The Arithmetic-Geometric mean inequality in an external formula. Yuki Seo. October 23, 2012
|
|
- Emory Merritt
- 5 years ago
- Views:
Transcription
1 Sc. Math. Japocae Vol. 00, No , The Arthmetc-Geometrc mea equalty a exteral formula Yuk Seo October 23, 2012 Abstract. The classcal Jese equalty ad ts reverse are dscussed by meas of terally dvdg pots. J.I. Fuj poted out that the cocavty s also expressed by exterally dvdg pots. I ths paper, we shall dscuss a exteral verso of the arthmetc-geometrc mea equalty: For postve real umbers x, y 0 for = 1, 2,, ad r 0 1 x1 + x2 + + x y1 + y2 + + y r x1 + x x 1+r y 1 y 2 y r. 1 Itroducto. The Jese qualty for cocave fuctos s oe of the most mportat equaltes the fuctoal aalyss. Let f be a real valued fucto o a terval J. The classcal Jese equalty s expressed by terally dvdg pots: If f s cocave o J, the 1.1 α fx f α x for all x J ad all α 0 = 1, 2,, such that α = 1, see [3, Theorem 1.1]. Mod-Pečarć [4] showed the followg operator verso of 1.1: If A s a selfadjot operator o a Hlbert space H wth the spectrum J, the 1.2 fax, x f Ax, x for every ut vector x H. I [1], J.I. Fuj poted out that the cocavty s also expressed by exterally dvdg pots: f s cocave o J f ad oly f 1.3 f1 x ry 1 fx rfy for all x, y J ad r > 0 wth 1 x ry J. Thus, a exteral verso of the classcal Jese equalty s as follows: f s cocave o J f ad oly f f α x β j y j 1 α f α α x β j fy j for all x, y j J ad α, β j 0 = 1,, ad j = 1,, k such that α k β j = 1, ad α x k β jy j J, also see [5, p83]. I ths paper, by vrture of a exteral formula, we shall dscuss the arthmetc-geometrc mea equalty. Moreover, we show reverses of the Jese operator equalty by meas of exterally dvdg pots Mathematcs Subject Classfcato. 47A63, 47A64. Key words ad phrases. Cocave fucto, Jese equalty, Reverse equalty, Postve operator, Arthmetc-Geometrc mea equalty.
2 2 Y. SEO 2 Arthmetc-Geometrc mea equalty. The arthmetc-geometrc mea equalty says that for o-egatve real umbers x 1, x 2,, x 2.1 x 1 x 2 x x 1 + x x. By vrture of a exteral formula 1.3, the equalty 2.1 s regarded as oe by meas of terally dvdg pots. I [6], Specht estmated the upper boudary of the arthmetc mea by the geometrc oe for postve umbers: For x 1, x 2,, x [m, M] wth 0 < m M, 2.2 x 1 + x x ad the Specht rato Sh s defed by Sh x 1 x 2 x, 2.3 Sh = h 1h 1 h 1 e log h h 1, h > 0 ad S1 = 1. We call 2.2 the Specht theorem, see [3, Theorem 2.49]. We also have the weghted Specht theorem: For x 0 ad ω 0 for = 1, 2,, wth ω = ω 1 x 1 + ω 2 x ω x Shx ω1 1 xω2 2 xω. Frst of all, we cosder the arthmetc-geometrc mea equalty by vrture of a exteral formula. Theorem 1. For postve real umbers x, y for = 1, 2,, ad r x1 + x x r y1 + y y Proof. We may assume that 1 x1+x2+ +x fucto log t s cocave, t follows that x1 + x x 1 log 1 = 1 log 1 x x log 1 x x = log 1 x x r y y log 1 x x r y y 1+r x1 + x x y 1 y 2 y r. r y1+y2+ +y r y y r y y > 0. Sce the logrthm 1 log y y y y 1 log y y log y log y ad ths mples 1+r x1 + + x log log y 1 y r log 1 x x r y y. By takg the expoet of both sdes, we have the desred equalty 2.5.
3 The Arthmetc-Geometrc mea equalty a exteral formula 3 Remark 2. Theorem 1 mples the arthmetc-geometrc mea equalty 2.1. I fact, f we put r = 1 ad x = y for = 1,, 2.5 of Theorem 1, the we have 2.1. Smlarly we have a exteral verso of the weghted arthmetc-geometrc mea equalty. Theorem 3. For postve real umbers x, y for = 1, 2,, ad ω 1 0 for = 1,, such that ω = 1, ω x r 1+r ω y ω x y ω r. By vrture of a exteral formula 1.3, oe mght expect that x1 + x x r y1 + y y x 1 x 2 x 1+r y 1 y 2 y r. However, the equalty 2.7 does ot hold geeral. By the Specht theorem, we have the followg complemetary equalty to the arthmetc-geometrc mea equalty a exteral formula. Theorem 4. Let x ad y be postve real umbers such that x, y [m, M] wth 0 < m M for = 1, 2,,, ad r 0. If 1 x1+x2+ +x r y1+y2+ +y > 0, the Sh 2r 1 x 1 x 2 x 1+r y 1 y 2 y r 1 x1 + x x r y1 + y y Sh 1+r x 1 x 2 x 1+r y 1 y 2 y r, ad the Specht rato Sh s defed as 2.3. Proof. By the Specht theorem 2.2 ad Theorem 1, t follows that 1 x1 + x x r y1 + y y 1+r x1 + x x y 1 y 2 y r Sh 1+r x 1 x 2 x 1+r y 1 y 2 y r ad ths mples the secod part of Theorem 4.
