q-analogues of Triple Series Reduction Formulas due to Srivastava and Panda with General Terms
|
|
- Ήρα Βιτάλης
- 5 χρόνια πριν
- Προβολές:
Transcript
1 Advances in Dynamical Systems and Applications ISSN , Volume 7, Numbe 1, pp ( analogues of Tiple Seies Reduction Fomulas due to Sivastava and Panda with Geneal Tems Thomas Enst Uppsala Univesity Depatment of Mathematics PO Box 480, SE Uppsala, Sweden Abstact We find six double -analogues of the eduction fomulas by Sivastava and Panda fo tiple seies, with geneal tems These ae then spezialized to summations fo -Kampé de Féiet functions AMS Subject Classifications: 33D15, 33D05, 33C65 Keywods: Tiple seies, geneal tems, eduction fomula, Γ -function 1 Intoduction The histoy of the theme in this aticle goes many yeas back It deals with eualities between uite geneal tiple and double sums The coefficients consist of -shifted factoials, Γ -functions and -binomial coefficients Sivastava and Panda [10] collected these fomulas, peviously scatteed in the liteatue into 6 fomulas of a slightly diffeent chaacte than we pesent hee (fo 1 In Section 2 we find double -foms of these, by the 2 -Vandemonde summation fomulas, ie, 12 -foms These ae then specialized, in Section 3, to six summation fomulas fo -Kampé de Féiet functions We will use the (; l; λ opeato, which is exploed in detail in [6] We tun to the fist definitions Definition 11 The powe function is defined by a e a log( Let δ > 0 be an abitay small numbe We will use the following banch of the logaithm: π + δ < Im (log π + δ This defines a simply connected space in the complex plane The vaiables a, b, c, C denote cetain paametes The vaiables i, j, k, l, m, n, p, will denote natual numbes except fo cetain cases whee it will be clea fom Received July 1, 2010; Accepted Novembe 22, 2011 Communicated by Lance Littlejohn
2 42 Thomas Enst the context that i will denote the imaginay unit Let the -shifted factoial be defined by 1, n 0; n 1 a; n (1 a+m (11 n 1, 2, m0 Since poducts of -shifted factoials occu so often, to simplify them we shall feuently use the moe compact notation The opeato is defined by By (13 it follows that a 1,, a m ; n : C Z C Z a a + m a j ; n (12 j1 πi log (13 n 1 a; n (1 + a+m, (14 m0 Assume that (m, l 1, ie, m and l elatively pime The opeato m l : C Z C Z is defined by a a + 2πim l log (15 We will also need anothe genealization of the tilde opeato n 1 k 1 k ã; n ( i(a+m (16 m0 i0 This leads to the following -analogue of [11, p22, (2] Theoem 12 (See [4] a; kn k 1 a + m a ; n k + m ; n (17 k k m0 The following notation will be convenient: QE(x x
3 -analogues of Tiple Seies Reduction Fomulas 43 Definition 13 (; l; λ; n l 1 λ + m λ ; n l + m ; n (18 l l m0 When λ is a vecto, we mean the coesponding poduct of vecto elements When λ is eplaced by a seuence of numbes sepaated by commas, we mean the coesponding poduct as in the case of -shifted factoials Definition 14 Let have dimensions Let (a, (b, (g i, (h i, (a, (b, (g i, (h i A, B, G i, H i, A, B, G i, H i 1 + B + B + H i + H i A A G i G i 0, i 1,, n Then the genealized -Kampé de Féiet function is defined by Φ A+A :G 1 +G 1 ;;Gn+G n B+B :H 1 +H 1 ;;Hn+H n m [ ] (a ˆ : (gˆ 1 ; ; (gˆ n (b ˆ : (hˆ 1 ; ; (hˆ n ; x (a : (g 1; ; (g n (b : (h 1; ; (h n (a; ˆ 0 m (a ( 0, m n j1 ( (g ˆ j ; j mj ((g j( j, m j x m j j (b; ˆ 0 m (b ( 0, m n j1 ( (h ˆ j ; j mj (h j ( j, m j 1; j mj ( 1 n j1 m j(1+h j +H j G j G j +B+B A A ( ( m n ( QE (B + B A A, 0 QE (1 + H j + H j G j G 2 j We assume that no factos in the denominato ae zeo We assume that j1 (a ( 0, m, (g j( j, m j, (b ( 0, m, (h j( j, m j contain factos of the fom ˆ a(k; k, (s; k, (s(k; k o QE (f( m ( mj 2, j (19 Definition 15 Let the Gauss -binomial coefficient be defined by ( n 1; n, k 0, 1,, n (110 k 1; k 1; n k Let the Γ -function be defined in the unit disk 0 < < 1 by Γ (x 1; x; (1 1 x (111
4 44 Thomas Enst 2 Cetain -summation Fomulas Sivastava [12] and Panda & Sivastava [10] have systematically collected and genealized a numbe of elated summation fomulas known fom the liteatue Ou task in this section is to find symmetic -analogues of these fomulas, which always occu in pais In cetain exceptional cases the convegence in the fomulas is not so good, we then eplace the euality sign by the sign fo fomal euality, We assume thoughout that M km + ln and {} is a seuence of bounded complex numbes Theoem 21 (Compae [10, (4 p 244] and [12, (9 p 28] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2 M xm y n 1 β; M NM 1 + α β; M 1 + α β N; M (21 Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β 1 β+; M ( 2 M 1 β; M ( α+β M+N Γ (α β α; N Γ (α 1 α; N 1 β; +M 1 α; N; 1; ( α+β M+N 1 β + M; 1 α; N; 1; xm y n 1 β; M α + β M; N 1 α; N xm y n 1 β; M NM 1 + α β; M 1 + α β N; M (22 The poof is complete
