DOCSLIB.ORG

Positivity Properties of Some Special Matrices

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Positivity Properties of Some Special Matrices Positivity properties of some special matrices Priyanka Grover ,∗ Veer Singh Panwar ,† A. Satyanarayana Reddy‡ Department of Mathematics Shiv Nadar University, Dadri U.P. 201314, India. Abstract 1 It is shown that for positive real numbers 0 < λ1 < ··· < λn, , where β(·, ·) denotes the beta function, is h β(λi,λj ) i 1 infinitely divisible and totally positive. For h β(i,j) i, the Cholesky decomposition and successive elementary bidiagonal decomposition are computed. Let w(n) be the nth Bell number. It is proved that [w(i + j)] is a totally positive matrix but is infinitely divisible only upto order 4. It is also shown that the symmetrized Stirling matrices are totally positive. AMS classification: 15B48, 42A82,15B36 Keywords : Bell numbers, infinitely divisible matrices, positive semidefinite matrices, Schur product, Stirling num- bers, totally positive matrices, the beta function. 1 Introduction Let Mn(C) be the set of all n n complex matrices. A matrix A Mn(C) is said to be positive × n ∈ n semidefinite if x∗Ax 0 for all x C and positive definite if x∗Ax > 0 for all x C , x = 0. Let ≥ ∈ ∈ 6 A = [aij ] and B = [bij ]. In this paper, 1 i, j n, unless otherwise specified. The Hadamard product or ≤ ≤ the Schur product of two matrices A and B is denoted by A B, where A B = [aij bij ]. For a nonnegative r r ◦ ◦ real number r, A◦ = [aij ]. r Let A = [aij ] be such that aij 0. The matrix A is called infinitely divisible if A◦ is positive semidefinite for every real number r≥ > 0. For examples and properties of infinitely divisible matrices, see [3, 7, 15]. The matrix A is called totally positive or totally nonnegative if all its minors are positive or nonnegative respectively. For more results on these, see [12]. The main objective of this paper is to explore the above mentioned properties for a few matrices which are constructed from interesting arXiv:2002.08703v4 [math.FA] 3 May 2020 functions. Many such matrices have been studied in [3, 7, 5, 6]. A famous example of such a matrix is 1 i+j n the Hilbert matrix i+j 1 . Another important example is the Pascal matrix = i i,j=0. Both − P of these are knownh to be infinitelyi divisible and totally positive. In [11], Cholesky decomposition of P was given, that is, a lower triangular matrix L was obtained such that = LL∗. Let = [ 1 ], where β( , ) is the beta function. We call as the betaP matrix. By definition, B β(i,j) · · B Γ(i)Γ(j) β(i, j)= , Γ(i + j) where Γ( ) is the Gamma function. Thus · (i + j 1)! = − . B (i 1)!(j 1)! − − ∗The research of this author is supported by INSPIRE Faculty Award IFA-14-MA52 of DST, India, and by Early Career Research Award ECR/2018/001784 of SERB, India. Email: [email protected] †Email: [email protected] ‡Email: [email protected] 1 Note that the entries of this matrix look similar to those of . Using the infinite divisibility and total positivity of , we show that is infinitely divisible and totallyP positive, respectively. We also compute the CholeskyP decomposition ofB . B Let Ei,j be the matrix whose (i, j)th entry is 1 and others are zero. For any complex numbers s, t, let Li(s)= I + sEi,i 1 and Uj (t)= I + tEj 1,j , − − where 2 i, j n. Matrices of the form Li(s) or Uj (t) are called elementary bidiagonal matrices. For ≤ ≤ a vector (d , d ,...,dn), let diag([di]) denote the