Wishart α-determinant, α-hafnian
|
|
- Βαυκις Ράγκος
- 6 χρόνια πριν
- Προβολές:
Transcript
1 Wishart α-determinant, α-hafnian (, JST CREST) (, JST CREST), Wishart,. ( )Wishart,. determinant Hafnian analogue., ( )Wishart,. 1 Introduction, Wishart. p ν M = (µ 1,..., µ ν ) = (µ ij ) i=1,...,p p p Σ = (σ i,j ) fix = M tm. j=1,...,ν X 1 = (x i1 ) 1 i p, X 2 = (x i2 ) 1 i p,..., X ν = (x iν ) 1 i p ν p, N p (µ 1, Σ),..., N p (µ ν, Σ),, N p (µ i, Σ) µ i, Σ. X (X 1,..., X p ) ν p, W = (w ij ) W = X tx., W, W Wishart W p (ν, Σ, )., mean square matrix, = 0 Wishart W p (ν, Σ).,, W = X t X Wishart CWp (ν, Σ, ).,. x = (x 1,..., x m ) D. n = (n 1,..., n m ), x n = x n 1 1 xn m m E[x n 1 1 xn m m ] D (n- ). D, D., Wishart, Wishart : E[e tr(θw ) ] = det(i 2ΘΣ) ν 2 e 1 2 tr(i (I 2ΘΣ) 1 )Ω,, Θ p p symmetric parameter matrix, Ω ΣΩ = 1
2 . Wishart ( [2, 11] ). Wishart E[w i1,i 2 ]., determinant analogue. E[w i1,i 2 ]. [5]. 2 Notation of graphs,.,. v w, v w {v, w}. v w, {v, w} = {w, v}., self loop {v, v}. V, U, K V K V,U : K V,U = { {v, u} v V, u U, v u }, K V = K V,V = { {v, u} v u V }. V E K V G = (V, E ), vertex(e ) = { v V {v, u} E for some u V }. (V, K ). E K (V, K ) : {v, u}, {v, u } E = u = u., ( ) 1. M (V, K ) (V, K )., (V, K V ) M (V, K V ) M (V )., (V, K ) E vertex(e ) = V, E perfect. ( 1.) P (V, K ) (V, K ) perfect matchings. P (V ) = P(V, K V ).. v w (v, u). v u, (v, u) (u, v).,, self loop (v, v)., self loop. V, U, K V K V,U K V,U = { (v, u) v V, u U }, K V = K V,V = { (v, u) v, u V }. V E K V G = (V, E), start(e) end(e) 2
3 : start(e) = { v V (v, u) E for some u V }, end(e) = { u V (v, u) E for some v V }. (V, K). 2 E K (V, K) : (v, u), (v, u ) E = u = u (v, u), (v, u) E = v = v., ( ) ( ) 1. (V, K) M(V, K). E M(V, K) start(e) = V end(e) = V, perfect, P(V, K) (V, K) perfect matching. Perfect matching 1., M(V ) = M(V, K V ), P(V ) = P(V, K V ). Remark 2.1. V Z, V = { v v V }, V = { v v V }, l = 2l 1, l = 2l., (V, E) 2 ( V, V, { { v, ü} (v, u) E })., M(V, K) 2 ( V, V, { { v, ü} (v, u) E }). M(V, K). 3 Definition of our polynomials l Z, l = 2l 1, l = 2l. n Z>0 fix, V, V fix : V = [n] = { 1,..., n }, V = [n] = { 1,..., ṅ }, V = [n] { = 1,..., n }, V = V V = [n] [n] = [2n].,. E M(V ). (V, E). len(e) (V, E). V \ start(e) (V, E), V \ end(e)., Ě : Ě = { (v, u) K V \start(e),v \end(e) E u v. } K V. Remark 3.1. E M(V ), Ě : Ě M(V ), Ě E =, Ě E P(E), (V, E) (V, Ě E). 3
4 Remark 3.2. E M(V ), len(e) : len(e) = ((V, E) ) Ě., E P(V ), (i, j) E σ E (i) = j n S n σ E, E, len(e) σ E. (V, E) x = (x i,j ), weight monomial x E x E = (v,u) E. : Definition 3.3. K K V, det α (x, y; K) det α (x; K) : det α (x, y; K) = α n len(e) x E yě, det α (x; K) = E M(V,K) E P(V,K) x v,u α n len(e) x E., det α (x, y) = det α (x, y; K V ), det α (x) = det α (x; K V ). Remark 3.4. det α (x; K) = det α (x, 0; K). {,. { 1, 1},..., {ṅ, n} } K V, E V 0. E M (V ), Ě len(e ) : { } Ě = {v, u} K V \vertex(e ) E E 0 v u. len(e ) = ((V, E E 0) ) Ě. Remark 3.5. E M (V ), Ě : Ě M (V ), Ě E =, Ě E P (E ), (V, E E 0) (V, Ě E E 0). Remark 3.6. E M (V ), (V, E E 0)., (V, E E 0). E V \ vertex(e ) (V, E E 0). (V, E E 0) Ě. 4
