Reachability Problems for Continuous Linear Dynamical Systems
|
|
- Angelica Blake
- 5 years ago
- Views:
Transcription
1 Reachability Problems for Continuous Linear Dynamical Systems James Worrell Department of Computer Science, Oxford University (Joint work with Ventsislav Chonev and Joël Ouaknine) CONCUR September 2nd 2015
2 Reachability for Continuous-Time Markov Chains Stagnant market Bull market 0.02 Bear market 0.3
3 Reachability for Continuous-Time Markov Chains Stagnant market Bull market 0.02 Bear market 0.3 Distribution P(t) at time t satisfies P (t) = P(t)Q, where Q = is the rate matrix.
4 Reachability for Continuous-Time Markov Chains Stagnant market Bull market 0.02 Bear market 0.3 Distribution P(t) at time t satisfies P (t) = P(t)Q, where Q = is the rate matrix. Is it ever more likely to be a Bear market than a Bull market? t (P(t) Bear P(t) Bull )
5 Reachability for Continuous-Time Markov Chains Stagnant market Bull market 0.02 Bear market 0.3 Distribution P(t) at time t satisfies P (t) = P(t)Q, where Q = is the rate matrix. Stationary distribution π = (0.885, 0.071, 0.044).
6 Cyber-Physical Systems To analyze a cyber-physical system, such as a pacemaker, we need to consider the discrete software controller interacting with the physical world, which is typically modelled by differential equations Rajeev Alur (CACM, 2013)
7 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k
8 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata
9 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata ẋ = c rectangular hybrid automata
10 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata ẋ = c rectangular hybrid automata ẋ = Ax linear hybrid automata
11 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata ẋ = c rectangular hybrid automata ẋ = Ax linear hybrid automata...
12 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata ẋ = c rectangular hybrid automata ẋ = Ax linear hybrid automata... o-minimal flows + strong resets reachability decidable
13 Hybrid Automata: Various Continuous Dynamics Hybrid automaton = states + variables x R k ẋ = 1 timed automata ẋ = c rectangular hybrid automata ẋ = Ax linear hybrid automata... o-minimal flows + strong resets reachability decidable Is this location a trap? x := 2 y := 4 ẋ = 3x y ẏ = x 5y x 10 y 2?
14 Reachability for Continuous Linear Dynamical Systems Is this location a trap? x := 2 y := 4 ẋ = 3x y ẏ = x 5y x 10 y 2? Stagnant market Is ever more likely to be a Bear Bull market Bear market market than a Bull market: t (P(t) Bear P(t) Bull )?
15 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax
16 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
17 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
18 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
19 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
20 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
21 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0)
22 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0) u
23 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0) f (t) = u T x(t)
24 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0) f (t) = u T x(t) f (k) (t) + a k 1 f (k 1) (t) a 1 f (t) + a 0 f (t) = 0
25 Reachability for Continuous Linear Dynamical Systems x : R 0 R k ẋ = Ax x(t) = exp(at)x(0) f (t) = u T x(t) f (k) (t) + a k 1 f (k 1) (t) a 1 f (t) + a 0 f (t) = 0 m f (t) = P j (t)e λ j t j=1
26 Reachability for Continuous Linear Dynamical Systems Let f : R 0 R be given as above, with all coefficients algebraic.
27 Reachability for Continuous Linear Dynamical Systems Let f : R 0 R be given as above, with all coefficients algebraic. BOUNDED-ZERO Problem Instance: f and bounded interval [a, b] Question: Is there t [a, b] such that f (t) = 0?
28 Reachability for Continuous Linear Dynamical Systems Let f : R 0 R be given as above, with all coefficients algebraic. BOUNDED-ZERO Problem Instance: f and bounded interval [a, b] Question: Is there t [a, b] such that f (t) = 0? ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0?
29 Reachability for Continuous Linear Dynamical Systems Let f : R 0 R be given as above, with all coefficients algebraic. BOUNDED-ZERO Problem Instance: f and bounded interval [a, b] Question: Is there t [a, b] such that f (t) = 0? ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0? Decidability open! [Bell, Delvenne, Jungers, Blondel 2010]
30 Related Work A lot of work since 1920s on the zeros of exponential polynomials f (z) = m P j (z)e λ j z j=1 (Polya, Ritt, Tamarkin, Kac, Voorhoeve, van der Poorten,... ) but mostly on distribution of complex zeros.
31 Related Work A lot of work since 1920s on the zeros of exponential polynomials f (z) = m P j (z)e λ j z j=1 (Polya, Ritt, Tamarkin, Kac, Voorhoeve, van der Poorten,... ) but mostly on distribution of complex zeros. CONTINUOUS-ORBIT Problem The problem of whether the trajectory x(t) = e At x(0) reaches a given target point was shown to be decidable by Hainry (2008) and in PTIME by Chen, Han and Yu (2015).
32 Reachability for Continuous Linear Dynamical Systems Theorem (Bell, Delvenne, Jungers, Blondel 2010) In dimension 2, BOUNDED-ZERO and ZERO are decidable.
33 Reachability for Continuous Linear Dynamical Systems Theorem (Bell, Delvenne, Jungers, Blondel 2010) In dimension 2, BOUNDED-ZERO and ZERO are decidable. Theorem (arxiv: , 2015) In dimension 3, BOUNDED-ZERO and ZERO are decidable.
34 Reachability for Continuous Linear Dynamical Systems Theorem (Bell, Delvenne, Jungers, Blondel 2010) In dimension 2, BOUNDED-ZERO and ZERO are decidable. Theorem (arxiv: , 2015) In dimension 3, BOUNDED-ZERO and ZERO are decidable. Theorem (arxiv: , 2015) Assuming Schanuel s Conjecture, BOUNDED-ZERO is decidable in all dimensions.