diagonal matrix diag(d ,...,dn). An n n matrix is 1 2 1 × totally positive if and only if it can be written as (Ln (lk) Ln 1 (lk 1) L2 (lk n+2)) − − ··· − (Ln (lk n+1) Ln 1 (lk n) L3 (lk 2n+4)) (Ln (l1))D(Un (u1))(Un 1 (u2) Un (u3)) (U2 (uk n+2) − − − ··· −n ··· − ··· − ··· Un 1 (uk 1) Un (uk)), where k = , li,uj > 0 for all i, j 1, 2,...,k and D = diag[(di)] is a di- − − 2 ∈ { } agonal matrix with all di > 0 [12, Corollary 2.2.3]. This particular factorization (with li,uj 0) is called successive elementary bidiagonal (SEB) factorization or Neville factorization. We obtain≥ the 1 SEB factorization for β(i,j) explicitly. Let (x)0 = 1 and forh a positivei integer k, let (x)k = x(x 1)(x 2) (x k +1). The Stirling numbers of first kind s(n, k) [9, p. 213] and the Stirling numbers− of second− ··· kind−S(n, k) [9, p. 207] are respectively defined as n k (x)n = s(n, k)x k X=0 and n n x = S(n, k)(x)k. k X=0 The unsigned Stirling matrix of first kind s = [sij ] is defined as i j ( 1) − s(i, j) if i j, sij = − ≥ (0 otherwise. The Stirling matrix of second kind = [ ij ] is defined as S S S(i, j) if i j, ij = ≥ S (0 otherwise. It is a well known fact that s and are totally nonnegative (see for example [13]). We consider the S matrices ss∗ and ∗ and call them symmetrized unsigned Stirling matrix of first kind and symmetrized Stirling matrix ofSS second kind, respectively. By definition, these are positive semidefinite and totally nonnegative. We show that both these matrices are in fact totally positive. Another matrix that we consider is formed by the well known Bell numbers. The sum w(n) = n k=0 S(n, k) is the number of partitions of a set of n objects and is known as the nth Bell number [9, B w n 1 n 1 p. 210]. Consider the matrix = [ (i + j)]i,j−=0. Let X = (xij )i,j−=0 be the lower triangular matrix definedP recursively by x00 =1, x0j = 0 for j > 0, and xij = xi 1,j 1 + (j + 1)xi 1,j + (j + 1)xi 1,j+1 for i 1. − − − − ≥ (Here xi, 1 = 0 for every i and xin = 0 for i = 0,...,n 2.) This is known as the Bell triangle [8]. − − n 1 − Lemma 2 in [1] gives the Cholesky decomposition of B as LL∗, where L = X diag √i! . It is i=0 B n 1 B h i also shown in [1] that det = i=0− i!. It is a known fact that is totally nonnegative [18]. We show that B is totally positive. We also show that B is infinitely divisible only upto order 4. In Section 2 we give our resultsQ for the beta matrix, namely, its infinite divisibility and total positivity, 1 r its Cholesky decomposition, its determinant, − , and its SEB factorization. We show that ◦ is in B B 2 fact totally positive for all r> 0. We end the discussion on the beta matrix by proving that for positive 1 1 real numbers λ1 < < λn and µ1 < <µn, is an infinitely divisible matrix and ··· ··· β(λi,λj ) β(λi,µj ) is a totally positive matrix. In section 3, we proveh that symmetrizedi Stirling matrices and B areh totallyi positive. For the first kind, we give the SEB factorization for s. For the second kind, we show that is triangular totally positive [12, p. 3]. We also show that B is infinitely divisible if and only if its orderS is less than or equal to 4. 2 The beta matrix The infinite divisibility and total positivity of are easy consequences of the corresponding results for . For 1 i, j n, let A(i, j) denote the submatrixB of A obtained by deleting ith row and jth column fromP A. Each≤ A≤(i,i) is infinitely divisible, if A is infinitely divisible, and each A(i, j) is totally positive, if A is totally positive. Theorem 2.1. The matrix = 1 is infinitely divisible and totally positive. B β(i,j) h i Proof. By definition, 1 (i + j 1)! ij(i + j)! 