# BERNSTEIN POLYNOMIALS

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful mathematcal tools as they are smply defned, can be calculated quckly on computer systems and represent a tremendous varety of functons. They can be dfferentated and ntegrated easly, and can be peced together to form splne curves that can approxmate any functon to any accuracy desred. Most students are ntroducted to polynomals at a very early stage n ther studes of mathematcs, and would probably recall them n the form below: p(t a n t n + a n 1 t n a 1 t + a 0 whch represents a polynomal as a lnear combnaton of certan elementary polynomals { (1, t, t 2,..., t n}. In general, any polynomal functon that has degree less than or equal to n, can be wrtten n ths way, and the reasons are smply The set of polynomals of degree less than or equal to n forms a vector space: polynomals can be added together, can be multpled by a scalar, and all the vector space propertes hold. The set of functons { 1, t, t 2,..., t n} form a bass for ths vector space that s, any polynomal of degree less than or equal to n can be unquely wrtten as a lnear combnatons of these functons. Ths bass, commonly called the power bass, s only one of an nfnte number of bases for the space of polynomals. In these notes we dscuss another of the commonly used bases for the space of polynomals, the Bernsten bass, and dscuss ts many useful propertes.

2 Bernsten Polynomals The Bernsten polynomals of degree n are defned by for 0, 1,..., n, where B,n (t ( n ( n t (1 t n n!!(n! There are n + 1 nth-degree Bernsten polynomals. For mathematcal convenence, we usually set B,n 0, f < 0 or > n. These polynomals are qute easy to wrte down: the coeffcents can be obtaned from Pascal s trangle; the exponents on the t term ncrease by one as ncreases; and the exponents on the (1 t term decrease by one as ncreases. In the smple cases, we obtan The Bernsten polynomals of degree 1 are and can be plotted for 0 t 1 as B 0,1 (t 1 t B 1,1 (t t 2

3 The Bernsten polynomals of degree 2 are B 0,2 (t (1 t 2 B 1,2 (t 2t(1 t B 2,2 (t t 2 and can be plotted for 0 t 1 as The Bernsten polynomals of degree 3 are B 0,3 (t (1 t 3 B 1,3 (t 3t(1 t 2 B 2,3 (t 3t 2 (1 t B 3,3 (t t 3 and can be plotted for 0 t 1 as 3

4 !"# "\$%& 4

5 A Recursve Defnton of the Bernsten Polynomals The Bernsten polynomals of degree n can be defned by blendng together two Bernsten polynomals of degree n 1. That s, the kth nth-degree Bernsten polynomal can be wrtten as B k,n (t (1 tb k,n 1 (t + tb k 1,n 1 (t To show ths, we need only use the defnton of the Bernsten polynomals and some smple algebra: ( ( n 1 n 1 (1 tb k,n 1 (t + tb k 1,n 1 (t (1 t t k (1 t n 1 k + t t k 1 (1 t n 1 (k 1 k k 1 ( ( n 1 n 1 t k (1 t n k + t k (1 t n k k k 1 [( ( ] n 1 n 1 + t k (1 t n k k k 1 ( n t k (1 t n k k B k,n (t The Bernsten Polynomals are All Non-Negatve A functon f(t s non-negatve over an nterval [a, b] f f(t 0 for t [a, b]. In the case of the Bernsten polynomals of degree n, each s non-negatve over the nterval [0, 1]. To show ths we use the recursve defnton property above and mathematcal nducton. It s easly seen that the functons B 0,1 (t 1 t and B 1,1 (t t are both non-negatve for 0 t 1. If we assume that all Bernsten polynomals of degree less than k are non-negatve, then by usng the recursve defnton of the Bernsten polynomal, we can wrte B,k (t (1 tb,k 1 (t + tb 1,k 1 (t and argue that B,k (t s also non-negatve for 0 t 1, snce all components on the rght-hand sde of the equaton are non-negatve components for 0 t 1. By nducton, all Bernsten polynomals are non-negatve for 0 t 1. In ths process, we have also shown that each of the Bernsten polynomals s postve when 0 < t < 1. 5