4 4 Y. SEO Sce the weghted verso of the Specht theorem 2.4 holds, we have 1 log x 1 x 2 x 1 log x 1 + x x 1 = 1 log 1 x x r y y 1 1 log Sh log 1 x x 1 log y y = log Sh 1+r + log 1 x x log Sh 1+r + log 1 x x log Sh + log y 1 + log y ad ths mples the frst part of Theorem 4. r y y r y y 1 r y y y y log y y Remark 5. If r = 0 ad x = y for = 1,, Theorem 4, the we have the Specht theorem 2.2. Smlarly we have the followg exteral verso of Theorem 4. Theorem 6. For postve real umbers x, y [m, M] for = 1, 2,, wth 0 < m M ad ω 1 0 for = 1,, such that ω = 1, 1+r r Sh 2r 1 1 ω x r ω y x ω Sh 1+r x ω y ω 1+r y ω r, where ad the Specht rato Sh s defed as Jese operator equalty. By vrture of a exteral formula, the Jese operator equalty 1.2 fax, x f Ax, x s regarded as a equalty a teral formula. I [2], we showed the followg exteral verso of the Jese operator equalty. Theorem A. Let f be a real valued fucto o a terval J. The f s cocave o J f ad oly f f Ax, x By, y x 2 f A x fby, y x for all x, y H such that x 2 y 2 = 1 ad for all selfadjot operators A ad B wth the spectra J such that Ax, x By, y J.
5 The Arthmetc-Geometrc mea equalty a exteral formula 5 To show the fluctuato of cocavty a exteral formula, we eed the followg wellkow lemma whch s regarded as a reverse of the Jese operator equalty a teral formula, also see [3, Remark 2.7]. Lemma 7. Let A be a selfadjot operator o H wth mi A MI for some scalars m M. If f s cocave o [m, M], the 3.1 x 2 f A x x fax, x + µm, M, f x 2 for all ozero vectors x H where the boud µm, M, f of cocavty s defed by { } fm fm µm, M, f = max ft t m + fm : t [m, M]. Proof. For readers covece, we gve a proof of Lemma 7. Put y = x/ x. Sce f s cocave, t follows that fm fm M m A + Mfm mfm M m I fa. Therefore we have fm fm Mfm mfm f Ay, y Ay, y + + µm, M, f fm fm Mfm mfm = A + y, y + µm, M, f fay, y + µm, M, f. If we replace y by x/ x, the we have the desred equalty 3.1. Though the equalty f Ax, x By, y fax, x fby, y does ot hold geeral for x 2 y 2 = 1, we show the fluctuato of cocavty a exteral formula by usg the boud µm, M, f of cocavty. Theorem 8. If f s cocave o [m, M], the µm, M, f x 2 + y 2 f Ax, x By, y fax, x fby, y µm, M, f x 2 for all x, y H such that x 2 y 2 = 1 ad for all selfadjot operators A ad B wth the spectra J such that Ax, x By, y J. Proof. By Theorem A ad Lemma 7 t follows that f Ax, x By, y x 2 f A x fby, y x fax, x + µm, M, f x 2 fby, y ad ths mples the secod part of Theorem 8. For the frst part of Theorem 8, t follows from Lemma 7 that fax, x x 2 f A x x x 2 1 y 2 f Ax, x By, y + 1+ y 2 1+ y 2 B y y, x 2 1 y 2 f Ax, x By, y + 1+ y 2 1+ y 2 f B y y, = f Ax, x By, y + y 2 f B y y, y y y + µm, M, f x 2 y y + µm, M, f y f Ax, x By, y + fby, y + µm, M, f y 2 +µm, M, f x 2
6 6 Y. SEO as desred. Remark 9. If y = 0 Theorem 8, the we have Lemma 7. exteso of 3.1. Hece Theorem 8 s a As a applcato of Theorem 8, we show the fluctuato of the logarthm fucto a exteral formula by meas of the Specht rato. Corollary 10. If x, y j [m, M] wth 0 < m M ad a, b j 0 for = 1,, ad j = 1,, k such that a k b j = 1 ad a x k b jy j > 0, the log Sh a + b j log a x b j y j a log x b j log y j log Sh a ad the Specht rato Sh s defed by Arthmetc-Geometrc mea operator equalty. Let A ad B be postve operators o H. For each α [0, 1], the weghted arthmetc meaa A α B s defed as A α B = 1 αa + αb ad the weghted geometrc mea A α B s defed as A α B = A 1 2 A 1 2 BA 1 2 α A 1 2. The the followg arthmetc-geometrc mea operator equalty holds: 4.1 A α B A α B for all postve operators A ad B, ad α [0, 1]. The equalty 4.1 s regarded as oe a teral formula. I the followg theorem we propose the arthmetc-geometrc mea operator equalty the exteral formula. Theorem 11. Let A, B, C ad D be postve operators o a Hlbert space H such that C, D are vertble. The for each α [0, 1] 2A α B C α D A α BC α D 1 A α B. Proof. Sce the arthmetc-geometrc mea operator equalty C α D C α D holds for each α [0, 1], t follows that A α BC α D 1 A α B 2A α B + C α D = C α D 1 2 A α B C α D 1 2 C α D 1 2 A α B C α D as desred. C α D + C α D Remark 12. Theorem 11 s a exteso of the arthmetc-geometrc mea operator equalty. I fact, f we put C = A ad D = B Theorem 11, the we have A α B A α B.