5 (V, E ) x = (x i,j ), weight monomial x E x E = {v,u} E x v,u. v, u V x v,u = x u,v, x E well-defined. Definition 3.7. K K V, Hf α(x, y; K ) Hf α (x; K ) : Hf α (x, y; K ) = α n len(e ) x E, yě Hf α (x; K ) = E M (V,K ) E P (V,K ) α n len(e ) x E. Hf α (x, y) = Hf α (x, y; K V ), Hf α(x) = Hf α (x; K V ). Remark 3.8., Hf α (x; K ) = Hf α (x, 0; K ). Remark 3.9. A = (a ij ), α-determinant (or α-permanent) : det α (A) = σ S n α n len(σ) a 1,σ(1) a 2,σ(2) a n,σ(n). determinant permanent α-analogue ;, α-determinant α = 1 determinant, α = 1 permanent. (See also [13, 14].) Remark 3.2, α-determinant det α (A). [9] α-pfaffian, skew-symmetric matrix A : Pf α (A) = E P (V ) ( α) n len(e ) sgn(e )A E. sgn(e )A E E = { {x 1, x 1 },..., {x ṅ, x n } } x S 2n sgn(x)a x 1,x 1 a xṅ,x n, A skew symmetric sgn(e )A E x S 2n. α-pfaffian Pfaffian α-analogue,, α = 1 α-pfaffian Pfaffian Pf(A), i.e., sgn(x)a x 1 x 1 a xṅx n,., symmetric matrix B Hf α (B) Hf α (B) = α n len(e ) B E E P (V ),, α = 1, Hafnian Hf(B) = b x 1 x 1 b xṅx n. Hafnian analogue. 5
6 4 Main results det α (x, y), Hf α (x, y) Wishart. Propsition 4.1. W = (w i,j ) W p (ν, Σ, ),, W Wishart W p (ν, Σ, ). A B : a u,v = σ u,v, b u,v = δ u,v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb). : Theorem 4.2. A B : a u,v = σ iu,i v, b u,v = δ iu,i v. W W p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n Hf ν 1(A, B) = Hf ν 1(νA, νb)., Wishart : Propsition 4.3. W = (w i,j ) CW p (ν, Σ, ), A, B : a u,v = σ u, v, b u,v = δ u, v. : E[w 1,2 w 3,4 w 2n 1,2n ] = E[w 1, 1 w 2, 2 w ṅ, n] = ν n det ν 1(A, B) = det ν 1(νA, νb)., : Theorem 4.4. A, B : a u,v = σ i u,i v, b u,v = δ i u,i v., W = (w i,j ) CW p (ν, Σ, ) : E[w i1,i 2 ] = E[w i w 1,i 1 i w 2,i 2 iṅ,i n ] = ν n det ν 1(A, B) = det ν 1(νA, νb). Remark 4.5.,, Wishart,. [5, 12]. Remark 4.6. Wishart,, Wishart, Lu, Richards [7]; Graczyk, Letac, Massam [3, 4]; Vere-Jones [13]., Letac, Massam [6] Wishart,., Wishart,, Matsumoto [10]. 6
7 [1] Bai, Z. D. (1999). Methodologies in spectral analysis of large dimensional random matrices, A review. Statist. Sinica, 9, [2] Goodman, N. R. (1963). Statistical analysis based on a certain multivariate complex Gaussian distribution (An introduction). Ann. Math. Statist., 34, [3] Graczyk, P., Letac, G. and Massam, H. (2003). The complex Wishart distribution and the symmetric groups. Ann. Statist., 31, [4] Graczyk, P., Letac, G. and Massam, H. (2005). The hyperoctahedral group, symmetric group representations and the moments of the real Wishart distribution. J. Theor. Probab., 18, [5] Kuriki, S. and Numata, Y. (2009). Graph representations for moments of noncentral Wishart distributions and their applications. Submitted. arxiv: v1 [6] Letac, G. and Massam, H. (2008). The noncentral Wishart as an exponential family, and its moments. J. Multivariate Anal., 99, [7] Lu, I-L. and Richards, D. St. P. (2001). MacMahon s master theorem, representation theory, and moments of Wishart distributions. Adv. Appl. Math., 27, [8] Maiwald, D. and Kraus, D. (2000). Calculation of moments of complex Wishart and complex inverse Wishart distributed matrices. IEE Proc.-Radar, Sonar Navig, 147, [9] Matsumoto, S. (2005) α-pfaffian, pfaffian point process and shifted Schur measure. Linear Algebra and its Applications, 403, [10] Matsumoto, S. (2010) General moments of the inverse real Wishart distribution and orthogonal Weingarten functions, arxiv: v2 [11] Muirhead, R. J. (1982). Aspects of Multivariate Statistical Theory. John Wiley & Sons [12] Numata, Y. and Kuriki, S. (2009). On formulas for moments of the Wishart distributions as weighted generating functions of matchings, DMTCS Proceedings, 22nd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2010), pp (online journal) [13] Vere-Jones, D. (1988). A generalization of permanents and determinants. Linear Algebra Appl., 111, [14] Vere-Jones, D. (1997). Alpha-permanents and their applications to multivariate gamma, negative binomial and ordinary binomial distributions. New Zealand J. Math., 26,
