Appendix B. Criterion of Riemann-Stieltjes Integrability
|
|
- Esther Ross
- 5 years ago
- Views:
Transcription
1 Appendx B. Crteron of Remann-Steltes Integrablty Ths note s complementary to [R, Ch. 6] and [T, Sec. 3.5]. The man result of ths note s Theorem B.3, whch provdes the necessary and suffcent condtons for Remann-Steltes ntegrablty of f wth respect to α n terms of sets of pont of dscontnuty of these functons. In an equvalent form, ths result s contaned n [H, Theorem C]. Here we gve a more drect proof, whch does not use explctly the Lebesgue measure. Let α = α(x) be a monotoncally non-decreasng functon on a fnte nterval [a, b], and let f = f(x) be a bounded real functon on [a, b]. For an arbtrary partton := {a = x 0 x 1... x n 1 x n = b} of [a, b], we defne the upper and lower sums as follows: (1) U(, f, α) := M α, L(, f, α) := where m α, (2) M := sup f m := nf f, α := α(x ) α(x 1 ) for = 1, 2,..., n. [x 1,x ] [x 1,x ] For any two parttons 1 and 2, ther common refnement := 1 2 satsfes (see [R, Theorem 6.4]) (3) U( 1, f, α) U(, f, α) L(, f, α) L( 2, f, α). Therefore, we always have (4) nf U(, f, α) sup L(, f, α). Defnton B.1. The functon f s Remann-Steltes ntegrable wth respect to α on [a, b] f both sdes of (4) are equal. In ths case, we wrte f R(α) on [a, b] and defne the Remann- Steltes ntegral (5) b a f dα := nf U(, f, α) = sup L(, f, α). Theorem B.2 ([R], Theorem 6.6). f R(α) on [a, b] f and only f for every ε > 0 there exsts a partton such that (6) U(, f, α) L(, f, α) < ε. roof. For every ε > 0 there are parttons 1 and 2 such that U( 1, f, α) < nf U(, f, α) + ε 2, L( 2, f, α) > sup U(, f, α) ε 2. If f R(α) on [a, b], then we have equalty n (4), whch mples 0 U( 1, f, α) L( 2, f, α) < ε 2 + ε 2 = ε, and (6) follows from (3) wth = := 1 2. On the other hand, f we have (6), then the dfference between nf and sup n (4) s less than ε. Snce ε > 0 s arbtrary, we must have the equalty,.e. f R(α) on [a, b]. B 1
2 Further, snce α(x) s non-decreasng on [a, b], there are one-sded lmts α(p ) := lm α(y), a < p b; α(p+) := lm α(y), a p < b, y p y p+ and α(p ) α(p) α(p+). Theorem B.3. Let f be a bounded real functon on [a, b]. Then f R(α) on [a, b] f and only f f and α satsfy both propertes (I) and (II) below. (I) () If α(p ) < α(p), a < p b, then f(p ) = f(p). () If α(p) < α(p+), a p < b, then f(p+) = f(p). (II) Let S f and S α denote the sets of ponts of dscontnuty of f and α correspondngly. Then for every ε > 0 there exsts a (fnte or countable) sequence of ntervals (a, b ), 1, such that (7) S := (S f \ S α ) ( (a, b ), and α(b ) α(a ) ) < ε. Here the ntervals (a, b ) are not necessarly contaned n [a, b]. We extend f f(a), α α(a) on (, a) and f f(b), α α(b) on (b, + ), so that the last expresson, and also the expresson n (9) below, are well defned n any case. Remark B.4. The property (I) smply says that f f R(α) on [a, b], then f and α cannot be both left-dscontnuous, or both rght-dscontnuous at same pont. Of course, ths property s redundant f α s contnuous on [a, b]. By change of varable ([R, Theorem 6.19]), ths case can be reduced to α(x) x. In ths partcular case, our theorem s contaned n [T, Theorem 3.5.6]. Defnton B.5. The llaton of f on a set A, (8) A f := sup A f nf f = sup f(x) f(y). A x,y A If f s defned on [a, b], then the llaton of f at a pont p [a, b], (9) ω f (p) := lm h 0+ [p h,p+h] Lemma B.6. () f s contnuous at p f and only f ω f (p) = 0; () f(p ) = f(p) f and only f f 0 as h 0+; () f(p+) = f(p) f and only f [p,p+h] f 0 as h 0+. We skp the proof, because t s very elementary (see [T, Theorem 3.5.2]). Lemma B.7. If f R(α) on [a, b], then f and α satsfy the propertes (I) n Theorem B.3. roof. Let p be a pont such that α(p ) < α(p), a < p b. By Theorem B.2, for every ε > 0 there s a partton := {a = x 0 x 1... x n = b} (dependng on α) such that (10) U(, f, α) L(, f, α) = (M m ) α < ε. Next, for small h (0, p a), the nterval (p h, p) does not contan pont x. From (3) (wth 1 = 2 = ) t follows that the refned partton := {p h, p} satsfes U(, f, α) L(, f, α) U(, f, α) L(, f, α) < ε B 2 f.
