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Let $\mathbb{R}$ be the field of real numbers. We consider the problem of computing the real isolated points of a real algebraic set in $\mathbb{R}^n$ given as the vanishing set of a polynomial system. This problem plays an important role…
Large language models (LLMs) may generate text that lacks consistency with human knowledge, leading to factual inaccuracies or \textit{hallucination}. Existing research for evaluating the factuality of LLMs involves extracting fact claims…
3D Shape representation has substantial effects on 3D shape reconstruction. Primitive-based representations approximate a 3D shape mainly by a set of simple implicit primitives, but the low geometrical complexity of the primitives limits…
One of the general problems in algebraic geometry is to determine algorithmically whether or not a given geometric object, defined by explicit polynomial equations (e.g. a curve or a surface), satisfies a given property (e.g. has…
The hamiltonian circuit polytope is the convex hull of feasible solutions for the circuit constraint, which provides a succinct formulation of the traveling salesman and other sequencing problems. We study the polytope by establishing its…
We study the rational approximation properties of special manifolds defined by a set of polynomials with rational coefficients. Mostly we will assume the case of all polynomials to depend on only one variable. In this case the manifold can…
The action of ring automorphisms of the polynomial ring in two variables over the real numbers on real plane curves is considered. The orbits containing degree-three polynomials are computed, with one representative per orbit being…
With the advantage of high mobility, Unmanned Aerial Vehicles (UAVs) are used to fuel numerous important applications in computer vision, delivering more efficiency and convenience than surveillance cameras with fixed camera angle, scale…
In this note, we initiate a study of the finite-dimensional representation theory of a class of algebras that correspond to noncommutative deformations of compact surfaces of arbitrary genus. Low dimensional representations are investigated…
The earlier approach is used for description of qubits and geometric phase parameters, the things critical in the area of topological quantum computing. The used tool, Geometric (Clifford) Algebra is the most convenient formalism for that…
We present a method for computing projective isomorphisms between rational surfaces that are given in terms of their parametrizations. The main idea is to reduce the computation of such projective isomorphisms to five base cases by…
A parameterized surface can be represented as a projection from a certain toric surface. This generalizes the classical homogeneous and bihomogeneous parameterizations. We extend to the toric case two methods for computing the implicit…
This survey gives a brief overview of theoretically and practically relevant algorithms to compute geodesic paths and distances on three-dimensional surfaces. The survey focuses on polyhedral three-dimensional surfaces.
The numbers of $\mathbb{F}_q$-points of nonsingular hypersurfaces of a fixed degree in an odd-dimensional projective space are investigated, and an upper bound for them is given. Also we give the complete list of nonsingular hypersurfaces…
This paper presents a method of capturing objects appearances from its environment and it also describes how to recognize unknown appearances creating an eigenspace. This representation and recognition can be done automatically taking…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
Open World Object Detection (OWOD) is a challenging computer vision problem that requires detecting unknown objects and gradually learning the identified unknown classes. However, it cannot distinguish unknown instances as multiple unknown…
In this work we present a new polynomial map $f:=(f_1,f_2):{\mathbb R}^2\to{\mathbb R}^2$ whose image is the open quadrant $\{x>0,y>0\}\subset{\mathbb R}^2$. The proof of this fact involves arguments of topological nature that avoid hard…
In this paper geometry is studied with a novel approach. Every geometrical object is defined as a symbol which satisfies some properties. These symbols are then coded into a class of numbers which are named here as many dots numbers (MDN).…
Student appreciation of a function is enhanced by understanding the graphical representation of that function. From the real graph of a polynomial, students can identify real-valued solutions to polynomial equations that correspond to the…