text: If the general quadratic form above is equated to 0, the resulting equation is that of an ellipse if the quadratic form is positive or negative-definite, a hyperbola if it is indefinite, and a parabola if From MathWorld--A As you might imagine, a quadratic form is positive de nite if and only if all of the eigenvalues of the associated matrix are positive. Any positive-definite form can be converted by a linear transformation to the representation $$\sum _ { i= } 1 ^ { n } x _ {i} ^ {2} . As an example, let + Now turn the parabola upside down with the vertex being its highest point, and hold it below the surface of the water. References . In linear algebra, a positive-definite matrix is a matrix that in many ways is analogous to a positive real number.The notion is closely related to a positive-definite symmetric bilinear form (or a sesquilinear form in the complex case).. Graphical analysis. "Determine if the following quadratic form is positive definite, negative definite or undefinite Q:\mathbb R^3\to \mathbb R, \,Q(u)=x_1^2+4x_1x_2-2x_2^2+2x_1x_3-2x_3^2"$$Q=\begin{bmatrix} 1&2&1 \\\ 2&-2&0 \\\ 1&0&-2 \end{bmatrix} Let denote a symmetric matrix and let denote an dimensional vector.. Then is called Quadratic Form.. For , we obtain , which can be considered as a paraboloid in 3-dimensional space, i.e., .This idea may be useful later on when discussing positive definiteness. {\displaystyle \in V} If the quadratic form, and hence A, is positive-definite, the second-order conditions for a minimum are met at this point. When x has only two elements, we can graphically represent Q in 3 di-mensions. Thus we have two types of definite quadratic forms: positive definite and negative definite. QUADRATIC OPTIMIZATION: THE POSITIVE DEFINITE CASE 677 Such functions can be conveniently deﬁned in the form P(x)=x>Axx>b, whereAisasymmetricn⇥nmatrix, andx,b,arevectors in Rn,viewedascolumnvectors. Positive Definite Quadratic form an expression of the form (where a ik = a ki) that assumes nonnegative values for all real values of x1, x2, …, xn and that vanishes only when x1 = x2 = … = xn = 0. A real quadratic form in variables is positive Since it opens downward it is negative definite. Ask Question Asked 10 years, 11 months ago. 2 If a quadratic form is positive definite on a convex set, is it convex on that set? (a) A positive-de nite form. }, The square of the Euclidean norm in n-dimensional space, the most commonly used measure of distance, is. {\displaystyle (x_{1},\cdots ,x_{n})^{\text{T}}} § Definition: A quadratic form Q is: a. positive definite if for all , b. negative definite if for all , c. indefinite if Q (x) assumes both positive and negative values. x of the general linear group , i.e., under − in other words, Portions of this entry contributed by David Ergebnisse der Mathematik und ihrer Grenzgebiete, https://en.wikipedia.org/w/index.php?title=Definite_quadratic_form&oldid=983701537, Creative Commons Attribution-ShareAlike License, This page was last edited on 15 October 2020, at 19:11. A quadratic form Q which depends on n variables is considered positive definite. . Conclusion: If a ≠ 0 the matrix is indefinite; if a = 0 it is positive semidefinite. ⋯ A Course in Computational Algebraic Number Theory. If all λ i are negative, the form is said to be negative definite.. 3. B When x has only two elements, we can graphically represent Q in 3 di-mensions. Proof . and almost-everywhere). 4x 2 − y 2. Hints help you try the next step on your own. We say that an S-universal form has minimal dimension if there are no S-universal forms of lower dimension. is then determined by , , and . Home; Courses; Groups; Login; Register; Quadratic Form in Linear Algebra. 3.2.3 Exercises on quadratic forms: conditions for semidefiniteness. POSITIVE DEFINITE n-REGULAR QUADRATIC FORMS BYEONG-KWEON OH Abstract. discriminant and the set 1 of are positive, These forms allows her to define the curvature lines on V 2. matrix is positive definite if all the principal minors in the top-left corner {\displaystyle c_{1}>0} QUADRATIC FORMS AND DEFINITE MATRICES 3 1.3. If a ij are real then quadratic form is called real quadratic form. Graphical analysis. Reading [SB], Ch. ) 1 If A is diagonal this is equivalent to a non-matrix form containing solely terms involving squared variables; but if A has any non-zero off-diagonal elements, the non-matrix form will also contain some terms involving products of two different variables. y Sometimes this condition can be confirmed from the definition of . 2 QUADRATIC FORMS AND DEFINITE MATRICES 3 1.3. , and if or . Therefore, the quadratic form is positive de nite i ker(A)=f~0g. 1 Generalization of this notion to two variables is the quadratic form Q(x1;x2) = a11x 2 1 +a12x1x2 +a21x2x1 +a22x 2 2: Here each term has degree 2 (the sum of exponents is 2 for all summands). Since it is, obviously, a symmetric matrix, it is (semi)positive definite iff all its main minors are (non-negative) positive, and indeed: First main minor: $\;A>0\;$ (given) Second main minor (the matrix determinant): $\;AC-B^2>0\;$ (given) Quadratic forms can be classified according to the nature of the eigenvalues of the matrix of the quadratic form:. T 0 14.1. A Quadratic Form is positive definite Iff every Eigenvalue of is Positive. Paris: Hermann, p. 38, 1983. Q A real quadratic form in variables is positive definite iff its canonical form is. Under the action Thus, the positive definite quadratic form is a special case of a quadratic form. R has the form f(x) = a ¢ x2.Generalization of this notion to two variables is the quadratic form Q(x1;x2) = a11x 2 1 +a12x1x2 +a21x2x1 +a22x 2 2: Here each term has degree 2 (the sum of … Graphical analysis. V Quadratic Form… This gives a nice graphical representation where the plane at x = 0 bounds the function from below. {\displaystyle V=\mathbb {R} ^{2}} Cohen, H. A Course in Computational Algebraic Number Theory. c number of the imaginary quadratic field is equal to the number of reduced It is positive or negative semidefinite if }, This bivariate quadratic form appears in the context of conic sections centered on the origin. This video clearly explains the nature of the quadratic form & Nature of roots. Gradshteyn, I. S. and Ryzhik, I. M. Tables of Integrals, Series, and Products, 6th ed. Quadratic forms can be classified according to the nature of the eigenvalues of the matrix of the quadratic form: 1. 1 Quadratic forms correspond one-to-one to symmetric bilinear forms over the same space. Positive or negative-definiteness or semi-definiteness, or indefiniteness, of this quadratic form is equivalent to the same property of A, which can be checked by considering all eigenvalues of A or by checking the signs of all of its principal minors. LEVELS OF POSITIVE DEFINITE TERNARY QUADRATIC FORMS 401 and giXx , X2 , X-}) = 3x2 + 1 lX2 + 1 lx2 - IOX2X3 - 2x1X3 - 2XiX2. Definite quadratic forms lend themselves readily to optimization problems. 1. This video clearly explains the nature of the quadratic form & Nature of roots. Determine whether each of the following quadratic forms in two variables is positive or negative definite or semidefinite, or indefinite. {\displaystyle c_{1}. y c Practice online or make a printable study sheet. 1 Quadratic functions can be described as definite or indefinite based on where they lie on the x-y plane. 1 3 . one. This gives a nice graphical representation where the plane at x = 0 bounds the function from below. Since q(~x)=jjA~xjj2 0 for all vectors ~x in Rn, this quadratic form is positive semide nite. = QUADRATIC OPTIMIZATION: THE POSITIVE DEFINITE CASE 677 Such functions can be conveniently deﬁned in the form P(x)=x>Axx>b, whereAisasymmetricn⇥nmatrix, andx,b,arevectors in Rn,viewedascolumnvectors. 1 0 If such a function takes only negative values for all x other than zero, then we call it, we just replace the word positive by the word negative, is negative definite. R A6. Representations of positive definite quadratic forms. Go to your Tickets dashboard to see if you won! easily computed by systematically constructing all binary quadratic forms of discriminant 2 Let f be a primitive positive definite integral binary quadratic form of discriminant − D and let π f (x) be the number of primes up to x which are represented by f.We prove several types of upper bounds for π f (x) within a constant factor of its asymptotic size: unconditional, conditional on the Generalized Riemann Hypothesis (GRH) and for almost all discriminants. If c1 > 0 and c2 > 0, the quadratic form Q is positive-definite, so Q evaluates to a positive number whenever Let's look at an equation of a p… ( 1 Quadratic Forms A quadratic function f: R ! and indefinite if in which not all elements are 0, superscript T denotes a transpose, and A is an n×n symmetric matrix. of two real variables is positive definite if it is for any , therefore if and the binary quadratic form discriminant . https://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html, Eigenvalues, In this case, the first-order principal minors are 1, 0, and 1; the second-order principal minors are 0, 0, and 0; and the third-order principal minor is 0. x A positive deﬁnite quadratic form will always be positive except at the point where x 2 -x_{1}^{2}-3 x_{2}^{2} The Study-to-Win Winning Ticket number has been announced! 1 2 c In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. discriminant , and consider It is definitely sometimes hyphenated, including in two of the examples of usage in the OED (the ones from 1904 and 1957): positive definite adj. {\displaystyle x_{1}} . A kernel that is semi-positive definite (non-negative definite) is one that satisfies $\int K(x, y) \phi (x) \overline{ {\phi (y) }}\; dx dy \geq 0$ Such a kernel is sometimes also simply called positive. An important example of such an optimization arises in multiple regression, in which a vector of estimated parameters is sought which minimizes the sum of squared deviations from a perfect fit within the dataset. New York: Springer-Verlag, ) Similarly, the form in Figure1e is called negative semi-denite. 2 Wolfram Web Resource. (c) An inde nite form. −x 2 + xy − 3y 2. x 2 − 6xy + 9y 2. 2. 2. 2 . 0 Explore anything with the first computational knowledge engine. ( This occurs when all of the eigenvalues of the associated matrix are negative. R has the form f(x) = a ¢ x2. POSITIVE DEFINITENESS OF A QUADRATIC FORM 241 A6.3 The Quadratic Form in the Entropy Representation Let us now examine under what circumstances (tTA q ~ be negative definite when the base function is chosen as y(O) = S(U, V, NA). 221-224, 1993. The first-order conditions for a maximum or minimum are found by setting the matrix derivative to the zero vector: assuming A is nonsingular. Knowledge-based programming for everyone. , if there exist nonzero and such that. (of a matrix) displaying the coefficients of a positive definite quadratic form… = In linear algebra, a symmetric. , Without loss of generality we put P ′ = ε ( x 1 2 + . 0 Go to your Tickets dashboard to see if you won! There is a remarkable relationship between lattice sphere packings and positive definite quadratic forms. 0. If all λ i are positive, the form is said to be positive definite. x ∈ c When x has only two elements, we can graphically represent Q in 3 di-mensions. Active 10 years, 11 months ago. Then this map is one-to-one and onto. A quadratic form is said to c Graphical analysis. > c containing the ideal . This complements and improves a result … San Diego, CA: Academic Press, This relationship plays an important role in determining the values of δ * (S n) and k * (S n) for small n.Let Λ be a lattice with a basis {a 1, 2 2, ..., a n}, where a i = (a i1, a i2, ..., a in), and write For example, if and has linearly independent columns then for . In this lesson, we will investigate what makes a quadratic definite or indefinite. Two equivalent conditions to being symmetric positive definite are. Terr, David and Weisstein, Eric W. "Positive Definite Quadratic Form." c A positive deﬁnite quadratic form will always be positive except at the point where x = 0. , − This example illustrates another point. Quadratic forms are frequently encountered in statistics. Generally, though, this condition is not easy to check. In two dimensions this means that the distance between two points is the square root of the sum of the squared distances along the Hsia , John S. ; Kitaoka , Y. ; Kneser , M. 1978-01-01 00:00:00 By John S. Hsia *) at Columbus, Yoshiyuki Kitaoka at Nagoya and Martin Kneser at G ttingen Let /(*!, . . A positive deﬁnite quadratic form will always be positive except at the point where x = 0. So then, q of x is positive definite. (b) A negative-de nite form. where b is an n×1 vector of constants. Quadratic Forms. Therefore, a positive-definite form is a quadratic form of special type. 0. A su cient condition for the convexity of a nite product of positive de nite quadratic forms is given in terms of the condition numbers of the underlying matrices. 2 c 2 Collection of teaching and learning tools built by Wolfram education experts: dynamic textbook, lesson plans, widgets, interactive Demonstrations, and more. c nombres remarquables. A square matrix is positive definite if pre-multiplying and post-multiplying it by the same vector always gives a positive number as a result, independently of how we choose the vector. Positive definite symmetric matrices have the property that all their eigenvalues are positive. x c Our goal now is to classify quadratic forms according to these ve categories. So then, q of x is positive definite. Suppose the matrix quadratic form is augmented with linear terms, as. This shows that q is a quadratic form, with symmetric matrix ATA. + Since the pool-noodle parabola doesn't touch or cross through the surface of the water it is still a definite quadratic. A positive deﬁnite quadratic form will always be positive except at the point where x = 0. x we say A is positive deﬁnite if xTAx > 0 for all x 6= 0 • denoted A > 0 • A > 0 if and only if λmin(A) > 0, i.e., all eigenvalues are positive Symmetric matrices, quadratic forms, matrix norm, and SVD 15–14 2- x 1 2 + 2x 2 2 + 3x 3 2 + 4x 1 x 2-6 x 2 x 3 +8 x 3 x 1 is a quadratic form in three variables x 1, x 2 and 3. ) In mathematics, a definite quadratic form is a quadratic form over some real vector space V that has the same sign (always positive or always negative) for every nonzero vector of V. According to that sign, the quadratic form is called positive-definite or negative-definite. 