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One dimensional metrical geometry may be developed in either an affine or projective setting over a general field using only algebraic ideas and quadratic forms. Some basic results of universal geometry are already present in this…
We consider higher-dimensional generalizations of the $\alpha$-Grushin plane, focusing on the problem of classification of geodesics that minimize length, also known as optimal synthesis. Solving Hamilton's equations on these spaces using…
We deal with a divisorial contraction in dimension 3 which contracts its exceptional divisor to a smooth point. We prove that any such contraction can be obtained by a suitable weighted blow-up.
In this note we sketch a proof of a fundamental conjecture, the codimension-three conjecture, for microdifferential holonomic systems with regular singularities. It states that any regular holonomic E-module extends beyond a…
We have derived an analytical formulation for estimating the volume of geometries enclosed by implicitly defined surfaces. The novelty of this work is due to two aspects. First we provide a general analytical formulation for all…
When testing for the mean vector in a high dimensional setting, it is generally assumed that the observations are independently and identically distributed. However if the data are dependent, the existing test procedures fail to preserve…
Motivated by applications in geomorphology, the aim of this paper is to extend Morse-Smale theory from smooth functions to the radial distance function (measured from an internal point), defining a convex polyhedron in 3-dimensional…
A well-known result in the study of convex polyhedra, due to Minkowski, is that a convex polyhedron is uniquely determined (up to translation) by the directions and areas of its faces. The theorem guarantees existence of the polyhedron…
Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.
We prove that under some assumptions on the mean curvature the set of umbilical points of an immersed surface in a $3$-dimensional space form has positive measure. In case of an immersed sphere our result can be seen as a generalization of…
We establish the second part of Milnor's conjecture on the volume of simplexes in hyperbolic and spherical spaces. A characterization of the closure of the space of the angle Gram matrices of simplexes is also obtained.
The problem of maximizing the average cross section through a point within a shape is introduced. This idea is extended into arbitrary dimensions. However, the average cross sectional volume cannot be maximized unless the cross sections…
Let $L_{a,b}$ be a line in the Euclidean plane with slope $a$ and intercept $b$. The dimension spectrum $\spec(L_{a,b})$ is the set of all effective dimensions of individual points on $L_{a,b}$. The dimension spectrum conjecture states…
We propose the Medial Skeletal Diagram, a novel skeletal representation that tackles the prevailing issues around skeleton sparsity and reconstruction accuracy in existing skeletal representations. Our approach augments the continuous…
We introduce a notion of $k$-convexity and explore polygons in the plane that have this property. Polygons which are \mbox{$k$-convex} can be triangulated with fast yet simple algorithms. However, recognizing them in general is a 3SUM-hard…
Structured-light three-dimensional (3D) imaging can achieve 3D shape of a stationary object via one or more pixelated array cameras with phase-shifting illumination. In order to extend 3D imaging to moving scenarios, we propose a 3D imaging…
We study the prime pair counting functions $\pi_{2k}(x),$ and their averages over $2k.$ We show that good results can be achieved with relatively little effort by considering averages. We prove an asymptotic relation for longer averages of…
We will prove the Brannan conjecture for particular values of the parameter. The basic tool of the study is an integral representation published in a recent work [3].
In 1999, K. Bezdek posed a conjecture stating that among all convex bodies in $\mathbb R^3$, ellipsoids and bodies of revolution are characterized by the fact that all their planar sections have an axis of reflection. We prove Bezdek's…
It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.