Hyperbolicity and Stable Polynomials in Combinatorics and Probability
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Factor of Safety and Probability of Failure
Factor of safety and probability of failure Introduction How does one assess the acceptability of an engineering design? Relying on judgement alone can lead to one of the two extremes illustrated in Figure 1. The first case is economically unacceptable while the example illustrated in the drawing on the right violates all normal safety standards. Figure 1: Rockbolting alternatives involving individual judgement. (Drawings based on a cartoon in a brochure on rockfalls published by the Department of Mines of Western Australia.) Sensitivity studies The classical approach used in designing engineering structures is to consider the relationship between the capacity C (strength or resisting force) of the element and the demand D (stress or disturbing force). The Factor of Safety of the structure is defined as F = C/D and failure is assumed to occur when F is less than unity. Factor of safety and probability of failure Rather than base an engineering design decision on a single calculated factor of safety, an approach which is frequently used to give a more rational assessment of the risks associated with a particular design is to carry out a sensitivity study. This involves a series of calculations in which each significant parameter is varied systematically over its maximum credible range in order to determine its influence upon the factor of safety. This approach was used in the analysis of the Sau Mau Ping slope in Hong Kong, described in detail in another chapter of these notes. It provided a useful means of exploring a range of possibilities and reaching practical decisions on some difficult problems. -
Homogeneous Polynomial Solutions to Constant Coefficient Pde’S
HOMOGENEOUS POLYNOMIAL SOLUTIONS TO CONSTANT COEFFICIENT PDE'S Bruce Reznick University of Illinois 1409 W. Green St. Urbana, IL 61801 [email protected] October 21, 1994 1. Introduction Suppose p is a homogeneous polynomial over a field K. Let p(D) be the differen- @ tial operator defined by replacing each occurence of xj in p by , defined formally @xj 2 in case K is not a subset of C. (The classical example is p(x1; · · · ; xn) = j xj , for which p(D) is the Laplacian r2.) In this paper we solve the equation p(DP)q = 0 for homogeneous polynomials q over K, under the restriction that K be an algebraically closed field of characteristic 0. (This restriction is mild, considering that the \natu- ral" field is C.) Our Main Theorem and its proof are the natural extensions of work by Sylvester, Clifford, Rosanes, Gundelfinger, Cartan, Maass and Helgason. The novelties in the presentation are the generalization of the base field and the removal of some restrictions on p. The proofs require only undergraduate mathematics and Hilbert's Nullstellensatz. A fringe benefit of the proof of the Main Theorem is an Expansion Theorem for homogeneous polynomials over R and C. This theorem is 2 2 2 trivial for linear forms, goes back at least to Gauss for p = x1 + x2 + x3, and has also been studied by Maass and Helgason. Main Theorem (See Theorem 4.1). Let K be an algebraically closed field of mj characteristic 0. Suppose p 2 K[x1; · · · ; xn] is homogeneous and p = j pj for distinct non-constant irreducible factors pj 2 K[x1; · · · ; xn]. -
Algebra Polynomial(Lecture-I)