3 Therefore, replacng by f necessary, we can assume that p h, p,.e. Then from (10) t follows p h = x 0 1 < p = x 0 for some 0 {1, 2,..., n}. f α 0 = ( ) M 0 m 0 α0 < ε. Snce α 0 = α(p) α(p h) α(p) α(p ) > 0, and ε > 0 can be chosen arbtrarly small, we conclude that f 0 as h 0+. By Lemma B.6(), we have f(p ) = f(p). The proof of part () n (I) s complete. art () can be proved qute smlarly. Lemma B.8. If f R(α) on [a, b], then f and α satsfy the property (II) n Theorem B.3. roof. By Lemma B.6(), the set of ponts of dscontnuty of f, (11) S f = { p [a, b] : ω f (p) > 0 } = F k, where F k := { p [a, b] : ω f (p) 2 k}. Fx ε > 0. By Theorem B.2, for every k = 1, 2,..., there exsts a partton := {a = x 0 x 1... x n = b} (dependng on k) such that (12) U(, f, α) L(, f, α) = (M m ) α < ε k := 4 k ε. Note that f p F k \, then for some {1, 2,..., n} we have p (x 1, x ), and M m ω f (p) 2 k. Let A k denote the set of all such ndces. Then (13) (F k \ ) (x x 1 ), and α 2 k (M m ) α < 2 k ε. A k A k A k Further, F k \ S α s contaned n (F k \ ) ( \ S α ). Snce α(x) s contnuous at every pont p \ S α, one can cover such pont by ntervals (p h, p + h) wth arbtrarly small α(p + h) α(p h). Together wth (x 1, x ), A k, these ntervals compose a fnte famly of ntervals (a k,, b k, ) such that (F k \ S α ) ( (a k,, b k, ), and α(bk, ) α(a k, ) ) < 2 k ε. Fnally, by vrtue of (11), (S f \ S α ) = (F k \ S α ) (a k,, b k, ), and ( α(bk, ) α(a k, ) ) < 2 k ε = ε. Snce the countable set of ntervals {(a k,, b k, )} can be renumbered as {(a, b )}, we get the desred property (7). The followng lemma, together wth the prevous Lemmas B.7 and B.8, completes the proof of Theorem B.3. Lemma B.9. Let f be a bounded functon on [a, b] satsfyng the propertes (I) and (II) n Theorem B.3. Then f R(α) on [a, b]. B 3
4 roof. Step 1. We have f M = const < on [a, b]. By Theorem B.2, t suffces to show that for an arbtrary ε > 0, there exsts a partton := {a = x 0 < x 1 <... < x n 1 < x n = b} of [a, b] satsfyng the nequalty (6) for gven f and α. Ths nequalty can be wrtten n the form (14) U(, f, α) L(, f, α) = f α < ε, where I := [x 1, x ]. Step 2. Fx a constant ε 1 > 0. Note that snce α(x) s a monotone functon, ts set of ponts of dscontnuty S α s at most countable: S α := {c 1, c 2,...}. From the assumpton (I) n Theorem B.3 t follows that for each = 1, 2,..., one can choose a small constant h > 0 such that (15) I 1, f α < 2 ε 1, I 1, I + 1, f α < 2 ε 1, for = 1, 2,..., I + 1, where I 1, := [c h, c ], I + 1, := [c, c + h ]. Obvously, we also have (16) S α := {c 1, c 2,...} V 1 := 1 I 1,, where I 1, := (a 1,, b 1, ) := (c h, c + h ). Step 3. Based on the constant ε 1 > 0, defne the set (17) F := {p [a, b] : ω f (p) ε 1 > 0}. We clam (as n [T, Lemma 3.5.4]) that F s compact. Indeed, f p F and p p 0 [a, b] as, then for an arbtrary h > 0 there s such that p p 0 < h/2. For such, we have (p h/2, p + h/2) (p 0 h, p 0 + h), hence by (8) and (9), the llaton of f, and f f ω f(p ) ε 1, [p 0 h,p 0 +h] [p h/2, p +h/2] ω f (p 0 ) := lm h 0+ Ths argument proves the compactness of F. f ε 1 > 0,.e. p 0 F. [p 0 h,p 0 +h] Step 4. Further, note that F S f the set of ponts of dscontnuty of f. Therefore, by our assumpton (II), for the gven constant ε 1 > 0, there exsts a sequence of ntervals I 2, := (a 