Integrals and Polygamma Representations for Binomial Sums

Size: px
Start display at page:

Download "Integrals and Polygamma Representations for Binomial Sums"

Transcription

1 Jounl of Intege Sequences, Vol. 3 (, Aticle..8 Integls nd Polygmm Repesenttions fo Binomil Sums Anthony Sofo School of Engineeing nd Science Victoi Univesity PO Box 448 Melboune City, VIC 8 Austli nthony.sofo@vu.edu.u Abstct We conside sums involving the poduct of ecipocl binomil coefficiend polynomil tems nd develop some double integl identities. In pticul cses it is possible to expess the sums in closed fom, give some genel esults, ecove some known esults in Coffey nd poduce new identities. Intoduction In ecent ppe Coffey [8] consides summtions ove digmm nd polygmm functions nd develops mny esults, nmely two of his popositions, in tems of the Riemnn zet function ζ (, e espectively equtions (59 nd (66 nd n ( n n p+ (n + ( p ( ln + n n p+ (n + ( p + p ( p+m ( m ζ (m + ( m p ( p+m ζ (m +. ( Coffey [8] lso constucts new integl epesenttions fo these sums. The mjo im of this ppe is to investigte genel binomil sums with vious pmetes tht then enbles one to give moe genel epesenttions of ( nd (, theeby genelizing the popositions of m

2 Coffey, both in closed fom in tems of zet functions nd digmm functions t possible tionl vlues of the gument, nd in double integl fom. The following definitions will be useful. The Psi, o digmm function ψ (z, is defined by ψ (z d dz log Γ (z Γ (z Γ (z ( n + γ n + z whee γ denotes the Eule-Mscheoni constnd Γ (z is the Gmm function. Similly nd ψ (z + n n ( n γ, (3 n + z ( ψ (z ψ (z + ψ z + + ln. (4 Sums of ecipocls of binomil coefficients ppe in the clcultion of mssive Feynmn digms [3] within sevel diffeent ppoches: fo instnce, s solutions of diffeentil equtions fo Feynmn mplitudes, though nive ε-expnsion of hypegeometic functions within Mellin-Bnes technique o in the fmewok of ecently poposed lgebic ppoch []. Thee hs ecently been enewed inteest in the study of seies involving binomil coefficients nd numbe of uthos hve obtined eithe closed fom epesenttion o integl epesenttion fo some pticul cses of these seies. The inteested ede is efeed to [,, 3, 4, 5, 6, 9,, 6, 7, 8, 9,,,, 3, 5, 6, 7]. The min esults The following Lemm nd well-known definition will be useful in the poof of the min theoem. Definition. Let z, m nd q,, 3,... Then z (q + m ( m (m! The next Lemm dels with two infinite sums. Lemm. Let nd be positive el numbes. Then n n (n + H n ( n n (n + { z y q (ln (y m dy. (5 zy ( H ( nd (6 } H (. (7

3 Poof. n n (n + ( n n + n [ ( ] γ + ψ + nd fom (3 H( ; hence (6 is ttined. n ( n n (n + [ n [ γ + ψ ( n ( + [ ( ψ + ψ ( n + n n + n n ( + γ ψ + ] ln ( + ] ln, ] n (n fom the definition (4 ( ψ + ( ( ψ + ψ + ln ; hence n ( n n (n + [ ( ψ [ H ( + γ H ( ( ] ψ + ] + γ, theefoe (7 follows. Remk 3. In the following Coollies nd emks we encounte hmonic numbes t possible tionl vlues of the gument, of the fom H (α whee,, 3,...,k, α,, 3,... nd k N. The polygmm function ψ (α (z is defined s: ψ (α (z dα+ dα [log Γ (z] [ψ (z], z {,,, 3,...}. dzα+ dzα To evlute H (α we hve vilble eltion in tems of the polygmm function ψ (α (z, fo tionl guments z, we lso define H (α+ H ( ( ζ (α + + ( α ψ (α α! + ( γ + ψ +, nd H (α. 3

4 The evlution of the polygmm function ψ ( (α t tionl vlues of the gument cn be explicitly done vi fomul s given by Kölbig [5], (see lso [4], o Choi nd Cvijovic [7] in tems of the polylogithmic o othe specil functions. Some specific vlues e given s, mny othes e listed in the book [4]: ( ψ (n ( n n! ( n+ ζ (n + H ( 3 H ( 4 4 π 3 π 3 3 ln 3 We now stte the following theoem. 3 ln (, H( 3 4, nd H( π 3 ln (, π 3 ln (3 ln (. Theoem 4. Let be positive el numbe, t, j, nd k N {}. Then S k+ (, j, t (j+t( k k! n n k+ ( n + j + j + ( x j x (ln(y k tx y dxdy, fo k ( x t j+ x dx, fo k tx (k+ tems {}}{,,...,, T +k+ F +k,,...,, + j + }{{} k tems whee pf q [ ] is the genelized hypegeometic function, nd B (, is the bet function. Poof. Conside ( tems {}}{ +,...,,..., + j + + } {{ } tems T t (j + B (j +, +, (8. t (9 n n k+ ( n + j + j + n (j + (j + Γ (j + Γ (n + n k+ Γ (n + j + n B (n +,j + nk+ 4

5 now eplcing the bet function with its integl epesenttion, we hve (j + n n B (n +,j + (j + k+ n k+ By justified chnging the ode of integtion nd summtion we hve, n x n ( x j dx. n n k+ ( n + j + j + (j + ( x j (tx n n n k+ dx (j + t( k k! ( x j x (ln (y k dxdy, fo k tx y upon utilizing Definition. The cse of k follows in simil wy so tht t S (, j, t ( n n n + j + n j + t ( x j+ x tx dx; hence the integls in (8 e ttined. By the considetion of the tio of successive tems whee U n+ U n we obtin the esult (9. U n n k+ ( n + j + j + The following inteesting coollies follow fom Theoem 4. Coolly 5. Let t nd >. Also let j nd k be integes. Then S k+ (, j, ( n n + j + n k+ j + (j + ( k k! ( x j x (ln (y k dxdy, fo k ( x y k j+ A s (j +!ζ (k + s + s ( +k+ ( ( k j + H ( ( 5

