The Central Limit Theorems for Sums of Powers of Function of Independent Random Variables
|
|
- Maurice McKinney
- 5 years ago
- Views:
Transcription
1 ScieceAsia 8 () : 55-6 The Ceal Limi Theoems fo Sums of Poes of Fucio of Idepede Radom Vaiables K Laipapo a ad K Neammaee b a Depame of Mahemaics Walailak Uivesiy Nakho Si Thammaa 86 Thailad b Depame of Mahemaics Chulalogko Uivesiy Bagkok 33 Thailad Coespodig auho k_eammaee@homailcom Received Feb Acceped Sep ABSTRACT Le (X ) k k ; be a double sequece of ifiiesimal adom vaiables hich ae oise idepede I his pape e give ecessay ad sufficie codiios fo he ( ( )) ( ( )) ( ) sequece of disibuio fucios of S g X L g X B ( ) o eakly covege o a limiig disibuio fucio F fo each aual umbe ad also fo covegece of (F ) KEYWORDS: ceal limi heoem ifiiely divisible Le vy s fomula Mahemaics Subjec Classificaio: ():6E7 6F5 6G5 INTRODUCTION Le (X ) k k ; be a double sequece of ifiiesimal adom vaiables hich ae oise idepede Le S X L X A hee A ae cosas ad le G be he disibuio fucios of S Necessay ad sufficie codiios fo (G ) o covege o a disibuio fucio G ae ko ad i paicula i is ell ko ha G is ifiiely divisible I 957 Shapio cosideed he limi disibuio () fucios of he sums X L X B hee B () ae suiably chose cosas ad N I Shapio -4 ad Temuipog 5 gave he codiios hich guaaees ha he disibuio fucios of he sums X X L X covege o a limi fo < I 998 Neammaee 6 gave he codiios fo covegece of disibuio fucios of l X l X L l X fo < MAIN OF OBJECTIVE I his ok e coside he disibuio fucios of he sums ( ) S g X g X B ( ) ( ( ) ) L ( ) ( ) hee N ad g: R R saisfies he folloig popeies: (g-) g() (g-) g is coiuous sicly deceasig o ( ] ad sicly iceasig o [ ) (g-3) hee eis posiive cosas ad c such ha g ( ) < cfo all ( ) (g-4) g( ) g( ) Sice g saisfies (g-) ad (g-) e ca ie g( ) if ; g ( ) g ( ) if < hee g : g : R R defied by g () g() ad g : R R defied by g () g() Sice g is coiuous a ad g() e ca assume he i (g-3) has popeies g( ) < ad g( ) < The folloigs ae eamples of g g ( ) c fo c> ad N si if ; g ( ) si if < So Shapio s esuls ae ou special case ( ) ( ) Fom o o fo N e le F F F be ( ) ad ( he disibuio fucios of S ) g( X ) X especively ad fo ifiiely divisible disibuio fucio F e le M N γ σ be M N γ σ i
2 56 ScieceAsia 8 () Le vy s fomula of F (Peov 7 chape II) The ecessay ad sufficie codiios fo covegece ( of he sequece of disibuio fucios of S ) ad he sequece of disibuio fucios F ae give i Theoem A ad Theoem B hich saed belo Theoem A Assume ha G G as The fo each N ad fo suiably chose ( cosas B( ) F ) F as if ad oly if k g lim limsup { ( g ( )) df ( ) ( g ( )) df ( ) ad k g g ( ( g ()) df ( ) ( g ()) df ( )) } σ < g k g lim lim if { ( g ( )) df ( ) ( g ( )) df ( ) k g g ( ( g ()) df ( ) ( g ()) df ( )) } < g σ ( ) Theoem B Le G G ad F F as fo all N The F H ad if ad oly if M( ) < g fo all ( ) N( ) > g fo all ( ) 3 lim σ ( σ ) hee he fucios M N ae fucios i Le vy s fomula of F ad σ is he cosa i Le vy s fomula of H Moeove e ko ha 4 if σ M is coiuous a g () ad N is coiuous a g () he H is degeeae 5 if σ M is coiuous a g () ad N is coiuous a g () he H is omal 6 if σ M is discoiuous a g () o N is discoiuous a g () he H ( - m) is Poisso fo some cosa m 7 if σ M is discoiuous a g () o N is discoiuous a g () he H is he disibuio fucio of he sum of o idepede adom vaiables oe of hich is omal ad he ohe is Poisso PROOFS OF MAIN RESULTS Befoe e pove he mai esuls e eed he folloig lemmas Lemma Le X ~ N(a σ ) ad Y ~ Poi(λ) If X ad Y ae idepede he Le vy s fomula of he chaaceisic fucio of X Y is λ i i log ϕ X Y( ) i( a ) σ ( e ) dk( ) hee K : R R is defied by K λ if < ; ( ) if > Poof Le ϕ X ad ϕ Y be he chaaceisic fucios of X ad Y especively Fom Lukacs 8 p93 e have log ϕx ( ) ia λ i i σ ad log ϕy ( ) i ( e ) dk( ) Sice X ad Y ae idepede log ϕ ( ) log ϕ ( ) ϕ ( ) X Y X Y log ϕ ( ) log ϕ ( ) X ia λ i i e i σ ( ) dk( ) Y λ ia i e i ( ) σ ( ) dk( ) # Lemma If G G as he fo evey N k ( ) lim F ( ) fo all < ad k k k ( ) lim ( F ( ) ) N( g ( )) M( g ( )) ae o ( ) ( Fuhemoe if F ) F fo evey N he fo each N e have 3 M o ( ) ad 4 N ( ) N( g ( )) M( g ( )) ae o ( ) hee M ad N ae fucios i Le vy s fomula of F