Algebra Polynomial(Lecture-I) Dr. Amal Kumar Adak Department of Mathematics, G.D.College, Begusarai March 27, 2020 Algebra Dr. Amal Kumar Adak Definition Function: Any mathematical expression involving a quantity (say x) which can take different values, is called a function of that quantity. The quantity (x) is called the variable and the function of it is denoted by f (x) ; F (x) etc. Example:f (x) = x2 + 5x + 6 + x6 + x3 + 4x5. Algebra Dr. Amal Kumar Adak Polynomial Definition Polynomial: A polynomial in x over a real or complex field is an expression of the form n n−1 n−3 p (x) = a0x + a1x + a2x + ::: + an,where a0; a1;a2; :::; an are constants (free from x) may be real or complex and n is a positive integer. Example: (i)5x4 + 4x3 + 6x + 1,is a polynomial where a0 = 5; a1 = 4; a2 = 0; a3 = 6; a4 = 1a0 = 4; a1 = 4; 2 1 3 3 2 1 (ii)x + x + x + 2 is not a polynomial, since 3 ; 3 are not integers. (iii)x4 + x3 cos (x) + x2 + x + 1 is not a polynomial for the presence of the term cos (x) Algebra Dr. Amal Kumar Adak Definition Zero Polynomial: A polynomial in which all the coefficients a0; a1; a2; :::; an are zero is called a zero polynomial. Definition Complete and Incomplete polynomials: A polynomial with non-zero coefficients is called a complete polynomial. Other-wise it is incomplete. Example of a complete polynomial:x5 + 2x4 + 3x3 + 7x2 + 9x + 1. Example of an incomplete polynomial: x5 + 3x3 + 7x2 + 9x + 1. -
Probabilistic Stability Analysis
TABLE OF CONTENTS Page A-7 Probabilistic Limit State Analysis ....................................................... A-7-1 A-7.1 Key Concepts ....................................................................... A-7-1 A-7.2 Example: FOSM and MC Analysis for Heave at the Toe of a Levee ................................................................. A-7-4 A-7.3 Example: RCC Gravity Dam Stability .............................. A-7-14 A-7.4 Example: Screening-Level Check of Embankment Post-Liquefaction Stability ............................................ A-7-19 A-7.5 Example: Foundation Rock Wedge Stability .................... A-7-23 A-7.6 Model Uncertainty ............................................................. A-7-26 A-7.7 References .......................................................................... A-7-28 Tables Page A-7-1 Variables in Levee Heave Analysis ................................................. A-7-7 A-7-2 FOSM Calculations for Water Surface at Levee Crest .................... A-7-8 A-7-3 Variable Distributions for MC Simulation .................................... A-7-11 A-7-4 Comparison of MC and FOSM Analysis Results .......................... A-7-12 A-7-5 Correlation Coefficients for Water Surface at the Levee Crest ..... A-7-13 A-7-6 Comparison of MC and FOSM Sensitivity Analysis Results ........ A-7-13 A-7-7 Summary of Concrete Input Properties .......................................... A-7-16 A-7-8 RCC Dam Sensitivity Rankings..................................................... A-7-18 A-7-9 Summary -
Quadratic Form - Wikipedia, the Free Encyclopedia
Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form Quadratic form From Wikipedia, the free encyclopedia In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. For example, is a quadratic form in the variables x and y. Quadratic forms occupy a central place in various branches of mathematics, including number theory, linear algebra, group theory (orthogonal group), differential geometry (Riemannian metric), differential topology (intersection forms of four-manifolds), and Lie theory (the Killing form). Contents 1 Introduction 2 History 3 Real quadratic forms 4 Definitions 4.1 Quadratic spaces 4.2 Further definitions 5 Equivalence of forms 6 Geometric meaning 7 Integral quadratic forms 7.1 Historical use 7.2 Universal quadratic forms 8 See also 9 Notes 10 References 11 External links Introduction Quadratic forms are homogeneous quadratic polynomials in n variables. In the cases of one, two, and three variables they are called unary, binary, and ternary and have the following explicit form: where a,…,f are the coefficients.[1] Note that quadratic functions, such as ax2 + bx + c in the one variable case, are not quadratic forms, as they are typically not homogeneous (unless b and c are both 0). The theory of quadratic forms and methods used in their study depend in a large measure on the nature of the 1 of 8 27/03/2013 12:41 Quadratic form - Wikipedia, the free encyclopedia http://en.wikipedia.org/wiki/Quadratic_form coefficients, which may be real or complex numbers, rational numbers, or integers. In linear algebra, analytic geometry, and in the majority of applications of quadratic forms, the coefficients are real or complex numbers. -
