On the L p -conjecture for locally compact groups

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "On the L p -conjecture for locally compact groups"

Transcription

1 Arch. Math. 89 (2007), c 2007 Birkhäuser Verlag Basel/Switzerlad 0003/889X/ , ublished olie DOI 0.007/s x Archiv der Mathematik O the L -cojecture for locally comact grous F. Abtahi, R. Nasr-Isfahai ad A. Rejali Abstract. Let be a locally comact grou. For <<, it is well-kow that f g exists ad belogs to L () for all f, g L () if ad oly if is comact. Here, for 2 <<, we show that f g exists for all f, g L () if ad oly if is comact. We also show that this result does ot remai true for < 2. Mathematics Subject Classificatio (2000). Primary: 43A5; Secodary: 43A20. eywords. Covolutio, locally comact grou, L sace, L cojecture.. Itroductio. Throughout the aer, let be a locally comact grou with a fixed left Haar measure λ. For <<, the Lebesgue sace L () with resect to λ is as defied i [5]; i.e. the equivalece classes of measurable fuctios o with ( / f := f(x) dλ(x)) <. For measurable fuctios f ad g o, the covolutio multilicatio (f g)(x) = f(y) g(y x) dλ(y) is defied at each oit x for which this makes sese; i.e. the fuctio y f(y) g(y x) is λ-itegrable. The f g is said to exist if (f g)(x) exists for almost all x. The L -cojecture asserts that if f g exists ad belogs to L () for all f,g L (), the is comact. This cojecture was first formulated by Rajagoala i his Ph.D. thesis i 963. However, the first result related to this cojecture is due to Zelazko [8] ad Urbaik [7] i 96; they roved that the cojecture is true for all locally

2 238 F. Abtahi, R. Nasr-Isfahai ad A. Rejali Arch. Math. comact abelia grous. The truth of the cojecture has bee established for >2 by Zelazko [9] ad Rajagoala [] ideedetly; see also Rajagoala s works [0] for the case where 2 ad is discrete, [] for the case where = 2 ad is totally discoected, ad [2] for the case where > ad is either ilotet or a semidirect roduct of two locally comact grous. I the joit work [3], they showed that the cojecture is true for > ad ameable grous; this result ca be also foud i reeleaf s book [4]. Rickert [5] cofirmed the cojecture for = 2. For related results o the subject see also Crombez [] ad [2], audet ad amle [3], Johso [6], uze ad Stei [7], Lohoue [8], Miles [9], Rickert [4], ad Zelazko [20]. Fially, i 990, Saeki [6] gave a affirmative aswer to the cojecture by a comletely self-cotaied roof. Motivated by the L -cojecture, we cosider oly the roerty that f g exists for all f,g L (), ad rove the followig result which is ideed the urose of this work. Theorem.. Let be a locally comact grou ad >2. Iff g exists for all f,g L (), the is comact. The followig examle shows that Theorem. is, i geeral, ot true for < 2. Examle.2. Let be a locally comact grou. (a) If is uimodular, it follows from Hölder s iequality that f g exists for all f,g L 2 (). (b) If < 2 ad is discrete, the f g exists for all f,g L (); this follows from (a) together with the fact that L () L 2 (). As a cosequece of these observatios together with the solutio of the L -cojecture ad Theorem., we have the followig corollary. Corollary.3. Let be a locally comact grou ad <<. Cosider the followig coditios. (a) is comact. (b) f g exists ad belogs to L () for all f,g L (). (c) f g exists for all f,g L (). The (a) (b) (c) ad also the followig assertios hold. (i) If <<2 ad is discrete, the (c) always holds. (ii) If =2ad is uimodular, the (c) always holds. (iii) If >2, the (a) (b) (c). This corollary leads us to the followig atural questio. Questio Let be a locally comact grou ad < 2. Doesf g exist for all f,g L ()?

3 Vol. 89 (2007) O the L -cojecture for locally comact grous The roof. Proof of Theorem.. Let be a fixed comact symmetric eighbourhood of the idetity elemet of. The 0 <λ() λ( 2 ) < ad there exists a costat C>0 such that (x) <C (x ), where deotes the modular fuctio of. Suose o the cotrary that is ot comact. The \ is a oemty symmetric subset of. Thus, there is a elemet a of \ with (a ). Iductively, we may fid a sequece (a )i with (a ) such that a \ m= The for every m, with m<, a m 4 ( 2). a m 2 a 2 = ad a m a =. For each A, let χ A deote the characteristic fuctio of A o, ad defie the fuctios f ad g o by ad f(x) = (x ) / g(x) = χ a (x) χ a 2(x) for all x. The f,g L (); ideed, for each wehave (x ) χ a (x) dλ(x) = (x ) χ (xa ) dλ(x) = (a ) (a x ) χ (x) dλ(x) = (a ) (a ) (x ) dλ(x) = (x ) dλ(x) Cλ()

4 240 F. Abtahi, R. Nasr-Isfahai ad A. Rejali Arch. Math. from which it follows that f(x) dλ(x) = Moreover, <. g(x) dλ(x) = Cλ() = = (x ) χ a (x) dλ(x) χ a2(x) dλ(x) = λ( 2 ) <. χ 2(a x) dλ(x) χ 2(x) dλ(x) We ext show that (f g)(x) = for all x. To that ed, recall that (a ) ad / > 0 ad hece (a ) / ( ). We thus have (y ) / χ a (y) dλ(y) = (y ) / χ (ya ) dλ(y) = (a ) (a y ) / χ (y) dλ(y) = (a ) (a ) / (y ) / χ (y) dλ(y) = (a ) / (y ) / dλ(y) (y ) / dλ(y) C / λ().