3 ScieceAsia 8 () 57 Poof Noe ha F ( ) if < ; ( ) PX ( ) if ; ad F F ( g ( )) F ( g ( ) ) if > ( ) if < ; ( ) ( ) F ( ) if () () So follos fom () To pove le N Sice G G by Theoem 8 of Peov 7 p8-8 e ko ha k k lim F ( ) M( ) ad lim ( F ( ) ) N( ) (3) k k fo all coiuiy pois of M ad N Fom () ad (3) k ( ) k lim ( F ( ) ) k lim { F ( g ( )) F ( g ( ) )} k k lim { F ( g ( )) } lim { F ( g ( ) ) } k k k N( g ( )) M( g ( ) ) ae o ( ) N( g ( )) M( g ( )) ae o ( ) ( No e suppose ha F ) F fo evey N By () () ad Theoem 8 of Peov 7 p8-8 e have (3) ad (4) # ( Lemma 3 Assume ha F ) F fo evey N The fo evey N M( ) o ( ) ad N( ) N( ) ae o ( ) Poof We use he same agume i povig of Lemma by usig () isead of () # Lemma 4 Assume ha ( ) fo evey N F F as ad F H as The H is oe of he folloig a degeeae disibuio fucio a Poisso disibuio fucio 3 a omal disibuio fucio 4 he disibuio fucio of he sum of o idepede adom vaiables oe of hich is omal ad he ohe is Poisso Poof Le be ay aual umbe The by Lemma 3 e have M o (- ) ad N( ) N ( ) ae o ( ) Sice F H as by Theoem 3 of Peov 7 p75 e have lim M ( ) M ( ) fo all coiuiy pois of M lim N ( ) N ( ) fo all coiuiy pois of N limγ γ ad lim limsup udm( u) σ udn( u) lim limif ( ) ( ) udm σ ( σ ) u udn u hee M N γ ad σ ae associaed ih H i Le vy s fomula This shos ha M N N N ( ) if ; ( ) lim ( ) > ad N( ) if < < Bu if > ; N ( ) so N( ) Thus N ( ) N( ) if < < Case σ ad N The H is degeeae Case σ ad N The H is omal Case 3 σ ad N akes oe jump If γ ( ) N he H is Poisso
4 58 ScieceAsia 8 () N ( ) γ N ( ) If γ le m e oe ha he chaaceisic fucio ϕ () m of H ( m ) im is e ϕ () hee ϕ is he chaaceisic fucio of H Hece log im ϕ ( ) log e ϕ ( ) m im log ϕ ( ) i i im iγ ( e ) dn ( ) N( ) i i i( ) ( e ) dn ( ) So H( - m) is Poisso Case 4 σ ad N akes oe jump By Lemma H is he disibuio fucio of he sum of o idepede adom vaiables oe of hich is a Poisso ad he ohe is a omal # ( ) Lemma 5 Assume ha F F as fo evey N ad G G as If F H as he M o ( ) if > ; N ( ) N( g ( )) M( g ( )) if < < o ( ) ad 3 M( g ( )) N( g ( )) hee M ad N ae fucios i Le vy s fomula of F ad Mad N ae fucios i Le vy s fomula of H Poof Use he same echique i fidig N ad M i Lemma 4 by usig Lemma isead of Lemma 3 # Poof of Theoem A Noe ha fo > e have ( ) ( ) df ( ) ( df ( )) < < d[ F ( g ( )) F ( g ( ) ) ] d[ F g F g ] ( ( ( )) ( ( ) ) ) g ( g ( )) df ( ) ( g ( )) df ( ) g g g ( ( g ( )) df ( ) ( g ( )) df ( ) ) g [ g ( ) ad g ( )] ( g ( )) df ( ) ( g ( )) df ( ) g g ( ( g ()) df () ( g ()) df ( )) g To pove ecessiy e suppose ha F (4) ( ) as The ad follo fom Theoem 8 of Peov 7 p8-8 ad (4) Fo sufficiecy e defie M :( ) R ad d N :( ) R by M ( ) ad N ( ) N( g ( )) M( g ( )) Clealy M ad N ae odeceasig ad M (- ) N ( ) By () ad () of Lemma e k k ( ) have lim F ( ) M ( ) ad (5) k ( ) lim ( F ( ) ) N( ) (6) k fo all coiuiy pois of M ad N By assumpios ad (4) e have k ( ) ( ) lim limsup df ( ) ( df ( )) k < < { } k ( ) ( ) lim limif { df ( ) ( df ( )) } σ < k < < F (7) By (5)-(7) ad Theoem 8 of Peov 7 p8-8 F F as # ( ) Poof of Theoem B Fo ad < < mi {(g ()) (g (-)) } e have ma{ g g } ad udm( u) udn( u)
5 ScieceAsia 8 () 59 ud[ N( g ( u)) M( g ( u)) ] (by Lemma (3) ad(4)) g ( ) g ( ) ( g ( )) dn( ) ( g ( )) dm( ) g ( ) [ g ( u ) ad g ( u )] ( g ( )) dn( ) ( g ( )) dm( ) g ( ) g ( ) ( ( g ()) dn()) ( ( g ()) dm()) g ( ) { ( g ()) dn() ( g ()) dm()} { ( g ()) dn() ( g ()) dm()} { g () () g () dn dm()} c { dn( ) dm( )} The Hece (by popey (g-3))(8) lim limsup { udm( u) udn( u) } lim limsup c { dn( ) dm( )} { } lim limsup udm( u) udn( u ) (9) Similaly e have lim limif udm( u) udn( u ) () { } To pove ecessiy e suppose ha F H as Sice G G by Theoem 8 of Peov 7 p8-8 k k e have lim F ( ) M( ) ad lim [ F ( ) ] N( ) k k fo all coiuiy pois of M ad N The () ad () follo fom Lemma 5(3) ad he fac ha M ad N ae odeceasig ad M(- ) N( ) No e ill sho (3) Sice F H by Theoem 3 of Peov 7 p75 e have lim limsup { udm( u) σ udn( u) } lim limif udm( u) σ udn( u) ( σ ) { } By (9) - () e see ha limsup σ ( σ ) ad limif σ ( σ ) So lim σ ( σ ) () To pove sufficiecy e assume ha () () ad (3) hold ( ) Sice G Gad F F as by Lemma M ad N( ) N( g ( )) M( g ( )) ae o ( ) Le N : R R be defied by N ( ) lim N ( ) ) M : ad R R be defied by M ( ) lim M( ) The M o (- ) ad by assumpios () ad if > ; () N ( ) N( g ( )) M( g ( )) if < < o ( ) Tha is M (- ) N( ) Fom assumpio (3) ad (9) e have lim limsup udm( u) σ udn( u ) lim σ ( σ ) { } Similaly e ca sho ha lim limif udm( u) σ udn( u ) ( σ ) { } By Theoem 3 of Peov 7 p75 e have lim F ( ) H( ) hee H is he ifiiely divisible disibuio deemied by M N γ ad (σ) By he same agume of Lemma 4 e have (4)-(7) # REFERENCES Shapio JM (957) Sums of Poes of Idepede Radom Vaiables The Ameica Mahemaical Sociey Shapio JM (975) Domai of aacio ecipocals of poe of adom vaiables Siam Joual Appl Mah
6 6 ScieceAsia 8 () 3 Shapio JM (977) O domias of omal aacio o sable disibuios Houso J Mah Shapio JM (988) Limi disibuios fo sums of ecipocals of idepede adom vaiables Houso J Mah Temuipog I (986) Limi Disibuios fo Sums of he Recipocal of a Posiive Poe of Idepede Radom Vaiables Ph D hesis Chulalogko Uiv 6 Neammaee K (998) Limi Disibuios fo Radom Sums of he Recipocals of Logaihms of Idepede Coiuous Radom Vaiables J Sci Res Chula Uiv 3 No 7 Peov VV (975) Sum of Idepede Radom Vaiables Spige-Valag NeYok 8 Lukacs E (97) Chaaceisic Fucios Hafe Publishig Compay NeYok
ABSOLUTE INDEXED SUMMABILITY FACTOR OF AN INFINITE SERIES USING QUASI-F-POWER INCREASING SEQUENCES