Solving Quadratic Equations in Many Variables
Snapshots of modern mathematics № 12/2017 from Oberwolfach Solving quadratic equations in many variables Jean-Pierre Tignol 1 Fields are number systems in which every linear equa- tion has a solution, such as the set of all rational numbers Q or the set of all real numbers R. All fields have the same properties in relation with systems of linear equations, but quadratic equations behave dif- ferently from field to field. Is there a field in which every quadratic equation in five variables has a solu- tion, but some quadratic equation in four variables has no solution? The answer is in this snapshot. 1 Fields No problem is more fundamental to algebra than solving polynomial equations. Indeed, the name algebra itself derives from a ninth century treatise [1] where the author explains how to solve quadratic equations like x2 +10x = 39. It was clear from the start that some equations with integral coefficients like x2 = 2 may not have any integral solution. Even allowing positive and negative numbers with infinite decimal expansion as solutions (these numbers are called real numbers), one cannot solve x2 = −1 because the square of every real number is positive or zero. However, it is only a small step from the real numbers to the solution 1 Jean-Pierre Tignol acknowledges support from the Fonds de la Recherche Scientifique (FNRS) under grant n◦ J.0014.15. He is grateful to Karim Johannes Becher, Andrew Dolphin, and David Leep for comments on a preliminary version of this text. 1 of every polynomial equation: one gives the solution of the equation x2 = −1 (or x2 + 1 = 0) the name i and defines complex numbers as expressions of the form a + bi, where a and b are real numbers. -
Degree Bounds for Gröbner Bases in Algebras of Solvable Type
Degree Bounds for Gröbner Bases in Algebras of Solvable Type Matthias Aschenbrenner University of California, Los Angeles (joint with Anton Leykin) Algebras of Solvable Type • . introduced by Kandri-Rody & Weispfenning (1990); • . form a class of associative algebras over fields which generalize • commutative polynomial rings; • Weyl algebras; • universal enveloping algebras of f. d. Lie algebras; • . are sometimes also called polynomial rings of solvable type or PBW-algebras (Poincaré-Birkhoff-Witt). Weyl Algebras Systems of linear PDE with polynomial coefficients can be represented by left ideals in the Weyl algebra An(C) = Chx1,..., xn, ∂1, . , ∂ni, the C-algebra generated by the xi , ∂j subject to the relations: xj xi = xi xj , ∂j ∂i = ∂i ∂j and ( xi ∂j if i 6= j ∂j xi = xi ∂j + 1 if i = j. The Weyl algebra acts naturally on C[x1,..., xn]: ∂f (∂i , f ) 7→ , (xi , f ) 7→ xi f . ∂xi Universal Enveloping Algebras Let g be a Lie algebra over a field K . The universal enveloping algebra U(g) of g is the K -algebra obtained by imposing the relations g ⊗ h − h ⊗ g = [g, h]g on the tensor algebra of the K -linear space g. Poincaré-Birkhoff-Witt Theorem: the canonical morphism g → U(g) is injective, and g generates the K -algebra U(g). If g corresponds to a Lie group G, then U(g) can be identified with the algebra of left-invariant differential operators on G. Affine Algebras and Monomials N Let R be a K -algebra, and x = (x1,..., xN ) ∈ R . Write α α1 αN N x := x1 ··· xN for a multi-index α = (α1, . -
