Some remarks on the paper Some elementary inequalities of G. Bennett
|
|
- Derek King
- 5 years ago
- Views:
Transcription
1 Soe rears o the paper Soe eleetary iequalities of G. Beett Dag Ah Tua ad Luu Quag Bay Vieta Natioal Uiversity - Haoi Uiversity of Sciece Abstract We give soe couterexaples ad soe rears of soe of the corollaries of Beett s paper, ivolvig the weighted ea atrices o l p. We also discuss Littlewood s proble. 1 Itroductio Let p be fixed ad 1 p. The l p spaces are the spaces of real-valued sequeces x x ) 1 with ors ) 1/p x p x p < with 1 p <, 1 x sup x <. N For 1 < p <, let p p/p 1) deote the cougate of p. A weighted ea atrix A a ),1 is defied by { a /A whe 1, a 0 otherwise, 1) where A a, a 0, a 1 > 0. 1 The atrix A fro l p ito l q has the or A p,q sup Ax q. x p 1 I [1], G. Beett gave soe corollaries that ivolve the boudedess of weighted ea atrices o l p. To show that the coverses of soe of the corollaries do ot hold, we prove soe couterexaples i sectio. We also preset soe propositios which give sufficiet coditios for a weighted ea atrix A ot to be bouded o l p. I [1], whe cosiderig Littlewood s proble, G. Beett gave a exaple to show that there does ot exist a positive costat K for reverse of Littlewood s iequality to be true. I sectio 3, by odifyig the exaple of G. Beett we show that the reverse of Littlewood s iequality is ot true i geeral. 1
2 Soe rears about weighted ea atrices I paper [1], G. Beett obtaied sufficiet coditios for a weighted ea atrix to be a bouded liear operator fro l p ito l q as show by followig theore. Theore 1 [1]). Let A be the weighted ea atrix give by 1). a) If 1 < p q < the followig coditios are equivalet : i) A aps l p ito l q. ii) For soe costat K 1 ad all 1,,... such that 1 1 ) q K 1 1 iii) For soe costat K ad all 1,,... such that A q ) 1/q 1 ) 1/p iv) For soe costat K 3 ad all 1,,... such that a A q K 3 b) If 1 q < p the A does ot ap l p ito l q. ) q/p. ) K. 3) A q /q. 4) I additio to applyig above theore, G. Beett also gave soe corollaries. Corollary 1 [1]). Let p be fixed ad 1 < p <. If the A is bouded o l p. Corollary [1]). Let p be fixed ad 1 < p <. If O ) a 1/p 1) A, 5) the A is bouded o l p. a OA 1 p ), 6) I the followig exaple we show that the coditio 5) is ot ecessary; i.e., A is bouded ad li sup a 1/p 1) A. 7)
3 Exaple 1. Let a { 1 if, if. The For < +1, we have I : A r0 a a 1/p 1) A r + 1 +, r0 [ rp + p 1 p 1 > 1 +1)p ] +. /p 1) + ) ) /p 1). + As, we get I, ad 7) is true. Next, we chec the boudedess of A. By Corollary we eed oly show that We have So Thus, A p 1 a OA 1 p ). A log. a p 1. p p. p p p 1 p 1 p p, ad A is bouded. I the followig propositio, we state a sufficiet coditio for A ot to be bouded. Propositio 1. Let p be fixed ad 1 < p <. If the ad A is ot bouded. li if sup N a 1/p 1) A +, 8) ) 3
4 Proof. Let K > 0. To show that 9) is true we have to fid a N such that ) p Case 1: 1 I : 1 M < +. We have 1 1 > K. A 1 a 1/p 1 ) 1/p. The, Thus, I : > 1 ) 1/p > ap /p I order to have I > K we eed to fid a such that i.e., 1 l M > K; > e KM/ap 1. 1 > ap 1 l M. Hece, we ca choose 0 [e KM/ap 1 ] + 1. The I > K. We have show that 9) is true. I additio, by part ii) of Theore 1, we deduce that A is ot bouded. Case : +. By 8), there exists 0 such that, for all > > K)1/p a 1 p. Thus, 1 > K. 4
5 For > 0 we have I : > > K K 1 1 I order to have I > K we eed to fid a such that K > K; i.e., Sice 1 +, the > 0 1 as.. 10) Thus we ca always fid a large eough such that 10) holds. Therefore we obtai 9) ad A is ot bouded. We ow provide a exaple i which A is bouded ad li sup a A 1 p +. 11) 5
6 Exaple. Taig For < +1, we have a { 1 if, if. 1) A a + +1 a r + ) + ) r0 + 1) +, ad r p + p r0 Thus + 1) A + log 1) log + log, + 1) A + log log + 1) +. It the follows that A p 1 a ) p 1 1 p 1 p 1) 1 p), p 1 ad 11) is true. To chec that A is bouded, usig Corollary 1, it is oly ecessary to show that Oa 1/p 1) A ). Fro the defiitio of a, Sice a 1/p 1) A p )/ + log +1 + / log +1 log li +1 0, + 1/. 6
7 there exists a C > 0 such that Therefore ad A is bouded. If +, the 1 log +1 < C. a 1/p 1) A C + 1, sup N ) 1/p 1 ) 1/p 1 ) 1/p a 1 +. Fro part iii) of Theore 1, A is ot bouded. Now, we have the propositio i the case < +. 1 Propositio. Let p be fixed ad 1 < p <. If the ad A is ot bouded. sup N li if a A 1 p a 1 < + ad +, 13) + 14) Proof. Let K > 0. Fro 13), there exists a 0 such that, for all > 0, Let > 0. The or 0 +1 a a 0 +1 a 0 +1 > K 1/p A 1 p. 7 > K 0 +1 > K.,