Available olie a h://sciog Egieeig Maheaics Lees 2 (23) No 56-66 ISSN 249-9337 ABSLUE INDEED SUMMABILIY FACR F AN INFINIE SERIES USING QUASI-F-WER INCREASING SEQUENCES SKAIKRAY * RKJAI 2 UKMISRA 3 NCSAH
More informationSpectrum of The Direct Sum of Operators. 1. Introduction
Specu of The Diec Su of Opeaos by E.OTKUN ÇEVİK ad Z.I.ISMILOV Kaadeiz Techical Uivesiy, Faculy of Scieces, Depae of Maheaics 6080 Tabzo, TURKEY e-ail adess : zaeddi@yahoo.co bsac: I his wok, a coecio
More informationON GENERALIZED FRACTIONAL INTEGRAL OPERATORS. ( ρ( x y ) T ρ f(x) := f(y) R x y n dy, R x y n ρ( y )(1 χ )
Scieiae Mahemaicae Japoicae Olie, Vol., 24), 37 38 37 ON GENERALIZED FRACTIONAL INTEGRAL OPERATORS ERIDANI, HENDRA GUNAWAN 2 AND EIICHI NAKAI 3 Received Augus 29, 23; evised Apil 7, 24 Absac. We pove he
More informationGENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS
GENERALIZED FRACTIONAL INTEGRAL OPERATORS AND THEIR MODIFIED VERSIONS HENDRA GUNAWAN Absac. Associaed o a fucio ρ :(, ) (, ), le T ρ be he opeao defied o a suiable fucio space by T ρ f(x) := f(y) dy, R
More informationSupplementary Information
Supplemeay Ifomaio No-ivasive, asie deemiaio of he coe empeaue of a hea-geeaig solid body Dea Ahoy, Daipaya Saka, Aku Jai * Mechaical ad Aeospace Egieeig Depame Uivesiy of Texas a Aligo, Aligo, TX, USA.
More informationON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES
M aheaical I equaliies & A pplicaios Volue 19, Nube 1 (216), 287 296 doi:1.7153/ia-19-21 ON POINTWISE APPROXIMATION OF FUNCTIONS BY SOME MATRIX MEANS OF FOURIER SERIES W. ŁENSKI AND B. SZAL (Couicaed by
More informationCalculus Limits. Limit of a function.. 1. One-Sided Limits...1. Infinite limits 2. Vertical Asymptotes...3. Calculating Limits Using the Limit Laws.
Limi of a fucio.. Oe-Sided..... Ifiie limis Verical Asympoes... Calculaig Usig he Limi Laws.5 The Squeeze Theorem.6 The Precise Defiiio of a Limi......7 Coiuiy.8 Iermediae Value Theorem..9 Refereces..
More informationConditional Convergence of Infinite Products
Coditioal Covegece of Ifiite Poducts William F. Tech Ameica Mathematical Mothly 106 1999), 646-651 I this aticle we evisit the classical subject of ifiite poducts. Fo stadad defiitios ad theoems o this
More informationCHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE
Fameal Joal of Mahemaic a Mahemaical Sciece Vol. 7 Ie 07 Page 5- Thi pape i aailable olie a hp://.fi.com/ Pblihe olie Jaa 0 07 CHATTERJEA CONTRACTION MAPPING THEOREM IN CONE HEPTAGONAL METRIC SPACE Caolo
More informationThe shortest path between two truths in the real domain passes through the complex domain. J. Hadamard
Complex Analysis R.G. Halbud R.Halbud@ucl.ac.uk Depamen of Mahemaics Univesiy College London 202 The shoes pah beween wo uhs in he eal domain passes hough he complex domain. J. Hadamad Chape The fis fundamenal
More informationS, we call the base curve and the director curve. The straight lines
Developable Ruled Sufaces wih Daboux Fame i iowsi -Space Sezai KIZILTUĞ, Ali ÇAKAK ahemaics Depame, Faculy of As ad Sciece, Ezica Uivesiy, Ezica, Tuey ahemaics Depame, Faculy of Sciece, Aau Uivesiy, Ezuum,
More informationOn a Z-Transformation Approach to a Continuous-Time Markov Process with Nonfixed Transition Rates
Ge. Mah. Noes, Vol. 24, No. 2, Ocobe 24, pp. 85-96 ISSN 229-784; Copyigh ICSRS Publicaio, 24 www.i-css.og Available fee olie a hp://www.gema.i O a Z-Tasfomaio Appoach o a Coiuous-Time Maov Pocess wih Nofixed
More informationUsing Difference Equations to Generalize Results for Periodic Nested Radicals
Usig Diffeece Equatios to Geealize Results fo Peiodic Nested Radicals Chis Lyd Uivesity of Rhode Islad, Depatmet of Mathematics South Kigsto, Rhode Islad 2 2 2 2 2 2 2 π = + + +... Vieta (593) 2 2 2 =
More informationRelations on the Apostol Type (p, q)-frobenius-euler Polynomials and Generalizations of the Srivastava-Pintér Addition Theorems
Tish Joal of Aalysis ad Nmbe Theoy 27 Vol 5 No 4 26-3 Available olie a hp://pbssciepbcom/ja/5/4/2 Sciece ad Edcaio Pblishig DOI:269/ja-5-4-2 Relaios o he Aposol Type (p -Fobeis-Ele Polyomials ad Geealizaios
More information7 Wave Equation in Higher Dimensions
7 Wave Equaion in Highe Dimensions We now conside he iniial-value poblem fo he wave equaion in n dimensions, u c u x R n u(x, φ(x u (x, ψ(x whee u n i u x i x i. (7. 7. Mehod of Spheical Means Ref: Evans,
More informationAvailable online at J. Math. Comput. Sci. 2 (2012), No. 4, ISSN:
Available olie a h://scik.og J. Mah. Comu. Sci. 2 (22), No. 4, 83-835 ISSN: 927-537 UNBIASED ESTIMATION IN BURR DISTRIBUTION YASHBIR SINGH * Deame of Saisics, School of Mahemaics, Saisics ad Comuaioal
More informationSTK4080/9080 Survival and event history analysis