6 whee Poof. By expnsion, [ { }] d s n k A s lim n s! dn s n k j+, s,,,...k. ( (n + n n k+ ( n + j + j + n n (j +! n k+ (n + j+ (j +! n [ k s (j +! n k+ j+ (n + n j+ A s n + k s B n + ], whee { } n + B lim j+ n ( (n + ( +k+ (j +! A s is defined by (. Hence, fte intechnging the sums, we hve ( n + j + n k+ j + n (j +! [ k j+ A s n + k+ s s n j+ k (j +! A s ζ (k + s + s B n ( ] n (n + ( +k+ ( ( k j +, k ( j + H ( upon utilizing Lemm, which is the esult (. The degenete cse, fo j, gives the known esult ζ (k +. nk+ n The integl ( follows fom the integl in (8. A simil esult is evident fo the cse t. 6

7 Coolly 6. Let t nd >. Also let j nd k be integes. Then ( n S k+ (, j, ( n n + j + n k+ j + (j + ( k+ k! ( x j x (ln (y k dxdy, fo k + x y k A s (j +! ( s k ζ (k + s (3 s j+ + ( +k+ ( k ( j + ( H ( H ( Poof. The poof, uses (7 nd follows the sme detils s tht of Coolly 5, nd will not be given hee. The ddition nd subtction of ( nd (3 gives us the following epesenttions. Remk 7. nd k n Let > nd let j nd k be integes. Then n k+ ( n + j + j + k j+ A s (j +! s k ζ (k + s + s n (n k+ ( n + j + j + ( +k+ ( k A s (j +! ( s k j+ ζ (k + s + s k ( j + ( +k+ ( ( H ( We give the following exmple to illustte some of the bove identities. Exmple 8. Let k 4, fom ( A A 3 (j +!, A H( j+ (j +!, A 3 ( + 3H ( ( 3 H ( j+ 6 (j +! j+ H( j+ + H(3 j+ 7 ( k j + ( ( H ( ( j+ + H j+ (j +! H ( H (.

8 theefoe nd n n Remk 9. n 5 ( n + j + j + ( n n 5 ( n + j + j + (j +! [A ζ (5 + A ζ (4 + A ζ (3 + A 3 ζ (] j+ + ( + ( ( 4 j + H ( [ (j +! 5A 6 ζ (5 7A 8 ζ (4 3A 4 ζ (3 A ] 3 ζ ( j+ + ( ( 4 ( j + ( H ( H (. The vey specil cse of nd j llows one to evlute [ { }] d s n k A s lim, s,,,...k n s! dn s n k (n + ( s, nd fom ( nd (3 we cn esily obtin ( nd (. A ecuence eltion fo degenete cse, j, of Theoem 4 is embodied in the following coolly. Coolly. Let the conditions of Theoem 4 hold with j nd put Sk+ : Sk+ (,t n k+ (n + n (k+ tems {}}{ t,,...,, + k+3f k+ +,,...,, }{{} + t, (k+ tems then with solution S k+ + S k Li k+ (t, fo k S k+ ( k S + ( k S + k ( t Φ (t,k +, k ( Li k+ (t 8

9 whee S (,t n n (n + t + [,, + 3F, + ] t nd Φ, Li e the Lech tnscendend the polylogithm espectively. Poof. We notice tht S k+ + S k nd hence the solution follows by itetion. n n k+ Li k+ (t, Relted esults my be seen in Coffey [, Lemms nd ]. Some exmples e s follows: Fo t, we know tht Li k+ ( ζ (k +. Hence S k+ (, ( k S (, + k ( ζ (k +, fo k. When, we obtin Coffey s [8] esult, by noting tht, fom (6, S (, When, S k+ (, ( k + S k+ k ( ζ (k +. (, 3 ( k + k+ k ( ζ (k +. Similly Sk+ (8, ( k 3(k+ ( k 3k π 3 + ( k 3k+ ln ( ( ( k 3k 3/ ln 3 + k + ( 3( ζ (k +. Fo t, Li k+ ( η (k + ( k ζ (k +, whee η ( is the Diichlet Et function. Hence k Sk+ (, ( k S (, + ( ( k ζ (k +, fo k. When, we obtin Coffey s [8] esult, by noting tht k Sk+ (, ( k ( ln + ( ( k ζ (k +. 9

10 When 4, ( Sk+ (4, ( k 4 ( ln + π ln + k ( ( ( k ζ (k +, nd similly S k+ ( 8, ( k 533 k ( ( 84 3k + k ζ (k + 8 (k+ tems {}}{ 8,,...,, 9 k+3f k+ 9,,...,,. }{{} (k+ tems Refeences [] H. Alze, D. Kynnkis nd H. M. Sivstv. Seies epesenttions fo some mthemticl constnts. J. Mth. Anl. Appl. 3 (6, [] H. Alze nd S. Koumndos. Seies nd poduct epesenttions fo some mthemticl constnts. Peiod. Mth. Hung.58(, (9, 7 8. [3] N. Bti. Integl epesenttions of some seies involving ( k k k n nd some elted seies. Appl. Mth. Comp.47 (4, [4] N. Bti. On the seies k ( 3k k k n x k. Poceedings of the Indin Acdemy of Sciences: Mthemticl Sciences, 5(4 (5, [5] J. M. Bowein nd R. Gigensohn. Evlutions of binomil seies. Aequtiones Mth., 7, (5, [6] J. M. Bowein, D. J. Bodhusd J. Kmnitze. Centl binomil sums, multiple Clusen vlues nd zet vlues. Expeiment. Mth.,, (, [7] J. Choi nd D. Cvijovic, Vlues of the polygmm functions t tionl guments, J. Phys. A: Mth. Theo. 4 (7, [8] M. W. Coffey. On one dimensionl digmm nd polygmm seies elted to the evlution of Feynmn digms. J. Comput. Appl. Mths., 83, (5, 84. [9] M. W. Coffey. On some seies epesenttions of the Huwitz zet function. J. Comput. Appl. Mths. 6, (8,

