Article
Vladimir school of number theory was studied quasiperiodic tilings for a long time. The uniform distribution problem of fractional parts on the torus come from here. It is important to find exact estimates of the remainder for this distribution. The paper is devoted to the important problem of number theory: bounded remainder sets. Relevance of the problem caused by the transition from the classical numerical and functional arithmetic structures to nonlinear arithmetic structures. Dynamical systems on bounded remainder sets generate balanced words,similar to words Sturmian and Rauzy words. Balanced words are important, for dynamical systems, coding theory,theory of communications and optimization problems, theory of languages and linguistics, recognition theory, statistical physics, etc. The purpose of our research is construction of multidimensional bounded remainder sets and finding exact estimates of the remainder for this sets. The solution to this problem we start from two-dimensional case. We construct three classes of three-parameter two-dimensional bounded remainder sets. For their construction, we use hexagonal toric development. Now we know bounded remainder intervals, obtained by Hecke, and two-dimensional bounded remainder sets. There is the question: can we construct a new multi-dimensional sets using known sets? We construct four classes of four-parameter threedimensional bounded remainder sets. We used for this the multiplication of toric developments. By multiplication of Hecke’s intervals and two-dimensional hexagonal developments we obtain three-dimensional hexagonal Fedorov’s prisms-developments. For all described sets we give exact estimates of the remainder and prove generalization of Hecke’s theorem to the multidimensional case. Also we obtain average values of the remainders, and fined sets with minimal value of the remainder. This paper is an expository of the author’s main results on bounded remainder sets.
The problem of description the structure of the discriminant set of a real polynomial often occurs in solving various applied problems, for example, for describing a set of stability of stationary points of multiparameter systems, for computing the normal form of a Hamiltonian system in vicinity of equilibrium in the case of multiple frequencies. This paper considers the structure of the discriminant set of a polynomial with real coefficients. There are two approaches to its study. The first approach is based on the study of zeroes of ideals formed by the set of subdiscriminants of the original polynomial. Different ways of computing subdiscriminants are given. There is proposed to investigate the singular points of the discriminant set in the second approach. By the methods of computer algebra it is shown that for small values of the degree of the original polynomial, both approaches are equivalent, but the first one is preferred because of smaller ideals. Proposed constructive algorithm for obtaining polynomial parameterization of the discriminant set in the space of coefficients of the polynomial. From the applied point of view the most interesting is the description of the components of codimension 1 of the discriminant set. It is this component divides the space of the coefficients into the domains with the same structure of the roots of the polynomial. The set of components of different dimensions of the discriminant set has a hierarchical structure. Each component of higher dimensions can be considered as some kind of tangent developable surface which is formed by linear varieties of respective dimension. The role of directrix of this component performs a variety of dimension one less than that on which the original polynomial has only multiple zero and the remaining zeroes are simple. Starting with a one-dimensional algebraic variety of dimension 1 on which the original polynomial has the unique zero of maximal multiplicity, in the next step of the algorithm we obtain the description of the variety on which the polynomial has a pair of zeroes: one simple and another multiple. Repeating sequentially the steps of the algorithm, the resulting parametric representation of components of codimension 1 of the discriminant set can be obtained. Examples of the discriminant set of a cubic and quartic polynomials are considered.
1. Simple generalization. Let three homogeneous real linear forms be given in a three-dimensional real space. Their moduli give a mapping of the space into another space. In the second space, we consider the convex hull of images of all integer points of the first space except its origin. This convex hull is called the modular polyhedron. The best integer approximations to the root subspaces of these forms are given by the integer points whose images lie on the boundary of the modular polyhedron. For the concret three linear forms, any part of the boundary of the modular polyhedron can be computed by means of any standard program for computation of a convex hull. The algorithm gives the best approximations, and it is periodic for cubic irrationalities with positive discriminant. It also allows to understand why matrix algorithms proposed by Euler, Jacobi, Dirichlet, Hermite, Poincare, Hurwitz, Brun, Guting and others are not universal: proper algorithm is composed from several different matrix algorithms. 2. Universal generalization. Let l linear forms and k quadratic forms (n = l + 2k) be given in the n-dimensional real space Rn. Absolute values of the forms define a map of the space Rn into the positive orthant S of the mdimensional real space Rm, where m = l + k. Here the integer lattice Zn in Rn is mapped into a set Z in S. The closure of the convex hull H of the set Z\0 is a polyhedral set. Integer points from Rn, which are mapped in the boundary ∂H of the polyhedron H, give the best Diophantine approximations to root subspaces of all given forms. In the algebraic case, when the given forms are connected with roots of a polynomial of degree n, we prove that the polyhedron H has m − 1 independent periods. It is a generalization of the Lagrange Theorem, that continued fractions of a square irrationality is periodic. For the certain set of the m forms, any part of the boundary ∂H of the polyhedron H can be computed by a program for computing convex hulls. 3. Main achievement. Best Diophantine approximations can be computed by a global algorithm using a standard program for computing convex hulls, instead of step-by-step computations as in the continued fraction algorithm. It gives a solution of the problem, that majority of main mathematicians of the XIX century tried to solve.