STK48/98 Survival ad eve hisory aalysis Marigales i discree ime Cosider a sochasic process The process M is a marigale if Lecure 3: Marigales ad oher sochasic processes i discree ime (recap) where (formally
More informationLower Bounds for Cover-Free Families
Loe Bouds fo Cove-Fee Families Ali Z. Abdi Covet of Nazaeth High School Gade, Abas 7, Haifa Nade H. Bshouty Dept. of Compute Sciece Techio, Haifa, 3000 Apil, 05 Abstact Let F be a set of blocks of a t-set
More informationComparing Different Estimators for Parameters of Kumaraswamy Distribution
Compaig Diffee Esimaos fo Paamees of Kumaaswamy Disibuio ا.م.د نذير عباس ابراهيم الشمري جامعة النهرين/بغداد-العراق أ.م.د نشات جاسم محمد الجامعة التقنية الوسطى/بغداد- العراق Absac: This pape deals wih compaig
More informationOn ARMA(1,q) models with bounded and periodically correlated solutions
Reseach Repot HSC/03/3 O ARMA(,q) models with bouded ad peiodically coelated solutios Aleksade Weo,2 ad Agieszka Wy oma ska,2 Hugo Steihaus Cete, Woc aw Uivesity of Techology 2 Istitute of Mathematics,
More informationFIXED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE
Mohia & Samaa, Vol. 1, No. II, December, 016, pp 34-49. ORIGINAL RESEARCH ARTICLE OPEN ACCESS FIED FUZZY POINT THEOREMS IN FUZZY METRIC SPACE 1 Mohia S. *, Samaa T. K. 1 Deparme of Mahemaics, Sudhir Memorial
More informationth m m m m central moment : E[( X X) ] ( X X) ( x X) f ( x)
1 Trasform Techiques h m m m m mome : E[ ] x f ( x) dx h m m m m ceral mome : E[( ) ] ( ) ( x) f ( x) dx A coveie wa of fidig he momes of a radom variable is he mome geeraig fucio (MGF). Oher rasform echiques
More informationMoment Generating Function
1 Mome Geeraig Fucio m h mome m m m E[ ] x f ( x) dx m h ceral mome m m m E[( ) ] ( ) ( x ) f ( x) dx Mome Geeraig Fucio For a real, M () E[ e ] e k x k e p ( x ) discree x k e f ( x) dx coiuous Example
More informationLecture 3 : Concentration and Correlation
Lectue 3 : Cocetatio ad Coelatio 1. Talagad s iequality 2. Covegece i distibutio 3. Coelatio iequalities 1. Talagad s iequality Cetifiable fuctios Let g : R N be a fuctio. The a fuctio f : 1 2 Ω Ω L Ω
More informationTaylor Transformations into G 2
Iteatioal Mathematical Foum, 5,, o. 43, - 3 Taylo Tasfomatios ito Mulatu Lemma Savaah State Uivesity Savaah, a 344, USA Lemmam@savstate.edu Abstact. Though out this pape, we assume that
More information( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to
Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special
More informationThe Multivariate-t distribution and the Simes Inequality. Abstract. Sarkar (1998) showed that certain positively dependent (MTP 2 ) random variables
The Multivaiate-t distibutio ad the Simes Iequality by Hey W. Block 1, Saat K. Saka 2, Thomas H. Savits 1 ad Jie Wag 3 Uivesity of ittsbugh 1,Temple Uivesity 2,Gad Valley State Uivesity 3 Abstact. Saka
More informationA note on random minimum length spanning trees
A ote o adom miimum legth spaig tees Ala Fieze Miklós Ruszikó Lubos Thoma Depatmet of Mathematical Scieces Caegie Mello Uivesity Pittsbugh PA15213, USA ala@adom.math.cmu.edu, usziko@luta.sztaki.hu, thoma@qwes.math.cmu.edu
More informationBy the end of this section you will be able to prove the Chinese Remainder Theorem apply this theorem to solve simultaneous linear congruences
Chapte : Theoy of Modula Aithmetic 8 Sectio D Chiese Remaide Theoem By the ed of this sectio you will be able to pove the Chiese Remaide Theoem apply this theoem to solve simultaeous liea cogueces The
More information1 Notes on Little s Law (l = λw)
Copyrigh c 26 by Karl Sigma Noes o Lile s Law (l λw) We cosider here a famous ad very useful law i queueig heory called Lile s Law, also kow as l λw, which assers ha he ime average umber of cusomers i
More informationA General Iterative Scheme for Variational Inequality Problems and Fixed Point Problems
A Geeral Iterative Scheme for Variatioal Iequality Problems ad Fixed Poit Problems Wicha Khogtham Abstract We itroduce a geeral iterative scheme for fidig a commo of the set solutios of variatioal iequality
More informationCounting Functions and Subsets
CHAPTER 1 Coutig Fuctios ad Subsets This chapte of the otes is based o Chapte 12 of PJE See PJE p144 Hee ad below, the efeeces to the PJEccles book ae give as PJE The goal of this shot chapte is to itoduce
More informationThe Nehari Manifold for a Class of Elliptic Equations of P-laplacian Type. S. Khademloo and H. Mohammadnia. afrouzi
Wold Alied cieces Joal (8): 898-95 IN 88-495 IDOI Pblicaios = h x g x x = x N i W whee is a eal aamee is a boded domai wih smooh boday i R N 3 ad< < INTRODUCTION Whee s ha is s = I his ae we ove he exisece
More informationMathematical Statistics. 1 Introduction to the materials to be covered in this course