5 Vol. 89 (2007) O the L -cojecture for locally comact grous 24 Now, let x ad ote that y x a 2 for all y a ad. Therefore (f g)(x) = f(y) g(y x) dλ(y) = (y ) / χ a (y) dλ(y) C / λ() =. It follows that f g does ot exist whereas f,g L (). This cotradictio comletes the roof. Remark. It follows from Theorem. that if is discrete, >2 ad f g exists for all f,g L (), the is fiite. Let us coclude this work with a direct roof of this fact. Suose o the cotrary that is ifiite. The there is a elemet a with {a,a } \{e}. Iductively, we may fid a sequece (a )i such that {a +,a + } \{e, a,a,..., a,a } for all. Defie the fuctio f o by f(a )=f(a )=/ for all ad f(x) = 0 for all x {a,a,a 2,a 2,... }. The f L (), but (f f)(e) does ot exist; ideed, (f f)(e) = f(a )f(a )= /. Sice is discrete, this shows that f f does ot exist, a cotradictio. Ackowledgemet. The authors would like to thak the referee of the aer for ivaluable commets. This research was artially suorted by the Ceters of Excellece for Mathematics at the Isfaha Uiversity of Techology ad the Uiversity of Isfaha. Refereces []. Crombez, A characterizatio of comact grous. Quart. J. Pure Al. Math. 53, 9 2 (979). [2]. Crombez, A elemetary roof about the order of the elemets i a discrete grou. Proc. Amer. Math. Soc. 85, (983). [3] R. J. audet ad J. L. amle, A elemetary roof of art of a classical cojecture. Bull. Austral. Math. Soc. 3, (970).

6 242 F. Abtahi, R. Nasr-Isfahai ad A. Rejali Arch. Math. [4] F. P. reeleaf, Ivariat meas o locally comact grous ad their alicatios. Math. Studies 6, Va Nostrad, New York, 969. [5] E. Hewitt ad. Ross, Abstract harmoic aalysis I. Sriger-Verlag, New York, 970. [6] D. L. Johso, A ew roof of the L -cojecture for locally comact grous. Colloq. Math. 47, 0 02 (982). [7] R. uze ad E. Stei, Uiformly bouded reresetatios ad harmoic aalysis of the 2 2 real uimodualr grou. Amer. J. Math. 82, 62 (960). [8] N. Lohoue, Estimatios L des coefficiets de reresetatio et oerateurs de covolutio. Adv. Math. 38, (980). [9] P. Miles, Covolutio of L fuctios o o-commutative grous. Caad. Math. Bull. 4, (97). [0] M. Rajagoala, Othel -saces of a discrete grou. Colloq. Math. 0, (963). [] M. Rajagoala, L -cojecture for locally comact grous I. Tras. Amer. Math. Soc. 25, (966). [2] M. Rajagoala, L -cojecture for locally comact grous II. Math. A. 69, (967). [3] M. Rajagoala ad W. Zelazko, L -cojecture for solvable locally comact grous. J. Idia Math. Soc. 29, (965). [4] N. W. Rickert, Covolutio of L fuctios. Proc. Amer. Math. Soc. 8, (967). [5] N. W. Rickert, Covolutio of L 2 fuctios. Colloq. Math. 9, (968). [6] S. Saeki, The L cojecture ad Youg s iequality. Illiois. J. Math. 34, (990). [7]. Urbaik, A roof of a theorem of Zelazko o L -algebras. Colloq. Math 8, 2 23 (96). [8] W. Zelazko, O the algebras L of a locally comact grou. Colloq. Math. 8, 2 20 (96). [9] W. Zelazko, A ote o L algebras. Colloq. Math. 0, (963). [20] W. Zelazko, O the Burside roblem for locally comact grous. Sym. Math. 6, (975). F. Abtahi, Deartmet of Mathematics, Uiversity of Isfaha, Isfaha, Ira R. Nasr-Isfahai, Deartmet of Mathematical Scieces, Isfaha Uiversity of Techology, Isfaha , Ira A. Rejali, Deartmet of Mathematics, Uiversity of Isfaha, Isfaha, Ira Received: 23 Aril 2006

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein)

F. ABTAHI and M. ZARRIN. (Communicated by J. Goldstein) Journal of Algerian Mathematical Society Vol. 1, pp. 1 6 1 CONCERNING THE l p -CONJECTURE FOR DISCRETE SEMIGROUPS F. ABTAHI and M. ZARRIN (Communicated by J. Goldstein) Abstract. For 2 < p