5 -analogues of Tiple Seies Reduction Fomulas 45 Theoem 22 (Compae [10, (4 p 244] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2+(α β N+1 xm y n 1 β; M N(1 β 1 + α β; M 1 + α β N; M (23 Poof We have LHS ( 1 Γ (α Γ (β (( QE + (α β N Γ (α Γ (α β α; N Γ (α 1 α; N Γ (α 1 β; M 1 β+; M 1 β; +M 1 α; N; 1; 1 β + M; 1 α; N; 1; xm y n 1 β; M α + β M; N 1 α; N N(1 β+m xm y n 1 β; M N(1 β 1 + α β; M 1 + α β N; M (24 The poof is complete Theoem 23 (Compae [10, (5 p 244] and [12, (8 p 28] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N 1 β+; M ( 2 1 α+; M xm y n 1 β; M 1 1 α + N; M (25
6 46 Thomas Enst Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β ( α+β+n Γ (α β α; N Γ (α 1 α; N 1 β+; M ( 2 1 α+; M 1 β; M 1 α; M 1 β; +M 1 α; +M N; 1; ( α+β+n 1 β + M; 1 α + M; N; 1; xm y n 1 β; M α + β; N 1 α; M 1 α + M; N xm y n 1 β; M 1 1 α + N; M (26 The poof is complete Theoem 24 (Compae [10, (5 p 244] ( 1 Γ (α 1 β+; M Γ (β 1 α+; M (( QE + (α β N β α; N Γ (α xm y n 1 β; M N(1 β+m 1 α; N 1 α + N; M (27
7 -analogues of Tiple Seies Reduction Fomulas 47 Poof We have LHS ( 1 Γ (α Γ (β (( QE + (α β N Γ (α 1 β; +M N; Γ (α 1 α; +M 1; 1 β + M; N; Γ (α 1 α + M; 1; 1 β+; M 1 α+; M 1 β; M 1 α; M xm y n 1 β; M α + β; N N(1 β+m 1 α; M 1 + α + M; N β α; N Γ (α xm y n 1 β; M N(1 β+m 1 α; N 1 α + N; M (28 The poof is complete Theoem 25 (Compae [10, (6 p 244] and [12, (10 p 29] ( 1 Γ (α α ; M Γ (β β ; M β α; NΓ (α NM α N; M 1 α; N β; M QE (( 2 (29
8 48 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α α; M Γ (α β; M ( α+β+n α; M Γ (α β; M β α; N Γ (α 1 α; N The poof is complete α ; M ( 2 β ; M α; M β; M N; 1; N 1 β M; N; 1 α M; 1; α + β; N 1; m 1; n 1 α M; N NM α N; M β; M Theoem 26 (Compae [10, (6 p 244] ( 1 Γ (α α ; M ( Γ (β β ; M β α; NΓ (α xm y n N(1 β α N; M 1 α; N β; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α α; M Γ (α β; M α; M Γ (α β; M β α; N Γ (α 1 α; N 2+(α β N+1 α ; M ( β ; M α; M β; M N; 1; (α β+1 (210 (211 2+(α β N+1 1 β M; N; 1 α M; 1; α + β; N N(1 β M 1 α M; N α N; M N(1 β β; M (212
9 -analogues of Tiple Seies Reduction Fomulas 49 The poof is complete Theoem 27 (Compae [10, (7 p 244] and [12, (11 p 29] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N Poof We have LHS Γ (α Γ (α ( 1 Γ (α Γ (β β; M ( 2 β ; M ( 2 β; M α + β + N; M α + β; M ( α+β+m+n Γ (α β α; N Γ (α 1 α; N ( 2 β ; M N; α ; 1; N β; M 1 β M; 1 α; N; 1; β; M α + β + M; N 1 α; N β; M α + β + N; M α + β; M (213 (214 The poof is complete Remak 28 The convegence in fomula (213 is not so good Howeve, this fomula woks well in a numbe of special cases One example is 1, 85, α 53, β 5543, N 4, x 2, y 176 (215 In this case the diffeence LHS-RHS in (213 is fo 0 m 60, 0 n 60 Theoem 29 (Compae [10, (7 p 244] ( 1 Γ (α Γ (β β α; N Γ (α 1 α; N (2+(1+α β M N β ; M N(1 β M β; M α + β + N; M α + β; M (216
10 50 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α β; M Γ (α β α; N Γ (α 1 α; N The poof is complete β ; M (1+α β M N ( 2+1 2M N; α ; 1; (1+α β M β; M 1 β M; 1 α; N; 1; N(1 β M β; M α + β + M; N 1 α; N N(1 β M β; M α + β + N; M α + β; M Theoem 210 (Compae [10, (8 p 244] and [12, (12 p 29] ( 1 Γ (α α ; M Γ (β β α; N Γ (α 1 α; N Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α QE xm y n α N; M 1 + α β; M 1 + α β N; M α ; M ( 2 α; M β α; N Γ (α 1 α; N (217 (( 2 β ; α; M N; 1; N α; M α + β M; N 1 M α; N (218 1 β; ( α+β M+N 1 α M; N; 1; α N; M 1 + α β; M 1 + α β N; M (219
11 -analogues of Tiple Seies Reduction Fomulas 51 The poof is complete Theoem 211 (Compae [10, (8 p 244] ( 1 Γ (α ( 2+(1+α β+m N α ; M Γ (β 1; m 1; n β α; N Γ (α xm y n 1 β; M N(1 β 1 + α β; M 1 α; N 1 + α β N; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α α; M β α; N Γ (α 1 α; N (220 α ; M ( 2+(1+α β+m N β ; α; M N; (1+α β+m 1; 1 β; 1 α M; N; 1; α; M N(1 β α + β M; N 1 M α; N The poof is complete α N; M N(1 β 1 + α β; M 1 + α β N; M (221 Remak 212 The convegence in fomula (220 is not so good Howeve, this fomula woks somehow in a numbe of special cases One example is 1, 99, α 43, β 5543, N 4, x 1, y 076 (222 In this case the diffeence LHS-RHS in (220 is fo 0 m 20, 0 n 20 Theoem 213 (Compae [10, (9 p 244] and [12, (13 p 29] ( 1 Γ (α ( 2+M Γ (β 1 α+; M β α; N Γ (α α + β + N; M 1 α; N 1 α + N, α + β; M (223
12 52 Thomas Enst Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α ( α+β+m+n Γ (α β α; N Γ (α 1 α; N The poof is complete 1 α; M ( 2+M 1 α+; M 1 β; 1 α; +M N; 1; ( α+β+m+n 1 β; 1 α + M; N; 1; 1 α; M α + β + M; N 1 α + M; N α + β + N; M 1 α + N, α + β; M Theoem 214 (Compae [10, (9 p 244] ( 1 Γ (α ( 2+(α β+1 N Γ (β 1 α+; M β α; N Γ (α NM α + β + N; M N(1 β 1 α; N 1 α + N, α + β; M Poof We have LHS ( 1 Γ (α Γ (β Γ (α Γ (α Γ (α β α; N Γ (α 1 α; N The poof is complete 1 α; M ( 2+(α β+1 N 1 α+; M 1 β; 1 α; +M N; 1; 1 β; 1 α + M; N; 1; 1 α; M N(1 β α + β + M; N 1 α + M; N xm y n N(1 β α + β + N; M 1 α + N, α + β; M (224 (225 (226