7 The Arthmetc-Geometrc mea equalty a exteral formula 7 The equalty 2A α B C α D A α BC α D 1 A α B does ot hold geeral. However, we have the followg theorem by vrture of the Specht theorem. Theorem 13. Let A, B, Cad D be postve vertble operators o H such that mi A, B MI for some scalars 0 < m M. The for each α [0, 1] 2A α B C α D Sh 2 A α BC α D 1 A α B, ad the Specht rato Sh s defed as 2.3. Proof. Sce t follows from [7] that the Specht theorem A α B Sh A α B holds for each α [0, 1], we have Sh 2 A α BC α D 1 A α B 2A α B + C α D = ShC α D 1 2 A α B C α D 1 2 ShC α D 1 2 A α B C α D as desred. + 2ShA α B 2A α B C α D + C α D Refereces [1] J.I. Fuj, A exteral verso of the Jese operator equalty, Sc. Math. Japo., Ole, e-2011, [2] J.I. Fuj, J. Pečarć ad Y. Seo, The Jese equalty a exteral formula, to appear J. Math. Iequal. [3] T. Furuta, J. Mćć Hot, J. Pečarć ad Y. Seo, Mod-Pečarć Method Operator Iequaltes, Moographs Iequaltes 1, Elemet, Zagreb, [4] B. Mod ad J. Pečarć, Covex equaltes Hlbert space, Housto J. Math., , [5] J. Pečarć, F. Proscha ad Y.L. Tog, Covex fuctos, Partal Ordergs, ad Statstcal Applcatos, Academc Press, Ic [6] W. Specht, Zur Theore der elemetare Mttel, Math. Z , [7] M. Tomaga, Specht s rato the Youg equalty, Sc. Math. Japo., , Departmet of Mathematcs Educato, Osaka Kyoku Uversty, Asahgaoka Kashwara Osaka Japa. E-mal address : yuks@cc.osaka-kyoku.ac.jp
Strong Convergence of Weighted Averaged Approximants of Asymptotically Nonexpansive Mappings in Banach Spaces without Uniform Convexity
BULLETIN of the MALAYSIAN MATHEMATICAL SCIENCES SOCIETY Bull. Malays. Math. Sc. Soc. () 7 (004), 5 35 Strog Covergece of Weghted Averaged Appromats of Asymptotcally Noepasve Mappgs Baach Spaces wthout
More informationComplete Convergence and Some Maximal Inequalities for Weighted Sums of Random Variables
Joural of Sceces, Islamc Republc of Ira 8(4): -6 (007) Uversty of Tehra, ISSN 06-04 http://sceces.ut.ac.r Complete Covergece ad Some Maxmal Iequaltes for Weghted Sums of Radom Varables M. Am,,* H.R. Nl
More informationONE GENERALIZED INEQUALITY FOR CONVEX FUNCTIONS ON THE TRIANGLE
Joural of Pure ad Appled Mathematcs: Advaces ad Applcatos Volume 4 Number 205 Pages 77-87 Avalable at http://scetfcadvaces.co. DOI: http://.do.org/0.8642/jpamaa_7002534 ONE GENERALIZED INEQUALITY FOR CONVEX
More informationarxiv:math/ v1 [math.gm] 8 Dec 2005
arxv:math/05272v [math.gm] 8 Dec 2005 A GENERALIZATION OF AN INEQUALITY FROM IMO 2005 NIKOLAI NIKOLOV The preset paper was spred by the thrd problem from the IMO 2005. A specal award was gve to Yure Boreko
More informationPROJECTION PROBLEM FOR REGULAR POLYGONS
Joural of Mathematcal Sceces: Advaces ad Applcatos Volume, Number, 008, Pages 95-50 PROJECTION PROBLEM FOR REGULAR POLYGONS College of Scece Bejg Forestry Uversty Bejg 0008 P. R. Cha e-mal: sl@bjfu.edu.c
More informationQ-analogue of a Linear Transformation Preserving Log-concavity
Iteratoal Joural of Algebra, Vol. 1, 2007, o. 2, 87-94 Q-aalogue of a Lear Trasformato Preservg Log-cocavty Daozhog Luo Departmet of Mathematcs, Huaqao Uversty Quazhou, Fua 362021, P. R. Cha ldzblue@163.com
More information#A27 INTEGERS 13 (2013) SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES
#A27 INTEGERS 3 (203) SOME WEIGHTED SUMS OF PRODUCTS OF LUCAS SEQUENCES Emrah Kılıç Mathematcs Departmet, TOBB Uversty of Ecoomcs ad Techology, Akara, Turkey eklc@etu.edu.tr Neşe Ömür Mathematcs Departmet,
More informationLecture 4 Sep 9, 2015
CS 388R: Radomzed Algorthms Fall 205 Prof. Erc Prce Lecture 4 Sep 9, 205 Scrbe: Xagru Huag & Chad Voegele Overvew I prevous lectures, we troduced some basc probablty, the Cheroff boud, the coupo collector
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationF. Inequalities. HKAL Pure Mathematics. 進佳數學團隊 Dr. Herbert Lam 林康榮博士. [Solution] Example Basic properties
進佳數學團隊 Dr. Herbert Lam 林康榮博士 HKAL Pure Mathematcs F. Ieualtes. Basc propertes Theorem Let a, b, c be real umbers. () If a b ad b c, the a c. () If a b ad c 0, the ac bc, but f a b ad c 0, the ac bc. Theorem