35 Reachability for Continuous Linear Dynamical Systems Theorem (Bell, Delvenne, Jungers, Blondel 2010) In dimension 2, BOUNDED-ZERO and ZERO are decidable. Theorem (arxiv: , 2015) In dimension 3, BOUNDED-ZERO and ZERO are decidable. Theorem (arxiv: , 2015) Assuming Schanuel s Conjecture, BOUNDED-ZERO is decidable in all dimensions. It turns out that this result (in fact, a powerful generalisation of it) had already been discovered (but never published) in the early 1990s by Macintyre and Wilkie! [Angus Macintyre, personal communication, July 2015]
36 Reachability for Continuous Linear Dynamical Systems Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO.
37 Reachability for Continuous Linear Dynamical Systems Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO. Theorem (arxiv: , 2015) In dimension 9 (and above), decidability of ZERO would entail major breakthroughs in Diophantine approximation the Diophantine approximation type of α would be computable to within arbitrary precision.
38 Schanuel s Conjecture Theorem (Lindemann-Weierstrass) If a 1,..., a n are algebraic numbers linearly independent over Q, then e a 1,..., e an are algebraically independent.
39 Schanuel s Conjecture Theorem (Lindemann-Weierstrass) If a 1,..., a n are algebraic numbers linearly independent over Q, then e a 1,..., e an are algebraically independent. Schanuel s Conjecture If z 1,..., z n are complex numbers linearly independent over Q then some n-element subset of {z 1,..., z n, e z 1,..., e zn } is algebraically independent.
40 Schanuel s Conjecture Theorem (Lindemann-Weierstrass) If a 1,..., a n are algebraic numbers linearly independent over Q, then e a 1,..., e an are algebraically independent. Schanuel s Conjecture If z 1,..., z n are complex numbers linearly independent over Q then some n-element subset of {z 1,..., z n, e z 1,..., e zn } is algebraically independent. Example By Schanuel s conjecture some two-element subset of {1, πi, e 1, e πi } is algebraically independent.
41 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1
42 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) t
43 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a b t
44 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a b t
45 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a t* b t non-trivial zero t transcendental
46 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a t* b t non-trivial zero t transcendental
47 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a b t
48 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a b t
49 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a t* b t
50 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a t* b t Can this situation arise?
51 The BOUNDED-ZERO Problem m Real-valued exponential polynomial f (t) = P j (t)e λ j t j=1 f(t) a t* b t Easily! For example, f (t) = 2 + e it + e it.
52 Laurent Polynomials and Factorisation Example Write f (t) = 2 + e it + e it in the form f (t) = P(e it ) for the Laurent polynomial P(z) = 2 + z + z 1.
53 Laurent Polynomials and Factorisation Example Write f (t) = 2 + e it + e it in the form f (t) = P(e it ) for the Laurent polynomial P(z) = 2 + z + z 1. Factorisation P(z) = (1 + z)(1 + z 1 ) induces a factorisation f (t) = (1 + e it ) (1 e it ) }{{}}{{} f 1 (t) f 2 (t)
54 Laurent Polynomials and Factorisation Example Write f (t) = 2 + e it + e it in the form f (t) = P(e it ) for the Laurent polynomial P(z) = 2 + z + z 1. Factorisation P(z) = (1 + z)(1 + z 1 ) induces a factorisation f (t) = (1 + e it ) (1 e it ) }{{}}{{} f 1 (t) f 2 (t) Common zeros of f 1 and f 2 are tangential zeros of f
55 Laurent Polynomials and Factorisation Example Write f (t) = 2 + e it + e it in the form f (t) = P(e it ) for the Laurent polynomial P(z) = 2 + z + z 1. Factorisation P(z) = (1 + z)(1 + z 1 ) induces a factorisation f (t) = (1 + e it ) (1 e it ) }{{}}{{} f 1 (t) f 2 (t) Common zeros of f 1 and f 2 are tangential zeros of f Idea: factorise f.
56 Laurent Polynomials and Factorisation Example Write f (t) = 2 + e it + e it in the form f (t) = P(e it ) for the Laurent polynomial P(z) = 2 + z + z 1. Factorisation P(z) = (1 + z)(1 + z 1 ) induces a factorisation f (t) = (1 + e it ) (1 e it ) }{{}}{{} f 1 (t) f 2 (t) Common zeros of f 1 and f 2 are tangential zeros of f Idea: factorise f. Noting that factors may be complex-valued!
57 The Real Case Any exponential polynomial f (t) can be written with f (t) = P(t, e a 1t,..., e amt ) P C[x, x ±1 1,..., x ±1 m ] and {a 1,..., a m } a set of real and imaginary algebraic numbers that is linearly independent over Q.
58 The Real Case Any exponential polynomial f (t) can be written with f (t) = P(t, e a 1t,..., e amt ) P C[x, x ±1 1,..., x ±1 m ] and {a 1,..., a m } a set of real and imaginary algebraic numbers that is linearly independent over Q. Lemma Assuming Schanuel s conjecture, if f is real valued and P is irreducible then f has no tangential zeros.
59 The Real Case Any exponential polynomial f (t) can be written with f (t) = P(t, e a 1t,..., e amt ) P C[x, x ±1 1,..., x ±1 m ] and {a 1,..., a m } a set of real and imaginary algebraic numbers that is linearly independent over Q. Lemma Assuming Schanuel s conjecture, if f is real valued and P is irreducible then f has no tangential zeros. Complex case requires some new ideas...
60 The Unbounded Case there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns the ones we don t know we don t know.
61 Continued Fractions Finite continued fractions: [3, 7, 15, 1, 292] =
62 Continued Fractions Finite continued fractions: [3, 7, 15, 1, 292] = =
63 Continued Fractions Finite continued fractions: [3, 7, 15, 1, 292] = 3 + Infinite continued fractions: = [a 0, a 1, a 2, a 3,...] = a a a 2 + a 3 +
64 Real Algebraic Numbers Theorem The continued fraction expansion of a real quadratic irrational number is periodic.
65 Real Algebraic Numbers Theorem The continued fraction expansion of a real quadratic irrational number is periodic. 389 = [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,...]