8 This above iequality is equivalet to 14) ad, fro part iv) of the Theore 1, A is ot bouded. Next, we cosider aother exaple to show that the coverse of two above Corollaries is false. Exaple 3. Let For < +1, we get For, 1 if, a if, is odd, if, is eve. A a +. A A 1 i0 1 r0 a i+1 + a r + a i + i1 i + 1) + i0 Siilarly, for + 1, we have A i + i1 1) ) ). 3 4 i + 1) + i0 i i1 + 1) ) ) Thus A { ) if, ) if ) 8
9 It is clear that If is eve A p 1 a > A A > > p) p [ 1 ], p 1 ad the left had side teds to ifiity as teds to ifiity. If is odd, the 1 +1 a 1/p 1) +1 A )/p 1) ) +1)p ) 1 1 The left had side of the above iequality also teds to ifiity as to ifiity. To show the boudedess of A, fro part ii) of the Theore 1, it is sufficiet to show that l l K 1 l Z + for soe costat K 1. For s l < s+1, O other had, The l l 1 s 1 s s. 16) +1 r p r0 s s 0 < +1 s ) p
10 Sice li 0, the there exists a K > 0 such that, for all, Hece l 1 Fro 16) ad 17), we get l 1 l < K. s K + < K + s+1. 17) 0 K + )p s+1 s s 4K +, ad hece A is bouded. We have show that the coverses of Corollaries 1 ad are ot true. We ow cosider aother Corollary i the paper of G. Beett [1]. Corollary 3 [1]). Let p be fixed, 1 < p <. If A is bouded o l p, the O ). 18) Exaple 4. I this exaple we show that the coverse of Corollary 3 does ot hold by taig 1 whe 1, a whe + 1 for soe Z + {0}, 0 otherwise. For < +1, the 19) A a 1 + l +1, l0 1 + l0 lp +1)p + p. 0) p 1 10
11 The so that )p l+1 +1)p ) + +1)p ) + l < l+1 l+1)p l+1 l < l+1 l+1 l l+1)p +1)p ) + p +1)1 p) 1 1 p +1)p ) + +1)1 p) [ p ) p ) + +1] +1)p, 1) p ) p ) + +1 p ) < p p. The a defied by 19) also satisfy 18). We shall ow show that A is ot bouded. It is sufficiet to show that ) 1/p ) 1/p sup. By 0) ad 1), we have ) 1/p ) 1/p [ ) p ) + +1] ) 1/p +1)p +1)p + p p p 1 ) 1/p > [ ) p ) + +1] 1/p +1) > 1/p 8, +1 11
12 ad A is ot bouded. Returig to Propositio 1 ad Propositio, with a as chose i Exaple 4, we have Hece I ap a 1/p 1) A +1)p 1 p 1) /p 1) p 1 I p 1 as. O other had, for a as i Exaple 4, li if 3 Littlewood s proble a A 1 p 0. 1 p 1). p )p We cosider the Littlewood s proble that is discussed i [1]; i.e., does there exists a costat K such that a A K a 4 A. ) a By applyig Holder s iequality ad Hardy s iequality G. Beett showed that, for decreasig sequece a, K. He also showed that there does ot exist a positive costat K such that a 4 A K a A. 3) by taig 1 1 a if > N, a 1 if r for r 1,,.., N, ɛ otherwise, where ɛ N / N. The geeral case of Littlewood s proble is preseted by the followig theore. Theore [1]). Let p, q, r 1. If a ) 1 is a sequece of o-egative ubers with partial su A a a, the ) r ) r pq + r) q a p A q a 1+p/q a p p A q ) 1+r/q. 5) 1 We ow cosider the reverse iequality of 5) for p, q, r 1; i.e., does there exist a positive costat K such that ) r a p A q ) 1+r/p K a p A q a 1+p/q? 6) )
13 To aswer this questio we cosider the followig cases. If r 1, p 1, q 1 the the left-had side of 6) becoes LHS a A q ) 1+1/q a 1+1/q A q+1, 1 ad the right-had side of 6) becoes RHS a A q 1 Thus, we eed oly to copare A q+1 1 a 1+1/q 1 1 a 1+1/q a A q. 1 ad a A q. For ay q 1, 1 a A q A q+1 q a A q. Thus 6) holds with K q, for p 1, r 1, q 1 ad the iverse of 6) is true for this case. I the other cases we shall show that there does ot exist a positive costat K such that 6) holds by taig a as i 4), ad with ɛ N α / N, for soe α > 1 to be chose later. For r < r+1, r A a s + a r + r)ɛ. s1 Estiatig the left had side of 6, 1 < s 1 LHS N 1 a p A q ) 1+r/q N a p saq + s s1 a 1 < N s p A q ) 1+r/q. LHS > > N [s + s s)ɛ] q+r s1 N [ 1 ɛ) q+r s q+r + sq+r) ɛ q+r] s1 N 1 ɛ) q+r s q+r + ɛ q+r s1 N sq+r) s1 > 1 ɛ) q+r N q+r + ɛ q+r Nq+r) > N αq+r) Sice ɛ N α N ). 13
14 Cosiderig the right had side of 6), RHS I 1 N 1 N s1 a p A q a p saq s I 1 + ɛ p I. < < N a 1+p/q N a 1+p/q s r r + 1 < N s p aa q N N [s + s s)ɛ] q [ N N s )ɛ 1+p/q] r s1 N s + s ɛ) q N N ɛ 1+p/q) r s1 N q s q + sq ɛ q ) [ r N + 1) r + Nr ɛ r1+p/q)] s1 < [ q+r N) r + Nr ɛ r1+p/q)] N s q + sq ɛ q ) s1 < [ N q+r N) r + Nr ɛ r1+p/q)] N q + ɛ q s1 N s1 sq ) a 1+p/q < [ q+r N) r + Nr ɛ r1+p/q)] ) N q+1 + ɛ q N+1)q q q 1 < q+r [ N) r + Nr ɛ r1+p/q)] N q+1 + ɛ q N+1)q q 1 ). r [ ) ] N I 1 < q+r r N r + Nr α r1+p/q) [ ] N q+1 + q N αq N [ ) ] N < q+r r N r + Nr α r1+p/q) N q+1 N αq q 1. q 1 Sice ɛ N α N ) Sice N αr1+p/q) li 0, N Np/q N r there exists a N 1 N 1 α, r, p, q) such that, for all N > N 1 ) N Nr α r1+p/q) < r N r. 7) N 14