11 [] M. W. Coffey. Some definite logithmic integls fom Eule sums, nd othe integtion esults, Jnuy 8. [] S. R. Finch. Mthemticl constnts. Encyclopedi of Mthemtics nd its pplictions, 94. Cmbidge Univ. Pess, New Yok, 3. [] A. P. Isev. Multi-loop Feynmn integls nd confoml quntum mechnics, Nucl. Phys. B66 (3, (3, [3] M. Yu. Klmykov, O. Veetin. Single-scle digms nd multiple binomil sums. Physics Lettes, B483, (, [4] K. Kölbig, The polygmm function ψ (x fo x /4 nd x 3/4, J. Comput. Appl. Mth. 75 (996, [5] K. Kölbig, The polygmm function nd the deivtives of the cotngent function fo tionl guments, CERN-IT-Repots CERN-CN., (996, [6] D. H. Lehme. Inteesting seies involving the centl binomil coefficient. Ame. Mth. Monthly, 9, (985, [7] L. Lewin. Polylogithms nd ssocited functions. Noth Hollnd, Amstedm, 98. [8] T. Shemn. Summtion of Glishe nd Apéy-like seies, vilble t [9] A. Sofo. Computtionl Techniques fo the Summtion of Seies, Kluwe Acdemic/Plenum Publishes, 3. [] A. Sofo. Genel popeties involving ecipocls of binomil coefficients, J. Intege Sequences, 9 (6, Aticle [] A. Sofo. Integl foms of sums ssocited with hmonic numbes, Appl. Mth. Comput., 7, (9, [] A. Sofo. Sums of deivtives of binomil coefficients, Advnces in Appl. Mth., 4, (9, [3] A. Sofo. Convexity popeties of Recipocls of binomil coefficients. In T. E. Simos, ed., Numeicl Anlysis nd Applied Mthemtics, AIP, Melville, New Yok, 7, pp [4] H. M. Sivstv nd J. Choi. Seies Associted with the Zed Relted Functions. Kluwe Acdemic Publishes, London,. [5] E. W. Weisstein. Nielsen genelized polylogithm. [6] E. W. Weisstein. Binomil sums.

12 [7] I. J. Zucke. On the seies k ( k k k n nd elted sums. J. Numbe Theoy,, (985, 9. Mthemtics Subject Clssifiction: Pimy 5A; Secondy B65, 5A9, 33C. Keywods: double integl, combintoil identity, hmonic numbe, polygmm function, ecuence. Received Decembe 7 9; evised vesion eceived Febuy 6. Published in Jounl of Intege Sequences, Febuy 3. Retun to Jounl of Intege Sequences home pge.

Fourier-Bessel Expansions with Arbitrary Radial Boundaries

Fourier-Bessel Expansions with Arbitrary Radial Boundaries Applied Mthemtics,,, - doi:./m.. Pulished Online My (http://www.scirp.og/jounl/m) Astct Fouie-Bessel Expnsions with Aity Rdil Boundies Muhmmd A. Mushef P. O. Box, Jeddh, Sudi Ai E-mil: mmushef@yhoo.co.uk

More information

About Some Inequalities for Isotonic Linear Functionals and Applications

About Some Inequalities for Isotonic Linear Functionals and Applications Applied Mthemticl Sciences Vol. 8 04 no. 79 8909-899 HIKARI Ltd www.m-hiki.com http://dx.doi.og/0.988/ms.04.40858 Aout Some Inequlities fo Isotonic Line Functionls nd Applictions Loedn Ciudiu Deptment

More information

Math 4318 : Real Analysis II Mid-Term Exam 1 14 February 2013

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

More information

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z)

defined on a domain can be expanded into the Taylor series around a point a except a singular point. Also, f( z) 08 Tylo eie nd Mcluin eie A holomophic function f( z) defined on domin cn be expnded into the Tylo eie ound point except ingul point. Alo, f( z) cn be expnded into the Mcluin eie in the open dik with diu

More information

This immediately suggests an inverse-square law for a "piece" of current along the line.

This immediately suggests an inverse-square law for a piece of current along the line. Electomgnetic Theoy (EMT) Pof Rui, UNC Asheville, doctophys on YouTube Chpte T Notes The iot-svt Lw T nvese-sque Lw fo Mgnetism Compe the mgnitude of the electic field t distnce wy fom n infinite line

More information

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by

Class Summary. be functions and f( D) , we define the composition of f with g, denoted g f by Clss Summy.5 Eponentil Functions.6 Invese Functions nd Logithms A function f is ule tht ssigns to ech element D ectly one element, clled f( ), in. Fo emple : function not function Given functions f, g:

More information

9.4 The response of equilibrium to temperature (continued)

9.4 The response of equilibrium to temperature (continued) 9.4 The esponse of equilibium to tempetue (continued) In the lst lectue, we studied how the chemicl equilibium esponds to the vition of pessue nd tempetue. At the end, we deived the vn t off eqution: d

More information

Friedmannien equations

Friedmannien equations ..6 Fiedmnnien equtions FLRW metic is : ds c The metic intevl is: dt ( t) d ( ) hee f ( ) is function which detemines globl geometic l popety of D spce. f d sin d One cn put it in the Einstein equtions

More information

Week 8. Topic 2 Properties of Logarithms

Week 8. Topic 2 Properties of Logarithms Week 8 Topic 2 Popeties of Logithms 1 Week 8 Topic 2 Popeties of Logithms Intoduction Since the esult of ithm is n eponent, we hve mny popeties of ithms tht e elted to the popeties of eponents. They e

More information

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1

RELATIVE KINEMATICS. q 2 R 12. u 1 O 2 S 2 S 1. r 1 O 1. Figure 1 RELAIVE KINEMAICS he equtions of motion fo point P will be nlyzed in two diffeent efeence systems. One efeence system is inetil, fixed to the gound, the second system is moving in the physicl spce nd the

More information

On Natural Partial Orders of IC-Abundant Semigroups

On Natural Partial Orders of IC-Abundant Semigroups Intentionl Jounl of Mthemtics nd Computtionl Science Vol. No. 05 pp. 5-9 http://www.publicsciencefmewok.og/jounl/ijmcs On Ntul Ptil Odes of IC-Abundnt Semigoups Chunhu Li Bogen Xu School of Science Est

More information

Discrete Model Parametrization

Discrete Model Parametrization Poceedings of Intentionl cientific Confeence of FME ession 4: Automtion Contol nd Applied Infomtics Ppe 9 Discete Model Pmetition NOKIEVIČ, Pet Doc,Ing,Cc Deptment of Contol ystems nd Instumenttion, Fculty

More information

Review of Mathematical Concepts

Review of Mathematical Concepts ENEE 322: Signls nd Systems view of Mthemticl Concepts This hndout contins ief eview of mthemticl concepts which e vitlly impotnt to ENEE 322: Signls nd Systems. Since this mteil is coveed in vious couses

More information

Right-indefinite half-linear Sturm Liouville problems

Right-indefinite half-linear Sturm Liouville problems Computes nd Mthemtics with Applictions 55 2008) 2554 2564 www.elsevie.com/locte/cmw Right-indefinite hlf-line Stum Liouville poblems Lingju Kong, Qingki Kong b, Deptment of Mthemtics, The Univesity of

More information

EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM

EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE BOUNDARY-VALUE PROBLEM Electonic Jounl of Diffeentil Eutions, Vol. 20 (20, No. 9, pp.. ISSN: 072-669. URL: http://ejde.mth.txstte.edu o http://ejde.mth.unt.edu ftp ejde.mth.txstte.edu EXISTENCE OF THREE SOLUTIONS FOR A KIRCHHOFF-TYPE

More information

FI 2201 Electromagnetism

FI 2201 Electromagnetism FI 1 Electomgnetism Alexnde A. Isknd, Ph.D. Physics of Mgnetism nd Photonics Resech Goup Electosttics ELECTRIC PTENTIALS 1 Recll tht we e inteested to clculte the electic field of some chge distiution.