4 QUADRATIC FORMS AND DEFINITE MATRICES F IGURE 2. where x is any n×1 Cartesian vector 0 answer correctly in detail please. the set of linear transformations of coordinates with integer More generally, these definitions apply to any vector space over an ordered field.[1]. to be a positive definite quadratic form, it is necessary and sufficient that Δ 1 > 0…..Δ n > 0, where. Viewed 5k times 1. This gives a nice graphical representation where the plane at x = 0 bounds the function from below. A quadratic form with a Hermitian If all λ i are positive, the form is said to be positive definite.. 2. 1 Join the initiative for modernizing math education. Show transcribed image text. where x = (x1, x2) Go to your Tickets dashboard to see if you won! 3 by looping over the coefficients and . When x has only two elements, we can graphically represent Q in 3 di-mensions. There is actually a simple test for whether a given symmetric matrix is positive de nite. c of is positive. be a reduced positive definite binary quadratic form with fundamental + ≠ z. Thus the quadratic form is positive semidefinite. z T M z. And if one of the constants is negative and the other is 0, then Q is negative semidefinite and always evaluates to either 0 or a negative number. x_{1}^{2}+x_{2}^{2} The Study-to-Win Winning Ticket number has been announced! quadratic form discriminant . § Also, Q is said to be positive semidefinite if for all x, and negative semidefinite if for all x. is said to be reduced if , {\displaystyle Q(x+y)=B(x+y,x+y)} ) You can learn the Rank, Index and Signature of the roots. {\displaystyle c_{1}c_{2}-{c_{3}}^{2}>0,} We say that the form in Figure1dis positive semi-denite, meaning that q(~x) 0 for all ~x, but that there is some nonzero vector ~xso that q(~x) = 0. QUADRATIC OPTIMIZATION: THE POSITIVE DEFINITE CASE 449 Such functions can be conveniently deﬁned in the form P(x)=x Ax−xb, whereAisasymmetricn×nmatrix, andx,b,arevectors in Rn,viewedascolumnvectors. would be complete if positive definite were hyphenated. Question: (c) [6 Marks] Let Q Be A Quadratic Form On A Real Vector Space V. Suppose That Q Is Not Positive Definite But Q(v) = 0 For All Non-zero V EV. If c1 > 0 and c2 < 0, or vice versa, then Q is indefinite and sometimes evaluates to a positive number and sometimes to a negative number. , by means of a linear transformation. Explore thousands of free applications across science, mathematics, engineering, technology, business, art, finance, social sciences, and more. The third coefficient = Since this form q in suitable coordinates is q »X,2 +. Fact 8.2.4 Eigenvalues and de niteness , Remember: positive or negative-definite is not a matrix property but it only applies to quadratic forms, which are naturally described only by symmetric matrices. Le Lionnais, F. Les and ≠ If one of the constants is positive and the other is 0, then Q is positive semidefinite and always evaluates to either 0 or a positive number. Curvature, and Quadratic Forms. There exists a one-to-one correspondence between the set of reduced quadratic forms with fundamental c [2] A symmetric bilinear form is also described as definite, semidefinite, etc. x_{1}^{2}+x_{2}^{2} The Study-to-Win Winning Ticket number has been announced! Classify the quadratic form as positive definite, negative definite, indefinite, positive semidefinite, or negative semidefinite. x A semidefinite (or semi-definite) quadratic form is defined in much the same way, except that "always positive" and "always negative" are replaced by "always nonnegative" and "always nonpositive", respectively. {\displaystyle z} of. ..+ X 2 , with matrix the identity matrix E, it follows that each positive definite A must be congruent to E, that is A * P T EP. The definition requires the positivity of the quadratic form . {\displaystyle c_{1}<0} More generally, ax2 1 + cx2 2 is positive definite whenever a > 0 and c > 0 be positive definite if for . definite iff its canonical form is, of two real variables is positive definite if it is for any , therefore x negative-definite if Figure 1: Plots of quadratic forms. When only two factors are involved the condition is also necessary. PositiveDefiniteMatrixQ [ m] gives True if m is explicitly positive definite, and False otherwise. 2 . 2 > 0 if (x1, x2) ≠ 0, so this quadratic form is positive definite. In other words, it may take on zero values. x binary quadratic forms of discriminant , which can be . A binary quadratic form is positive definite If all λ i are nonnegative (positive or zero), the form is said to be positive semidefinite.. 4. x A Quadratic Form with a Hermitian Matrix is positive definite if all the principal minors in the top-left corner of are Positive, in other words (4) (5) (6) See also Indefinite Quadratic Form, Positive Semidefinite Quadratic Form. The analogy with what I said above about ancient-Indic, dark-blue, etc. 4x 2 + 8xy + 5y 2. An indefinite quadratic form takes on both positive and negative values and is called an isotropic quadratic form. ( This question hasn't been answered yet Ask an expert. , and consider the quadratic form. ) with the sign of the semidefiniteness coinciding with the sign of < A quadratic form is said to be positive definite if for . A quadratic form Q and its associated symmetric bilinear form B are related by the following equations: The latter formula arises from expanding Note that q(~x) = 0 i ~x is in the kernel of A. What is Quadratic form and how it can be used to check positive definiteness If the quadratic form is > 0, then it’s positive definite. Actually, for reasons that will be clear shortly, it is prefer-able to put a factor 1 2 in front of the quadratic term, so that P(x)= 1 2 x>Axx>b. according to its associated quadratic form. p. 1106, 2000. ( , Expert Answer . • inequalities for quadratic forms • positive semideﬁnite matrices • norm of a matrix • singular value decomposition 15–1. Figure 1 showsa positive deﬁnite quadratic form. The question is, under what conditions (on … {\displaystyle (x_{1},x_{2})\neq (0,0).} c M. {\displaystyle M} is said to be positive-definite if the scalar. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share … the map which maps the form to the ideal class Terr. In general a quadratic form in two variables will also involve a cross-product term in x1x2: This quadratic form is positive-definite if Classify the quadratic form as positive definite, negative definite, indefinite, positive semidefinite, or negative semidefinite. Prove That The Form (-9) Is Positive Definite. (A6.3h We have A= Suu Svv @A 9 SVV SVA 9 9 SAA (A6.3)2 2 Let $$f = [a, b, c]$$ be a reduced positive definite binary quadratic form. 3 1 axis and the Figure 2 shows a negative definite quadratic form. definite, if it is the matrix of a positive quadratic form q. positive definite (quadratic form) Example sentences with "positive definite (quadratic form)", translation memory A metric tensor is called positive-definite if it assigns a positive value g (v, v) > 0 to every nonzero vector v. A manifold equipped with a positive-definite … 1 Thus, the class − I can run simply x where x is a vector if this function takes positive values for all x other than zero. Representations of positive definite quadratic forms. Positive Definite Quadratic Form. Actually, for reasons that will be clear shortly, it is prefer-able to put a factor 1 2 in front of the quadratic term, so that P(x)= 1 2 xAx−xb. , Integers represented by positive-de nite quaternary quadratic forms and Petersson inner products Jeremy Rouse Emory University Algebra Seminar November 5, 2019 Jeremy Rouse Quadratic forms 1/45. You can learn the Rank, Index and Signature of the roots. − 1. n QUADRATIC FORMS AND DEFINITE MATRICES 3 1.3. {\displaystyle x_{2}} A positive integer n is said to be locally represented by Q if there is a solution to Q(~x) = n with ~x2Zr p for every p. Theorem (Tartakowski) If r 5, then a positive-de nite form Q represents every su ciently large locally represented positive integer n. Jeremy Rouse Quadratic forms 3/45 A quadratic form is said to be positive semidefinite if it is never .However, unlike a positive definite quadratic form, there may exist a such that the form is zero. = ( 3 Similarly, a quadratic form is negative de nite if f(x) 0 for all x 2Rn. if and the binary 2 positive definiteness. Optimizing these quadratic forms isn’t very interesting at this point; a positive de nite form will have a global minimum at (0;0) and grow without bound, while a negative de nite will have a global maximum at (0;0). of classes of fractional ideals of the unique Posted By: Bindeshwar S. Kushwaha December 9, 2019. 1 2 2 Positive Definite Quadratic Form 3x12 + 3x22 10 x2 5 0 -5 -10 600 400 Q 200 0 -10 -5 0 x1 5 10 Similarly, a negative definite quadratic form is bounded above by the plane x = 0. Tables of Integrals, Series, and Products, 6th ed. definite iff every eigenvalue Hub and Network of Posts Blog, Forum and Social Media Platform. . c By Theorem 1, an immediate and stunning conclusion is that the set of integers represented by a positive-deﬁnite integral quadratic form is determined by a unique ﬁnite subset of this set. ( 0 The quadratic form q is positive definite (resp., negative definite) if q(v) > 0 (resp., q(v) < 0) for every nonzero vector v. When q ( v ) assumes both positive and negative values, q … 0 a unique reduced positive definite binary quadratic form equivalent to any given A quadratic form Q which depends on n variables is considered positive definite. .1- Positive definite if all eigen values of matrix A in quadratic form are positive.2-Negative definite if all eigen values of matrix A are negative. c A CONDITION FOR CONVEXITY OF A PRODUCT OF POSITIVE DEFINITE QUADRATIC FORMS MINGHUA LIN AND GORD SINNAMONy Abstract. Unlimited random practice problems and answers with built-in Step-by-step solutions. Theorem: The equivalence class of a positive definite binary quadratic contains exactly one reduced form. ) But / has discriminant 64 while that of g is 1024. QUADRATIC FORMS AND DEFINITE MATRICES 3 1.3. c The quadratic form, written in the form , is positive semidefinite iff every eigenvalue of is nonnegative., is positive semidefinite iff every eigenvalue of is nonnegative. The #1 tool for creating Demonstrations and anything technical. {\displaystyle c_{1}c_{2}-{c_{3}}^{2}=0,} ) Consider a real symmetric matrix A ∈ R n × n. The associated quadratic form x T A x is a convex function on all of R n iff A is positive semidefinite, i.e., if x T A x ≥ 0 for all x ∈ R n. Now suppose we have a convex subset Φ of R n such that x ∈ Φ implies x T A x ≥ 0. form is simply called a universal quadratic form. {\displaystyle c_{1}c_{2}-{c_{3}}^{2}=0. https://mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html. Math. , {\displaystyle c_{1}c_{2}-{c_{3}}^{2}>0,} n × n. {\displaystyle n\times n} real matrix. + x n 2 ) , ε > 0 , where c does not depend on x. Examples of Quadratic Form. V y Figure 1 shows a positive deﬁnite quadratic form. In order for. A positive kernel $\neq 0$ in the latter sense has at least one eigen value $> 0$ 1-x 1 2 +x 2 2 + 6 x 1 x 2 is a quadratic form in variables x 1 and x 2. − 16.1-16.3, p. 375-393 1 Quadratic Forms A quadratic function f: R ! ( 3 Abstract. x FINITENESS THEOREMS FOR POSITIVE DEFINITE n-REGULAR QUADRATIC FORMS WAI KIU CHAN AND BYEONG-KWEON OH Abstract. 1 tool for creating Demonstrations and anything technical we will investigate what makes a quadratic form always. Is still a definite quadratic Form… Representations of positive definite quadratic form. has minimal if... Any positive definite quadratic form can be written in terms of matrices as matrix is positive to quadratic... Setting the matrix quadratic form is negative de nite if f ( x 1 2... Q » X,2 + positive-definite, the form in Figure1e is called quadratic. 9Y 2 3 } } Mz } is said to be positive definite binary quadratic is... } ^ { 2 } =0 we put P ′ = ε ( x 1 +x... The square of the Euclidean norm in n-dimensional space, the form in variables x 1 2... We have two types of definite quadratic forms 10 years, 11 months ago a ij are then. Form discriminant David and Weisstein, Eric W.  positive definite the Curvature lines V. Negative-Definite, the form., Index and Signature of the variables, where c does not depend on.! The eigenvalues of the roots makes a quadratic form, positive definite quadratic form Products 6th! Is not easy to check a special case of a when x has two. Elements, we can graphically represent q in 3 di-mensions linear Algebra columns then for forms allows her to the. Context of conic sections centered on the x-y plane when x has only two,. { T } } ^ { 2 } +x_ { 2 } ) (... Can be written in terms of matrices as this Question has n't been answered yet an! Form. universal quadratic form. dashboard to see if you won measure of distance, is it convex that... A special case of a quadratic function f: R most commonly used of! ) positive for all x 2Rn, and hence a, is }, x_ { }. An isotropic quadratic form can be confirmed from the definition requires the positivity of the roots clearly explains nature... +X 2 2 + definition of × n. { \displaystyle ( x_ { }... The x-y plane + 2xy − 3y 2. x 2 − 6xy 9y... The water it is for any, therefore if and has linearly independent columns for. Zero ), the square of the following quadratic forms according to these ve categories terr David!, positive semidefinite.. 4 water it is for any, therefore if and the binary quadratic form a. } real matrix ) positive for all x 2Rn linear Algebra be written in terms of as. A vector if this function takes positive values for all x other than zero } +x_ { 2 } {! To check given symmetric matrix ATA has linearly independent columns then for on the..: //mathworld.wolfram.com/PositiveDefiniteQuadraticForm.html, eigenvalues, Curvature, and if or n't been answered yet an. These definitions apply to any vector space over an ordered field. [ 1 ] ) ≠ 0, this. Walk through homework problems step-by-step from beginning to end Integrals, Series and. A is nonsingular q of x is positive definite put P ′ ε... Creating Demonstrations and anything technical conic sections centered on the x-y plane \neq... P. 375-393 1 quadratic forms in two variables is positive semide nite 3/2 forms of level and! No S-universal forms of level 32 and trivial character say that an S-universal form has minimal dimension there! ) displaying the coefficients of a quadratic form is positive de nite i ker ( a ).... Between lattice sphere packings and positive definite iff every eigenvalue of is positive or zero ) ε... 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# Time spent in eclipse of an object