The classical theorem of M. Lazar (see [1]) on the structure of the ring of coefficients of the universal formal group is a key result of the theory of onedimensional formal groups. The discovery of the formal group of geometric cobordisms ([2], [3]) and D. Quillen’s theorem ([4]) that it can be identified with the universal formal group allowed to introduce the theory of formal groups in the apparatus of algebraic topology, including the apparatus of the theory of Hirzebruch genera. Due to this there has been a widely-known fundamental mutual penetration of methods and results of algebraic topology, (see [5]), algebraic geometry, the theory of functional equations and mathematical physics. Important applications in algebraic topology found results of the theory of elliptic functions and Baker–Akhiezer functions, which play a fundamental role in the modern theory of integrable systems. The construction of universal formal groups of given form, with exponents given by these functions, became actual. Known results in this direction use both classic and recently obtained addition theorems, that determine the form of formal groups. In this paper we solved a long standing problem: we have found the form of universal formal group the exponent of which is the elliptic function of level 3. We have obtained results on the coefficient ring of this group and described its relationship with known universal formal groups.
In this paper the problems of construction and description of cones and polyhedra of finite quasi-metrics are considered. These objects are asymmetrical analogs of classical finite metrics. The introduction presents the historical background and examples of applications of metrics and quasi-metrics. In particular, the questions connected with maximum cut problem are represented. In the first section definitions of finite metrics and semi-metrics are given, and also their major special cases are considered: cuts, muluticuts and hypersemimetrics. Cones and polyhedrons of the specified objects are constructed; their properties are investigated. Connections of the cut cone with metric l1-spaces are indicated. The special attention is paid to symmetries of the constructed cones which consist of permutations and so-called switchings; transformation of a switching serves the basis for a choice of the inequalities defining the corresponding polyhedron. In the second section finite quasi-metrics and quasi-semimetrics are considered. They are asymmetrical analogs of the usual finite metrics and semimetrics. Definition of the oriented cuts and oriented multicuts are given: they are the most important special cases of the quasi-semimetrics. Concept of weightable quasi-metrics and related to them partial metrics is introduced. Cones and polyhedrons of these objects are constructed; their properties are investigated. Connections of the oriented cut cone with quasi-metric l1-space are considered. The special attention is paid to symmetries of the constructed cones, which consist of permutations and oriented switchings; as well as in symmetric case, transformation of the oriented switching serves the basis for a choice of the inequalities defining the corresponding polyhedron. Different approaches to creation of a cone and a polyhedron of asymmetrical hypersemimetrics are considered. In the last section results of the calculations devoted to cones and to polyhedrons of quasi-semimetrics, the oriented cuts, the oriented multicuts, weighed quasimetrics and partial metrics for 3, 4, 5 and 6 points are considered. In fact, the dimension of an object, the number of its extreme rays (vertices) and their orbits, the number of its facets and their orbits, the diameters of the skeleton and the the ridge graph of the constructed cones and polyhedrons are specified.
