5. Taylor and Laurent series Complex sequences and series

5. Taylor and Laurent series Complex sequences and series An infinite sequence of complex numbers, denoted by {zn}, can be considered as a function defined on a set of positive integers into the unextended complex plane. For example, we take zn= n+ 1 2n so that the complex sequence is {zn} = ˆ1 + i 2, 2 + i 22, 3 + i 23 ...

  Series, Sequence, Sequences and series

Matrix Algebra for Engineers - Hong Kong University of ...

The mathematics in this matrix algebra course is at the level of an advanced high school student, but typically students would take this course after completing a university-level single variable calculus course. There are no derivatives and integrals in this course, but student’s are expected to …

  Matrix, Algebra, Matrix algebra

Differential Equations - Department of Mathematics, HKUST

used textbook “Elementary differential equations and boundary value problems” by Boyce & DiPrima (John Wiley & Sons, Inc., Seventh Edition, c 2001). Many of the examples presented in these notes may be found in this book. The material of Chapter 7 is adapted from the textbook “Nonlinear dynamics and chaos” by Steven

  Differential, Equations, Differential equations

Vector Calculus for Engineers - Hong Kong University of ...

These are the lecture notes for my online Coursera course,Vector Calculus for Engineers. Students who take this course are expected to already know single-variable differential and integral calculus to the level of an introductory college calculus course. Students should also be familiar with matrices,

  Vector, Introductory, Calculus, Vector calculus

Differential Equations for Engineers

equations 1. By checking all that apply, classify the following differential equation: d3y dx3 +y d2y dx2 = 0 a)first order b)second order c)third order d)ordinary e)partial f)linear g)nonlinear 2. By checking all that apply, classify the following differential equation: 1 x2 d dx x2 dy dx = e y a)first order b)second order c)ordinary d ...

  Engineer, Differential, Equations, Nonlinear, Differential equations for engineers

Portfolio Selection Harry Markowitz The Journal of Finance ...

vary with risk. The hypothesis (or maxim) that the investor does (or should) maximize discounted return must be rejected. If we ignore market im- perfections the foregoing rule never implies that there is a diversified portfolio which is preferable to all non-diversified portfolios. Diversi-

  Risks, Portfolio, Markowitz

4. Complex integration: Cauchy integral theorem and …

4. Complex integration: Cauchy integral theorem and Cauchy integral formulas Definite integral of a complex-valued function of a real variable Consider a complex valued function f(t) of a real variable t: f(t) = u(t) + iv(t), which is assumed to be a piecewise continuous function defined in the closed interval a ≤ t ≤ b.

  Formula, Integration, Complex, Relating, Theorem, Cauchy, Complex integration, Cauchy integral theorem and

(2.1) Markowitz’s mean-variance formulation (2.2) Two …

frontier portfolios need only invest in combinations of these two funds. Remark Any convex combination (that is, weights are non-negative) of ef-ficient portfolios is an efficient portfolio. Let αi ≥ 0 be the weight of Fund i whose rate of return is Ri f. Since E h Ri f i …

  Name, Portfolio, Variance, Formulation, Markowitz, Markowitz s mean variance formulation

Numerical Methods - Hong Kong University of Science and ...

In MATLAB, single(224) has the same value as single(224 +1). Since single(224 +1) is exactly halfway between the two consecutive machine numbers 224 and 224 +2, MATLAB rounds to the number with a final zero-bit in f, which is 224. 1.10Machine epsilon Machine epsilon (e mach) is the distance between 1 and the next largest number. If

  Numerical

Systems of Linear Equations - Hong Kong University of ...

