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A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature
A survey of some arithmetic applications of ergodic theory in negative curvature Jouni Parkkonen Frédéric Paulin January 12, 2015 Abstract This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in R, C and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition is based on lectures at the conference “Chaire Jean Morlet: Géométrie et systèmes dynamiques”, at the CIRM, Luminy, 2014. We thank B. Hasselblatt for his strong encouragements to write this survey. 1 1 Introduction For several decades, tools from dynamical systems, and in particular ergodic theory, have been used to derive arithmetic and number theoretic, in particular Diophantine approxi- mation results, see for instance the works of Furstenberg, Margulis, Sullivan, Dani, Klein- bock, Clozel, Oh, Ullmo, Lindenstrauss, Einsiedler, Michel, Venkatesh, Marklof, Green- Tao, Elkies-McMullen, Ratner, Mozes, Shah, Gorodnik, Ghosh, Weiss, Hersonsky-Paulin, Parkkonen-Paulin and many others, and the references [Kle2, Lin, Kle1, AMM, Ath, GorN, EiW, PaP5]. arXiv:1501.02072v1 [math.NT] 9 Jan 2015 In Subsection 2.2 of this survey, we introduce a general framework of Diophantine approximation in measured metric spaces, in which most of our arithmetic corollaries are inserted (see the end of Subsection 2.2 for references concerning this framework). -
Algorithmic Factorization of Polynomials Over Number Fields
Rose-Hulman Institute of Technology Rose-Hulman Scholar Mathematical Sciences Technical Reports (MSTR) Mathematics 5-18-2017 Algorithmic Factorization of Polynomials over Number Fields Christian Schulz Rose-Hulman Institute of Technology Follow this and additional works at: https://scholar.rose-hulman.edu/math_mstr Part of the Number Theory Commons, and the Theory and Algorithms Commons Recommended Citation Schulz, Christian, "Algorithmic Factorization of Polynomials over Number Fields" (2017). Mathematical Sciences Technical Reports (MSTR). 163. https://scholar.rose-hulman.edu/math_mstr/163 This Dissertation is brought to you for free and open access by the Mathematics at Rose-Hulman Scholar. It has been accepted for inclusion in Mathematical Sciences Technical Reports (MSTR) by an authorized administrator of Rose-Hulman Scholar. For more information, please contact [email protected]. Algorithmic Factorization of Polynomials over Number Fields Christian Schulz May 18, 2017 Abstract The problem of exact polynomial factorization, in other words expressing a poly- nomial as a product of irreducible polynomials over some field, has applications in algebraic number theory. Although some algorithms for factorization over algebraic number fields are known, few are taught such general algorithms, as their use is mainly as part of the code of various computer algebra systems. This thesis provides a summary of one such algorithm, which the author has also fully implemented at https://github.com/Whirligig231/number-field-factorization, along with an analysis of the runtime of this algorithm. Let k be the product of the degrees of the adjoined elements used to form the algebraic number field in question, let s be the sum of the squares of these degrees, and let d be the degree of the polynomial to be factored; then the runtime of this algorithm is found to be O(d4sk2 + 2dd3). -
Gaussian Integers1
FORMALIZED MATHEMATICS Vol. 21, No. 2, Pages 115–125, 2013 DOI: 10.2478/forma-2013-0013 degruyter.com/view/j/forma Gaussian Integers1 Yuichi Futa Hiroyuki Okazaki Japan Advanced Institute Shinshu University of Science and Technology Nagano, Japan Ishikawa, Japan Daichi Mizushima2 Yasunari Shidama Shinshu University Shinshu University Nagano, Japan Nagano, Japan Summary. Gaussian integer is one of basic algebraic integers. In this artic- le we formalize some definitions about Gaussian integers [27]. We also formalize ring (called Gaussian integer ring), Z-module and Z-algebra generated by Gaus- sian integer mentioned above. Moreover, we formalize some definitions about Gaussian rational numbers and Gaussian rational number field. Then we prove that the Gaussian rational number field and a quotient field of the Gaussian integer ring are isomorphic. MSC: 11R04 03B35 Keywords: formalization of Gaussian integers; algebraic integers MML identifier: GAUSSINT, version: 8.1.02 5.17.1179 The notation and terminology used in this paper have been introduced in the following articles: [5], [1], [2], [6], [12], [11], [7], [8], [18], [24], [23], [16], [19], [21], [3], [9], [20], [14], [4], [28], [25], [22], [26], [15], [17], [10], and [13]. 1. Gaussian Integer Ring Now we state the proposition: (1) Let us consider natural numbers x, y. If x + y = 1, then x = 1 and y = 0 or x = 0 and y = 1. Proof: x ¬ 1. 1This work was supported by JSPS KAKENHI 21240001 and 22300285. 2This research was presented during the 2012 International Symposium on Information Theory and its Applications (ISITA2012) in Honolulu, USA. c 2013 University of Białystok CC-BY-SA License ver. -