Sho Matsumoto Graduate School of Mathematics, Nagoya University. Tomoyuki Shirai Institute of Mathematics for Industry, Kyushu University
Sho Matsumoto Graduate School of Mathematics, Nagoya University Tomoyuki Shirai Institute of Mathematics for Industry, Kyushu University. Kac f n (t) = n k=0 a kt k ({a k } n k=0 i.i.d. ) N n E[N n ] =
Διαβάστε περισσότεραHorizontal and Vertical Recurrence Relations for Exponential Riordan Matrices and Their Applications
4th RART, July 17-20 2017, Universidad Complutense de Madrid Horizontal and Vertical Recurrence Relations for Exponential Riordan Matrices and Their Applications Ji-Hwan Jung Sungkyunkwan University, Korea
Διαβάστε περισσότεραJ. of Math. (PRC) Banach, , X = N(T ) R(T + ), Y = R(T ) N(T + ). Vol. 37 ( 2017 ) No. 5
Vol. 37 ( 2017 ) No. 5 J. of Math. (PRC) 1,2, 1, 1 (1., 225002) (2., 225009) :. I +AT +, T + = T + (I +AT + ) 1, T +. Banach Hilbert Moore-Penrose.. : ; ; Moore-Penrose ; ; MR(2010) : 47L05; 46A32 : O177.2
Διαβάστε περισσότεραFundamentals of Probability: A First Course. Anirban DasGupta
Fundamentals of Probability: A First Course Anirban DasGupta Contents 1 Introducing Probability 5 1.1 ExperimentsandSampleSpaces... 6 1.2 Set Theory Notation and Axioms of Probability........... 7 1.3
Διαβάστε περισσότεραg-selberg integrals MV Conjecture An A 2 Selberg integral Summary Long Live the King Ole Warnaar Department of Mathematics Long Live the King
Ole Warnaar Department of Mathematics g-selberg integrals The Selberg integral corresponds to the following k-dimensional generalisation of the beta integral: D Here and k t α 1 i (1 t i ) β 1 1 i
Διαβάστε περισσότεραON NEGATIVE MOMENTS OF CERTAIN DISCRETE DISTRIBUTIONS
Pa J Statist 2009 Vol 25(2), 135-140 ON NEGTIVE MOMENTS OF CERTIN DISCRETE DISTRIBUTIONS Masood nwar 1 and Munir hmad 2 1 Department of Maematics, COMSTS Institute of Information Technology, Islamabad,
Διαβάστε περισσότερα[I2], [IK1], [IK2], [AI], [AIK], [INA], [IN], [IK2], [IA1], [I3], [IKP], [BIK], [IA2], [KB]
(Akihiko Inoue) Graduate School of Science, Hiroshima University (Yukio Kasahara) Graduate School of Science, Hokkaido University Mohsen Pourahmadi, Department of Statistics, Texas A&M University 1, =
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 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 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 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 περισσότερα255 (log-normal distribution) 83, 106, 239 (malus) 26 - (Belgian BMS, Markovian presentation) 32 (median premium calculation principle) 186 À / Á (goo
(absolute loss function)186 - (posterior structure function)163 - (a priori rating variables)25 (Bayes scale) 178 (bancassurance)233 - (beta distribution)203, 204 (high deductible)218 (bonus)26 ( ) (total
Διαβάστε περισσότεραBundle Adjustment for 3-D Reconstruction: Implementation and Evaluation
3 2 3 2 3 undle Adjustment or 3-D Reconstruction: Implementation and Evaluation Yuuki Iwamoto, Yasuyuki Sugaya 2 and Kenichi Kanatani We describe in detail the algorithm o bundle adjustment or 3-D reconstruction
Διαβάστε περισσότερα( ) 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K.