Systems of Linear Equations Beifang Chen 1 Systems of linear equations Linear systems A linear equation in variables x1;x2;:::;xn is an equation of the form a1x1 +a2x2 +¢¢¢+anxn = b; where a1;a2;:::;an and b are constant real or complex numbers. The constant ai is called the coe–cient of xi; and b is called the constant term of the equation. A system of linear equations …

  System, Linear, Equations, Systems of linear equations

Restricted Cubic Spline Regression: A Brief Introduction

Cubic splines tend to be poorly behaved at the two tails (before the first knot and after the last knot). To avoid this, restricted cubic splines are used. A r estricted cubic spline is a cubic spline in which the splines are constrained to be linear in the two tails. This generally provides a better fit to the data, and also has

  Regression, Cubic, Spline, Cubic splines, Cubic spline regression

IEEE TRANSACTIONS Cubic Convolution Interpolation for ...

as the method of cubic splines. The spline interpolation kernel is not zero for nonzero integers. As a result, the ck’s must be determined by solving a matrix problem. In addition to being 0 or 1 at the interpolation nodes, the in- terpolation kernel must …

  Cubic, Spline

Solving a tridiagonal linear system

can be used to de ne an n n tridiagonal matrix associated with interpolation by cubic splines. The functions A=spline matrix(g) and [a,b,c]=spline sparse matrix(g) return dense and sparse versions of this matrix. In order to manufacture a problem, we choose a simple solution vector x and compute f = Ax. However,

  Cubic, Spline, Tridiagonal

METODE NUMERIK - Universitas Brawijaya

Interpolasi Cubic Spline dimana S i adalah polinomial berderajat 3: p(x i) = d i + (x-x i) c i + (x-x i)2 b i + (x-x i)3 a i, i=1,2, …, n-1 Syarat: S i(x i) = S i+1(x i), S i’(x i) = S i+1’(x i), S i’’(x i) = S i+1’’(x i)

  Cubic, Spline, Cubic splines

list of some useful R functions - Columbia University

smooth.spline() - Fits a cubic smoothing spline jitter() - Add a small amount of noise to a numeric vector pdf()/ png() / jpeg() - send plot to .pdf / .png / .jpeg le

  University, Functions, Useful, Some, Columbia university, Columbia, Cubic, Spline, Of some useful r functions

Bezier Curves and Splines - MIT OpenCourseWare

• Express what happens when a spline curve is transformed by an affine transform (rotation, translation, etc.) • Cool simple example of non-trivial vector space • Important to understand for advanced methods ... • Cubic polynomials also compose a vector space

  Cubic, Curves, Spline, Mit opencourseware, Opencourseware, Curves and splines

MATLAB Examples - Interpolation and Curve Fitting

– spline – cubic – etc. • Type “help interp1” in order to read more about the different options. Interpolation x y 0 15 1 10 2 9 3 6 4 2 5 0 Given the following Data Points: Problem: Assume we want to find the interpolated value for, e.g., ! = 3.5 x=0:5; y=[15, 10, 9, 6, 2, 0]; plot(x,y ,'o') grid (Logged Data from a given

  Cubic, Spline

Cubic Spline Interpolation - College of the Redwoods

Cubic Spline Interpolation Sky McKinley and Megan Levine Math 45: Linear Algebra Abstract.An introduction into the theory and application of cubic splines with accompanying Matlab m -file cspline .m Introduction Real world numerical data is usually difficult to analyze .

  Cubic, Spline, Interpolation, Cubic spline interpolation

2000, 4000, 6000, and 10,000 Series Hydraulic Motors

• 1-1/4 inch Dia. Splined 14 T with 38,1 [1.50] Min. Full Spline Length and 53,1 [2.09] Max. Coupling Length • 3/4 inch 4 Bolt Split Flange with 7/16-20 O-ring Case Drain and Check Valve • 1-1/2 inch Dia. Splined 17 T with 31,2 [1.23] Min. Full Spline Length • 40 mm Dia. Straight with Straight Key, M12 x 1,75-6H Threaded Hole

  Spline

「定量」分析法である p - TOYAKU

3 13 H2SO4 + 2NaOH →Na2SO4 + H2O 滴定に要したNaOH のモル数：H2SO4 のモル数= 2 ：1 この容量分析用標準液の真のNaOH 濃度は 0.1 mol/L NaOH (f = ＊＊＊)14 指示薬 指示薬 標準試薬 KIO3 0.1 mol/L Na2S2O3 直接法による標定