A Unified Formula for The nth Derivative and The nth Anti-Derivative of the Bessel Function of Real Orders
|
|
|
- Claire Gallagher
- 6 years ago
- Views:
Transcription
1 Aec Joul of Aled Mthetc d Stttc 5 Vol 3 No 3-4 Avlble ole t htt://ubceubco/j/3/3/3 Scece d Educto Publhg DOI:69/j A Ufed Foul fo The th Devtve d The th At-Devtve of the eel Fucto of Rel Ode Mhe M eghobl * Motel QC Cd *Coeodg utho: beghobl@glco Receved Al 5; Reved My 5; Acceted Jue 9 5 Abtct A colete oluto to the oble of fdg the th devtve d the th t-devtve of eleety d ecl fucto h bee gve It del wth the oble of fdg foul fo the th devtve d the th t-devtve of eleety d ecl fucto We do ot lt to be tege t c be el ube I geel the oluto gve though ufed foul te of the Fox H-fucto d the Meje G- fucto whch y ce c be lfed to le geel fucto Th tu ke the ft el ue of thee two ecl fucto the ltetue d how the eed of uch fucto I th tlk we would lke to eet the de o the eel fucto whch well kow ecl fucto Oe of the key ot th wok tht the och doe ot deed o tegto techque We dot the clcl defto fo geelzto of dffeetto d tegto Nely the th ode of dffeetto foud ccodg to the Re-Louvlle defto k d x k f ( x ( x t f ( t dt ( k k whee (k < < k d k dx The geelzed Cuchy -fold tegl doted fo the th ode of tegto Keywod: Fot Mcooft Wod Telte Style Iet Telte ( x f ( x ( x t f ( t dt > Cte Th Atcle: Mhe M eghobl A Ufed Foul fo The th Devtve d The th At- Devtve of the eel Fucto of Rel Ode Aec Joul of Aled Mthetc d Stttc vol 3 o (5: -4 do: 69/j ( Itoducto The otvto of th wok coe fo the e of ybolc coutto well the e of clcl d fctol clculu The de tht: Gve fucto f vble x c we o coute lgeb yte (CAS fd foul fo the th devtve the th tdevtve o both? Th ehce the owe of tegto d dffeetto of CAS I Mle the foul coeod to vokg the cod dff(f(xx$ fo the th devtve d t(f(x x$ fo the th t-devtve The ltte cod wok oly fo The coutto of fctol devtve d fctol tegl volve evlutg oe tye of o eleety tegl I fct tegto techque do ot wok uully Th h led of thkg bout dffeet oche to evlute the A ee of e h bee etblhed [3] to toduce dffeet oche to evlutg th tye of tegl Th wok cotuto of th ee I ddto to tht ecet book [4] by the utho The eel Fucto The eel fucto of ode ν (whee ν cott oluto of the eel dffeetl equto of ode ν ( ν d y dy x x x y ( dx dx The oluto of the bove dffeetl equto c be foud ug the Fobeu ee The oluto y(x whch eeet the eel fucto gve by the owe ee ( x k k ν ( x ( k k! ( k ν! Fo futhe edg bout the eel fucto we c efe the ede to [5] 3 Re-Louvlle Fctol Devtve Defto The ot wdely kow defto of the fctol devtve The Re-Louvlle defto (RLFD [678] It e eult of ufcto of the oto of tege-ode tegto d dffeetto The defto gve by
2 Aec Joul of Aled Mthetc d Stttc k q d x k q f ( x t f ( t dt ( k q k (3 dx whee k < q < k k q d f(x fucto wth wek gulty ove the tevl of tegto If f(x cotuou ove the tevl [x] the by lettg q k oe get f (k (x 3 The th Devtve of (x of Rel Ode We e teeted the fctol devtve (the th devtve of el ode of the fucto f x x (4 becue of t ltte ue Subttutg (4 (3 yeld d x α ( x ( x y ( y dy k dx ( α ( ( x k (5 The bove foul gve the th devtve of el ode of the fucto (x 4 The Geelzed Cuchy -fold Itegl Defto The geelzed Cuchy -fold tegl geelzto of the Cuchy -fold tegl of tege ode ( x f ( x ( x t f ( t dt ( Relxg the codto o the bove defto fo tege to el > doe ot ffect the covegece of the tegl f(x fucto wth wek gulty ove the tevl of tegto 4 The th At-Devtve of (x of Rel Ode Ag fo ou eed of the th t-devtve of the fucto (x we gve t th ecto It the evluto of the tegl x ( x t ( t dt ( Clcultg the tegl eult the deed foul ( ( x 5 The Mell Tfo (6 A tte of fct we eed the ell tfo of the eel fucto t wll be exled lte to fd the deed ufed foul So we toduce the Mell tfo d t vee though the followg two defto Fo oe dcuo of the Mell tfo; ee [9] Defto : The Mell tfo [9] of loclly tegble fucto f(x o ( defed by [ ] ( α < < β M f ; F x f x dx whee α d β e el cott The t α < ( < β kow the t of lytcty Defto : The vee Mell tfo [9] gve by f ( x c [ ; ] M f x d α c β π < < c whee α d β e the e oe the lt defto Ug the Mell tfo defto d evlutg the coeodg tegl exlct exeo fo the Mell tfo of the eel fucto c be foud ν ν G whee ( the g fucto d defed (7 t ( t e dt Re( > (8 6 The H-fucto The H-fucto vey geel fucto tht ecoe the ot of ecl fucto cludg the Meje G-fucto d the geelzed hyegeoetc fucto; ee [] Notto H ( A ( A ( b ( b ( A ( b z Hq z Hq z H z Defto 3 The H-fucto defed by the Mell- e tegl [9] H ( z h( z d π C whee h( gve by ( b j j ( j A j j j h( (9 q ( b j j j j ( j A j fo Fo the bove we eque tht ( A j d j e otve ube ( j d b j e colex ube uch tht ( ν ( λ Aj bk k j