More information

Topics for Review for Final Exam in Calculus 16A

Topics for Review for Final Exam in Calculus 16A Topics fo Review fo Finl Em in Clculus 16A Instucto: Zvezdelin Stnkov Contents 1. Definitions 1. Theoems nd Poblem Solving Techniques 1 3. Eecises to Review 5 4. Chet Sheet 5 1. Definitions Undestnd the

More information

EECE 260 Electrical Circuits Prof. Mark Fowler

EECE 260 Electrical Circuits Prof. Mark Fowler EECE 60 Electicl Cicuits Pof. Mk Fowle Complex Numbe Review /6 Complex Numbes Complex numbes ise s oots of polynomils. Definition of imginy # nd some esulting popeties: ( ( )( ) )( ) Recll tht the solution

More information

Research Article Hermite-Hadamard-Type Inequalities for r-preinvex Functions

Research Article Hermite-Hadamard-Type Inequalities for r-preinvex Functions Hindwi Publishing Copotion Jounl of Applied Mthemtics Volume 3, Aticle ID 6457, 5 pges http://dx.doi.og/.55/3/6457 Resech Aticle Hemite-Hdmd-Type Inequlities fo -Peinvex Functions Wsim Ul-Hq nd Jved Iqbl

More information

#A11 INTEGERS 11 (2011) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT

#A11 INTEGERS 11 (2011) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT #A INTEGERS (20) NEW SEQUENCES THAT CONVERGE TO A GENERALIZATION OF EULER S CONSTANT Alin Sîntămărin Deprtment of Mthemtics, Technicl University of Cluj-Npoc, Cluj-Npoc, Romni Alin.Sintmrin@mth.utcluj.ro

More information

Two dimensional polar coordinate system in airy stress functions

Two dimensional polar coordinate system in airy stress functions I J C T A, 9(9), 6, pp. 433-44 Intentionl Science Pess Two dimensionl pol coodinte system in iy stess functions S. Senthil nd P. Sek ABSTRACT Stisfy the given equtions, boundy conditions nd bihmonic eqution.in

More information

Michael Rotkowitz 1,2

Michael Rotkowitz 1,2 Novembe 23, 2006 edited Line Contolles e Unifomly Optiml fo the Witsenhusen Counteexmple Michel Rotkowitz 1,2 IEEE Confeence on Decision nd Contol, 2006 Abstct In 1968, Witsenhusen intoduced his celebted

More information

Quadratic Harmonic Number Sums

Quadratic Harmonic Number Sums Applied Matheatics E-Notes, (), -7 c ISSN 67-5 Available fee at io sites of http//www.ath.nthu.edu.tw/aen/ Quadatic Haonic Nube Sus Anthony Sofo y and Mehdi Hassani z Received July Abstact In this pape,

More information

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x

( ) D x ( s) if r s (3) ( ) (6) ( r) = d dr D x SIO 22B, Rudnick dpted fom Dvis III. Single vile sttistics The next few lectues e intended s eview of fundmentl sttistics. The gol is to hve us ll speking the sme lnguge s we move to moe dvnced topics.

More information

π,π is the angle FROM a! TO b

π,π is the angle FROM a! TO b Mth 151: 1.2 The Dot Poduct We hve scled vectos (o, multiplied vectos y el nume clled scl) nd dded vectos (in ectngul component fom). Cn we multiply vectos togethe? The nswe is YES! In fct, thee e two

More information

Important design issues and engineering applications of SDOF system Frequency response Functions

Important design issues and engineering applications of SDOF system Frequency response Functions Impotnt design issues nd engineeing pplictions of SDOF system Fequency esponse Functions The following desciptions show typicl questions elted to the design nd dynmic pefomnce of second-ode mechnicl system

More information

Radial geodesics in Schwarzschild spacetime

Radial geodesics in Schwarzschild spacetime Rdil geodesics in Schwzschild spcetime Spheiclly symmetic solutions to the Einstein eqution tke the fom ds dt d dθ sin θdϕ whee is constnt. We lso hve the connection components, which now tke the fom using

More information

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018.

SPA7010U/SPA7010P: THE GALAXY. Solutions for Coursework 1. Questions distributed on: 25 January 2018. SPA7U/SPA7P: THE GALAXY Solutions fo Cousewok Questions distibuted on: 25 Jnuy 28. Solution. Assessed question] We e told tht this is fint glxy, so essentilly we hve to ty to clssify it bsed on its spectl

More information

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems.

Physics 505 Fall 2005 Midterm Solutions. This midterm is a two hour open book, open notes exam. Do all three problems. Physics 55 Fll 5 Midtem Solutions This midtem is two hou open ook, open notes exm. Do ll thee polems. [35 pts] 1. A ectngul ox hs sides of lengths, nd c z x c [1] ) Fo the Diichlet polem in the inteio

More information

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s:

Chapter 7. Kleene s Theorem. 7.1 Kleene s Theorem. The following theorem is the most important and fundamental result in the theory of FA s: Chpte 7 Kleene s Theoem 7.1 Kleene s Theoem The following theoem is the most impotnt nd fundmentl esult in the theoy of FA s: Theoem 6 Any lnguge tht cn e defined y eithe egul expession, o finite utomt,

More information

10 Statistical Distributions Solutions

10 Statistical Distributions Solutions Communictions Engineeing MSc - Peliminy Reding 1 Sttisticl Distiutions Solutions 1) Pove tht the vince of unifom distiution with minimum vlue nd mximum vlue ( is ) 1. The vince is the men of the sques

More information

Available online at ScienceDirect. Procedia Engineering 91 (2014 ) 32 36

Available online at   ScienceDirect. Procedia Engineering 91 (2014 ) 32 36 Aville online t wwwsciencediectcom ScienceDiect Pocedi Engineeing 91 (014 ) 3 36 XXIII R-S-P semin Theoeticl Foundtion of Civil Engineeing (3RSP) (TFoCE 014) Stess Stte of Rdil Inhomogeneous Semi Sphee