13 -analogues of Tiple Seies Reduction Fomulas 53 3 Specializations Specializing the pevious fomulas leads to the following fomulas fo two vaiables, whee we have put θ(n; α, β; ; ( 1 ( 2 Γ (α Γ (β ; ω(n; α, β; β α; NΓ (α 1 α; N (31 We have assumed that B B G G in all fomulas The conditions fo A and E ae given sepaately in each case Theoem 31 (Compae [10, (16 p 246] We assume that A + 2l E [ ] (; l; 1 β+, (a : (b; (b θ(n; α, β; ; Φ A+2l:B E:G (e : (g; (g ; x l(n, y l(n [ ] (; l; 1 β, 1 + α β, (a : (b; (b ω(n; α, β; Φ A+4l:B (; l; 1 + α β N, (e : (g; (g ; x, y (32 Poof Put (a; m+n (b; m (b ; n (e; m+n (g; m (g ; n Nl(m+n (33 in (21 We have assumed that k l In the following poofs we use the value and assume that k l (a; m+n (b; m (b ; n (e; m+n (g; m (g ; n (34 Theoem 32 (Compae [10, (19 p 247] We assume that A E [ ] (; l; 1 β+, (a : (b; (b θ(n; α, β; ; Φ A+2l:B (; l; 1 α+, (e : (g; (g ; x, y [ ] ω(n; α, β; Φ A+2l:B (; l; 1 β, (a : (b; (b (; l; 1 α + N, (e : (g; (g ; x, y Poof Use (25 Theoem 33 (Compae [10, (20 p 247] We assume that A E [ ] (; l; α, (a : (b; (b θ(n; α, β; ; Φ A+2l:B (; l; β, (e : (g; (g ; x, y [ ] (; l; α N, (a : (b; (b ω(n; α, β; Φ A+2l:B (; l; β, (e : (g; (g ; x Nl, y Nl (35 (36
14 54 Thomas Enst Poof Use (29 Theoem 34 (Compae [10, (21 p 247] We assume that A E + 2l [ θ(n; α, β; ; (1+α β N Φ A:B ω(n; α, β; N(1 β Φ A+2l:B E+4l:G Poof Use (216 ] (a : (b; (b (; l; β, (e : (g; (g ; x l, y l [ (; l; β α + N, (a : (b; (b (; l; β, α + β, (e : (g; (g ; x Nl, y Nl (37 ] Theoem 35 (Compae [10, (22 p 248] We assume that A + 2l E θ(n; α, β; ; Φ A+2l:B E:G ω(n; α, β; Φ A+4l:B [ (; l; α, (a : (b; (b (e : (g; (g [ (; l; α N, 1 + α β, (a : (b; (b (; l; 1 + α β N, (e : (g; (g ] ; x, y ] ; x, y (38 Poof Use (218 Remak 36 In [9, (31 p 439] Kandu tied to deive a simila fomula Howeve, in this aticle the definitions ae insufficient Remak 37 Fomula (38 is a -analogue of [8] Theoem 38 (Compae [10, (23 p 248] We assume that A E + 2l [ θ(n; α, β; ; Φ A:B ω(n; α, β; Φ A+2l:B E+4l:G [ ] (a : (b; (b (; l; 1 α+, (e : (g; (g ; xl, y l (; l; α + β + N, (a : (b; (b (; l; 1 α + N, α + β, (e : (g; (g ] ; x, y (39 Poof Use (223 4 Conclusion We expect that these fomulas will be of geatest value when looking fo -analogues of eductions fo tiple -seies This is an investigation which has only stated, and hopefully will continue in the next yeas The connection with Γ -functions is uite inteesting, as is manifested in the book [7]
15 -analogues of Tiple Seies Reduction Fomulas 55 Refeences [1] T Enst, A method fo -calculus J nonlinea Math Physics 10 No4 (2003, [2] T Enst, Some esults fo -functions of many vaiables Rendiconti di Padova 112 (2004, [3] T Enst, A enaissance fo a -umbal calculus Poceedings of the Intenational Confeence Munich, Gemany July 2005 Wold Scientific, (2007 [4] T Enst, Some new fomulas involving Γ functions Rendiconti di Padova 118 (2007, [5] T Enst, The diffeent tongues of -calculus, Poceedings of Estonian Academy of Sciences 57 no 2 (2008, [6] T Enst, -analogues of geneal eduction fomulas by Buschman and Sivastava, togethe with the impotant (l; λ opeato eminding of MacRobet Demonstatio Mathematica XLIV (2011, [7] T Enst, A compehensive teatment of -calculus, Bikhäuse, 2012 [8] KC Gupta and A Sivastava, Cetain esults involving Kampé de Féiet s function Indian J Math 15 (1973, [9] D Kandu, Cetain expansions involving basic hypegeometic functions of two vaiables Indian J Pue Appl Math 18, no 5, (1987, [10] R Panda and HMSivastava, Some ecusion fomulas associated with multiple hypegeometic functions Bull Acad Polon Sci Se Sci Math Astonom Phys , no 3 (1975, [11] ED Rainville, Special functions Repint of 1960 fist edition (Chelsea Publishing Co, Bonx, NY, 1971 [12] H MSivastava, Cetain summation fomulas involving genealized hypegeometic functions Comment Math Univ St Paul 21, (1972/ ,
(a,b) Let s review the general definitions of trig functions first. (See back cover of your book) sin θ = b/r cos θ = a/r tan θ = b/a, a 0
TRIGONOMETRIC IDENTITIES (a,b) Let s eview the geneal definitions of tig functions fist. (See back cove of you book) θ b/ θ a/ tan θ b/a, a 0 θ csc θ /b, b 0 sec θ /a, a 0 cot θ a/b, b 0 By doing some
Διαβάστε περισσότεραAnalytical Expression for Hessian
Analytical Expession fo Hessian We deive the expession of Hessian fo a binay potential the coesponding expessions wee deived in [] fo a multibody potential. In what follows, we use the convention that
Διαβάστε περισσότεραMatrix Hartree-Fock Equations for a Closed Shell System
atix Hatee-Fock Equations fo a Closed Shell System A single deteminant wavefunction fo a system containing an even numbe of electon N) consists of N/ spatial obitals, each occupied with an α & β spin has
Διαβάστε περισσότερα2 Composition. Invertible Mappings
Arkansas Tech University MATH 4033: Elementary Modern Algebra Dr. Marcel B. Finan Composition. Invertible Mappings In this section we discuss two procedures for creating new mappings from old ones, namely,
Διαβάστε περισσότεραCurvilinear Systems of Coordinates
A Cuvilinea Systems of Coodinates A.1 Geneal Fomulas Given a nonlinea tansfomation between Catesian coodinates x i, i 1,..., 3 and geneal cuvilinea coodinates u j, j 1,..., 3, x i x i (u j ), we intoduce