Well consider an orbit - I'm trying to calculate the exact time spent in the shadow of the body you orbit around. An explanation of "shadow" (sun is positioned to the far left):

For a circular orbit this is quite easy: One just calculates the orbit radius and solve it using a simple sine ($T$ is the orbital period):

$$R_{earth} = r \cdot\sin(\theta)$$ $$t = T \cdot \frac{\theta}{\pi}$$

(notice division by $\pi$ since $\theta$ represents half the time in shadow.)

However this is for the specific case of a circular orbit. - I'm wondering how to do it for an (highly) eccentric orbit.

The simple equation above becomes slightly more complicated, since $r$ is no longer constant: $$r = a \frac{1-e^2}{1 + e \cdot \cos(\theta)}$$ For the point around the periapsis, Filling that in to the equation above results (wolfram alpha) into something I do not particularly like. Solving it at a random point it even becomes worse: $$r = a \frac{1-e^2}{1 + e \cdot \cos(\theta_{avg} \pm \theta)}$$

Once I have the true anomalies ($\theta$) I could use some straight forward solution: Eccentric anomaly -> mean anomaly -> time

Before I wish to start this, a question pops up: does the time in shadow actually depend on the position? (true anomaly/radius). When we're close to the planet the planet overshadows a higher angle of the orbit. However due to Kepler's law an object also moves faster at this point.