3 Aec Joul of Aled Mthetc d Stttc ν λ ; j k ; Tht e the ole of (b k k fo k d ( j A j fo j do ot cocde ( The cotou C ete the ole eultg fo (b k k ( k fo thoe of ( j A j ( j 6 Extece Codto fo The H-fucto Fo extece codto of the H-fucto we efe the ede to [] 6 Poete of The H-fucto The followg e oe oete of the H-fucto tht e ueful fo ou uoe (ee [] Poety ( α ( α A A A α z H z H z b b Poety A ( A α H z Hq zα > ( b α b α Poety 3 ( ( A b H z H b A z 7 Meje G-fucto The G-fucto ecl ce of the H-fucto A lge ube of ecl fucto e ecl ce of th fucto I th ecto we gve oe defto of the fucto wthout y oof Fo detled dcuo of the G-fucto we efe the ede to [] Notto G z G z Gq ( z G( z b b b Thee e the tdd otto ued the ltetue I the followg defto ety oduct teeted uty d q The Meje G -fucto wth the ete d b b q defed Mell-e tye tegl follow [] Defto 4 g Gq z g ( z d b b L π ( bj ( j j ( bj ( j j q j j It cle tht the Meje G-fucto ecl ce of the H-fucto d t deved fo the ltte by lettg A j d j equl oe 7 The th Devtve d The th At- Devtve of The H-fucto The cloue oety of the H-fucto ude el ode of dffeetto d tegto ke t vey oweful tool fo fdg ufed foul fo the th devtve d the th t-devtve of eleety d ecl fucto I othe wod oe c exe el ode devtve d tegl of H-fucto te of ew H-fucto Th vey ce oety of the H- fucto whch ot oeed by othe ecl fucto The followg le llutte the de Le The foul ( ( H z ( z H q ( A A ( ( b ( gve ( devtve of bty ode f > ( tegl of bty ode f < of the H-fucto H ( A ( b z z ( Poof: We gve oof fo t ( The oof fo t ( l to t( d oly oe eed to ue foul (6 fo fctol ode tegto Recllg the defto of the H-fucto (9 H z h z d π C whee h( gve by (9 Oe c dffeette both de of the bove equto ovded the bove tegl covege ufoly fo oe z ug the foul gve whee ( ( d x dx x ( ( H z h z d π C ( ( h h h( gve by equto (9 Ug the otto of the H- fucto the bove c be wtte ( ( A ( ( b ( H z z x H q z
4 Aec Joul of Aled Mthetc d Stttc 3 Poety ( of the H-fucto lfe the lt equto to oe coct fo ( ( A A ( ( b ( H z z H q z If C tke th ( the extece codto fo the bove H-fucto q π g ( z < Aj j Aj j j j j j Fo Pt( oe eed foul (6 ( ( whch deved fo the geelzed Cuchy foul fo the -fold tegl The et of the oof exctly l to t ( Foul ( vey ott fo fdg tege d bty ode ybolc devtve d tegl of both eleety d ecl fucto log they e eeetble te of the H-fucto 8 A Ufed Foul Fo The th Devtve d The th t-devtve of The eel Fucto of Rel Ode Th ecto devoted to toduce the colete oluto to the oble of ybolc dffeetto d tegto of the eel fucto of el ode We ly dow the theoe d t oof The oof gve te tht ke t cle d to the ot Theoe Let f ( x J ( x b ν be eel fucto of ode ν whee b R the the foul ( ( ( J ν x b ( ν H 3 ν / ( gve ( Devtve of y ode f > (b At-Devtve of y ode f < (c The ogl fucto f Poof: The ft te ou oof eque tkg the Mell tfo of ou eel fucto whch we hve ledy foud ecto (5 d t gve by equto (7 ν ν G ( Reeetg the eel fucto te of t vee Mell tfo gve ν ( ( x b d π C ν whee C utble cotou Relcg by the lt equto yeld ν ( ( x b d π C ν wth tkg codeto the chge the cotou C Ug the H-fucto otto we get the H-fucto eeetto of ou eel fucto ( H ν ν y exlotg oety ( of the H-fucto we c lfy the bove H-fucto to ( H ν ν / Alyg le ( fo el ode devtve d t-devtve to the bove fucto gve ufed foul fo the th devtve d the th t-devtve of the eel fucto ( ( ( x b x b ( H3 ν ν / ( (3 A futhe lfcto c be de to ou ufed foul by elg to oety ( ( ( ( J ν x b ( ν H 3 ν / λ (4
5 4 Aec Joul of Aled Mthetc d Stttc A extece codto fo the ufed foul c be foud d t gve by (x b Exle : A ufed foul fo the th devtve d the th t-devtve fo the eel fucto ( 3/ x b c be foud by ubttutg b b d d 3/ the ufed foul ( Slfyg the eultg H-fucto to le geel fucto ely the Meje G-fucto yeld the ufed foul ( ( ( 3/ J x b ν bx d (5 4 3 bg 35 ν ν ( bx d 3 The bove G-fucto educe to the ogl fucto f They gve devtve of y ode f > d tdevtve of y ode f < 9 Cocluo The cheveet th wok c be uzed the followg tteet A colete oluto to the oble of fdg the th devtve d the th t-devtve of the eel fucto of el ode obted Th c be codeed bg bekthough the e of clcl d fctol clculu well the e of ybolc coutto The dffculty h bee the evluto of o eleety tegl The gol h bee eched wthout elg to tegto techque The Fox H-fucto h bee toduced tool to gve colete oluto to the oble codeto Aothe ott fct thee ufed foul gve ufcto fo the two clcl defto of geelzed dffeetto d tegto I the e of ybolc coutto thee foul ehce the owe of