More informationSome identities involving the partial sum of q-binomial coefficients
Some dettes volvg the partal sum of -bomal coeffcets Bg He Departmet of Mathematcs, Shagha Key Laboratory of PMMP East Cha Normal Uversty 500 Dogchua Road, Shagha 20024, People s Republc of Cha yuhe00@foxmal.com
More informationD KL (P Q) := p i ln p i q i
Cheroff-Bouds 1 The Geeral Boud Let P 1,, m ) ad Q q 1,, q m ) be two dstrbutos o m elemets, e,, q 0, for 1,, m, ad m 1 m 1 q 1 The Kullback-Lebler dvergece or relatve etroy of P ad Q s defed as m D KL
More informationAssignment 5/MATH 247/Winter Due: Friday, February 19 in class (!) (answers will be posted right after class)
Assgmet 5/MATH 7/Wter 00 Due: Frday, February 9 class (!) (aswers wll be posted rght after class) As usual, there are peces of text, before the questos [], [], themselves. Recall: For the quadratc form
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More information. The set of these sums. be a partition of [ ab, ]. Consider the sum f( x) f( x 1)
Chapter 7 Fuctos o Bouded Varato. Subject: Real Aalyss Level: M.Sc. Source: Syed Gul Shah (Charma, Departmet o Mathematcs, US Sargodha Collected & Composed by: Atq ur Rehma (atq@mathcty.org, http://www.mathcty.org
More informationExtend the Borel-Cantelli Lemma to Sequences of. Non-Independent Random Variables
ppled Mathematcal Sceces, Vol 4, 00, o 3, 637-64 xted the Borel-Catell Lemma to Sequeces of No-Idepedet Radom Varables olah Der Departmet of Statstc, Scece ad Research Campus zad Uversty of Tehra-Ira der53@gmalcom
More informationAbout k-perfect numbers
DOI: 0.47/auom-04-0005 A. Şt. Uv. Ovdus Costaţa Vol.,04, 45 50 About k-perfect umbers Mhály Becze Abstract ABSTRACT. I ths paper we preset some results about k-perfect umbers, ad geeralze two equaltes
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationChapter 8: Statistical Analysis of Simulated Data
Marquette Uversty MSCS600 Chapter 8: Statstcal Aalyss of Smulated Data Dael B. Rowe, Ph.D. Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 08 by Marquette Uversty MSCS600 Ageda 8. The Sample
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More information9 U-STATISTICS. Eh =(m!) 1 Eh(X (1),..., X (m ) ) i.i.d
9 U-STATISTICS Suppose,,..., are P P..d. wth CDF F. Our goal s to estmate the expectato t (P)=Eh(,,..., m ). Note that ths expectato requres more tha oe cotrast to E, E, or Eh( ). Oe example s E or P((,
More informationBounds for the Connective Eccentric Index
It. J. Cotemp. Math. Sceces, Vol. 7, 0, o. 44, 6-66 Bouds for the Coectve Eccetrc Idex Nlaja De Departmet of Basc Scece, Humates ad Socal Scece (Mathematcs Calcutta Isttute of Egeerg ad Maagemet Kolkata,
More informationUNIT 2 SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS
Numercal Computg -I UNIT SOLUTION OF ALGEBRAIC AND TRANSCENDENTAL EQUATIONS Structure Page Nos..0 Itroducto 6. Objectves 7. Ital Approxmato to a Root 7. Bsecto Method 8.. Error Aalyss 9.4 Regula Fals Method
More informationResearch Article Multidimensional Hilbert-Type Inequalities with a Homogeneous Kernel
Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 29, Artcle ID 3958, 2 pages do:.55/29/3958 Research Artcle Multdmesoal Hlbert-Type Iequaltes wth a Homogeeous Kerel Predrag Vuovć Faculty
More informationJournal Of Inequalities And Applications, 2008, v. 2008, p
Ttle O verse Hlbert-tye equaltes Authors Chagja, Z; Cheug, WS Ctato Joural Of Iequaltes Ad Alcatos, 2008, v. 2008,. 693248 Issued Date 2008 URL htt://hdl.hadle.et/0722/56208 Rghts Ths work s lcesed uder
More informationLebesgue Measure of Generalized Cantor Set
Aals of Pure ad Appled Mathematcs Vol., No.,, -8 ISSN: -8X P), -888ole) Publshed o 8 May www.researchmathsc.org Aals of Lebesgue Measure of Geeralzed ator Set Md. Jahurul Islam ad Md. Shahdul Islam Departmet
More informationRademacher Complexity. Examples