Specifically does Kepler's second law of motion prove that the time in shadow is independent on the mean true anomaly of the shadow? Kepler's law:

$$\frac{dA}{dt} = \tfrac{1}{2} r^2 \frac{d\theta}{dt}$$

I have a feeling that through kepler's law above problem could be reduced a lot..

Now once I know the position of the longest-time I could solve above equations (wondering if I should try to fill in the equations or solve it numerically).

I think that instead of considering the amount of time spent in the shadow of a planet with a significant radius $R_{earth}$ that it may make the analysis clearer and simpler if you instead consider the limit of some very small object with an infinitesimal radius casting a shadow. Let's call the radius $\text{$\Delta $R}$. What you're then asking is if it takes the satellite an equal amount of time to traverse a distance $\text{$\Delta $R}$ as seen from Earth at any point in its orbit. In other words, if you take the satellite's velocity vector and only consider the component of this velocity vector perpendicular to a vector pointing from Earth to the satellite, does this perpendicular component of the velocity vector change as the satellite orbits the Earth?
In terms of the angle $\theta$ you defined and r, the distance from the Earth to the satellite, this perpendicular component of the satellite's velocity vector is simply $r\frac{\text{d$\theta $}}{\text{dt}}$. Your hypothesis is that this quantity is a constant. However, as you noted, Kepler's $2^{nd}$ law states that $\frac{\text{dA}}{\text{dt}}=\frac{1}{2} r^2 \frac{\text{d$\theta $} }{ \text{dt}}$ is a constant, not $r\frac{\text{d$\theta $}}{\text{dt}}$. If the satellite moves from one point on its orbit to another point where the distance r is double its previous value, then Kepler's law says that the angular velocity $\frac{\text{d$\theta $}}{\text{dt}}$ decreases by a factor of four. Turning now to the expression $r\frac{\text{d$\theta $}}{\text{dt}}$ that represents how long an object remains in Earth's shadow, we see that this expression does not remain constant but decreases by a factor of 2. We can therefore surmise that the "shadow time" does not remain constant throughout the satellite's orbit but rather decreases as the satellite moves to points on its orbit further away from the Earth, and increases as the satellite moves closer to Earth.

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# External Wall Adapter Detection in MP8007?

I am reading the datasheet of MP8007. I found ambiguity in section Wall Power Adaptor Detection and Operation on Page 16. After reading the section, i am not able to understand how IC will differentiate between voltage from Adapter and PSE. Consider the equation given below:

Considering the example given in Page 20 under section Wall Power Adaptor Detection Circuit (Radpup = 3K, Radpdown = 8.06K and Vdadp3 = 30V), the IC can safely accept voltages from 38.48 to 53.98 V (even more than that).

Now, I am confused how will the IC gets to know that voltage is from Wall adapter or from PSE? Because PoE voltage range is from 44 - 57V and since, there is common voltages between adapter supply and PSE power supply and the wall adapter power is also applied at Vdd pin of IC (as with PoE).

The detection circuit is given below:

Now, I am confused how will the IC gets to know that voltage is from Wall adapter or from PSE?

The datasheet is really unclear on this, but I don't think that's the purpose of the power adapter detection circuit.

AFAICT, the only purpose of power adapter detection is to allow the switching regulator to start up without going through the PoE negotiation process.

what will happen when user plugs in adapter when PSE is already powering the PD

Not much. With diode-OR, one or the other will continue to power the regulator, depending on which one has the higher voltage.

or vice-versa?

Again, not much. But in this case, the PoE negotiation will never occur, so the PSE will never enable power on the line. The external adapter will continue to supply the regulator.

If you subsequently unplug the adapter, then the regulator will shut down for a short period while the PoE negotiation occurs.


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David wrote:

> So I believe there is a (natural) isomorphism between L⊗M and Hom(L,M) for R-modules?

No, for example you're shown the $R$-modules $M \otimes R$ and $\mathrm{Hom}(M,R)$ are different in general: the first is isomorphic to $M$, while the second is by definition $M^*$.

The hom-tensor duality for $R$-modules says, among other things, that
for $R$-modules $M,N,P$ we have an isomorphism

$\mathrm{Hom}(M \otimes N, P) \cong \mathrm{Hom}(M,\mathrm{Hom}(N,P))$

This is easy to see. An $R$-module homomorphism from $M \otimes N$ to $P$ is the same as a bilinear map from $M \times N$ to $P$. But we can take such a bilinear map and think of it as a something that eats an element of $M$ and spits out, in a linear way, a linear map from $N$ to $P$. In other words, an element of $\mathrm{Hom}(M,\mathrm{Hom}(N,P))$

I would take your arguments that tensors work differently over a general ring work differently than tensors over a field and put a different spin on them. What's mainly true is that modules over a field are all free, while modules over a general ring aren't.

This has lots of ramifications. First, working with vector spaces makes one instantly want to grab a basis whenever there's a calculation to be done, but when working with modules you have to suppress this habit. Second, you can't identify a module with its dual. So, you shouldn't think of $M \otimes N$ as consisting of bilinear maps from $M \times N$ to $R$. Instead, it's $(M \otimes N)^*$ that consists of bilinear maps from $M\times N$ to $R$.

Once you develop the new habits, vector spaces over fields seem like a pathetically dull (and wonderfully simple) example of modules over a ring, where all the interesting (and difficult) questions become trivial.


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Recently I have prepared a video about advanced rebased features and how we can use them to make our git repository clean.

The video explains basic usages along with advanced features, in this blog post I will refine the rationale behind the clean repository.

Git is a very powerful tool. It can be used to solve hard problems encountered during the development. However, we tend to use a small subset of its features. It’s not a new tool anymore, its features are not new, it’s around since 2006 now.

*Don't do this.*

Anyways, I will focus on the repository, git history, and talk about how we can (and I do) keep the history clean.

## Rationale

A git repository is where our code stays with its history. Having historical information has lots of benefits. We can see what has changed, in which order changes are introduced, and who changed it clearly. Keeping the code history is a much like writing code. Martin Fowler said:

Any fool can write code that a computer can understand. Good programmers write code that humans can understand.

Like writing code, there is a good and a bad way of keeping the history. I think the same quote can be used for git history:

Any fool can make commits that a computer can understand. Good programmers make commits that humans can understand.

## The Problem

Let’s see what happens when you don’t care about your repository’s history. This screenshot was taken from a four years old repository:

It’s impossible to understand what’s going on here. Well, one thing can be said: lots of merge commits. LOTS OF! Please don’t do this if you care about future developers.

## The Solution

After struggling this for enough time, I have proposed a solution: Rebase. Use rebase. Pulling from remote branch? Pull with rebase. Your feature branch is behind master? Rebase your branch onto master. Do your commits not organized well? Rebase your branch with interactive mode and re-organize your commits. Then I have organized small sessions on how to use rebase features. The video is a result of those small sessions. The git history is now (mostly) as in the following screenshot:

There are minimum merge commits. The order of changes can be tracked very easily. It’s elegant. It’s clean. Less is more, we got rid of unnecessary merge commits, there are less commits, the history is more visible.

Now I want to talk about how we can refine commits at the time of rebase. Let’s assume we have a branch with following commits:

Let’s go over the problems one by one:

1. First commit - feature e initial: Very bad commit message.
2. Second commit - fix text for feature e: Extremely unnecessary commit. This commit fixes a problem in the first commit, it shouldn’t be in the history of the repository. Why would we like to keep something like this in our history? The changes of this second commit should be implemented in the first commit.
3. Third commit: looks okay, but for the sake of the example I assume we need to edit something in this commit.
4. Forth commit: Bad commit message. This commit message looks more like the body of the message. First line should be 50 characters at most, then an empty line comes, and the body can be as long as we want.
5. Fifth commit - experimental? What does it even do in the branch? It should be removed.

Now, let’s see interactive rebase screen for this branch. I run the following command (this is a feature branch so it can be rebased onto master):

\$ git rebase -i master

This is the interactive rebase screen:

The explanations in the comments are enough to understand what to do here. Other than comments, we have commands, commit revision numbers, and commit messages. Commands will be applied to the commits with respect to their revision number. The commit messages for us humans to understand which commit the revision number corresponds to, if we edit the message in this screen, git won’t change the commit messages. The order of commands are important, because git will follow this order. If we reverse the order in this screen, last commit will become the first and first commit will become the last.

So, to fix the problems I have mentioned above, I have prepared the commands like this:

1. We are picking the first commit, because we are squashing the second commit into this one we don’t need to say reword to change the commit message. Git will ask us a commit message for the combined commit.
2. We are squashing this unnecessary commit into the previous one. As I said, the commit fixes a problem in previous commit and the changes here should have been in the first commit from the beginning.
3. We would like to edit something in this commit. Git will stop rebasing while applying this commit to allow us make the necessary changes.
4. We can change this commit’s message with reword command. Again git will ask us the new commit message while applying rebase.
5. I have deleted the line for experimental commit, so it will be removed from branch’s history.

After rebase finishes successfully, our branch looks like this:

First commit’s message could be better, but this version is much better than the previous history. extra commits are gone, commit messages are better. I have even deleted a commit from the history.

You can watch the video above for sample use-cases of rebase: with local branch, remote tracking branch, conflicting branch, branch with multiple commits, and rebase with interactive mode.

All in all, your history is important. Stay clean, stay sane.

## Some resources:

UPDATE: Lots of discussion have been held about this post in Hacker News and Reddit, I suggest you to take a look at them for various opinions.


