Related papers: A property of the bidimensional sphere

We prove an extension of the theorem of Drinfeld, Grinberg and Kazhdan to arcs with arbitrary residue field. As an application we show that the embedding codimension is generically constant on each irreducible subset of the arc space which…

A (positive definite and non-classic integral) quadratic form is called strongly $s$-regular if it satisfies a strong regularity property on the number of representations of squares of integers. In this article, we prove that for any…

Let Delta be a random spherical triangle (meaning that vertices are independent and uniform on the unit sphere). A closed-form expression for the area density of Delta has been known since 1867; a complicated integral expression for the…

We introduce the notion of strongly orthogonal set relative to an element in the sense of Birkhoff-James in a normed linear space to find a necessary and sufficient condition for an element $ x $ of the unit sphere $ S_{X}$ to be an exposed…

It is well-known that if a curve is a geodesic line of the tangent (sphere) bundle with Sasaki metric of a locally symmetric Riemannian manifold then the projected curve has all its geodesic curvatures constant. In this paper we consider…

Let K denote an algebraically closed field with characteristic 0. Let V denote a vector space over K with finite positive dimension and let A,B denote a tridiagonal pair on V. We make an assumption about this pair. Let q denote a nonzero…

Given R\subset N, an (R,k)$-sphere is a k-regular map on the sphere whose faces have gonalities i\in R. The most interesting/useful are (geometric) fullerenes, i.e., (\{5,6\},3)$-spheres. Call \kappa_i=1 + \frac{i}{k} - \frac{i}{2} the…

We establish a lower bound for the surface area of a closed, convex hypersurface in Euclidean space in terms of its displacement under continuous maps. As a result, a hypothesized lower bound for the volume of a Riemannian $n$-sphere,…

We provide lower and upper bounds on the minimum size of a maximum stable set over graphs of flag spheres, as a function of the dimension of the sphere and the number of vertices. Further, we use stable sets to obtain an improved Lower…

If we label the vertices of a triangle with 1, 2 and 4, and the orthocentre with 7, then any of the four numbers 1, 2, 4, 7 is the nim-sum of the other three and is their orthocentre. Regard the triangle as an orthocentric quadrangle.…

We give a short and simple proof of Cauchy's surface area formula, which states that the average area of a projection of a convex body is equal to its surface area up to a multiplicative constant in the dimension.

The convex hull on three points in two dimensional euclidean space of three flat edges (trihedron) was studied. The Bohr-Sommerfeld quantization of the area of space is performed. It is shown that it reproduces exactly the equidistant…

In this paper, if prime $p\equiv 3\pmod 4$ is sufficiently large then we prove an upper bound on the number of occurences of any arbitrary pattern of quadratic residues and nonresidues of length $k$ as $k$ tends to $\lceil \log_2 p\rceil$.…

We study a volume/area preserving curvature flow of hypersurfaces that are convex by horospheres in the hyperbolic space, with velocity given by a generic positive, increasing function of the mean curvature, not necessarly homogeneous. For…

Let A \subseteq [1,..,N]^2 be a set of density at least 1/(log log log N)^c, where c some constant c>0. We prove that A contains a so-called right-angle triangle, i.e. a triple of the form {(k,m), (k+d,m), (k,m+d)}, where d>0.

Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that…

A regular (i.e., singularity-free) cycling cosmological model is advanced. In the model, there are only two constants: the gravitational constant (or the Planck time) and the cosmic period. The radius of the universe is a simple periodic…

A bounded subset of a normed linear space is said to be (diametrically) complete if it cannot be enlarged without increasing the diameter. A complete super set of a bounded set $K$ having the same diameter as $K$ is called a completion of…

A three-dimensional orthoscheme is defined as a tetrahedron whose base is a right-angled triangle and an edge joining the apex and a non-right-angled vertex is perpendicular to the base. A generalization, called complete orthoschemes, of…

Stabilizer-free $P_k$ virtual elements are constructed on polygonal and polyhedral meshes. Here the interpolating space is the space of continuous $P_k$ polynomials on a triangular-subdivision of each polygon, or a tetrahedral-subdivision…