More information

Optimization. x = 22 corresponds to local maximum by second derivative test

Optimization. x = 22 corresponds to local maximum by second derivative test Optimiztion Lectue 17 discussed the exteme vlues of functions. This lectue will pply the lesson fom Lectue 17 to wod poblems. In this section, it is impotnt to emembe we e in Clculus I nd e deling one-vible

More information

Mark Scheme (Results) January 2008

Mark Scheme (Results) January 2008 Mk Scheme (Results) Jnuy 00 GCE GCE Mthemtics (6679/0) Edecel Limited. Registeed in Englnd nd Wles No. 4496750 Registeed Office: One90 High Holbon, London WCV 7BH Jnuy 00 6679 Mechnics M Mk Scheme Question

More information

Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations

Qualitative Analysis for Solutions of a Class of. Nonlinear Ordinary Differential Equations Adv. Theo. Appl. Mech., Vol. 7, 2014, no. 1, 1-7 HIKARI Ltd, www.m-hiki.com http://dx.doi.og/10.12988/tm.2014.458 Qulittive Anlysis fo Solutions of Clss of Nonline Odiny Diffeentil Equtions Juxin Li *,

More information

Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut

Solution of fuzzy multi-objective nonlinear programming problem using interval arithmetic based alpha-cut Intentionl Jounl of Sttistics nd Applied Mthemtics 016; 1(3): 1-5 ISSN: 456-145 Mths 016; 1(3): 1-5 016 Stts & Mths www.mthsounl.com Received: 05-07-016 Accepted: 06-08-016 C Lognthn Dept of Mthemtics

More information

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS

TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS TRAPEZOIDAL TYPE INEQUALITIES FOR n TIME DIFFERENTIABLE FUNCTIONS S.S. DRAGOMIR AND A. SOFO Abstrct. In this pper by utilising result given by Fink we obtin some new results relting to the trpezoidl inequlity

More information

Computing the first eigenpair of the p-laplacian in annuli

Computing the first eigenpair of the p-laplacian in annuli J. Mth. Anl. Appl. 422 2015277 1307 Contents lists vilble t ScienceDiect Jounl of Mthemticl Anlysis nd Applictions www.elsevie.com/locte/jm Computing the fist eigenpi of the p-lplcin in nnuli Gey Ecole,,

More information

A comparison principle for nonlinear heat Rockland operators on graded groups

A comparison principle for nonlinear heat Rockland operators on graded groups Bull. London Mth. Soc. 00 2018 1 6 doi:10.1112/blms.12178 A compison pinciple fo nonline het ocklnd opetos on gded goups Michel uzhnsky nd Duvudkhn Sugn Abstct In this note we show compison pinciple fo

More information

Plane Wave Expansion Method (PWEM)

Plane Wave Expansion Method (PWEM) /15/18 Instucto D. Rymond Rumpf (915) 747 6958 cumpf@utep.edu EE 5337 Computtionl Electomgnetics Lectue #19 Plne Wve Expnsion Method (PWEM) Lectue 19 These notes my contin copyighted mteil obtined unde

More information

The Formulas of Vector Calculus John Cullinan

The Formulas of Vector Calculus John Cullinan The Fomuls of Vecto lculus John ullinn Anlytic Geomety A vecto v is n n-tuple of el numbes: v = (v 1,..., v n ). Given two vectos v, w n, ddition nd multipliction with scl t e defined by Hee is bief list

More information

On Certain Classes of Analytic and Univalent Functions Based on Al-Oboudi Operator

On Certain Classes of Analytic and Univalent Functions Based on Al-Oboudi Operator Boig Itetiol Joul o t Miig, Vol, No, Jue 0 6 O Ceti Clsses o Alytic d Uivlet Fuctios Bsed o Al-Oboudi Opeto TV Sudhs d SP Viylkshmi Abstct--- Followig the woks o [, 4, 7, 9] o lytic d uivlet uctios i this

More information

A Study of Some Integral Problems Using Maple

A Study of Some Integral Problems Using Maple Mthemtis n Sttistis (): -, 0 DOI: 0.89/ms.0.000 http://www.hpub.og A Stuy of Some Integl Poblems Ug Mple Chii-Huei Yu Deptment of Mngement n Infomtion, Nn Jeon Univesity of Siene n Tehnology, Tinn City,

More information

1 Using Integration to Find Arc Lengths and Surface Areas

1 Using Integration to Find Arc Lengths and Surface Areas Novembe 9, 8 MAT86 Week Justin Ko Using Integtion to Find Ac Lengths nd Sufce Aes. Ac Length Fomul: If f () is continuous on [, b], then the c length of the cuve = f() on the intevl [, b] is given b s

More information

The Clamped Plate Equation for the Limaçon

The Clamped Plate Equation for the Limaçon A. Dll Acqu G. Swees The Clmped Plte Eqution fo the Limçon Received: dte / in finl fom: dte Abstct. Hdmd climed in 907 tht the clmped plte eqution is positivity peseving fo domins which e bounded by Limçon

More information

ITI Introduction to Computing II

ITI Introduction to Computing II ITI 1121. Intoduction to Computing II Mcel Tucotte School of Electicl Engineeing nd Compute Science Abstct dt type: Stck Stck-bsed lgoithms Vesion of Febuy 2, 2013 Abstct These lectue notes e ment to be

More information

On the Eötvös effect

On the Eötvös effect On the Eötvös effect Mugu B. Răuţ The im of this ppe is to popose new theoy bout the Eötvös effect. We develop mthemticl model which loud us bette undestnding of this effect. Fom the eqution of motion

More information

New Expansion and Infinite Series

New Expansion and Infinite Series Interntionl Mthemticl Forum, Vol. 9, 204, no. 22, 06-073 HIKARI Ltd, www.m-hikri.com http://dx.doi.org/0.2988/imf.204.4502 New Expnsion nd Infinite Series Diyun Zhng College of Computer Nnjing University

More information

STUDY OF THE UNIFORM MAGNETIC FIELD DOMAINS (3D) IN THE CASE OF THE HELMHOLTZ COILS

STUDY OF THE UNIFORM MAGNETIC FIELD DOMAINS (3D) IN THE CASE OF THE HELMHOLTZ COILS STUDY OF THE UNIFORM MAGNETIC FIED DOMAINS (3D) IN THE CASE OF THE HEMHOTZ COIS FORIN ENACHE, GHEORGHE GAVRIĂ, EMI CAZACU, Key wods: Unifom mgnetic field, Helmholt coils. Helmholt coils e used to estblish