Let r(i, X1) be the number of points in the Sα-orbit of the length i with respect to a rotation Sα : T1 −→ T1 of the unit circle T1 = R/Z by an angle α hit the X1. Denote by δ(i, X1) = r(i, X1)−i|X1| the deviation of the function r(i, X1) from its average value i|X1|, where |X1| is the length of X1 . In 1921 E. Hecke had proved the theorem: if X1 has the length |X1| = hα + b, where h ∈ N, b ∈ Z, then the inequality |δ(i, X1)| � h для всех i = 0, 1, 2, . . . holds for all i = 0, 1, 2, . . . In 1981 г. I. Oren was able to generalize the Hecke theorem to the case of a finite union of intervals X1. He proved the estimation δ(i, X1) = O(1) as i → ∞. In the general case, if Xd belongs to the d-dimensional torus Td = Rd/Zd and there is δ(i, Xd) = O(1) as i → ∞, then Xd is called a bounded remainder set. Global approach to search of bounded remainder sets was proposed by V.G. Zhuravlev in 2011 when, instead of separate sets Xd on the torus Td, k the complete toric decompositions Td = X0 d ⊔ Xd . . . ⊔ Xd with parameters c,λ 1 ⊔ s c, λ began to be considered. The main idea was to determine a lifting π−1 : R α maps d of the torus Td into the covering space Rd T ֒→ so the rotation S d ′ ′ ′ to a rearrangement Sv of the corresponding sets X0, X1, . . . , X in Rd. In the s ′ case s + 1 � d + 1, each set Xd = π(X ) is a bounded remainder set and the k k ′ ′ ′ union Td = X ⊔ X ⊔ . . . ⊔ X in Rd is a toric development for Td. These c,λ 0 1 s developments Td were built with the help of rearrangement parallelohedra, and the parallelohedra obtained as the Minkowskii sums of the unit cube Cd and intervals. If d = 3, 4 we have the Voronoi parallelohedra and the Fedorov rhombic dodecahedron. In the present paper, by using tilings of multidimensional tori, bounded remainder sets are constructed. The tilings consist of a finite combination of convex polyhedra. A multi-dimension version of Hecke theorem with respect to the uniform distribution of fractional parts on the unit circle is proved for these sets.
The article deals with an algorithm of solving polynomial equations in a ring D[x], where D is an arbitrary order of field Q(ω) and ω is an algebraic integer. The algorithm develops Kurt Hensel’s idea published in 1904 which was named Hensel’s lifting lemma later. The algorithm described is based on the following theoretical results. Firstly, basis of order D expansion coefficients of the equation roots are estimated, i. e. an estimate for the polynomial equation roots height in an algebraic number field arbitrary order is derived. Secondly, an iterative formula for the corresponding polynomial congruence solution quadratic lifting modulo power of prime in the order is obtained. This formula does not contain any divisions. Thirdly, an effective bound for prime power the congruence solution needs to be lifted to obtain the exact solution of the original equation is derived. Notice that a prime p which is used for lifting needs to satisfy certain conditions for the algorithm correct work. In particular, p should not derive the original polynomial and its derivative resultant norm and also p should not derive discriminant of an algebraic integer ω. Also the algorithm complexity is decreased if it is possible to choose prime p which in addition to two previous conditions has the following property: the minimal polynomial of ω which coefficients are reduced modulo p is irreducible over finite field Fp. 4 In this case the algorithm complexity is O(m + m γi ) 3 ln m ln(max0�i�m + m3 ln(max0�i�m γi ) ln ln(max0�i�m γi ) arithmetic operations. Here m is the original polynomial degree, γi, 0 � i � m are its coefficients and γ is the algebraic numbers conjugated to γ absolute values maximum. If it is impossible to choose prime number p such that minimal polynomial of ω is irreducible 4 over Fp then the algorithm complexity is O(m + m3 ln m ln(max0�i�m γi ) + m3 ln(max0�i�m γi ) ln ln(max0�i�m γi ) + md ln ln(max0�i�m γi )) arithmetic operations. Here d is the minimal polynomial of ω irreducible factors over Fp number. The algorithm includes strategy to select a prime p to achieve lower computational complexity.
the realization of a random graph by geometric graphs in the space Rd. In the case of graphs of diameters we prove asymptotic behavior for the threshold probability on the plane, as well as the exact expression in the case d � 3.
There is a numeric algorithm for finding non-trivial zeros of regular Dirichlet L-functions. This algorithm is based on a construction of Dirichlet polynomials which approximate these L-functions in any rectangle in the critical strip with exponential speed. This result does not hold for Dirichlet L-functions in number fields, because if it did, a power series with the same coefficients as the Dirichlet series defining the L-function would converge to a function which is holomorphic at 1, however, it is known that that such power series in case of a number field different from the field of rational numbers can’t be continued analytically past its convergence boundary. Consequently, we need to develop a new numerical algorithm for finding non-trivial zeros of Dirichlet L-functions in number fields. This problem is discussed in this paper. We show that there exists a sequence of Dirichlet polynomials which approximate a Dirichlet L-function in a number field faster than any power function in any rectangle inside the critical strip. We also provide an explicit construction of approximating Dirichlet polynomials, whose zeros coincide with those of a Dirichlet L-function in the specified rectangle, for an L-function, if it can be split into a product of classical L-functions. Additionally we discuss some questions related to the construction of such polynomials for arbitrary Dirichlet L-functions.