2,, b 2, ) such that (18) (F \ S α ) (S f \ S α ) V 2 := ( I 2,, and α(b2, ) α(a 2, ) ) < ε 1. Step 5. From (16) and (18) t follows F (V 1 V 2 ), so that the compact set F s covered by the unon of two famles of open ntervals {I 1, } and {I 2, }. Therefore, one can choose fnte subfamles {I 1,} {I 1, } and {I 2,} {I 2, } such that (19) F (V 1 V 2), where V 1 := I 1,, V 2 := I 2,. Consder another compact set F := [a, b] \ (V 1 V 2). Snce F does not ntersect F, we have ω f (p) < ε 1 for every p F. By defnton of ω f (p) n (9), (20) [p h,p+h] f < ε 1 for every p F wth some h = h(p) > 0. B 4
5 The famly of the correspondng open ntervals {(p h, p + h), p F } covers the compact F. Therefore, ths famly contans a fnte subfamly {I 3, := (a 3,, b 3, )} such that (21) F V 3 := I 3,, and [a 3,,b 3, ] f < ε 1 for each. Step 6. It s easy to see that (19) and (21) mply [a, b] (V 1 V 2 V 3), so that [a, b] s covered by the unon of three fnte famles of open ntervals {I 1,}, {I 2,}, and {I 3,}. Let := {a = x 0 < x 1 <... < x n 1 < x n = b} be a partton of [a, b], whch ncludes the pont a, b, all the endponts of ntervals I 1,, I 2,, I 3,, and also the centers c of the ntervals I 1, := (c h, c + h ), whch belong to (a, b). Denote I := [x 1, x ] for = 1, 2,..., n. Note that I are closed ntervals, whereas I 1,, I 2,, I 3, are open. However, all the estmates (15), (18), and (21), hold true for closed ntervals. Let A 1 denote the set of all ndces {1, 2,..., } such that I V 1, A 2 the set of all / A 1 such that I 2 V 2, and A 3 the set of all the remanng, for whch we automatcally have I V 3, because [a, b] (V 1 V 2 V 3). For each A 1, we have ether I I 1, or I I + 1, for some, hence by vrtue of (15), (22) f α < 2 A 1 2 ε 1 = 2ε 1. Smlarly, snce f M, we have f 2M, and the last nequalty n (18) mples (23) f α 2M α < 2M ε 1. I A 2 A 2 Fnally, from (21) and monotoncty of α t follows (24) f α ε 1 α ( α(b) α(a) ) ε 1. I A 3 A 3 Snce A 1 A 2 A 3 = {1, 2,..., n}, the estmates (22) (24) yeld f α ( 2 + 2M + α(b) α(a) ) ε 1 < ε, provded 0 < ε 1 < ( 2 + 2M + α(b) α(a) ) 1 ε. Thus we have the desred estmate (14) and lemma s proved. =1 References [H] H. J. Ter Horst, Remann-Steltes and Lebesgue-Steltes Integrablty, Amer. Math. Monthly, vol. 91, 1984, pp [R] W. Rudn, rncples of Mathematcal Analyss, 3rd edton. [T] W. F. Trench, Introducton to real analyss. B 5
More metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationREAL ANALYSIS I HOMEWORK 1
REAL ANALYSIS I HOMEWORK CİHAN BAHRAN The questons are from Tao s text. Exercse 0.0.. If (x α ) α A s a collecton of numbers x α [0, + ] such that x α
More informationTHE WEIGHTED WEAK TYPE INEQUALITY FOR THE STRONG MAXIMAL FUNCTION
THE WEIGHTED WEAK TYPE INEQUALITY FO THE STONG MAXIMAL FUNCTION THEMIS MITSIS Abstract. We prove the natural Fefferman-Sten weak type nequalty for the strong maxmal functon n the plane, under the assumpton
More informationMath 702 Midterm Exam Solutions
Math 702 Mdterm xam Solutons The terms measurable, measure, ntegrable, and almost everywhere (a.e.) n a ucldean space always refer to Lebesgue measure m. Problem. [6 pts] In each case, prove the statement
More informationLinear, affine, and convex sets and hulls In the sequel, unless otherwise specified, X will denote a real vector space.