Algorthmc Foudatos of Learg Lecture 3 Rademacher Complexty. Examples Lecturer: Patrck Rebesch Verso: October 16th 018 3.1 Itroducto I the last lecture we troduced the oto of Rademacher complexty ad showed
More informationResearch Article Gauss-Lobatto Formulae and Extremal Problems
Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 2008 Artcle ID 624989 0 pages do:055/2008/624989 Research Artcle Gauss-Lobatto Formulae ad Extremal Problems wth Polyomals Aa Mara Acu ad
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationInternational Journal of Mathematical Archive-5(8), 2014, Available online through ISSN
Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad
More informationChapter 14 Logistic Regression Models
Chapter 4 Logstc Regresso Models I the lear regresso model X β + ε, there are two types of varables explaatory varables X, X,, X k ad study varable y These varables ca be measured o a cotuous scale as
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationResearch Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings
Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationOn Eccentricity Sum Eigenvalue and Eccentricity Sum Energy of a Graph
Aals of Pure ad Appled Mathematcs Vol. 3, No., 7, -3 ISSN: 79-87X (P, 79-888(ole Publshed o 3 March 7 www.researchmathsc.org DOI: http://dx.do.org/.7/apam.3a Aals of O Eccetrcty Sum Egealue ad Eccetrcty
More informationOn the convergence of derivatives of Bernstein approximation
O the covergece of dervatves of Berste approxmato Mchael S. Floater Abstract: By dfferetatg a remader formula of Stacu, we derve both a error boud ad a asymptotc formula for the dervatves of Berste approxmato.
More informationHájek-Rényi Type Inequalities and Strong Law of Large Numbers for NOD Sequences
Appl Math If Sc 7, No 6, 59-53 03 59 Appled Matheatcs & Iforato Sceces A Iteratoal Joural http://dxdoorg/0785/as/070647 Háje-Réy Type Iequaltes ad Strog Law of Large Nuers for NOD Sequeces Ma Sogl Departet
More informationDimensionality Reduction and Learning
CMSC 35900 (Sprg 009) Large Scale Learg Lecture: 3 Dmesoalty Reducto ad Learg Istructors: Sham Kakade ad Greg Shakharovch L Supervsed Methods ad Dmesoalty Reducto The theme of these two lectures s that
More informationON THE LOGARITHMIC INTEGRAL
Hacettepe Joural of Mathematcs ad Statstcs Volume 39(3) (21), 393 41 ON THE LOGARITHMIC INTEGRAL Bra Fsher ad Bljaa Jolevska-Tueska Receved 29:9 :29 : Accepted 2 :3 :21 Abstract The logarthmc tegral l(x)
More informationANALYSIS ON THE NATURE OF THE BASIC EQUATIONS IN SYNERGETIC INTER-REPRESENTATION NETWORK
Far East Joural of Appled Mathematcs Volume, Number, 2008, Pages Ths paper s avalable ole at http://www.pphm.com 2008 Pushpa Publshg House ANALYSIS ON THE NATURE OF THE ASI EQUATIONS IN SYNERGETI INTER-REPRESENTATION
More informationLecture 02: Bounding tail distributions of a random variable
CSCI-B609: A Theorst s Toolkt, Fall 206 Aug 25 Lecture 02: Boudg tal dstrbutos of a radom varable Lecturer: Yua Zhou Scrbe: Yua Xe & Yua Zhou Let us cosder the ubased co flps aga. I.e. let the outcome
More informationhp calculators HP 30S Statistics Averages and Standard Deviations Average and Standard Deviation Practice Finding Averages and Standard Deviations
HP 30S Statstcs Averages ad Stadard Devatos Average ad Stadard Devato Practce Fdg Averages ad Stadard Devatos HP 30S Statstcs Averages ad Stadard Devatos Average ad stadard devato The HP 30S provdes several
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationMaximum Walk Entropy Implies Walk Regularity
Maxmum Walk Etropy Imples Walk Regularty Eresto Estraa, a José. e la Peña Departmet of Mathematcs a Statstcs, Uversty of Strathclye, Glasgow G XH, U.K., CIMT, Guaajuato, Mexco BSTRCT: The oto of walk etropy
More informationTHE PUBLISHING HOUSE PROCEEDINGS OF THE ROMANIAN ACADEMY, Series A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/2008, pp