Geometric Stable Laws Through Series Representations
Serdica Math. J. 25 (1999), 241-256 GEOMETRIC STABLE LAWS THROUGH SERIES REPRESENTATIONS Tomasz J. Kozubowski, Krzysztof Podg´orski Communicated by S. T. Rachev Abstract. Let (Xi) be a sequence of i.i.d. random variables, and let N be a geometric random variable independent of (Xi). Geometric stable distributions are weak limits of (normalized) geometric compounds, SN = X + + XN , when the mean of N converges to infinity. By an appro- 1 · · · priate representation of the individual summands in SN we obtain series representation of the limiting geometric stable distribution. In addition, we [Nt] study the asymptotic behavior of the partial sum process SN (t) = Xi, i=1 and derive series representations of the limiting geometric stable processP and the corresponding stochastic integral. We also obtain strong invariance principles for stable and geometric stable laws. 1. Introduction. An increasing interest has been seen recently in geo- metric stable (GS) distributions: the class of limiting laws of appropriately nor- malized random sums of i.i.d. random variables, (1) S = X + + X , N 1 · · · N 1991 Mathematics Subject Classification: 60E07, 60F05, 60F15, 60F17, 60G50, 60H05 Key words: geometric compound, invariance principle, Linnik distribution, Mittag-Leffler distribution, random sum, stable distribution, stochastic integral 242 Tomasz J. Kozubowski, Krzysztof Podg´orski where the number of terms is geometrically distributed with mean 1/p, and p 0 → (see, e.g., [7], [8], [10], [11], [12], [13], [14] and [21]). These heavy-tailed distribu- tions provide useful models in mathematical finance (see, e.g., [1], [20], [13]), as well as in a variety of other fields (see, e.g., [6] for examples, applications, and extensive references for geometric compounds (1)). -
Resultant and Discriminant of Polynomials
RESULTANT AND DISCRIMINANT OF POLYNOMIALS SVANTE JANSON Abstract. This is a collection of classical results about resultants and discriminants for polynomials, compiled mainly for my own use. All results are well-known 19th century mathematics, but I have not inves- tigated the history, and no references are given. 1. Resultant n m Definition 1.1. Let f(x) = anx + ··· + a0 and g(x) = bmx + ··· + b0 be two polynomials of degrees (at most) n and m, respectively, with coefficients in an arbitrary field F . Their resultant R(f; g) = Rn;m(f; g) is the element of F given by the determinant of the (m + n) × (m + n) Sylvester matrix Syl(f; g) = Syln;m(f; g) given by 0an an−1 an−2 ::: 0 0 0 1 B 0 an an−1 ::: 0 0 0 C B . C B . C B . C B C B 0 0 0 : : : a1 a0 0 C B C B 0 0 0 : : : a2 a1 a0C B C (1.1) Bbm bm−1 bm−2 ::: 0 0 0 C B C B 0 bm bm−1 ::: 0 0 0 C B . C B . C B C @ 0 0 0 : : : b1 b0 0 A 0 0 0 : : : b2 b1 b0 where the m first rows contain the coefficients an; an−1; : : : ; a0 of f shifted 0; 1; : : : ; m − 1 steps and padded with zeros, and the n last rows contain the coefficients bm; bm−1; : : : ; b0 of g shifted 0; 1; : : : ; n−1 steps and padded with zeros. In other words, the entry at (i; j) equals an+i−j if 1 ≤ i ≤ m and bi−j if m + 1 ≤ i ≤ m + n, with ai = 0 if i > n or i < 0 and bi = 0 if i > m or i < 0. -
Algebraic Combinatorics and Resultant Methods for Polynomial System Solving Anna Karasoulou