Διαβάστε περισσότεραLaplace s Equation in Spherical Polar Coördinates
Laplace s Equation in Spheical Pola Coödinates C. W. David Dated: Januay 3, 001 We stat with the pimitive definitions I. x = sin θ cos φ y = sin θ sin φ z = cos θ thei inveses = x y z θ = cos 1 z = z cos1
Διαβάστε περισσότεραe t e r Cylindrical and Spherical Coordinate Representation of grad, div, curl and 2
Cylindical and Spheical Coodinate Repesentation of gad, div, cul and 2 Thus fa, we have descibed an abitay vecto in F as a linea combination of i, j and k, which ae unit vectos in the diection of inceasin,
Διαβάστε περισσότεραExample 1: THE ELECTRIC DIPOLE
Example 1: THE ELECTRIC DIPOLE 1 The Electic Dipole: z + P + θ d _ Φ = Q 4πε + Q = Q 4πε 4πε 1 + 1 2 The Electic Dipole: d + _ z + Law of Cosines: θ A B α C A 2 = B 2 + C 2 2ABcosα P ± = 2 ( + d ) 2 2
Διαβάστε περισσότεραSpace Physics (I) [AP-3044] Lecture 1 by Ling-Hsiao Lyu Oct Lecture 1. Dipole Magnetic Field and Equations of Magnetic Field Lines
Space Physics (I) [AP-344] Lectue by Ling-Hsiao Lyu Oct. 2 Lectue. Dipole Magnetic Field and Equations of Magnetic Field Lines.. Dipole Magnetic Field Since = we can define = A (.) whee A is called the
Διαβάστε περισσότεραOn a four-dimensional hyperbolic manifold with finite volume
BULETINUL ACADEMIEI DE ŞTIINŢE A REPUBLICII MOLDOVA. MATEMATICA Numbers 2(72) 3(73), 2013, Pages 80 89 ISSN 1024 7696 On a four-dimensional hyperbolic manifold with finite volume I.S.Gutsul Abstract. In
Διαβάστε περισσότεραC.S. 430 Assignment 6, Sample Solutions
C.S. 430 Assignment 6, Sample Solutions Paul Liu November 15, 2007 Note that these are sample solutions only; in many cases there were many acceptable answers. 1 Reynolds Problem 10.1 1.1 Normal-order
Διαβάστε περισσότεραExample Sheet 3 Solutions
Example Sheet 3 Solutions. i Regular Sturm-Liouville. ii Singular Sturm-Liouville mixed boundary conditions. iii Not Sturm-Liouville ODE is not in Sturm-Liouville form. iv Regular Sturm-Liouville note
Διαβάστε περισσότεραSection 8.3 Trigonometric Equations
99 Section 8. Trigonometric Equations Objective 1: Solve Equations Involving One Trigonometric Function. In this section and the next, we will exple how to solving equations involving trigonometric functions.
Διαβάστε περισσότερα4.2 Differential Equations in Polar Coordinates
Section 4. 4. Diffeential qations in Pola Coodinates Hee the two-dimensional Catesian elations of Chapte ae e-cast in pola coodinates. 4.. qilibim eqations in Pola Coodinates One wa of epesg the eqations
Διαβάστε περισσότεραTutorial Note - Week 09 - Solution
Tutoial Note - Week 9 - Solution ouble Integals in Pola Coodinates. a Since + and + 5 ae cicles centeed at oigin with adius and 5, then {,θ 5, θ π } Figue. f, f cos θ, sin θ cos θ sin θ sin θ da 5 69 5
Διαβάστε περισσότεραProduct of two generalized pseudo-differential operators involving fractional Fourier transform
J. Pseudo-Diffe. Ope. Appl. 2011 2:355 365 DOI 10.1007/s11868-011-0034-5 Poduct of two genealized pseudo-diffeential opeatos involving factional Fouie tansfom Akhilesh Pasad Manish Kuma eceived: 21 Febuay
Διαβάστε περισσότεραSCITECH Volume 13, Issue 2 RESEARCH ORGANISATION Published online: March 29, 2018
Journal of rogressive Research in Mathematics(JRM) ISSN: 2395-028 SCITECH Volume 3, Issue 2 RESEARCH ORGANISATION ublished online: March 29, 208 Journal of rogressive Research in Mathematics www.scitecresearch.com/journals
Διαβάστε περισσότεραA NEW CLASS OF MODULAR EQUATIONS IN RAMANUJAN S ALTERNATIVE THEORY OF ELLIPTIC FUNCTIONS OF SIGNATURE 4 AND SOME NEW P-Q ETA-FUNCTION IDENTITIES
A NEW CLASS OF MODULAR EQUATIONS IN RAMANUJAN S ALTERNATIVE THEORY OF ELLIPTIC FUNCTIONS OF SIGNATURE AND SOME NEW P-Q ETA-FUNCTION IDENTITIES S. Bhagava Chasheka Adiga M. S. Mahadeva Naika. Depatent of
Διαβάστε περισσότερα6.3 Forecasting ARMA processes
122 CHAPTER 6. ARMA MODELS 6.3 Forecasting ARMA processes The purpose of forecasting is to predict future values of a TS based on the data collected to the present. In this section we will discuss a linear
Διαβάστε περισσότεραCHAPTER 25 SOLVING EQUATIONS BY ITERATIVE METHODS
CHAPTER 5 SOLVING EQUATIONS BY ITERATIVE METHODS EXERCISE 104 Page 8 1. Find the positive root of the equation x + 3x 5 = 0, correct to 3 significant figures, using the method of bisection. Let f(x) =
Διαβάστε περισσότεραTL-Moments and L-Moments Estimation for the Generalized Pareto Distribution
Applied Mathematical Sciences, Vol. 3, 2009, no. 1, 43-52 TL-Moments L-Moments Estimation fo the Genealized Paeto Distibution Ibahim B. Abdul-Moniem Madina Highe Institute fo Management Technology Madina
Διαβάστε περισσότεραCRASH COURSE IN PRECALCULUS
CRASH COURSE IN PRECALCULUS Shiah-Sen Wang The graphs are prepared by Chien-Lun Lai Based on : Precalculus: Mathematics for Calculus by J. Stuwart, L. Redin & S. Watson, 6th edition, 01, Brooks/Cole Chapter
Διαβάστε περισσότερα2. THEORY OF EQUATIONS. PREVIOUS EAMCET Bits.