1 i + j = − = = . (1) β(i, j) (i 1)!(j 1)! (i + j)i!j! 1 + 1 i − − i j 1 Thus = C (1, 1), where C = 1 + 1 is a Cauchy matrix. Both C and (1, 1) are infinitely divisible B ◦P i j P [3]. Since Hadamard product of infinitelyh i divisible matrices is infinitely divisible, we get is infinitely divisible. B Again, note that 1 (i + j 1)! (i + j 1)! (i 1)+ j = − = j − = − j. (2) β(i, j) (i 1)!(j 1)! (i 1)!(j!) i 1 − − − − So is the product of the totally positive matrix (n +1, 1) with the positive diagonal matrix diag([i]). HenceB is totally positive. P B The below remarks were suggested by the anonymous referee. r r Remark 2.2. Since for each r> 0, C◦ and ◦ are positive definite (see p. 183 of [4]), (1) gives that r n P r ◦ is positive definite. Let = [(i + j)!]i,j=0. Then is a Hankel matrix. Since ◦ is congruent to B r G r n rG G r ◦ via the positive diagonal matrix diag([i! ])i=0, ◦ is positive definite. The matrix ◦ (n +1, 1) = P r r G r G r [(i + j 1)! ] is congruent to ◦ , via the positive diagonal matrix diag [(i 1)! ]. So ◦ (n +1, 1) is − B r − G r also positive definite. Hence ◦ is totally positive, by Theorem 4.4 in [19]. This shows that ◦ is r G P totally positive. By (2), ◦ is also totally positive. B Remark 2.3. Another proof for infinite divisibility of can be given as follows.
Load more

Recommended publications
  • Stirling Matrix Via Pascal Matrix
    LinearAlgebraanditsApplications329(2001)49–59 www.elsevier.com/locate/laa StirlingmatrixviaPascalmatrix Gi-SangCheon a,∗,Jin-SooKim b aDepartmentofMathematics,DaejinUniversity,Pocheon487-711,SouthKorea bDepartmentofMathematics,SungkyunkwanUniversity,Suwon440-746,SouthKorea Received11May2000;accepted17November2000 SubmittedbyR.A.Brualdi Abstract ThePascal-typematricesobtainedfromtheStirlingnumbersofthefirstkinds(n,k)and ofthesecondkindS(n,k)arestudied,respectively.Itisshownthatthesematricescanbe factorizedbythePascalmatrices.AlsotheLDU-factorizationofaVandermondematrixof theformVn(x,x+1,...,x+n−1)foranyrealnumberxisobtained.Furthermore,some well-knowncombinatorialidentitiesareobtainedfromthematrixrepresentationoftheStirling numbers,andthesematricesaregeneralizedinoneortwovariables.©2001ElsevierScience Inc.Allrightsreserved. AMSclassification:05A19;05A10 Keywords:Pascalmatrix;Stirlingnumber;Stirlingmatrix 1.Introduction Forintegersnandkwithnk0,theStirlingnumbersofthefirstkinds(n,k) andofthesecondkindS(n,k)canbedefinedasthecoefficientsinthefollowing expansionofavariablex(see[3,pp.271–279]): n n−k k [x]n = (−1) s(n,k)x k=0 and ∗ Correspondingauthor. E-mailaddresses:[email protected](G.-S.Cheon),[email protected](J.-S.Kim). 0024-3795/01/$-seefrontmatter2001ElsevierScienceInc.Allrightsreserved. PII:S0024-3795(01)00234-8 50 G.-S. Cheon, J.-S. Kim / Linear Algebra and its Applications 329 (2001) 49–59 n n x = S(n,k)[x]k, (1.1) k=0 where x(x − 1) ···(x − n + 1) if n 1, [x] = (1.2) n 1ifn = 0. It is known that for an n, k 0, the s(n,k), S(n,k) and [n]k satisfy the following Pascal-type recurrence relations: s(n,k) = s(n − 1,k− 1) + (n − 1)s(n − 1,k), S(n,k) = S(n − 1,k− 1) + kS(n − 1,k), (1.3) [n]k =[n − 1]k + k[n − 1]k−1, where s(n,0) = s(0,k)= S(n,0) = S(0,k)=[0]k = 0ands(0, 0) = S(0, 0) = 1, and moreover the S(n,k) satisfies the following formula known as ‘vertical’ recur- rence relation: n− 1 n − 1 S(n,k) = S(l,k − 1).