Mahemaical Saisics Iroducio o he maerials o be covered i his course. Uivariae & Mulivariae r.v s 2. Borl-Caelli Lemma Large Deviaios. e.g. X,, X are iid r.v s, P ( X + + X where I(A) is a umber depedig
More informationSome Integral Mean Estimates for Polynomials
Iteatioal Mathematical Foum, Vol. 8, 23, o., 5-5 HIKARI Ltd, www.m-hikai.com Some Itegal Mea Estimates fo Polyomials Abdullah Mi, Bilal Ahmad Da ad Q. M. Dawood Depatmet of Mathematics, Uivesity of Kashmi
More informationBEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS
BEST LINEAR FORECASTS VS. BEST POSSIBLE FORECASTS Opimal ear Forecasig Alhough we have o meioed hem explicily so far i he course, here are geeral saisical priciples for derivig he bes liear forecas, ad
More informationON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS
Joual of Pue ad Alied Mathematics: Advaces ad Alicatios Volume 0 Numbe 03 Pages 5-58 ON EUCLID S AND EULER S PROOF THAT THE NUMBER OF PRIMES IS INFINITE AND SOME APPLICATIONS ALI H HAKAMI Deatmet of Mathematics
More informationA NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS
Discussioes Mathematicae Gaph Theoy 28 (2008 335 343 A NOTE ON DOMINATION PARAMETERS IN RANDOM GRAPHS Athoy Boato Depatmet of Mathematics Wilfid Lauie Uivesity Wateloo, ON, Caada, N2L 3C5 e-mail: aboato@oges.com
More informationResearch Article On Pointwise Approximation of Conjugate Functions by Some Hump Matrix Means of Conjugate Fourier Series
Hidawi Publishig Copoaio Joual of Fucio Spaces Volue 5, Aicle ID 475, 9 pages hp://dx.doi.og/.55/5/475 Reseach Aicle O Poiwise Appoxiaio of Cojugae Fucios by Soe Hup Maix Meas of Cojugae Fouie Seies W.
More informationRange Symmetric Matrices in Minkowski Space
BULLETIN of the Bull. alaysia ath. Sc. Soc. (Secod Seies) 3 (000) 45-5 LYSIN THETICL SCIENCES SOCIETY Rae Symmetic atices i ikowski Space.R. EENKSHI Depatmet of athematics, amalai Uivesity, amalaiaa 608
More informationDegree of Approximation of Fourier Series
Ieaioal Mahemaical Foum Vol. 9 4 o. 9 49-47 HIARI Ld www.m-hiai.com h://d.doi.og/.988/im.4.49 Degee o Aoimaio o Fouie Seies by N E Meas B. P. Padhy U.. Misa Maheda Misa 3 ad Saosh uma Naya 4 Deame o Mahemaics
More informationThe analysis of the method on the one variable function s limit Ke Wu
Ieraioal Coferece o Advaces i Mechaical Egieerig ad Idusrial Iformaics (AMEII 5) The aalysis of he mehod o he oe variable fucio s i Ke Wu Deparme of Mahemaics ad Saisics Zaozhuag Uiversiy Zaozhuag 776
More informationMath 6710, Fall 2016 Final Exam Solutions
Mah 67, Fall 6 Fial Exam Soluios. Firs, a sude poied ou a suble hig: if P (X i p >, he X + + X (X + + X / ( evaluaes o / wih probabiliy p >. This is roublesome because a radom variable is supposed o be
More informationMATH Midterm Solutions
MATH 2113 - Midtem Solutios Febuay 18 1. A bag of mables cotais 4 which ae ed, 4 which ae blue ad 4 which ae gee. a How may mables must be chose fom the bag to guaatee that thee ae the same colou? We ca
More informationGeneralized Fibonacci-Lucas Sequence
Tuish Joual of Aalysis ad Numbe Theoy, 4, Vol, No 6, -7 Available olie at http://pubssciepubcom/tjat//6/ Sciece ad Educatio Publishig DOI:6/tjat--6- Geealized Fiboacci-Lucas Sequece Bijeda Sigh, Ompaash
More informationFIXED POINT AND HYERS-ULAM-RASSIAS STABILITY OF A QUADRATIC FUNCTIONAL EQUATION IN BANACH SPACES
IJRRAS 6 () July 0 www.apapess.com/volumes/vol6issue/ijrras_6.pdf FIXED POINT AND HYERS-UAM-RASSIAS STABIITY OF A QUADRATIC FUNCTIONA EQUATION IN BANACH SPACES E. Movahedia Behbaha Khatam Al-Abia Uivesity
More informationOn imploding cylindrical and spherical shock waves in a perfect gas
J. Fluid Mech. (2006), vol. 560, pp. 103 122. c 2006 Cambidge Uivesiy Pess doi:10.1017/s0022112006000590 Pied i he Uied Kigdom 103 O implodig cylidical ad spheical shock waves i a pefec gas By N. F. PONCHAUT,
More informationA Note on Random k-sat for Moderately Growing k
A Noe o Radom k-sat for Moderaely Growig k Ju Liu LMIB ad School of Mahemaics ad Sysems Sciece, Beihag Uiversiy, Beijig, 100191, P.R. Chia juliu@smss.buaa.edu.c Zogsheg Gao LMIB ad School of Mahemaics
More informationA TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY
U.P.B. Sci. Bull., Series A, Vol. 78, Iss. 2, 206 ISSN 223-7027 A TAUBERIAN THEOREM FOR THE WEIGHTED MEAN METHOD OF SUMMABILITY İbrahim Çaak I his paper we obai a Tauberia codiio i erms of he weighed classical
More informationCHAPTER 5 : SERIES. 5.2 The Sum of a Series Sum of Power of n Positive Integers Sum of Series of Partial Fraction Difference Method
CHAPTER 5 : SERIES 5.1 Seies 5. The Sum of a Seies 5..1 Sum of Powe of Positive Iteges 5.. Sum of Seies of Patial Factio 5..3 Diffeece Method 5.3 Test of covegece 5.3.1 Divegece Test 5.3. Itegal Test 5.3.3
More informationFujii, Takao; Hayashi, Fumiaki; Iri Author(s) Oguro, Kazumasa.