6 The Bernsten Polynomals form a Partton of Unty A set of functons f (t s sad to partton unty f they sum to one for all values of t. The k+1 Bernsten polynomals of degree k form a partton of unty n that they all sum to one. To show that ths s true, t s easest to frst show a slghtly dfferent fact: for each k, the sum of the k + 1 Bernsten polynomals of degree k s equal to the sum of the k Bernsten polynomals of degree k 1. That s, k B,k (t k 1 B,k 1 (t Ths calculaton s straghtforward, usng the recursve defnton and cleverly rearrangng the sums: k B,k (t k [(1 tb,k 1 (t + tb 1,k 1 (t] (1 t ] [ k ] B,k 1 (t + B k,k 1 (t + t B 1,k 1 (t + B 1,k 1 (t [ k 1 k 1 (1 t B,k 1 (t + t k B 1,k 1 (t 1 k 1 k 1 (1 t B,k 1 (t + t B,k 1 (t k 1 B,k 1 (t (where we have utlzed B k,k 1 (t B 1,k 1 (t 0. Once we have establshed ths equalty, t s smple to wrte 1 B,n (t n 1 B,n 1 (t n 2 B,n 2 (t 1 B,1 (t (1 t + t 1 The partton of unty s a very mportant property when utlzng Bernsten polynomals n geometrc modelng and computer graphcs. In partcular, for any set of ponts P 0, P 1,..., P n, n three-dmensonal space, and for any t, the expresson P(t P 0 B 0,n (t + P 1 B 1,n (t + + P n B n,n (t 6

7 s an affne combnaton of the set of ponts P 0, P 1,..., P n and f 0 t 1, t s a convex combnaton of the ponts. Degree Rasng Any of the lower-degree Bernsten polynomals (degree < n can be expressed as a lnear combnaton of Bernsten polynomals of degree n. In partcular, any Bernsten polynomal of degree n 1 can be wrtten as a lnear combnaton of Bernsten polynomals of degree n. We frst note that ( n tb,n (t t +1 (1 t n ( n t +1 (1 t (n+1 (+1 +1 B +1,n+1 (t n + 1 B +1,n+1(t and ( n (1 tb,n (t t (1 t n+1 +1 B,n+1 (t n + 1 n + 1 B,n+1(t and fnally 1 B,n (t + 1 B +1,n (t t (1 t n + t +1 (1 t n (+1 +1 t (1 t n 1 ((1 t + t t (1 t n B,n 1 (t Usng ths fnal equaton, we can wrte an arbtrary Bernsten polynomal n terms of Bernsten polynomals 7

8 of hgher degree. That s, 1 B,n 1 (t n [ 1 (n ] B,n (t + 1 B +1,n (t B,n (t + ( + 1 n +1 B +1,n (t whch expresses a Bernsten polynomal of degree n 1 n terms of a lnear combnaton of Bernsten polynomals of degree n. We can easly extend ths to show that any Bernsten polynomal of degree k (less than n can be wrtten as a lnear combnaton of Bernsten polynomals of degree n e.g., a Bernsten polynomal of degree n 2 can be expressed as a lnear combnaton of two Bernsten polynomals of degree n 1, each of whch can be expressed as a lnear combnaton of two Bernsten polynomals of degree n, etc. Convertng from the Bernsten Bass to the Power Bass Snce the power bass {1, t, t 2,..., t n } forms a bass for the space of polynomals of degree less than or equal to n, any Bernsten polynomal of degree n can be wrtten n terms of the power bass. Ths can be drectly calculated usng the defnton of the Bernsten polynomals and the bnomal theorem, as follows: B k,n (t k k t k (1 t n k t k n k n k ( 1 k ( n k ( 1 t ( n k t +k ( n k ( 1 k k k ( ( n ( 1 k t k where we have used the bnomal theorem to expand (1 t n k. k k To show that each power bass element can be wrtten as a lnear combnaton of Bernsten Polynomals, t 8