Octonion Multiplication and Heawood's
CONFLUENTES MATHEMATICI Bruno SÉVENNEC Octonion multiplication and Heawood’s map Tome 5, no 2 (2013), p. 71-76. <http://cml.cedram.org/item?id=CML_2013__5_2_71_0> © Les auteurs et Confluentes Mathematici, 2013. Tous droits réservés. L’accès aux articles de la revue « Confluentes Mathematici » (http://cml.cedram.org/), implique l’accord avec les condi- tions générales d’utilisation (http://cml.cedram.org/legal/). Toute reproduction en tout ou partie de cet article sous quelque forme que ce soit pour tout usage autre que l’utilisation á fin strictement personnelle du copiste est constitutive d’une infrac- tion pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Confluentes Math. 5, 2 (2013) 71-76 OCTONION MULTIPLICATION AND HEAWOOD’S MAP BRUNO SÉVENNEC Abstract. In this note, the octonion multiplication table is recovered from a regular tesse- lation of the equilateral two timensional torus by seven hexagons, also known as Heawood’s map. Almost any article or book dealing with Cayley-Graves algebra O of octonions (to be recalled shortly) has a picture like the following Figure 0.1 representing the so-called ‘Fano plane’, which will be denoted by Π, together with some cyclic ordering on each of its ‘lines’. The Fano plane is a set of seven points, in which seven three-point subsets called ‘lines’ are specified, such that any two points are contained in a unique line, and any two lines intersect in a unique point, giving a so-called (combinatorial) projective plane [8,7]. -
Applications of Hyperbolic Geometry to Continued Fractions and Diophantine Approximation
Applications of hyperbolic geometry to continued fractions and Diophantine approximation Robert Hines University of Colorado, Boulder April 1, 2019 A picture Summary Our goal is to generalize features of the preceding picture to some nearby settings: H2 H3 (H2)r × (H3)s Hn P 1(R) P 1(C) P 1(R)r × P 1(C)s Sn−1 SL2(R) SL2(C) SL2(F ⊗ R) SVn−1(R) SL2(Z) SL2(O) " SV (O) ideal right-angled triangles ideal polyhedra horoball bounded geodesic " neighborhoods trajectories quad. quad./Herm. " " forms forms closed geodesics closed surfaces aniso. subgroups " Ingredients Ingredients Upper half-space models in dimensions two and three Hyperbolic two-space: 2 H = fz = x + iy 2 C : y > 0g; 2 1 @H = P (R); 2 Isom(H ) = P GL2(R); az + b az¯ + b g · z = ; (det g = ±1); cz + d cz¯ + d + ∼ Stab (i) = SO2(R)={±1g = SO2(R): Hyperbolic three-space: 3 H = fζ = z + jt 2 H : t > 0; z 2 Cg; 3 1 @H = P (C); 3 Isom(H ) = P SL2(C) o hτi; g · ζ = (aζ + b)(cζ + d)−1; τ(ζ) =z ¯ + jt; + ∼ Stab (j) = SU2(C)={±1g = SO3(R): Binary quadratic and Hermitian forms Hyperbolic two- and three-space are the Riemannian symmetric spaces associated to G = SL2(R), SL2(C). The points can be identified with roots of binary forms: 2 SL2(R)=SO2(R) ! fdet. 1 pos. def. bin. quadratic formsg ! H p −b + b2 − 4ac g 7! ggt = ax2 + bxy + cy2 = Q 7! =: Z(Q) 2a 3 SL2(C)=SU2(C) ! fdet. -
An Introduction to Pythagorean Arithmetic
AN INTRODUCTION TO PYTHAGOREAN ARITHMETIC by JASON SCOTT NICHOLSON B.Sc. (Pure Mathematics), University of Calgary, 1993 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE in THE FACULTY OF GRADUATE STUDIES Department of Mathematics We accept this thesis as conforming tc^ the required standard THE UNIVERSITY OF BRITISH COLUMBIA March 1996 © Jason S. Nicholson, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my i department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia Vancouver, Canada Dale //W 39, If96. DE-6 (2/88) Abstract This thesis provides a look at some aspects of Pythagorean Arithmetic. The topic is intro• duced by looking at the historical context in which the Pythagoreans nourished, that is at the arithmetic known to the ancient Egyptians and Babylonians. The view of mathematics that the Pythagoreans held is introduced via a look at the extraordinary life of Pythagoras and a description of the mystical mathematical doctrine that he taught. His disciples, the Pythagore• ans, and their school and history are briefly mentioned. Also, the lives and works of some of the authors of the main sources for Pythagorean arithmetic and thought, namely Euclid and the Neo-Pythagoreans Nicomachus of Gerasa, Theon of Smyrna, and Proclus of Lycia, are looked i at in more detail. -