    [Show full text]
  • Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected]
    Pascal Matrices Alan Edelman and Gilbert Strang Department of Mathematics, Massachusetts Institute of Technology [email protected] and [email protected] Every polynomial of degree n has n roots; every continuous function on [0, 1] attains its maximum; every real symmetric matrix has a complete set of orthonormal eigenvectors. “General theorems” are a big part of the mathematics we know. We can hardly resist the urge to generalize further! Remove hypotheses, make the theorem tighter and more difficult, include more functions, move into Hilbert space,. It’s in our nature. The other extreme in mathematics might be called the “particular case”. One specific function or group or matrix becomes special. It obeys the general rules, like everyone else. At the same time it has some little twist that connects familiar objects in a neat way. This paper is about an extremely particular case. The familiar object is Pascal’s triangle. The little twist begins by putting that triangle of binomial coefficients into a matrix. Three different matrices—symmetric, lower triangular, and upper triangular—can hold Pascal’s triangle in a convenient way. Truncation produces n by n matrices Sn and Ln and Un—the pattern is visible for n = 4: 1 1 1 1 1 1 1 1 1 1 2 3 4 1 1 1 2 3 S4 = L4 = U4 = . 1 3 6 10 1 2 1 1 3 1 4 10 20 1 3 3 1 1 We mention first a very specific fact: The determinant of every Sn is 1. (If we emphasized det Ln = 1 and det Un = 1, you would write to the Editor.
    [Show full text]
  • Arxiv:1904.01037V3 [Math.GR]
    AN EFFECTIVE LIE–KOLCHIN THEOREM FOR QUASI-UNIPOTENT MATRICES THOMAS KOBERDA, FENG LUO, AND HONGBIN SUN Abstract. We establish an effective version of the classical Lie–Kolchin Theo- rem. Namely, let A, B P GLmpCq be quasi–unipotent matrices such that the Jordan Canonical Form of B consists of a single block, and suppose that for all k ě 0 the matrix ABk is also quasi–unipotent. Then A and B have a common eigenvector. In particular, xA, Byă GLmpCq is a solvable subgroup. We give applications of this result to the representation theory of mapping class groups of orientable surfaces. 1. Introduction Let V be a finite dimensional vector space over an algebraically closed field. In this paper, we study the structure of certain subgroups of GLpVq which contain “sufficiently many” elements of a relatively simple form. We are motivated by the representation theory of the mapping class group of a surface of hyperbolic type. If S be an orientable surface of genus 2 or more, the mapping class group ModpS q is the group of homotopy classes of orientation preserving homeomor- phisms of S . The group ModpS q is generated by certain mapping classes known as Dehn twists, which are defined for essential simple closed curves of S . Here, an essential simple closed curve is a free homotopy class of embedded copies of S 1 in S which is homotopically essential, in that the homotopy class represents a nontrivial conjugacy class in π1pS q which is not the homotopy class of a boundary component or a puncture of S .