More information

Irreducible polynomials with consecutive zero coefficients

Irreducible polynomials with consecutive zero coefficients Irreducible polyomials with cosecutive zero coefficiets Theodoulos Garefalakis Departmet of Mathematics, Uiversity of Crete, 71409 Heraklio, Greece Abstract Let q be a prime power. We cosider the problem

More information

Factors of sums of powers of binomial coefficients

Factors of sums of powers of binomial coefficients ACTA ARITHMETICA LXXXVI.1 (1998) Factors of sums of powers of biomial coefficiets by Neil J. Cali (Clemso, S.C.) Dedicated to the memory of Paul Erdős 1. Itroductio. It is well ow that if ( ) a f,a = the

More information

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE

THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE THE LEAST COMMON MULTIPLE OF A QUADRATIC SEQUENCE JAVIER CILLERUELO Abstract. We obtai, for ay irreducible quadratic olyomial f(x = ax 2 + bx + c, the asymtotic estimate log l.c.m. {f(1,..., f(} log. Whe

More information

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series

Our aim is to show that under reasonable assumptions a given 2π-periodic function f can be represented as convergent series 8 Fourier Series Our aim is to show that uder reasoable assumptios a give -periodic fuctio f ca be represeted as coverget series f(x) = a + (a cos x + b si x). (8.) By defiitio, the covergece of the series

More information

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008 I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

More information

Infinite Sequences and Series

Infinite Sequences and Series CHAPTER 4 Ifiite Sequeces ad Series 4.1. Sequeces A sequece is a ifiite ordered list of umbers, for example the sequece of odd positive itegers: 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29...

More information

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx

SAMPLE QUESTIONS FOR FINAL EXAM. (1) (2) (3) (4) Find the following using the definition of the Riemann integral: (2x + 1)dx SAMPLE QUESTIONS FOR FINAL EXAM REAL ANALYSIS I FALL 006 3 4 Fid the followig usig the defiitio of the Riema itegral: a 0 x + dx 3 Cosider the partitio P x 0 3, x 3 +, x 3 +,......, x 3 3 + 3 of the iterval

More information

Asymptotic Growth of Functions

Asymptotic Growth of Functions CMPS Itroductio to Aalysis of Algorithms Fall 3 Asymptotic Growth of Fuctios We itroduce several types of asymptotic otatio which are used to compare the performace ad efficiecy of algorithms As we ll

More information

Section 11.3: The Integral Test

Section 11.3: The Integral Test Sectio.3: The Itegral Test Most of the series we have looked at have either diverged or have coverged ad we have bee able to fid what they coverge to. I geeral however, the problem is much more difficult

More information

1. MATHEMATICAL INDUCTION

1. MATHEMATICAL INDUCTION 1. MATHEMATICAL INDUCTION EXAMPLE 1: Prove that for ay iteger 1. Proof: 1 + 2 + 3 +... + ( + 1 2 (1.1 STEP 1: For 1 (1.1 is true, sice 1 1(1 + 1. 2 STEP 2: Suppose (1.1 is true for some k 1, that is 1

More information

Theorems About Power Series

Theorems About Power Series Physics 6A Witer 20 Theorems About Power Series Cosider a power series, f(x) = a x, () where the a are real coefficiets ad x is a real variable. There exists a real o-egative umber R, called the radius

More information

A note on the boundary behavior for a modified Green function in the upper-half space

A note on the boundary behavior for a modified Green function in the upper-half space Zhag ad Pisarev Boudary Value Problems (015) 015:114 DOI 10.1186/s13661-015-0363-z RESEARCH Ope Access A ote o the boudary behavior for a modified Gree fuctio i the upper-half space Yulia Zhag1 ad Valery

More information

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER?

WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? WHEN IS THE (CO)SINE OF A RATIONAL ANGLE EQUAL TO A RATIONAL NUMBER? JÖRG JAHNEL 1. My Motivatio Some Sort of a Itroductio Last term I tought Topological Groups at the Göttige Georg August Uiversity. This

More information

Acta Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p <

Acta Acad. Paed. Agriensis, Sectio Mathematicae 29 (2002) 77 87. ALMOST SURE FUNCTIONAL LIMIT THEOREMS IN L p( ]0, 1[ ), WHERE 1 p < Acta Acad. Paed. Agriesis, Sectio Mathematicae 29 22) 77 87 ALMOST SUR FUNCTIONAL LIMIT THORMS IN L ], [ ), WHR < József Túri Nyíregyháza, Hugary) Dedicated to the memory of Professor Péter Kiss Abstract.