EAMCET-. THEORY OF EQUATIONS PREVIOUS EAMCET Bits. Each of the roots of the equation x 6x + 6x 5= are increased by k so that the new transformed equation does not contain term. Then k =... - 4. - Sol.
Διαβάστε περισσότεραHomework 3 Solutions
Homework 3 Solutions Igor Yanovsky (Math 151A TA) Problem 1: Compute the absolute error and relative error in approximations of p by p. (Use calculator!) a) p π, p 22/7; b) p π, p 3.141. Solution: For
Διαβάστε περισσότεραOther Test Constructions: Likelihood Ratio & Bayes Tests
Other Test Constructions: Likelihood Ratio & Bayes Tests Side-Note: So far we have seen a few approaches for creating tests such as Neyman-Pearson Lemma ( most powerful tests of H 0 : θ = θ 0 vs H 1 :
Διαβάστε περισσότεραUniform Convergence of Fourier Series Michael Taylor
Uniform Convergence of Fourier Series Michael Taylor Given f L 1 T 1 ), we consider the partial sums of the Fourier series of f: N 1) S N fθ) = ˆfk)e ikθ. k= N A calculation gives the Dirichlet formula
Διαβάστε περισσότεραStatistical Inference I Locally most powerful tests
Statistical Inference I Locally most powerful tests Shirsendu Mukherjee Department of Statistics, Asutosh College, Kolkata, India. shirsendu st@yahoo.co.in So far we have treated the testing of one-sided
Διαβάστε περισσότεραReminders: linear functions
Reminders: linear functions Let U and V be vector spaces over the same field F. Definition A function f : U V is linear if for every u 1, u 2 U, f (u 1 + u 2 ) = f (u 1 ) + f (u 2 ), and for every u U
Διαβάστε περισσότεραOrdinal Arithmetic: Addition, Multiplication, Exponentiation and Limit
Ordinal Arithmetic: Addition, Multiplication, Exponentiation and Limit Ting Zhang Stanford May 11, 2001 Stanford, 5/11/2001 1 Outline Ordinal Classification Ordinal Addition Ordinal Multiplication Ordinal
Διαβάστε περισσότεραInverse trigonometric functions & General Solution of Trigonometric Equations. ------------------ ----------------------------- -----------------
Inverse trigonometric functions & General Solution of Trigonometric Equations. 1. Sin ( ) = a) b) c) d) Ans b. Solution : Method 1. Ans a: 17 > 1 a) is rejected. w.k.t Sin ( sin ) = d is rejected. If sin
Διαβάστε περισσότεραFourier Series. MATH 211, Calculus II. J. Robert Buchanan. Spring Department of Mathematics
Fourier Series MATH 211, Calculus II J. Robert Buchanan Department of Mathematics Spring 2018 Introduction Not all functions can be represented by Taylor series. f (k) (c) A Taylor series f (x) = (x c)
Διαβάστε περισσότεραMock Exam 7. 1 Hong Kong Educational Publishing Company. Section A 1. Reference: HKDSE Math M Q2 (a) (1 + kx) n 1M + 1A = (1) =
Mock Eam 7 Mock Eam 7 Section A. Reference: HKDSE Math M 0 Q (a) ( + k) n nn ( )( k) + nk ( ) + + nn ( ) k + nk + + + A nk... () nn ( ) k... () From (), k...() n Substituting () into (), nn ( ) n 76n 76n
Διαβάστε περισσότεραThe Simply Typed Lambda Calculus
Type Inference Instead of writing type annotations, can we use an algorithm to infer what the type annotations should be? That depends on the type system. For simple type systems the answer is yes, and
Διαβάστε περισσότεραHomework 8 Model Solution Section
MATH 004 Homework Solution Homework 8 Model Solution Section 14.5 14.6. 14.5. Use the Chain Rule to find dz where z cosx + 4y), x 5t 4, y 1 t. dz dx + dy y sinx + 4y)0t + 4) sinx + 4y) 1t ) 0t + 4t ) sinx
Διαβάστε περισσότεραLecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3
Lecture 2: Dirac notation and a review of linear algebra Read Sakurai chapter 1, Baym chatper 3 1 State vector space and the dual space Space of wavefunctions The space of wavefunctions is the set of all
Διαβάστε περισσότεραANSWERSHEET (TOPIC = DIFFERENTIAL CALCULUS) COLLECTION #2. h 0 h h 0 h h 0 ( ) g k = g 0 + g 1 + g g 2009 =?
Teko Classes IITJEE/AIEEE Maths by SUHAAG SIR, Bhopal, Ph (0755) 3 00 000 www.tekoclasses.com ANSWERSHEET (TOPIC DIFFERENTIAL CALCULUS) COLLECTION # Question Type A.Single Correct Type Q. (A) Sol least
Διαβάστε περισσότεραST5224: Advanced Statistical Theory II
ST5224: Advanced Statistical Theory II 2014/2015: Semester II Tutorial 7 1. Let X be a sample from a population P and consider testing hypotheses H 0 : P = P 0 versus H 1 : P = P 1, where P j is a known
Διαβάστε περισσότεραEvery set of first-order formulas is equivalent to an independent set
Every set of first-order formulas is equivalent to an independent set May 6, 2008 Abstract A set of first-order formulas, whatever the cardinality of the set of symbols, is equivalent to an independent
Διαβάστε περισσότεραFundamental Equations of Fluid Mechanics
Fundamental Equations of Fluid Mechanics 1 Calculus 1.1 Gadient of a scala s The gadient of a scala is a vecto quantit. The foms of the diffeential gadient opeato depend on the paticula geomet of inteest.