15 Therefore We also have I < < < 1 < N s N 1 q A I 1 < q+r+ q 1 N αq+r for N > N 1. 8) N A q s0 s << s+1 N 1 s0 N 1 s0 s A q s+1 N a 1+p/q 1 N a 1+p/q r a 1+p/q r s [ s s+1 ɛ ] q [ N + N ɛ 1+p/q] r N 1 s [ q s + 1) q + qs+1) ɛ q] [ r N r + Nr ɛ r1+p/q)] s0 < [ [ ] q+r N r + Nr ɛ r1+p/q)] N 1 N 1 s s + 1) q + ɛ q s qs+1) s0 r < [ q+r N r + Nr ɛ r1+p/q)] N N q + ɛ q Nq+1)) [ ) ] N < q+r N r + Nr α r1+p/q) N N q + N N αq) Sice ɛ N α N ). N Sice 7) ad α > 1, If p > 1, r 1, the I < q+r 1 + r )N r N+1 N αq for N > N 1. 9) li N q+r r ) N αp+αq+r 0, Np 1) so there exists a N N α, r, p, q) > N 1 such that, for all N > N, s0 We ay choose α. If p 1, r > 1, the RHS I 1 + ɛ p I < q+r+3 q 1 N αq+r. ɛ p I < q+r r )N α+αq+r for N > N 1, ad LHS > N αq+r), 15
16 RHS I 1 + ɛ p I < q+r+ q 1 N αq+r + q+r r )N α+αq+r for N > N 1. Thus, we choose α > r r 1, for istace α r + 1 r 1. Suary, for p r 1, q 1, we obtai the Littlewood s iequality ad reverse Littlewood s iequality. For other cases, we oly get the Littlewood s iequality ad we does ot fid the positive costat K such that the reverse Littlewood s iequality holds. Acowledgets We would lie to express y sicere tha to Haoi Uiversity of Sciece. Moreover, the first author give thafuless to the fud of VIASM. Last, we especially wish to tha Professor B. E. Rhoades. Refereces [1] G. Beett, Soe eleetary iequalities, Quart. J. Math. Oxford Ser. ) ),
Bertrand s postulate Chapter 2
Bertrad s postulate Chapter We have see that the sequece of prie ubers, 3, 5, 7,... is ifiite. To see that the size of its gaps is ot bouded, let N := 3 5 p deote the product of all prie ubers that are
More informationLecture 10: Bounded Linear Operators and Orthogonality in Hilbert Spaces
Lecture : Bouded Liear Operators ad Orthogoality i Hilbert Spaces 34 Bouded Liear Operator Let ( X, ), ( Y, ) i i be ored liear vector spaces ad { } X Y The, T is said to be bouded if a real uber c such
More informationOn Summability Factors for N, p n k
Advaces i Dyamical Systems ad Applicatios. ISSN 0973-532 Volume Number 2006, pp. 79 89 c Research Idia Publicatios http://www.ripublicatio.com/adsa.htm O Summability Factors for N, p B.E. Rhoades Departmet
More information1 Lecture 2: Sequence, Series and power series (8/14/2012)
Summer Jump-Start Program for Aalysis, 202 Sog-Yig Li Lecture 2: Sequece, Series ad power series (8/4/202). More o sequeces Example.. Let {x } ad {y } be two bouded sequeces. Show lim sup (x + y ) lim
More informationTopics. Homework Problems. MATH 301 Introduction to Analysis Chapter Four Sequences. 1. Definition of convergence of sequences.
MATH 301 Itroductio to Aalysis Chapter Four Sequeces Topics 1. Defiitio of covergece of sequeces. 2. Fidig ad provig the limit of sequeces. 3. Bouded covergece theorem: Theorem 4.1.8. 4. Theorems 4.1.13
More informationsin(n) + 2 cos(2n) n 3/2 3 sin(n) 2cos(2n) n 3/2 a n =
60. Ratio ad root tests 60.1. Absolutely coverget series. Defiitio 13. (Absolute covergece) A series a is called absolutely coverget if the series of absolute values a is coverget. The absolute covergece
More information1. By using truth tables prove that, for all statements P and Q, the statement
Author: Satiago Salazar Problems I: Mathematical Statemets ad Proofs. By usig truth tables prove that, for all statemets P ad Q, the statemet P Q ad its cotrapositive ot Q (ot P) are equivalet. I example.2.3
More informationChapter 6 Infinite Series
Chapter 6 Ifiite Series I the previous chapter we cosidered itegrals which were improper i the sese that the iterval of itegratio was ubouded. I this chapter we are goig to discuss a topic which is somewhat
More informationM17 MAT25-21 HOMEWORK 5 SOLUTIONS
M17 MAT5-1 HOMEWORK 5 SOLUTIONS 1. To Had I Cauchy Codesatio Test. Exercise 1: Applicatio of the Cauchy Codesatio Test Use the Cauchy Codesatio Test to prove that 1 diverges. Solutio 1. Give the series
More informationIf a subset E of R contains no open interval, is it of zero measure? For instance, is the set of irrationals in [0, 1] is of measure zero?