More information

Convexity, Inequalities, and Norms

Convexity, Inequalities, and Norms Covexity, Iequalities, ad Norms Covex Fuctios You are probably familiar with the otio of cocavity of fuctios. Give a twicedifferetiable fuctio ϕ: R R, We say that ϕ is covex (or cocave up) if ϕ (x) 0 for

More information

A Note on Sums of Greatest (Least) Prime Factors

A Note on Sums of Greatest (Least) Prime Factors It. J. Cotemp. Math. Scieces, Vol. 8, 203, o. 9, 423-432 HIKARI Ltd, www.m-hikari.com A Note o Sums of Greatest (Least Prime Factors Rafael Jakimczuk Divisio Matemática, Uiversidad Nacioal de Luá Bueos

More information

THE ABRACADABRA PROBLEM

THE ABRACADABRA PROBLEM THE ABRACADABRA PROBLEM FRANCESCO CARAVENNA Abstract. We preset a detailed solutio of Exercise E0.6 i [Wil9]: i a radom sequece of letters, draw idepedetly ad uiformly from the Eglish alphabet, the expected

More information

4.3. The Integral and Comparison Tests

4.3. The Integral and Comparison Tests 4.3. THE INTEGRAL AND COMPARISON TESTS 9 4.3. The Itegral ad Compariso Tests 4.3.. The Itegral Test. Suppose f is a cotiuous, positive, decreasig fuctio o [, ), ad let a = f(). The the covergece or divergece

More information

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE

ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE Proceedigs of the Iteratioal Coferece o Theory ad Applicatios of Mathematics ad Iformatics ICTAMI 3, Alba Iulia ON AN INTEGRAL OPERATOR WHICH PRESERVE THE UNIVALENCE by Maria E Gageoea ad Silvia Moldoveau

More information

Chapter 7 Methods of Finding Estimators

Chapter 7 Methods of Finding Estimators Chapter 7 for BST 695: Special Topics i Statistical Theory. Kui Zhag, 011 Chapter 7 Methods of Fidig Estimators Sectio 7.1 Itroductio Defiitio 7.1.1 A poit estimator is ay fuctio W( X) W( X1, X,, X ) of

More information

Analysis Notes (only a draft, and the first one!)

Analysis Notes (only a draft, and the first one!) Aalysis Notes (oly a draft, ad the first oe!) Ali Nesi Mathematics Departmet Istabul Bilgi Uiversity Kuştepe Şişli Istabul Turkey aesi@bilgi.edu.tr Jue 22, 2004 2 Cotets 1 Prelimiaries 9 1.1 Biary Operatio...........................

More information

Chapter 5 O A Cojecture Of Erdíos Proceedigs NCUR VIII è1994è, Vol II, pp 794í798 Jeærey F Gold Departmet of Mathematics, Departmet of Physics Uiversity of Utah Do H Tucker Departmet of Mathematics Uiversity

More information

Class Meeting # 16: The Fourier Transform on R n

Class Meeting # 16: The Fourier Transform on R n MATH 18.152 COUSE NOTES - CLASS MEETING # 16 18.152 Itroductio to PDEs, Fall 2011 Professor: Jared Speck Class Meetig # 16: The Fourier Trasform o 1. Itroductio to the Fourier Trasform Earlier i the course,

More information

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variance, the law of large numbers and the Monte-Carlo method Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

More information

ON THE DENSE TRAJECTORY OF LASOTA EQUATION

ON THE DENSE TRAJECTORY OF LASOTA EQUATION UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLIII 2005 ON THE DENSE TRAJECTORY OF LASOTA EQUATION by Atoi Leo Dawidowicz ad Najemedi Haribash Abstract. I preseted paper the dese trajectory

More information

Department of Computer Science, University of Otago

Department of Computer Science, University of Otago Departmet of Computer Sciece, Uiversity of Otago Techical Report OUCS-2006-09 Permutatios Cotaiig May Patters Authors: M.H. Albert Departmet of Computer Sciece, Uiversity of Otago Micah Colema, Rya Fly

More information

3. Greatest Common Divisor - Least Common Multiple

3. Greatest Common Divisor - Least Common Multiple 3 Greatest Commo Divisor - Least Commo Multiple Defiitio 31: The greatest commo divisor of two atural umbers a ad b is the largest atural umber c which divides both a ad b We deote the greatest commo gcd

More information

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp. 12-19 ISSN 2219-7184; Copyright ICSRS Publicatio, 2012 www.i-csrs.org Available free olie at http://www.gema.i Degree of Approximatio of Cotiuous Fuctios

More information

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK

THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for

More information

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009)

Lecture 13. Lecturer: Jonathan Kelner Scribe: Jonathan Pines (2009) 18.409 A Algorithmist s Toolkit October 27, 2009 Lecture 13 Lecturer: Joatha Keler Scribe: Joatha Pies (2009) 1 Outlie Last time, we proved the Bru-Mikowski iequality for boxes. Today we ll go over the

More information

Soving Recurrence Relations

Soving Recurrence Relations Sovig Recurrece Relatios Part 1. Homogeeous liear 2d degree relatios with costat coefficiets. Cosider the recurrece relatio ( ) T () + at ( 1) + bt ( 2) = 0 This is called a homogeeous liear 2d degree

More information

3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2

3 Energy. 3.3. Non-Flow Energy Equation (NFEE) Internal Energy. MECH 225 Engineering Science 2 MECH 5 Egieerig Sciece 3 Eergy 3.3. No-Flow Eergy Equatio (NFEE) You may have oticed that the term system kees croig u. It is ecessary, therefore, that before we start ay aalysis we defie the system that