Διαβάστε περισσότεραω ω ω ω ω ω+2 ω ω+2 + ω ω ω ω+2 + ω ω+1 ω ω+2 2 ω ω ω ω ω ω ω ω+1 ω ω2 ω ω2 + ω ω ω2 + ω ω ω ω2 + ω ω+1 ω ω2 + ω ω+1 + ω ω ω ω2 + ω
0 1 2 3 4 5 6 ω ω + 1 ω + 2 ω + 3 ω + 4 ω2 ω2 + 1 ω2 + 2 ω2 + 3 ω3 ω3 + 1 ω3 + 2 ω4 ω4 + 1 ω5 ω 2 ω 2 + 1 ω 2 + 2 ω 2 + ω ω 2 + ω + 1 ω 2 + ω2 ω 2 2 ω 2 2 + 1 ω 2 2 + ω ω 2 3 ω 3 ω 3 + 1 ω 3 + ω ω 3 +
Διαβάστε περισσότεραPractice Exam 2. Conceptual Questions. 1. State a Basic identity and then verify it. (a) Identity: Solution: One identity is csc(θ) = 1
Conceptual Questions. State a Basic identity and then verify it. a) Identity: Solution: One identity is cscθ) = sinθ) Practice Exam b) Verification: Solution: Given the point of intersection x, y) of the
Διαβάστε περισσότεραThe Laplacian in Spherical Polar Coordinates
Univesity of Connecticut DigitalCommons@UConn Chemisty Education Mateials Depatment of Chemisty -6-007 The Laplacian in Spheical Pola Coodinates Cal W. David Univesity of Connecticut, Cal.David@uconn.edu
Διαβάστε περισσότεραDerivation of Optical-Bloch Equations
Appendix C Derivation of Optical-Bloch Equations In this appendix the optical-bloch equations that give the populations and coherences for an idealized three-level Λ system, Fig. 3. on page 47, will be
Διαβάστε περισσότεραApproximation of distance between locations on earth given by latitude and longitude
Approximation of distance between locations on earth given by latitude and longitude Jan Behrens 2012-12-31 In this paper we shall provide a method to approximate distances between two points on earth
Διαβάστε περισσότεραA summation formula ramified with hypergeometric function and involving recurrence relation
South Asian Journal of Mathematics 017, Vol. 7 ( 1): 1 4 www.sajm-online.com ISSN 51-151 RESEARCH ARTICLE A summation formula ramified with hypergeometric function and involving recurrence relation Salahuddin
Διαβάστε περισσότεραPhys460.nb Solution for the t-dependent Schrodinger s equation How did we find the solution? (not required)
Phys460.nb 81 ψ n (t) is still the (same) eigenstate of H But for tdependent H. The answer is NO. 5.5.5. Solution for the tdependent Schrodinger s equation If we assume that at time t 0, the electron starts
Διαβάστε περισσότεραThe Neutrix Product of the Distributions r. x λ
ULLETIN u. Maaysia Math. Soc. Secod Seies 22 999 - of the MALAYSIAN MATHEMATICAL SOCIETY The Neuti Poduct of the Distibutios ad RIAN FISHER AND 2 FATMA AL-SIREHY Depatet of Matheatics ad Copute Sciece
Διαβάστε περισσότεραFinite Field Problems: Solutions
Finite Field Problems: Solutions 1. Let f = x 2 +1 Z 11 [x] and let F = Z 11 [x]/(f), a field. Let Solution: F =11 2 = 121, so F = 121 1 = 120. The possible orders are the divisors of 120. Solution: The
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραA General Note on δ-quasi Monotone and Increasing Sequence
International Mathematical Forum, 4, 2009, no. 3, 143-149 A General Note on δ-quasi Monotone and Increasing Sequence Santosh Kr. Saxena H. N. 419, Jawaharpuri, Badaun, U.P., India Presently working in
Διαβάστε περισσότεραEE512: Error Control Coding
EE512: Error Control Coding Solution for Assignment on Finite Fields February 16, 2007 1. (a) Addition and Multiplication tables for GF (5) and GF (7) are shown in Tables 1 and 2. + 0 1 2 3 4 0 0 1 2 3
Διαβάστε περισσότεραMath221: HW# 1 solutions
Math: HW# solutions Andy Royston October, 5 7.5.7, 3 rd Ed. We have a n = b n = a = fxdx = xdx =, x cos nxdx = x sin nx n sin nxdx n = cos nx n = n n, x sin nxdx = x cos nx n + cos nxdx n cos n = + sin
Διαβάστε περισσότερα4.6 Autoregressive Moving Average Model ARMA(1,1)
84 CHAPTER 4. STATIONARY TS MODELS 4.6 Autoregressive Moving Average Model ARMA(,) This section is an introduction to a wide class of models ARMA(p,q) which we will consider in more detail later in this
Διαβάστε περισσότεραSection 7.6 Double and Half Angle Formulas
09 Section 7. Double and Half Angle Fmulas To derive the double-angles fmulas, we will use the sum of two angles fmulas that we developed in the last section. We will let α θ and β θ: cos(θ) cos(θ + θ)
Διαβάστε περισσότεραHomomorphism in Intuitionistic Fuzzy Automata
International Journal of Fuzzy Mathematics Systems. ISSN 2248-9940 Volume 3, Number 1 (2013), pp. 39-45 Research India Publications http://www.ripublication.com/ijfms.htm Homomorphism in Intuitionistic
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 6/5/2006
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Ολοι οι αριθμοί που αναφέρονται σε όλα τα ερωτήματα είναι μικρότεροι το 1000 εκτός αν ορίζεται διαφορετικά στη διατύπωση του προβλήματος. Διάρκεια: 3,5 ώρες Καλή
Διαβάστε περισσότερα( y) Partial Differential Equations
Partial Dierential Equations Linear P.D.Es. contains no owers roducts o the deendent variables / an o its derivatives can occasionall be solved. Consider eamle ( ) a (sometimes written as a ) we can integrate
Διαβάστε περισσότεραHOMEWORK 4 = G. In order to plot the stress versus the stretch we define a normalized stretch:
HOMEWORK 4 Problem a For the fast loading case, we want to derive the relationship between P zz and λ z. We know that the nominal stress is expressed as: P zz = ψ λ z where λ z = λ λ z. Therefore, applying
Διαβάστε περισσότεραMATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutions to Problems on Matrix Algebra