    [Show full text]
  • The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials
    The Pascal Matrix Function and Its Applications to Bernoulli Numbers and Bernoulli Polynomials and Euler Numbers and Euler Polynomials Tian-Xiao He ∗ Jeff H.-C. Liao y and Peter J.-S. Shiue z Dedicated to Professor L. C. Hsu on the occasion of his 95th birthday Abstract A Pascal matrix function is introduced by Call and Velleman in [3]. In this paper, we will use the function to give a unified approach in the study of Bernoulli numbers and Bernoulli poly- nomials. Many well-known and new properties of the Bernoulli numbers and polynomials can be established by using the Pascal matrix function. The approach is also applied to the study of Euler numbers and Euler polynomials. AMS Subject Classification: 05A15, 65B10, 33C45, 39A70, 41A80. Key Words and Phrases: Pascal matrix, Pascal matrix func- tion, Bernoulli number, Bernoulli polynomial, Euler number, Euler polynomial. ∗Department of Mathematics, Illinois Wesleyan University, Bloomington, Illinois 61702 yInstitute of Mathematics, Academia Sinica, Taipei, Taiwan zDepartment of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada, 89154-4020. This work is partially supported by the Tzu (?) Tze foundation, Taipei, Taiwan, and the Institute of Mathematics, Academia Sinica, Taipei, Taiwan. The author would also thank to the Institute of Mathematics, Academia Sinica for the hospitality. 1 2 T. X. He, J. H.-C. Liao, and P. J.-S. Shiue 1 Introduction A large literature scatters widely in books and journals on Bernoulli numbers Bn, and Bernoulli polynomials Bn(x). They can be studied by means of the binomial expression connecting them, n X n B (x) = B xn−k; n ≥ 0: (1) n k k k=0 The study brings consistent attention of researchers working in combi- natorics, number theory, etc.
    [Show full text]
  • Q-Pascal and Q-Bernoulli Matrices, an Umbral Approach Thomas Ernst
    U.U.D.M. Report 2008:23 q-Pascal and q-Bernoulli matrices, an umbral approach Thomas Ernst Department of Mathematics Uppsala University q- PASCAL AND q-BERNOULLI MATRICES, AN UMBRAL APPROACH THOMAS ERNST Abstract A q-analogue Hn,q ∈ Mat(n)(C(q)) of the Polya-Vein ma- trix is used to define the q-Pascal matrix. The Nalli–Ward–AlSalam (NWA) q-shift operator acting on polynomials is a commutative semi- group. The q-Cauchy-Vandermonde matrix generalizing Aceto-Trigiante is defined by the NWA q-shift operator. A new formula for a q-Cauchy- Vandermonde determinant with matrix elements equal to q-Ward num- bers is found. The matrix form of the q-derivatives of the q-Bernoulli polynomials can be expressed in terms of the Hn,q. With the help of a new q-matrix multiplication certain special q-analogues of Aceto- Trigiante and Brawer-Pirovino are found. The q-Cauchy-Vandermonde matrix can be expressed in terms of the q-Bernoulli matrix. With the help of the Jackson-Hahn-Cigler (JHC) q-Bernoulli polynomials, the q-analogue of the Bernoulli complementary argument theorem is ob- tained. Analogous results for q-Euler polynomials are obtained. The q-Pascal matrix is factorized by the summation matrices and the so- called q-unit matrices. 1. Introduction In this paper we are going to find q-analogues of matrix formulas from two pairs of authors: L. Aceto, & D. Trigiante [1], [2] and R. Brawer & M.Pirovino [5]. The umbral method of the author [11] is used to find natural q-analogues of the Pascal- and the Cauchy-Vandermonde matrices.
    [Show full text]
  • Relation Between Lah Matrix and K-Fibonacci Matrix
    International Journal of Mathematics Trends and Technology Volume 66 Issue 10, 116-122, October 2020 ISSN: 2231 – 5373 /doi:10.14445/22315373/IJMTT-V66I10P513 © 2020 Seventh Sense Research Group® Relation between Lah matrix and k-Fibonacci Matrix Irda Melina Zet#1, Sri Gemawati#2, Kartini Kartini#3 #Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Riau Bina Widya Campus, Pekanbaru 28293, Indonesia Abstract. - The Lah matrix is represented by 퐿푛, is a matrix where each entry is Lah number. Lah number is count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. k-Fibonacci matrix, 퐹푛(푘) is a matrix which all the entries are k-Fibonacci numbers. k-Fibonacci numbers are consist of the first term being 0, the second term being 1 and the next term depends on a natural number k. In this paper, a new matrix is defined namely 퐴푛 where it is not commutative to multiplicity of two matrices, so that another matrix 퐵푛 is defined such that 퐴푛 ≠ 퐵푛. The result is two forms of factorization from those matrices. In addition, the properties of the relation of Lah matrix and k- Fibonacci matrix is yielded as well. Keywords — Lah numbers, Lah matrix, k-Fibonacci numbers, k-Fibonacci matrix. I. INTRODUCTION Guo and Qi [5] stated that Lah numbers were introduced in 1955 and discovered by Ivo. Lah numbers are count the number of ways a set of n elements can be partitioned into k non-empty linearly ordered subsets. Lah numbers [9] is denoted by 퐿(푛, 푘) for every 푛, 푘 are elements of integers with initial value 퐿(0,0) = 1.