More information

A probabilistic proof of a binomial identity

A probabilistic proof of a binomial identity A probabilistic proof of a biomial idetity Joatho Peterso Abstract We give a elemetary probabilistic proof of a biomial idetity. The proof is obtaied by computig the probability of a certai evet i two

More information

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM

PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM PROCEEDINGS OF THE YEREVAN STATE UNIVERSITY Physical ad Mathematical Scieces 2015, 1, p. 15 19 M a t h e m a t i c s AN ALTERNATIVE MODEL FOR BONUS-MALUS SYSTEM A. G. GULYAN Chair of Actuarial Mathematics

More information

I. Chi-squared Distributions

I. Chi-squared Distributions 1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

More information

Sequences and Series

Sequences and Series CHAPTER 9 Sequeces ad Series 9.. Covergece: Defiitio ad Examples Sequeces The purpose of this chapter is to itroduce a particular way of geeratig algorithms for fidig the values of fuctios defied by their

More information

Metric, Normed, and Topological Spaces

Metric, Normed, and Topological Spaces Chapter 13 Metric, Normed, ad Topological Spaces A metric space is a set X that has a otio of the distace d(x, y) betwee every pair of poits x, y X. A fudametal example is R with the absolute-value metric

More information

Lecture 5: Span, linear independence, bases, and dimension

Lecture 5: Span, linear independence, bases, and dimension Lecture 5: Spa, liear idepedece, bases, ad dimesio Travis Schedler Thurs, Sep 23, 2010 (versio: 9/21 9:55 PM) 1 Motivatio Motivatio To uderstad what it meas that R has dimesio oe, R 2 dimesio 2, etc.;

More information

Overview on S-Box Design Principles

Overview on S-Box Design Principles Overview o S-Box Desig Priciples Debdeep Mukhopadhyay Assistat Professor Departmet of Computer Sciece ad Egieerig Idia Istitute of Techology Kharagpur INDIA -721302 What is a S-Box? S-Boxes are Boolea

More information

Chapter 5: Inner Product Spaces

Chapter 5: Inner Product Spaces Chapter 5: Ier Product Spaces Chapter 5: Ier Product Spaces SECION A Itroductio to Ier Product Spaces By the ed of this sectio you will be able to uderstad what is meat by a ier product space give examples

More information

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem

Lecture 4: Cauchy sequences, Bolzano-Weierstrass, and the Squeeze theorem Lecture 4: Cauchy sequeces, Bolzao-Weierstrass, ad the Squeeze theorem The purpose of this lecture is more modest tha the previous oes. It is to state certai coditios uder which we are guarateed that limits

More information

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006

UC Berkeley Department of Electrical Engineering and Computer Science. EE 126: Probablity and Random Processes. Solutions 9 Spring 2006 Exam format UC Bereley Departmet of Electrical Egieerig ad Computer Sciece EE 6: Probablity ad Radom Processes Solutios 9 Sprig 006 The secod midterm will be held o Wedesday May 7; CHECK the fial exam

More information

a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x

a 4 = 4 2 4 = 12. 2. Which of the following sequences converge to zero? n 2 (a) n 2 (b) 2 n x 2 x 2 + 1 = lim x n 2 + 1 = lim x 0 INFINITE SERIES 0. Sequeces Preiary Questios. What is a 4 for the sequece a? solutio Substitutig 4 i the expressio for a gives a 4 4 4.. Which of the followig sequeces coverge to zero? a b + solutio

More information

Building Blocks Problem Related to Harmonic Series

Building Blocks Problem Related to Harmonic Series TMME, vol3, o, p.76 Buildig Blocks Problem Related to Harmoic Series Yutaka Nishiyama Osaka Uiversity of Ecoomics, Japa Abstract: I this discussio I give a eplaatio of the divergece ad covergece of ifiite

More information

2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS

2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 2 MATH 101B: ALGEBRA II, PART D: REPRESENTATIONS OF GROUPS 1. The group rig k[g] The mai idea is that represetatios of a group G over a field k are the same as modules over the group rig k[g]. First I

More information

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5

0.7 0.6 0.2 0 0 96 96.5 97 97.5 98 98.5 99 99.5 100 100.5 96.5 97 97.5 98 98.5 99 99.5 100 100.5 Sectio 13 Kolmogorov-Smirov test. Suppose that we have a i.i.d. sample X 1,..., X with some ukow distributio P ad we would like to test the hypothesis that P is equal to a particular distributio P 0, i.e.

