Related papers: The Full Pythagorean Theorem
The classical honeycomb conjecture asserts that any partition of the plane into regions of equal area has perimeter at least that of the regular hexagonal honeycomb tiling. Pappus discusses this problem in his preface to Book V. This paper…
We begin with recalling the correspond theorem of induced modules and global sections of vector bundles. After that, we give a generalization of this theorem. Finally, we apply the result to branching laws, and give some concrete examples.
Motivated by the analysis and geometry of metric-measure structures in infinite dimensions, we study the category of extended metric-topological spaces, along with many of its distinguished subcategories (such as the one of compact spaces).…
Since $n$-dimensional $\lambda$-hypersurfaces in the Euclidean space $\mathbb {R}^{n+1}$ are critical points of the weighted area functional for the weighted volume-preserving variations, in this paper, we study the rigidity properties of…
A special theorem related to the Fagnano's problem is proved and an example of the theorem is shown in a golden rectangle.
Given a trapezoid dissected into triangles, the area of any triangle determined by either diagonal of the trapezoid is integral over the ring generated by the areas of the triangles in the dissection. Given a parallelogram dissected into…
In this paper, we give a detailed account of Goldfeld's proof of Siegel's theorem. Particularly, we present complete proofs of the nontrivial assumptions made in his paper.
Here I present several theorems about trapezoids tilings. The first one is related to trapezoids with rational base relation, the other ones are related to those with base relation from quadratic number field.
In this work, we prove a synthetic splitting theorem for globally hyperbolic Lorentzian length spaces with global non-negative timelike curvature containing a complete timelike line. Just like in the case of smooth spacetimes, we construct…
Starting with univariate polynomial interpolation we arrive to a natural generalization of fundamental theorem of algebra for certain systems of multivariate algebraic equations.
We give a short proof of a strengthening of the Maximal Ergodic Theorem which also immediately yields the Pointwise Ergodic Theorem.
The initial techniques developed in Euclid's Elements, well before the use of the parallel postulate, are reexamined in order to clarify even the most obscure details, particularly those related to equality, superposition and angle…
In this note we give a detailed proof of a theorem of Aubin.
We introduce a new approach for the study of the Problem of Iterates using the theory on general ultradifferentiable structures developed in the last years. Our framework generalizes many of the previous settings including the Gevrey case…
The purpose of this note is to rephrase Speyer's elegant topological proof for Kasteleyn's Theorem in a simple graph theoretical manner.
In the paper we consider an extension of Mobius-Pompeiu theorem of the elementary geometry over metric spaces. We specially take into consideration Ptolemaic metric spaces.
We give a geometric proof of the Routh's theorem for tetrahedra.
We prove a version of the fundamental theorems of Morse Theory in the setting of finite spaces or partially ordered sets. By using these results we extend Forman's discrete Morse theory to more general cell complexes and derive the…
The leading idea of the paper is to treat the theorem of Wigner with methods inspired by geometry. The exercise mentionned in the title has two functions: On the one hand it can serve as a pedagogical text in order to make the reader…
Complete complex parabolic geometries (including projective connections and conformal connections) are flat and homogeneous. This is the first global theorem on parabolic geometries.