9 we use the degree elevaton formulas and nducton to calculate: t k t(t k 1 t ( k 1 k 1 k 1 1 k 1 1 k k 1 ( n 1 k 1 k 1 k 1 n 1 k 1 ( k B,n 1 (t tb 1,n 1 (t n B,n(t kb,n (t, where the nducton hypothess was used n the second step. Dervatves Dervatves of the nth degree Bernsten polynomals are polynomals of degree n 1. Usng the defnton of the Bernsten polynomal we can show that ths dervatve can be wrtten as a lnear combnaton of Bernsten polynomals. In partcular d dt B k,n(t n(b k 1,n 1 (t B k,n 1 (t for 0 k n. Ths can be shown by drect dfferentaton d dt B k,n(t d dt ( n t k (1 t n k k kn! k!(n k! tk 1 (1 t n k + n(n 1! (k 1!(n k! tk 1 (1 t n k + ( (n 1! n (k 1!(n k! tk 1 (1 t n k + n (B k 1,n 1 (t B k,n 1 (t (n kn! k!(n k! tk (1 t n k 1 n(n 1! k!(n k 1! tk (1 t n k 1 (n 1! k!(n k 1! tk (1 t n k 1 9

10 That s, the dervatve of a Bernsten polynomal can be expressed as the degree of the polynomal, multpled by the dfference of two Bernsten polynomals of degree n 1. The Bernsten Polynomals as a Bass Why do the Bernsten polynomals of order n form a bass for the space of polynomals of degree less than or equal to n? 1. They span the space of polynomals any polynomal of degree less than or equal to n can be wrtten as a lnear combnaton of the Bernsten polynomals. Ths s easly seen f one realzes that The power bass spans the space of polynomals and any member of the power bass can be wrtten as a lnear combnaton of Bernsten polynomals. 2. They are lnearly ndependent that s, f there exst constants c 0, c 1,..., c n so that the dentty 0 c 0 B 0,n (t + c 1 B 1,n (t + + c n B n,n (t holds for all t, then all the c s must be zero. If ths were true, then we could wrte 0 c 0 B 0,n (t + c 1 B 1,n (t + + c n B n,n (t ( ( n ( ( n c 0 ( 1 t + c 1 ( 1 1 t + + c n [ 1 ( ( ] [ ( ( ] n 1 n n c 0 + c t c t n 1 1 n n Snce the power bass s a lnearly ndependent set, we must have that ( ( n ( 1 n t n n c ( ( n 1 c ( ( n n c 0 n n whch mples that c 0 c 1 c n 0 (c 0 s clearly zero, substtutng ths n the second equaton 10

11 gves c 1 0, substtutng these two nto the thrd equaton gves... 11

12 A Matrx Representaton for Bernsten Polynomals In many applcatons, a matrx formulaton for the Bernsten polynomals s useful. These are straghtforward to develop f one only looks at a lnear combnaton n terms of dot products. Gven a polynomal wrtten as a lnear combnaton of the Bernsten bass functons B(t c 0 B 0,n (t + c 1 B 1,n (t + + c n B n,n (t It s easy to wrte ths as a dot product of two vectors B(t [ B 0,n (t B 1,n (t B n,n (t ] c 0 c 1. c n We can convert ths to B(t [ 1 t t 2 ] t n b 0, b 1,0 b 1,1 0 0 b 2,0 b 2,1 b 2, c 0 c 1 c 2. b n,0 b n,1 b n,2 b n,n c n where the b,j are the coeffcents of the power bass that are used to determne the respectve Bernsten polynomals. We note that the matrx n ths case s lower trangular. In the quadratc case (n 2, the matrx representaton s [ ] B(t 1 t t c 0 c 1 c 2 12

13 and n the cubc case (n 3, the matrx representaton s B(t [ 1 t t 2 ] t c 0 c 1 c 2 c 3 All contents copyrght (c 1996, 1997, 1998, 1999, 2000 Computer Scence Department, Unversty of Calforna, Davs All rghts reserved. 13

### Recurrence. 1 Definitions and main statements

Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

### 8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

### Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

### greatest common divisor

4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

### Support Vector Machines

Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

### n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

### Generalizing the degree sequence problem

Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts

### v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

### 6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

### QUANTUM MECHANICS, BRAS AND KETS

PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

### Ring structure of splines on triangulations

www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

### Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

### Fast degree elevation and knot insertion for B-spline curves

Computer Aded Geometrc Desgn 22 (2005) 183 197 www.elsever.com/locate/cagd Fast degree elevaton and knot nserton for B-splne curves Q-Xng Huang a,sh-mnhu a,, Ralph R. Martn b a Department of Computer Scence