coute lgeb yte ce they e lug foul CAS uch Mle d Mthetc hve ledy the Meje G-fucto leeted O the othe hd the Fox H-fucto h ot bee leeted yet Refeece [] Mhe M eghobl A ufed foul fo bty ode ybolc devtve d tegl of the owe-exoetl cl Itetol Joul of Pue d Aled Mthetc 37(3: [] Mhe M eghobl Ufed foul fo tege d fctol ode ybolc devtve d tegl of the owevee tgooetc cl I Itetol Joul of Pue d Aled Mthetc 4(: [3] Mhe M eghobl A ufed foul fo the th devtve d the th-t devtve of the owe-logthc cl Poceedg of the Itetol Cofeece of Coutg Egeeg Scece d Ifoto Clfo Stte Uvety Fulleto ClfoIEEE ge [4] Mhe eghobl Fctol Dffeetl Equto & Sybolc Devtve d Itegl Schol Pe Gey 4 [5] M Abowtz d IA Stegu Hdbook of Mthetcl Fucto wth Foul Gh d Mthetcl Tble Dove 97 [6] Keth Oldh d Jeoe Se The Fctol Clculu Acdc Pe 974 [7] SE Sko AA Klb d OI Mchev Fctol Itegl d Devtve theoy d lcto Godo d ech Scece Publhe 996 [8] Igo Podluby Fctol Dffeetl Equto Acdec Pe S Dego Clfo 999 [9] R P d D Kk Aytotc d Mell-e Itegl Cbdge [] Chle Fox The G d H-fucto yetcl foue keel T Ae Mth Soc 98 ge [] AM Mth d RK Sxe The H-fucto wth Alcto Stttc d Othe Dcle Joh Wley d So 978 [] AM Mth A Hdbook of Geelzed Secl Fucto fo Stttcl d Phycl Scece Oxfod Scece Publcto 993
Certain Expansion Formulae Involving a Basic Analogue of Fox s H-Function
vlle t htt:vu.edu l. l. Mth. ISSN: 93-9466 Vol. 3 Iue Jue 8. 8 36 Pevouly Vol. 3 No. lcto d led Mthetc: Itetol Joul M Cet Exo Foule Ivolvg c logue o Fox -Fucto S.. Puoht etet o c-scece Mthetc College o
The shifted Jacobi polynomial integral operational matrix for solving Riccati differential equation of fractional order
Avlble t htt://vuedu/ Al Al Mth SSN: 9-966 Vol ue Decebe 5 878-89 Alcto d Aled Mthetc: A tetol oul AAM he hted cob olyol tegl oetol t o olvg Rcct deetl euto o ctol ode A Nety B Aghel d R D Detet o Mthetc
Studying the Problems of Multiple Integrals with Maple Chii-Huei Yu
Itetol Joul of Resech (IJR) e-issn: 2348-6848, - ISSN: 2348-795X Volume 3, Issue 5, Mch 26 Avlble t htt://tetoljoulofesechog Studyg the Poblems of Multle Itegls wth Mle Ch-Hue Yu Detmet of Ifomto Techology,
Fredholm Type Integral Equations with Aleph-Function. and General Polynomials
Iteto Mthetc Fou Vo. 8 3 o. 989-999 HIKI Ltd.-h.co Fedho Te Iteg uto th eh-fucto d Gee Poo u J K.J. o Ittute o Mgeet tude & eech Mu Id u5@g.co Kt e K.J. o Ittute o Mgeet tude & eech Mu Id dehuh_3@hoo.co
NONDIFFERENTIABLE MATHEMATICAL PROGRAMS. OPTIMALITY AND HIGHER-ORDER DUALITY RESULTS
HE PUBLISHING HOUSE PROCEEDINGS OF HE ROMANIAN ACADEMY, See A, OF HE ROMANIAN ACADEMY Volue 9, Nube 3/8,. NONDIFFERENIABLE MAHEMAICAL PROGRAMS. OPIMALIY AND HIGHER-ORDER DUALIY RESULS Vale PREDA Uvety
Chapter Linear Regression
Chpte 6.3 Le Regesso Afte edg ths chpte, ou should be ble to. defe egesso,. use sevel mmzg of esdul cte to choose the ght cteo, 3. deve the costts of le egesso model bsed o lest sques method cteo,. use
Moments of Generalized Order Statistics from a General Class of Distributions
ISSN 684-843 Jol of Sttt Vole 5 28. 36-43 Moet of Geelzed Ode Sttt fo Geel l of Dtto Att Mhd Fz d Hee Ath Ode ttt eod le d eel othe odel of odeed do le e ewed el e of geelzed ode ttt go K 995. I th e exlt
University of Pavia, Pavia, Italy. North Andover MA 01845, USA
Iteatoal Joual of Optmzato: heoy, Method ad Applcato 27-5565(Pt) 27-6839(Ole) wwwgph/otma 29 Global Ifomato Publhe (HK) Co, Ltd 29, Vol, No 2, 55-59 η -Peudoleaty ad Effcecy Gogo Gog, Noma G Rueda 2 *
ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY)
ANOTHER INTEGER NUMBER ALGORITHM TO SOLVE LINEAR EQUATIONS (USING CONGRUENCY) Floet Smdche, Ph D Aocte Pofeo Ch of Deptmet of Mth & Scece Uvety of New Mexco 2 College Rod Gllup, NM 873, USA E-ml: md@um.edu
SUBSEQUENCE CHARACTERIZAT ION OF UNIFORM STATISTICAL CONVERGENCE OF DOUBLE SEQUENCE
Reseach ad Coucatos atheatcs ad atheatcal ceces Vol 9 Issue 7 Pages 37-5 IN 39-6939 Publshed Ole o Novebe 9 7 7 Jyot cadec Pess htt//yotacadecessog UBEQUENCE CHRCTERIZT ION OF UNIFOR TTITIC CONVERGENCE
Difference Sets of Null Density Subsets of
dvces Pue Mthetcs 95-99 http://ddoog/436/p37 Pulshed Ole M (http://wwwscrpog/oul/p) Dffeece Sets of Null Dest Susets of Dwoud hd Dsted M Hosse Deptet of Mthetcs Uvest of Gul Rsht I El: hd@gulc h@googlelco
Chapter #2 EEE State Space Analysis and Controller Design
Chpte EEE8- Chpte # EEE8- Stte Spce Al d Cotolle Deg Itodcto to tte pce Obevblt/Cotollblt Modle ede: D D Go - d.go@cl.c.k /4 Chpte EEE8-. Itodcto Ae tht we hve th ode te: f, ', '',.... Ve dffclt to td