2 Lebesgue Measure I Chapter 1 we defied the cocept of a set of measure zero, ad we have observed that every coutable set is of measure zero. Here are some atural questios: If a subset E of R cotais a
More informationSolutions to Tutorial 5 (Week 6)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial 5 (Wee 6 MATH2962: Real ad Complex Aalysis (Advaced Semester, 207 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationn=1 a n is the sequence (s n ) n 1 n=1 a n converges to s. We write a n = s, n=1 n=1 a n
Series. Defiitios ad first properties A series is a ifiite sum a + a + a +..., deoted i short by a. The sequece of partial sums of the series a is the sequece s ) defied by s = a k = a +... + a,. k= Defiitio
More informationFall 2013 MTH431/531 Real analysis Section Notes
Fall 013 MTH431/531 Real aalysis Sectio 8.1-8. Notes Yi Su 013.11.1 1. Defiitio of uiform covergece. We look at a sequece of fuctios f (x) ad study the coverget property. Notice we have two parameters
More informationSeries III. Chapter Alternating Series
Chapter 9 Series III With the exceptio of the Null Sequece Test, all the tests for series covergece ad divergece that we have cosidered so far have dealt oly with series of oegative terms. Series with
More informationSOME SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS
ARCHIVU ATHEATICU BRNO Tomus 40 2004, 33 40 SOE SEQUENCE SPACES DEFINED BY ORLICZ FUNCTIONS E. SAVAŞ AND R. SAVAŞ Abstract. I this paper we itroduce a ew cocept of λ-strog covergece with respect to a Orlicz
More informationALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS
It. J. Cotep. Math. Sci., Vol. 1, 2006, o. 1, 39-43 ALMOST CONVERGENCE AND SOME MATRIX TRANSFORMATIONS Qaaruddi ad S. A. Mohiuddie Departet of Matheatics, Aligarh Musli Uiversity Aligarh-202002, Idia sdqaar@rediffail.co,
More informationMAT1026 Calculus II Basic Convergence Tests for Series
MAT026 Calculus II Basic Covergece Tests for Series Egi MERMUT 202.03.08 Dokuz Eylül Uiversity Faculty of Sciece Departmet of Mathematics İzmir/TURKEY Cotets Mootoe Covergece Theorem 2 2 Series of Real
More informationMatrix Algebra 2.2 THE INVERSE OF A MATRIX Pearson Education, Inc.
2 Matrix Algebra 2.2 THE INVERSE OF A MATRIX MATRIX OPERATIONS A matrix A is said to be ivertible if there is a matrix C such that CA = I ad AC = I where, the idetity matrix. I = I I this case, C is a
More information1+x 1 + α+x. x = 2(α x2 ) 1+x
Math 2030 Homework 6 Solutios # [Problem 5] For coveiece we let α lim sup a ad β lim sup b. Without loss of geerality let us assume that α β. If α the by assumptio β < so i this case α + β. By Theorem
More informationIt is often useful to approximate complicated functions using simpler ones. We consider the task of approximating a function by a polynomial.
Taylor Polyomials ad Taylor Series It is ofte useful to approximate complicated fuctios usig simpler oes We cosider the task of approximatig a fuctio by a polyomial If f is at least -times differetiable
More informationSolutions to Tutorial 3 (Week 4)
The Uiversity of Sydey School of Mathematics ad Statistics Solutios to Tutorial Week 4 MATH2962: Real ad Complex Aalysis Advaced Semester 1, 2017 Web Page: http://www.maths.usyd.edu.au/u/ug/im/math2962/
More informationConvergence of random variables. (telegram style notes) P.J.C. Spreij
Covergece of radom variables (telegram style otes).j.c. Spreij this versio: September 6, 2005 Itroductio As we kow, radom variables are by defiitio measurable fuctios o some uderlyig measurable space
More informationJacobi symbols. p 1. Note: The Jacobi symbol does not necessarily distinguish between quadratic residues and nonresidues. That is, we could have ( a
Jacobi sybols efiitio Let be a odd positive iteger If 1, the Jacobi sybol : Z C is the costat fuctio 1 1 If > 1, it has a decopositio ( as ) a product of (ot ecessarily distict) pries p 1 p r The Jacobi
More informationProblem. Consider the sequence a j for j N defined by the recurrence a j+1 = 2a j + j for j > 0
GENERATING FUNCTIONS Give a ifiite sequece a 0,a,a,, its ordiary geeratig fuctio is A : a Geeratig fuctios are ofte useful for fidig a closed forula for the eleets of a sequece, fidig a recurrece forula,
More informationSelf-normalized deviation inequalities with application to t-statistic
Self-ormalized deviatio iequalities with applicatio to t-statistic Xiequa Fa Ceter for Applied Mathematics, Tiaji Uiversity, 30007 Tiaji, Chia Abstract Let ξ i i 1 be a sequece of idepedet ad symmetric
More informationAssignment 5: Solutions
McGill Uiversity Departmet of Mathematics ad Statistics MATH 54 Aalysis, Fall 05 Assigmet 5: Solutios. Let y be a ubouded sequece of positive umbers satisfyig y + > y for all N. Let x be aother sequece
More informationSequences and Series of Functions
Chapter 6 Sequeces ad Series of Fuctios 6.1. Covergece of a Sequece of Fuctios Poitwise Covergece. Defiitio 6.1. Let, for each N, fuctio f : A R be defied. If, for each x A, the sequece (f (x)) coverges
More informationSome Mean Inequalities
Irish Math. Soc. Bulleti 57 (2006), 69 79 69 Some Mea Iequalities FINBARR HOLLAND Dedicated to Trevor West o the occasio of his retiremet. Abstract. Let P deote the collectio of positive sequeces defied
More informationSome Tauberian Conditions for the Weighted Mean Method of Summability
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. 3 Some Tauberia Coditios for the Weighted Mea Method of Summability Ümit Totur İbrahim Çaak Received: 2.VIII.204 / Accepted: 6.III.205 Abstract
More informationMath 104: Homework 2 solutions
Math 04: Homework solutios. A (0, ): Sice this is a ope iterval, the miimum is udefied, ad sice the set is ot bouded above, the maximum is also udefied. if A 0 ad sup A. B { m + : m, N}: This set does
More informationPlease do NOT write in this box. Multiple Choice. Total
Istructor: Math 0560, Worksheet Alteratig Series Jauary, 3000 For realistic exam practice solve these problems without lookig at your book ad without usig a calculator. Multiple choice questios should
More informationBinomial transform of products
Jauary 02 207 Bioial trasfor of products Khristo N Boyadzhiev Departet of Matheatics ad Statistics Ohio Norther Uiversity Ada OH 4580 USA -boyadzhiev@ouedu Abstract Give the bioial trasfors { b } ad {
More informationf(1), and so, if f is continuous, f(x) = f(1)x.