( ),.,,, 1, [17]. 1. 1.1. (2 ),,.,.,.,,,,,.,,,,.,,., K, K. 1.2. Σ g g. M g, Σ g. g 1 Σ g,, Σ g Σ g. Σ g, M g,, Σ g.. g = 1, M 1 M 1, SL(2, Z). Q. g = 2, 2000 M 2 (Korkmaz [20], Bigelow Budney [5])., Bigelow
Διαβάστε περισσότεραDETERMINANT AND PFAFFIAN OF SUM OF SKEW SYMMETRIC MATRICES. 1. Introduction
Unspecified Journal Volume 00, Number 0, Pages 000 000 S????-????(XX)0000-0 DETERMINANT AND PFAFFIAN OF SUM OF SKEW SYMMETRIC MATRICES TIN-YAU TAM AND MARY CLAIR THOMPSON Abstract. We completely describe
Διαβάστε περισσότεραDeterminant and inverse of a Gaussian Fibonacci skew-hermitian Toeplitz matrix
Available online at wwwisr-publicationscom/nsa J Nonlinear Sci Appl 1 (17) 3694 377 Research Article Journal Homepage: wwwtnsacom - wwwisr-publicationscom/nsa Determinant and inverse of a Gaussian Fibonacci
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 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 περισσότεραΒιογραφικό Σημείωμα. Γεωργίου Κ. Ελευθεράκη ΓΕΝΙΚΑ ΣΤΟΙΧΕΙΑ EKΠΑΙΔΕΥΣΗ
Βιογραφικό Σημείωμα Γεωργίου Κ. Ελευθεράκη ΓΕΝΙΚΑ ΣΤΟΙΧΕΙΑ Ημερομηνία Γέννησης: 23 Δεκεμβρίου 1962. Οικογενειακή Κατάσταση: Έγγαμος με δύο παιδιά. EKΠΑΙΔΕΥΣΗ 1991: Πτυχίο Οικονομικού Τμήματος Πανεπιστημίου
Διαβάστε περισσότεραComputable error bounds for asymptotic expansions formulas of distributions related to gamma functions
Computable error bounds for asymptotic expansions formulas of distributions related to gamma functions Hirofumi Wakaki (Math. of Department, Hiroshima Univ.) 20.7. Hiroshima Statistical Group Meeting at
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 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 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 περισσότεραOn the Galois Group of Linear Difference-Differential Equations
On the Galois Group of Linear Difference-Differential Equations Ruyong Feng KLMM, Chinese Academy of Sciences, China Ruyong Feng (KLMM, CAS) Galois Group 1 / 19 Contents 1 Basic Notations and Concepts
Διαβάστε περισσότεραDiscriminantal arrangement
Discriminantal arrangement YAMAGATA So C k n arrangement C n discriminantal arrangement 1989 Manin-Schectman Braid arrangement Discriminantal arrangement Gr(3, n) S.Sawada S.Settepanella 1 A arrangement
Διαβάστε περισσότεραJordan Form of a Square Matrix
Jordan Form of a Square Matrix Josh Engwer Texas Tech University josh.engwer@ttu.edu June 3 KEY CONCEPTS & DEFINITIONS: R Set of all real numbers C Set of all complex numbers = {a + bi : a b R and i =
Διαβάστε περισσότεραStatistics 104: Quantitative Methods for Economics Formula and Theorem Review
Harvard College Statistics 104: Quantitative Methods for Economics Formula and Theorem Review Tommy MacWilliam, 13 tmacwilliam@college.harvard.edu March 10, 2011 Contents 1 Introduction to Data 5 1.1 Sample
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 περισσότεραA Bonus-Malus System as a Markov Set-Chain. Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics
A Bonus-Malus System as a Markov Set-Chain Małgorzata Niemiec Warsaw School of Economics Institute of Econometrics Contents 1. Markov set-chain 2. Model of bonus-malus system 3. Example 4. Conclusions
Διαβάστε περισσότεραCOMPUTING MULTIVARIATE STATISTICS
COMPUTING MULTIVARIATE STATISTICS PLAMEN KOEV Abstract. Many multivariate statistics are expressed as functions of the hypergeometric function of a matrix argument, or more generally, as series of Jac
Διαβάστε περισσότεραTakeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, RIMS
Takeaki Yamazaki (Toyo Univ.) 山崎丈明 ( 東洋大学 ) Oct. 24, 2017 @ RIMS Contents Introduction Generalized Karcher equation Ando-Hiai inequalities Problem Introduction PP: The set of all positive definite operators
Διαβάστε περισσότεραProbability and Random Processes (Part II)
Probability and Random Processes (Part II) 1. If the variance σ x of d(n) = x(n) x(n 1) is one-tenth the variance σ x of a stationary zero-mean discrete-time signal x(n), then the normalized autocorrelation
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 περισσότεραSolution Series 9. i=1 x i and i=1 x i.
Lecturer: Prof. Dr. Mete SONER Coordinator: Yilin WANG Solution Series 9 Q1. Let α, β >, the p.d.f. of a beta distribution with parameters α and β is { Γ(α+β) Γ(α)Γ(β) f(x α, β) xα 1 (1 x) β 1 for < x
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 περισσότεραThe k-α-exponential Function
Int Journal of Math Analysis, Vol 7, 213, no 11, 535-542 The --Exponential Function Luciano L Luque and Rubén A Cerutti Faculty of Exact Sciences National University of Nordeste Av Libertad 554 34 Corrientes,
Διαβάστε περισσότεραΒιογραφικό Σημείωμα. Διεύθυνση επικοινωνίας: Τμήμα Μαθηματικών, Πανεπιστήμιο Πατρών