TileDesigig a Opimal Public Pesio Fujii, Takao; Hayashi, Fumiaki; Ii Auho(s) Oguo, Kazumasa Ciaio Issue 3- Dae Type Techical Repo Tex Vesio publishe URL hp://hdl.hadle.e/86/54 Righ Hiosubashi Uivesiy Reposioy
More informationActuarial Society of India
Acuarial Sociey of Idia EXAMINAIONS Jue 5 C4 (3) Models oal Marks - 5 Idicaive Soluio Q. (i) a) Le U deoe he process described by 3 ad V deoe he process described by 4. he 5 e 5 PU [ ] PV [ ] ( e ).538!
More informationGeneralized Fibonacci-Type Sequence and its Properties
Geelized Fibocci-Type Sequece d is Popeies Ompsh Sihwl shw Vys Devshi Tuoil Keshv Kuj Mdsu (MP Idi Resech Schol Fculy of Sciece Pcific Acdemy of Highe Educio d Resech Uivesiy Udipu (Rj Absc: The Fibocci
More informationMATH /19: problems for supervision in week 08 SOLUTIONS
MATH10101 2018/19: poblems fo supevisio i week 08 Q1. Let A be a set. SOLUTIONS (i Pove that the fuctio c: P(A P(A, defied by c(x A \ X, is bijective. (ii Let ow A be fiite, A. Use (i to show that fo each
More information2 f(x) dx = 1, 0. 2f(x 1) dx d) 1 4t t6 t. t 2 dt i)
Mah PracTes Be sure o review Lab (ad all labs) There are los of good quesios o i a) Sae he Mea Value Theorem ad draw a graph ha illusraes b) Name a impora heorem where he Mea Value Theorem was used i he
More informationDavid Randall. ( )e ikx. k = u x,t. u( x,t)e ikx dx L. x L /2. Recall that the proof of (1) and (2) involves use of the orthogonality condition.
! Revised April 21, 2010 1:27 P! 1 Fourier Series David Radall Assume ha u( x,) is real ad iegrable If he domai is periodic, wih period L, we ca express u( x,) exacly by a Fourier series expasio: ( ) =
More informationSolution. 1 Solutions of Homework 6. Sangchul Lee. April 28, Problem 1.1 [Dur10, Exercise ]
Soluio Sagchul Lee April 28, 28 Soluios of Homework 6 Problem. [Dur, Exercise 2.3.2] Le A be a sequece of idepede eves wih PA < for all. Show ha P A = implies PA i.o. =. Proof. Noice ha = P A c = P A c
More informationLecture 6: October 16, 2017
Ifomatio ad Codig Theoy Autum 207 Lectue: Madhu Tulsiai Lectue 6: Octobe 6, 207 The Method of Types Fo this lectue, we will take U to be a fiite uivese U, ad use x (x, x 2,..., x to deote a sequece of
More informationExistence and Smoothness of Solution of Navier-Stokes Equation on R 3
Ieaioal Joual of Mode Noliea Theoy ad Applicaio, 5, 4, 7-6 Published Olie Jue 5 i SciRes. hp://www.scip.og/joual/ijma hp://dx.doi.og/.436/ijma.5.48 Exisece ad Smoohess of Soluio of Navie-Sokes Equaio o
More informationTHE ANALYTIC LARGE SIEVE
THE ANALYTIC LAGE SIEVE 1. The aalytic lage sieve I the last lectue we saw how to apply the aalytic lage sieve to deive a aithmetic fomulatio of the lage sieve, which we applied to the poblem of boudig
More informationA two-sided Iterative Method for Solving
NTERNATONAL JOURNAL OF MATHEMATCS AND COMPUTERS N SMULATON Volume 9 0 A two-sided teative Method fo Solvig * A Noliea Matix Equatio X= AX A Saa'a A Zaea Abstact A efficiet ad umeical algoithm is suggested
More informationMath 2414 Homework Set 7 Solutions 10 Points
Mah Homework Se 7 Soluios 0 Pois #. ( ps) Firs verify ha we ca use he iegral es. The erms are clearly posiive (he epoeial is always posiive ad + is posiive if >, which i is i his case). For decreasig we
More information336 ERIDANI kfk Lp = sup jf(y) ; f () jj j p p whee he supemum is aken ove all open balls = (a ) inr n, jj is he Lebesgue measue of in R n, () =(), f
TAMKANG JOURNAL OF MATHEMATIS Volume 33, Numbe 4, Wine 2002 ON THE OUNDEDNESS OF A GENERALIED FRATIONAL INTEGRAL ON GENERALIED MORREY SPAES ERIDANI Absac. In his pape we exend Nakai's esul on he boundedness
More informationODEs II, Supplement to Lectures 6 & 7: The Jordan Normal Form: Solving Autonomous, Homogeneous Linear Systems. April 2, 2003
ODEs II, Suppleme o Lecures 6 & 7: The Jorda Normal Form: Solvig Auoomous, Homogeeous Liear Sysems April 2, 23 I his oe, we describe he Jorda ormal form of a marix ad use i o solve a geeral homogeeous
More informationSOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES
#A17 INTEGERS 9 2009), 181-190 SOME ARITHMETIC PROPERTIES OF OVERPARTITION K -TUPLES Deick M. Keiste Depatmet of Mathematics, Pe State Uivesity, Uivesity Pak, PA 16802 dmk5075@psu.edu James A. Selles Depatmet
More informationLecture 24: Observability and Constructibility
ectue 24: Obsevability ad Costuctibility 7 Obsevability ad Costuctibility Motivatio: State feedback laws deped o a kowledge of the cuet state. I some systems, xt () ca be measued diectly, e.g., positio
More informationVariance and Covariance Processes
Vaiance and Covaiance Pocesses Pakash Balachandan Depamen of Mahemaics Duke Univesiy May 26, 2008 These noes ae based on Due s Sochasic Calculus, Revuz and Yo s Coninuous Maingales and Bownian Moion, Kaazas
More informationSupplement for SADAGRAD: Strongly Adaptive Stochastic Gradient Methods"
Suppleme for SADAGRAD: Srogly Adapive Sochasic Gradie Mehods" Zaiyi Che * 1 Yi Xu * Ehog Che 1 iabao Yag 1. Proof of Proposiio 1 Proposiio 1. Le ɛ > 0 be fixed, H 0 γi, γ g, EF (w 1 ) F (w ) ɛ 0 ad ieraio
More informationCS623: Introduction to Computing with Neural Nets (lecture-10) Pushpak Bhattacharyya Computer Science and Engineering Department IIT Bombay
CS6: Iroducio o Compuig ih Neural Nes lecure- Pushpak Bhaacharyya Compuer Sciece ad Egieerig Deparme IIT Bombay Tilig Algorihm repea A kid of divide ad coquer sraegy Give he classes i he daa, ru he percepro
More informationSums of Involving the Harmonic Numbers and the Binomial Coefficients
Ameica Joual of Computatioal Mathematics 5 5 96-5 Published Olie Jue 5 i SciRes. http://www.scip.og/oual/acm http://dx.doi.og/.46/acm.5.58 Sums of Ivolvig the amoic Numbes ad the Biomial Coefficiets Wuyugaowa
More informationThe Pigeonhole Principle 3.4 Binomial Coefficients
Discete M athematic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Ageda Ch 3. The Pigeohole Piciple
More informationHighly connected coloured subgraphs via the Regularity Lemma
Highly coeced coloued subgaphs via he Regulaiy Lemma Hey Liu 1 Depame of Mahemaics, Uivesiy College Lodo, Gowe See, Lodo WC1E 6BT, Uied Kigdom Yuy Peso 2 Isiu fü Ifomaik, Humbold-Uivesiä zu Beli, Ue de
More informationBINOMIAL THEOREM An expression consisting of two terms, connected by + or sign is called a
BINOMIAL THEOREM hapte 8 8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4 5y, etc., ae all biomial epessios. 8.. Biomial theoem If
More informationPrinciples of Communications Lecture 1: Signals and Systems. Chih-Wei Liu 劉志尉 National Chiao Tung University
Priciples of Commuicaios Lecure : Sigals ad Sysems Chih-Wei Liu 劉志尉 Naioal Chiao ug Uiversiy cwliu@wis.ee.cu.edu.w Oulies Sigal Models & Classificaios Sigal Space & Orhogoal Basis Fourier Series &rasform
More informationLacunary Almost Summability in Certain Linear Topological Spaces
BULLETIN of te MLYSİN MTHEMTİCL SCİENCES SOCİETY Bull. Malays. Mat. Sci. Soc. (2) 27 (2004), 27 223 Lacuay lost Suability i Cetai Liea Topological Spaces BÜNYMIN YDIN Cuuiyet Uivesity, Facutly of Educatio,
More informationDANIEL YAQUBI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD
MIXED -STIRLING NUMERS OF THE SEOND KIND DANIEL YAQUI, MADJID MIRZAVAZIRI AND YASIN SAEEDNEZHAD Abstact The Stilig umbe of the secod id { } couts the umbe of ways to patitio a set of labeled balls ito
More informationBINOMIAL THEOREM NCERT An expression consisting of two terms, connected by + or sign is called a
8. Oveview: 8.. A epessio cosistig of two tems, coected by + o sig is called a biomial epessio. Fo eample, + a, y,,7 4, etc., ae all biomial 5y epessios. 8.. Biomial theoem BINOMIAL THEOREM If a ad b ae
More informationCh 3.4 Binomial Coefficients. Pascal's Identit y and Triangle. Chapter 3.2 & 3.4. South China University of Technology
Disc ete Mathem atic Chapte 3: Coutig 3. The Pigeohole Piciple 3.4 Biomial Coefficiets D Patic Cha School of Compute Sciece ad Egieeig South Chia Uivesity of Techology Pigeohole Piciple Suppose that a
More informationAuchmuty High School Mathematics Department Sequences & Series Notes Teacher Version
equeces ad eies Auchmuty High chool Mathematics Depatmet equeces & eies Notes Teache Vesio A sequece takes the fom,,7,0,, while 7 0 is a seies. Thee ae two types of sequece/seies aithmetic ad geometic.