This paper provides an overview of the results (with varying degrees of detail) in three different directions. The main Central direction refers to recurrent sequences, primarily to their base (in a different sense) sets. Another direction is related to new combinatorial objects (v, k1, k2)-confi gurations encountered on the way of weakening the determinants of well-known combinatorial objects (v, k, λ)-configuration. The third direction deals with invariant differentials of higher orders from several smooth functions of one real variable. In each of these themes the issues associated with combinatorial configurations in the form of finite planes, and the results obtained through the same type of views, points of the corresponding configurations of points in multidimensional locally Euclidean spaces. In the case of invariant differentials of these representations arise naturally, and in the case of recurrent sequences and (v, k1, k2)-configurations are introduced by analogy, but in an artificial way.
The work contains the research of tetrahedrons of a hyperbolic space H�3 of positive curvature. The space H�3 is realized by on external domain of projective three-dimensional space with respect to the oval hyperquadric, т. e. on ideal domain of the Lobachevskii space. All lines of the space H�3 on existence of the common points with the absolute can be elliptic, parabolic or hyperbolic. All planes of the space H�3 depending on position with respect to the absolute belong to the three types: elliptic, coeuclidean and hyperbolic of positive curvature. The angles of the elliptic plane of one type. The angles of the coeuclidean plane of three types. The angles of the hyperbolic plane of positive curvature of fifteen types. Also all dihedral angles of the space H�3 belong to fifteen types. Various sets of types of sides define in the space H�3 fifteen types of tetrahedrons. In work the classification of tetrahedrons with not hyperbolic sides is carried out. All such tetrahedrons belong to five types. It is proved that each edge of a tetrahedron with not hyperbolic sides belongs to the elliptic line. The elliptic line is the closed line in the space H�3 . In further classification of tetrahedrons with not hyperbolic sides we use concept α-sides of the tetrahedron. In the space H�3 the cone of tangents to absolute oval hyperquadric is connected with each point. This cone we called a light cone of a point. The curve of crossing of the side plane with a light cone of tetrahedron top, opposite to this side, is called the light curve of a side of a tetrahedron. The tetrahedron side in the space H�3 is called a α-side if it contains completely the light curve. The following theorem is proved. The tetrahedron with not hyperbolic sides in the space H�3 or doesn’t contain the α-sides, or supports one α-side, or all its sides are the α-sides. Quantity α-sides and their types define classes and sorts of the tetrahedrons with not hyperbolic sides. In work the types of dihedral angles in a tetrahedron of each class (sort) are established.
The paper deals with the symmetry properties of the associated closed convex polyhedra in three-dimensional Euclidean space. Themes work relates in part to the problem of generalization class of regular (Platonic) polyhedra. Historically, the first such generalizations are equiangularly-semiregular (Archimedean) polyhedra. The direction of generalization of regular polyhedra, considered by the author in this paper due to the symmetry axes of the convex polyhedron. A convex polyhedron is called symmetric if it has at least one non-trivial symmetry axis. All the axis of symmetry of the polyhedron intersect at one point called the center of the polyhedron. All considered the polyhedra are polyhedra with the center. Previously we listed all polyhedra, strongly symmetrical with respect to rotation of faces, as well as their dual-metrically polyhedra strongly symmetric with respect to the rotation polyhedral angles [9]–[15]. It is interesting to note that among the highly symmetric polyhedra there are exactly eight of which are not even equivalent to combinatorial Archimedean or equiangularly semiregular polyhedra. By definition, the property of strong symmetry polyhedron requires a global symmetry of the polyhedron with respect to each axis of symmetry perpendicular to the faces of the polyhedron. It is therefore of interest to find weaker conditions on the symmetry elements of the polyhedron. We give a new proof of the local criterion of strong symmetry of the polyhedron, which is based on the properties of the axes of two consecutive rotations. We also consider two classes of polyhedra that generalize the concept of a strongly rotationally symmetrical faces of: a class of polyhedra with isolated asymmetrical faces and the class of polyhedra with isolated asymmetrical zone. It is proved that every polyhedron with isolated asymmetrical faces can be obtained by cutting off the vertices or edges of a polyhedron, highly symmetrical with respect to rotation faces; and each polyhedron with isolated asymmetrical zone by build axially symmetric truncated pyramids on some facets of one of the highly symmetric with respect to rotation of the faces of the polyhedron.In each of these classes there are the largest number of polyhedron faces excluding two infinite series: truncated prisms; truncated on two apex and elongated bipyramid.
The paper presents the fundamentals of the theory of arithmetic sums and oscillatory integrals of polynomials Bernoulli, an argument that is the real function of a certain differential properties. Drawing an analogy with the method of trigonometric sums I. M. Vinogradov. The introduction listed problems in number theory and mathematical analysis, which deal the study of the above mentioned sums and integrals. Research arithmetic sums essentially uses a functional equation type Gauss theorem for multiplication of the Euler gamma function. Estimations of the individual arithmetic the amounts found indicators of convergence of their averages. In particular, the problems are solved analogues Hua Loo-Keng for one-dimensional integrals and sums.