Lnear, affne, and convex sets and hulls In the sequel, unless otherwse specfed, X wll denote a real vector space. Lnes and segments. Gven two ponts x, y X, we defne xy = {x + t(y x) : t R} = {(1 t)x +
More informationON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EQUATION
Advanced Mathematcal Models & Applcatons Vol.3, No.3, 2018, pp.215-222 ON A DETERMINATION OF THE INITIAL FUNCTIONS FROM THE OBSERVED VALUES OF THE BOUNDARY FUNCTIONS FOR THE SECOND-ORDER HYPERBOLIC EUATION
More informationFACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP
C O L L O Q U I U M M A T H E M A T I C U M VOL. 80 1999 NO. 1 FACTORIZATION IN KRULL MONOIDS WITH INFINITE CLASS GROUP BY FLORIAN K A I N R A T H (GRAZ) Abstract. Let H be a Krull monod wth nfnte class
More informationExercise Solutions to Real Analysis
xercse Solutons to Real Analyss Note: References refer to H. L. Royden, Real Analyss xersze 1. Gven any set A any ɛ > 0, there s an open set O such that A O m O m A + ɛ. Soluton 1. If m A =, then there
More informationAnother converse of Jensen s inequality
Another converse of Jensen s nequalty Slavko Smc Abstract. We gve the best possble global bounds for a form of dscrete Jensen s nequalty. By some examples ts frutfulness s shown. 1. Introducton Throughout
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationarxiv: v1 [math.co] 1 Mar 2014
Unon-ntersectng set systems Gyula O.H. Katona and Dánel T. Nagy March 4, 014 arxv:1403.0088v1 [math.co] 1 Mar 014 Abstract Three ntersecton theorems are proved. Frst, we determne the sze of the largest
More informationFirst day August 1, Problems and Solutions
FOURTH INTERNATIONAL COMPETITION FOR UNIVERSITY STUDENTS IN MATHEMATICS July 30 August 4, 997, Plovdv, BULGARIA Frst day August, 997 Problems and Solutons Problem. Let {ε n } n= be a sequence of postve
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationarxiv: v1 [math.ca] 31 Jul 2018
LOWE ASSOUAD TYPE DIMENSIONS OF UNIFOMLY PEFECT SETS IN DOUBLING METIC SPACE HAIPENG CHEN, MIN WU, AND YUANYANG CHANG arxv:80769v [mathca] 3 Jul 08 Abstract In ths paper, we are concerned wth the relatonshps
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More information10-801: Advanced Optimization and Randomized Methods Lecture 2: Convex functions (Jan 15, 2014)
0-80: Advanced Optmzaton and Randomzed Methods Lecture : Convex functons (Jan 5, 04) Lecturer: Suvrt Sra Addr: Carnege Mellon Unversty, Sprng 04 Scrbes: Avnava Dubey, Ahmed Hefny Dsclamer: These notes
More information20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The first idea is connectedness.
20. Mon, Oct. 13 What we have done so far corresponds roughly to Chapters 2 & 3 of Lee. Now we turn to Chapter 4. The frst dea s connectedness. Essentally, we want to say that a space cannot be decomposed
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
More informationSTEINHAUS PROPERTY IN BANACH LATTICES
DEPARTMENT OF MATHEMATICS TECHNICAL REPORT STEINHAUS PROPERTY IN BANACH LATTICES DAMIAN KUBIAK AND DAVID TIDWELL SPRING 2015 No. 2015-1 TENNESSEE TECHNOLOGICAL UNIVERSITY Cookevlle, TN 38505 STEINHAUS
More informationMATH 5707 HOMEWORK 4 SOLUTIONS 2. 2 i 2p i E(X i ) + E(Xi 2 ) ä i=1. i=1
MATH 5707 HOMEWORK 4 SOLUTIONS CİHAN BAHRAN 1. Let v 1,..., v n R m, all lengths v are not larger than 1. Let p 1,..., p n [0, 1] be arbtrary and set w = p 1 v 1 + + p n v n. Then there exst ε 1,..., ε
More informationMath 205A Homework #2 Edward Burkard. Assume each composition with a projection is continuous. Let U Y Y be an open set.