Multiplication Fact Strategies Assessment Directions and Analysis
Multiplication Fact Strategies Wichita Public Schools Curriculum and Instructional Design Mathematics Revised 2014 KCCRS version Table of Contents Introduction Page Research Connections (Strategies) 3 Making Meaning for Operations 7 Assessment 9 Tools 13 Doubles 23 Fives 31 Zeroes and Ones 35 Strategy Focus Review 41 Tens 45 Nines 48 Squared Numbers 54 Strategy Focus Review 59 Double and Double Again 64 Double and One More Set 69 Half and Then Double 74 Strategy Focus Review 80 Related Equations (fact families) 82 Practice and Review 92 Wichita Public Schools 2014 2 Research Connections Where Do Fact Strategies Fit In? Adapted from Randall Charles Fact strategies are considered a crucial second phase in a three-phase program for teaching students basic math facts. The first phase is concept learning. Here, the goal is for students to understand the meanings of multiplication and division. In this phase, students focus on actions (i.e. “groups of”, “equal parts”, “building arrays”) that relate to multiplication and division concepts. An important instructional bridge that is often neglected between concept learning and memorization is the second phase, fact strategies. There are two goals in this phase. First, students need to recognize there are clusters of multiplication and division facts that relate in certain ways. Second, students need to understand those relationships. These lessons are designed to assist with the second phase of this process. If you have students that are not ready, you will need to address the first phase of concept learning. The third phase is memorization of the basic facts. Here the goal is for students to master products and quotients so they can recall them efficiently and accurately, and retain them over time. -
On a Certain Non-Split Cubic Surface
ON A CERTAIN NON-SPLIT CUBIC SURFACE R. DE LA BRETECHE,` K. DESTAGNOL, J. LIU, J. WU & Y. ZHAO Abstract. In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of 3 bounded height on the singular cubic surface of PQ 2 2 3 x0(x1 + x2)= x3 in agreement with the Manin-Peyre conjectures. 1. Introduction and results 3 Let V ⊂ PQ be the cubic surface defined by 2 2 3 x0(x1 + x2) − x3 =0. The surface V has three singular points ξ1 = [1 : 0 : 0 : 0], ξ2 = [0 : 1 : i : 0] and ξ3 = [0 : 1 : −i : 0]. It is easy to see that the only three lines contained in VQ = V ×Spec(Q) Spec(Q) are ℓ1 := {x3 = x1 − ix2 =0}, ℓ2 := {x3 = x1 + ix2 =0}, and ℓ3 := {x3 = x0 =0}. Clearly both ℓ1 and ℓ2 pass through ξ1, which is actually the only rational point lying on these two lines. Let U = V r {ℓ1 ∪ ℓ2 ∪ ℓ3}, and B a parameter that can approach infinity. In this note we are concerned with the behavior of the counting arXiv:1709.09476v2 [math.NT] 12 Dec 2018 function NU (B) = #{x ∈ U(Q): H(x) 6 B}, where H is the anticanonical height function on V defined by 2 2 H(x) := max |x0|, x1 + x2, |x3| (1.1) n q o where each xj ∈ Z and gcd(x0, x1, x2, x3) = 1. The main result of this note is the following. Theorem 1.1. There exists a constant ϑ > 0 and a polynomial Q ∈ R[X] of degree 3 such that 1−ϑ NU (B)= BQ(log B)+ O(B ). -
Some Mathematics
Copyright Cambridge University Press 2003. On-screen viewing permitted. Printing not permitted. http://www.cambridge.org/0521642981 You can buy this book for 30 pounds or $50. See http://www.inference.phy.cam.ac.uk/mackay/itila/ for links. C Some Mathematics C.1 Finite field theory Most linear codes are expressed in the language of Galois theory Why are Galois fields an appropriate language for linear codes? First, a defi- nition and some examples. AfieldF is a set F = {0,F} such that 1. F forms an Abelian group under an addition operation ‘+’, with + 01 · 01 0 being the identity; [Abelian means all elements commute, i.e., 0 01 0 00 satisfy a + b = b + a.] 1 10 1 01 2. F forms an Abelian group under a multiplication operation ‘·’; multiplication of any element by 0 yields 0; Table C.1. Addition and 3. these operations satisfy the distributive rule (a + b) · c = a · c + b · c. multiplication tables for GF (2). For example, the real numbers form a field, with ‘+’ and ‘·’denoting ordinary addition and multiplication. + 01AB 0 01AB AGaloisfieldGF q q ( ) is a field with a finite number of elements . 1 10BA A unique Galois field exists for any q = pm,wherep is a prime number A AB 01 and m is a positive integer; there are no other finite fields. B BA10 · 01AB GF (2). The addition and multiplication tables for GF (2) are shown in ta- 0 00 0 0 ble C.1. These are the rules of addition and multiplication modulo 2. 1 01AB A AB GF (p). -