More information

MARTINGALES AND A BASIC APPLICATION

MARTINGALES AND A BASIC APPLICATION MARTINGALES AND A BASIC APPLICATION TURNER SMITH Abstract. This paper will develop the measure-theoretic approach to probability i order to preset the defiitio of martigales. From there we will apply this

More information

Ekkehart Schlicht: Economic Surplus and Derived Demand

Ekkehart Schlicht: Economic Surplus and Derived Demand Ekkehart Schlicht: Ecoomic Surplus ad Derived Demad Muich Discussio Paper No. 2006-17 Departmet of Ecoomics Uiversity of Muich Volkswirtschaftliche Fakultät Ludwig-Maximilias-Uiversität Müche Olie at http://epub.ub.ui-mueche.de/940/

More information

A Recursive Formula for Moments of a Binomial Distribution

A Recursive Formula for Moments of a Binomial Distribution A Recursive Formula for Momets of a Biomial Distributio Árpád Béyi beyi@mathumassedu, Uiversity of Massachusetts, Amherst, MA 01003 ad Saverio M Maago smmaago@psavymil Naval Postgraduate School, Moterey,

More information

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix

FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. 1. Powers of a matrix FIBONACCI NUMBERS: AN APPLICATION OF LINEAR ALGEBRA. Powers of a matrix We begi with a propositio which illustrates the usefuless of the diagoalizatio. Recall that a square matrix A is diogaalizable if

More information

Ramsey-type theorems with forbidden subgraphs

Ramsey-type theorems with forbidden subgraphs Ramsey-type theorems with forbidde subgraphs Noga Alo Jáos Pach József Solymosi Abstract A graph is called H-free if it cotais o iduced copy of H. We discuss the followig questio raised by Erdős ad Hajal.

More information

PART TWO. Measure, Integration, and Differentiation

PART TWO. Measure, Integration, and Differentiation PART TWO Measure, Itegratio, ad Differetiatio Émile Félix-Édouard-Justi Borel (1871 1956 Émile Borel was bor at Sait-Affrique, Frace, o Jauary 7, 1871, the third child of Hooré Borel, a Protestat miister,

More information

A sharp Trudinger-Moser type inequality for unbounded domains in R n

A sharp Trudinger-Moser type inequality for unbounded domains in R n A sharp Trudiger-Moser type iequality for ubouded domais i R Yuxiag Li ad Berhard Ruf Abstract The Trudiger-Moser iequality states that for fuctios u H, 0 (Ω) (Ω R a bouded domai) with Ω u dx oe has Ω

More information

Properties of MLE: consistency, asymptotic normality. Fisher information.

Properties of MLE: consistency, asymptotic normality. Fisher information. Lecture 3 Properties of MLE: cosistecy, asymptotic ormality. Fisher iformatio. I this sectio we will try to uderstad why MLEs are good. Let us recall two facts from probability that we be used ofte throughout

More information

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing

Perfect Packing Theorems and the Average-Case Behavior of Optimal and Online Bin Packing SIAM REVIEW Vol. 44, No. 1, pp. 95 108 c 2002 Society for Idustrial ad Applied Mathematics Perfect Packig Theorems ad the Average-Case Behavior of Optimal ad Olie Bi Packig E. G. Coffma, Jr. C. Courcoubetis

More information

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length

A Faster Clause-Shortening Algorithm for SAT with No Restriction on Clause Length Joural o Satisfiability, Boolea Modelig ad Computatio 1 2005) 49-60 A Faster Clause-Shorteig Algorithm for SAT with No Restrictio o Clause Legth Evgey Datsi Alexader Wolpert Departmet of Computer Sciece

More information

2-3 The Remainder and Factor Theorems

2-3 The Remainder and Factor Theorems - The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

More information

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13

Discrete Mathematics and Probability Theory Spring 2014 Anant Sahai Note 13 EECS 70 Discrete Mathematics ad Probability Theory Sprig 2014 Aat Sahai Note 13 Itroductio At this poit, we have see eough examples that it is worth just takig stock of our model of probability ad may

More information

Central Limit Theorem and Its Applications to Baseball

Central Limit Theorem and Its Applications to Baseball Cetral Limit Theorem ad Its Applicatios to Baseball by Nicole Aderso A project submitted to the Departmet of Mathematical Scieces i coformity with the requiremets for Math 4301 (Hoours Semiar) Lakehead

More information

Maximum Likelihood Estimators.

Maximum Likelihood Estimators. Lecture 2 Maximum Likelihood Estimators. Matlab example. As a motivatio, let us look at oe Matlab example. Let us geerate a radom sample of size 00 from beta distributio Beta(5, 2). We will lear the defiitio

More information

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is 0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

More information

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

More information

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas:

Chapter 7 - Sampling Distributions. 1 Introduction. What is statistics? It consist of three major areas: Chapter 7 - Samplig Distributios 1 Itroductio What is statistics? It cosist of three major areas: Data Collectio: samplig plas ad experimetal desigs Descriptive Statistics: umerical ad graphical summaries

More information

5 Boolean Decision Trees (February 11)

5 Boolean Decision Trees (February 11) 5 Boolea Decisio Trees (February 11) 5.1 Graph Coectivity Suppose we are give a udirected graph G, represeted as a boolea adjacecy matrix = (a ij ), where a ij = 1 if ad oly if vertices i ad j are coected

More information

The Stable Marriage Problem

The Stable Marriage Problem The Stable Marriage Problem William Hut Lae Departmet of Computer Sciece ad Electrical Egieerig, West Virgiia Uiversity, Morgatow, WV William.Hut@mail.wvu.edu 1 Itroductio Imagie you are a matchmaker,