In this note we provide some analogues of our numerous recent results on traces and distances in terms of Martinelly — Bochner integrals and kernels. These are first results of this type in terms of such kernels. Some assertions for Martinelly — Bochner integrals related with Holder classes and Lebegues points will be also discussed. In recent years various new sharp results on traces and distances were provided in a big series of papers of the first author. In all these papers properties of Bergman-type kernels and Bergman-type integral representations are playing a critical role. The intension of this paper to find some analogues of these results in terms of or with the help of more general integral representations and more general kernels in analytic function spaces in higher dimension so-called Martinelly — Bochner integral representations and Martinelly — Bochner kernels in Cn. Our work consists of three parts. In the first part we partially generalize our results on traces. In the second part we provide estimates of distance function in terms of Martinelly — Bochner kernels and Martinelly — Bochner integrals. In the third part we present results on Martinelly — Bochner integrals related with Holder classes and Lebegues points. This type of issues arise naturally in view of recent series of papers and new results of the first author on multifunctional analytic spaces and related issues. In our proofs we modify the methods of earlier results and theorems for the case of Martinelli-Bochner integrals and kernels.
Rational and irrational rotations for the set of rational directions in the flat point lattice are considered. It is proved that in the case of rational rotations an order of noncrystallographic turn can be only 8 or 12. The set of rational 2 2 directions in the rectangular point lattice with metric quadratic form x +λ2 y and arbitrary its centering has irrational rotation if and only if the number λ2 is rational.
When solving geometric problems, writing manuals and books on geometry for secondary school and University we have to deal with technical drawings. And even if the drawing is represented quite clearly, move its from the "head" to a paper is quite difficult for many people. To this can help a variety of graphic editors, for example, GeoGebra — the free system of a graphics and calculations which is used for study and teaching of mathematics in schools. However, there is another approach. Geometric drawings can be created using the TikZ system [2], which is an extension package of TEX/LATEX. Using TikZ, not walking out of LATEX, and without resorting to third-party graphical editors, easily write code to output both simple and very complex diagrams, charts, graphs and geometrical drawings. The article discusses the specific of writing code of fragments TikZ to output the drawings for solving typical tasks on planimetry associated with remarkable points in a triangle. Namely, when creating some geometric drawings on those or other data often necessary to calculate and display remarkable points of a triangle, which include: "the centroid (center of mass, centre of gravity)" — the point of intersection of medians; "the orthocenter" — the point crossing heights; "the circle circumscribed around triangle" — the point of intersection "middle" perpendiculars (perpendicular to the midpoints of the sides of the triangle); "the incenter" — the center of the inscribed circle, which is the point of intersection of the bisectors. The following shows how are calculated and returned these points with tikz-code. Also discussed codes for the solution of some auxiliary tasks such as conducting: bisector of an angle; a line which passing through the given point and parallel to the line; a circle with a center at a particular point tangent to a given straight line, etc.
This article establishes the connection between the spatial inversion and the inversion of time and the transition from one inertial reference system (IRS) to another one within the framework of the special theory of relativity. It turns out that if two events occurs that in the case of the transition from one IRS to another two variants are possible. The first one is when their sequence in time is the same in both IRS. The second variant is when their sequence in time is mutually opposite. In the latter case we speak about the inversion of time. The conditions under which the inversion of time takes place have been drawn. The results arising from the introduced concepts are being analyzed. The idea is that on the whole in the Universe there is no single direction of time, the chaos of time reigns in it. Therefore such notions as the age of the Universe and some others notions lose their sense. Thus, in non-relativistic velocities and relatively short distances we return to the usual point of view of the unidirectional flow of time. The calculations and the results introduced in this work are based on the special theory of relativity, and therefore they are considered obvious. Nevertheless, it would be interesting to carry out an experiment to test the inversion of time. The scheme of the experiment is the following. One IRS is connected with the Earth, the other IRS – with two satellites which have the equal velocities and which located at some distance from each other. Signals are sent from satellite to Earth at first from the first satellite and then from the other one. On Earth, these signals are received in the reverse order. The parameters of such experiment have been calculated and it has been proved that it can be carried out at the present moment. At the end of the paper brief summery are given and possible ways of continuing of this work are considered. On these ways is the analysis of the changes which must be made in the proposed presentation taking into account non-inertial of the real objects of the Universe and gravitational effects.