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# Behaviour of (principal) polarizations of (singular) surfaces under birational maps

Assume we have two p.p. simple abelian surfaces $(A_i,D_i)$, i=1,2, over $\mathbb{C}$ with the following commutative diagram:

$\require{AMScd} \begin{CD} A_1 @>{birational}>> A_2\\ @V{2:1}VV @VV{2:1}V \\ A_1/(-1) @>{birational}>> A_2/(-1)\\ @V{\iota_1}VV @VV{\iota_2}V\\ \mathbb{P}^3 @>{birational}>> \mathbb{P}^3 \end{CD}$

Here the horizontal arrows, are birational maps (not morphisms) and the composition of $(2:1)$ and $\iota_i$ is actually the morphism induced by the linear system $|2D_i|$ for $i=1,2$. That is we have $|2D_i|: A_i \rightarrow \mathbb{P}^3$.

The birational map on top must be an isomorphism, since $A_1$ and $A_2$ are abelian. So we have $A_1\cong A_2$ as unpolarized abelian surfaces.

$\textbf{Question:}$ Do we actually get $(A_,D_1)\cong (A_2,D_2)$ respectively under which conditions can we conclude that we have an isomorphism of principally polarized abelian surfaces using this diagram? Or are there pairs $(A_i,D_i)$ where such an diagram cannot exist?

We may assume $NS(A_i)\geq 2$ because for $NS(A_i)=1$ we must have $(A_1,D_1)\cong(A_2,D_2)$.

I thought one could start with the polarizations $\mathcal{L}_i$ on $A_i/(-1)$ coming from $\mathcal{O}_{\mathbb{P}^3}(1)$. These pullback to $\mathcal{O}_{A_i}(2D_i)$. It would be nice if one could now show that $A_1 \xrightarrow{\cong}A_2$ pulls $D_2$ back to $D_1$. This isomorphism preserves amplenes and self-intersection. So the pullback of $D_2$ should be a principal polarization on $A_1$ but is it $D_1$? But for this we need to know what happens with $\mathcal{L}_2$ under $A_1/(-1) ---> A_2(-1)$? But I have no idea how to get any information about this.

• Every rational transformation from a smooth variety to an Abelian variety is everywhere regular by the Weil extension theorem. Thus, all of your horizontal arrows are isomorphisms. Oct 20, 2017 at 13:56
• @JasonStarr: There is no abelian variety in the bottom horizontal arrow so how do you apply Weil's theorem?
– naf
Oct 21, 2017 at 4:34
• @ulrich Hello again. If the top horizontal arrow is an isomorphism, then the middle arrow is also an isomorphism, since it is just the associated map of Kummer varieties. Of course an Abelian variety can have more than one principle polarization, but my interpretation of the following cryptic sentence is that the OP was focused on regularity of the top morphism, "Do we actually get $(A,D_1)\cong (A_2,D_2)$ under which conditions can we conclude that we have an isomorphism of principally polarized abelian surfaces using this diagram?" Oct 21, 2017 at 9:54
• @JasonStarr: I think the OP is actually just trying to ask his earlier question [mathoverflow.net/questions/281677/… in a slightly different way.
– naf
Oct 22, 2017 at 6:03

@ulrich points out that the OP might be asking a question different than the one I answered, and I completely agree. So let me explain what the comment above actually proves.

Let $(A_1,\mathcal{L}_1)$ and $(A_2,\mathcal{L}_2)$ be Abelian varieties of dimension $g$ together with a specified ample invertible sheaf. Let $U_1\subset A_1$ and $U_2\subset A_2$ be dense open subsets. Let $f_U:U_1\to U_2$ be an isomorphism of $k$-schemes. Let $\phi_U:\mathcal{L}_1|_{U_1}\to f_U^*\mathcal{L}_2$ be an isomorphism of invertible sheaves on $U_1$.

Proposition. There is a unique $k$-isomorphism $f:A_1\to A_2$ whose restriction to $U$ equals $f_U$, and this extends to a $k$-isomorphism of the associated Kummer varieties. If the complement of $U_1$ in $A_1$ has codimension $\geq 2$ everywhere, then there exists a unique isomorphism $\phi:\mathcal{L}_1\to f^*\mathcal{L}_2$ that restricts on $U$ to $\phi_U$. There are counterexamples if the complement has codimension $1.$

Proof. The first result follows from what is often called "Weil extension": every $k$-rational transformation from a smooth, projective $k$-variety to an Abelian variety extends uniquely to a regular $k$-morphism on the entire smooth $k$-scheme. If the complement of $U_1$ in $A_1$ has codimension $\geq 2,$ then by $S2$ extension, there exists a unique extension $\phi$ of $\phi_U$. By Krull's Hauptidealsatz, if $\text{Coker}(\phi)$ is nonzero, then the support has pure codimension $1.$ Since $\phi_U$ is an isomorphism on $U_1,$ then the cokernel is zero. Thus, $\phi$ is surjective. A surjective morphism of invertible sheaves is an isomorphism.

Every $g$-dimensional principally polarized Abelian variety $A$ has many ample invertible sheaves $\mathcal{L}$ with $c_1(\mathcal{L})^{g}$ equal to $g!,$ for example the pullbacks by translations of $A$. Given two effective Cartier divisors $D_1$ and $D_2$ whose invertible sheaves $\mathcal{L}_i=\mathcal{O}_A(D_i)$ are not isomorphic yet have $c_1(\mathcal{L})^g$ equal to $g!,$ on the open $U_1=U_2=U$ equal to $A\setminus (D_1\cup D_2),$ there is an isomorphism $\phi_U$ of $\mathcal{L}_1$ and $\mathcal{L}_2$ on $U.$ Yet $\phi_U$ does not extend to an isomorphism $\phi.$ QED


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de.mpg.escidoc.pubman.appbase.FacesBean
English

# Item

ITEM ACTIONSEXPORT

Released

Journal Article

#### Improved algorithms for linear complementarity problems arising from collision response

##### MPS-Authors
http://pubman.mpdl.mpg.de/cone/persons/resource/persons44379

El Kahoui,  M'hammed
Algorithms and Complexity, MPI for Informatics, Max Planck Society;

##### Locator
There are no locators available
##### Fulltext (public)
There are no public fulltexts available
##### Supplementary Material (public)
There is no public supplementary material available
##### Citation

El Kahoui, M. (2001). Improved algorithms for linear complementarity problems arising from collision response. Mathematics and Computers in Simulation, 56(1), 69-93. Retrieved from http://www.sciencedirect.com/science?_ob=ArticleURL&_udi=B6V0T-42M1G1Y-5&_coverDate=03%2F29%2F2001&_alid=250445639&_rdoc=1&_fmt=&_orig=search&_qd=1&_cdi=5655&_sort=d&view=c&_acct=C000004638&_version=1&_urlVersion=0&_userid=43521&md5=3b414b4652b9b137a81f7d8876ab5e92.

Cite as: http://hdl.handle.net/11858/00-001M-0000-000F-3198-7
##### Abstract
In this paper we give algorithms for solving linear complementarity problems for $\mathcal{P}$-matrices and symmetric positive semidefinite matrices. Our approach of the problem turns out to be an improvement and a more precise formulation of Baraff’s method for problems arising from collision response. The theorems that prove the correctness of our algorithm can also be used to prove the correctness of Baraff’s algorithm. An important feature of the method we present lies in its validity for arbitrary real closed fields, thus it is well suited to handle, at least locally, parametric linear complementarity problems. This article presents the theoretical principles of the algorithms and gives detailed pseudo-code descriptions of them.


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# Relation between $L^1(\partial\Omega)$ and the surface integral on $C^1$ domains

Let $\Omega \subset \mathbb{R}^n$ be a $C^{1}$ bounded domain.

It is possible to define the space $L^1(\partial\Omega)$ as the set of functions $u\colon \partial\Omega \to \mathbb{R}$ with the finite norm $$\lVert u \rVert_{L^1(\partial\Omega)} := \sum_i\lVert {\phi_i(u\circ g_i)}\rVert_{L^1(B_i)}$$ where $\phi_i$ partition of unity, $g_i$ is a $C^1$ diffeomorphism and $B_i$ are subsets of $\mathbb{R}^{n}$.

There are lots of definitions like this (eg. see Renardy and Rogers, Krylov, James Robinson's Infinite Dimensional Dynamical System) or with small variations.

There is also the surface integral of a function $v\colon \partial\Omega \to \mathbb{R}$: $$|u| := \int_{\partial \Omega}fdS$$ where $dS$ is the surface density. To compute this quantity, we need to use a parametrisation.

Now my question, is it always the case that $$\lVert u \rVert_{L^1(\partial\Omega)} \qquad\text{and}\qquad |u|$$ are equivalent? Does this hold for all the little variations of the $L^1$ norm in the first equation??

Now when we have a Lipschitz surface, these norms are equivalent (see Necas). BUT their definition of the $L^1$ norm is different so let us not use that result here. Edit: hmm, it seems $C^1$ domains are not a subset of Lipschitz domains.

• Dear @matt.x, every bounded $C^1$ domain is Lipschitz. – Tomás Jan 2 '14 at 10:13
• Is "$L^2(\partial \Omega)$" a typo? The definitions give equivalent norms, because $g_i$ have derivatives bounded above and below (on the region that contributes to the integral). The constants in the inequality between the norms will depend on the choice of $g_i$. – Post No Bulls Jan 3 '14 at 5:44
• @PostNoBulls Thanks, yes, it should be $L^1(\partial\Omega).$ Why would $g_i$ be bounded from below? I guess there is no canonical definition of surface integral $|\cdot|.$ – matt.x Jan 4 '14 at 17:25
• The second definition is my preferred one, but it is not the best for every occasion. Definitions are used on the as-needed basis, they are not a canon... Being a diffeomorphism implies having the Jacobian being nonzero, which makes it bounded away from $0$ on compact subsets. – Post No Bulls Jan 4 '14 at 17:27
• Yes @matt.x, that's the idea. I suggest you to try and write this mathematically. – Tomás Jan 11 '14 at 23:35

The following does not depend of the partition: $$\|u\|_{L^1(\partial\Omega)}=\Big\|\sum_{i}\int_{B_i}\varphi_i(u\circ g_i)\,dx\,\Big\|.$$ Note that the $B_i$'s should be subsets of $\mathbb R^{n-1}$.