2. Elementary Linear Algebra Problems
. Eleety e lge Pole. BS: B e lge Suoute (Pog pge wth PCK) Su of veto opoet:. Coputto y f- poe: () () () (3) N 3 4 5 3 6 4 7 8 Full y tee Depth te tep log()n Veto updte the f- poe wth N : ) ( ) ( ) ( )
Chapter 17. Least Square Regression
The Islmc Uvest of Gz Fcult of Egeeg Cvl Egeeg Deptmet Numecl Alss ECIV 336 Chpte 7 Lest que Regesso Assocte Pof. Mze Abultef Cvl Egeeg Deptmet, The Islmc Uvest of Gz Pt 5 - CURVE FITTING Descbes techques
AN ALGEBRAIC APPROACH TO M-BAND WAVELETS CONSTRUCTION
AN ALGEBRAIC APPROACH TO -BAN WAELETS CONSTRUCTION Toy L Qy S Pewe Ho Ntol Lotoy o e Peeto Pe Uety Be 8 P. R. C Att T e eet le o to ott - otool welet e. A yte of ott eto ote fo - otool flte te olto e o
SGN Audio and Speech Processing. Linear Prediction
SGN-46 Audo d Seech Poceg Le Pedcto Slde fo th lectue e bed o thoe ceted by Kt Mhoe fo TUT coue Puheeättely meetelmät Sg 3. Othe ouce: K. Koe: Puheeättely meetelmät, lectue mtel, TUT, htt://www.c.tut.f/coue/sgn-4/g4.df
On Almost Increasing Sequences For Generalized Absolute Summability
Joul of Applied Mthetic & Bioifotic, ol., o., 0, 43-50 ISSN: 79-660 (pit), 79-6939 (olie) Itetiol Scietific Pe, 0 O Alot Iceig Sequece Fo Geelized Abolute Subility W.. Suli Abtct A geel eult coceig bolute
African Journal of Science and Technology (AJST) Science and Engineering Series Vol. 4, No. 2, pp GENERALISED DELETION DESIGNS
Af Joul of See Tehology (AJST) See Egeeg See Vol. 4, No.,. 7-79 GENERALISED DELETION DESIGNS Mhel Ku Gh Joh Wylff Ohbo Dee of Mhe, Uvey of Nob, P. O. Bo 3097, Nob, Key ABSTRACT:- I h e yel gle ele fol
Journal of Engineering and Natural Sciences Mühendislik ve Fen Bilimleri Dergisi SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A WEYL SPACE
Jou of Eee d Ntu Scece Mühed e Fe Be De S 5/4 SOME PROPERTIES CONCERNING THE HYPERSURFACES OF A EYL SPACE N KOFOĞLU M S Güze St Üete, Fe-Edeyt Füte, Mtet Böüü, Beştş-İSTANBUL Geş/Receed:..4 Ku/Accepted:
E-Companion: Mathematical Proofs
E-omnon: Mthemtcl Poo Poo o emm : Pt DS Sytem y denton o t ey to vey tht t ncee n wth d ncee n We dene } ] : [ { M whee / We let the ttegy et o ech etle n DS e ]} [ ] [ : { M w whee M lge otve nume oth
Professor Wei Zhu. 1. Sampling from the Normal Population
AMS570 Pofesso We Zhu. Samplg fom the Nomal Populato *Example: We wsh to estmate the dstbuto of heghts of adult US male. It s beleved that the heght of adult US male follows a omal dstbuto N(, ) Def. Smple
ECONOMETRIC ANALYSIS ON EFFICIENCY OF ESTIMATOR ABSTRACT
ECOOMETRIC LYSIS O EFFICIECY OF ESTIMTOR M. Khohev, Lectue, Gffth Uvet, School of ccoutg d Fce, utl F. K, tt Pofeo, Mchuett Ittute of Techolog, Deptet of Mechcl Egeeg, US; cuetl t Shf Uvet, I. Houl P.
Chapter 2: Descriptive Statistics
Chapte : Decptve Stattc Peequte: Chapte. Revew of Uvaate Stattc The cetal teecy of a oe o le yetc tbuto of a et of teval, o hghe, cale coe, ofte uaze by the athetc ea, whch efe a We ca ue the ea to ceate
Describes techniques to fit curves (curve fitting) to discrete data to obtain intermediate estimates.
CURVE FITTING Descbes techques to ft cuves (cuve fttg) to dscete dt to obt temedte estmtes. Thee e two geel ppoches fo cuve fttg: Regesso: Dt ehbt sgfct degee of sctte. The stteg s to deve sgle cuve tht
10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
On Eigenvalues of Nonlinear Operator Pencils with Many Parameters
Ope Scece Joual of Matheatc ad Applcato 5; 3(4): 96- Publhed ole Jue 5 (http://wwwopececeoleco/oual/oa) O Egevalue of Nolea Opeato Pecl wth May Paaete Rakhhada Dhabaadeh Guay Salaova Depatet of Fuctoal
2.Decision Theory of Dependence
.Deciio Theoy of Depedece Theoy :I et of vecto if thee i uet which i liely depedet the whole et i liely depedet too. Coolly :If the et i liely idepedet y oepty uet of it i liely idepedet. Theoy : Give
Born-Oppenheimer Approximation. Kaito Takahashi
o-oppehee ppoato Kato Takahah toc Ut Fo quatu yte uch a ecto ad olecule t eae to ue ut that ft the=tomc UNT Ue a of ecto (ot kg) Ue chage of ecto (ot coulob) Ue hba fo agula oetu (ot kg - ) Ue 4pe 0 fo
The use of linear parametric approximation in numerical solving of nonlinear non-smooth Fuzzy equations
vlle ole t www.choleechl.co chve o ppled Scece Reech 5 6:49-6 http://choleechl.co/chve.htl ISSN 975-58X CODEN US SRC9 The ue o le petc ppoto uecl olvg o ole o-ooth uzz equto Mjd Hllj e Ze d l Vhd Kd Nehu
are positive, and the pair A, B is controllable. The uncertainty in is introduced to model control failures.