Βιογραφικό Σημείωμα Προσωπικά στοιχεία Όνομα: Σταύρος Επώνυμο: Κουρούκλης Έτος γέννησης: 1952 Τόπος γέννησης: Ληξούρι Κεφαλλονιάς Στρατιωτική θητεία: Φεβρουάριος 2002 Οκτώβριος 2003 Οικογενειακή κατάσταση:
Διαβάστε περισσότεραMatrices and vectors. Matrix and vector. a 11 a 12 a 1n a 21 a 22 a 2n A = b 1 b 2. b m. R m n, b = = ( a ij. a m1 a m2 a mn. def
Matrices and vectors Matrix and vector a 11 a 12 a 1n a 21 a 22 a 2n A = a m1 a m2 a mn def = ( a ij ) R m n, b = b 1 b 2 b m Rm Matrix and vectors in linear equations: example E 1 : x 1 + x 2 + 3x 4 =
Διαβάστε περισσότεραˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ Ä Œμ Ìμ. ±É- É Ê ± μ Ê É Ò Ê É É, ±É- É Ê, μ Ö
ˆ ˆŠ Œ ˆ ˆ Œ ƒ Ÿ 2017.. 48.. 5.. 740Ä744 ˆ Œˆ ƒ Š Œ ˆ Œˆ ˆŸ ˆ ˆ ˆŸ ˆˆ ƒ ˆ Šˆ ˆ.. Œμ Ìμ ±É- É Ê ± μ Ê É Ò Ê É É, ±É- É Ê, μ Ö ±μ³ ² ± ÒÌ ³μ ʲÖÌ Ð É Ò³ ² ³ Š² ËËμ Î É μ - ³ μ É Ò Ë ³ μ Ò ³ Ò Å ²μ ÉÉ. Ì
Διαβάστε περισσότεραN. P. Mozhey Belarusian State University of Informatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS
Òðóäû ÁÃÒÓ 07 ñåðèÿ ñ. 9 54.765.... -. -. -. -. -. : -. N. P. Mozhey Belarusian State University of Inforatics and Radioelectronics NORMAL CONNECTIONS ON SYMMETRIC MANIFOLDS In this article we present
Διαβάστε περισσότεραThe Spiral of Theodorus, Numerical Analysis, and Special Functions
Theo p. / The Spiral of Theodorus, Numerical Analysis, and Special Functions Walter Gautschi wxg@cs.purdue.edu Purdue University Theo p. 2/ Theodorus of ca. 46 399 B.C. Theo p. 3/ spiral of Theodorus 6
Διαβάστε περισσότεραProperties of Matrix Variate Hypergeometric Function Distribution
Applied Mathematical Sciences, Vol. 11, 2017, no. 14, 677-692 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.7254 Properties of Matrix Variate Hypergeometric Function Distribution Daya
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 περισσότεραMantel & Haenzel (1959) Mantel-Haenszel
Mantel-Haenszel 2008 6 12 1 / 39 1 (, (, (,,, pp719 730 2 2 2 3 1 4 pp730 746 2 2, i j 3 / 39 Mantel & Haenzel (1959 Mantel N, Haenszel W Statistical aspects of the analysis of data from retrospective
Διαβάστε περισσότεραNotations. Primary definition. Specific values. General characteristics. Traditional name. Traditional notation. Mathematica StandardForm notation
KelvinKei Notations Traditional name Kelvin function of the second kind Traditional notation kei Mathematica StandardForm notation KelvinKei Primary definition 03.5.0.000.0 kei kei 0 Specific values Values
Διαβάστε περισσότεραStabilization of stock price prediction by cross entropy optimization
,,,,,,,, Stabilization of stock prediction by cross entropy optimization Kazuki Miura, Hideitsu Hino and Noboru Murata Prediction of series data is a long standing important problem Especially, prediction
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 περισσότεραOptimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices
Optimal Parameter in Hermitian and Skew-Hermitian Splitting Method for Certain Two-by-Two Block Matrices Chi-Kwong Li Department of Mathematics The College of William and Mary Williamsburg, Virginia 23187-8795
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 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 περισσότεραNotations. Primary definition. Specific values. General characteristics. Series representations. Traditional name. Traditional notation
Pi Notations Traditional name Π Traditional notation Π Mathematica StandardForm notation Pi Primary definition.3... Π Specific values.3.3.. Π 3.5965358979338663383795889769399375589795937866868998683853
Διαβάστε περισσότεραBayesian statistics. DS GA 1002 Probability and Statistics for Data Science.
Bayesian statistics DS GA 1002 Probability and Statistics for Data Science http://www.cims.nyu.edu/~cfgranda/pages/dsga1002_fall17 Carlos Fernandez-Granda Frequentist vs Bayesian statistics In frequentist
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 περισσότεραThe Jordan Form of Complex Tridiagonal Matrices
The Jordan Form of Complex Tridiagonal Matrices Ilse Ipsen North Carolina State University ILAS p.1 Goal Complex tridiagonal matrix α 1 β 1. γ T = 1 α 2........ β n 1 γ n 1 α n Jordan decomposition T =
Διαβάστε περισσότερα= λ 1 1 e. = λ 1 =12. has the properties e 1. e 3,V(Y
Stat 50 Homework Solutions Spring 005. (a λ λ λ 44 (b trace( λ + λ + λ 0 (c V (e x e e λ e e λ e (λ e by definition, the eigenvector e has the properties e λ e and e e. (d λ e e + λ e e + λ e e 8 6 4 4
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 περισσότεραCyclic or elementary abelian Covers of K 4
Cyclic or elementary abelian Covers of K 4 Yan-Quan Feng Mathematics, Beijing Jiaotong University Beijing 100044, P.R. China Summer School, Rogla, Slovenian 2011-06 Outline 1 Question 2 Main results 3
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 περισσότεραIntroduction to Time Series Analysis. Lecture 16.