More information

Part - I. Mathematics

Part - I. Mathematics Part - I Mathematics CHAPTER Set Theory. Objectives. Itroductio. Set Cocept.. Sets ad Elemets. Subset.. Proper ad Improper Subsets.. Equality of Sets.. Trasitivity of Set Iclusio.4 Uiversal Set.5 Complemet

More information

Lipschitz maps and nets in Euclidean space

Lipschitz maps and nets in Euclidean space Lipschitz maps ad ets i Euclidea space Curtis T. McMulle 1 April, 1997 1 Itroductio I this paper we discuss the followig three questios. 1. Give a real-valued fuctio f L (R ) with if f(x) > 0, is there

More information

THE HEIGHT OF q-binary SEARCH TREES

THE HEIGHT OF q-binary SEARCH TREES THE HEIGHT OF q-binary SEARCH TREES MICHAEL DRMOTA AND HELMUT PRODINGER Abstract. q biary search trees are obtaied from words, equipped with the geometric distributio istead of permutatios. The average

More information

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley

Cooley-Tukey. Tukey FFT Algorithms. FFT Algorithms. Cooley Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Cosider a legth- sequece x[ with a -poit DFT X[ where Represet the idices ad as +, +, Cooley Cooley-Tuey Tuey FFT Algorithms FFT Algorithms Usig these

More information

INFINITE SERIES KEITH CONRAD

INFINITE SERIES KEITH CONRAD INFINITE SERIES KEITH CONRAD. Itroductio The two basic cocepts of calculus, differetiatio ad itegratio, are defied i terms of limits (Newto quotiets ad Riema sums). I additio to these is a third fudametal

More information

ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS

ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS ON THE EDGE-BANDWIDTH OF GRAPH PRODUCTS JÓZSEF BALOGH, DHRUV MUBAYI, AND ANDRÁS PLUHÁR Abstract The edge-badwidth of a graph G is the badwidth of the lie graph of G We show asymptotically tight bouds o

More information

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract

The Gompertz Makeham coupling as a Dynamic Life Table. Abraham Zaks. Technion I.I.T. Haifa ISRAEL. Abstract The Gompertz Makeham couplig as a Dyamic Life Table By Abraham Zaks Techio I.I.T. Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 32000, Haifa, Israel Abstract A very famous

More information

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

More information

Quaderni di Dipartimento. Rate of Convergence of Predictive Distributions for Dependent Data. Patrizia Berti (Università di Modena e Reggio Emilia)

Quaderni di Dipartimento. Rate of Convergence of Predictive Distributions for Dependent Data. Patrizia Berti (Università di Modena e Reggio Emilia) Quaderi di Dipartimeto Rate of Covergece of Predictive Distributios for Depedet Data Patrizia Berti Uiversità di Modea e Reggio Emilia Iree Crimaldi Uiversità di Bologa Luca Pratelli Accademia Navale di

More information

CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homework 4 Solutions CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

More information

Sampling Distribution And Central Limit Theorem

Sampling Distribution And Central Limit Theorem () Samplig Distributio & Cetral Limit Samplig Distributio Ad Cetral Limit Samplig distributio of the sample mea If we sample a umber of samples (say k samples where k is very large umber) each of size,

More information

Estimating Probability Distributions by Observing Betting Practices

Estimating Probability Distributions by Observing Betting Practices 5th Iteratioal Symposium o Imprecise Probability: Theories ad Applicatios, Prague, Czech Republic, 007 Estimatig Probability Distributios by Observig Bettig Practices Dr C Lych Natioal Uiversity of Irelad,

More information

Incremental calculation of weighted mean and variance

Incremental calculation of weighted mean and variance Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

More information

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations

CS103A Handout 23 Winter 2002 February 22, 2002 Solving Recurrence Relations CS3A Hadout 3 Witer 00 February, 00 Solvig Recurrece Relatios Itroductio A wide variety of recurrece problems occur i models. Some of these recurrece relatios ca be solved usig iteratio or some other ad

More information

Review: Classification Outline

Review: Classification Outline Data Miig CS 341, Sprig 2007 Decisio Trees Neural etworks Review: Lecture 6: Classificatio issues, regressio, bayesia classificatio Pretice Hall 2 Data Miig Core Techiques Classificatio Clusterig Associatio

More information

AP Calculus BC 2003 Scoring Guidelines Form B

AP Calculus BC 2003 Scoring Guidelines Form B AP Calculus BC Scorig Guidelies Form B The materials icluded i these files are iteded for use by AP teachers for course ad exam preparatio; permissio for ay other use must be sought from the Advaced Placemet

More information

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem

Tangent circles in the ratio 2 : 1. Hiroshi Okumura and Masayuki Watanabe. In this article we consider the following old Japanese geometry problem 116 Taget circles i the ratio 2 : 1 Hiroshi Okumura ad Masayuki Wataabe I this article we cosider the followig old Japaese geometry problem (see Figure 1), whose statemet i [1, p. 39] is missig the coditio

More information

4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph.

4. Trees. 4.1 Basics. Definition: A graph having no cycles is said to be acyclic. A forest is an acyclic graph. 4. Trees Oe of the importat classes of graphs is the trees. The importace of trees is evidet from their applicatios i various areas, especially theoretical computer sciece ad molecular evolutio. 4.1 Basics

More information

Lecture 4: Cheeger s Inequality

Lecture 4: Cheeger s Inequality Spectral Graph Theory ad Applicatios WS 0/0 Lecture 4: Cheeger s Iequality Lecturer: Thomas Sauerwald & He Su Statemet of Cheeger s Iequality I this lecture we assume for simplicity that G is a d-regular

More information

Overview of some probability distributions.

Overview of some probability distributions. Lecture Overview of some probability distributios. I this lecture we will review several commo distributios that will be used ofte throughtout the class. Each distributio is usually described by its probability

More information

LECTURE 13: Cross-validation

LECTURE 13: Cross-validation LECTURE 3: Cross-validatio Resampli methods Cross Validatio Bootstrap Bias ad variace estimatio with the Bootstrap Three-way data partitioi Itroductio to Patter Aalysis Ricardo Gutierrez-Osua Texas A&M

More information

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n

THE REGRESSION MODEL IN MATRIX FORM. For simple linear regression, meaning one predictor, the model is. for i = 1, 2, 3,, n We will cosider the liear regressio model i matrix form. For simple liear regressio, meaig oe predictor, the model is i = + x i + ε i for i =,,,, This model icludes the assumptio that the ε i s are a sample

More information

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients

BINOMIAL EXPANSIONS 12.5. In this section. Some Examples. Obtaining the Coefficients 652 (12-26) Chapter 12 Sequeces ad Series 12.5 BINOMIAL EXPANSIONS I this sectio Some Examples Otaiig the Coefficiets The Biomial Theorem I Chapter 5 you leared how to square a iomial. I this sectio you

More information

Entropy of bi-capacities

Entropy of bi-capacities Etropy of bi-capacities Iva Kojadiovic LINA CNRS FRE 2729 Site école polytechique de l uiv. de Nates Rue Christia Pauc 44306 Nates, Frace iva.kojadiovic@uiv-ates.fr Jea-Luc Marichal Applied Mathematics

More information

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS

Unit 8: Inference for Proportions. Chapters 8 & 9 in IPS Uit 8: Iferece for Proortios Chaters 8 & 9 i IPS Lecture Outlie Iferece for a Proortio (oe samle) Iferece for Two Proortios (two samles) Cotigecy Tables ad the χ test Iferece for Proortios IPS, Chater

More information

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL.

Annuities Under Random Rates of Interest II By Abraham Zaks. Technion I.I.T. Haifa ISRAEL and Haifa University Haifa ISRAEL. Auities Uder Radom Rates of Iterest II By Abraham Zas Techio I.I.T. Haifa ISRAEL ad Haifa Uiversity Haifa ISRAEL Departmet of Mathematics, Techio - Israel Istitute of Techology, 3000, Haifa, Israel I memory

More information

Permutations, the Parity Theorem, and Determinants

Permutations, the Parity Theorem, and Determinants 1 Permutatios, the Parity Theorem, ad Determiats Joh A. Guber Departmet of Electrical ad Computer Egieerig Uiversity of Wiscosi Madiso Cotets 1 What is a Permutatio 1 2 Cycles 2 2.1 Traspositios 4 3 Orbits

More information

Descriptive Statistics

Descriptive Statistics Descriptive Statistics We leared to describe data sets graphically. We ca also describe a data set umerically. Measures of Locatio Defiitio The sample mea is the arithmetic average of values. We deote

More information

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal

Study on the application of the software phase-locked loop in tracking and filtering of pulse signal Advaced Sciece ad Techology Letters, pp.31-35 http://dx.doi.org/10.14257/astl.2014.78.06 Study o the applicatio of the software phase-locked loop i trackig ad filterig of pulse sigal Sog Wei Xia 1 (College

More information

Hypothesis testing. Null and alternative hypotheses

Hypothesis testing. Null and alternative hypotheses Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

More information

Elementary Theory of Russian Roulette

Elementary Theory of Russian Roulette Elemetary Theory of Russia Roulette -iterestig patters of fractios- Satoshi Hashiba Daisuke Miematsu Ryohei Miyadera Itroductio. Today we are goig to study mathematical theory of Russia roulette. If some

More information

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling

Taking DCOP to the Real World: Efficient Complete Solutions for Distributed Multi-Event Scheduling Taig DCOP to the Real World: Efficiet Complete Solutios for Distributed Multi-Evet Schedulig Rajiv T. Maheswara, Milid Tambe, Emma Bowrig, Joatha P. Pearce, ad Pradeep araatham Uiversity of Souther Califoria

More information

On Formula to Compute Primes. and the n th Prime

On Formula to Compute Primes. and the n th Prime Applied Mathematical cieces, Vol., 0, o., 35-35 O Formula to Compute Primes ad the th Prime Issam Kaddoura Lebaese Iteratioal Uiversity Faculty of Arts ad cieces, Lebao issam.kaddoura@liu.edu.lb amih Abdul-Nabi

More information