Lectue 4 8. MRAC Desg fo Affe--Cotol MIMO Systes I ths secto, we cosde MRAC desg fo a class of ult-ut-ult-outut (MIMO) olea systes, whose lat dyacs ae lealy aaetezed, the ucetates satsfy the so-called
3. REVIEW OF PROPERTIES OF EIGENVALUES AND EIGENVECTORS
. REVIEW OF PROPERTIES OF EIGENVLUES ND EIGENVECTORS. EIGENVLUES ND EIGENVECTORS We hll ow revew ome bc fct from mtr theory. Let be mtr. clr clled egevlue of f there et ozero vector uch tht Emle: Let 9
Spectral Problems of Two-Parameter System of Operators
Pue ad Appled Matheatc Joual 5; 4(4-: 33-37 Publhed ole Augut, 5 (http://wwwcecepublhggoupco//pa do: 648/pa5447 ISSN: 36-979 (Pt; ISSN: 36-98 (Ole Spectal Poble of Two-Paaete Syte of Opeato Rahhada Dhabaadeh
On The Circulant K Fibonacci Matrices
IOSR Jou of Mthetcs (IOSR-JM) e-issn: 78-578 p-issn: 39-765X. Voue 3 Issue Ve. II (M. - Ap. 07) PP 38-4 www.osous.og O he Ccut K bocc Mtces Sego co (Deptet of Mthetcs Uvesty of Ls Ps de G C Sp) Abstct:
Partition and the Perfect Codes in the Additive Channel
Oe Joual of Dcete Matheatc 3 3 - htt://dxdoog/36/od333 Publhed Ole July 3 (htt://wwwcog/oual/od) Patto ad the Pefect ode the Addtve hael Gab Movya BI Gou Mocow Rua Eal: gab@fbtu Receved Mach 33; eved May
Linear Open Loop Systems
Colordo School of Me CHEN43 Trfer Fucto Ler Ope Loop Sytem Ler Ope Loop Sytem... Trfer Fucto for Smple Proce... Exmple Trfer Fucto Mercury Thermometer... 2 Derblty of Devto Vrble... 3 Trfer Fucto for Proce
Exponential Generating Functions - J. T. Butler
Epoetal Geeatg Fuctos - J. T. Butle Epoetal Geeatg Fuctos Geeatg fuctos fo pemutatos. Defto: a +a +a 2 2 + a + s the oday geeatg fucto fo the sequece of teges (a, a, a 2, a, ). Ep. Ge. Fuc.- J. T. Butle
= y and Normed Linear Spaces
304-50 LINER SYSTEMS Lectue 8: Solutos to = ad Nomed Lea Spaces 73 Fdg N To fd N, we eed to chaacteze all solutos to = 0 Recall that ow opeatos peseve N, so that = 0 = 0 We ca solve = 0 ecusvel backwads
Some Integrals Pertaining Biorthogonal Polynomials and Certain Product of Special Functions
Global Joual o Scece Fote Reeach atheatc ad Deco Scece Volue Iue Veo Te : Double Bld ee Reewed Iteatoal Reeach Joual ublhe: Global Joual Ic SA Ole ISSN: 49-466 & t ISSN: 975-5896 Soe Itegal etag Bothogoal
Theory of Finsler spaces with ( λβ, ) Metric
Theoy of Fsle sces wth ( λβ ) Metc Dhed Thu Kll Multle us Thuv Uvesty Kll DhdhNel E-l: dhedthuc@lco ABTRAT The of ths e s to toduce d study the cocet of ( ) theoes hve ee woout fo ( ) etc whee (x)y s oe
Super-Mixed Multiple Attribute Group Decision Making Method Based on Hybrid Fuzzy Grey Relation Approach Degree *
Supe-Med Multple Attbute Goup Decso Mkg Method Bsed o Hybd Fuzzy Gey Relto Appoch Degee Gol K Fe Ye b Cete of Ntul Scece vesty of Sceces Pyogyg DPR Koe b School of Busess Adstto South Ch vesty of Techology
THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION
Advce d Alcto Stttc Volue Nue Pge 5-55 Th e vlle ole t htt://hco/oul/dht Puh Pulhg Houe THE WEIBULL NEGATIVE BINOMIAL DISTRIBUTION CRISTIANE RODRIGUES GAUSS M CORDEIRO CLARICE G B DEMÉTRIO d EDWIN M M
SOME REMARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOMIAL ASYMPTOTE
D I D A C T I C S O F A T H E A T I C S No (4) 3 SOE REARKS ON HORIZONTAL, SLANT, PARABOLIC AND POLYNOIAL ASYPTOTE Tdeusz Jszk Abstct I the techg o clculus, we cosde hozotl d slt symptote I ths ppe the
such that for 1 From the definition of the k-fibonacci numbers, the firsts of them are presented in Table 1. Table 1: First k-fibonacci numbers F 1
Scholas Joual of Egeeg ad Techology (SJET) Sch. J. Eg. Tech. 0; (C):669-67 Scholas Academc ad Scetfc Publshe (A Iteatoal Publshe fo Academc ad Scetfc Resouces) www.saspublshe.com ISSN -X (Ole) ISSN 7-9
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge og to the ut whh the e fst toue. Thus te sttg ut m e eque to use the fomule tht wee toue peeg ut e.g. tes sttg C mght e epete to use fomule
SOLVING SYSTEMS OF EQUATIONS, DIRECT METHODS
ELM Numecl Alyss D Muhem Mecmek SOLVING SYSTEMS OF EQUATIONS DIRECT METHODS ELM Numecl Alyss Some of the cotets e dopted fom Luee V. Fusett Appled Numecl Alyss usg MATLAB. Petce Hll Ic. 999 ELM Numecl
Maximum likelihood estimate of phylogeny. BIOL 495S/ CS 490B/ MATH 490B/ STAT 490B Introduction to Bioinformatics April 24, 2002
Mmm lkelhood eme of phylogey BIO 9S/ S 90B/ MH 90B/ S 90B Iodco o Bofomc pl 00 Ovevew of he pobblc ppoch o phylogey o k ee ccodg o he lkelhood d ee whee d e e of eqece d ee by ee wh leve fo he eqece. he
A Study on New Sequence of Functions Involving the Generalized Contour Integral
Globl Jourl of Scece Froter Reerch Mthetc d Deco Scece Volue 3 Iue Vero. Yer 23 Type : Double Bld Peer Revewed Itertol Reerch Jourl Publher: Globl Jourl Ic. (USA Ole ISS: 2249-4626 & Prt ISS: 975-5896
this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]
![this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]](/thumbs/78/76749440.jpg "this is the indefinite integral Since integration is the reverse of differentiation we can check the previous by [ ]")
Atervtves The Itegrl Atervtves Ojectve: Use efte tegrl otto for tervtves. Use sc tegrto rules to f tervtves. Aother mportt questo clculus s gve ervtve f the fucto tht t cme from. Ths s the process kow
On EPr Bimatrices II. ON EP BIMATRICES A1 A Hence x. is said to be EP if it satisfies the condition ABx
Iteatoal Joual of Mathematcs ad Statstcs Iveto (IJMSI) E-ISSN: 3 4767 P-ISSN: 3-4759 www.jms.og Volume Issue 5 May. 4 PP-44-5 O EP matces.ramesh, N.baas ssocate Pofesso of Mathematcs, ovt. ts College(utoomous),Kumbakoam.