MATH 38061/MATH48061/MATH68061: MULTIVARIATE STATISTICS Solutios to Poblems o Matix Algeba 1 Let A be a squae diagoal matix takig the fom a 11 0 0 0 a 22 0 A 0 0 a pp The ad So, log det A t log A t log
Διαβάστε περισσότεραA Note on Intuitionistic Fuzzy. Equivalence Relation
International Mathematical Forum, 5, 2010, no. 67, 3301-3307 A Note on Intuitionistic Fuzzy Equivalence Relation D. K. Basnet Dept. of Mathematics, Assam University Silchar-788011, Assam, India dkbasnet@rediffmail.com
Διαβάστε περισσότεραSPECIAL FUNCTIONS and POLYNOMIALS
SPECIAL FUNCTIONS and POLYNOMIALS Gerard t Hooft Stefan Nobbenhuis Institute for Theoretical Physics Utrecht University, Leuvenlaan 4 3584 CC Utrecht, the Netherlands and Spinoza Institute Postbox 8.195
Διαβάστε περισσότεραAreas and Lengths in Polar Coordinates
Kiryl Tsishchanka Areas and Lengths in Polar Coordinates In this section we develop the formula for the area of a region whose boundary is given by a polar equation. We need to use the formula for the
Διαβάστε περισσότεραk A = [k, k]( )[a 1, a 2 ] = [ka 1,ka 2 ] 4For the division of two intervals of confidence in R +
Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b
Διαβάστε περισσότεραFractional Colorings and Zykov Products of graphs
Fractional Colorings and Zykov Products of graphs Who? Nichole Schimanski When? July 27, 2011 Graphs A graph, G, consists of a vertex set, V (G), and an edge set, E(G). V (G) is any finite set E(G) is
Διαβάστε περισσότεραConcrete Mathematics Exercises from 30 September 2016
Concrete Mathematics Exercises from 30 September 2016 Silvio Capobianco Exercise 1.7 Let H(n) = J(n + 1) J(n). Equation (1.8) tells us that H(2n) = 2, and H(2n+1) = J(2n+2) J(2n+1) = (2J(n+1) 1) (2J(n)+1)
Διαβάστε περισσότερα3.4 SUM AND DIFFERENCE FORMULAS. NOTE: cos(α+β) cos α + cos β cos(α-β) cos α -cos β
3.4 SUM AND DIFFERENCE FORMULAS Page Theorem cos(αβ cos α cos β -sin α cos(α-β cos α cos β sin α NOTE: cos(αβ cos α cos β cos(α-β cos α -cos β Proof of cos(α-β cos α cos β sin α Let s use a unit circle
Διαβάστε περισσότεραΚΥΠΡΙΑΚΗ ΕΤΑΙΡΕΙΑ ΠΛΗΡΟΦΟΡΙΚΗΣ CYPRUS COMPUTER SOCIETY ΠΑΓΚΥΠΡΙΟΣ ΜΑΘΗΤΙΚΟΣ ΔΙΑΓΩΝΙΣΜΟΣ ΠΛΗΡΟΦΟΡΙΚΗΣ 19/5/2007
Οδηγίες: Να απαντηθούν όλες οι ερωτήσεις. Αν κάπου κάνετε κάποιες υποθέσεις να αναφερθούν στη σχετική ερώτηση. Όλα τα αρχεία που αναφέρονται στα προβλήματα βρίσκονται στον ίδιο φάκελο με το εκτελέσιμο
Διαβάστε περισσότερα6.1. Dirac Equation. Hamiltonian. Dirac Eq.
6.1. Dirac Equation Ref: M.Kaku, Quantum Field Theory, Oxford Univ Press (1993) η μν = η μν = diag(1, -1, -1, -1) p 0 = p 0 p = p i = -p i p μ p μ = p 0 p 0 + p i p i = E c 2 - p 2 = (m c) 2 H = c p 2
Διαβάστε περισσότεραCommutative Monoids in Intuitionistic Fuzzy Sets
Commutative Monoids in Intuitionistic Fuzzy Sets S K Mala #1, Dr. MM Shanmugapriya *2 1 PhD Scholar in Mathematics, Karpagam University, Coimbatore, Tamilnadu- 641021 Assistant Professor of Mathematics,
Διαβάστε περισσότεραDiracDelta. Notations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation
DiracDelta Notations Traditional name Dirac delta function Traditional notation x Mathematica StandardForm notation DiracDeltax Primary definition 4.03.02.000.0 x Π lim ε ; x ε0 x 2 2 ε Specific values
Διαβάστε περισσότεραPARTIAL NOTES for 6.1 Trigonometric Identities
PARTIAL NOTES for 6.1 Trigonometric Identities tanθ = sinθ cosθ cotθ = cosθ sinθ BASIC IDENTITIES cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ PYTHAGOREAN IDENTITIES sin θ + cos θ =1 tan θ +1= sec θ 1 + cot
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π, π ], the restrict function. y = sin x, π 2 x π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότεραMath 6 SL Probability Distributions Practice Test Mark Scheme
Math 6 SL Probability Distributions Practice Test Mark Scheme. (a) Note: Award A for vertical line to right of mean, A for shading to right of their vertical line. AA N (b) evidence of recognizing symmetry
Διαβάστε περισσότεραJesse Maassen and Mark Lundstrom Purdue University November 25, 2013
Notes on Average Scattering imes and Hall Factors Jesse Maassen and Mar Lundstrom Purdue University November 5, 13 I. Introduction 1 II. Solution of the BE 1 III. Exercises: Woring out average scattering
Διαβάστε περισσότεραIf we restrict the domain of y = sin x to [ π 2, π 2
Chapter 3. Analytic Trigonometry 3.1 The inverse sine, cosine, and tangent functions 1. Review: Inverse function (1) f 1 (f(x)) = x for every x in the domain of f and f(f 1 (x)) = x for every x in the
Διαβάστε περισσότερα2. Let H 1 and H 2 be Hilbert spaces and let T : H 1 H 2 be a bounded linear operator. Prove that [T (H 1 )] = N (T ). (6p)
Uppsala Universitet Matematiska Institutionen Andreas Strömbergsson Prov i matematik Funktionalanalys Kurs: F3B, F4Sy, NVP 2005-03-08 Skrivtid: 9 14 Tillåtna hjälpmedel: Manuella skrivdon, Kreyszigs bok
Διαβάστε περισσότεραDIRECT PRODUCT AND WREATH PRODUCT OF TRANSFORMATION SEMIGROUPS
GANIT J. Bangladesh Math. oc. IN 606-694) 0) -7 DIRECT PRODUCT AND WREATH PRODUCT OF TRANFORMATION EMIGROUP ubrata Majumdar, * Kalyan Kumar Dey and Mohd. Altab Hossain Department of Mathematics University
Διαβάστε περισσότεραMath 446 Homework 3 Solutions. (1). (i): Reverse triangle inequality for metrics: Let (X, d) be a metric space and let x, y, z X.