More information

Some Solutions to the Fractional and Relativistic Schrödinger Equations

Some Solutions to the Fractional and Relativistic Schrödinger Equations Intentionl Jounl of Theoeticl nd Mthemticl Physics 5, 5(5): 87- DOI:.593/j.ijtmp.555.3 Some Solutions to the Fctionl nd Reltivistic Schödinge Equtions Yuchun Wei Deptment of Rdition Oncology, Wke Foest

More information

THEORY OF EQUATIONS OBJECTIVE PROBLEMS. If the eqution x 6x 0 0 ) - ) 4) -. If the sum of two oots of the eqution k is -48 ) 6 ) 48 4) 4. If the poduct of two oots of 4 ) -4 ) 4) - 4. If one oot of is

More information

Lecture 10. Solution of Nonlinear Equations - II

Lecture 10. Solution of Nonlinear Equations - II Fied point Poblems Lectue Solution o Nonline Equtions - II Given unction g : R R, vlue such tht gis clled ied point o the unction g, since is unchnged when g is pplied to it. Whees with nonline eqution

More information

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface

General Physics II. number of field lines/area. for whole surface: for continuous surface is a whole surface Genel Physics II Chpte 3: Guss w We now wnt to quickly discuss one of the moe useful tools fo clculting the electic field, nmely Guss lw. In ode to undestnd Guss s lw, it seems we need to know the concept

More information

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0

STD: XI MATHEMATICS Total Marks: 90. I Choose the correct answer: ( 20 x 1 = 20 ) a) x = 1 b) x =2 c) x = 3 d) x = 0 STD: XI MATHEMATICS Totl Mks: 90 Time: ½ Hs I Choose the coect nswe: ( 0 = 0 ). The solution of is ) = b) = c) = d) = 0. Given tht the vlue of thid ode deteminnt is then the vlue of the deteminnt fomed

More information

A Mathematica package to cope with partially ordered sets

A Mathematica package to cope with partially ordered sets A Mthemtic pckge to cope with ptill odeed sets P. Cod Diptimento di Infomtic e Comunicione, Univesità degli Studi di Milno Abstct Mthemtic offes, b w of the pckge Combintoics, mn useful functions to wok

More information

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system

Previously. Extensions to backstepping controller designs. Tracking using backstepping Suppose we consider the general system 436-459 Advnced contol nd utomtion Extensions to bckstepping contolle designs Tcking Obseves (nonline dmping) Peviously Lst lectue we looked t designing nonline contolles using the bckstepping technique

More information

6. Numbers. The line of numbers: Important subsets of IR:

6. Numbers. The line of numbers: Important subsets of IR: 6. Nubes We do not give n xiotic definition of the el nubes hee. Intuitive ening: Ech point on the (infinite) line of nubes coesponds to el nube, i.e., n eleent of IR. The line of nubes: Ipotnt subsets

More information

Cheeger Gromoll type metrics on the tangent bundle

Cheeger Gromoll type metrics on the tangent bundle Cheege Gomoll type metics on the tngent bundle Min Ion MUNTEANU Abstct In this ppe we study Riemnin metic on the tngent bundle T M) of Riemnnin mnifold M which genelizes the Cheege Gomoll metic nd comptible

More information

Study of Electromagnetic Wave Propagation in Periodic Dielectric Structure; MathCAD Analysis

Study of Electromagnetic Wave Propagation in Periodic Dielectric Structure; MathCAD Analysis Communictions in Applied Sciences ISSN -737 Volume Nume 3-9 Stud of lectomgnetic Wve Popgtion in Peiodic Dielectic Stuctue; MthCAD Anlsis Ugwu mmnuel.i Ieogu C. nd chi M.I Deptment of Industil phsics oni

More information

Euler-Maclaurin Summation Formula 1

Euler-Maclaurin Summation Formula 1 Jnury 9, Euler-Mclurin Summtion Formul Suppose tht f nd its derivtive re continuous functions on the closed intervl [, b]. Let ψ(x) {x}, where {x} x [x] is the frctionl prt of x. Lemm : If < b nd, b Z,

More information

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2.

Data Structures. Element Uniqueness Problem. Hash Tables. Example. Hash Tables. Dana Shapira. 19 x 1. ) h(x 4. ) h(x 2. ) h(x 3. h(x 1. x 4. x 2. Element Uniqueness Poblem Dt Stuctues Let x,..., xn < m Detemine whethe thee exist i j such tht x i =x j Sot Algoithm Bucket Sot Dn Shpi Hsh Tbles fo (i=;i

More information

On Polynomials Construction

On Polynomials Construction Intenational Jounal of Mathematical Analysis Vol., 08, no. 6, 5-57 HIKARI Ltd, www.m-hikai.com https://doi.og/0.988/ima.08.843 On Polynomials Constuction E. O. Adeyefa Depatment of Mathematics, Fedeal

More information

7.5-Determinants in Two Variables

7.5-Determinants in Two Variables 7.-eteminnts in Two Vibles efinition of eteminnt The deteminnt of sque mti is el numbe ssocited with the mti. Eve sque mti hs deteminnt. The deteminnt of mti is the single ent of the mti. The deteminnt

More information

PROGRESSION AND SERIES

PROGRESSION AND SERIES INTRODUCTION PROGRESSION AND SERIES A gemet of umbes {,,,,, } ccodig to some well defied ule o set of ules is clled sequece Moe pecisely, we my defie sequece s fuctio whose domi is some subset of set of

More information

On Some Hadamard-Type Inequalıtıes for Convex Functıons

On Some Hadamard-Type Inequalıtıes for Convex Functıons Aville t htt://vuedu/ Al Al Mth ISSN: 93-9466 Vol 9, Issue June 4, 388-4 Alictions nd Alied Mthetics: An Intentionl Jounl AAM On Soe Hdd-Tye Inequlıtıes o, Convex Functıons M Ein Özdei Detent o Mthetics

More information

Numerical approximation to ζ(2n+1)

Numerical approximation to ζ(2n+1) Illinois Wesleyan Univesity Fom the SelectedWoks of Tian-Xiao He 6 Numeical appoximation to ζ(n+1) Tian-Xiao He, Illinois Wesleyan Univesity Michael J. Dancs Available at: https://woks.bepess.com/tian_xiao_he/6/