Fractional Integrals Involving Generalized Polynomials And Multivariable Function
IOSR Joual of ateatcs (IOSRJ) ISSN: 78-578 Volue, Issue 5 (Jul-Aug 0), PP 05- wwwosoualsog Factoal Itegals Ivolvg Geealzed Poloals Ad ultvaable Fucto D Neela Pade ad Resa Ka Deatet of ateatcs APS uvest
Fairing of Parametric Quintic Splines
ISSN 46-69 Eglad UK Joual of Ifomato ad omputg Scece Vol No 6 pp -8 Fag of Paametc Qutc Sples Yuau Wag Shagha Isttute of Spots Shagha 48 ha School of Mathematcal Scece Fuda Uvesty Shagha 4 ha { P t )}
FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL SEQUENCES
Joual of Appled Matheatcs ad Coputatoal Mechacs 7, 6(), 59-7 www.ac.pcz.pl p-issn 99-9965 DOI:.75/jac.7..3 e-issn 353-588 FIBONACCI-LIKE SEQUENCE ASSOCIATED WITH K-PELL, K-PELL-LUCAS AND MODIFIED K-PELL
GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS
GCE AS/A Level MATHEMATICS GCE AS/A Level FURTHER MATHEMATICS FORMULA BOOKLET Fom Septembe 07 Issued 07 Mesuto Pue Mthemtcs Sufce e of sphee = 4 Ae of cuved sufce of coe = slt heght Athmetc Sees S l d
Transmuted Generalized Lindley Distribution
Itetol Joul of Memtcs Teds d Techology- olume9 Numbe Juy 06 Tsmuted Geelzed Ldley Dstbuto M. Elghy, M.Rshed d A.W.Shwk 3, Buydh colleges, Deptmet of Memtcl Sttstcs, KSA.,, 3 Isttute of Sttstcl Studes d
The formulae in this booklet have been arranged according to the unit in which they are first
Fomule Booklet Fomule Booklet The fomule ths ooklet hve ee ge ccog to the ut whch the e fst touce. Thus cte sttg ut m e eque to use the fomule tht wee touce peceg ut e.g. ctes sttg C mght e epecte to use
Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]
![Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]](/thumbs/83/87614710.jpg "Area and the Definite Integral. Area under Curve. The Partition. y f (x) We want to find the area under f (x) on [ a, b ]")
Are d the Defte Itegrl 1 Are uder Curve We wt to fd the re uder f (x) o [, ] y f (x) x The Prtto We eg y prttog the tervl [, ] to smller su-tervls x 0 x 1 x x - x -1 x 1 The Bsc Ide We the crete rectgles
Occurrences of ordered patterns in rectangular space filling curve through homomorphism
AUSTRALIAN JOURNAL OF BASI AND APPLIED SIENES ISSN:99- EISSN: 9- Jo hoe ge: www.bweb.co Occece o odeed tte ectg ce g cve thogh hoooh K.Nveeth K. Thg S. Jeybhth Po Nt Reech cho Ph.D-B JUL-69 Detet o Mthetc
ROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K
ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu
Week 10: DTMC Applications Ranking Web Pages & Slotted ALOHA. Network Performance 10-1
Week : DTMC Alictions Rnking Web ges & Slotted ALOHA etwok efonce - Outline Aly the theoy of discete tie Mkov chins: Google s nking of web-ges Wht ge is the use ost likely seching fo? Foulte web-gh s Mkov
Available online through
Avlble ole through wwwmfo FIXED POINTS FOR NON-SELF MAPPINGS ON CONEX ECTOR METRIC SPACES Susht Kumr Moht* Deprtmet of Mthemtcs West Begl Stte Uverst Brst 4 PrgsNorth) Kolt 76 West Begl Id E-ml: smwbes@yhoo
Control of industrial robots. Robot dynamics
Coto of dut oot Root dy of. oo Roo (oo.oo@o.t) oteo d Mo Dteto d Eetto, Ifozoe e Bogege Itoduto Wth thee de we w deve the dy ode of the uto he dy ode out fo the eto etwee the oue of oto (foe d oet) d the
( m is the length of columns of A ) spanned by the columns of A : . Select those columns of B that contain a pivot; say those are Bi
Assgmet /MATH 47/Wte Due: Thusday Jauay The poblems to solve ae umbeed [] to [] below Fst some explaatoy otes Fdg a bass of the colum-space of a max ad povg that the colum ak (dmeso of the colum space)
Adjacent Vertex Distinguishing Edge Colouring of Cactus Graphs
ISSN: 77-374 ISO 9:8 etfed Itetol Joul of Egeeg d Iote Techology (IJEIT) Volue 3 Issue 4 Octobe 3 Adcet Vetex Dstgushg Edge oloug of ctus Ghs Nsee Kh Mdhugl Pl Detet of Mthetcs Globl Isttute of Mgeet d
Analysis of torque cam mechanism
Aly o toque c ech Iy He,*, Pet Hej, Šte Mch, Mt Svood, Joe Souku culty o Puducto Techology d Mgeet J Evgelt Pukyě Uvety Útí d Le, Detet o Mche d Mechc, 400 96, Puteov, Útí /L, Czech Reulc Atct: The tcle
Application of Higher Order Derivatives of Lyapunov Functions in Stability Analysis of Nonlinear Homogeneous Systems
Poceedgs of the Itetol MultCofeece of Egees d Comute Scetsts 009 Vol II IMECS 009, Mch 8-0, 009, Hog Kog Alcto of Hghe Ode Detes of Lyuo Fuctos Stblty Alyss of Nole Homogeeous Systems Vhd Megol, d S. K.