    [Show full text]
  • Factorizations for Q-Pascal Matrices of Two Variables
    Spec. Matrices 2015; 3:207–213 Research Article Open Access Thomas Ernst Factorizations for q-Pascal matrices of two variables DOI 10.1515/spma-2015-0020 Received July 15, 2015; accepted September 14, 2015 Abstract: In this second article on q-Pascal matrices, we show how the previous factorizations by the sum- mation matrices and the so-called q-unit matrices extend in a natural way to produce q-analogues of Pascal matrices of two variables by Z. Zhang and M. Liu as follows 0 ! 1n−1 i Φ x y xi−j yi+j n,q( , ) = @ j A . q i,j=0 We also find two different matrix products for 0 ! 1n−1 i j Ψ x y i j + xi−j yi+j n,q( , ; , ) = @ j A . q i,j=0 Keywords: q-Pascal matrix; q-unit matrix; q-matrix multiplication 1 Introduction Once upon a time, Brawer and Pirovino [2, p.15 (1), p. 16(3)] found factorizations of the Pascal matrix and its inverse by the summation and difference matrices. In another article [7] we treated q-Pascal matrices and the corresponding factorizations. It turns out that an analoguous reasoning can be used to find q-analogues of the two variable factorizations by Zhang and Liu [13]. The purpose of this paper is thus to continue the q-analysis- matrix theme from our earlier papers [3]-[4] and [6]. To this aim, we define two new kinds of q-Pascal matrices, the lower triangular Φn,q matrix and the Ψn,q, both of two variables. To be able to write down addition and subtraction formulas for the most important q-special functions, i.e.
    [Show full text]
  • Euler Matrices and Their Algebraic Properties Revisited
    Appl. Math. Inf. Sci. 14, No. 4, 1-14 (2020) 1 Applied Mathematics & Information Sciences An International Journal http://dx.doi.org/10.18576/amis/AMIS submission quintana ramirez urieles FINAL VERSION Euler Matrices and their Algebraic Properties Revisited Yamilet Quintana1,∗, William Ram´ırez 2 and Alejandro Urieles3 1 Department of Pure and Applied Mathematics, Postal Code: 89000, Caracas 1080 A, Simon Bol´ıvar University, Venezuela 2 Department of Natural and Exact Sciences, University of the Coast - CUC, Barranquilla, Colombia 3 Faculty of Basic Sciences - Mathematics Program, University of Atl´antico, Km 7, Port Colombia, Barranquilla, Colombia Received: 2 Sep. 2019, Revised: 23 Feb. 2020, Accepted: 13 Apr. 2020 Published online: 1 Jul. 2020 Abstract: This paper addresses the generalized Euler polynomial matrix E (α)(x) and the Euler matrix E . Taking into account some properties of Euler polynomials and numbers, we deduce product formulae for E (α)(x) and define the inverse matrix of E . We establish some explicit expressions for the Euler polynomial matrix E (x), which involves the generalized Pascal, Fibonacci and Lucas matrices, respectively. From these formulae, we get some new interesting identities involving Fibonacci and Lucas numbers. Also, we provide some factorizations of the Euler polynomial matrix in terms of Stirling matrices, as well as a connection between the shifted Euler matrices and Vandermonde matrices. Keywords: Euler polynomials, Euler matrix, generalized Euler matrix, generalized Pascal matrix, Fibonacci matrix,
    [Show full text]
  • An Effective Lie–Kolchin Theorem for Quasi-Unipotent Matrices