More information

Chapter Direct Method of Interpolation More Examples Mechanical Engineering

Chapter Direct Method of Interpolation More Examples Mechanical Engineering Chpte 5 iect Method o Intepoltion Moe Exmples Mechnicl Engineeing Exmple Fo the pupose o shinking tunnion into hub, the eduction o dimete o tunnion sht by cooling it though tempetue chnge o is given by

More information

u(r, θ) = 1 + 3a r n=1

u(r, θ) = 1 + 3a r n=1 Mth 45 / AMCS 55. etuck Assignment 8 ue Tuesdy, Apil, 6 Topics fo this week Convegence of Fouie seies; Lplce s eqution nd hmonic functions: bsic popeties, computions on ectngles nd cubes Fouie!, Poisson

More information

Electric Potential. and Equipotentials

Electric Potential. and Equipotentials Electic Potentil nd Euipotentils U Electicl Potentil Review: W wok done y foce in going fom to long pth. l d E dl F W dl F θ Δ l d E W U U U Δ Δ l d E W U U U U potentil enegy electic potentil Potentil

More information

Using Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu

Using Laplace Transform to Evaluate Improper Integrals Chii-Huei Yu Available at https://edupediapublicationsog/jounals Volume 3 Issue 4 Febuay 216 Using Laplace Tansfom to Evaluate Impope Integals Chii-Huei Yu Depatment of Infomation Technology, Nan Jeon Univesity of

More information

Chapter 6 Frequency Response & System Concepts

Chapter 6 Frequency Response & System Concepts hpte 6 Fequency esponse & ystem oncepts Jesung Jng stedy stte (fequency) esponse Phso nottion Filte v v Foced esponse by inusoidl Excittion ( t) dv v v dv v cos t dt dt ince the focing fuction is sinusoid,

More information

Review of Gaussian Quadrature method

Review of Gaussian Quadrature method Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge

More information

Electronic Supplementary Material

Electronic Supplementary Material Electonic Supplementy Mteil On the coevolution of socil esponsiveness nd behvioul consistency Mx Wolf, G Snde vn Doon & Fnz J Weissing Poc R Soc B 78, 440-448; 0 Bsic set-up of the model Conside the model

More information

Comparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers

Comparative Studies of Law of Gravity and General Relativity. No.1 of Comparative Physics Series Papers Comptive Studies of Lw of Gvity nd Genel Reltivity No. of Comptive hysics Seies pes Fu Yuhu (CNOOC Resech Institute, E-mil:fuyh945@sin.com) Abstct: As No. of comptive physics seies ppes, this ppe discusses

More information

MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4

MATHEMATICS IV 2 MARKS. 5 2 = e 3, 4 MATHEMATICS IV MARKS. If + + 6 + c epesents cicle with dius 6, find the vlue of c. R 9 f c ; g, f 6 9 c 6 c c. Find the eccenticit of the hpeol Eqution of the hpeol Hee, nd + e + e 5 e 5 e. Find the distnce

More information

International Journal of Pure and Applied Sciences and Technology

International Journal of Pure and Applied Sciences and Technology Int. J. Pue l. Sci. Technol. () (0). -6 Intentionl Jounl of Pue nd lied Sciences nd Technology ISSN 9-607 vilble online t www.ijost.in Resech Pe Rdil Vibtions in Mico-Isotoic Mico-Elstic Hollow Shee R.

More information

B.A. (PROGRAMME) 1 YEAR MATHEMATICS

B.A. (PROGRAMME) 1 YEAR MATHEMATICS Gdute Couse B.A. (PROGRAMME) YEAR MATHEMATICS ALGEBRA & CALCULUS PART B : CALCULUS SM 4 CONTENTS Lesson Lesson Lesson Lesson Lesson Lesson Lesson : Tngents nd Nomls : Tngents nd Nomls (Pol Co-odintes)

More information

Summary: Binomial Expansion...! r. where

Summary: Binomial Expansion...! r. where Summy: Biomil Epsio 009 M Teo www.techmejcmth-sg.wes.com ) Re-cp of Additiol Mthemtics Biomil Theoem... whee )!!(! () The fomul is ville i MF so studets do ot eed to memoise it. () The fomul pplies oly

More information

f (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1)

f (a) + f (b) f (λx + (1 λ)y) max {f (x),f (y)}, x, y [a, b]. (1.1) TAMKANG JOURNAL OF MATHEMATICS Volume 41, Number 4, 353-359, Winter 1 NEW INEQUALITIES OF HERMITE-HADAMARD TYPE FOR FUNCTIONS WHOSE SECOND DERIVATIVES ABSOLUTE VALUES ARE QUASI-CONVEX M. ALOMARI, M. DARUS

More information

Solution to HW 3, Ma 1a Fall 2016

Solution to HW 3, Ma 1a Fall 2016 Solution to HW 3, Ma a Fall 206 Section 2. Execise 2: Let C be a subset of the eal numbes consisting of those eal numbes x having the popety that evey digit in the decimal expansion of x is, 3, 5, o 7.

More information

Research Article Modeling of Thermal Distributions around a Barrier at the Interface of Coating and Substrate

Research Article Modeling of Thermal Distributions around a Barrier at the Interface of Coating and Substrate Abstct nd Applied Anlysis Volume 23, Aticle ID 968464, 8 pges http://dx.doi.og/.55/23/968464 Resech Aticle Modeling of Theml Distibutions ound Bie t the Intefce of Coting nd Substte Ali Shin Deptment of

More information

U>, and is negative. Electric Potential Energy

U>, and is negative. Electric Potential Energy Electic Potentil Enegy Think of gvittionl potentil enegy. When the lock is moved veticlly up ginst gvity, the gvittionl foce does negtive wok (you do positive wok), nd the potentil enegy (U) inceses. When

More information

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007

School of Electrical and Computer Engineering, Cornell University. ECE 303: Electromagnetic Fields and Waves. Fall 2007 School of Electicl nd Compute Engineeing, Conell Univesity ECE 303: Electomgnetic Fields nd Wves Fll 007 Homewok 4 Due on Sep. 1, 007 by 5:00 PM Reding Assignments: i) Review the lectue notes. ii) Relevnt

More information

D-STABLE ROBUST RELIABLE CONTROL FOR UNCERTAIN DELTA OPERATOR SYSTEMS

D-STABLE ROBUST RELIABLE CONTROL FOR UNCERTAIN DELTA OPERATOR SYSTEMS Jounl of Theoeticl nd Applied nfomtion Technology 8 th Febuy 3. Vol. 48 No.3 5-3 JATT & LLS. All ights eseved. SSN: 99-8645 www.jtit.og E-SSN: 87-395 D-STABLE ROBUST RELABLE CONTROL FOR UNCERTAN DELTA