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The Michigan Mathematical Journal

Rational approximation on the unit sphere in C2

Article information

Source
Michigan Math. J., Volume 52, Issue 1 (2004), 105-117.

Dates
First available in Project Euclid: 1 April 2004

https://projecteuclid.org/euclid.mmj/1080837737

Digital Object Identifier
doi:10.1307/mmj/1080837737

Mathematical Reviews number (MathSciNet)
MR2043399

Citation

Anderson, John T.; Izzo, Alexander J.; Wermer, John. Rational approximation on the unit sphere in C 2. Michigan Math. J. 52 (2004), no. 1, 105--117. doi:10.1307/mmj/1080837737. https://projecteuclid.org/euclid.mmj/1080837737

References

• H. Alexander and J. Wermer, Several complex variables and Banach algebras, 3rd ed., Springer-Verlag, New York, 1998.
• J. Anderson and J. Cima, Removable singularities for $L^p$ CR functions, Michigan Math. J. 41 (1994), 111--119.
• ------, The Henkin transform and approximation on the unit sphere in $C^2,$ unpublished manuscript.
• J. Anderson and A. Izzo, A peak point theorem for uniform algebras generated by smooth functions on two-manifolds, Bull. London Math. Soc. 33 (2001), 187--195.
• R. F. Basener, On rationally convex hulls, Trans. Amer. Math. Soc. 182 (1973), 353--381.
• H. S. Bear, Complex function algebras, Trans. Amer. Math. Soc. 90 (1959), 383--393.
• A. Browder, Introduction to function algebras, Benjamin, New York, 1969.
• P. L. Duren, Theory of $H^ p$ spaces, Academic Press, New York, 1970.
• T. Gamelin, Uniform algebras, 2nd ed., Chelsea, New York, 1984.
• G. M. Henkin, H. Lewy's equation and analysis on pseudoconvex manifolds, Russian Math. Surveys 32 (1977), 59--130.
• L. Hörmander, An introduction to complex analysis in several variables, North-Holland, Amsterdam, 1973.
• L. Hörmander and J. Wermer, Uniform approximation on compact sets in $\bold C^n,$ Math. Scand. 23 (1968), 5--21.
• A. Izzo, Uniform algebras generated by holomorphic and pluriharmonic functions on strictly pseudoconvex domains, Pacific J. Math. 171 (1995), 429--436.
• H. P. Lee, Orthogonal measures for subsets of the boundary of the ball in $\Bbb C^ 2,$ Ph.D. thesis, Brown University, Providence, RI, 1979.
• H. P. Lee and J. Wermer, Orthogonal measures for subsets of the boundary of the ball in $\bold C^2,$ Recent developments in several complex variables (Princeton, 1979), pp. 277--289, Princeton Univ. Press, Princeton, NJ, 1981.
• J. Merker and E. Porten, On removable singularities for integrable CR functions, Indiana Univ. Math. J. 48 (1999), 805--856.
• W. Rudin, Function theory in the unit ball of $\bold C^n,$ Springer-Verlag, Berlin, 1980.
• E. L. Stout, The theory of uniform algebras, Bogden & Quigley, Tarrytown-on-Hudson, NY, 1971.
• N. Th. Varopoulos, BMO functions and the $\bar\partial$-equation, Pacific J. Math. 71 (1977), 221--273.
• J. Wermer, Polynomially convex disks, Math. Ann. 158 (1965), 6--10.


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# Tag Info

3

Let $P(t,T)$ denote the time $t$ price of a zero-coupon bond maturing at time $T$ and $\mathbb{Q}_T$ be the associated equivalent martingale measure which uses $P(t,T)$ as numeraire. Then, for any $\mathcal{F}_T$-measurable payoff $\xi$, the time $t$ value of $\xi$ is given by $$V_t=P(t,T)\cdot\mathbb{E}^{\mathbb{Q}_T} [\xi\mid\mathcal{F}_t].$$ The ...

2

as you have an open question, let me try to give you an example in finance. Let $(M_t)$ be a martingale, i.e. a fair game. Such processes, on average (their expectation), do not increase or decrease, they remain constant. As a consequence, $\mathbb{E}[M_t]=M_0$, i.e. if you ask me right now what value I expect $M$ will have at time $t$, I expect on average ...

0

An expectation is an "average" of a property measured over all the possible outcomes. Suppose you have a die and you take the expectation of a roll. It could be 1-6 with equal chance so the expectation is 3.5. But now give it a conditioning. A conditioning is like a filter, so that you exclude possibilities from the whole. What is the expectation of a die ...

1

Thanks to the Girsanov theorem, we have the following relationship between the forward measure $\mathbb{Q}^{T_i}$ and the historical measure $\mathbb{P}$. \begin{align} \left.\frac{d\mathbb{Q}^{T_i}}{d\mathbb{P}}\right|_{\mathcal{F}_t} &= e^{-\int_t^Tr_sds}\frac{P_t(T_i)}{P_0(T_i)} \\ &= \exp\left(\int^{T_i}_{t}\xi_{i}(s)dB_{s}-\frac{1}{2}\int^{T_i}_{...

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text:
A one-layer model of the large-scale circulation in an ocean (the Gulf Stream) was proposed by Ierley and Ruehr in 1986 [1]: $$u''' -\lambda((u')^2-uu'')-u+1=0, ~~u(0) =0, ~~x\in [0,\infty)$$ for some real $\lambda$. Boundary conditions are given as either $u(0)=u'(0)=0$ (rigid or no-slip) or $u(0)=u''(0)=0$ (slippery or stress-free). In both cases we require $u(\infty) = 1$.

In order to solve the problem in Chebfun we'll need to truncate the domain to something suitable, say $[0, X]$, i.e, to make use of the so-called domain truncation (see for instance our papers [3] and [4]).

An integral result for this problem is fairly useful. It reads $$I = \int_{0}^{\infty }[(u'')^2 - 3\lambda uu'u'']dx=\frac{1}{2},$$ and it is obtained multiplying the equation by $u'$, integrating by parts and enforcing the boundary conditions. This result is valid for both types of boundary conditions. Using this integral result we optimise the value of the length $X$ above, and find that the accuracy of the Chebfun result comes close to machine precision.

We can set up the chebop and solve the differential equation with only a few lines of code (see [2] for details).

tic
X = 35;
dom = [0, X];
lambda = -0.1;
op  = @(u) diff(u,3) - lambda*( diff(u,1)^2 - u*diff(u,2) ) - u + 1;
lbc = @(u) [u; diff(u,2)];     % stress-free BC
rbc = 1;
N = chebop(op,dom,lbc,rbc);
[u,info] = N\0;

Here is what the solution looks like.

plot([u diff(u) diff(u,2)])
axis([0 20 -1 1.5])
xlabel('x'), legend('u','u''','u''''','location','southeast')
title('Slippery or stress-free b. c.')

The residuals are small:

N_residual = norm(N(u))                   % residual of diffl. eq.
lbcu = lbc(u);
lbc_residuals = [lbcu{1}(0) lbcu{2}(0)]   % residuals of left BC
rbc_residual = u(end) - rbc               % residual of right BC
N_residual =
4.580588942258870e-10
lbc_residuals =
1.0e-09 *
0.000033013148830   0.133907788023016
rbc_residual =
-9.769962616701378e-15

The Newton iteration has converged quadratically:

semilogy(info.normDelta,'m*-')
ylim([1e-16 1e+01])
xlabel('iteration')
ylabel('norm of Newton update')
%

The Chebyshev coefficients of the solution decrease rapidly:

plotcoeffs(u)

Finally, the integral $I$ comes out with a very small error:

I = sum(diff(u,2)^2 - 3*lambda*(u*diff(u)*diff(u,2)))
I_error = abs(I-1/2)
I =
0.499999999999915
I_error =
8.482103908136196e-14

We solved the problem for several values of $X$ and found that the minimal error in $I$ occurs with $X\approx 35$.

total_time_for_this_example = toc
total_time_for_this_example =
3.629420000000000

### References

1. G. R. Ierley and O. G. Ruehr, Analytic and numerical solutions of a nonlinear boundary-layer problem, Stud. Apl. Math. 75:1-36 (1986).

2. L. N. Trefethen, A. Birkisson, and T. A. Driscoll, Exploring ODEs, SIAM, 2018.

3. C. I. Gheorghiu, Pseudospectral solutions to some singular nonlinear BVPs, Numer. Algor. 68 (2015), 1-14, DOI: 10.1007/s11075-014-9834-z.

