Rational Coordinates Research Articles

A dimensional mass transference principle from ball to rectangles for projections of Gibbs measures and applications

Mass transference principle are tools designed to provide estimates for the Hausdorff dimension of points approximable at a certain rate by a specific sequence of points (for instance rationals). Usually, such results are established provided that ambiant space is equipped with an Ahlfors regular measure (see [7,28] for instance). In the case of balls, Barral and Seuret established that mass transference principles still holds provided that the ambiant measure is a Gibbs measure or a self-similar measure satisfying the open set condition ([3]). In this article, we establish a mass transference principle “from ball to rectangle” assuming that the ambiant measure is the projection of Gibbs measure on the dyadic grid. Such measures are not Ahlfors regular in general and this result extends in particular the mass transference principle from ball to rectangle in the case of the Lebesgue measure, established in [28], and partially the result of Barral and Seuret [3]. As an application, given b∈N, we estimate the dimension of weighted approximation of points of [0,1]2 approximable by points with rational coordinates whose expansion in base b has prescribed frequencies. These results are new and surprising in the sens that, although the Lebesgue measure “scales” naturally rectangles, this needs not be the case for projection of Gibbs measures.

Algorithm of the electronic sign-code signature on the basis of the composition of existing complexities

The article developed a new algorithm for electronic digital signature in the composition of existing difficulties: discrete logarithm in a finite field, addition of points with rational coordinates of the elliptic curve. Based on a combination of the complexities of a discrete logarithm on a finite field with a large number characteristic, decomposition of a sufficiently large odd number into prime factors, and addition of points of an elliptic curve, an electronic digital signature algorithm is developed for generating. The generally recognized model of electronic sign-code signature covers three processes: generation of digital signature keys; ESS formation; confirmation of authenticity of electronic sign-code signature. Firmness offered AESS is based on several complexities: calculations of the discrete logarithming in a finite field, decisions of the task of expansion of enough big odd number on simple factors, executions of operation of addition of points of the elliptic curve set in a finite field. It considerably raises crypto algorithm strength.

From positional representation of numbers to positional representation of vectors

To represent real m-dimensional vectors, a positional vector system given by a non-singular matrix M ∈ ℤm×m and a digit set Ɗ ⊂ ℤm is used. If m = 1, the system coincides with the well known numeration system used to represent real numbers. We study some properties of the vector systems which are transformable from the case m = 1 to higher dimensions. We focus on an algorithm for parallel addition and on systems allowing an eventually periodic representation of vectors with rational coordinates.

The equivariant Ehrhart theory of polytopes with order-two symmetries

We study the equivariant Ehrhart theory of families of polytopes that are invariant under a non-trivial action of the group with order two. We study families of polytopes whose equivariant H ∗ H^\ast -polynomial both succeed and fail to be effective, in particular, the symmetric edge polytopes of cycle graphs and the rational cross-polytope. The latter provides a counterexample to the effectiveness conjecture if the requirement that the vertices of the polytope have integral coordinates is loosened to allow rational coordinates. Moreover, we exhibit such a counterexample whose Ehrhart function has period one and coincides with the Ehrhart function of a lattice polytope.

Pythagorean Triples, Complex Numbers, Abelian Groups and Prime Numbers

Pythagorean triples can be represented by points of the unit circle with rational coordinates. These points form an abelian group, and we describe its structure. This structural description yields an enumeration of the normalized Pythagorean triples with a given hypotenuse. It also gives rise to an effective method for producing all such triples, and this effective method seems to be new. This paper is intended for the general mathematical audience, including undergraduate mathematics students. Therefore it contains background material, explanations, examples and drawings; and the proofs are complete.

The Art Gallery Problem is ∃ℝ-complete

The Art Gallery Problem (AGP) is a classic problem in computational geometry, introduced in 1973 by Victor Klee. Given a simple polygon 풫 and an integer k , the goal is to decide if there exists a set G of k guards within 풫 such that every point p ∈ 풫 is seen by at least one guard g ∈ G . Each guard corresponds to a point in the polygon 풫, and we say that a guard g sees a point p if the line segment pg is contained in 풫. We prove that the AGP is ∃ ℝ-complete, implying that (1) any system of polynomial equations over the real numbers can be encoded as an instance of the AGP, and (2) the AGP is not in the complexity class NP unless NP = ∃ ℝ. As a corollary of our construction, we prove that for any real algebraic number α, there is an instance of the AGP where one of the coordinates of the guards equals α in any guard set of minimum cardinality. That rules out many natural geometric approaches to the problem, as it shows that any approach based on constructing a finite set of candidate points for placing guards has to include points with coordinates being roots of polynomials with arbitrary degree. As an illustration of our techniques, we show that for every compact semi-algebraic set S ⊆ [0, 1] 2 , there exists a polygon with corners at rational coordinates such that for every p ∈ [0, 1] 2 , there is a set of guards of minimum cardinality containing p if and only if p ∈ S . In the ∃ ℝ-hardness proof for the AGP, we introduce a new ∃ ℝ-complete problem ETR-INV. We believe that this problem is of independent interest, as it has already been used to obtain ∃ ℝ-hardness proofs for other problems.