    Linear Algebra and its Applications 581 (2019) 304–323 Contents lists available at ScienceDirect Linear Algebra and its Applications www.elsevier.com/locate/laa An effective Lie–Kolchin Theorem for quasi-unipotent matrices Thomas Koberda a, Feng Luo b,∗, Hongbin Sun b a Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA b Department of Mathematics, Rutgers University, Hill Center – Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA a r t i c l e i n f oa b s t r a c t Article history: We establish an effective version of the classical Lie–Kolchin Received 27 April 2019 Theorem. Namely, let A, B ∈ GLm(C)be quasi-unipotent Accepted 17 July 2019 matrices such that the Jordan Canonical Form of B consists Available online 23 July 2019 of a single block, and suppose that for all k 0the Submitted by V.V. Sergeichuk matrix ABk is also quasi-unipotent. Then A and B have a A, B < C MSC: common eigenvector. In particular, GLm( )is a primary 20H20, 20F38 solvable subgroup. We give applications of this result to the secondary 20F16, 15A15 representation theory of mapping class groups of orientable surfaces. Keywords: © 2019 Elsevier Inc. All rights reserved. Lie–Kolchin theorem Unipotent matrices Solvable groups Mapping class groups * Corresponding author. E-mail addresses: [email protected] (T. Koberda), fl[email protected] (F. Luo), [email protected] (H. Sun). URLs: http://faculty.virginia.edu/Koberda/ (T. Koberda), http://sites.math.rutgers.edu/~fluo/ (F. Luo), http://sites.math.rutgers.edu/~hs735/ (H.
    [Show full text]
  • Package 'Matrixcalc'
    Package ‘matrixcalc’ July 28, 2021 Version 1.0-5 Date 2021-07-27 Title Collection of Functions for Matrix Calculations Author Frederick Novomestky <[email protected]> Maintainer S. Thomas Kelly <[email protected]> Depends R (>= 2.0.1) Description A collection of functions to support matrix calculations for probability, econometric and numerical analysis. There are additional functions that are comparable to APL functions which are useful for actuarial models such as pension mathematics. This package is used for teaching and research purposes at the Department of Finance and Risk Engineering, New York University, Polytechnic Institute, Brooklyn, NY 11201. Horn, R.A. (1990) Matrix Analysis. ISBN 978-0521386326. Lancaster, P. (1969) Theory of Matrices. ISBN 978-0124355507. Lay, D.C. (1995) Linear Algebra: And Its Applications. ISBN 978-0201845563. License GPL (>= 2) Repository CRAN Date/Publication 2021-07-28 08:00:02 UTC NeedsCompilation no R topics documented: commutation.matrix . .3 creation.matrix . .4 D.matrix . .5 direct.prod . .6 direct.sum . .7 duplication.matrix . .8 E.matrices . .9 elimination.matrix . 10 entrywise.norm . 11 1 2 R topics documented: fibonacci.matrix . 12 frobenius.matrix . 13 frobenius.norm . 14 frobenius.prod . 15 H.matrices . 17 hadamard.prod . 18 hankel.matrix . 19 hilbert.matrix . 20 hilbert.schmidt.norm . 21 inf.norm . 22 is.diagonal.matrix . 23 is.idempotent.matrix . 24 is.indefinite . 25 is.negative.definite . 26 is.negative.semi.definite . 28 is.non.singular.matrix . 29 is.positive.definite . 31 is.positive.semi.definite . 32 is.singular.matrix . 34 is.skew.symmetric.matrix . 35 is.square.matrix . 36 is.symmetric.matrix .