Math 446 Homework 3 Solutions. (1). (i): Reverse triangle inequalit for metrics: Let (X, d) be a metric space and let x,, z X. Prove that d(x, z) d(, z) d(x, ). (ii): Reverse triangle inequalit for norms:
Διαβάστε περισσότεραCongruence Classes of Invertible Matrices of Order 3 over F 2
International Journal of Algebra, Vol. 8, 24, no. 5, 239-246 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/.2988/ija.24.422 Congruence Classes of Invertible Matrices of Order 3 over F 2 Ligong An and
Διαβάστε περισσότεραSCHOOL OF MATHEMATICAL SCIENCES G11LMA Linear Mathematics Examination Solutions
SCHOOL OF MATHEMATICAL SCIENCES GLMA Linear Mathematics 00- Examination Solutions. (a) i. ( + 5i)( i) = (6 + 5) + (5 )i = + i. Real part is, imaginary part is. (b) ii. + 5i i ( + 5i)( + i) = ( i)( + i)
Διαβάστε περισσότεραNew bounds for spherical two-distance sets and equiangular lines
New bounds for spherical two-distance sets and equiangular lines Michigan State University Oct 8-31, 016 Anhui University Definition If X = {x 1, x,, x N } S n 1 (unit sphere in R n ) and x i, x j = a
Διαβάστε περισσότεραSolutions to Exercise Sheet 5
Solutions to Eercise Sheet 5 jacques@ucsd.edu. Let X and Y be random variables with joint pdf f(, y) = 3y( + y) where and y. Determine each of the following probabilities. Solutions. a. P (X ). b. P (X
Διαβάστε περισσότεραTMA4115 Matematikk 3
TMA4115 Matematikk 3 Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet Trondheim Spring 2010 Lecture 12: Mathematics Marvellous Matrices Andrew Stacey Norges Teknisk-Naturvitenskapelige Universitet
Διαβάστε περισσότερα1 3D Helmholtz Equation
Deivation of the Geen s Funtions fo the Helmholtz and Wave Equations Alexande Miles Witten: Deembe 19th, 211 Last Edited: Deembe 19, 211 1 3D Helmholtz Equation A Geen s Funtion fo the 3D Helmholtz equation
Διαβάστε περισσότεραOn mixing generalized poison with Generalized Gamma distribution
34 WALFORD I.E. CHUKWU(*) and DEVENDRA GUPTA (**) On mixing genealized poison with Genealized Gamma distibution CONTENTS: Intoduction Mixtue. Refeences. Summay. Riassunto. Key wods (*) Walfod I.E. Chukwu
Διαβάστε περισσότεραTridiagonal matrices. Gérard MEURANT. October, 2008
Tridiagonal matrices Gérard MEURANT October, 2008 1 Similarity 2 Cholesy factorizations 3 Eigenvalues 4 Inverse Similarity Let α 1 ω 1 β 1 α 2 ω 2 T =......... β 2 α 1 ω 1 β 1 α and β i ω i, i = 1,...,
Διαβάστε περισσότεραMatrices and Determinants
Matrices and Determinants SUBJECTIVE PROBLEMS: Q 1. For what value of k do the following system of equations possess a non-trivial (i.e., not all zero) solution over the set of rationals Q? x + ky + 3z
Διαβάστε περισσότεραEcon 2110: Fall 2008 Suggested Solutions to Problem Set 8 questions or comments to Dan Fetter 1
Eon : Fall 8 Suggested Solutions to Problem Set 8 Email questions or omments to Dan Fetter Problem. Let X be a salar with density f(x, θ) (θx + θ) [ x ] with θ. (a) Find the most powerful level α test
Διαβάστε περισσότεραAnswer sheet: Third Midterm for Math 2339
Answer sheet: Third Midterm for Math 339 November 3, Problem. Calculate the iterated integrals (Simplify as much as possible) (a) e sin(x) dydx y e sin(x) dydx y sin(x) ln y ( cos(x)) ye y dx sin(x)(lne
Διαβάστε περισσότεραPartial Differential Equations in Biology The boundary element method. March 26, 2013
The boundary element method March 26, 203 Introduction and notation The problem: u = f in D R d u = ϕ in Γ D u n = g on Γ N, where D = Γ D Γ N, Γ D Γ N = (possibly, Γ D = [Neumann problem] or Γ N = [Dirichlet
Διαβάστε περισσότεραThe ε-pseudospectrum of a Matrix
The ε-pseudospectrum of a Matrix Feb 16, 2015 () The ε-pseudospectrum of a Matrix Feb 16, 2015 1 / 18 1 Preliminaries 2 Definitions 3 Basic Properties 4 Computation of Pseudospectrum of 2 2 5 Problems
Διαβάστε περισσότεραProblem Set 3: Solutions
CMPSCI 69GG Applied Information Theory Fall 006 Problem Set 3: Solutions. [Cover and Thomas 7.] a Define the following notation, C I p xx; Y max X; Y C I p xx; Ỹ max I X; Ỹ We would like to show that C
Διαβάστε περισσότεραSOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM
SOLUTIONS TO MATH38181 EXTREME VALUES AND FINANCIAL RISK EXAM Solutions to Question 1 a) The cumulative distribution function of T conditional on N n is Pr T t N n) Pr max X 1,..., X N ) t N n) Pr max
Διαβάστε περισσότεραAn Inventory of Continuous Distributions
Appendi A An Inventory of Continuous Distributions A.1 Introduction The incomplete gamma function is given by Also, define Γ(α; ) = 1 with = G(α; ) = Z 0 Z 0 Z t α 1 e t dt, α > 0, >0 t α 1 e t dt, α >
Διαβάστε περισσότεραw o = R 1 p. (1) R = p =. = 1
Πανεπιστήµιο Κρήτης - Τµήµα Επιστήµης Υπολογιστών ΗΥ-570: Στατιστική Επεξεργασία Σήµατος 205 ιδάσκων : Α. Μουχτάρης Τριτη Σειρά Ασκήσεων Λύσεις Ασκηση 3. 5.2 (a) From the Wiener-Hopf equation we have:
Διαβάστε περισσότεραMathCity.org Merging man and maths
MathCity.org Merging man and maths Exercise 10. (s) Page Textbook of Algebra and Trigonometry for Class XI Available online @, Version:.0 Question # 1 Find the values of sin, and tan when: 1 π (i) (ii)
Διαβάστε περισσότεραThe k-bessel Function of the First Kind
International Mathematical Forum, Vol. 7, 01, no. 38, 1859-186 The k-bessel Function of the First Kin Luis Guillermo Romero, Gustavo Abel Dorrego an Ruben Alejanro Cerutti Faculty of Exact Sciences National
Διαβάστε περισσότεραderivation of the Laplacian from rectangular to spherical coordinates
derivation of the Laplacian from rectangular to spherical coordinates swapnizzle 03-03- :5:43 We begin by recognizing the familiar conversion from rectangular to spherical coordinates (note that φ is used
Διαβάστε περισσότεραChapter 6: Systems of Linear Differential. be continuous functions on the interval
Chapter 6: Systems of Linear Differential Equations Let a (t), a 2 (t),..., a nn (t), b (t), b 2 (t),..., b n (t) be continuous functions on the interval I. The system of n first-order differential equations
Διαβάστε περισσότεραBounding Nonsplitting Enumeration Degrees
Bounding Nonsplitting Enumeration Degrees Thomas F. Kent Andrea Sorbi Università degli Studi di Siena Italia July 18, 2007 Goal: Introduce a form of Σ 0 2-permitting for the enumeration degrees. Till now,
Διαβάστε περισσότερα