The solvability cardinal of the class of polynomials

Let G be an Abelian group, and let {{mathbb {C}}}^G denote the set of complex valued functions defined on G. A map D: {{mathbb {C}}}^G rightarrow {{mathbb {C}}}^G is a difference operator, if there are complex numbers a_i and elements b_i in G(i=1,ldots , n) such that (Df)(x)=sum _{i=1}^n a_i f(x+b_i) for every fin {{mathbb {C}}}^G and xin G. By a system of difference equations we mean a set of equations { D_i f=g_i : iin I}, where I is an arbitrary set of indices, D_i is a difference operator and g_i in {{mathbb {C}}}^G is a given function for every iin I, and f is the unknown function. The solvability cardinal mathrm{sc} ,({{mathcal {F}}}) of a class of functions {{mathcal {F}}} subset {{mathbb {C}}}^G is the smallest cardinal number kappa with the following property: whenever S is a system of difference equations on G such that each subsystem of S of cardinality <kappa has a solution in {{mathcal {F}}}, then S itself has a solution in {{mathcal {F}}}. The behaviour of mathrm{sc} ,({{mathcal {F}}}) is rather erratic, even for classes of functions defined on {{mathbb {R}}}. For example, mathrm{sc} ,({{mathbb {C}}}[x])=3, but mathrm{sc} ,({mathcal {TP}}) =omega _1, where {mathcal {TP}} is the set of trigonometric polynomials; mathrm{sc} ,({{mathbb {C}}}^{{mathbb {R}}})=omega , but mathrm{sc} ,({mathcal {DF}}) =(2^omega )^+, where {mathcal {DF}} is the set of functions having the Darboux property. Our aim is to determine or to estimate the solvability cardinal of the class of polynomials defined on {{{mathbb {R}}}}^n, on normed linear spaces and, in general, on topological Abelian groups. Let {{mathcal {P}}}_G denote the class of polynomials defined on the group G. After presenting some general estimates we prove that mathrm{sc} ,({{mathbb {C}}}[x_1 ,ldots ,x_n ])=omega if 2le n<infty , and mathrm{sc} ,({{mathcal {P}}}_X)=omega _1 if X is a normed linear space of infinite dimension. For discrete Abelian groups we show that mathrm{sc} ,({{mathcal {P}}}_G)=3 if r_0 (G)le 1, mathrm{sc} ,({{mathcal {P}}}_G)=omega if 2le r_0 (G)<infty , and mathrm{sc} ,({{mathcal {P}}}_G)ge omega _1 if r_0 (G) is infinite, where r_0 (G) denotes the torsion free rank of G. The solvability of systems of difference equations is closely connected to the existence of projections of function classes commuting with translations (see Theorem 7.1). As an application we construct a projection from {{mathbb {C}}}^{{{{mathbb {R}}}}^n} onto {{mathbb {C}}}[x_1 ,ldots ,x_n ] commuting with translations by vectors having rational coordinates (Theorem 7.4).

Algorithm Of The Electronic Digital Subscript On The Basis Of The Composition Of Computing Complexities

In article the new algorithm of a digital signature in composition of the existing difficulties is developed: discrete logarithming in a final field, decomposition of rather large natural number on simple multipliers, additions of points with rational coordinates of an elliptic curve. On the basis of a combination of difficulties of discrete logarithming on a final field with the characteristic of large number, decomposition of rather large odd number on simple multipliers and additions of points of an elliptic curve develops algorithm of a digital signature for formation. The conventional scheme (model) of a digital signature covers three processes: generation of keys of EDS; formation of EDS; check (authenticity confirmation) of EDS. The idea of a design of the offered algorithm allows modifying and increasing crypto stability with addition to other computing difficulties. It is intended for use in systems of information processing of different function during forming and confirmation of authenticity of digital signature.

The complete enumeration of 4-polytopes and 3-spheres with nine vertices

We describe an algorithm to enumerate polytopes. This algorithm is then implemented to give a complete classification of combinatorial spheres of dimension 3 with 9 vertices and decide polytopality of those spheres. In order to decide polytopality, we generate polytopes by adding suitable points to polytopes with less than 9 vertices and therefore realize as many as possible of the combinatorial spheres as polytopes. For the rest, we prove non-realizability with techniques from oriented matroid theory. This yields a complete enumeration of all combinatorial types of 4-dimensional polytopes with 9 vertices. It is shown that all of those combinatorial types are rational: They can be realized with rational coordinates. We find 316 014 combinatorial spheres on 9 vertices. Of those, 274 148 can be realized as the boundary complex of a four-dimensional polytope and the remaining 41 866 are non-polytopal.