More informationApproximately Quasi Inner Generalized Dynamics on Modules. { } t t R
Joural of Scieces, Islamic epublic of Ira 23(3): 245-25 (22) Uiversiy of Tehra, ISSN 6-4 hp://jscieces.u.ac.ir Approximaely Quasi Ier Geeralized Dyamics o Modules M. Mosadeq, M. Hassai, ad A. Nikam Deparme
More informationEVALUATION OF SUMS INVOLVING GAUSSIAN q-binomial COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS
EVALUATION OF SUMS INVOLVING GAUSSIAN -BINOMIAL COEFFICIENTS WITH RATIONAL WEIGHT FUNCTIONS EMRAH KILIÇ AND HELMUT PRODINGER Abstact We coside sums of the Gaussia -biomial coefficiets with a paametic atioal
More informationThe Eigen Function of Linear Systems
1/25/211 The Eige Fucio of Liear Sysems.doc 1/7 The Eige Fucio of Liear Sysems Recall ha ha we ca express (expad) a ime-limied sigal wih a weighed summaio of basis fucios: v ( ) a ψ ( ) = where v ( ) =
More information10.3 Autocorrelation Function of Ergodic RP 10.4 Power Spectral Density of Ergodic RP 10.5 Normal RP (Gaussian RP)
ENGG450 Probabiliy ad Saisics for Egieers Iroducio 3 Probabiliy 4 Probabiliy disribuios 5 Probabiliy Desiies Orgaizaio ad descripio of daa 6 Samplig disribuios 7 Ifereces cocerig a mea 8 Comparig wo reames
More informationTechnical Report: Bessel Filter Analysis
Sasa Mahmoodi 1 Techical Repot: Bessel Filte Aalysis 1 School of Electoics ad Compute Sciece, Buildig 1, Southampto Uivesity, Southampto, S17 1BJ, UK, Email: sm3@ecs.soto.ac.uk I this techical epot, we
More informationConsider the time-varying system, (14.1)
Leue 4 // Oulie Moivaio Equivale Defiiios fo Lyapuov Sabiliy Uifomly Sabiliy ad Uifomly Asympoial Sabiliy 4 Covese Lyapuov Theoem 5 Ivaiae- lie Theoem 6 Summay Moivaio Taig poblem i ool, Suppose ha x (
More informationMATH 507a ASSIGNMENT 4 SOLUTIONS FALL 2018 Prof. Alexander. g (x) dx = g(b) g(0) = g(b),
MATH 57a ASSIGNMENT 4 SOLUTIONS FALL 28 Prof. Alexader (2.3.8)(a) Le g(x) = x/( + x) for x. The g (x) = /( + x) 2 is decreasig, so for a, b, g(a + b) g(a) = a+b a g (x) dx b so g(a + b) g(a) + g(b). Sice
More informationThe Moment Approximation of the First Passage Time for the Birth Death Diffusion Process with Immigraton to a Moving Linear Barrier
America Joural of Applied Mahemaics ad Saisics, 015, Vol. 3, No. 5, 184-189 Available olie a hp://pubs.sciepub.com/ajams/3/5/ Sciece ad Educaio Publishig DOI:10.1691/ajams-3-5- The Mome Approximaio of
More informationThis web appendix outlines sketch of proofs in Sections 3 5 of the paper. In this appendix we will use the following notations: c i. j=1.
Web Appedix: Supplemetay Mateials fo Two-fold Nested Desigs: Thei Aalysis ad oectio with Nopaametic ANOVA by Shu-Mi Liao ad Michael G. Akitas This web appedix outlies sketch of poofs i Sectios 3 5 of the
More informationBasic Results in Functional Analysis
Preared by: F.. ewis Udaed: Suday, Augus 7, 4 Basic Resuls i Fucioal Aalysis f ( ): X Y is coiuous o X if X, (, ) z f( z) f( ) f ( ): X Y is uiformly coiuous o X if i is coiuous ad ( ) does o deed o. f
More informationAdditional Tables of Simulation Results
Saisica Siica: Suppleme REGULARIZING LASSO: A CONSISTENT VARIABLE SELECTION METHOD Quefeg Li ad Ju Shao Uiversiy of Wiscosi, Madiso, Eas Chia Normal Uiversiy ad Uiversiy of Wiscosi, Madiso Supplemeary
More informationOn randomly generated non-trivially intersecting hypergraphs
O adomly geeated o-tivially itesectig hypegaphs Balázs Patkós Submitted: May 5, 009; Accepted: Feb, 010; Published: Feb 8, 010 Mathematics Subject Classificatio: 05C65, 05D05, 05D40 Abstact We popose two
More informationIdeal Amplifier/Attenuator. Memoryless. where k is some real constant. Integrator. System with memory
Liear Time-Ivaria Sysems (LTI Sysems) Oulie Basic Sysem Properies Memoryless ad sysems wih memory (saic or dyamic) Causal ad o-causal sysems (Causaliy) Liear ad o-liear sysems (Lieariy) Sable ad o-sable
More informationProgression. CATsyllabus.com. CATsyllabus.com. Sequence & Series. Arithmetic Progression (A.P.) n th term of an A.P.
Pogessio Sequece & Seies A set of umbes whose domai is a eal umbe is called a SEQUENCE ad sum of the sequece is called a SERIES. If a, a, a, a 4,., a, is a sequece, the the expessio a + a + a + a 4 + a
More informationMATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES
MATHS FOR ENGINEERS ALGEBRA TUTORIAL 8 MATHEMATICAL PROGRESSIONS AND SERIES O completio of this ttoial yo shold be able to do the followig. Eplai aithmetical ad geometic pogessios. Eplai factoial otatio
More informationInference for Stochastic Processes 4. Lévy Processes. Duke University ISDS, USA. Poisson Process. Limits of Simple Compound Poisson Processes
Poisso Process Iferece for Sochasic Processes 4. Lévy Processes τ = δ j, δ j iid Ex X sup { Z + : τ }, < By ober L. Wolper Duke Uiversiy ISDS, USA [X j+ X j ] id Po [ j+ j ],... < evised: Jue 8, 5 E[e
More informationLecture 8: Convergence of transformations and law of large numbers
Lecture 8: Covergece of trasformatios ad law of large umbers Trasformatio ad covergece Trasformatio is a importat tool i statistics. If X coverges to X i some sese, we ofte eed to check whether g(x ) coverges
More informationSections 3.1 and 3.4 Exponential Functions (Growth and Decay)
Secions 3.1 and 3.4 Eponenial Funcions (Gowh and Decay) Chape 3. Secions 1 and 4 Page 1 of 5 Wha Would You Rahe Have... $1million, o double you money evey day fo 31 days saing wih 1cen? Day Cens Day Cens
More informationExtremal problems for t-partite and t-colorable hypergraphs
Exemal poblems fo -paie and -coloable hypegaphs Dhuv Mubayi John Talbo June, 007 Absac Fix ineges and an -unifom hypegaph F. We pove ha he maximum numbe of edges in a -paie -unifom hypegaph on n veices
More informationA Statistical Integral of Bohner Type. on Banach Space
Applied Mathematical cieces, Vol. 6, 202, o. 38, 6857-6870 A tatistical Itegal of Bohe Type o Baach pace Aita Caushi aita_caushi@yahoo.com Ago Tato agtato@gmail.com Depatmet of Mathematics Polytechic Uivesity
More informationFinite q-identities related to well-known theorems of Euler and Gauss. Johann Cigler
Fiite -idetities elated to well-ow theoems of Eule ad Gauss Joha Cigle Faultät fü Mathemati Uivesität Wie A-9 Wie, Nodbegstaße 5 email: oha.cigle@uivie.ac.at Abstact We give geealizatios of a fiite vesio
More information