Math 205A Homework #2 Edward Burkard Problem - Determne whether the topology T = fx;?; fcg ; fa; bg ; fa; b; cg ; fa; b; c; dgg s Hausdor. Choose the two ponts a; b 2 X. Snce there s no two dsjont open
More informationTHE FUNDAMENTAL THEOREM OF CALCULUS FOR MULTIDIMENSIONAL BANACH SPACE-VALUED HENSTOCK VECTOR INTEGRALS
Real Analyss Exchange Vol.,, pp. 469 480 Márca Federson, Insttuto de Matemátca e Estatístca, Unversdade de São Paulo, R. do Matão 1010, SP, Brazl, 05315-970. e-mal: marca@me.usp.br THE FUNDAMENTAL THEOREM
More informationFoundations of Arithmetic
Foundatons of Arthmetc Notaton We shall denote the sum and product of numbers n the usual notaton as a 2 + a 2 + a 3 + + a = a, a 1 a 2 a 3 a = a The notaton a b means a dvdes b,.e. ac = b where c s an
More informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationSome basic inequalities. Definition. Let V be a vector space over the complex numbers. An inner product is given by a function, V V C
Some basc nequaltes Defnton. Let V be a vector space over the complex numbers. An nner product s gven by a functon, V V C (x, y) x, y satsfyng the followng propertes (for all x V, y V and c C) (1) x +
More informationTHERE ARE INFINITELY MANY FIBONACCI COMPOSITES WITH PRIME SUBSCRIPTS
Research and Communcatons n Mathematcs and Mathematcal Scences Vol 10, Issue 2, 2018, Pages 123-140 ISSN 2319-6939 Publshed Onlne on November 19, 2018 2018 Jyot Academc Press http://jyotacademcpressorg
More informationGeneral viscosity iterative method for a sequence of quasi-nonexpansive mappings
Avalable onlne at www.tjnsa.com J. Nonlnear Sc. Appl. 9 (2016), 5672 5682 Research Artcle General vscosty teratve method for a sequence of quas-nonexpansve mappngs Cuje Zhang, Ynan Wang College of Scence,
More informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationBOUNDEDNESS OF THE RIESZ TRANSFORM WITH MATRIX A 2 WEIGHTS
BOUNDEDNESS OF THE IESZ TANSFOM WITH MATIX A WEIGHTS Introducton Let L = L ( n, be the functon space wth norm (ˆ f L = f(x C dx d < For a d d matrx valued functon W : wth W (x postve sem-defnte for all
More informationComplete subgraphs in multipartite graphs
Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
More informationGenericity of Critical Types
Genercty of Crtcal Types Y-Chun Chen Alfredo D Tllo Eduardo Fangold Syang Xong September 2008 Abstract Ely and Pesk 2008 offers an nsghtful characterzaton of crtcal types: a type s crtcal f and only f
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationSupplement to Clustering with Statistical Error Control
Supplement to Clusterng wth Statstcal Error Control Mchael Vogt Unversty of Bonn Matthas Schmd Unversty of Bonn In ths supplement, we provde the proofs that are omtted n the paper. In partcular, we derve
More informationSL n (F ) Equals its Own Derived Group
Internatonal Journal of Algebra, Vol. 2, 2008, no. 12, 585-594 SL n (F ) Equals ts Own Derved Group Jorge Macel BMCC-The Cty Unversty of New York, CUNY 199 Chambers street, New York, NY 10007, USA macel@cms.nyu.edu
More informationCurvature and isoperimetric inequality
urvature and sopermetrc nequalty Julà ufí, Agustí Reventós, arlos J Rodríguez Abstract We prove an nequalty nvolvng the length of a plane curve and the ntegral of ts radus of curvature, that has as a consequence
More informationON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES. 1. Introduction
ON THE EXTENDED HAAGERUP TENSOR PRODUCT IN OPERATOR SPACES TAKASHI ITOH AND MASARU NAGISA Abstract We descrbe the Haagerup tensor product l h l and the extended Haagerup tensor product l eh l n terms of
More informationDirichlet s Theorem In Arithmetic Progressions
Drchlet s Theorem In Arthmetc Progressons Parsa Kavkan Hang Wang The Unversty of Adelade February 26, 205 Abstract The am of ths paper s to ntroduce and prove Drchlet s theorem n arthmetc progressons,
More informationFINITELY-GENERATED MODULES OVER A PRINCIPAL IDEAL DOMAIN
FINITELY-GENERTED MODULES OVER PRINCIPL IDEL DOMIN EMMNUEL KOWLSKI Throughout ths note, s a prncpal deal doman. We recall the classfcaton theorem: Theorem 1. Let M be a fntely-generated -module. (1) There
More informationPROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 7, July 1997, Pages 2119{2125 S (97) THE STRONG OPEN SET CONDITION
PROCDINGS OF TH AMRICAN MATHMATICAL SOCITY Volume 125, Number 7, July 1997, Pages 2119{2125 S 0002-9939(97)03816-1 TH STRONG OPN ST CONDITION IN TH RANDOM CAS NORBRT PATZSCHK (Communcated by Palle. T.
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationMAT 578 Functional Analysis
MAT 578 Functonal Analyss John Qugg Fall 2008 Locally convex spaces revsed September 6, 2008 Ths secton establshes the fundamental propertes of locally convex spaces. Acknowledgment: although I wrote these
More informationREGULAR POSITIVE TERNARY QUADRATIC FORMS. 1. Introduction
REGULAR POSITIVE TERNARY QUADRATIC FORMS BYEONG-KWEON OH Abstract. A postve defnte quadratc form f s sad to be regular f t globally represents all ntegers that are represented by the genus of f. In 997
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationLecture 17 : Stochastic Processes II
: Stochastc Processes II 1 Contnuous-tme stochastc process So far we have studed dscrete-tme stochastc processes. We studed the concept of Makov chans and martngales, tme seres analyss, and regresson analyss
More informationSociété de Calcul Mathématique SA
Socété de Calcul Mathématque SA Outls d'ade à la décson Tools for decson help Probablstc Studes: Normalzng the Hstograms Bernard Beauzamy December, 202 I. General constructon of the hstogram Any probablstc
More informationResearch Article A Generalized Sum-Difference Inequality and Applications to Partial Difference Equations
Hndaw Publshng Corporaton Advances n Dfference Equatons Volume 008, Artcle ID 695495, pages do:0.55/008/695495 Research Artcle A Generalzed Sum-Dfference Inequalty and Applcatons to Partal Dfference Equatons
More information3 Basic boundary value problems for analytic function in the upper half plane
3 Basc boundary value problems for analytc functon n the upper half plane 3. Posson representaton formulas for the half plane Let f be an analytc functon of z throughout the half plane Imz > 0, contnuous
More informationIntegration and Expectation
Chapter 9 Integraton and Expectaton fter Lebesgue s nvestgatons, the analogy between the measure of a set and the probablty of an event, as well as between the ntegral of a functon and the mathematcal
More informationFUNCTIONAL ANALYSIS DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR
FUNCTIONAL ANALYSIS DR. RITU AGARWAL MALAVIYA NATIONAL INSTITUTE OF TECHNOLOGY JAIPUR Contents 1. About 1 2. Syllabus 1 References 1 2.1. Orgn 2 3. Abstract space 2 4. Metrc Space 2 5. Sem-Metrc 6 6. Separable
More informationA note on almost sure behavior of randomly weighted sums of φ-mixing random variables with φ-mixing weights
ACTA ET COMMENTATIONES UNIVERSITATIS TARTUENSIS DE MATHEMATICA Volume 7, Number 2, December 203 Avalable onlne at http://acutm.math.ut.ee A note on almost sure behavor of randomly weghted sums of φ-mxng
More informationMATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS
MATH 241B FUNCTIONAL ANALYSIS - NOTES EXAMPLES OF C ALGEBRAS These are nformal notes whch cover some of the materal whch s not n the course book. The man purpose s to gve a number of nontrval examples
More informationThe internal structure of natural numbers and one method for the definition of large prime numbers
The nternal structure of natural numbers and one method for the defnton of large prme numbers Emmanul Manousos APM Insttute for the Advancement of Physcs and Mathematcs 3 Poulou str. 53 Athens Greece Abstract
More informationA SURVEY OF PROPERTIES OF FINITE HORIZON DIFFERENTIAL GAMES UNDER ISAACS CONDITION. Contents
A SURVEY OF PROPERTIES OF FINITE HORIZON DIFFERENTIAL GAMES UNDER ISAACS CONDITION BOTAO WU Abstract. In ths paper, we attempt to answer the followng questons about dfferental games: 1) when does a two-player,
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationAsymptotics of the Solution of a Boundary Value. Problem for One-Characteristic Differential. Equation Degenerating into a Parabolic Equation
Nonl. Analyss and Dfferental Equatons, ol., 4, no., 5 - HIKARI Ltd, www.m-har.com http://dx.do.org/.988/nade.4.456 Asymptotcs of the Soluton of a Boundary alue Problem for One-Characterstc Dfferental Equaton
More informationLINEAR INTEGRAL EQUATIONS OF VOLTERRA CONCERNING HENSTOCK INTEGRALS
Real Analyss Exchange Vol. (),, pp. 389 418 M. Federson and R. Bancon, Insttute of Mathematcs and Statstcs, Unversty of São Paulo, CP 66281, 05315-970. e-mal: federson@cmc.sc.usp.br and bancon@me.usp.br
More informationMaximizing the number of nonnegative subsets
Maxmzng the number of nonnegatve subsets Noga Alon Hao Huang December 1, 213 Abstract Gven a set of n real numbers, f the sum of elements of every subset of sze larger than k s negatve, what s the maxmum
More informationGoogle PageRank with Stochastic Matrix
Google PageRank wth Stochastc Matrx Md. Sharq, Puranjt Sanyal, Samk Mtra (M.Sc. Applcatons of Mathematcs) Dscrete Tme Markov Chan Let S be a countable set (usually S s a subset of Z or Z d or R or R d
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More informationMath 426: Probability MWF 1pm, Gasson 310 Homework 4 Selected Solutions
Exercses from Ross, 3, : Math 26: Probablty MWF pm, Gasson 30 Homework Selected Solutons 3, p. 05 Problems 76, 86 3, p. 06 Theoretcal exercses 3, 6, p. 63 Problems 5, 0, 20, p. 69 Theoretcal exercses 2,
More informationSELECTED SOLUTIONS, SECTION (Weak duality) Prove that the primal and dual values p and d defined by equations (4.3.2) and (4.3.3) satisfy p d.
SELECTED SOLUTIONS, SECTION 4.3 1. Weak dualty Prove that the prmal and dual values p and d defned by equatons 4.3. and 4.3.3 satsfy p d. We consder an optmzaton problem of the form The Lagrangan for ths
More informationCSCE 790S Background Results
CSCE 790S Background Results Stephen A. Fenner September 8, 011 Abstract These results are background to the course CSCE 790S/CSCE 790B, Quantum Computaton and Informaton (Sprng 007 and Fall 011). Each
More informationModelli Clamfim Equazioni differenziali 7 ottobre 2013
CLAMFIM Bologna Modell 1 @ Clamfm Equazon dfferenzal 7 ottobre 2013 professor Danele Rtell danele.rtell@unbo.t 1/18? Ordnary Dfferental Equatons A dfferental equaton s an equaton that defnes a relatonshp
More informationMATH 829: Introduction to Data Mining and Analysis The EM algorithm (part 2)
1/16 MATH 829: Introducton to Data Mnng and Analyss The EM algorthm (part 2) Domnque Gullot Departments of Mathematcal Scences Unversty of Delaware Aprl 20, 2016 Recall 2/16 We are gven ndependent observatons
More informationMULTIPLE LEBESGUE INTEGRATION ON TIME SCALES
MULTIPLE LEBESGUE INTEGRATION ON TIME SCALES MARTIN BOHNER AND GUSEIN SH. GUSEINO Receved 26 January 2006; Revsed 17 Aprl 2006; Accepted 18 Aprl 2006 We study the process of multple Lebesgue ntegraton
More information5 The Rational Canonical Form
5 The Ratonal Canoncal Form Here p s a monc rreducble factor of the mnmum polynomal m T and s not necessarly of degree one Let F p denote the feld constructed earler n the course, consstng of all matrces
More informationSUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) M(B) := # ( B Z N)
SUCCESSIVE MINIMA AND LATTICE POINTS (AFTER HENK, GILLET AND SOULÉ) S.BOUCKSOM Abstract. The goal of ths note s to present a remarably smple proof, due to Hen, of a result prevously obtaned by Gllet-Soulé,
More informationEngineering Risk Benefit Analysis
Engneerng Rsk Beneft Analyss.55, 2.943, 3.577, 6.938, 0.86, 3.62, 6.862, 22.82, ESD.72, ESD.72 RPRA 2. Elements of Probablty Theory George E. Apostolaks Massachusetts Insttute of Technology Sprng 2007
More informationTHE RING AND ALGEBRA OF INTUITIONISTIC SETS
Hacettepe Journal of Mathematcs and Statstcs Volume 401 2011, 21 26 THE RING AND ALGEBRA OF INTUITIONISTIC SETS Alattn Ural Receved 01:08 :2009 : Accepted 19 :03 :2010 Abstract The am of ths study s to
More informationExpected Value and Variance
MATH 38 Expected Value and Varance Dr. Neal, WKU We now shall dscuss how to fnd the average and standard devaton of a random varable X. Expected Value Defnton. The expected value (or average value, or
More informationOn some variants of Jensen s inequality
On some varants of Jensen s nequalty S S DRAGOMIR School of Communcatons & Informatcs, Vctora Unversty, Vc 800, Australa EMMA HUNT Department of Mathematcs, Unversty of Adelade, SA 5005, Adelade, Australa
More informationChapter Twelve. Integration. We now turn our attention to the idea of an integral in dimensions higher than one. Consider a real-valued function f : D
Chapter Twelve Integraton 12.1 Introducton We now turn our attenton to the dea of an ntegral n dmensons hgher than one. Consder a real-valued functon f : R, where the doman s a nce closed subset of Eucldean
More informationOn the set of natural numbers
On the set of natural numbers by Jalton C. Ferrera Copyrght 2001 Jalton da Costa Ferrera Introducton The natural numbers have been understood as fnte numbers, ths wor tres to show that the natural numbers
More informationThe Second Eigenvalue of Planar Graphs
Spectral Graph Theory Lecture 20 The Second Egenvalue of Planar Graphs Danel A. Spelman November 11, 2015 Dsclamer These notes are not necessarly an accurate representaton of what happened n class. The
More informationY. Guo. A. Liu, T. Liu, Q. Ma UDC
UDC 517. 9 OSCILLATION OF A CLASS OF NONLINEAR PARTIAL DIFFERENCE EQUATIONS WITH CONTINUOUS VARIABLES* ОСЦИЛЯЦIЯ КЛАСУ НЕЛIНIЙНИХ ЧАСТКОВО РIЗНИЦЕВИХ РIВНЯНЬ З НЕПЕРЕРВНИМИ ЗМIННИМИ Y. Guo Graduate School
More informationEXPANSIVE MAPPINGS. by W. R. Utz
Volume 3, 978 Pages 6 http://topology.auburn.edu/tp/ EXPANSIVE MAPPINGS by W. R. Utz Topology Proceedngs Web: http://topology.auburn.edu/tp/ Mal: Topology Proceedngs Department of Mathematcs & Statstcs
More informationEnergy of flows on Z 2 percolation clusters
Energy of flows on Z 2 percolaton clusters Chrstopher Hoffman 1,2 Abstract We show that f p > p c (Z 2 ), then the unque nfnte percolaton cluster supports a nonzero flow f wth fnte q energy for all q >
More informationLectures on Stochastic Stability. Sergey FOSS. Heriot-Watt University. Lecture 5. Monotonicity and Saturation Rule
Lectures on Stochastc Stablty Sergey FOSS Herot-Watt Unversty Lecture 5 Monotoncty and Saturaton Rule Introducton The paper of Loynes [8] was the frst to consder a system (sngle server queue, and, later,
More informationj) = 1 (note sigma notation) ii. Continuous random variable (e.g. Normal distribution) 1. density function: f ( x) 0 and f ( x) dx = 1
Random varables Measure of central tendences and varablty (means and varances) Jont densty functons and ndependence Measures of assocaton (covarance and correlaton) Interestng result Condtonal dstrbutons
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationRepresentation theory and quantum mechanics tutorial Representation theory and quantum conservation laws
Representaton theory and quantum mechancs tutoral Representaton theory and quantum conservaton laws Justn Campbell August 1, 2017 1 Generaltes on representaton theory 1.1 Let G GL m (R) be a real algebrac
More informationarxiv: v1 [math.ca] 9 Apr 2019
HAUSDORFF DIMENSION OF THE LARGE VALUES OF WEYL SUMS arxv:1904.04457v1 [math.ca] 9 Apr 2019 CHANGHAO CHEN AND IGOR E. SHPARLINSKI Abstract. The authors have recently obtaned a lower bound of the Hausdorffdmenson
More informationCase Study of Markov Chains Ray-Knight Compactification
Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and
More informationSemilattices of Rectangular Bands and Groups of Order Two.
1 Semlattces of Rectangular Bs Groups of Order Two R A R Monzo Abstract We prove that a semgroup S s a semlattce of rectangular bs groups of order two f only f t satsfes the dentty y y, y y, y S 1 Introducton
More informationSolutions to the 71st William Lowell Putnam Mathematical Competition Saturday, December 4, 2010
Solutons to the 7st Wllam Lowell Putnam Mathematcal Competton Saturday, December 4, 2 Kran Kedlaya and Lenny Ng A The largest such k s n+ 2 n 2. For n even, ths value s acheved by the partton {,n},{2,n
More informationAffine and Riemannian Connections
Affne and Remannan Connectons Semnar Remannan Geometry Summer Term 2015 Prof Dr Anna Wenhard and Dr Gye-Seon Lee Jakob Ullmann Notaton: X(M) space of smooth vector felds on M D(M) space of smooth functons
More informationSpectral Graph Theory and its Applications September 16, Lecture 5
Spectral Graph Theory and ts Applcatons September 16, 2004 Lecturer: Danel A. Spelman Lecture 5 5.1 Introducton In ths lecture, we wll prove the followng theorem: Theorem 5.1.1. Let G be a planar graph
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationTHE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS.
THE ORNSTEIN-WEISS LEMMA FOR DISCRETE AMENABLE GROUPS FABRICE KRIEGER Abstract In ths note we prove a convergence theorem for nvarant subaddtve functons defned on the fnte subsets of a dscrete amenable
More informationAnti-van der Waerden numbers of 3-term arithmetic progressions.
Ant-van der Waerden numbers of 3-term arthmetc progressons. Zhanar Berkkyzy, Alex Schulte, and Mchael Young Aprl 24, 2016 Abstract The ant-van der Waerden number, denoted by aw([n], k), s the smallest
More informationVector Norms. Chapter 7 Iterative Techniques in Matrix Algebra. Cauchy-Bunyakovsky-Schwarz Inequality for Sums. Distances. Convergence.
Vector Norms Chapter 7 Iteratve Technques n Matrx Algebra Per-Olof Persson persson@berkeley.edu Department of Mathematcs Unversty of Calforna, Berkeley Math 128B Numercal Analyss Defnton A vector norm
More informationConvergence of option rewards for multivariate price processes
Mathematcal Statstcs Stockholm Unversty Convergence of opton rewards for multvarate prce processes Robn Lundgren Dmtr Slvestrov Research Report 2009:10 ISSN 1650-0377 Postal address: Mathematcal Statstcs
More information12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA. 4. Tensor product
12 MATH 101A: ALGEBRA I, PART C: MULTILINEAR ALGEBRA Here s an outlne of what I dd: (1) categorcal defnton (2) constructon (3) lst of basc propertes (4) dstrbutve property (5) rght exactness (6) localzaton
More information