Fast tetrahedral meshing in the wild

We propose a new tetrahedral meshing method, fTetWild, to convert triangle soups into high-quality tetrahedral meshes. Our method builds on the TetWild algorithm, replacing the rational triangle insertion with a new incremental approach to construct and optimize the output mesh, interleaving triangle insertion and mesh optimization. Our approach makes it possible to maintain a valid floating-point tetrahedral mesh at all algorithmic stages, eliminating the need for costly constructions with rational numbers used by TetWild, while maintaining full robustness and similar output quality. This allows us to improve on TetWild in two ways. First, our algorithm is significantly faster, with running time comparable to less robust Delaunay-based tetrahedralization algorithms. Second, our algorithm is guaranteed to produce a valid tetrahedral mesh with floating-point vertex coordinates, while TetWild produces a valid mesh with rational coordinates which is not guaranteed to be valid after floating-point conversion. As a trade-off, our algorithm no longer guarantees that all input triangles are present in the output mesh, but in practice, as confirmed by our tests on the Thingi10k dataset, the algorithm always succeeds in inserting all input triangles.

On selection of optimal stop position and spatial orientation of mobile robot

The paper analyses the problem of finding optimal coordinates of a mobile robot equipped with a manipulator. An optimization criterion is proposed and validated, and a method for calculating most rational coordinates of the robot is described.

Exploring the Isoptics of Fermat Curves in the Affine Plane Using DGS and CAS

Isoptic curves of plane curves are a live domain of study, mostly for closed, smooth, strictly convex curves. A technology-rich environment allows for a two-fold development: dynamical geometry systems enable us to perform experiments and to derive conjectures, and computer algebra systems (CAS) are the appropriate environments for an algebraic approach for determining isoptics, with its Grobner bases packages, amongst others. Closed Fermat curves in the affine plane present a specific problem: the variables of the involved polynomials represent the coordinates of points on the Fermat curve, which avoid the dense set of points with two rational coordinates (excepted 4 “trivial” points). Therefore, automated computation has to be performed over the field $$\mathbb {R}$$ of the real numbers, with which the Grobner bases packages do not work. Instead, other packages implemented in CASs have to be used. First, we present a study of the orthoptics of closed Fermat curves of even order. Then we proceed to an algebraic study of these curves using a CAS. The generalization to angles other than $$90^\circ $$ is performed afterwards, with an algebraic approach using support functions, then using numerical methods.

On Parametrization of the Symplectic Quotient of the Cartesian Product of Coadjoint Orbits of the Complex General Linear Group with Respect to its Diagonal Action

We consider the problem of coordinatizing the manifold constructed via the Marsden–Weinstein quotient. Rational canonical coordinates on the symplectic reduction with respect to the diagonal action of the general linear group on the Cartesian product of coadjoint orbits in the case of the complex general linear group are constructed on an algebraically open subset of the quotient space. The method is based on an iterative process of constructing projection-flag coordinates, and works if the matrices constituting the orbits have a sufficiently rich set of invariant subspaces.

A universal Banach space with a $K$-unconditional basis

‎For a constant $K\geq 1$‎, ‎let $\mathfrak{B}_K$ be the class of pairs $(X,(\mathbf e_n)_{n\in\omega})$ consisting of a Banach space $X$ and an unconditional Schauder basis $(\mathbf e_n)_{n\in\omega}$ for $X$‎, ‎having the unconditional basic constant $K_u\le K$‎. ‎Such pairs are called $K$-based Banach spaces‎. ‎A based Banach space $X$ is rational if the unit ball of any finite-dimensional subspace spanned by finitely many basic vectors is a polyhedron whose vertices have rational coordinates in the Schauder basis of $X$‎. Using the technique of Fraïssé theory‎, ‎we construct a rational $K$-based Banach space $\big(\mathbb U_K,(\mathbf e_n)_{n\in\omega}\big)$ which is $\mathfrak{RI}_K$-universal in the sense that each basis preserving isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional rational $K$-based Banach space $A$ extends to a basis preserving isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$‎. ‎We also prove that the $K$-based Banach space $\mathbb U_K$ is almost $\mathfrak{FI}_1$-universal in the sense that any base preserving‎ ‎$\varepsilon$-isometry $f:\Lambda\to\mathbb U_K$ defined on a based subspace $\Lambda$ of a finite-dimensional $1$-based Banach space $A$ extends to a base preserving $\varepsilon$-isometry $\bar f:A\to\mathbb U_K$ of the based Banach space $A$‎. ‎On the other hand‎, ‎we show that no almost $\mathfrak{FI}_K$-universal based Banach space exists for $K>1$‎.‎ The Banach space $\mathbb U_K$ is isomorphic to the complementably universal Banach space for the class of Banach spaces with an unconditional Schauder basis‎, ‎constructed by Pełczyński in 1969‎.