2.2.35: Let f be a additive fuctio. i Clearly fx = fx ad therefore f x = fx for all Z+ ad x R. Hece, for ay, Z +, f = f, ad so, if f is cotiuous, fx = fx. ii Suppose that f is bouded o soe o-epty ope set.
More informationSeries: Infinite Sums
Series: Ifiite Sums Series are a way to mae sese of certai types of ifiitely log sums. We will eed to be able to do this if we are to attai our goal of approximatig trascedetal fuctios by usig ifiite degree
More informationInfinite Sequences and Series
Chapter 6 Ifiite Sequeces ad Series 6.1 Ifiite Sequeces 6.1.1 Elemetary Cocepts Simply speakig, a sequece is a ordered list of umbers writte: {a 1, a 2, a 3,...a, a +1,...} where the elemets a i represet
More informationCarleton College, Winter 2017 Math 121, Practice Final Prof. Jones. Note: the exam will have a section of true-false questions, like the one below.
Carleto College, Witer 207 Math 2, Practice Fial Prof. Joes Note: the exam will have a sectio of true-false questios, like the oe below.. True or False. Briefly explai your aswer. A icorrectly justified
More informationCSI 2101 Discrete Structures Winter Homework Assignment #4 (100 points, weight 5%) Due: Thursday, April 5, at 1:00pm (in lecture)
CSI 101 Discrete Structures Witer 01 Prof. Lucia Moura Uiversity of Ottawa Homework Assigmet #4 (100 poits, weight %) Due: Thursday, April, at 1:00pm (i lecture) Program verificatio, Recurrece Relatios
More informationGENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES
J. Nuber Theory 0, o., 9-9. GENERALIZED LEGENDRE POLYNOMIALS AND RELATED SUPERCONGRUENCES Zhi-Hog Su School of Matheatical Scieces, Huaiyi Noral Uiversity, Huaia, Jiagsu 00, PR Chia Eail: zhihogsu@yahoo.co
More informationSome Tauberian theorems for weighted means of bounded double sequences
A. Ştiiţ. Uiv. Al. I. Cuza Iaşi. Mat. N.S. Tomul LXIII, 207, f. Some Tauberia theorems for weighted meas of bouded double sequeces Cemal Bele Received: 22.XII.202 / Revised: 24.VII.203/ Accepted: 3.VII.203
More informationx !1! + 1!2!
4 Euler-Maclauri Suatio Forula 4. Beroulli Nuber & Beroulli Polyoial 4.. Defiitio of Beroulli Nuber Beroulli ubers B (,,3,) are defied as coefficiets of the followig equatio. x e x - B x! 4.. Expreesio
More informationMath 4707 Spring 2018 (Darij Grinberg): midterm 2 page 1. Math 4707 Spring 2018 (Darij Grinberg): midterm 2 with solutions [preliminary version]
Math 4707 Sprig 08 Darij Griberg: idter page Math 4707 Sprig 08 Darij Griberg: idter with solutios [preliiary versio] Cotets 0.. Coutig first-eve tuples......................... 3 0.. Coutig legal paths
More informationTHE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION
MATHEMATICA MONTISNIGRI Vol XXVIII (013) 17-5 THE GREATEST ORDER OF THE DIVISOR FUNCTION WITH INCREASING DIMENSION GLEB V. FEDOROV * * Mechaics ad Matheatics Faculty Moscow State Uiversity Moscow, Russia
More information(A sequence also can be thought of as the list of function values attained for a function f :ℵ X, where f (n) = x n for n 1.) x 1 x N +k x N +4 x 3
MATH 337 Sequeces Dr. Neal, WKU Let X be a metric space with distace fuctio d. We shall defie the geeral cocept of sequece ad limit i a metric space, the apply the results i particular to some special
More informationTR/46 OCTOBER THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION A. TALBOT
TR/46 OCTOBER 974 THE ZEROS OF PARTIAL SUMS OF A MACLAURIN EXPANSION by A. TALBOT .. Itroductio. A problem i approximatio theory o which I have recetly worked [] required for its solutio a proof that the
More informationdistinct distinct n k n k n! n n k k n 1 if k n, identical identical p j (k) p 0 if k > n n (k)
THE TWELVEFOLD WAY FOLLOWING GIAN-CARLO ROTA How ay ways ca we distribute objects to recipiets? Equivaletly, we wat to euerate equivalece classes of fuctios f : X Y where X = ad Y = The fuctios are subject
More informationMath Solutions to homework 6
Math 175 - Solutios to homework 6 Cédric De Groote November 16, 2017 Problem 1 (8.11 i the book): Let K be a compact Hermitia operator o a Hilbert space H ad let the kerel of K be {0}. Show that there
More informationThe log-behavior of n p(n) and n p(n)/n
Ramauja J. 44 017, 81-99 The log-behavior of p ad p/ William Y.C. Che 1 ad Ke Y. Zheg 1 Ceter for Applied Mathematics Tiaji Uiversity Tiaji 0007, P. R. Chia Ceter for Combiatorics, LPMC Nakai Uivercity
More informationMAS111 Convergence and Continuity
MAS Covergece ad Cotiuity Key Objectives At the ed of the course, studets should kow the followig topics ad be able to apply the basic priciples ad theorems therei to solvig various problems cocerig covergece
More informationA Proof of Birkhoff s Ergodic Theorem
A Proof of Birkhoff s Ergodic Theorem Joseph Hora September 2, 205 Itroductio I Fall 203, I was learig the basics of ergodic theory, ad I came across this theorem. Oe of my supervisors, Athoy Quas, showed
More informationInternational Journal of Multidisciplinary Research and Modern Education (IJMRME) ISSN (Online): (www.rdmodernresearch.