66 Real Algebraic Numbers Theorem The continued fraction expansion of a real quadratic irrational number is periodic. 389 = [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,...] What about numbers of degree 3?
67 Real Algebraic Numbers Theorem The continued fraction expansion of a real quadratic irrational number is periodic. 389 = [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,...] What about numbers of degree 3? 3 2 = [1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1,...]
68 Real Algebraic Numbers Theorem The continued fraction expansion of a real quadratic irrational number is periodic. 389 = [19, 1, 2, 1, 1, 1, 1, 2, 1, 38, 1, 2, 1, 1, 1, 1, 2, 1, 38,...] What about numbers of degree 3? 3 2 = [1, 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10, 2, 1, 4, 12, 2, 3, 2, 1 3, 4, 1, 1, 2, 14, 3, 12, 1, 15, 3, 1, 4, 534, 1, 1, 5, 1, 1,...] Lang and Trotter: no significant departure from random behaviour
69 An Open Problem [... ] no continued fraction development of an algebraic number of higher degree than the second is known. It is not even known if such a development has bounded elements. A. Khinchin
70 An Open Problem [... ] no continued fraction development of an algebraic number of higher degree than the second is known. It is not even known if such a development has bounded elements. A. Khinchin Is there an algebraic number of degree higher than two whose simple continued fraction has unbounded partial quotients? Does every such number have unbounded partial quotients? R. K. Guy, 2004
71 A Mathematical Obstacle at Dimension 9 Given x = [a 0, a 1, a 2,...], define S(x) = sup n N a n.
72 A Mathematical Obstacle at Dimension 9 Given x = [a 0, a 1, a 2,...], define S(x) = sup n N a n. Theorem (arxiv: , 2015) If the ZERO PROBLEM is decidable at dimension 9 then {x R A : S(x) < } is recursively enumerable.
73 A Mathematical Obstacle at Dimension 9 Given x = [a 0, a 1, a 2,...], define S(x) = sup n N a n. Theorem (arxiv: , 2015) If the ZERO PROBLEM is decidable at dimension 9 then {x R A : S(x) < } is recursively enumerable. Remark Perhaps this set is recursive it may even be or R A. However proving recursive enumerability would be a significant achievement.
74 Diophantine Approximation How well can one approximate a real number x with rationals? x m n
75 Diophantine Approximation How well can one approximate a real number x with rationals? x m n Theorem (Dirichlet 1842) There are infinitely many integers m, n such that x m < 1 n n 2.
76 Diophantine Approximation How well can one approximate a real number x with rationals? x m n Theorem (Dirichlet 1842) There are infinitely many integers m, n such that x m < 1 n n 2. S(x) < if and only if there exists ε > 0 such that x m < ε n n 2 has no solutions.
77 Diophantine Approximation How well can one approximate a real number x with rationals? x m n Theorem (Dirichlet 1842) There are infinitely many integers m, n such that x m < 1 n n 2. S(x) < if and only if there exists ε > 0 such that x m < ε n n 2 has no solutions. Relate this to the existence of zeros of order-9 exponential polynomial f (t) with terms e ixt and e it.
78 The ZERO Problem ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0?
79 The ZERO Problem ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0? Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO.
80 The ZERO Problem ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0? Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO. Diophantine approximation Kronecker s Theorem on simultaneous Diophantine approximation.
81 The ZERO Problem ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0? Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO. Diophantine approximation Kronecker s Theorem on simultaneous Diophantine approximation. Baker s Theorem on lower bounds for linear forms in logarithms of algebraic numbers.
82 The ZERO Problem ZERO Problem Instance: f Question: Is there t R 0 such that f (t) = 0? Theorem (arxiv: , 2015) In dimension 8 or less, ZERO reduces to BOUNDED-ZERO. Diophantine approximation Kronecker s Theorem on simultaneous Diophantine approximation. Baker s Theorem on lower bounds for linear forms in logarithms of algebraic numbers. Model theory of the reals o-minimality of (R, <, +,, e x, 0, 1).
83 Conclusion and Perspectives
84 The Discrete Case A linear recurrence sequence is a sequence u 0, u 1, u 2,... of integers such that there exist constants a 1,..., a k, such that for all n 0. u n+k = a 1 u n+k 1 + a 2 u n+k a k u n
85 The Discrete Case A linear recurrence sequence is a sequence u 0, u 1, u 2,... of integers such that there exist constants a 1,..., a k, such that for all n 0. u n+k = a 1 u n+k 1 + a 2 u n+k a k u n Theorem (Skolem 1934; Mahler 1935, 1956; Lech 1953) The set of zeros of a linear recurrence sequence is semi-linear: {n : u n = 0} = F A 1... A l where F is finite and each A i is a full arithmetic progression.
86 The Discrete Case A linear recurrence sequence is a sequence u 0, u 1, u 2,... of integers such that there exist constants a 1,..., a k, such that for all n 0. u n+k = a 1 u n+k 1 + a 2 u n+k a k u n Theorem (Skolem 1934; Mahler 1935, 1956; Lech 1953) The set of zeros of a linear recurrence sequence is semi-linear: {n : u n = 0} = F A 1... A l where F is finite and each A i is a full arithmetic progression. Theorem (Berstel and Mignotte 1976) In Skolem-Mahler-Lech, the infinite part (arithmetic progressions A 1,..., A l ) is fully constructive.
87 The Skolem Problem Skolem Problem Does n such that u n = 0?
88 The Skolem Problem Skolem Problem Does n such that u n = 0? It is faintly outrageous that this problem is still open; it is saying that we do not know how to decide the Halting Problem even for linear automata! Terence Tao
89 The Skolem Problem Skolem Problem Does n such that u n = 0? It is faintly outrageous that this problem is still open; it is saying that we do not know how to decide the Halting Problem even for linear automata! Terence Tao... a mathematical embarrassment... Richard Lipton
90 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0?
91 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0? Not a mathematical embarrassment!
92 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0? Not a mathematical embarrassment! Even the bounded problem is hard (apparently).
93 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0? Not a mathematical embarrassment! Even the bounded problem is hard (apparently). Formidable mathematical obstacle at dimension 9 in the unbounded case.
94 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0? Not a mathematical embarrassment! Even the bounded problem is hard (apparently). Formidable mathematical obstacle at dimension 9 in the unbounded case. The infinite-zeros problem is also hard.
95 Wrapping Things Up Continuous Skolem Problem Does t such that f (t) = 0? Not a mathematical embarrassment! Even the bounded problem is hard (apparently). Formidable mathematical obstacle at dimension 9 in the unbounded case. The infinite-zeros problem is also hard. Diophantine-approximation techniques unavoidable.
Reachability Problems for Continuous Linear Dynamical Systems
Reachability Problems for Continuous Linear Dynamical Systems James Worrell Department of Computer Science, Oxford University (Joint work with Ventsislav Chonev and Joël Ouaknine) FICS 2015 September 12th,
More informationDecision Problems for Linear Recurrence Sequences
Decision Problems for Linear Recurrence Sequences Joël Ouaknine Department of Computer Science, Oxford University (Joint work with James Worrell and Matt Daws) RP 2012 Bordeaux, September 2012 Termination
More informationReachability problems for Markov chains
1 Reachability problems for Markov chains S Akshay Dept of CSE, IIT Bombay 1 st CMI Alumni Conference, Chennai 10 Jan 2015 2 Markov chains: a basic model for probabilistic systems 1 3 2 3 3 1 2 1 1 2 2
More informationLINEAR FORMS IN LOGARITHMS
LINEAR FORMS IN LOGARITHMS JAN-HENDRIK EVERTSE April 2011 Literature: T.N. Shorey, R. Tijdeman, Exponential Diophantine equations, Cambridge University Press, 1986; reprinted 2008. 1. Linear forms in logarithms
More informationRECENT RESULTS ON LINEAR RECURRENCE SEQUENCES
RECENT RESULTS ON LINEAR RECURRENCE SEQUENCES Jan-Hendrik Evertse (Leiden) General Mathematics Colloquium, Delft May 19, 2005 1 INTRODUCTION A linear recurrence sequence U = {u n } n=0 (in C) is a sequence
More informationOn the Polytope Escape Problem for Continuous Linear Dynamical Systems
On the Polytope Escape Problem for Continuous Linear Dynamical Systems Joël Ouaknine MPI-SWS and Oxford U joel@mpi-swsorg João Sousa-Pinto Oxford U jspinto@csoxacuk James Worrell Oxford U jbw@csoxacuk
More informationQUARTIC POWER SERIES IN F 3 ((T 1 )) WITH BOUNDED PARTIAL QUOTIENTS. Alain Lasjaunias
QUARTIC POWER SERIES IN F 3 ((T 1 )) WITH BOUNDED PARTIAL QUOTIENTS Alain Lasjaunias 1991 Mathematics Subject Classification: 11J61, 11J70. 1. Introduction. We are concerned with diophantine approximation
More informationP -adic root separation for quadratic and cubic polynomials
P -adic root separation for quadratic and cubic polynomials Tomislav Pejković Abstract We study p-adic root separation for quadratic and cubic polynomials with integer coefficients. The quadratic and reducible
More informationModeling & Control of Hybrid Systems. Chapter 7 Model Checking and Timed Automata
Modeling & Control of Hybrid Systems Chapter 7 Model Checking and Timed Automata Overview 1. Introduction 2. Transition systems 3. Bisimulation 4. Timed automata hs check.1 1. Introduction Model checking
More informationEffective Positivity Problems for Simple Linear Recurrence Sequences
Effective Positivity Problems for Simple Linear Recurrence Sequences Joël Ouaknine and James Worrell Department of Computer Science, Oxford University, UK {joel,jbw}@cs.ox.ac.uk Abstract. We consider two
More informationEffective Divergence Analysis for Linear Recurrence Sequences
Effective Divergence Analysis for Linear Recurrence Sequences Shaull Almagor Department of Computer Science, Oxford University, UK shaull.almagor@cs.ox.ac.uk Brynmor Chapman MIT CSAIL brynmor@mit.edu Mehran
More informationCONSEQUENCES OF POWER SERIES REPRESENTATION
CONSEQUENCES OF POWER SERIES REPRESENTATION 1. The Uniqueness Theorem Theorem 1.1 (Uniqueness). Let Ω C be a region, and consider two analytic functions f, g : Ω C. Suppose that S is a subset of Ω that
More informationEffective results for unit equations over finitely generated domains. Jan-Hendrik Evertse
Effective results for unit equations over finitely generated domains Jan-Hendrik Evertse Universiteit Leiden Joint work with Kálmán Győry (Debrecen) Paul Turán Memorial Conference, Budapest August 22,
More information) = nlog b ( m) ( m) log b ( ) ( ) = log a b ( ) Algebra 2 (1) Semester 2. Exponents and Logarithmic Functions
Exponents and Logarithmic Functions Algebra 2 (1) Semester 2! a. Graph exponential growth functions!!!!!! [7.1]!! - y = ab x for b > 0!! - y = ab x h + k for b > 0!! - exponential growth models:! y = a(
More informationThe Orbit Problem in Zero and One Dimensions
The Orbit Problem in Zero and One Dimensions Master's dissertation of Ventsislav K. Chonev Supervised by Joël Ouaknine and James Worrell 1 Contents Contents 2 1 Introduction 3 1.1 Verication of linear
More informationNOTES ON IRRATIONALITY AND TRANSCENDENCE
NOTES ON IRRATIONALITY AND TRANSCENDENCE Frits Beukers September, 27 Introduction. Irrationality Definition.. Let α C. We call α irrational when α Q. Proving irrationality and transcendence of numbers
More informationLecture Notes. Advanced Discrete Structures COT S
Lecture Notes Advanced Discrete Structures COT 4115.001 S15 2015-01-13 Recap Divisibility Prime Number Theorem Euclid s Lemma Fundamental Theorem of Arithmetic Euclidean Algorithm Basic Notions - Section
More informationMCPS Algebra 2 and Precalculus Standards, Categories, and Indicators*
Content Standard 1.0 (HS) Patterns, Algebra and Functions Students will algebraically represent, model, analyze, and solve mathematical and real-world problems involving functional patterns and relationships.
More informationarxiv: v3 [cs.cc] 22 Jun 2016
0 On the Complexity of the Orbit Problem VENTSISLAV CHONEV, Institute of Science and Technology Austria JOËL OUAKNINE, University of Oxford JAMES WORRELL, University of Oxford arxiv:1303.2981v3 [cs.cc]
More informationDIOPHANTINE EQUATIONS AND DIOPHANTINE APPROXIMATION
DIOPHANTINE EQUATIONS AND DIOPHANTINE APPROXIMATION JAN-HENDRIK EVERTSE 1. Introduction Originally, Diophantine approximation is the branch of number theory dealing with problems such as whether a given
More informationDay 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x x 2-9x x 2 + 6x + 5
Day 6: 6.4 Solving Polynomial Equations Warm Up: Factor. 1. x 2-2x - 15 2. x 2-9x + 14 3. x 2 + 6x + 5 Solving Equations by Factoring Recall the factoring pattern: Difference of Squares:...... Note: There
More informationOn a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set
On a Question of Wintner Concerning the Sequence of Integers Composed of Primes from a Given Set by Jeongsoo Kim A thesis presented to the University of Waterloo in fulfillment of the thesis requirement
More informationNORMAL NUMBERS AND UNIFORM DISTRIBUTION (WEEKS 1-3) OPEN PROBLEMS IN NUMBER THEORY SPRING 2018, TEL AVIV UNIVERSITY
ORMAL UMBERS AD UIFORM DISTRIBUTIO WEEKS -3 OPE PROBLEMS I UMBER THEORY SPRIG 28, TEL AVIV UIVERSITY Contents.. ormal numbers.2. Un-natural examples 2.3. ormality and uniform distribution 2.4. Weyl s criterion
More informationAutomata-theoretic analysis of hybrid systems
Automata-theoretic analysis of hybrid systems Madhavan Mukund SPIC Mathematical Institute 92, G N Chetty Road Chennai 600 017, India Email: madhavan@smi.ernet.in URL: http://www.smi.ernet.in/~madhavan
More informationBadly approximable numbers over imaginary quadratic fields
Badly approximable numbers over imaginary quadratic fields Robert Hines University of Colorado, Boulder March 11, 2018 Simple continued fractions: Definitions 1 The Euclidean algorthim a = ba 0 + r 0,
More informationL-Polynomials of Curves over Finite Fields
School of Mathematical Sciences University College Dublin Ireland July 2015 12th Finite Fields and their Applications Conference Introduction This talk is about when the L-polynomial of one curve divides
More informationOn Computably Enumerable Sets over Function Fields
On Computably Enumerable Sets over Function Fields Alexandra Shlapentokh East Carolina University Joint Meetings 2017 Atlanta January 2017 Some History Outline 1 Some History A Question and the Answer
More informationP -ADIC ROOT SEPARATION FOR QUADRATIC AND CUBIC POLYNOMIALS. Tomislav Pejković
RAD HAZU. MATEMATIČKE ZNANOSTI Vol. 20 = 528 2016): 9-18 P -ADIC ROOT SEPARATION FOR QUADRATIC AND CUBIC POLYNOMIALS Tomislav Pejković Abstract. We study p-adic root separation for quadratic and cubic
More informationZero estimates for polynomials inspired by Hilbert s seventh problem
Radboud University Faculty of Science Institute for Mathematics, Astrophysics and Particle Physics Zero estimates for polynomials inspired by Hilbert s seventh problem Author: Janet Flikkema s4457242 Supervisor:
More informationEcon Slides from Lecture 1
Econ 205 Sobel Econ 205 - Slides from Lecture 1 Joel Sobel August 23, 2010 Warning I can t start without assuming that something is common knowledge. You can find basic definitions of Sets and Set Operations
More informationINDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC
INDEX UNIT 3 TSFX REFERENCE MATERIALS 2014 ALGEBRA AND ARITHMETIC Surds Page 1 Algebra of Polynomial Functions Page 2 Polynomial Expressions Page 2 Expanding Expressions Page 3 Factorising Expressions
More informationNotes on uniform convergence
Notes on uniform convergence Erik Wahlén erik.wahlen@math.lu.se January 17, 2012 1 Numerical sequences We begin by recalling some properties of numerical sequences. By a numerical sequence we simply mean
More informationPade Approximations and the Transcendence
Pade Approximations and the Transcendence of π Ernie Croot March 9, 27 1 Introduction Lindemann proved the following theorem, which implies that π is transcendental: Theorem 1 Suppose that α 1,..., α k
More information1. If 1, ω, ω 2, -----, ω 9 are the 10 th roots of unity, then (1 + ω) (1 + ω 2 ) (1 + ω 9 ) is A) 1 B) 1 C) 10 D) 0
4 INUTES. If, ω, ω, -----, ω 9 are the th roots of unity, then ( + ω) ( + ω ) ----- ( + ω 9 ) is B) D) 5. i If - i = a + ib, then a =, b = B) a =, b = a =, b = D) a =, b= 3. Find the integral values for
More informationCollege Algebra. Basics to Theory of Equations. Chapter Goals and Assessment. John J. Schiller and Marie A. Wurster. Slide 1
College Algebra Basics to Theory of Equations Chapter Goals and Assessment John J. Schiller and Marie A. Wurster Slide 1 Chapter R Review of Basic Algebra The goal of this chapter is to make the transition
More informationOn the Total Variation Distance of Labelled Markov Chains
On the Total Variation Distance of Labelled Markov Chains Taolue Chen Stefan Kiefer Middlesex University London, UK University of Oxford, UK CSL-LICS, Vienna 4 July 04 Labelled Markov Chains (LMCs) a c
More informationFactoring univariate polynomials over the rationals
Factoring univariate polynomials over the rationals Tommy Hofmann TU Kaiserslautern November 21, 2017 Tommy Hofmann Factoring polynomials over the rationals November 21, 2017 1 / 31 Factoring univariate
More information1 Prologue (X 111 X 14 )F = X 11 (X 100 X 3 )F = X 11 (X 200 X 6 ) = X 211 X 17.
Prologue In this chapter we give a couple of examples where the method of auxiliary polynomials is used for problems that have no diophantine character. Thus we are not following Sam Goldwyn s advice to
More informationMahler measure as special values of L-functions
Mahler measure as special values of L-functions Matilde N. Laĺın Université de Montréal mlalin@dms.umontreal.ca http://www.dms.umontreal.ca/~mlalin CRM-ISM Colloquium February 4, 2011 Diophantine equations
More informationTRANSCENDENTAL NUMBERS AND PERIODS. Contents
TRANSCENDENTAL NUMBERS AND PERIODS JAMES CARLSON Contents. Introduction.. Diophantine approximation I: upper bounds 2.2. Diophantine approximation II: lower bounds 4.3. Proof of the lower bound 5 2. Periods
More informationThe Schanuel paradigm
The Schanuel paradigm Jonathan Pila University of Oxford 2014 Clay Research Conference, Oxford Themes Schanuel s conjecture, analogues, in particular Functional analogues and their connections with certain
More informationA video College Algebra course & 6 Enrichment videos
A video College Algebra course & 6 Enrichment videos Recorded at the University of Missouri Kansas City in 1998. All times are approximate. About 43 hours total. Available on YouTube at http://www.youtube.com/user/umkc
More informationRamsey theory. Andrés Eduardo Caicedo. Graduate Student Seminar, October 19, Department of Mathematics Boise State University
Andrés Eduardo Department of Mathematics Boise State University Graduate Student Seminar, October 19, 2011 Thanks to the NSF for partial support through grant DMS-0801189. My work is mostly in set theory,
More informationA NON-ARCHIMEDEAN AX-LINDEMANN THEOREM
A NON-ARCHIMEDEAN AX-LINDEMANN THEOREM François Loeser Sorbonne University, formerly Pierre and Marie Curie University, Paris Model Theory of Valued fields Institut Henri Poincaré, March 8, 2018. P. 1
More informationDefining Valuation Rings
East Carolina University, Greenville, North Carolina, USA June 8, 2018 Outline 1 What? Valuations and Valuation Rings Definability Questions in Number Theory 2 Why? Some Questions and Answers Becoming
More informationComputing coefficients of modular forms
Computing coefficients of modular forms (Work in progress; extension of results of Couveignes, Edixhoven et al.) Peter Bruin Mathematisch Instituut, Universiteit Leiden Théorie des nombres et applications
More informationNotes on State Minimization
U.C. Berkeley CS172: Automata, Computability and Complexity Handout 1 Professor Luca Trevisan 2/3/2015 Notes on State Minimization These notes present a technique to prove a lower bound on the number of
More informationIntroduction to Diophantine methods Michel Waldschmidt miw/courshcmuns2007.html. Sixth course: september 21, 2007.
Introduction to Diophantine methods Michel Waldschmidt http://www.math.jussieu.fr/ miw/courshcmuns2007.html Sixth course: september 2, 2007. 8 2.2.2 Diophantine approximation and applications Diophantine
More informationA remark about the positivity problem of fourth order linear recurrence sequences
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 18, Number 1, June 2014 Available online at http://acutm.math.ut.ee A remark about the positivity problem of fourth order linear recurrence
More informationINSPECT Algebra I Summative Assessment Summary
and Quantity The Real System Quantities Seeing Structure in Use properties of rational and irrational numbers. Reason quantitatively and use units to solve problems. Interpret the structure of expressions.
More informationO-minimality and Diophantine Geometry
O-minimality and Diophantine Geometry Jonathan Pila University of Oxford ICM 2014, Seoul Themes Rational points on definable sets in o-minimal structures and applications to some Diophantine problems comprehended
More informationCLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS
CLOSED FORM CONTINUED FRACTION EXPANSIONS OF SPECIAL QUADRATIC IRRATIONALS DANIEL FISHMAN AND STEVEN J. MILLER ABSTRACT. We derive closed form expressions for the continued fractions of powers of certain
More informationTranscendental Number Theory: Recent Results and Conjectures
Days of Mathematics Oulu, January 9, 2004 Transcendental Number Theory: Recent Results and Conjectures Michel Waldschmidt http://www.math.jussieu.fr/ miw/ http://math.oulu.fi/matematiikanpaivat/matematiikanpaivat.html
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationTest 2. Monday, November 12, 2018
Test 2 Monday, November 12, 2018 Instructions. The only aids allowed are a hand-held calculator and one cheat sheet, i.e. an 8.5 11 sheet with information written on one side in your own handwriting. No
More informationAlgebra II. A2.1.1 Recognize and graph various types of functions, including polynomial, rational, and algebraic functions.
Standard 1: Relations and Functions Students graph relations and functions and find zeros. They use function notation and combine functions by composition. They interpret functions in given situations.
More informationApproximation exponents for algebraic functions in positive characteristic
ACTA ARITHMETICA LX.4 (1992) Approximation exponents for algebraic functions in positive characteristic by Bernard de Mathan (Talence) In this paper, we study rational approximations for algebraic functions
More informationAutomorphisms of varieties and critical density. Jason Bell (U. Michigan)
Automorphisms of varieties and critical density Jason Bell (U. Michigan) belljp@umich.edu 1 Let X be a topological space and let S be an infinite subset of X. We say that S is critically dense if every
More informationOn values of Modular Forms at Algebraic Points
On values of Modular Forms at Algebraic Points Jing Yu National Taiwan University, Taipei, Taiwan August 14, 2010, 18th ICFIDCAA, Macau Hermite-Lindemann-Weierstrass In value distribution theory the exponential
More informationIntroduction to Techniques for Counting
Introduction to Techniques for Counting A generating function is a device somewhat similar to a bag. Instead of carrying many little objects detachedly, which could be embarrassing, we put them all in
More informationOn repdigits as product of consecutive Lucas numbers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 24, 2018, No. 3, 5 102 DOI: 10.7546/nntdm.2018.24.3.5-102 On repdigits as product of consecutive Lucas numbers
More informationSolving Third Order Linear Differential Equations in Terms of Second Order Equations
Solving Third Order Linear Differential Equations in Terms of Second Order Equations Mark van Hoeij (Florida State University) ISSAC 2007 Talk presented by: George Labahn (University of Waterloo) Notations.
More informationRobust computation with dynamical systems
Robust computation with dynamical systems Olivier Bournez 1 Daniel S. Graça 2 Emmanuel Hainry 3 1 École Polytechnique, LIX, Palaiseau, France 2 DM/FCT, Universidade do Algarve, Faro & SQIG/Instituto de
More informationNotes on Complex Analysis
Michael Papadimitrakis Notes on Complex Analysis Department of Mathematics University of Crete Contents The complex plane.. The complex plane...................................2 Argument and polar representation.........................
More informationRoot separation for irreducible integer polynomials
Root separation for irreducible integer polynomials Yann Bugeaud and Andrej Dujella 1 Introduction The height H(P ) of an integer polynomial P (x) is the maximum of the absolute values of its coefficients.
More informationCharacterizing planar polynomial vector fields with an elementary first integral
Characterizing planar polynomial vector fields with an elementary first integral Sebastian Walcher (Joint work with Jaume Llibre and Chara Pantazi) Lleida, September 2016 The topic Ultimate goal: Understand
More informationSimultaneous Diophantine Approximation with Excluded Primes. László Babai Daniel Štefankovič
Simultaneous Diophantine Approximation with Excluded Primes László Babai Daniel Štefankovič Dirichlet (1842) Simultaneous Diophantine Approximation Given reals integers α,,...,, 1 α 2 α n Q r1,..., r n
More informationComplexity of Reachability, Mortality and Freeness Problems for Matrix Semigroups
Complexity of Reachability, Mortality and Freeness Problems for Matrix Semigroups Paul C. Bell Department of Computer Science Loughborough University P.Bell@lboro.ac.uk Co-authors for todays topics: V.
More informationExercises Chapter II.
Page 64 Exercises Chapter II. 5. Let A = (1, 2) and B = ( 2, 6). Sketch vectors of the form X = c 1 A + c 2 B for various values of c 1 and c 2. Which vectors in R 2 can be written in this manner? B y
More informationThe hyperbolic Ax-Lindemann-Weierstraß conjecture
The hyperbolic Ax-Lindemann-Weierstraß conjecture B. Klingler (joint work with E.Ullmo and A.Yafaev) Université Paris 7 ICM Satellite Conference, Daejeon Plan of the talk: (1) Motivation: Hodge locus and
More informationResults and open problems related to Schmidt s Subspace Theorem. Jan-Hendrik Evertse
Results and open problems related to Schmidt s Subspace Theorem Jan-Hendrik Evertse Universiteit Leiden 29 ièmes Journées Arithmétiques July 6, 2015, Debrecen Slides can be downloaded from http://pub.math.leidenuniv.nl/
More informationarxiv: v1 [math.nt] 13 Mar 2017
MAHLER TAKES A REGULAR VIEW OF ZAREMBA MICHAEL COONS arxiv:170304287v1 mathnt] 13 Mar 2017 Abstract In the theory of continued fractions, Zaremba s conjecture states that there is a positive integer M
More informationAutomatic Sequences and Transcendence of Real Numbers
Automatic Sequences and Transcendence of Real Numbers Wu Guohua School of Physical and Mathematical Sciences Nanyang Technological University Sendai Logic School, Tohoku University 28 Jan, 2016 Numbers
More informationHTP over Algebraic Extensions of Q: Normforms vs. Elliptic Curves
HTP over Algebraic Extensions of Q: Normforms vs. Elliptic Curves Alexandra Shlapentokh East Carolina University Number Theory and Computability, ICMS, June 2007 A Talk in Two Rounds Round I: Normforms
More informationDiscrete abstractions of hybrid systems for verification
Discrete abstractions of hybrid systems for verification George J. Pappas Departments of ESE and CIS University of Pennsylvania pappasg@ee.upenn.edu http://www.seas.upenn.edu/~pappasg DISC Summer School
More informationOn Reachability for Hybrid Automata over Bounded Time
On Reachability for Hybrid Automata over Bounded Time Thomas Brihaye, Laurent Doyen 2, Gilles Geeraerts 3, Joël Ouaknine 4, Jean-François Raskin 3, and James Worrell 4 Université de Mons, Belgium 2 LSV,
More informationMATH 1040 Objectives List
MATH 1040 Objectives List Textbook: Calculus, Early Transcendentals, 7th edition, James Stewart Students should expect test questions that require synthesis of these objectives. Unit 1 WebAssign problems
More informationPre-Calculus and Trigonometry Capacity Matrix
Review Polynomials A1.1.4 A1.2.5 Add, subtract, multiply and simplify polynomials and rational expressions Solve polynomial equations and equations involving rational expressions Review Chapter 1 and their
More informationLECTURE 22: COUNTABLE AND UNCOUNTABLE SETS
LECTURE 22: COUNTABLE AND UNCOUNTABLE SETS 1. Introduction To end the course we will investigate various notions of size associated to subsets of R. The simplest example is that of cardinality - a very
More informationConference on Diophantine Analysis and Related Fields 2006 in honor of Professor Iekata Shiokawa Keio University Yokohama March 8, 2006
Diophantine analysis and words Institut de Mathématiques de Jussieu + CIMPA http://www.math.jussieu.fr/ miw/ March 8, 2006 Conference on Diophantine Analysis and Related Fields 2006 in honor of Professor
More informationMETRIC HEIGHTS ON AN ABELIAN GROUP
ROCKY MOUNTAIN JOURNAL OF MATHEMATICS Volume 44, Number 6, 2014 METRIC HEIGHTS ON AN ABELIAN GROUP CHARLES L. SAMUELS ABSTRACT. Suppose mα) denotes the Mahler measure of the non-zero algebraic number α.
More informationPart IA Numbers and Sets
Part IA Numbers and Sets Definitions Based on lectures by A. G. Thomason Notes taken by Dexter Chua Michaelmas 2014 These notes are not endorsed by the lecturers, and I have modified them (often significantly)
More informationRamanujan s first letter to Hardy: 5 + = 1 + e 2π 1 + e 4π 1 +
Ramanujan s first letter to Hardy: e 2π/ + + 1 = 1 + e 2π 2 2 1 + e 4π 1 + e π/ 1 e π = 2 2 1 + 1 + e 2π 1 + Hardy: [These formulas ] defeated me completely. I had never seen anything in the least like
More informationHybrid systems and computer science a short tutorial
Hybrid systems and computer science a short tutorial Eugene Asarin Université Paris 7 - LIAFA SFM 04 - RT, Bertinoro p. 1/4 Introductory equations Hybrid Systems = Discrete+Continuous SFM 04 - RT, Bertinoro
More informationSolution of the 8 th Homework
Solution of the 8 th Homework Sangchul Lee December 8, 2014 1 Preinary 1.1 A simple remark on continuity The following is a very simple and trivial observation. But still this saves a lot of words in actual
More informationReport on some recent advances in Diophantine approximation
Report on some recent advances in Diophantine approximation Michel Waldschmidt Université Pierre et Marie Curie Paris 6, UMR 7586 IMJ Institut de Mathématiques de Jussieu, 4, Place Jussieu, Paris, F 75005
More informationNotes on Equidistribution
otes on Equidistribution Jacques Verstraëte Department of Mathematics University of California, San Diego La Jolla, CA, 92093. E-mail: jacques@ucsd.edu. Introduction For a real number a we write {a} for
More informationMA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES
MA257: INTRODUCTION TO NUMBER THEORY LECTURE NOTES 2018 57 5. p-adic Numbers 5.1. Motivating examples. We all know that 2 is irrational, so that 2 is not a square in the rational field Q, but that we can
More informationRon Paul Curriculum Mathematics 8 Lesson List
Ron Paul Curriculum Mathematics 8 Lesson List 1 Introduction 2 Algebraic Addition 3 Algebraic Subtraction 4 Algebraic Multiplication 5 Week 1 Review 6 Algebraic Division 7 Powers and Exponents 8 Order
More informationAuxiliary functions in transcendence proofs
AWS Lecture 4 Tucson, Monday, March 17, 2008 Auxiliary functions in transcendence proofs Abstract Transcendence proofs most often involve an auxiliary function. Such functions can take several forms. Historically,
More informationRecent results on Timed Systems
Recent results on Timed Systems Time Petri Nets and Timed Automata Béatrice Bérard LAMSADE Université Paris-Dauphine & CNRS berard@lamsade.dauphine.fr Based on joint work with F. Cassez, S. Haddad, D.
More informationFURTHER NUMBER THEORY
FURTHER NUMBER THEORY SANJU VELANI. Dirichlet s Theorem and Continued Fractions Recall a fundamental theorem in the theory of Diophantine approximation. Theorem. (Dirichlet 842). For any real number x
More informationAppendix: a brief history of numbers
Appendix: a brief history of numbers God created the natural numbers. Everything else is the work of man. Leopold Kronecker (1823 1891) Fundamentals of Computing 2017 18 (2, appendix) http://www.dcs.bbk.ac.uk/~michael/foc/foc.html
More informationCritical density, automorphisms, and Harm Derksen. Jason Bell
Critical density, automorphisms, and Harm Derksen Jason Bell 1 Let X be a topological space and let S be an infinite subset of X. We say that S is critically dense if every infinite subset of S is dense
More informationTrigonometry Self-study: Reading: Red Bostock and Chandler p , p , p
Trigonometry Self-study: Reading: Red Bostock Chler p137-151, p157-234, p244-254 Trigonometric functions be familiar with the six trigonometric functions, i.e. sine, cosine, tangent, cosecant, secant,
More informationMahler measure and special values of L-functions
Mahler measure and special values of L-functions Matilde N. Laĺın University of Alberta mlalin@math.ualberta.ca http://www.math.ualberta.ca/~mlalin October 24, 2008 Matilde N. Laĺın (U of A) Mahler measure
More informationExistence of a Limit on a Dense Set, and. Construction of Continuous Functions on Special Sets
Existence of a Limit on a Dense Set, and Construction of Continuous Functions on Special Sets REU 2012 Recap: Definitions Definition Given a real-valued function f, the limit of f exists at a point c R
More informationObservations Homework Checkpoint quizzes Chapter assessments (Possibly Projects) Blocks of Algebra
September The Building Blocks of Algebra Rates, Patterns and Problem Solving Variables and Expressions The Commutative and Associative Properties The Distributive Property Equivalent Expressions Seeing
More informationFINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016
FINITE GROUPS AND EQUATIONS OVER FINITE FIELDS A PROBLEM SET FOR ARIZONA WINTER SCHOOL 2016 PREPARED BY SHABNAM AKHTARI Introduction and Notations The problems in Part I are related to Andrew Sutherland
More informationUNIVERSITY OF CAMBRIDGE
UNIVERSITY OF CAMBRIDGE DOWNING COLLEGE MATHEMATICS FOR ECONOMISTS WORKBOOK This workbook is intended for students coming to Downing College Cambridge to study Economics 2018/ 19 1 Introduction Mathematics
More information