Related papers: Simplifying the axiomatization for the order affin…
Adomian decomposition method is used for solving the seventh order boundary value problems. The approximate solutions of the problems are calculated in the form of a rapid convergent series and not at grid points. Two numerical examples…
In this paper, we extend the classical arithmetic defined over the set of natural numbers N, to the set of all finite directed connected multigraphs having a pair of distinct distinguished vertices. Specifically, we introduce a model F on…
We consider the problem of decomposing the edges of a digraph into as few paths as possible. A natural lower bound for the number of paths in any path decomposition of a digraph $D$ is $\frac{1}{2}\sum_{v\in V(D)}|d^+(v)-d^-(v)|$; any…
Constructive-deductive method for plane Euclidean geometry is proposed and formalized within Coq Proof Assistant. This method includes both postulates that describe elementary constructions by idealized geometric tools (pencil, straightedge…
This paper introduces ordered skew fields that result from the construction of a skew field over an ordered line in a Desargues affine plane. A special case of a finite ordered skew field in the construction of a skew field over an ordered…
A classical result of von Staudt states that if eight planes osculate a twisted cubic curve and we divide them into two groups of four, then the eight vertices of the corresponding tetrahedra lie on a twisted cubic curve. In the current…
Finite-order invariants of knots in arbitrary 3-manifolds (including non-orientable ones) are constructed and studied by methods of the topology of discriminant sets. Obstructions to the integrability of admissible weight systems to…
We study natural additional structures on real algebraic surfaces with trivial first homology mod 2 of the complexification. If the set of real points realizes the zero of the second homology mod 2 of the complexification, then the set of…
A classical theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to…
In this paper we detail a number of properties of the affine line of a derivator, including a number of morphisms between $\mathbb{D}$ and $\mathbb{A}^1_{\mathbb{D}}$, a monoidal structure on $\mathbb{A}^1_{\mathbb{D}}$ if $\mathbb{D}$ is…
We propose a simple, geometrically-motivated construction of smooth random paths in the plane. The construction is such that, with probability one, the paths have finite curvature everywhere (and the realizations are visually pleasing when…
We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with…
Based on results of Harding, Heunen, Lindenhovius and Navara, (2019), we give a connection between the category of AW*-algebras and their normal Jordan homomorphisms and a category COG of orthogemetries, which are structures that are…
We establish a fundamental theorem of orders (FTO) which allows us to express all orders uniquely as an intersection of `irreducible orders' along which the index and the conductor distributes multiplicatively. We define a subclass of…
We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call…
Axiomatizing mathematical structures is a goal of Mathematical Logic. Axiomatizability of the theories of some structures have turned out to be quite difficult and challenging, and some remain open. However axiomatization of some…
Automated theorem proving in Euclidean geometry, particularly for International Mathematical Olympiad (IMO) level problems, remains a major challenge and an important research focus in Artificial Intelligence. In this paper, we present a…
Convex geometries (Edelman and Jamison, 1985) are finite combinatorial structures dual to union-closed antimatroids or learning spaces. We define an operation of resolution for convex geometries, which replaces each element of a base convex…
This work is devoted to study orientation theory in arithmetic geometric within the motivic homotopy theory of Morel and Voevodsky. The main tool is a formulation of the absolute purity property for an \emph{arithmetic cohomology theory},…
A geometric first-order axiomatization of differentially closed fields of characteristic zero with several commuting derivations, in the spirit of Pierce-Pillay, is formulated in terms of a relative notion of prolongation for Kolchin-closed…