Introduction to Time Series Analysis. Lecture 16. 1. Review: Spectral density 2. Examples 3. Spectral distribution function. 4. Autocovariance generating function and spectral density. 1 Review: Spectral
Διαβάστε περισσότεραJean Bourgain Institute for Advanced Study Princeton, NJ 08540
Jean Bourgain Institute for Advanced Study Princeton, NJ 08540 1 PRIMES IN LINEAR GROUPS Joint work with A Gamburd, A Kontorovich, P Sarnak 2 Primes and pseudo-primes in orbits of groups acting on Z n
Διαβάστε περισσότεραExponential Families
Exponential Families Robert L. Wolpert Department of Statistical Science Duke University, Durham, NC, USA Surprisingly many of the distributions we use in statistics for random variables taking value in
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 περισσότεραOn the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University)
On the summability of divergent power series solutions for certain first-order linear PDEs Masaki HIBINO (Meijo University) 1 1 Introduction (E) {1+x 2 +β(x,y)}y u x (x,y)+{x+b(x,y)}y2 u y (x,y) +u(x,y)=f(x,y)
Διαβάστε περισσότεραCHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS
CHAPTER 48 APPLICATIONS OF MATRICES AND DETERMINANTS EXERCISE 01 Page 545 1. Use matrices to solve: 3x + 4y x + 5y + 7 3x + 4y x + 5y 7 Hence, 3 4 x 0 5 y 7 The inverse of 3 4 5 is: 1 5 4 1 5 4 15 8 3
Διαβάστε περισσότεραExpIntegralE. Notations. Primary definition. Specific values. Traditional name. Traditional notation. Mathematica StandardForm notation
ExpIntegralE Notations Traditional name Exponential integral E Traditional notation E Mathematica StandardForm notation ExpIntegralE, Primary definition 06.34.0.000.0 E t t t ; Re 0 Specific values Specialied
Διαβάστε περισσότεραCovariance and Pseudo-Covariance of Complex Uncertain Variables
Covariance and Pseudo-Covariance of Complex Uncertain Variables Rong Gao 1, Hamed Ahmadzade 2, Mojtaba Esfahani 3 1. School of Economics and Management, Hebei University of Technology, Tianjin 341, China
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 περισσότερα11 Drinfeld. k( ) = A/( ) A K. [Hat1, Hat2] k M > 0. Γ 1 (M) = γ SL 2 (Z) f : H C. ( ) az + b = (cz + d) k f(z) ( z H, γ = cz + d Γ 1 (M))
Drinfeld Drinfeld 29 8 8 11 Drinfeld [Hat3] 1 p q > 1 p A = F q [t] A \ F q d > 0 K A ( ) k( ) = A/( ) A K Laurent F q ((1/t)) 1/t C Drinfeld Drinfeld p p p [Hat1, Hat2] 1.1 p 1.1.1 k M > 0 { Γ 1 (M) =
Διαβάστε περισσότεραGaussian related distributions
Gaussian related distributions Santiago Aja-Fernández June 19, 009 1 Gaussian related distributions 1. Gaussian: ormal PDF: MGF: Main moments:. Rayleigh: PDF: MGF: Raw moments: Main moments: px = 1 σ π
Διαβάστε περισσότεραDivergence for log concave functions
Divergence or log concave unctions Umut Caglar The Euler International Mathematical Institute June 22nd, 2013 Joint work with C. Schütt and E. Werner Outline 1 Introduction 2 Main Theorem 3 -divergence
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 περισσότεραMagnetically Coupled Circuits
DR. GYURCSEK ISTVÁN Magnetically Coupled Circuits Sources and additional materials (recommended) Dr. Gyurcsek Dr. Elmer: Theories in Electric Circuits, GlobeEdit, 2016, ISBN:978-3-330-71341-3 Ch. Alexander,
Διαβάστε περισσότεραThe Determinant of a Hypergeometric Period Matrix and a Generalization of Selberg s Integral
The Determinant of a Hypergeometric Period Matrix and a Generalization of Selberg s Integral Donald Richards a,,1 Qifu Zheng b, a Department of Statistics, Penn State University, University Park, PA 1680,
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 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 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 περισσότεραGlobal nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl
Around Vortices: from Cont. to Quantum Mech. Global nonlinear stability of steady solutions of the 3-D incompressible Euler equations with helical symmetry and with no swirl Maicon José Benvenutti (UNICAMP)
Διαβάστε περισσότεραHOSVD. Higher Order Data Classification Method with Autocorrelation Matrix Correcting on HOSVD. Junichi MORIGAKI and Kaoru KATAYAMA