From the Maxwell s equations to the String Theory: new possible mathematical connections
o the Mwell equto to the Stg Theoy: ew oble thetcl coecto Mchele Ndell,, Roo Tuco Dteto d Sceze dell Te Uvetà degl Stud d Nol edeco, Lgo S. Mcello, 88 Nol, tly Dteto d Mtetc ed lczo R. Cccool Uvetà degl
Generalisation on the Zeros of a Family of Complex Polynomials
Ieol Joul of hemcs esech. ISSN 976-584 Volume 6 Numbe 4. 93-97 Ieol esech Publco House h://www.house.com Geelso o he Zeos of Fmly of Comlex Polyomls Aee sgh Neh d S.K.Shu Deme of hemcs Lgys Uvesy Fdbd-
Distribution of Geometrically Weighted Sum of Bernoulli Random Variables
Appled Mathematc, 0,, 8-86 do:046/am095 Publhed Ole Novembe 0 (http://wwwscrpog/oual/am) Dtbuto of Geometcally Weghted Sum of Beoull Radom Vaable Abtact Deepeh Bhat, Phazamle Kgo, Ragaath Naayaachaya Ratthall
The Geometric Proof of the Hecke Conjecture
The Geometc Poof of the Hecke Cojectue Kada Sh Depatmet of Mathematc Zhejag Ocea Uvety Zhouha Cty 6 Zhejag Povce Cha Atact Begg fom the eoluto of Dchlet fucto ug the e poduct fomula of two fte-dmeoal vecto
A Deterministic Model for Channel Capacity with Utility
CAPTER 6 A Detestc Model fo Chel Cct wth tlt 6. todcto Chel cct s tl oeto ssocted wth elble cocto d defed s the hghest te t whch foto c be set ove the chel wth btl sll obblt of eo. Chel codg theoes d the
The Linear Probability Density Function of Continuous Random Variables in the Real Number Field and Its Existence Proof
MATEC Web of Cofeeces ICIEA 06 600 (06) DOI: 0.05/mateccof/0668600 The ea Pobablty Desty Fucto of Cotuous Radom Vaables the Real Numbe Feld ad Its Estece Poof Yya Che ad Ye Collee of Softwae, Taj Uvesty,
Sequences and summations
Lecture 0 Sequeces d summtos Istructor: Kgl Km CSE) E-ml: kkm0@kokuk.c.kr Tel. : 0-0-9 Room : New Mleum Bldg. 0 Lb : New Egeerg Bldg. 0 All sldes re bsed o CS Dscrete Mthemtcs for Computer Scece course
On Stability of a Class of Fractional Differential Equations
Proceedgs of the Pkst Acdey of Sceces 49 (: 39-43 (0 Coyrght Pkst Acdey of Sceces SSN: 0377-969 Pkst Acdey of Sceces Orgl Artcle O Stblty of Clss of Frctol Dfferetl Equtos Rbh W. brh* sttute of Mthetcl
International Journal of Mathematics Trends and Technology (IJMTT) Volume 47 Number 1 July 2017
Iteatioal Joual of Matheatics Teds ad Techology (IJMTT) Volue 47 Nube July 07 Coe Metic Saces, Coe Rectagula Metic Saces ad Coo Fixed Poit Theoes M. Sivastava; S.C. Ghosh Deatet of Matheatics, D.A.V. College
The Solution of Heat Conduction Equation with Mixed Boundary Conditions
Joul of Mhec d Sc (: 346-35, 6 ISSN 549-3644 6 Scece Publco The Soluo of He Coduco Equo wh Mxed Bou Codo Ne Abdelzq Dee of Bc d Aled Scece, Tfl Techcl Uvey PO Box 79, Tfl, Jod Abc: The u devoed o deee
RECAPITULATION & CONDITIONAL PROBABILITY. Number of favourable events n E Total number of elementary events n S
Fomulae Fo u Pobablty By OP Gupta [Ida Awad We, +91-9650 350 480] Impotat Tems, Deftos & Fomulae 01 Bascs Of Pobablty: Let S ad E be the sample space ad a evet a expemet espectvely Numbe of favouable evets
A convex hull characterization
Pue d ppled Mthets Joul 4; (: 4-48 Pulshed ole My 4 (http://www.seepulshggoup.o//p do:.648/.p.4. ove hull htezto Fo Fesh Gov Qut Deptet DISG Uvesty of Se Itly El ddess: fesh@us.t (F. Fesh qut@us.t (G.
X-Ray Notes, Part III
oll 6 X-y oe 3: Pe X-Ry oe, P III oe Deeo Coe oupu o x-y ye h look lke h: We efe ue of que lhly ffee efo h ue y ovk: Co: C ΔS S Sl o oe Ro: SR S Co o oe Ro: CR ΔS C SR Pevouly, we ee he SR fo ye hv pxel
Lecture 9-3/8/10-14 Spatial Description and Transformation
Letue 9-8- tl Deton nd nfomton Homewo No. Due 9. Fme ngement onl. Do not lulte...8..7.8 Otonl et edt hot oof tht = - Homewo No. egned due 9 tud eton.-.. olve oblem:.....7.8. ee lde 6 7. e Mtlb on. f oble.
ASYMPTOTICS OF THE GENERALIZED STATISTICS FOR TESTING THE HYPOTHESIS UNDER RANDOM CENSORING
IJRRAS 3 () Novembe www.apape.com/volume/vol3iue/ijrras_3.pdf ASYMPOICS OF HE GENERALIZE SAISICS FOR ESING HE HYPOHESIS UNER RANOM CENSORING A.A. Abduhukuov & N.S. Numuhamedova Natoal Uvety of Uzbekta
A METHOD FOR THE RAPID NUMERICAL CALCULATION OF PARTIAL SUMS OF GENERALIZED HARMONICAL SERIES WITH PRESCRIBED ACCURACY
UPB c Bull, eres D, Vol 8, No, 00 A METHOD FOR THE RAPD NUMERAL ALULATON OF PARTAL UM OF GENERALZED HARMONAL ERE WTH PRERBED AURAY BERBENTE e roue o etodă ouă etru clculul rd l suelor rţle le serlor roce
Introduction to mathematical Statistics
Itroducto to mthemtcl ttstcs Fl oluto. A grou of bbes ll of whom weghed romtely the sme t brth re rdomly dvded to two grous. The bbes smle were fed formul A; those smle were fed formul B. The weght gs
In Calculus I you learned an approximation method using a Riemann sum. Recall that the Riemann sum is
Mth Sprg 08 L Approxmtg Dete Itegrls I Itroducto We hve studed severl methods tht llow us to d the exct vlues o dete tegrls However, there re some cses whch t s ot possle to evlute dete tegrl exctly I
ON NILPOTENCY IN NONASSOCIATIVE ALGEBRAS
Jourl of Algebr Nuber Theory: Advces d Applctos Volue 6 Nuber 6 ges 85- Avlble t http://scetfcdvces.co. DOI: http://dx.do.org/.864/t_779 ON NILOTENCY IN NONASSOCIATIVE ALGERAS C. J. A. ÉRÉ M. F. OUEDRAOGO
under the curve in the first quadrant.