THE PUBLISHIN HOUSE PROCEEDINS OF THE ROMANIAN ACADEMY, Seres A, OF THE ROMANIAN ACADEMY Volume 9, Number 3/8, THE UNITS IN Stela Corelu ANDRONESCU Uversty of Pteşt, Deartmet of Mathematcs, Târgu Vale
More informationSTK4011 and STK9011 Autumn 2016
STK4 ad STK9 Autum 6 Pot estmato Covers (most of the followg materal from chapter 7: Secto 7.: pages 3-3 Secto 7..: pages 3-33 Secto 7..: pages 35-3 Secto 7..3: pages 34-35 Secto 7.3.: pages 33-33 Secto
More informationA tighter lower bound on the circuit size of the hardest Boolean functions
Electroc Colloquum o Computatoal Complexty, Report No. 86 2011) A tghter lower boud o the crcut sze of the hardest Boolea fuctos Masak Yamamoto Abstract I [IPL2005], Fradse ad Mlterse mproved bouds o the
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More informationA Remark on the Uniform Convergence of Some Sequences of Functions
Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationChain Rules for Entropy
Cha Rules for Etroy The etroy of a collecto of radom varables s the sum of codtoal etroes. Theorem: Let be radom varables havg the mass robablty x x.x. The...... The roof s obtaed by reeatg the alcato
More informationn -dimensional vectors follow naturally from the one
B. Vectors ad sets B. Vectors Ecoomsts study ecoomc pheomea by buldg hghly stylzed models. Uderstadg ad makg use of almost all such models requres a hgh comfort level wth some key mathematcal sklls. I
More informationThe internal structure of natural numbers, one method for the definition of large prime numbers, and a factorization test
Fal verso The teral structure of atural umbers oe method for the defto of large prme umbers ad a factorzato test Emmaul Maousos APM Isttute for the Advacemet of Physcs ad Mathematcs 3 Poulou str. 53 Athes
More informationv 1 -periodic 2-exponents of SU(2 e ) and SU(2 e + 1)
Joural of Pure ad Appled Algebra 216 (2012) 1268 1272 Cotets lsts avalable at ScVerse SceceDrect Joural of Pure ad Appled Algebra joural homepage: www.elsever.com/locate/jpaa v 1 -perodc 2-expoets of SU(2
More informationINEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS
Joural of Mathematcal Scece: Advace ad Alcato Volume 24, 23, Page 29-46 INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS ZLATKO PAVIĆ Mechacal Egeerg Faculty Slavok Brod Uverty of Ojek
More informationOn the construction of symmetric nonnegative matrix with prescribed Ritz values
Joural of Lear ad Topologcal Algebra Vol. 3, No., 14, 61-66 O the costructo of symmetrc oegatve matrx wth prescrbed Rtz values A. M. Nazar a, E. Afshar b a Departmet of Mathematcs, Arak Uversty, P.O. Box
More informationGENERALIZATIONS OF CEVA S THEOREM AND APPLICATIONS
GENERLIZTIONS OF CEV S THEOREM ND PPLICTIONS Floret Smaradache Uversty of New Mexco 200 College Road Gallup, NM 87301, US E-mal: smarad@um.edu I these paragraphs oe presets three geeralzatos of the famous
More informationCOV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.
c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs,
More informationNumerical Analysis Formulae Booklet
Numercal Aalyss Formulae Booklet. Iteratve Scemes for Systems of Lear Algebrac Equatos:.... Taylor Seres... 3. Fte Dfferece Approxmatos... 3 4. Egevalues ad Egevectors of Matrces.... 3 5. Vector ad Matrx
More informationMAX-MIN AND MIN-MAX VALUES OF VARIOUS MEASURES OF FUZZY DIVERGENCE
merca Jr of Mathematcs ad Sceces Vol, No,(Jauary 0) Copyrght Md Reader Publcatos wwwjouralshubcom MX-MIN ND MIN-MX VLUES OF VRIOUS MESURES OF FUZZY DIVERGENCE RKTul Departmet of Mathematcs SSM College
More informationComplete Convergence for Weighted Sums of Arrays of Rowwise Asymptotically Almost Negative Associated Random Variables
A^VÇÚO 1 32 ò 1 5 Ï 2016 c 10 Chese Joural of Appled Probablty ad Statstcs Oct., 2016, Vol. 32, No. 5, pp. 489-498 do: 10.3969/j.ss.1001-4268.2016.05.005 Complete Covergece for Weghted Sums of Arrays of
More informationb. There appears to be a positive relationship between X and Y; that is, as X increases, so does Y.