More information

A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS ABSTRACT

A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS ABSTRACT A COMPARISON OF MEMBRANE SHELL THEORIES OF HYBRID ANISOTROPIC MATERIALS S. W. Chung* School of Achitectue Univesity of Uth Slt Lke City, Uth, USA S.G. Hong Deptment of Achitectue Seoul Ntionl Univesity

More information

Math& 152 Section Integration by Parts

Math& 152 Section Integration by Parts Mth& 5 Section 7. - Integrtion by Prts Integrtion by prts is rule tht trnsforms the integrl of the product of two functions into other (idelly simpler) integrls. Recll from Clculus I tht given two differentible

More information

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION

KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION Fixed Point Theory, 13(2012), No. 1, 285-291 http://www.mth.ubbcluj.ro/ nodecj/sfptcj.html KRASNOSEL SKII TYPE FIXED POINT THEOREM FOR NONLINEAR EXPANSION FULI WANG AND FENG WANG School of Mthemtics nd

More information

dx was area under f ( x ) if ( ) 0

dx was area under f ( x ) if ( ) 0 13. Line Integls Line integls e simil to single integl, f ( x) dx ws e unde f ( x ) if ( ) 0 Insted of integting ove n intevl [, ] (, ) f xy ds f x., we integte ove cuve, (in the xy-plne). **Figue - get

More information

Electricity & Magnetism Lecture 6: Electric Potential

Electricity & Magnetism Lecture 6: Electric Potential Electicity & Mgnetism Lectue 6: Electic Potentil Tody s Concept: Electic Potenl (Defined in tems of Pth Integl of Electic Field) Electicity & Mgnesm Lectue 6, Slide Stuff you sked bout:! Explin moe why

More information

Double Sums of Binomial Coefficients

Double Sums of Binomial Coefficients Itertiol Mthemticl Forum, 3, 008, o. 3, 50-5 Double Sums of Biomil Coefficiets Athoy Sofo School of Computer Sciece d Mthemtics Victori Uiversity, PO Box 448 Melboure, VIC 800, Austrli thoy.sofo@vu.edu.u

More information

Journal of Inequalities in Pure and Applied Mathematics

Journal of Inequalities in Pure and Applied Mathematics Journl of Inequlities in Pure nd Applied Mthemtics GENERALIZATIONS OF THE TRAPEZOID INEQUALITIES BASED ON A NEW MEAN VALUE THEOREM FOR THE REMAINDER IN TAYLOR S FORMULA volume 7, issue 3, rticle 90, 006.

More information

On Error Sum Functions Formed by Convergents of Real Numbers

On Error Sum Functions Formed by Convergents of Real Numbers 3 47 6 3 Journl of Integer Sequences, Vol. 4 (), Article.8.6 On Error Sum Functions Formed by Convergents of Rel Numbers Crsten Elsner nd Mrtin Stein Fchhochschule für die Wirtschft Hnnover Freundllee

More information

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II

ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II STUDIA UNIV. BABEŞ BOLYAI, MATHEMATICA, Volume LV, Number 3, September 2010 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE POINTS IN CERTAIN MEAN VALUE THEOREMS. II TIBERIU TRIF Dedicted to Professor Grigore Ştefn

More information

Electronic Companion for Optimal Design of Co-Productive Services: Interaction and Work Allocation

Electronic Companion for Optimal Design of Co-Productive Services: Interaction and Work Allocation Submitted to Mnufctuing & Sevice Oetions Mngement mnuscit Electonic Comnion fo Otiml Design of Co-Poductive Sevices: Intection nd Wok Alloction Guillume Roels UCLA Andeson School of Mngement, 110 Westwood

More information

WENJUN LIU AND QUÔ C ANH NGÔ

WENJUN LIU AND QUÔ C ANH NGÔ AN OSTROWSKI-GRÜSS TYPE INEQUALITY ON TIME SCALES WENJUN LIU AND QUÔ C ANH NGÔ Astrct. In this pper we derive new inequlity of Ostrowski-Grüss type on time scles nd thus unify corresponding continuous

More information

Production Mechanism of Quark Gluon Plasma in Heavy Ion Collision. Ambar Jain And V.Ravishankar

Production Mechanism of Quark Gluon Plasma in Heavy Ion Collision. Ambar Jain And V.Ravishankar Poduction Mechnism of Quk Gluon Plsm in Hevy Ion Collision Amb Jin And V.Rvishnk Pimy im of theoeticlly studying URHIC is to undestnd Poduction of quks nd gluons tht fom the bulk of the plsm ( ) t 0 Thei

More information

Hadamard and Caputo-Hadamard FDE s with Three Point Integral Boundary Conditions

Hadamard and Caputo-Hadamard FDE s with Three Point Integral Boundary Conditions Nonline Anlyi nd Diffeentil Eqution, Vol. 5, 07, no. 6, 7-8 HIKARI Ltd, www.m-hiki.com http://doi.og/0.988/nde.07.796 Hdmd nd Cputo-Hdmd FDE with hee Point Integl Boundy Condition N.I. Mhmudov, M. Awdll

More information

Newton s Shell Theorem via Archimedes s Hat Box and Single-Variable Calculus

Newton s Shell Theorem via Archimedes s Hat Box and Single-Variable Calculus Newton s Shell Theoem vi Achimees s Ht Box n Single-Vible Clculus Pete McGth Pete McGth (pjmcgt@upenn.eu, MRID955520) eceive his Ph.D. fom Bown Univesity n is cuently Hns Remche Instucto t the Univesity

More information

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function

Asymptotically Lacunary Statistical Equivalent Sequence Spaces Defined by Ideal Convergence and an Orlicz Function "Science Stays Tue Hee" Jounal of Mathematics and Statistical Science, 335-35 Science Signpost Publishing Asymptotically Lacunay Statistical Equivalent Sequence Spaces Defined by Ideal Convegence and an

More information

ab b. c 3. y 5x. a b 3ab. x xy. p q pq. a b. x y) + 2a. a ab. 6. Simplify the following expressions. (a) (b) (c) (4x

ab b. c 3. y 5x. a b 3ab. x xy. p q pq. a b. x y) + 2a. a ab. 6. Simplify the following expressions. (a) (b) (c) (4x . Simplif the following epessions. 8 c c d. Simplif the following epessions. 6b pq 0q. Simplif the following epessions. ( ) q( m n) 6q ( m n) 7 ( b c) ( b c) 6. Simplif the following epessions. b b b p

More information