(wwwrdoderresearchco) Volue II, Issue II, 2016 PRODUC OPERAION ON FUZZY RANSIION MARICES V Chiadurai*, S Barkavi**, S Vayabalaji*** & J Parthiba**** * Departet of Matheatics, Aaalai Uiversity, Aaalai Nagar,
More informationBounds for the Extreme Eigenvalues Using the Trace and Determinant
ISSN 746-7659, Eglad, UK Joural of Iformatio ad Computig Sciece Vol 4, No, 9, pp 49-55 Bouds for the Etreme Eigevalues Usig the Trace ad Determiat Qi Zhog, +, Tig-Zhu Huag School of pplied Mathematics,
More informationarxiv: v1 [math.nt] 26 Feb 2014
FROBENIUS NUMBERS OF PYTHAGOREAN TRIPLES BYUNG KEON GIL, JI-WOO HAN, TAE HYUN KIM, RYUN HAN KOO, BON WOO LEE, JAEHOON LEE, KYEONG SIK NAM, HYEON WOO PARK, AND POO-SUNG PARK arxiv:1402.6440v1 [ath.nt] 26
More informationChapter 3. Strong convergence. 3.1 Definition of almost sure convergence
Chapter 3 Strog covergece As poited out i the Chapter 2, there are multiple ways to defie the otio of covergece of a sequece of radom variables. That chapter defied covergece i probability, covergece i
More informationTHE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS
THE ASYMPTOTIC COMPLEXITY OF MATRIX REDUCTION OVER FINITE FIELDS DEMETRES CHRISTOFIDES Abstract. Cosider a ivertible matrix over some field. The Gauss-Jorda elimiatio reduces this matrix to the idetity
More informationBeurling Integers: Part 2
Beurlig Itegers: Part 2 Isomorphisms Devi Platt July 11, 2015 1 Prime Factorizatio Sequeces I the last article we itroduced the Beurlig geeralized itegers, which ca be represeted as a sequece of real umbers
More informationBIRKHOFF ERGODIC THEOREM
BIRKHOFF ERGODIC THEOREM Abstract. We will give a proof of the poitwise ergodic theorem, which was first proved by Birkhoff. May improvemets have bee made sice Birkhoff s orgial proof. The versio we give
More informationAdvanced Analysis. Min Yan Department of Mathematics Hong Kong University of Science and Technology
Advaced Aalysis Mi Ya Departmet of Mathematics Hog Kog Uiversity of Sciece ad Techology September 3, 009 Cotets Limit ad Cotiuity 7 Limit of Sequece 8 Defiitio 8 Property 3 3 Ifiity ad Ifiitesimal 8 4
More informationA sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece,, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet as
More informationIJITE Vol.2 Issue-11, (November 2014) ISSN: Impact Factor
IJITE Vol Issue-, (November 4) ISSN: 3-776 ATTRACTIVITY OF A HIGHER ORDER NONLINEAR DIFFERENCE EQUATION Guagfeg Liu School of Zhagjiagag Jiagsu Uiversit of Sciece ad Techolog, Zhagjiagag, Jiagsu 56,PR
More information42 Dependence and Bases
42 Depedece ad Bases The spa s(a) of a subset A i vector space V is a subspace of V. This spa ay be the whole vector space V (we say the A spas V). I this paragraph we study subsets A of V which spa V
More informationDifferent kinds of Mathematical Induction
Differet ids of Mathematical Iductio () Mathematical Iductio Give A N, [ A (a A a A)] A N () (First) Priciple of Mathematical Iductio Let P() be a propositio (ope setece), if we put A { : N p() is true}
More informationZeros of Polynomials
Math 160 www.timetodare.com 4.5 4.6 Zeros of Polyomials I these sectios we will study polyomials algebraically. Most of our work will be cocered with fidig the solutios of polyomial equatios of ay degree
More informationAPPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS
Hacettepe Joural of Mathematics ad Statistics Volume 32 (2003), 1 5 APPROXIMATION BY BERNSTEIN-CHLODOWSKY POLYNOMIALS E. İbili Received 27/06/2002 : Accepted 17/03/2003 Abstract The weighted approximatio
More informationSequences A sequence of numbers is a function whose domain is the positive integers. We can see that the sequence
Sequeces A sequece of umbers is a fuctio whose domai is the positive itegers. We ca see that the sequece 1, 1, 2, 2, 3, 3,... is a fuctio from the positive itegers whe we write the first sequece elemet
More information#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I
#A18 INTEGERS 11 (2011) THE (EXPONENTIAL) BIPARTITIONAL POLYNOMIALS AND POLYNOMIAL SEQUENCES OF TRINOMIAL TYPE: PART I Miloud Mihoubi 1 Uiversité des Scieces et de la Techologie Houari Bouediee Faculty
More informationON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES
Publ. Math. Debrece 8504, o. 3-4, 85 95. ON MONOTONICITY OF SOME COMBINATORIAL SEQUENCES QING-HU HOU*, ZHI-WEI SUN** AND HAOMIN WEN Abstract. We cofirm Su s cojecture that F / F 4 is strictly decreasig
More informationREVIEW 1, MATH n=1 is convergent. (b) Determine whether a n is convergent.