.46. a. The frst varable (X) s the frst umber the par ad s plotted o the horzotal axs, whle the secod varable (Y) s the secod umber the par ad s plotted o the vertcal axs. The scatterplot s show the fgure
More informationJournal of Mathematical Analysis and Applications
J. Math. Aal. Appl. 365 200) 358 362 Cotets lsts avalable at SceceDrect Joural of Mathematcal Aalyss ad Applcatos www.elsever.com/locate/maa Asymptotc behavor of termedate pots the dfferetal mea value
More informationTHE PROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION
Joural of Scece ad Arts Year 12, No. 3(2), pp. 297-32, 212 ORIGINAL AER THE ROBABILISTIC STABILITY FOR THE GAMMA FUNCTIONAL EQUATION DOREL MIHET 1, CLAUDIA ZAHARIA 1 Mauscrpt receved: 3.6.212; Accepted
More informationMA 524 Homework 6 Solutions
MA 524 Homework 6 Solutos. Sce S(, s the umber of ways to partto [] to k oempty blocks, ad c(, s the umber of ways to partto to k oempty blocks ad also the arrage each block to a cycle, we must have S(,
More informationMultivariate Transformation of Variables and Maximum Likelihood Estimation
Marquette Uversty Multvarate Trasformato of Varables ad Maxmum Lkelhood Estmato Dael B. Rowe, Ph.D. Assocate Professor Departmet of Mathematcs, Statstcs, ad Computer Scece Copyrght 03 by Marquette Uversty
More information4 Inner Product Spaces
11.MH1 LINEAR ALGEBRA Summary Notes 4 Ier Product Spaces Ier product s the abstracto to geeral vector spaces of the famlar dea of the scalar product of two vectors or 3. I what follows, keep these key
More informationAN EULER-MC LAURIN FORMULA FOR INFINITE DIMENSIONAL SPACES
AN EULER-MC LAURIN FORMULA FOR INFINITE DIMENSIONAL SPACES Jose Javer Garca Moreta Graduate Studet of Physcs ( Sold State ) at UPV/EHU Address: P.O 6 890 Portugalete, Vzcaya (Spa) Phoe: (00) 3 685 77 16
More informationMultiple Linear Regression Analysis
LINEA EGESSION ANALYSIS MODULE III Lecture - 4 Multple Lear egresso Aalyss Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Cofdece terval estmato The cofdece tervals multple
More informationQualifying Exam Statistical Theory Problem Solutions August 2005
Qualfyg Exam Statstcal Theory Problem Solutos August 5. Let X, X,..., X be d uform U(,),
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More informationTESTS BASED ON MAXIMUM LIKELIHOOD
ESE 5 Toy E. Smth. The Basc Example. TESTS BASED ON MAXIMUM LIKELIHOOD To llustrate the propertes of maxmum lkelhood estmates ad tests, we cosder the smplest possble case of estmatg the mea of the ormal
More information02/15/04 INTERESTING FINITE AND INFINITE PRODUCTS FROM SIMPLE ALGEBRAIC IDENTITIES
0/5/04 ITERESTIG FIITE AD IFIITE PRODUCTS FROM SIMPLE ALGEBRAIC IDETITIES Thomas J Osler Mathematcs Departmet Rowa Uversty Glassboro J 0808 Osler@rowaedu Itroducto The dfferece of two squares, y = + y
More information5 Short Proofs of Simplified Stirling s Approximation
5 Short Proofs of Smplfed Strlg s Approxmato Ofr Gorodetsky, drtymaths.wordpress.com Jue, 20 0 Itroducto Strlg s approxmato s the followg (somewhat surprsg) approxmato of the factoral,, usg elemetary fuctos:
More informationAlgorithms Theory, Solution for Assignment 2
Juor-Prof. Dr. Robert Elsässer, Marco Muñz, Phllp Hedegger WS 2009/200 Algorthms Theory, Soluto for Assgmet 2 http://lak.formatk.u-freburg.de/lak_teachg/ws09_0/algo090.php Exercse 2. - Fast Fourer Trasform
More informationOn Submanifolds of an Almost r-paracontact Riemannian Manifold Endowed with a Quarter Symmetric Metric Connection
Theoretcal Mathematcs & Applcatos vol. 4 o. 4 04-7 ISS: 79-9687 prt 79-9709 ole Scepress Ltd 04 O Submafolds of a Almost r-paracotact emaa Mafold Edowed wth a Quarter Symmetrc Metrc Coecto Mob Ahmad Abdullah.
More informationGeneralization of the Dissimilarity Measure of Fuzzy Sets
Iteratoal Mathematcal Forum 2 2007 o. 68 3395-3400 Geeralzato of the Dssmlarty Measure of Fuzzy Sets Faramarz Faghh Boformatcs Laboratory Naobotechology Research Ceter vesa Research Isttute CECR Tehra
More informationmeans the first term, a2 means the term, etc. Infinite Sequences: follow the same pattern forever.