NOTES 5: INTEGRALS Nme: Dte: Perod: LESSON 5. AREAS AND DISTANCES Are uder the curve Are uder f( ), ove the -s, o the dom., Prctce Prolems:. f ( ). Fd the re uder the fucto, ove the - s, etwee,.. f ( )
Generating Function for Partitions with Parts in A.P
Geetig Fuctio fo Ptitio wi Pt i AP Hum Reddy K # K Jkmm * # Detmet of Memtic Hidu Coege Gutu 50 AP Idi * Detmet of Memtic 8 Mi AECS Lyout B BLOCK Sigd Bgoe 5604 Idi Abtct: I i e we deive e geetig fuctio
ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE
O The Covegece Theoems... (Muslm Aso) ON THE CONVERGENCE THEOREMS OF THE McSHANE INTEGRAL FOR RIESZ-SPACES-VALUED FUNCTIONS DEFINED ON REAL LINE Muslm Aso, Yosephus D. Sumato, Nov Rustaa Dew 3 ) Mathematcs
Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013
Mth 4318 : Rel Anlysis II Mid-Tem Exm 1 14 Febuy 2013 Nme: Definitions: Tue/Flse: Poofs: 1. 2. 3. 4. 5. 6. Totl: Definitions nd Sttements of Theoems 1. (2 points) Fo function f(x) defined on (, b) nd fo
On Several Inequalities Deduced Using a Power Series Approach
It J Cotemp Mth Sceces, Vol 8, 203, o 8, 855-864 HIKARI Ltd, wwwm-hrcom http://dxdoorg/02988/jcms2033896 O Severl Iequltes Deduced Usg Power Seres Approch Lored Curdru Deprtmet of Mthemtcs Poltehc Uversty
Quasi-Rational Canonical Forms of a Matrix over a Number Field
Avace Lea Algeba & Matx Theoy, 08, 8, -0 http://www.cp.og/joual/alamt ISSN Ole: 65-3348 ISSN Pt: 65-333X Qua-Ratoal Caocal om of a Matx ove a Numbe el Zhueg Wag *, Qg Wag, Na Q School of Mathematc a Stattc,
Spectral Continuity: (p, r) - Α P And (p, k) - Q
IOSR Joul of Mthemtcs (IOSR-JM) e-issn: 2278-5728, p-issn: 2319-765X Volume 11, Issue 1 Ve 1 (J - Feb 215), PP 13-18 wwwosjoulsog Spectl Cotuty: (p, ) - Α P Ad (p, k) - Q D Sethl Kum 1 d P Mhesw Nk 2 1
Efficient Estimator for Population Variance Using Auxiliary Variable
Amec Joul o Opetol eech 6 6): 9-5 DOI:.59/j.jo.66. Ecet Etmto o Populto Vce Ug Aul Vble ubhh Kum Yv heel M.. Mh * Deptmet o Mthemtc tttc A Cete o Ecellece) D. ML Avh Uvet Fzb U.P. I Deptmet o tttc Uvet
On Probability Density Function of the Quotient of Generalized Order Statistics from the Weibull Distribution
ISSN 684-843 Joua of Sac Voue 5 8 pp. 7-5 O Pobaby Dey Fuco of he Quoe of Geeaed Ode Sac fo he Webu Dbuo Abac The pobaby dey fuco of Muhaad Aee X k Y k Z whee k X ad Y k ae h ad h geeaed ode ac fo Webu
Minimum Hyper-Wiener Index of Molecular Graph and Some Results on Szeged Related Index
Joual of Multdscplay Egeeg Scece ad Techology (JMEST) ISSN: 359-0040 Vol Issue, Febuay - 05 Mmum Hype-Wee Idex of Molecula Gaph ad Some Results o eged Related Idex We Gao School of Ifomato Scece ad Techology,
Some Equivalent Forms of Bernoulli s Inequality: A Survey *
Ale Mthets 3 4 7-93 htt://oog/436/34746 Pulshe Ole Jul 3 (htt://wwwsog/joul/) Soe Euvlet Fos of Beoull s Ieult: A Suve * u-chu L Cheh-Chh eh 3 Detet of Ale Mthets Ntol Chug-Hsg Uvest Tw Detet of Mthets
Regularization of the Divergent Integrals I. General Consideration
Zozuly / Electoc Joul o Bouy Eleets ol 4 No pp 49-57 6 Reulzto o the Dveet Itels I Geel Coseto Zozuly Ceto e Ivestco Cetc e Yuct AC Clle 43 No 3 Colo Chubuá e Hlo C 97 Mé Yuctá Méco E-l: zozuly@ccy Abstct
XII. Addition of many identical spins
XII. Addto of may detcal sps XII.. ymmetc goup ymmetc goup s the goup of all possble pemutatos of obects. I total! elemets cludg detty opeato. Each pemutato s a poduct of a ceta fte umbe of pawse taspostos.
Chapter 2 Intro to Math Techniques for Quantum Mechanics
Wter 3 Chem 356: Itroductory Qutum Mechcs Chpter Itro to Mth Techques for Qutum Mechcs... Itro to dfferetl equtos... Boudry Codtos... 5 Prtl dfferetl equtos d seprto of vrbles... 5 Itroducto to Sttstcs...
χ be any function of X and Y then
We have show that whe we ae gve Y g(), the [ ] [ g() ] g() f () Y o all g ()() f d fo dscete case Ths ca be eteded to clude fuctos of ay ube of ado vaables. Fo eaple, suppose ad Y ae.v. wth jot desty fucto,
Phys 2310 Fri. Oct. 23, 2017 Today s Topics. Begin Chapter 6: More on Geometric Optics Reading for Next Time
Py F. Oct., 7 Today Topc Beg Capte 6: Moe o Geometc Optc eadg fo Next Tme Homewok t Week HW # Homewok t week due Mo., Oct. : Capte 4: #47, 57, 59, 6, 6, 6, 6, 67, 7 Supplemetal: Tck ee ad e Sytem Pcple