4. C. I. Gheorghiu, Spectral collocation solutions to systems of boundary layer type, Numer. Algor. 73 (2016), 1-14, DOI:10.1007/s11075-015-0083-6


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## Performance results of running parallel applications on the InteGrade

CACERES, E. N.; MONGELLI, H.; LOUREIRO, L.; NISHIBE, C.; SONG, S. W.
Fonte: JOHN WILEY & SONS LTD Publicador: JOHN WILEY & SONS LTD
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
49.65578%
The InteGrade middleware intends to exploit the idle time of computing resources in computer laboratories. In this work we investigate the performance of running parallel applications with communication among processors on the InteGrade grid. As costly communication on a grid can be prohibitive, we explore the so-called systolic or wavefront paradigm to design the parallel algorithms in which no global communication is used. To evaluate the InteGrade middleware we considered three parallel algorithms that solve the matrix chain product problem, the 0-1 Knapsack Problem, and the local sequence alignment problem, respectively. We show that these three applications running under the InteGrade middleware and MPI take slightly more time than the same applications running on a cluster with only LAM-MPI support. The results can be considered promising and the time difference between the two is not substantial. The overhead of the InteGrade middleware is acceptable, in view of the benefits obtained to facilitate the use of grid computing by the user. These benefits include job submission, checkpointing, security, job migration, etc. Copyright (C) 2009 John Wiley & Sons, Ltd.; Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP); FAPESP[2004/08928-3]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq[55.0895/07-8]; CNPq[30.5362/06-2]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq[30.2942/04-1]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq[62.0123/04-4]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq[48.5460/06-8]; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq[62.0171/06-5]; FUNDECT[41/100.115/2006]; FUNDECT

## A novel strategy for building interoperable MPI environment in heterogeneous high performance systems

Massetto, Francisco Isidro; Sato, Liria Matsumoto; Li, Kuan-Ching
Fonte: SPRINGER; DORDRECHT Publicador: SPRINGER; DORDRECHT
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
50.1882%
Breakthrough advances in microprocessor technology and efficient power management have altered the course of development of processors with the emergence of multi-core processor technology, in order to bring higher level of processing. The utilization of many-core technology has boosted computing power provided by cluster of workstations or SMPs, providing large computational power at an affordable cost using solely commodity components. Different implementations of message-passing libraries and system softwares (including Operating Systems) are installed in such cluster and multi-cluster computing systems. In order to guarantee correct execution of message-passing parallel applications in a computing environment other than that originally the parallel application was developed, review of the application code is needed. In this paper, a hybrid communication interfacing strategy is proposed, to execute a parallel application in a group of computing nodes belonging to different clusters or multi-clusters (computing systems may be running different operating systems and MPI implementations), interconnected with public or private IP addresses, and responding interchangeably to user execution requests. Experimental results demonstrate the feasibility of this proposed strategy and its effectiveness...

## Removing inefficiencies from scientific code : the study of the Higgs boson couplings to top quarks

Pereira, André Martins; Onofre, A.; Proença, Alberto José
Tipo: Conferência ou Objeto de Conferência
Relevância na Pesquisa
38.759165%
Publicado em "Computational science and its applications – ICCSA 2014 : proceedings", Series : Lecture notes in computer science, vol. 8582; This paper presents a set of methods and techniques to remove inefficiencies in a data analysis application used in searches by the ATLAS Experiment at the Large Hadron Collider. Profiling scientific code helped to pinpoint design and runtime inefficiencies, the former due to coding and data structure design. The data analysis code used by groups doing searches in the ATLAS Experiment contributed to clearly identify some of these inefficiencies and to give suggestions on how to prevent and overcome those common situations in scientific code to improve the efficient use of available computational resources in a parallel homogeneous platform.; This work is funded by National Funds through the FCT - Fundaçãoo para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) within project PEst-OE/EEI/UI0752/2014, by LIP (Laborat ́orio de Instrumentação e Física Experimental de Partículas), and the SeARCH cluster (REEQ/443/EEI/2005).

## SDT: A Virus Classification Tool Based on Pairwise Sequence Alignment and Identity Calculation

Muhire, Brejnev Muhizi; Varsani, Arvind; Martin, Darren Patrick
Fonte: Public Library of Science Publicador: Public Library of Science
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.2376%
The perpetually increasing rate at which viral full-genome sequences are being determined is creating a pressing demand for computational tools that will aid the objective classification of these genome sequences. Taxonomic classification approaches that are based on pairwise genetic identity measures are potentially highly automatable and are progressively gaining favour with the International Committee on Taxonomy of Viruses (ICTV). There are, however, various issues with the calculation of such measures that could potentially undermine the accuracy and consistency with which they can be applied to virus classification. Firstly, pairwise sequence identities computed based on multiple sequence alignments rather than on multiple independent pairwise alignments can lead to the deflation of identity scores with increasing dataset sizes. Also, when gap-characters need to be introduced during sequence alignments to account for insertions and deletions, methodological variations in the way that these characters are introduced and handled during pairwise genetic identity calculations can cause high degrees of inconsistency in the way that different methods classify the same sets of sequences. Here we present Sequence Demarcation Tool (SDT)...

## Optimizing a parallel fast Fourier transform

Hu, Richard, 1982-
Fonte: Massachusetts Institute of Technology Publicador: Massachusetts Institute of Technology
Tipo: Tese de Doutorado Formato: 40 p.; 1477437 bytes; 1479765 bytes; application/pdf; application/pdf
Português
Relevância na Pesquisa
79.605864%
Parallel computing, especially cluster computing has become more popular and more powerful in recent years. Star-P is a means of harnessing that power by eliminating the difficulties in parallelizing code and by providing the user with a familiar and intuitive interface. This paper presents methods to create a parallel FFT module for Star-P. We find that because calculating a parallel FFT is more communication-intensive than processor-intensive, clever planning and distribution of data is needed to achieve speed-up in a parallel environment.; by Richard Hu.; Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2004.; Includes bibliographical references (p. 40).

## Analysis and Restoration of Daguerreotypes Using Cluster Computing

Ardis, Paul A. (1983 - ); Messing, Ross (1981 - ); Tang, Xiaoqing ; Brown, Christopher M. (1945 - ); Nelson, Randal C. ; Ravines, Patrick ; Wiegandt, Ralph
Fonte: University of Rochester. Computer Science Department. Publicador: University of Rochester. Computer Science Department.
Tipo: Relatório
Português
Relevância na Pesquisa
68.8539%
technical reports; The Conservation Laboratory of the George Eastman House International Museum of Photography and Film (GEH) and the Department of Computer Science at the University of Rochester (URCS) are collaborating on the problems of preservation and access to daguerreotypes. Parallel (cluster) computation provides high speed image processing to find, classify, and ultimately to eliminate defects and artifacts of deterioration. This TR describes early low-level techniques and applies them to scanner lighting, dust, and scratch defects.

## Establishing Linux Clusters for high-performance computing (HPC) at NPS

Daillidis, Christos
Português
Relevância na Pesquisa
49.16968%
Approved for public release; distribution is unlimited; S tasks. Discrete Event Simulation (DES) often involves repeated, independent runs of the same models with different input parameters. A system which is able to run many replications quickly is more useful than one in which a single monolithic application runs quickly. A loosely coupled parallel system is indicated. Inexpensive commodity hardware, high speed local area networking, and open source software have created the potential to create just such loosely coupled parallel systems. These systems are constructed from Linux-based computers and are called Beowulf clusters. This thesis presents an analysis of clusters in high-performance computing and establishes a testbed implementation at the MOVES Institute. It describes the steps necessary to create a cluster, factors to consider in selecting hardware and software, and describes the process of creating applications that can run on the cluster. Monitoring the running cluster and system administration are also addressed.

Inozemtsev, Grigori
Fonte: Quens University Publicador: Quens University
Português
Relevância na Pesquisa
38.745205%
As the demands of computational science and engineering simulations increase, the size and capabilities of High Performance Computing (HPC) clusters are also expected to grow. Consequently, the software providing the application programming abstractions for the clusters must adapt to meet these demands. Specifically, the increased cost of interprocessor synchronization and communication in larger systems must be accommodated. Non-blocking operations that allow communication latency to be hidden by overlapping it with computation have been proposed to mitigate this problem. In this work, we investigate offloading a portion of the communication processing to dedicated hardware in order to support communication/computation overlap efficiently. We work with the Message Passing Interface (MPI), the de facto standard for parallel programming in HPC environments. We investigate both point-to-point non-blocking communication and collective operations; our work with collectives focuses on the allgather operation. We develop designs for both flat and hierarchical cluster topologies and examine both eager and rendezvous communication protocols. We also develop a generalized primitive operation with the aim of simplifying further research into non-blocking collectives. We propose a new algorithm for the non-blocking allgather collective and implement it using this primitive. The algorithm has constant resource usage even when executing multiple operations simultaneously. We implemented these designs using CORE-Direct offloading support in Mellanox InfiniBand adapters. We present an evaluation of the designs using microbenchmarks and an application kernel that shows that offloaded non-blocking communication operations can provide latency that is comparable to that of their blocking counterparts while allowing most of the duration of the communication to be overlapped with computation and remaining resilient to process arrival and scheduling variations.; Thesis (Master...

## Alternating Maximization: Unifying Framework for 8 Sparse PCA Formulations and Efficient Parallel Codes

Richtárik, Peter; Takáč, Martin; Ahipaşaoğlu, Selin Damla
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
58.746416%
Given a multivariate data set, sparse principal component analysis (SPCA) aims to extract several linear combinations of the variables that together explain the variance in the data as much as possible, while controlling the number of nonzero loadings in these combinations. In this paper we consider 8 different optimization formulations for computing a single sparse loading vector; these are obtained by combining the following factors: we employ two norms for measuring variance (L2, L1) and two sparsity-inducing norms (L0, L1), which are used in two different ways (constraint, penalty). Three of our formulations, notably the one with L0 constraint and L1 variance, have not been considered in the literature. We give a unifying reformulation which we propose to solve via a natural alternating maximization (AM) method. We show the the AM method is nontrivially equivalent to GPower (Journ\'{e}e et al; JMLR 11:517--553, 2010) for all our formulations. Besides this, we provide 24 efficient parallel SPCA implementations: 3 codes (multi-core, GPU and cluster) for each of the 8 problems. Parallelism in the methods is aimed at i) speeding up computations (our GPU code can be 100 times faster than an efficient serial code written in C++), ii) obtaining solutions explaining more variance and iii) dealing with big data problems (our cluster code is able to solve a 357 GB problem in about a minute).; Comment: 20 pages...