    [Show full text]
  • POST QUANTUM CRYPTOGRAPHY: IMPLEMENTING ALTERNATIVE PUBLIC KEY SCHEMES on EMBEDDED DEVICES Preparing for the Rise of Quantum Computers
    POST QUANTUM CRYPTOGRAPHY: IMPLEMENTING ALTERNATIVE PUBLIC KEY SCHEMES ON EMBEDDED DEVICES Preparing for the Rise of Quantum Computers DISSERTATION for the degree of Doktor-Ingenieur of the Faculty of Electrical Engineering and Information Technology at the Ruhr-University Bochum, Germany by Stefan Heyse Bochum, October 2013 Post Quantum Cryptography: Implementing Alternative Public Key Schemes on Embedded Devices Copyright © 2013 by Stefan Heyse. All rights reserved. Printed in Germany. F¨ur Mandy, Captain Chaos und den B¨oarn. Author’s contact information: [email protected] www.schnufff.de Thesis Advisor: Prof. Dr.-Ing. Christof Paar Ruhr-University Bochum, Germany Secondary Referee: Prof. Paulo S. L. M. Barreto Universidade de S˜ao Paulo, Brasil Tertiary Referee: Prof. Tim E. G¨uneysu Ruhr-University Bochum, Germany Thesis submitted: October 8, 2013 Thesis defense: November 26, 2013. v “If you want to succeed, double your failure rate.” (Tom Watson, IBM). “Kaffee dehydriert den K¨orper nicht. Ich w¨are sonst schon Staub.” (Franz Kafka) vii Abstract Almost all of today’s security systems rely on cryptographic primitives as core c components which are usually considered the most trusted part of the system. The realization of these primitives on the underlying platform plays a crucial role for any real-world deployment. In this thesis, we discuss new primitives in public-key cryptography that could serve as alterna- tives to the currently used RSA, ECC and discrete logarithm cryptosystems. Analyzing these primitives in the first part of this thesis from an implementer’s perspective, we show advantages of the new primitives. Moreover, by implementing them on embedded systems with restricted resources, we investigate if these schemes have already evolved into real alternatives to the current cryptosystems.
    [Show full text]
  • Semi-Exact Control Functionals from Sard's Method Arxiv:2002.00033V4
    Semi-Exact Control Functionals From Sard's Method Leah F. South1, Toni Karvonen2, Chris Nemeth3, Mark Girolami4;2, Chris. J. Oates5;2 1Queensland University of Technology, Australia 2The Alan Turing Institute, UK 3Lancaster University, UK 4University of Cambridge, UK 5Newcastle University, UK May 7, 2021 Abstract The numerical approximation of posterior expected quantities of interest is considered. A novel control variate technique is proposed for post-processing of Markov chain Monte Carlo output, based both on Stein's method and an approach to numerical integration due to Sard. The resulting estimators are proven to be polynomially exact in the Gaussian context, while empiri- cal results suggest the estimators approximate a Gaussian cubature method near the Bernstein-von-Mises limit. The main theoretical result establishes a bias-correction property in settings where the Markov chain does not leave the posterior invariant. Empirical results are presented across a selection of Bayesian inference tasks. All methods used in this paper are available in the R package ZVCV. Keywords: control variate; Stein operator; variance reduction. arXiv:2002.00033v4 [stat.CO] 6 May 2021 1 Introduction This paper focuses on the numerical approximation of integrals of the form Z I(f) = f(x)p(x)dx; where f is a function of interest and p is a positive and continuously differentiable probability density on Rd, under the restriction that p and its gradient can only be evaluated pointwise up to an intractable normalisation constant. The standard approach to computing I(f) in this context is to simulate the first n steps of a 1 (i) 1 p-invariant Markov chain (x )i=1, possibly after an initial burn-in period, and to take the average along the sample path as an approximation to the integral: n 1 X I(f) I (f) = f(x(i)): (1) MC n ≈ i=1 See Chapters 6{10 of Robert and Casella (2013) for background.
    [Show full text]