On a Certain Non-Split Cubic Surface Régis De La Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao
On a certain non-split cubic surface Régis de la Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao To cite this version: Régis de la Bretèche, Kévin Destagnol, Jianya Liu, Jie Wu, Yongqiang Zhao. On a certain non- split cubic surface. Science China Mathematics, Science China Press, 2019, 62 (12), pp.2435-2446. 10.1007/s11425-018-9543-8. hal-02468799 HAL Id: hal-02468799 https://hal.sorbonne-universite.fr/hal-02468799 Submitted on 6 Feb 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ON A CERTAIN NON-SPLIT CUBIC SURFACE R. DE LA BRETECHE,` K. DESTAGNOL, J. LIU, J. WU & Y. ZHAO Abstract. In this note, we establish an asymptotic formula with a power-saving error term for the number of rational points of 3 bounded height on the singular cubic surface of PQ 2 2 3 x0(x1 + x2)= x3 in agreement with the Manin-Peyre conjectures. 1. Introduction and results 3 Let V ⊂ PQ be the cubic surface defined by 2 2 3 x0(x1 + x2) − x3 =0. The surface V has three singular points ξ1 = [1 : 0 : 0 : 0], ξ2 = [0 : 1 : i : 0] and ξ3 = [0 : 1 : −i : 0]. -
Algebraic Number Theory and Its Application to Rational Theory
Journal of Shanghai Jiaotong University ISSN:1007-1172 Algebraic Number Theory And Its Application To Rational Theory Nitish Kumar Bharadwaj1 & Ajay Kumar2 1. Research Scholar, University Department of Mathematics, T. M. Bhagalpur University, Bhagalpur, Email: [email protected] 2. Research Scholar, University Department of Mathematics, T. M. Bhagalpur University, Bhagalpur, Email: [email protected] ABSTRACT In this paper we recall some nation connected with algebraic number theory and indicate some of its applications in the Gaussian field namely K(i) = √(–1). A Gaussian rational number is complex number of the form a+bi, where a and b both are rational numbers. The set of all Gaussian rationals forms the Gaussian rational field, denoted by K(i). Introduction Numbers have fascinated man from a very early period of human civilization. Pythagoreans studied many properties of natural numbers 1, 2, 3 ......... The famous theorem of Pythagoras (6th century B.C) through geometrical has a pronounced number theoretic content. The early Babylonians had noted many Pythagorean triads e.g. 3, 4, 5; 5, 12, 13 which are natural number a, b, c satisfying the equation. a2 + b2 = c2 ............. (1) In about 250 A.D Diophantus of Alexandria wrote treatise on polynomial equations which studied solution in fractions. Particular cases of these equations with natural number have been called Diophantine equations to this day. The study of algebra developed over centuries. The Hindu mathematicians deals with and introduced the nation of negative numbers and zero. In the 16th century and owned negative and imaginary numbers were used with increasing confidence and flexibility. Meanwhile the study of the theory of natural numbers went on side by side. -
First Moment of Distances Between Centres of Ford Spheres
First Moment of Distances Between Centres of Ford Spheres Kayleigh Measures (York) May 2018 Abstract This paper aims to develop the theory of Ford spheres in line with the current theory for Ford circles laid out in a recent paper by S. Chaubey, A. Malik and A. Zaharescu. As a first step towards this goal, we establish an asymptotic estimate for the first moment X 1 1 M (S) = + ; 1;I2 2jsj2 2js0j2 r r0 s ; 0 2GS consecs where the sum is taken over pairs of fractions associated with `consecutive' Ford 1 spheres of radius less than or equal to 2S2 . 1 Introduction and Motivation Ford spheres were first introduced by L. R. Ford in [3] alongside their two dimensional p analogues, Ford circles. For a Farey fraction q , its Ford circle is a circle in the upper 1 p half-plane of radius 2q2 which is tangent to the real line at q . Similarly, for a fraction r s with r and s Gaussian integers, its Ford sphere is a sphere in the upper half-space of 1 r radius 2jsj2 tangent to the complex plane at s . This paper studies moments of distances between centres of Ford spheres, in line with the moment calculations for Ford circles produced by S. Chaubey, A. Malik and A. Zaharescu. In [2], Chaubey et al. consider those Farey fractions in FQ which lie in a fixed interval I := [α; β] ⊆ [0; 1] for rationals α and β. They call the set of Ford circles corresponding to these fractions FI;Q, and its cardinality is denoted NI (Q).