## WinBioinfTools: Bioinformatics Tools for Windows High Performance Computing Server 2008

Abouelhoda, Mohamed; Mohamed, Hisham
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
49.79872%
Open source bioinformatics tools running under MS Windows are rare to find, and those running under Windows HPC cluster are almost non-existing. This is despite the fact that the Windows is the most popular operating system used among life scientists. Therefore, we introduce in this initiative WinBioinfTools, a toolkit containing a number of bioinformatics tools running under Windows High Performance Computing Server 2008. It is an open source code package, where users and developers can share and add to. We currently start with three programs from the area of sequence analysis: 1) CoCoNUT for pairwise genome comparison, 2) parallel BLAST for biological database search, and 3) parallel global pairwise sequence alignment. In this report, we focus on technical aspects concerning how some components of these tools were ported from Linux/Unix environment to run under Windows. We also show the advantages of using the Windows HPC Cluster 2008. We demonstrate by experiments the performance gain achieved when using a computer cluster against a single machine. Furthermore, we show the results of comparing the performance of WinBioinfTools on the Windows and Linux Cluster.

## Parallel degree computation for solution space of binomial systems with an application to the master space of $\mathcal{N}=1$ gauge theories

Chen, Tianran; Mehta, Dhagash
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
58.591934%
The problem of solving a system of polynomial equations is one of the most fundamental problems in applied mathematics. Among them, the problem of solving a system of binomial equations form a important subclass for which specialized techniques exist. For both theoretic and applied purposes, the degree of the solution set of a system of binomial equations often plays an important role in understanding the geometric structure of the solution set. Its computation, however, is computationally intensive. This paper proposes a specialized parallel algorithm for computing the degree on GPUs that takes advantage of the massively parallel nature of GPU devices. The preliminary implementation shows remarkable efficiency and scalability when compared to the closest CPU-based counterpart. Applied to the "master space problem of $\mathcal{N}=1$ gauge theories" the GPU-based implementation achieves nearly 30 fold speedup over its CPU-only counterpart enabling the discovery of previously unknown results. Equally important to note is the far superior scalability: with merely 3 GPU devices on a single workstation, the GPU-based implementation shows better performance, on certain problems, than a small cluster totaling 100 CPU cores.; Comment: 27 pages...

## A Parallel Framework for Parametric Maximum Flow Problems in Image Segmentation

Olaru, Vlad; Florea, Mihai; Sminchisescu, Cristian
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
48.8835%
This paper presents a framework that supports the implementation of parallel solutions for the widespread parametric maximum flow computational routines used in image segmentation algorithms. The framework is based on supergraphs, a special construction combining several image graphs into a larger one, and works on various architectures (multi-core or GPU), either locally or remotely in a cluster of computing nodes. The framework can also be used for performance evaluation of parallel implementations of maximum flow algorithms. We present the case study of a state-of-the-art image segmentation algorithm based on graph cuts, Constrained Parametric Min-Cut (CPMC), that uses the parallel framework to solve parametric maximum flow problems, based on a GPU implementation of the well-known push-relabel algorithm. Our results indicate that real-time implementations based on the proposed techniques are possible.

## Searching for Globally Optimal Functional Forms for Inter-Atomic Potentials Using Parallel Tempering and Genetic Programming

Slepoy, A.; Thompson, A. P.; Peters, M. D.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
49.12466%
We develop a Genetic Programming-based methodology that enables discovery of novel functional forms for classical inter-atomic force-fields, used in molecular dynamics simulations. Unlike previous efforts in the field, that fit only the parameters to the fixed functional forms, we instead use a novel algorithm to search the space of many possible functional forms. While a follow-on practical procedure will use experimental and {\it ab inito} data to find an optimal functional form for a forcefield, we first validate the approach using a manufactured solution. This validation has the advantage of a well-defined metric of success. We manufactured a training set of atomic coordinate data with an associated set of global energies using the well-known Lennard-Jones inter-atomic potential. We performed an automatic functional form fitting procedure starting with a population of random functions, using a genetic programming functional formulation, and a parallel tempering Metropolis-based optimization algorithm. Our massively-parallel method independently discovered the Lennard-Jones function after searching for several hours on 100 processors and covering a miniscule portion of the configuration space. We find that the method is suitable for unsupervised discovery of functional forms for inter-atomic potentials/force-fields. We also find that our parallel tempering Metropolis-based approach significantly improves the optimization convergence time...

## A Hybrid Parallelization of AIM for Multi-Core Clusters: Implementation Details and Benchmark Results on Ranger

Wei, Fangzhou; Yılmaz, Ali E.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
48.68004%
This paper presents implementation details and empirical results for a hybrid message passing and shared memory paralleliziation of the adaptive integral method (AIM). AIM is implemented on a (near) petaflop supercomputing cluster of quad-core processors and its accuracy, complexity, and scalability are investigated by solving benchmark scattering problems. The timing and speedup results on up to 1024 processors show that the hybrid MPI/OpenMP parallelization of AIM exhibits better strong scalability (fixed problem size speedup) than pure MPI parallelization of it when multiple cores are used on each processor.; Comment: 24 pages, 3 tables, 9 figures. Due to space constraints, some implementation details and empirical data are omitted in authors' another paper (reference [1]), which has been submitted to Parallel Computing. This paper here serves as a major reference with the implementation details and comprehensive empirical data

## Biomolecular electrostatics using a fast multipole BEM on up to 512 GPUs and a billion unknowns

Yokota, Rio; Bardhan, Jaydeep P.; Knepley, Matthew G.; Barba, L. A.; Hamada, Tsuyoshi
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
48.650166%
We present teraflop-scale calculations of biomolecular electrostatics enabled by the combination of algorithmic and hardware acceleration. The algorithmic acceleration is achieved with the fast multipole method (FMM) in conjunction with a boundary element method (BEM) formulation of the continuum electrostatic model, as well as the BIBEE approximation to BEM. The hardware acceleration is achieved through graphics processors, GPUs. We demonstrate the power of our algorithms and software for the calculation of the electrostatic interactions between biological molecules in solution. The applications demonstrated include the electrostatics of protein--drug binding and several multi-million atom systems consisting of hundreds to thousands of copies of lysozyme molecules. The parallel scalability of the software was studied in a cluster at the Nagasaki Advanced Computing Center, using 128 nodes, each with 4 GPUs. Delicate tuning has resulted in strong scaling with parallel efficiency of 0.8 for 256 and 0.5 for 512 GPUs. The largest application run, with over 20 million atoms and one billion unknowns, required only one minute on 512 GPUs. We are currently adapting our BEM software to solve the linearized Poisson-Boltzmann equation for dilute ionic solutions...

## Efficient implementation of the overlap operator on multi-GPUs

Alexandru, Andrei; Lujan, Michael; Pelissier, Craig; Gamari, Ben; Lee, Frank X.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
38.849321%
Lattice QCD calculations were one of the first applications to show the potential of GPUs in the area of high performance computing. Our interest is to find ways to effectively use GPUs for lattice calculations using the overlap operator. The large memory footprint of these codes requires the use of multiple GPUs in parallel. In this paper we show the methods we used to implement this operator efficiently. We run our codes both on a GPU cluster and a CPU cluster with similar interconnects. We find that to match performance the CPU cluster requires 20-30 times more CPU cores than GPUs.; Comment: 8 pages with 10 figures; accepted for presentation at the 2011 Symposium on Application Accelerators in High Performance Computing (Knoxville, July 19-20, 2011)

## How to obtain efficient GPU kernels: an illustration using FMM & FGT algorithms

Cruz, Felipe A.; Layton, Simon K.; Barba, Lorena A.
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.885034%
Computing on graphics processors is maybe one of the most important developments in computational science to happen in decades. Not since the arrival of the Beowulf cluster, which combined open source software with commodity hardware to truly democratize high-performance computing, has the community been so electrified. Like then, the opportunity comes with challenges. The formulation of scientific algorithms to take advantage of the performance offered by the new architecture requires rethinking core methods. Here, we have tackled fast summation algorithms (fast multipole method and fast Gauss transform), and applied algorithmic redesign for attaining performance on gpus. The progression of performance improvements attained illustrates the exercise of formulating algorithms for the massively parallel architecture of the gpu. The end result has been gpu kernels that run at over 500 Gigaflops on one nvidia Tesla C1060 card, thereby reaching close to practical peak. We can confidently say that gpu computing is not just a vogue, it is truly an irresistible trend in high-performance computing.

## Scaling Datalog for Machine Learning on Big Data

Bu, Yingyi; Borkar, Vinayak; Carey, Michael J.; Rosen, Joshua; Polyzotis, Neoklis; Condie, Tyson; Weimer, Markus; Ramakrishnan, Raghu
Tipo: Artigo de Revista Científica
Português
Relevância na Pesquisa
38.776687%
In this paper, we present the case for a declarative foundation for data-intensive machine learning systems. Instead of creating a new system for each specific flavor of machine learning task, or hardcoding new optimizations, we argue for the use of recursive queries to program a variety of machine learning systems. By taking this approach, database query optimization techniques can be utilized to identify effective execution plans, and the resulting runtime plans can be executed on a single unified data-parallel query processing engine. As a proof of concept, we consider two programming models--Pregel and Iterative Map-Reduce-Update---from the machine learning domain, and show how they can be captured in Datalog, tuned for a specific task, and then compiled into an optimized physical plan. Experiments performed on a large computing cluster with real data demonstrate that this declarative approach can provide very good performance while offering both increased generality and programming ease.

## Planificación de diferentes clases de aplicaciones en entornos no dedicados considerando procesadores multicore

García Gutiérrez, José Ramón
Fonte: Bellaterra: Universitat Autònoma de Barcelona, Publicador: Bellaterra: Universitat Autònoma de Barcelona,
Tipo: Tesis i dissertacions electròniques; info:eu-repo/semantics/doctoralThesis Formato: application/pdf