DEIM Forum 2010 D1-4 HOSVD 191-0065 6-6 E-mail: j.morigaki@gmail.com, katayama@tmu.ac.jp Lathauwer (HOSVD) (Tensor) HOSVD Savas HOSVD Sun HOSVD,, Higher Order Data Classification Method with Autocorrelation
Διαβάστε περισσότεραCBC MATHEMATICS DIVISION MATH 2412-PreCalculus Exam Formula Sheets
System of Equations and Matrices 3 Matrix Row Operations: MATH 41-PreCalculus Switch any two rows. Multiply any row by a nonzero constant. Add any constant-multiple row to another Even and Odd functions
Διαβάστε περισσότεραΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΕΦΑΡΜΟΣΜΕΝΩΝ ΜΑΘΗΜΑΤΙΚΩΝ ΚΑΙ ΦΥΣΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΚΑΤΕΥΘΥΝΣΗ ΜΑΘΗΜΑΤΙΚΟΥ ΕΦΑΡΜΟΓΩΝ ΡΟΕΣ ΣΤΑΤΙΣΤΙΚΗΣ ΑΝΑΛΥΣΗΣ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ του Γεωργίου Π. Νίνη «Η Θεωρία Ομάδων και
Διαβάστε περισσότεραACTA MATHEMATICAE APPLICATAE SINICA Nov., ( µ ) ( (
35 Þ 6 Ð Å Vol. 35 No. 6 2012 11 ACTA MATHEMATICAE APPLICATAE SINICA Nov., 2012 È ÄÎ Ç ÓÑ ( µ 266590) (E-mail: jgzhu980@yahoo.com.cn) Ð ( Æ (Í ), µ 266555) (E-mail: bbhao981@yahoo.com.cn) Þ» ½ α- Ð Æ Ä
Διαβάστε περισσότεραΚΑΘΟΡΙΣΜΟΣ ΠΑΡΑΓΟΝΤΩΝ ΠΟΥ ΕΠΗΡΕΑΖΟΥΝ ΤΗΝ ΠΑΡΑΓΟΜΕΝΗ ΙΣΧΥ ΣΕ Φ/Β ΠΑΡΚΟ 80KWp
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΣΥΣΤΗΜΑΤΩΝ ΜΕΤΑΔΟΣΗΣ ΠΛΗΡΟΦΟΡΙΑΣ ΚΑΙ ΤΕΧΝΟΛΟΓΙΑΣ ΥΛΙΚΩΝ ΚΑΘΟΡΙΣΜΟΣ ΠΑΡΑΓΟΝΤΩΝ ΠΟΥ ΕΠΗΡΕΑΖΟΥΝ ΤΗΝ ΠΑΡΑΓΟΜΕΝΗ ΙΣΧΥ
Διαβάστε περισσότερα6. MAXIMUM LIKELIHOOD ESTIMATION
6 MAXIMUM LIKELIHOOD ESIMAION [1] Maximum Likelihood Estimator (1) Cases in which θ (unknown parameter) is scalar Notational Clarification: From now on, we denote the true value of θ as θ o hen, view θ
Διαβάστε περισσότεραL p approach to free boundary problems of the Navier-Stokes equation
L p approach to free boundary problems of the Navier-Stokes equation e-mail address: yshibata@waseda.jp 28 4 1 e-mail address: ssshimi@ipc.shizuoka.ac.jp Ω R n (n 2) v Ω. Ω,,,, perturbed infinite layer,
Διαβάστε περισσότεραarxiv: v1 [math.ra] 19 Dec 2017
TWO-DIMENSIONAL LEFT RIGHT UNITAL ALGEBRAS OVER ALGEBRAICALLY CLOSED FIELDS AND R HAHMED UBEKBAEV IRAKHIMOV 3 arxiv:7673v [mathra] 9 Dec 7 Department of Math Faculty of Science UPM Selangor Malaysia &
Διαβάστε περισσότεραAn Automatic Modulation Classifier using a Frequency Discriminator for Intelligent Software Defined Radio
C IEEJ Transactions on Electronics, Information and Systems Vol.133 No.5 pp.910 915 DOI: 10.1541/ieejeiss.133.910 a) An Automatic Modulation Classifier using a Frequency Discriminator for Intelligent Software
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 περισσότεραComplex Hadamard Matrices and Combinatorial Structures
Complex Hadamard Matrices and Combinatorial Structures Ada Chan Abstract Forty years ago, Goethals and Seidel showed that if the adjacency algebra of a strongly regular graph X contains a Hadamard matrix
Διαβάστε περισσότερα794 Appendix A:Tables
Appendix A Tables A Table Contents Page A.1 Random numbers 794 A.2 Orthogonal polynomial trend contrast coefficients 800 A.3 Standard normal distribution 801 A.4 Student s t-distribution 802 A.5 Chi-squared
Διαβάστε περισσότεραPROPERTIES OF CERTAIN INTEGRAL OPERATORS. a n z n (1.1)
GEORGIAN MATHEMATICAL JOURNAL: Vol. 2, No. 5, 995, 535-545 PROPERTIES OF CERTAIN INTEGRAL OPERATORS SHIGEYOSHI OWA Abstract. Two integral operators P α and Q α for analytic functions in the open unit disk
Διαβάστε περισσότεραarxiv:math/ v1 [math.rt] 30 Oct 2006
On Regular Locally Scalar Representations of Graph D in Hilbert Spaces arxiv:math/06093v math.rt 30 Oct 006 S. A. Kruglyak, L. A. Nazarova, A. V. Roiter. Institute of Mathematics of National Academy of
Διαβάστε περισσότεραΖΩΝΟΠΟΙΗΣΗ ΤΗΣ ΚΑΤΟΛΙΣΘΗΤΙΚΗΣ ΕΠΙΚΙΝΔΥΝΟΤΗΤΑΣ ΣΤΟ ΟΡΟΣ ΠΗΛΙΟ ΜΕ ΤΗ ΣΥΜΒΟΛΗ ΔΕΔΟΜΕΝΩΝ ΣΥΜΒΟΛΟΜΕΤΡΙΑΣ ΜΟΝΙΜΩΝ ΣΚΕΔΑΣΤΩΝ
EΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΕΙΟ Τμήμα Μηχανικών Μεταλλείων-Μεταλλουργών ΖΩΝΟΠΟΙΗΣΗ ΤΗΣ ΚΑΤΟΛΙΣΘΗΤΙΚΗΣ ΕΠΙΚΙΝΔΥΝΟΤΗΤΑΣ ΜΕ ΤΗ ΣΥΜΒΟΛΗ ΔΕΔΟΜΕΝΩΝ ΣΥΜΒΟΛΟΜΕΤΡΙΑΣ ΜΟΝΙΜΩΝ ΣΚΕΔΑΣΤΩΝ ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ Κιτσάκη Μαρίνα
Διαβάστε περισσότεραBiostatistics for Health Sciences Review Sheet
Biostatistics for Health Sciences Review Sheet http://mathvault.ca June 1, 2017 Contents 1 Descriptive Statistics 2 1.1 Variables.............................................. 2 1.1.1 Qualitative........................................