REVIEW, MATH 00. Let a = +. a) Determie whether the sequece a ) is coverget. b) Determie whether a is coverget.. Determie whether the series is coverget or diverget. If it is coverget, fid its sum. a)
More informationOn Order of a Function of Several Complex Variables Analytic in the Unit Polydisc
ISSN 746-7659, Eglad, UK Joural of Iforatio ad Coutig Sciece Vol 6, No 3, 0, 95-06 O Order of a Fuctio of Several Colex Variables Aalytic i the Uit Polydisc Rata Kuar Dutta + Deartet of Matheatics, Siliguri
More informationAn Introduction to Randomized Algorithms
A Itroductio to Radomized Algorithms The focus of this lecture is to study a radomized algorithm for quick sort, aalyze it usig probabilistic recurrece relatios, ad also provide more geeral tools for aalysis
More informationApproximation theorems for localized szász Mirakjan operators
Joural of Approximatio Theory 152 (2008) 125 134 www.elsevier.com/locate/jat Approximatio theorems for localized szász Miraja operators Lise Xie a,,1, Tigfa Xie b a Departmet of Mathematics, Lishui Uiversity,
More informationBETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS. 1. Introduction. Throughout the paper we denote by X a linear space and by Y a topological linear
BETWEEN QUASICONVEX AND CONVEX SET-VALUED MAPPINGS Abstract. The aim of this paper is to give sufficiet coditios for a quasicovex setvalued mappig to be covex. I particular, we recover several kow characterizatios
More informationDisjoint Systems. Abstract
Disjoit Systems Noga Alo ad Bey Sudaov Departmet of Mathematics Raymod ad Beverly Sacler Faculty of Exact Scieces Tel Aviv Uiversity, Tel Aviv, Israel Abstract A disjoit system of type (,,, ) is a collectio
More informationConcavity of weighted arithmetic means with applications
Arch. Math. 69 (1997) 120±126 0003-889X/97/020120-07 $ 2.90/0 Birkhäuser Verlag, Basel, 1997 Archiv der Mathematik Cocavity of weighted arithmetic meas with applicatios By ARKADY BERENSTEIN ad ALEK VAINSHTEIN*)
More informationECE 901 Lecture 4: Estimation of Lipschitz smooth functions
ECE 9 Lecture 4: Estiatio of Lipschitz sooth fuctios R. Nowak 5/7/29 Cosider the followig settig. Let Y f (X) + W, where X is a rado variable (r.v.) o X [, ], W is a r.v. o Y R, idepedet of X ad satisfyig
More informationTHE INTEGRAL TEST AND ESTIMATES OF SUMS
THE INTEGRAL TEST AND ESTIMATES OF SUMS. Itroductio Determiig the exact sum of a series is i geeral ot a easy task. I the case of the geometric series ad the telescoig series it was ossible to fid a simle
More information[ 11 ] z of degree 2 as both degree 2 each. The degree of a polynomial in n variables is the maximum of the degrees of its terms.
[ 11 ] 1 1.1 Polyomial Fuctios 1 Algebra Ay fuctio f ( x) ax a1x... a1x a0 is a polyomial fuctio if ai ( i 0,1,,,..., ) is a costat which belogs to the set of real umbers ad the idices,, 1,...,1 are atural
More informationLesson 10: Limits and Continuity
www.scimsacademy.com Lesso 10: Limits ad Cotiuity SCIMS Academy 1 Limit of a fuctio The cocept of limit of a fuctio is cetral to all other cocepts i calculus (like cotiuity, derivative, defiite itegrals
More informationMATH301 Real Analysis (2008 Fall) Tutorial Note #7. k=1 f k (x) converges pointwise to S(x) on E if and
MATH01 Real Aalysis (2008 Fall) Tutorial Note #7 Sequece ad Series of fuctio 1: Poitwise Covergece ad Uiform Covergece Part I: Poitwise Covergece Defiitio of poitwise covergece: A sequece of fuctios f
More informationMA131 - Analysis 1. Workbook 3 Sequences II
MA3 - Aalysis Workbook 3 Sequeces II Autum 2004 Cotets 2.8 Coverget Sequeces........................ 2.9 Algebra of Limits......................... 2 2.0 Further Useful Results........................
More informationOn groups of diffeomorphisms of the interval with finitely many fixed points II. Azer Akhmedov
O groups of diffeomorphisms of the iterval with fiitely may fixed poits II Azer Akhmedov Abstract: I [6], it is proved that ay subgroup of Diff ω +(I) (the group of orietatio preservig aalytic diffeomorphisms
More informationConcavity Solutions of Second-Order Differential Equations
Proceedigs of the Paista Academy of Scieces 5 (3): 4 45 (4) Copyright Paista Academy of Scieces ISSN: 377-969 (prit), 36-448 (olie) Paista Academy of Scieces Research Article Cocavity Solutios of Secod-Order
More informationLecture 19. sup y 1,..., yn B d n
STAT 06A: Polyomials of adom Variables Lecture date: Nov Lecture 19 Grothedieck s Iequality Scribe: Be Hough The scribes are based o a guest lecture by ya O Doell. I this lecture we prove Grothedieck s
More informationNew Inequalities For Convex Sequences With Applications
It. J. Ope Problems Comput. Math., Vol. 5, No. 3, September, 0 ISSN 074-87; Copyright c ICSRS Publicatio, 0 www.i-csrs.org New Iequalities For Covex Sequeces With Applicatios Zielaâbidie Latreuch ad Beharrat
More informationMi-Hwa Ko and Tae-Sung Kim
J. Korea Math. Soc. 42 2005), No. 5, pp. 949 957 ALMOST SURE CONVERGENCE FOR WEIGHTED SUMS OF NEGATIVELY ORTHANT DEPENDENT RANDOM VARIABLES Mi-Hwa Ko ad Tae-Sug Kim Abstract. For weighted sum of a sequece
More informationA Hadamard-type lower bound for symmetric diagonally dominant positive matrices
A Hadamard-type lower boud for symmetric diagoally domiat positive matrices Christopher J. Hillar, Adre Wibisoo Uiversity of Califoria, Berkeley Jauary 7, 205 Abstract We prove a ew lower-boud form of
More informationEigenvalues and Eigenvectors
5 Eigevalues ad Eigevectors 5.3 DIAGONALIZATION DIAGONALIZATION Example 1: Let. Fid a formula for A k, give that P 1 1 = 1 2 ad, where Solutio: The stadard formula for the iverse of a 2 2 matrix yields
More informationMDIV. Multiple divisor functions
MDIV. Multiple divisor fuctios The fuctios τ k For k, defie τ k ( to be the umber of (ordered factorisatios of ito k factors, i other words, the umber of ordered k-tuples (j, j 2,..., j k with j j 2...
More informationMONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY
MONOTONICITY OF SEQUENCES INVOLVING GEOMETRIC MEANS OF POSITIVE SEQUENCES WITH LOGARITHMICAL CONVEXITY FENG QI AND BAI-NI GUO Abstract. Let f be a positive fuctio such that x [ f(x + )/f(x) ] is icreasig
More informationProblem 4: Evaluate ( k ) by negating (actually un-negating) its upper index. Binomial coefficient
Problem 4: Evaluate by egatig actually u-egatig its upper idex We ow that Biomial coefficiet r { where r is a real umber, is a iteger The above defiitio ca be recast i terms of factorials i the commo case
More informationMath 341 Lecture #31 6.5: Power Series
Math 341 Lecture #31 6.5: Power Series We ow tur our attetio to a particular kid of series of fuctios, amely, power series, f(x = a x = a 0 + a 1 x + a 2 x 2 + where a R for all N. I terms of a series
More informationLargest families without an r-fork
Largest families without a r-for Aalisa De Bois Uiversity of Salero Salero, Italy debois@math.it Gyula O.H. Katoa Réyi Istitute Budapest, Hugary ohatoa@reyi.hu Itroductio Let [] = {,,..., } be a fiite
More informationMath 113 Exam 3 Practice
Math Exam Practice Exam will cover.-.9. This sheet has three sectios. The first sectio will remid you about techiques ad formulas that you should kow. The secod gives a umber of practice questios for you
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More information10.1 Sequences. n term. We will deal a. a n or a n n. ( 1) n ( 1) n 1 2 ( 1) a =, 0 0,,,,, ln n. n an 2. n term.
0. Sequeces A sequece is a list of umbers writte i a defiite order: a, a,, a, a is called the first term, a is the secod term, ad i geeral eclusively with ifiite sequeces ad so each term Notatio: the sequece
More informationA LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II 1. INTRODUCTION
A LIMITED ARITHMETIC ON SIMPLE CONTINUED FRACTIONS - II C. T. LONG J. H. JORDAN* Washigto State Uiversity, Pullma, Washigto 1. INTRODUCTION I the first paper [2 ] i this series, we developed certai properties
More informationQ-BINOMIALS AND THE GREATEST COMMON DIVISOR. Keith R. Slavin 8474 SW Chevy Place, Beaverton, Oregon 97008, USA.
INTEGERS: ELECTRONIC JOURNAL OF COMBINATORIAL NUMBER THEORY 8 2008, #A05 Q-BINOMIALS AND THE GREATEST COMMON DIVISOR Keith R. Slavi 8474 SW Chevy Place, Beaverto, Orego 97008, USA slavi@dsl-oly.et Received:
More information1 Convergence in Probability and the Weak Law of Large Numbers
36-752 Advaced Probability Overview Sprig 2018 8. Covergece Cocepts: i Probability, i L p ad Almost Surely Istructor: Alessadro Rialdo Associated readig: Sec 2.4, 2.5, ad 4.11 of Ash ad Doléas-Dade; Sec
More informationSieve Estimators: Consistency and Rates of Convergence
EECS 598: Statistical Learig Theory, Witer 2014 Topic 6 Sieve Estimators: Cosistecy ad Rates of Covergece Lecturer: Clayto Scott Scribe: Julia Katz-Samuels, Brado Oselio, Pi-Yu Che Disclaimer: These otes
More informationOn the behavior at infinity of an integrable function
O the behavior at ifiity of a itegrable fuctio Emmauel Lesige To cite this versio: Emmauel Lesige. O the behavior at ifiity of a itegrable fuctio. The America Mathematical Mothly, 200, 7 (2), pp.75-8.
More informationMa 530 Infinite Series I
Ma 50 Ifiite Series I Please ote that i additio to the material below this lecture icorporated material from the Visual Calculus web site. The material o sequeces is at Visual Sequeces. (To use this li
More information