9.4 Sequeces ad Seres Pre Calculus 9.4 SEQUENCES AND SERIES Learg Targets:. Wrte the terms of a explctly defed sequece.. Wrte the terms of a recursvely defed sequece. 3. Determe whether a sequece s arthmetc,
More informationMarcinkiewicz strong laws for linear statistics of ρ -mixing sequences of random variables
Aas da Academa Braslera de Cêcas 2006 784: 65-62 Aals of the Brazla Academy of Sceces ISSN 000-3765 www.scelo.br/aabc Marckewcz strog laws for lear statstcs of ρ -mxg sequeces of radom varables GUANG-HUI
More informationX X X E[ ] E X E X. is the ()m n where the ( i,)th. j element is the mean of the ( i,)th., then
Secto 5 Vectors of Radom Varables Whe workg wth several radom varables,,..., to arrage them vector form x, t s ofte coveet We ca the make use of matrx algebra to help us orgaze ad mapulate large umbers
More informationρ < 1 be five real numbers. The
Lecture o BST 63: Statstcal Theory I Ku Zhag, /0/006 Revew for the prevous lecture Deftos: covarace, correlato Examples: How to calculate covarace ad correlato Theorems: propertes of correlato ad covarace
More informationIFYMB002 Mathematics Business Appendix C Formula Booklet
Iteratoal Foudato Year (IFY IFYMB00 Mathematcs Busess Apped C Formula Booklet Related Documet: IFY Mathematcs Busess Syllabus 07/8 IFYMB00 Maths Busess Apped C Formula Booklet Cotets lease ote that the
More informationThe Primitive Idempotents in
Iteratoal Joural of Algebra, Vol, 00, o 5, 3 - The Prmtve Idempotets FC - I Kulvr gh Departmet of Mathematcs, H College r Jwa Nagar (rsa)-5075, Ida kulvrsheora@yahoocom K Arora Departmet of Mathematcs,
More informationGeneralized Convex Functions on Fractal Sets and Two Related Inequalities
Geeralzed Covex Fuctos o Fractal Sets ad Two Related Iequaltes Huxa Mo, X Su ad Dogya Yu 3,,3School of Scece, Bejg Uversty of Posts ad Telecommucatos, Bejg,00876, Cha, Correspodece should be addressed
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationLower Bounds of the Kirchhoff and Degree Kirchhoff Indices
SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 7, (205), 25-3. Lower Bouds of the Krchhoff ad Degree Krchhoff Idces I. Ž. Mlovaovć, E. I. Mlovaovć,
More informationOPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2
OPERATOR POWER MEANS DUE TO LAWSON-LIM-PÁLFIA FOR 1 < t < 2 YUKI SEO Abstract. For 1 t 1, Lm-Pálfa defned a new famly of operator power means of postve defnte matrces and subsequently by Lawson-Lm ther
More informationExercises for Square-Congruence Modulo n ver 11
Exercses for Square-Cogruece Modulo ver Let ad ab,.. Mark True or False. a. 3S 30 b. 3S 90 c. 3S 3 d. 3S 4 e. 4S f. 5S g. 0S 55 h. 8S 57. 9S 58 j. S 76 k. 6S 304 l. 47S 5347. Fd the equvalece classes duced
More informationYuki Seo. Received May 23, 2010; revised August 15, 2010
Scietiae Mathematicae Japoicae Olie, e-00, 4 45 4 A GENERALIZED PÓLYA-SZEGÖ INEQUALITY FOR THE HADAMARD PRODUCT Yuki Seo Received May 3, 00; revised August 5, 00 Abstract. I this paper, we show a geeralized
More informationHypersurfaces with Constant Scalar Curvature in a Hyperbolic Space Form
Hypersurfaces wth Costat Scalar Curvature a Hyperbolc Space Form Lu Xm ad Su Wehog Abstract Let M be a complete hypersurface wth costat ormalzed scalar curvature R a hyperbolc space form H +1. We prove
More informationSTRONG CONSISTENCY FOR SIMPLE LINEAR EV MODEL WITH v/ -MIXING
Joural of tatstcs: Advaces Theory ad Alcatos Volume 5, Number, 6, Pages 3- Avalable at htt://scetfcadvaces.co. DOI: htt://d.do.org/.864/jsata_7678 TRONG CONITENCY FOR IMPLE LINEAR EV MODEL WITH v/ -MIXING
More information1 0, x? x x. 1 Root finding. 1.1 Introduction. Solve[x^2-1 0,x] {{x -1},{x 1}} Plot[x^2-1,{x,-2,2}] 3
Adrew Powuk - http://www.powuk.com- Math 49 (Numercal Aalyss) Root fdg. Itroducto f ( ),?,? Solve[^-,] {{-},{}} Plot[^-,{,-,}] Cubc equato https://e.wkpeda.org/wk/cubc_fucto Quartc equato https://e.wkpeda.org/wk/quartc_fucto
More informationIntroduction to Probability
Itroducto to Probablty Nader H Bshouty Departmet of Computer Scece Techo 32000 Israel e-mal: bshouty@cstechoacl 1 Combatorcs 11 Smple Rules I Combatorcs The rule of sum says that the umber of ways to choose
More informationON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES
M atheatcal I equaltes & A pplcatos Volue 19, Nuber 4 16, 195 137 do:1.7153/a-19-95 ON WEIGHTED INTEGRAL AND DISCRETE OPIAL TYPE INEQUALITIES MAJA ANDRIĆ, JOSIP PEČARIĆ AND IVAN PERIĆ Coucated by C. P.
More informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationα1 α2 Simplex and Rectangle Elements Multi-index Notation of polynomials of degree Definition: The set P k will be the set of all functions:
Smplex ad Rectagle Elemets Mult-dex Notato = (,..., ), o-egatve tegers = = β = ( β,..., β ) the + β = ( + β,..., + β ) + x = x x x x = x x β β + D = D = D D x x x β β Defto: The set P of polyomals of degree
More information18.413: Error Correcting Codes Lab March 2, Lecture 8
18.413: Error Correctg Codes Lab March 2, 2004 Lecturer: Dael A. Spelma Lecture 8 8.1 Vector Spaces A set C {0, 1} s a vector space f for x all C ad y C, x + y C, where we take addto to be compoet wse
More information