Διαβάστε περισσότεραΕξοικονόμηση Ενέργειας σε Εγκαταστάσεις Δρόμων, με Ρύθμιση (Dimming) ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ
ΕΘΝΙΚΟ ΜΕΤΣΟΒΙΟ ΠΟΛΥΤΕΧΝΕΙΟ ΣΧΟΛΗ ΗΛΕΚΤΡΟΛΟΓΩΝ ΜΗΧΑΝΙΚΩΝ ΚΑΙ ΜΗΧΑΝΙΚΩΝ ΥΠΟΛΟΓΙΣΤΩΝ ΤΟΜΕΑΣ ΗΛΕΚΤΡΙΚΗΣ ΙΣΧΥΟΣ Εξοικονόμηση Ενέργειας σε Εγκαταστάσεις Δρόμων, με Ρύθμιση (Dimming) ΔΙΠΛΩΜΑΤΙΚΗ ΕΡΓΑΣΙΑ Νικηφόρος
Διαβάστε περισσότερα«ΑΝΑΠΣΤΞΖ ΓΠ ΚΑΗ ΥΩΡΗΚΖ ΑΝΑΛΤΖ ΜΔΣΔΩΡΟΛΟΓΗΚΩΝ ΓΔΓΟΜΔΝΩΝ ΣΟΝ ΔΛΛΑΓΗΚΟ ΥΩΡΟ»
ΓΔΩΠΟΝΗΚΟ ΠΑΝΔΠΗΣΖΜΗΟ ΑΘΖΝΩΝ ΣΜΗΜΑ ΑΞΙΟΠΟΙΗΗ ΦΤΙΚΩΝ ΠΟΡΩΝ & ΓΕΩΡΓΙΚΗ ΜΗΥΑΝΙΚΗ ΣΟΜΕΑ ΕΔΑΦΟΛΟΓΙΑ ΚΑΙ ΓΕΩΡΓΙΚΗ ΥΗΜΕΙΑ ΕΙΔΙΚΕΤΗ: ΕΦΑΡΜΟΓΕ ΣΗ ΓΕΩΠΛΗΡΟΦΟΡΙΚΗ ΣΟΤ ΦΤΙΚΟΤ ΠΟΡΟΤ «ΑΝΑΠΣΤΞΖ ΓΠ ΚΑΗ ΥΩΡΗΚΖ ΑΝΑΛΤΖ ΜΔΣΔΩΡΟΛΟΓΗΚΩΝ
Διαβάστε περισσότερα5.4 The Poisson Distribution.
The worst thing you can do about a situation is nothing. Sr. O Shea Jackson 5.4 The Poisson Distribution. Description of the Poisson Distribution Discrete probability distribution. The random variable
Διαβάστε περισσότεραSupplementary Material For Testing Homogeneity of. High-dimensional Covariance Matrices
Supplementary Material For Testing Homogeneity of High-dimensional Covariance Matrices Shurong Zheng, Ruitao Lin, Jianhua Guo, and Guosheng Yin 3 School of Mathematics & Statistics and KLAS, Northeast
Διαβάστε περισσότεραAPPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 651 APPENDIX B. BIBLIOGRAPHY 677 APPENDIX C. ANSWERS TO SELECTED EXERCISES 679
APPENDICES APPENDIX A. STATISTICAL TABLES AND CHARTS 1 Table I Summary of Common Probability Distributions 2 Table II Cumulative Standard Normal Distribution Table III Percentage Points, 2 of the Chi-Squared
Διαβάστε περισσότεραLecture 10 - Representation Theory III: Theory of Weights
Lecture 10 - Representation Theory III: Theory of Weights February 18, 2012 1 Terminology One assumes a base = {α i } i has been chosen. Then a weight Λ with non-negative integral Dynkin coefficients Λ
Διαβάστε περισσότεραCoefficient Inequalities for a New Subclass of K-uniformly Convex Functions
International Journal of Computational Science and Mathematics. ISSN 0974-89 Volume, Number (00), pp. 67--75 International Research Publication House http://www.irphouse.com Coefficient Inequalities for
Διαβάστε περισσότερα