Conics from symmetric Pythagorean triple preserving matrices

The aim of this paper is to introduce and study the class of conics provided by symmetric Pythagorean triple preserving matrices. This class depends on three real parameters and various relationships between these parameters give special subclasses. A symmetric matrix of Barning and its associated hyperbola are carefully analyzed through a pair of points of rational coordinates. We transform and extend this latter hyperbola to a class of hyperbolas containing integral points (k; k + 3), (k + 3; k). A complex approach is also included.

THE CHOICE OF AN ELLIPTIC CURVE AND THE BASE POINT IN THE DEVELOPMENT OF AN ALGORITHM FOR ADDING ITS POINTS WITH RATIONAL COORDINATES ON A FINITE FIELD

In article the problems of solving the tasks of determination and calculation the parameters of an elliptic curve (EC) for the correct realization of asymmetric cryptoalgorithms are probed.Asymmetric cryptographic algorithms are constructed on the basis of the decomposition of a sufficiently large natural number into prime factors, of discrete logarithm on a finite field with a sufficiently large characteristic, of addition of points with rational coordinates EC on a finite field.The algorithms on the EC require the determination of the coefficients of the curve itself, the operation on the finite field by the characteristic, preferably by a prime number, the finding of a base point by rational coordinates by the order of a prime number, complex calculations related to the specific nature of the algorithm model.A condition is given for the choice of the coefficients by the sign of the discriminant value of the cubic equation to ensure the efficiency of algorithms application on the EC. Being used by Vieta formulas, for roots of polynoms, it is given a method of a choice of coefficients. The interval of a choice of a basic point is specified.The formulas of the tangent to the base point and the location of the point of intersection of the tangent with the EC are determined.A recurrence formula is obtained for the addition of a base point with other points of EC with rational coordinates, which is a generalized formula for the addition of any points of EC with rational coordinates.

Periods of Ehrhart Coefficients of Rational Polytopes

Let $\mathcal{P} \subset \mathbb{R}^{n}$ be a polytope whose vertices have rational coordinates. By a seminal result of E. Ehrhart, the number of integer lattice points in the $k$th dilate of $\mathcal{P}$ ($k$ a positive integer) is a quasi-polynomial function of $k$ — that is, a "polynomial" in which the coefficients are themselves periodic functions of $k$. It is an open problem to determine which quasi-polynomials are the Ehrhart quasi-polynomials of rational polytopes. As partial progress on this problem, we construct families of polytopes in which the periods of the coefficient functions take on various prescribed values.

The discrete Pompeiu problem on the plane

We say that a finite subset E of the Euclidean plane $$\mathbb {R}^2$$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $$f:\mathbb {R}^2\rightarrow \mathbb {C}$$ is such that the sum of the values of f on any congruent (similar) copy of E is zero, then f is identically zero. We show that every parallelogram and every quadrangle with rational coordinates has the discrete Pompeiu property with respect to isometries. We also present a family of quadrangles depending on a continuous parameter having the same property. We investigate the weighted version of the discrete Pompeiu property as well, and show that every finite linear set with commensurable distances has the weighted discrete Pompeiu property with respect to isometries, and every finite set has the weighted discrete Pompeiu property with respect to similarities.

Automatic estimation of bias parameters of RPC model and ground coordinates from multi-directional satellite images

Advanced Land Observing Satellite (ALOS)/Panchromatic Remote-sensing Instrument for Stereo Mapping (PRISM) measured triplet images at forward, nadir and backward view directions, and the digital surface model (DSM) is generated from set of triplet images. Existing methods generate multiple DSMs from individual triplet images, and eliminate the effects of outliers by taking average. The proposed method in this paper solves multiple observation equations from all triplet images, and simultaneously determines bias parameters contaminated in the rational polynomial coefficient (RPC) model and ground coordinates. The experimental results showed that errors of the coordinates estimated by using multi-temporal triplet images were almost as small as those estimated with manually determined bias parameters in case of 4 or more sets of triplet images. As a result, we conclude that the proposed method is effective for stably generating accurate DSMs from multi-temporal triplet images.

Rational hyperbolic triangles and a quartic model of elliptic curves

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle.