Rate of growth of Dfrequently hypercyclic functions


 Gladys Bridges
 3 years ago
 Views:
Transcription
1 Rate of growth of Dfrequently hypercyclic functions A. Bonilla Departamento de Análisis Matemático Universidad de La Laguna
2 Hypercyclic Definition A (linear and continuous) operator T in a topological vector space X is said to be hypercyclic if there exists a vector x X, also called hypercyclic, whose orbit {T n x : n N} is dense in X.
3 Hypercyclic Definition A (linear and continuous) operator T in a topological vector space X is said to be hypercyclic if there exists a vector x X, also called hypercyclic, whose orbit {T n x : n N} is dense in X. Examples: T a Birkhoff (1929) D MacLane (1952/53)
4 Frequently hypercyclic Definition (Bayart and Grivaux, 2006) Let X be a topological vector space and T : X X a operator. Then a vector x X is called frequently hypercyclic for T if, for every nonempty open subset U of X, dens{n N : T n x U} > 0. The operator T is called frequently hypercyclic if it possesses a frequently hypercyclic vector.
5 Frequently hypercyclic Definition (Bayart and Grivaux, 2006) Let X be a topological vector space and T : X X a operator. Then a vector x X is called frequently hypercyclic for T if, for every nonempty open subset U of X, dens{n N : T n x U} > 0. The operator T is called frequently hypercyclic if it possesses a frequently hypercyclic vector. The lower density of a subset A of N is defined by #{n A : n N} dens(a) = liminf N, N where # denotes the cardinality of a set.
6 Remark A vector x X is frequently hypercyclic for an operator T in X if and only if for every nonempty open subset U of X, there is a strictly increasing sequence (n k ) of positive integers and some C > 0 such that n k Ck and T n k x U para todo k N.
7 Remark A vector x X is frequently hypercyclic for an operator T in X if and only if for every nonempty open subset U of X, there is a strictly increasing sequence (n k ) of positive integers and some C > 0 such that n k Ck and T n k x U para todo k N. Theorem (Bayart and Grivaux, 2006) D is frequently hypercyclic.
8 Differences between hypercyclic and frequently hypercyclic vectors
9 Differences between hypercyclic and frequently hypercyclic vectors Let T : X X be, X a Frechet space HC(T ) = { set of hypercyclic vectors } FHC(T ) = { set of frequently hypercyclic vectors }
10 Differences between hypercyclic and frequently hypercyclic vectors Let T : X X be, X a Frechet space HC(T ) = { set of hypercyclic vectors } FHC(T ) = { set of frequently hypercyclic vectors } Then HC(T ) = k 1 n 0 {x X;T n x B k } where {B k } k 1 is a numerable basis of open sets.
11 Differences between hypercyclic and frequently hypercyclic vectors Let T : X X be, X a Frechet space Then HC(T ) = { set of hypercyclic vectors } FHC(T ) = { set of frequently hypercyclic vectors } HC(T ) = k 1 n 0 {x X;T n x B k } where {B k } k 1 is a numerable basis of open sets. Hence if T is hypercyclic, HC(T ) is a dense G δ set in X, thus it is residual.
12 Differences between hypercyclic and frequently hypercyclic vectors Let T : X X be, X a Frechet space Then HC(T ) = { set of hypercyclic vectors } FHC(T ) = { set of frequently hypercyclic vectors } HC(T ) = k 1 n 0 {x X;T n x B k } where {B k } k 1 is a numerable basis of open sets. Hence if T is hypercyclic, HC(T ) is a dense G δ set in X, thus it is residual. In the known examples of FHC operators, in particular, D and T a FHC(T ) is a dense set of first category.
13 Order of growth and Dfrequently hypercyclic function Given a entire function f is defined for 1 p <, ( 1 2π 1/p,r M p (f,r) = f (re it ) dt) p > 0. 2π 0
14 Order of growth and Dfrequently hypercyclic function Given a entire function f is defined for 1 p <, ( 1 2π 1/p,r M p (f,r) = f (re it ) dt) p > 0. 2π Theorem (K.G. GrosseErdmann, 1990) 0 Let 1 p. (a) For any function ϕ : R + R + with ϕ(r) as r there is a Dhypercyclic entire function f with M p (f,r) ϕ(r) er r for r > 0 sufficiently large. (b) There is no Dhypercyclic entire function f that satisfies M p (f,r) C er r for r > 0, where C > 0.
15 [Bayart and Grivaux, 2006] D is frequently hypercyclic.
16 [Bayart and Grivaux, 2006] D is frequently hypercyclic. [ B. and GrosseErdmann, 2006] ε > 0 there exist entire functions Dfrequently hypercyclic with f (z) Ce (1+ε)r for z = r.
17 Does there exist an entire function Dfrequently hypercyclic f with M (f,r) er r a for z = r sufficiently large?
18 Does there exist an entire function Dfrequently hypercyclic f with M (f,r) er r a for z = r sufficiently large? a 1 2 : No
19 Does there exist an entire function Dfrequently hypercyclic f with M (f,r) er r a for z = r sufficiently large? a 1 2 : No a < 0: Yes
20 Does there exist an entire function Dfrequently hypercyclic f with M (f,r) er r a for z = r sufficiently large? a 1 2 : No a < 0: Yes 0 a < 1 2 :?
21 Does there exist an entire function Dfrequently hypercyclic f with M p (f,r) er r a for z = r sufficiently large?
22 Does there exist an entire function Dfrequently hypercyclic f with M p (f,r) er r a for z = r sufficiently large?
23 Theorem (BlascoBGrosseErdmann, PEMS 2009) Let ψ : R + R + be a function with ψ(r) 0 as r. a)if 1 p 2, there is no Dfrequently hypercyclic entire function f that satisfies M p (f,r) ψ(r) er r 1/2p for r > 0 sufficiently large. b)if 2 < p, there is no Dfrequently hypercyclic entire function f that satisfies M p (f,r) ψ(r) er r 1/4 for r > 0 sufficiently large.
24 Lemma Let ψ : R + R + be a function with ψ(r) 0 as r, 1 < p 2 and f is an entire function that satisfies M p (f,r) ψ(r) er r 1/2p for r > 0 sufficiently large. Then m 1 m f (n) (0) q 0. n=0 where q is the conjugate exponent of p.
25 Proof of theorem For p=1, is consequence of GrosseErdmann s theorem. (There is no Dhypercyclic entire function f that satisfies where C > 0.) M p (f,r) C er r for r > 0,
26 Proof of theorem For p=1, is consequence of GrosseErdmann s theorem. (There is no Dhypercyclic entire function f that satisfies where C > 0.) M p (f,r) C er r for r > 0, For 1 < p 2 is consequence of Lemma. Because given the open set U = {g H(C) : g(0) > 1}
27 Proof of theorem For p=1, is consequence of GrosseErdmann s theorem. (There is no Dhypercyclic entire function f that satisfies where C > 0.) M p (f,r) C er r for r > 0, For 1 < p 2 is consequence of Lemma. Because given the open set U = {g H(C) : g(0) > 1} dens{n N : f (n) 1 U} = liminf m m #{n m : f (n) (0) > 1} m 1 liminf m m f (n) (0) q = 0, n=0
28 Proof of theorem For p=1, is consequence of GrosseErdmann s theorem. (There is no Dhypercyclic entire function f that satisfies where C > 0.) M p (f,r) C er r for r > 0, For 1 < p 2 is consequence of Lemma. Because given the open set U = {g H(C) : g(0) > 1} dens{n N : f (n) 1 U} = liminf m m #{n m : f (n) (0) > 1} m 1 liminf m m f (n) (0) q = 0, n=0 which shows that f is not frequently hypercyclic for the differentiation operator.
29 Proof of the lemma a)let p=2 be. Assume that ψ is decreasing.
30 Proof of the lemma a)let p=2 be. Assume that ψ is decreasing. Then ( M 2 (f,r) = n n=0 a 2 r 2n) 1/2 e r ψ(r) r 1/4 for r > 0 sufficiently large.
31 Proof of the lemma a)let p=2 be. Assume that ψ is decreasing. Then ( M 2 (f,r) = n n=0 a 2 r 2n) 1/2 e r ψ(r) r 1/4 for r > 0 sufficiently large. Hence, for big values of r, f (n) (0) 2 r 2n+1/2 e 2r n=0 ψ(r) 2 (n!) 2 C
32 Fix m 1. Using Stirling formula, we see that the function r r 2n+1/2 e 2r (n!) 2 has its maximum at n + 1/4 of order 1/ n and an inflection point at n n + 1/4 ± Hence, if m n < 2m then
33 Fix m 1. Using Stirling formula, we see that the function r r 2n+1/2 e 2r (n!) 2 has its maximum at n + 1/4 of order 1/ n and an inflection point at n n + 1/4 ± Hence, if m n < 2m then
34 Fix m 1. Using Stirling formula, we see that the function r r 2n+1/2 e 2r (n!) 2 has its maximum at n + 1/4 of order 1/ n and an inflection point at n n + 1/4 ± Hence, if m n < 2m then 3m m r 2n+1/2 e 2r ψ(r) 2 (n!) 2 C 1 ψ(m) 2 1 n n C 1 ψ(m) 2.
35 Integrating f (n) (0) 2 r 2n+1/2 e 2r n=0 ψ(r) 2 (n!) 2 on [m,3m] we obtain for m sufficiently large that C 2m 1 m f (n) (0) 2 Cψ(m) 2, n=m
36 Integrating f (n) (0) 2 r 2n+1/2 e 2r n=0 ψ(r) 2 (n!) 2 on [m,3m] we obtain for m sufficiently large that C 2m 1 m f (n) (0) 2 Cψ(m) 2, n=m hence 2m 1 m f (n) (0) 2 0. n=m
37 Integrating f (n) (0) 2 r 2n+1/2 e 2r n=0 ψ(r) 2 (n!) 2 on [m,3m] we obtain for m sufficiently large that C 2m 1 m f (n) (0) 2 Cψ(m) 2, n=m hence This implies 2m 1 m f (n) (0) 2 0. n=m m 1 m f (n) (0) 2 0. n=0
38 If 1 p < 2, using HausdorffYoung inequality ( a n q r qn) 1 ( ) q M p n z n=0 n=0a n,r and the same ideas for p = 2.
39
40
41 Theorem (BonetB., CAOT 2011) For any function ϕ : R + R + with ϕ(r) as r. If 1 p, there is an entire function Dfrequently hypercyclic f with M p (f,r) ϕ(r) er for z = r sufficiently large. r 1/2p
42 For 1 p and a weight function v, and B p, = B p, (C,v) := {f H(C) : supv(r)m p (f,r) < } r>0 B p,0 = B p,0 (C,v) := {f H(C) : lim r v(r)m p (f,r) = 0}. These spaces are Banach spaces with the norm f p,v := supv(r)m p (f,r). r>0
43 Remarks 1.The polynomials are contained and dense in B p,0 for all 1 p.
44 Remarks 1.The polynomials are contained and dense in B p,0 for all 1 p. 2.B p,0 is separable.
45 Remarks 1.The polynomials are contained and dense in B p,0 for all 1 p. 2.B p,0 is separable. 3.The inclusion B p, H(C) is continuous.
46 Lemma v(r) Let v be a weight function such that sup <. Then the r>0 v(r + 1) differentiation operators D : B p, B p, and D : B p,0 B p,0 are continuous. Example: v(r) = e ar,r > 0,a > 0.
47 Theorem (Grivaux, 2011) Let T be a bounded operator on a complex separable Banach space X. Suppose that, for any countable set A T, span{ker(t λi) : λ T \ A} is dense in X. Then T is frequently hypercyclic.
48 Lemma The following conditions are equivalent for a weight v and 1 p < : (i) {e θz : θ = 1} B p,0. (ii) There is θ C, θ = 1, such that e θz B p,0. (iii) lim r v(r) er r 1 2p = 0
49 Proof of the Lemma Consider f (z) = e z,z C, and write z = r(cost + isint). Now, we can apply the Laplace methods for integrals, for r > 0, 2π ( π ) 1/2e ( 1 ) 2πM p (e z,r) p = e rp cost dt = rp + e rp O. 0 2rp rp
50 Proof of the Lemma Consider f (z) = e z,z C, and write z = r(cost + isint). Now, we can apply the Laplace methods for integrals, for r > 0, 2π ( π ) 1/2e ( 1 ) 2πM p (e z,r) p = e rp cost dt = rp + e rp O. 0 2rp rp This yields, for a certain constant c p > 0 depending only on p, M p (f,r) = c p e r r 1 2p ( 1 ) + e r O. r p 1
51 Proof of the Lemma Consider f (z) = e z,z C, and write z = r(cost + isint). Now, we can apply the Laplace methods for integrals, for r > 0, 2π ( π ) 1/2e ( 1 ) 2πM p (e z,r) p = e rp cost dt = rp + e rp O. 0 2rp rp This yields, for a certain constant c p > 0 depending only on p, M p (f,r) = c p e r r 1 2p ( 1 ) + e r O. r p 1 This implies that for each 1 p < there are d p,d p > 0 and r 0 > 0 such that for each θ = 1 and each r > r 0 d p e r r 1 2p M p (e θz,r) D p e r r 1 2p (1)
52 Theorem Let v be a weight function such that lim r v(r) er r 1 2p = 0 for some 1 p. If the differentiation operator D : B p,0 B p,0 is continuous, then D is frequently hypercyclic.
53 Idea of the proof We must to prove for any countable set A T, E T\A = span({e θz : θ = 1,θ T \ A}) is contained and dense in B p,0.
54 Idea of the proof We must to prove for any countable set A T, E T\A = span({e θz : θ = 1,θ T \ A}) is contained and dense in B p,0. To prove the density, we define the following vector valued functions on the closed unit disc D: H : D B p,0, H(ζ )(z) := e ζ z, ζ D. The function H is well defined, H is holomorphic on D and H : D B p,0 is continuous.
55 We proceed with the proof that E T\A is dense in B p,0 and apply the HahnBanach theorem.
56 We proceed with the proof that E T\A is dense in B p,0 and apply the HahnBanach theorem. Assume that u (B p,0 ) vanishes on E T\A. We must show u = 0.
57 We proceed with the proof that E T\A is dense in B p,0 and apply the HahnBanach theorem. Assume that u (B p,0 ) vanishes on E T\A. We must show u = 0. Since the function u H is holomorphic in D, continuous at the boundary and vanishes at the points ζ T \ A, it is zero in D. In particular (u H) (n) (0) = u(h (n) (0)) = u(z n ) = 0, hence u vanishes on the polynomials. As the polynomials are dense in B p,0, we conclude u = 0.
58 Corollary Let ϕ(r) be a positive function with lim r ϕ(r) =. For each 1 p there is an entire function f such that M p (f,r) ϕ(r) er that is frequently hypercyclic for the differentiation operator D on H(C). r 1 2p
59 Proof It is possible to find a positive increasing continuous function ψ(r + 1) ψ(r) ϕ(r) with lim r ψ(r) = and sup <. r>0 ψ(r)
60 Proof It is possible to find a positive increasing continuous function ψ(r + 1) ψ(r) ϕ(r) with lim r ψ(r) = and sup <. r>0 ψ(r) Define v(r) = r 1 2p ψ(r)e r for r r 0, with r 0 large enough to ensure that v(r) is non increasing on [r 0, [, and v(r) = v(r 0 ) on [0,r 0 ]. One can take r 0 = 1 2p.
61 The following diagram represents our knowledge of possible or impossible growth rates e r /r a for frequent hypercyclicity with respect to the differentiation operator D two weeks ago.
62 The following diagram represents our knowledge of possible or impossible growth rates e r /r a for frequent hypercyclicity with respect to the differentiation operator D two weeks ago.
63 Theorem (DrasinSaksman, in a couple weeks in ArXiv) For any C > 0 there is an entire function Dfrequently hypercyclic f with M p (f,r) C er r a(p) where a(p) = for p [2, ] and a(p) = 2p for p (1,2]
64 Theorem (DrasinSaksman, in a couple weeks in ArXiv) For any C > 0 there is an entire function Dfrequently hypercyclic f with M p (f,r) C er r a(p) where a(p) = for p [2, ] and a(p) = 2p for p (1,2] The construction is direct with no functional analysis, but uses remarkable polynomials of RudinShapiro and de la Vallee Poussin.(Conference in honour of P. Gauthier and K. Gowrisankaran, june 2023, 2011)
Chapter 5: Application: Fourier Series
321 28 9 Chapter 5: Application: Fourier Series For lack of time, this chapter is only an outline of some applications of Functional Analysis and some proofs are not complete. 5.1 Definition. If f L 1
More informationFUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES: QUOTIENT SPACES CHRISTOPHER HEIL 1. Cosets and the Quotient Space Any vector space is an abelian group under the operation of vector addition. So, if you are have studied
More informationSOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS. Abdelaziz Tajmouati Mohammed El berrag. 1. Introduction
italian journal of pure and applied mathematics n. 35 2015 (487 492) 487 SOME RESULTS ON HYPERCYCLICITY OF TUPLE OF OPERATORS Abdelaziz Tajmouati Mohammed El berrag Sidi Mohamed Ben Abdellah University
More informationFinite Sets. Theorem 5.1. Two nonempty finite sets have the same cardinality if and only if they are equivalent.
MATH 337 Cardinality Dr. Neal, WKU We now shall prove that the rational numbers are a countable set while R is uncountable. This result shows that there are two different magnitudes of infinity. But we
More informationHOMEWORK 5 SOLUTIONS. n!f n (1) lim. ln x n! + xn x. 1 = G n 1 (x). (2) k + 1 n. (n 1)!
Math 7 Fall 205 HOMEWORK 5 SOLUTIONS Problem. 2008 B2 Let F 0 x = ln x. For n 0 and x > 0, let F n+ x = 0 F ntdt. Evaluate n!f n lim n ln n. By directly computing F n x for small n s, we obtain the following
More informationWeak topologies. David Lecomte. May 23, 2006
Weak topologies David Lecomte May 3, 006 1 Preliminaries from general topology In this section, we are given a set X, a collection of topological spaces (Y i ) i I and a collection of maps (f i ) i I such
More information6. Metric spaces. In this section we review the basic facts about metric spaces. d : X X [0, )
6. Metric spaces In this section we review the basic facts about metric spaces. Definitions. A metric on a nonempty set X is a map with the following properties: d : X X [0, ) (i) If x, y X are points
More informationx pn x qn 0 as n. Every convergent sequence is Cauchy. Not every Cauchy sequence in a normed space E converges to a vector in
78 CHAPTER 3. BANACH SPACES 3.2 Banach Spaces Cauchy Sequence. A sequence of vectors (x n ) in a normed space is a Cauchy sequence if for everyε > 0 there exists M N such that for all n, m M, x m x n
More informationCHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e.
CHAPTER II THE LIMIT OF A SEQUENCE OF NUMBERS DEFINITION OF THE NUMBER e. This chapter contains the beginnings of the most important, and probably the most subtle, notion in mathematical analysis, i.e.,
More informationMATH 289 PROBLEM SET 1: INDUCTION. 1. The induction Principle The following property of the natural numbers is intuitively clear:
MATH 89 PROBLEM SET : INDUCTION The induction Principle The following property of the natural numbers is intuitively clear: Axiom Every nonempty subset of the set of nonnegative integers Z 0 = {0,,, 3,
More informationRi and. i=1. S i N. and. R R i
The subset R of R n is a closed rectangle if there are n nonempty closed intervals {[a 1, b 1 ], [a 2, b 2 ],..., [a n, b n ]} so that R = [a 1, b 1 ] [a 2, b 2 ] [a n, b n ]. The subset R of R n is an
More informationMath 317 HW #7 Solutions
Math 17 HW #7 Solutions 1. Exercise..5. Decide which of the following sets are compact. For those that are not compact, show how Definition..1 breaks down. In other words, give an example of a sequence
More informationNOTES ON THE DYNAMICS OF LINEAR OPERATORS
NOTES ON THE DYNAMICS OF LINEAR OPERATORS JOEL H. SHAPIRO Abstract. These are notes for a series of lectures that illustrate how continuous linear transformations of infinite dimensional topological vector
More informationCylinder Maps and the Schwarzian
Cylinder Maps and the Schwarzian John Milnor Stony Brook University (www.math.sunysb.edu) Bremen June 16, 2008 Cylinder Maps 1 work with Araceli Bonifant Let C denote the cylinder (R/Z) I. (.5, 1) fixed
More informationNotes on weak convergence (MAT Spring 2006)
Notes on weak convergence (MAT4380  Spring 2006) Kenneth H. Karlsen (CMA) February 2, 2006 1 Weak convergence In what follows, let denote an open, bounded, smooth subset of R N with N 2. We assume 1 p
More informationON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE. 1. Introduction
ON COMPLETELY CONTINUOUS INTEGRATION OPERATORS OF A VECTOR MEASURE J.M. CALABUIG, J. RODRÍGUEZ, AND E.A. SÁNCHEZPÉREZ Abstract. Let m be a vector measure taking values in a Banach space X. We prove that
More informationGROUPS SUBGROUPS. Definition 1: An operation on a set G is a function : G G G.
Definition 1: GROUPS An operation on a set G is a function : G G G. Definition 2: A group is a set G which is equipped with an operation and a special element e G, called the identity, such that (i) the
More information1. Let P be the space of all polynomials (of one real variable and with real coefficients) with the norm
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 0050615 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationThe Goldberg Rao Algorithm for the Maximum Flow Problem
The Goldberg Rao Algorithm for the Maximum Flow Problem COS 528 class notes October 18, 2006 Scribe: Dávid Papp Main idea: use of the blocking flow paradigm to achieve essentially O(min{m 2/3, n 1/2 }
More informationNotes on metric spaces
Notes on metric spaces 1 Introduction The purpose of these notes is to quickly review some of the basic concepts from Real Analysis, Metric Spaces and some related results that will be used in this course.
More informationditional classes, depending on his or her own personal preference. In Chapter 5 we discuss the spectral properties of hypercyclic and chaotic
Preface According to a widely held view, chaos is intimately linked to nonlinearity. It is usually taken to be selfevident that a linear system behaves in a predictable manner. However, as early as 1929,
More informationSequences of Functions
Sequences of Functions Uniform convergence 9. Assume that f n f uniformly on S and that each f n is bounded on S. Prove that {f n } is uniformly bounded on S. Proof: Since f n f uniformly on S, then given
More informationBANACH AND HILBERT SPACE REVIEW
BANACH AND HILBET SPACE EVIEW CHISTOPHE HEIL These notes will briefly review some basic concepts related to the theory of Banach and Hilbert spaces. We are not trying to give a complete development, but
More informationSystems with Persistent Memory: the Observation Inequality Problems and Solutions
Chapter 6 Systems with Persistent Memory: the Observation Inequality Problems and Solutions Facts that are recalled in the problems wt) = ut) + 1 c A 1 s ] R c t s)) hws) + Ks r)wr)dr ds. 6.1) w = w +
More informationOn the Spaces of λconvergent and Bounded Sequences
Thai Journal of Mathematics Volume 8 2010 Number 2 : 311 329 www.math.science.cmu.ac.th/thaijournal Online ISSN 16860209 On the Spaces of λconvergent and Bounded Sequences M. Mursaleen 1 and A.K. Noman
More information0 <β 1 let u(x) u(y) kuk u := sup u(x) and [u] β := sup
456 BRUCE K. DRIVER 24. Hölder Spaces Notation 24.1. Let Ω be an open subset of R d,bc(ω) and BC( Ω) be the bounded continuous functions on Ω and Ω respectively. By identifying f BC( Ω) with f Ω BC(Ω),
More informationINTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES
INTEGRAL OPERATORS ON THE PRODUCT OF C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. We study and characterize the integral multilinear operators on a product of C(K) spaces in terms of the
More informationP (A) = lim P (A) = N(A)/N,
1.1 Probability, Relative Frequency and Classical Definition. Probability is the study of random or nondeterministic experiments. Suppose an experiment can be repeated any number of times, so that we
More informationCONTRIBUTIONS TO ZERO SUM PROBLEMS
CONTRIBUTIONS TO ZERO SUM PROBLEMS S. D. ADHIKARI, Y. G. CHEN, J. B. FRIEDLANDER, S. V. KONYAGIN AND F. PAPPALARDI Abstract. A prototype of zero sum theorems, the well known theorem of Erdős, Ginzburg
More informationFUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES
FUNCTIONAL ANALYSIS LECTURE NOTES CHAPTER 3. BANACH SPACES CHRISTOPHER HEIL 1. Elementary Properties and Examples Notation 1.1. Throughout, F will denote either the real line R or the complex plane C.
More informationMore Mathematical Induction. October 27, 2016
More Mathematical Induction October 7, 016 In these slides... Review of ordinary induction. Remark about exponential and polynomial growth. Example a second proof that P(A) = A. Strong induction. Least
More informationON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING. 0. Introduction
ON SEQUENTIAL CONTINUITY OF COMPOSITION MAPPING Abstract. In [1] there was proved a theorem concerning the continuity of the composition mapping, and there was announced a theorem on sequential continuity
More informationTheorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that. a = bq + r and 0 r < b.
Theorem (The division theorem) Suppose that a and b are integers with b > 0. There exist unique integers q and r so that a = bq + r and 0 r < b. We re dividing a by b: q is the quotient and r is the remainder,
More informationS = k=1 k. ( 1) k+1 N S N S N
Alternating Series Last time we talked about convergence of series. Our two most important tests, the integral test and the (limit) comparison test, both required that the terms of the series were (eventually)
More informationChapter 5. Banach Spaces
9 Chapter 5 Banach Spaces Many linear equations may be formulated in terms of a suitable linear operator acting on a Banach space. In this chapter, we study Banach spaces and linear operators acting on
More information9 More on differentiation
Tel Aviv University, 2013 Measure and category 75 9 More on differentiation 9a Finite Taylor expansion............... 75 9b Continuous and nowhere differentiable..... 78 9c Differentiable and nowhere monotone......
More informationThe Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
More information1. Prove that the empty set is a subset of every set.
1. Prove that the empty set is a subset of every set. Basic Topology Written by MenGen Tsai email: b89902089@ntu.edu.tw Proof: For any element x of the empty set, x is also an element of every set since
More informationON PARAMETRIC LIMIT SUPERIOR OF A SEQUENCE OF ANALYTIC SETS
Real Analysis Exchange ISSN:01471937 Vol. 31(1), 2005/2006, pp. 1 5 Szymon G l ab, Mathematical Institute, Polish Academy of Science, Śniadeckich 8, 00956 Warszawa, Poland.email: szymon glab@yahoo.com
More information1 Norms and Vector Spaces
008.10.07.01 1 Norms and Vector Spaces Suppose we have a complex vector space V. A norm is a function f : V R which satisfies (i) f(x) 0 for all x V (ii) f(x + y) f(x) + f(y) for all x,y V (iii) f(λx)
More informationNOTES ON MEASURE THEORY. M. Papadimitrakis Department of Mathematics University of Crete. Autumn of 2004
NOTES ON MEASURE THEORY M. Papadimitrakis Department of Mathematics University of Crete Autumn of 2004 2 Contents 1 σalgebras 7 1.1 σalgebras............................... 7 1.2 Generated σalgebras.........................
More informationApplications of Fourier series
Chapter Applications of Fourier series One of the applications of Fourier series is the evaluation of certain infinite sums. For example, n= n,, are computed in Chapter (see for example, Remark.4.). n=
More informationExample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum. asin. k, a, and b. We study stability of the origin x
Lecture 4. LaSalle s Invariance Principle We begin with a motivating eample. Eample 4.1 (nonlinear pendulum dynamics with friction) Figure 4.1: Pendulum Dynamics of a pendulum with friction can be written
More informationON SUPERCYCLICITY CRITERIA. Nuha H. Hamada Business Administration College Al Ain University of Science and Technology 5th st, Abu Dhabi, 112612, UAE
International Journal of Pure and Applied Mathematics Volume 101 No. 3 2015, 401405 ISSN: 13118080 (printed version); ISSN: 13143395 (online version) url: http://www.ijpam.eu doi: http://dx.doi.org/10.12732/ijpam.v101i3.7
More informationDynamics for the solutions of the waterhammer equations
Dynamics for the solutions of the waterhammer equations J. Alberto Conejero (IUMPAUniversitat Politècnica de València). Joint work with C. Lizama (Universidad Santiado de Chile) and F. Ródenas (IUMPAUniversitat
More informationRandom graphs with a given degree sequence
Sourav Chatterjee (NYU) Persi Diaconis (Stanford) Allan Sly (Microsoft) Let G be an undirected simple graph on n vertices. Let d 1,..., d n be the degrees of the vertices of G arranged in descending order.
More informationProperties of BMO functions whose reciprocals are also BMO
Properties of BMO functions whose reciprocals are also BMO R. L. Johnson and C. J. Neugebauer The main result says that a nonnegative BMOfunction w, whose reciprocal is also in BMO, belongs to p> A p,and
More informationCHARACTERISTIC PROPERTIES OF THE GURARIY SPACE. V. P. Fonf BenGurion University of the Negev, Isreal
CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE. V. P. Fonf BenGurion University of the Negev, Isreal 1 2 CHARACTERISTIC PROPERTIES OF THE GURARIY SPACE An infinitedimensional Banach space X is called
More informationSMALL SKEW FIELDS CÉDRIC MILLIET
SMALL SKEW FIELDS CÉDRIC MILLIET Abstract A division ring of positive characteristic with countably many pure types is a field Wedderburn showed in 1905 that finite fields are commutative As for infinite
More informationE3: PROBABILITY AND STATISTICS lecture notes
E3: PROBABILITY AND STATISTICS lecture notes 2 Contents 1 PROBABILITY THEORY 7 1.1 Experiments and random events............................ 7 1.2 Certain event. Impossible event............................
More informationElements of probability theory
2 Elements of probability theory Probability theory provides mathematical models for random phenomena, that is, phenomena which under repeated observations yield di erent outcomes that cannot be predicted
More informationFUNCTIONAL ANALYSIS PIOTR HAJ LASZ
FUNCTIONAL ANALYSIS PIOTR HAJ LASZ 1. Banach and Hilbert spaces In what follows K will denote R of C. Definition. A normed space is a pair (X, ), where X is a linear space over K and : X [0, ) is a function,
More informationCourse 221: Analysis Academic year , First Semester
Course 221: Analysis Academic year 200708, First Semester David R. Wilkins Copyright c David R. Wilkins 1989 2007 Contents 1 Basic Theorems of Real Analysis 1 1.1 The Least Upper Bound Principle................
More informationFurther Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1
Further Study on Strong Lagrangian Duality Property for Invex Programs via Penalty Functions 1 J. Zhang Institute of Applied Mathematics, Chongqing University of Posts and Telecommunications, Chongqing
More informationMetric Spaces. Chapter 7. 7.1. Metrics
Chapter 7 Metric Spaces A metric space is a set X that has a notion of the distance d(x, y) between every pair of points x, y X. The purpose of this chapter is to introduce metric spaces and give some
More informationA MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY
Guidetti, D. and Lorenzi, A. Osaka J. Math. 44 (27), 579 613 A MIXED TYPE IDENTIFICATION PROBLEM RELATED TO A PHASEFIELD MODEL WITH MEMORY DAVIDE GUIDETTI and ALFREDO LORENZI (Received January 23, 26,
More informationMath 563 Measure Theory Project 1 (Funky Functions Group) Luis Zerón, Sergey Dyachenko
Math 563 Measure Theory Project (Funky Functions Group) Luis Zerón, Sergey Dyachenko 34 Let C and C be any two Cantor sets (constructed in Exercise 3) Show that there exists a function F: [,] [,] with
More informationPrimality  Factorization
Primality  Factorization Christophe Ritzenthaler November 9, 2009 1 Prime and factorization Definition 1.1. An integer p > 1 is called a prime number (nombre premier) if it has only 1 and p as divisors.
More information1 if 1 x 0 1 if 0 x 1
Chapter 3 Continuity In this chapter we begin by defining the fundamental notion of continuity for real valued functions of a single real variable. When trying to decide whether a given function is or
More informationFINITE SEMIVARIATION AND REGULATED FUNCTIONS BY MEANS OF BILINEAR MAPS
1 Oscar Blasco*, Departamento de Análisis Matemático, Universidad de Valencia, 46100 Burjassot, Valencia, Spain, (email: oscar.blasco@uv.es) Pablo Gregori*, Departamento de Análisis Matemático, Universidad
More information1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution.
Name : Instructor: Marius Ionescu Instructions: 1. Please write your name in the blank above, and sign & date below. 2. Please use the space provided to write your solution. 3. If you need extra pages
More informationScalar Valued Functions of Several Variables; the Gradient Vector
Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,
More informationPUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
More informationGeometrical Characterization of RNoperators between Locally Convex Vector Spaces
Geometrical Characterization of RNoperators between Locally Convex Vector Spaces OLEG REINOV St. Petersburg State University Dept. of Mathematics and Mechanics Universitetskii pr. 28, 198504 St, Petersburg
More informationCONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian. Pasquale Candito and Giuseppina D Aguí
Opuscula Math. 34 no. 4 2014 683 690 http://dx.doi.org/10.7494/opmath.2014.34.4.683 Opuscula Mathematica CONSTANTSIGN SOLUTIONS FOR A NONLINEAR NEUMANN PROBLEM INVOLVING THE DISCRETE plaplacian Pasquale
More informationMatrix Methods for Linear Systems of Differential Equations
Matrix Methods for Linear Systems of Differential Equations We now present an application of matrix methods to linear systems of differential equations. We shall follow the development given in Chapter
More informationarxiv:math/ v2 [math.nt] 10 Nov 2007
Nontrivial solutions to a linear equation in integers arxiv:math/0703767v2 [math.nt] 10 Nov 2007 Boris Bukh Abstract For k 3 let A [1, N] be a set not containing a solution to a 1 x 1 + + a k x k = a
More informationConvex analysis and profit/cost/support functions
CALIFORNIA INSTITUTE OF TECHNOLOGY Division of the Humanities and Social Sciences Convex analysis and profit/cost/support functions KC Border October 2004 Revised January 2009 Let A be a subset of R m
More informationNonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results
Proceedings of Symposia in Applied Mathematics Volume 00, 1997 NonArbitrage and the Fundamental Theorem of Asset Pricing: Summary of Main Results Freddy Delbaen and Walter Schachermayer Abstract. The
More informationTHE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS
THE FUNDAMENTAL THEOREM OF ALGEBRA VIA PROPER MAPS KEITH CONRAD 1. Introduction The Fundamental Theorem of Algebra says every nonconstant polynomial with complex coefficients can be factored into linear
More informationk, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
More information4.3 Limit of a Sequence: Theorems
4.3. LIMIT OF A SEQUENCE: THEOREMS 5 4.3 Limit of a Sequence: Theorems These theorems fall in two categories. The first category deals with ways to combine sequences. Like numbers, sequences can be added,
More informationMeasuring sets with translation invariant Borel measures
Measuring sets with translation invariant Borel measures University of Warwick Fractals and Related Fields III, 21 September 2015 Measured sets Definition A Borel set A R is called measured if there is
More information4. Expanding dynamical systems
4.1. Metric definition. 4. Expanding dynamical systems Definition 4.1. Let X be a compact metric space. A map f : X X is said to be expanding if there exist ɛ > 0 and L > 1 such that d(f(x), f(y)) Ld(x,
More informationNON SINGULAR MATRICES. DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that
NON SINGULAR MATRICES DEFINITION. (Non singular matrix) An n n A is called non singular or invertible if there exists an n n matrix B such that AB = I n = BA. Any matrix B with the above property is called
More informationMATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set.
MATH 304 Linear Algebra Lecture 9: Subspaces of vector spaces (continued). Span. Spanning set. Vector space A vector space is a set V equipped with two operations, addition V V (x,y) x + y V and scalar
More informationNotes on Factoring. MA 206 Kurt Bryan
The General Approach Notes on Factoring MA 26 Kurt Bryan Suppose I hand you n, a 2 digit integer and tell you that n is composite, with smallest prime factor around 5 digits. Finding a nontrivial factor
More informationSequences and Convergence in Metric Spaces
Sequences and Convergence in Metric Spaces Definition: A sequence in a set X (a sequence of elements of X) is a function s : N X. We usually denote s(n) by s n, called the nth term of s, and write {s
More informationRAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS PART A
RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I  ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:
More informationSimilarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
More informationStructure of Measurable Sets
Structure of Measurable Sets In these notes we discuss the structure of Lebesgue measurable subsets of R from several different points of view. Along the way, we will see several alternative characterizations
More informationMath 317 HW #5 Solutions
Math 317 HW #5 Solutions 1. Exercise 2.4.2. (a) Prove that the sequence defined by x 1 = 3 and converges. x n+1 = 1 4 x n Proof. I intend to use the Monotone Convergence Theorem, so my goal is to show
More information1. Introduction. PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1
Publ. Mat. 45 (2001), 69 77 PROPER HOLOMORPHIC MAPPINGS BETWEEN RIGID POLYNOMIAL DOMAINS IN C n+1 Bernard Coupet and Nabil Ourimi Abstract We describe the branch locus of proper holomorphic mappings between
More informationLecture 5 Rational functions and partial fraction expansion
S. Boyd EE102 Lecture 5 Rational functions and partial fraction expansion (review of) polynomials rational functions polezero plots partial fraction expansion repeated poles nonproper rational functions
More information1.3 Induction and Other Proof Techniques
4CHAPTER 1. INTRODUCTORY MATERIAL: SETS, FUNCTIONS AND MATHEMATICAL INDU 1.3 Induction and Other Proof Techniques The purpose of this section is to study the proof technique known as mathematical induction.
More informationIntroduction to Algebraic Geometry. Bézout s Theorem and Inflection Points
Introduction to Algebraic Geometry Bézout s Theorem and Inflection Points 1. The resultant. Let K be a field. Then the polynomial ring K[x] is a unique factorisation domain (UFD). Another example of a
More informationProf. Girardi, Math 703, Fall 2012 Homework Solutions: Homework 13. in X is Cauchy.
Homework 13 Let (X, d) and (Y, ρ) be metric spaces. Consider a function f : X Y. 13.1. Prove or give a counterexample. f preserves convergent sequences if {x n } n=1 is a convergent sequence in X then
More informationOn an isomorphic BanachMazur rotation problem and maximal norms in Banach spaces
On an isomorphic BanachMazur rotation problem and maximal norms in Banach spaces B. Randrianantoanina (joint work with Stephen Dilworth) Department of Mathematics Miami University Ohio, USA Conference
More informationChapter 6 Finite sets and infinite sets. Copyright 2013, 2005, 2001 Pearson Education, Inc. Section 3.1, Slide 1
Chapter 6 Finite sets and infinite sets Copyright 013, 005, 001 Pearson Education, Inc. Section 3.1, Slide 1 Section 6. PROPERTIES OF THE NATURE NUMBERS 013 Pearson Education, Inc.1 Slide Recall that denotes
More informationAn example of a capacity for which all positive Borel sets are thick
An example of a capacity for which all positive Borel sets are thick Micha l Morayne, Szymon Żeberski Wroc law University of Technology Winter School in Abstract Analysis 2016 section Set Theory & Topology
More informationPOWER SETS AND RELATIONS
POWER SETS AND RELATIONS L. MARIZZA A. BAILEY 1. The Power Set Now that we have defined sets as best we can, we can consider a sets of sets. If we were to assume nothing, except the existence of the empty
More information1. Let X and Y be normed spaces and let T B(X, Y ).
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: NVP, Frist. 20050314 Skrivtid: 9 11.30 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
More informationMA651 Topology. Lecture 6. Separation Axioms.
MA651 Topology. Lecture 6. Separation Axioms. This text is based on the following books: Fundamental concepts of topology by Peter O Neil Elements of Mathematics: General Topology by Nicolas Bourbaki Counterexamples
More informationA CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT
Bol. Soc. Mat. Mexicana (3) Vol. 12, 2006 A CHARACTERIZATION OF C k (X) FOR X NORMAL AND REALCOMPACT F. MONTALVO, A. A. PULGARÍN AND B. REQUEJO Abstract. We present some internal conditions on a locally
More informationThe BanachTarski Paradox
University of Oslo MAT2 Project The BanachTarski Paradox Author: Fredrik Meyer Supervisor: Nadia S. Larsen Abstract In its weak form, the BanachTarski paradox states that for any ball in R, it is possible
More informationINDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS
INDISTINGUISHABILITY OF ABSOLUTELY CONTINUOUS AND SINGULAR DISTRIBUTIONS STEVEN P. LALLEY AND ANDREW NOBEL Abstract. It is shown that there are no consistent decision rules for the hypothesis testing problem
More informationAlgebra 2. Rings and fields. Finite fields. A.M. Cohen, H. Cuypers, H. Sterk. Algebra Interactive
2 Rings and fields A.M. Cohen, H. Cuypers, H. Sterk A.M. Cohen, H. Cuypers, H. Sterk 2 September 25, 2006 1 / 20 For p a prime number and f an irreducible polynomial of degree n in (Z/pZ)[X ], the quotient
More informationLecture 12 Basic Lyapunov theory
EE363 Winter 200809 Lecture 12 Basic Lyapunov theory stability positive definite functions global Lyapunov stability theorems Lasalle s theorem converse Lyapunov theorems finding Lyapunov functions 12
More informationEXERCISES FOR THE COURSE MATH 570, FALL 2010
EXERCISES FOR THE COURSE MATH 570, FALL 2010 EYAL Z. GOREN (1) Let G be a group and H Z(G) a subgroup such that G/H is cyclic. Prove that G is abelian. Conclude that every group of order p 2 (p a prime
More informationDEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS
DEGREES OF ORDERS ON TORSIONFREE ABELIAN GROUPS ASHER M. KACH, KAREN LANGE, AND REED SOLOMON Abstract. We construct two computable presentations of computable torsionfree abelian groups, one of isomorphism
More information