### 1 Example 1: Axis-aligned rectangles

COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

### We are now ready to answer the question: What are the possible cardinalities for finite fields?

Chapter 3 Fnte felds We have seen, n the prevous chapters, some examples of fnte felds. For example, the resdue class rng Z/pZ (when p s a prme) forms a feld wth p elements whch may be dentfed wth the

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

### The eigenvalue derivatives of linear damped systems

Control and Cybernetcs vol. 32 (2003) No. 4 The egenvalue dervatves of lnear damped systems by Yeong-Jeu Sun Department of Electrcal Engneerng I-Shou Unversty Kaohsung, Tawan 840, R.O.C e-mal: yjsun@su.edu.tw

### + + + - - This circuit than can be reduced to a planar circuit

MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

### where the coordinates are related to those in the old frame as follows.

Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

### 8 Algorithm for Binary Searching in Trees

8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

### 21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

### ABC. Parametric Curves & Surfaces. Overview. Curves. Many applications in graphics. Parametric curves. Goals. Part 1: Curves Part 2: Surfaces

arametrc Curves & Surfaces Adam Fnkelsten rnceton Unversty COS 46, Sprng Overvew art : Curves art : Surfaces rzemyslaw rusnkewcz Curves Splnes: mathematcal way to express curves Motvated by loftsman s

### Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

### An Alternative Way to Measure Private Equity Performance

An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

### The Mathematical Derivation of Least Squares

Pscholog 885 Prof. Federco The Mathematcal Dervaton of Least Squares Back when the powers that e forced ou to learn matr algera and calculus, I et ou all asked ourself the age-old queston: When the hell

### REGULAR MULTILINEAR OPERATORS ON C(K) SPACES

REGULAR MULTILINEAR OPERATORS ON C(K) SPACES FERNANDO BOMBAL AND IGNACIO VILLANUEVA Abstract. The purpose of ths paper s to characterze the class of regular contnuous multlnear operators on a product of

### 2.4 Bivariate distributions

page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

### benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

### On Leonid Gurvits s proof for permanents

On Leond Gurvts s proof for permanents Monque Laurent and Alexander Schrver Abstract We gve a concse exposton of the elegant proof gven recently by Leond Gurvts for several lower bounds on permanents.

### SIMPLE LINEAR CORRELATION

SIMPLE LINEAR CORRELATION Smple lnear correlaton s a measure of the degree to whch two varables vary together, or a measure of the ntensty of the assocaton between two varables. Correlaton often s abused.

### Loop Parallelization

- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

### FINANCIAL MATHEMATICS. A Practical Guide for Actuaries. and other Business Professionals

FINANCIAL MATHEMATICS A Practcal Gude for Actuares and other Busness Professonals Second Edton CHRIS RUCKMAN, FSA, MAAA JOE FRANCIS, FSA, MAAA, CFA Study Notes Prepared by Kevn Shand, FSA, FCIA Assstant

### Yves Genin, Yurii Nesterov, Paul Van Dooren. CESAME, Universite Catholique de Louvain. B^atiment Euler, Avenue G. Lema^tre 4-6

Submtted to ECC 99 as a regular paper n Lnear Systems Postve transfer functons and convex optmzaton 1 Yves Genn, Yur Nesterov, Paul Van Dooren CESAME, Unverste Catholque de Louvan B^atment Euler, Avenue

### PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

### On Robust Network Planning

On Robust Network Plannng Al Tzghadam School of Electrcal and Computer Engneerng Unversty of Toronto, Toronto, Canada Emal: al.tzghadam@utoronto.ca Alberto Leon-Garca School of Electrcal and Computer Engneerng

### Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks

Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz

### A Probabilistic Theory of Coherence

A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

### The k-binomial Transforms and the Hankel Transform

1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 9 (2006, Artcle 06.1.1 The k-bnomal Transforms and the Hankel Transform Mchael Z. Spvey Department of Mathematcs and Computer Scence Unversty of Puget

### Forecasting the Direction and Strength of Stock Market Movement

Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

### On Lockett pairs and Lockett conjecture for π-soluble Fitting classes

On Lockett pars and Lockett conjecture for π-soluble Fttng classes Lujn Zhu Department of Mathematcs, Yangzhou Unversty, Yangzhou 225002, P.R. Chna E-mal: ljzhu@yzu.edu.cn Nanyng Yang School of Mathematcs