Algebraic combinatorics and resultant methods for polynomial system solving Anna Karasoulou To cite this version: Anna Karasoulou. Algebraic combinatorics and resultant methods for polynomial system solving. Computer Science [cs]. National and Kapodistrian University of Athens, Greece, 2017. English. tel-01671507 HAL Id: tel-01671507 https://hal.inria.fr/tel-01671507 Submitted on 5 Jan 2018 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. ΕΘΝΙΚΟ ΚΑΙ ΚΑΠΟΔΙΣΤΡΙΑΚΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΑΘΗΝΩΝ ΣΧΟΛΗ ΘΕΤΙΚΩΝ ΕΠΙΣΤΗΜΩΝ ΤΜΗΜΑ ΠΛΗΡΟΦΟΡΙΚΗΣ ΚΑΙ ΤΗΛΕΠΙΚΟΙΝΩΝΙΩΝ ΠΡΟΓΡΑΜΜΑ ΜΕΤΑΠΤΥΧΙΑΚΩΝ ΣΠΟΥΔΩΝ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ Μελέτη και επίλυση πολυωνυμικών συστημάτων με χρήση αλγεβρικών και συνδυαστικών μεθόδων Άννα Ν. Καρασούλου ΑΘΗΝΑ Μάιος 2017 NATIONAL AND KAPODISTRIAN UNIVERSITY OF ATHENS SCHOOL OF SCIENCES DEPARTMENT OF INFORMATICS AND TELECOMMUNICATIONS PROGRAM OF POSTGRADUATE STUDIES PhD THESIS Algebraic combinatorics and resultant methods for polynomial system solving Anna N. Karasoulou ATHENS May 2017 ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ Μελέτη και επίλυση πολυωνυμικών συστημάτων με -
Extreme Value Theory (Or How to Go Beyond of the Range Data)
Extreme Value Theory (or how to go beyond of the range data) Sept 2007, Romania Philippe Naveau Laboratoire des Sciences du Climat et l’Environnement (LSCE) Gif-sur-Yvette, France Katz et al., Statistics of extremes in hydrology, Advances in Water Resources 25 (2002) 1287-1304 Motivation Univariate EVT Non-stationary extremes Spatial extremes Conclusions Extreme quotes 1 “Man can believe the impossible, but man can never believe the improbable” Oscar Wilde (Intentions, 1891) 2 “Il est impossible que l’improbable n’arrive jamais” Emil Julius Gumbel (1891-1966) Extreme events ? ... a probabilistic concept linked to the tail behavior : low frequency of occurrence, large uncertainty and sometimes strong amplitude. Region of interest Motivation Univariate EVT Non-stationary extremes Spatial extremes Conclusions Important issues in Extreme Value Theory Applied statistics An asymptotic probabilistic concept Non-stationarity A statistical modeling approach Identifying clearly assumptions Univariate Multivariate Non-parametric Parametric Assessing uncertainties Goodness of fit and model selection Independence Theoritical probability Motivation Univariate EVT Non-stationary extremes Spatial extremes Conclusions Outline 1 Motivation Heavy rainfalls Three applications 2 Univariate EVT Asymptotic result Historical perspective GPD Parameters estimation Brief summary of univariate iid EVT 3 Non-stationary extremes Spatial interpolation of return levels Downscaling of heavy rainfalls 4 Spatial extremes Assessing spatial dependences among maxima 5 Conclusions -
Spherical Harmonics and Homogeneous Har- Monic Polynomials
SPHERICAL HARMONICS AND HOMOGENEOUS HAR- MONIC POLYNOMIALS 1. The spherical Laplacean. Denote by S ½ R3 the unit sphere. For a function f(!) de¯ned on S, let f~ denote its extension to an open neighborhood N of S, constant along normals to S (i.e., constant along rays from the origin). We say f 2 C2(S) if f~ is a C2 function in N , and for such functions de¯ne a di®erential operator ¢S by: ¢Sf := ¢f;~ where ¢ on the right-hand side is the usual Laplace operator in R3. With a little work (omitted here) one may derive the expression for ¢ in polar coordinates (r; !) in R2 (r > 0;! 2 S): 2 1 ¢u = u + u + ¢ u: rr r r r2 S (Here ¢Su(r; !) is the operator ¢S acting on the function u(r; :) in S, for each ¯xed r.) A homogeneous polynomial of degree n ¸ 0 in three variables (x; y; z) is a linear combination of `monomials of degree n': d1 d2 d3 x y z ; di ¸ 0; d1 + d2 + d3 = n: This de¯nes a vector space (over R) denoted Pn. A simple combinatorial argument (involving balls and separators, as most of them do), seen in class, yields the dimension: 1 d := dim(P ) = (n + 1)(n + 2): n n 2 Writing a polynomial p 2 Pn in polar coordinates, we necessarily have: n p(r; !) = r f(!); f = pjS; where f is the restriction of p to S. This is an injective linear map p 7! f, but the functions on S so obtained are rather special (a dn-dimensional subspace of the in¯nite-dimensional space C(S) of continuous functions-let's call it Pn(S)) We are interested in the subspace Hn ½ Pn of homogeneous harmonic polynomials of degree n (¢p = 0).