order to compensate this shortage we solve the following exercise: {\em Determine three cycles of the hyperbolic plane so that each of them touches the two others moreover touches two of three given cycles of the hyperbolic plane.}

math.MG↗

arXiv2013

On the volume of the convex hull of two convex bodies

In this note we examine the volume of the convex hull of two congruent copies of a convex body in Euclidean $n$-space, under some subsets of the isometry group of the space. We prove inequalities for this volume if the two bodies are translates, or reflected copies of each other about a common point or a hyperplane containing it. In particular, we give a proof of a related conjecture of Rogers and Shephard.

math.MG↗

arXiv2013

On an extremal problem connected with simplices

In this note we investigate the behavior of the volume that the convex hull of two congruent and intersecting simplices in Euclidean $n$-space can have. We prove some useful equalities and inequalities on this volume. For the regular simplex we determine the maximal possible volume for the case when the two simplices are related to each other via reflection at a hyperplane intersecting them.

math.MG↗

arXiv2012

Generalized Minkowski space with changing shape

In earlier papers we changed the concept of the inner product to a more general one, to the so-called Minkowski product. This product changes on the tangent space hence we could investigate a more general structure than a Riemannian manifold. Particularly interesting such a model when the negative direct component has dimension one and the model shows certain space-time character. We will discuss this case here. We give a deterministic and a non-deterministic (random) variant of a such a model. As we showed, the deterministic model can be defined also with a "shape function".

physics.gen-ph↗

arXiv2011

Conics in normed planes

We study the generalized analogues of conics for normed planes by using the following natural approach: It is well known that there are different metrical definitions of conics in the Euclidean plane. We investigate how these definitions extend to normed planes, and we show that in this more general framework these different definitions yield, in almost all cases, different classes of curves.

math.MG↗

arXiv2010

Premanifolds

The tangent hyperplanes of the "manifolds" of this paper equipped a so-called Minkowski product. It is neither symmetric nor bilinear. We give a method to handing such an object as a locally hypersurface of a generalized space-time model and define the main tools of its differential geometry: its fundamental forms, its curvatures and so on. In the case, when the fixed space-time component of the embedding structure is a continuously differentiable semi-inner product space, we get a natural generalization of some important semi-Riemann manifolds as the hyperbolic space, the de Sitter sphere and the light cone of a Minkowski-Lorenz space, respectively.

math.DG↗

arXiv2009

Projection pencil of quadrics and Ivory theorem

We rewrite the property of confocality with respect to a pseudo-Euclidean space. Our observation is that the generalized definition of confocality in $\cite{stachel}$ does not give back to the original definition of confocality of Euclidean conics. The "confocality property" of $\cite{stachel}$ can be get from our definition in the case when the projection transformation is regular, it is the identity transformation of the space. Our examination concentrate to the singular cases, too because they are playing important role in the investigations of essential examples.

math.MG↗