### Mean Value Coordinates for Closed Triangular Meshes

Mean Value Coordnates for Closed Trangular Meshes Tao Ju, Scott Schaefer, Joe Warren Rce Unversty (a) (b) (c) (d) Fgure : Orgnal horse model wth enclosng trangle control mesh shown n black (a). Several

### Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

### Cautiousness and Measuring An Investor s Tendency to Buy Options

Cautousness and Measurng An Investor s Tendency to Buy Optons James Huang October 18, 2005 Abstract As s well known, Arrow-Pratt measure of rsk averson explans a ratonal nvestor s behavor n stock markets

### An Overview of Financial Mathematics

An Overvew of Fnancal Mathematcs Wllam Benedct McCartney July 2012 Abstract Ths document s meant to be a quck ntroducton to nterest theory. It s wrtten specfcally for actuaral students preparng to take

### Cluster algebras were introduced by Fomin and Zelevinsky (1)

Greedy bases n rank quantum cluster algebras Kyungyong Lee a,ll b, Dylan Rupel c,, and Andre Zelensky c, a Department of Mathematcs, Wayne State Unersty, Detrot, MI 480; b Department of Mathematcs and

### THE METHOD OF LEAST SQUARES THE METHOD OF LEAST SQUARES

The goal: to measure (determne) an unknown quantty x (the value of a RV X) Realsaton: n results: y 1, y 2,..., y j,..., y n, (the measured values of Y 1, Y 2,..., Y j,..., Y n ) every result s encumbered

### In our example i = r/12 =.0825/12 At the end of the first month after your payment is received your amount in the account, the balance, is

Payout annutes: Start wth P dollars, e.g., P = 100, 000. Over a 30 year perod you receve equal payments of A dollars at the end of each month. The amount of money left n the account, the balance, earns

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### "Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

### Solution: Let i = 10% and d = 5%. By definition, the respective forces of interest on funds A and B are. i 1 + it. S A (t) = d (1 dt) 2 1. = d 1 dt.

Chapter 9 Revew problems 9.1 Interest rate measurement Example 9.1. Fund A accumulates at a smple nterest rate of 10%. Fund B accumulates at a smple dscount rate of 5%. Fnd the pont n tme at whch the forces

### Using Series to Analyze Financial Situations: Present Value

2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

### THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

### Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

### 1. Math 210 Finite Mathematics

1. ath 210 Fnte athematcs Chapter 5.2 and 5.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

### Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

### A. Te densty matrx and densty operator In general, te many-body wave functon (q 1 ; :::; q 3N ; t) s far too large to calculate for a macroscopc syste

G25.2651: Statstcal Mecancs Notes for Lecture 13 I. PRINCIPLES OF QUANTUM STATISTICAL MECHANICS Te problem of quantum statstcal mecancs s te quantum mecancal treatment of an N-partcle system. Suppose te

### Thursday, December 10, 2009 Noon - 1:50 pm Faraday 143

1. ath 210 Fnte athematcs Chapter 5.2 and 4.3 Annutes ortgages Amortzaton Professor Rchard Blecksmth Dept. of athematcal Scences Northern Illnos Unversty ath 210 Webste: http://math.nu.edu/courses/math210

### Simple Interest Loans (Section 5.1) :

Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

### The OC Curve of Attribute Acceptance Plans

The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

### SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

### Section 2 Introduction to Statistical Mechanics

Secton 2 Introducton to Statstcal Mechancs 2.1 Introducng entropy 2.1.1 Boltzmann s formula A very mportant thermodynamc concept s that of entropy S. Entropy s a functon of state, lke the nternal energy.

### A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

### Natural hp-bem for the electric field integral equation with singular solutions

Natural hp-bem for the electrc feld ntegral equaton wth sngular solutons Alexe Bespalov Norbert Heuer Abstract We apply the hp-verson of the boundary element method (BEM) for the numercal soluton of the

### Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

### Formula of Total Probability, Bayes Rule, and Applications

1 Formula of Total Probablty, Bayes Rule, and Applcatons Recall that for any event A, the par of events A and A has an ntersecton that s empty, whereas the unon A A represents the total populaton of nterest.

### Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

### CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

### What is Candidate Sampling

What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

### Forecasting Irregularly Spaced UHF Financial Data: Realized Volatility vs UHF-GARCH Models

Forecastng Irregularly Spaced UHF Fnancal Data: Realzed Volatlty vs UHF-GARCH Models Franços-Érc Raccot *, LRSP Département des scences admnstratves, UQO Raymond Théoret Département Stratége des affares,

### OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL. Thomas S. Ferguson and C. Zachary Gilstein UCLA and Bell Communications May 1985, revised 2004

OPTIMAL INVESTMENT POLICIES FOR THE HORSE RACE MODEL Thomas S. Ferguson and C. Zachary Glsten UCLA and Bell Communcatons May 985, revsed 2004 Abstract. Optmal nvestment polces for maxmzng the expected

### x f(x) 1 0.25 1 0.75 x 1 0 1 1 0.04 0.01 0.20 1 0.12 0.03 0.60

BIVARIATE DISTRIBUTIONS Let be a varable that assumes the values { 1,,..., n }. Then, a functon that epresses the relatve frequenc of these values s called a unvarate frequenc functon. It must be true

### Logistic Regression. Steve Kroon

Logstc Regresson Steve Kroon Course notes sectons: 24.3-24.4 Dsclamer: these notes do not explctly ndcate whether values are vectors or scalars, but expects the reader to dscern ths from the context. Scenaro

### A Simple Economic Model about the Teamwork Pedagogy

Appled Mathematcal Scences, Vol. 6, 01, no. 1, 13-0 A Smple Economc Model about the Teamwork Pedagog Gregor L. Lght Department of Management, Provdence College Provdence, Rhode Island 0918, USA glght@provdence.edu

### Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

### Performance Management and Evaluation Research to University Students

631 A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 46, 2015 Guest Edtors: Peyu Ren, Yancang L, Hupng Song Copyrght 2015, AIDIC Servz S.r.l., ISBN 978-88-95608-37-2; ISSN 2283-9216 The Italan Assocaton

### New bounds in Balog-Szemerédi-Gowers theorem

New bounds n Balog-Szemeréd-Gowers theorem By Tomasz Schoen Abstract We prove, n partcular, that every fnte subset A of an abelan group wth the addtve energy κ A 3 contans a set A such that A κ A and A

### 8.4. Annuities: Future Value. INVESTIGATE the Math. 504 8.4 Annuities: Future Value

8. Annutes: Future Value YOU WILL NEED graphng calculator spreadsheet software GOAL Determne the future value of an annuty earnng compound nterest. INVESTIGATE the Math Chrstne decdes to nvest \$000 at

### Series Solutions of ODEs 2 the Frobenius method. The basic idea of the Frobenius method is to look for solutions of the form 3

Royal Holloway Unversty of London Department of Physs Seres Solutons of ODEs the Frobenus method Introduton to the Methodology The smple seres expanson method works for dfferental equatons whose solutons

### Matrix Multiplication I

Matrx Multplcaton I Yuval Flmus February 2, 2012 These notes are based on a lecture gven at the Toronto Student Semnar on February 2, 2012. The materal s taen mostly from the boo Algebrac Complexty Theory

### Section 5.4 Annuities, Present Value, and Amortization

Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

### On some special nonlevel annuities and yield rates for annuities

On some specal nonlevel annutes and yeld rates for annutes 1 Annutes wth payments n geometrc progresson 2 Annutes wth payments n Arthmetc Progresson 1 Annutes wth payments n geometrc progresson 2 Annutes

### Global stability of Cohen-Grossberg neural network with both time-varying and continuous distributed delays

Global stablty of Cohen-Grossberg neural network wth both tme-varyng and contnuous dstrbuted delays José J. Olvera Departamento de Matemátca e Aplcações and CMAT, Escola de Cêncas, Unversdade do Mnho,

### Description of the Force Method Procedure. Indeterminate Analysis Force Method 1. Force Method con t. Force Method con t

Indeternate Analyss Force Method The force (flexblty) ethod expresses the relatonshps between dsplaceents and forces that exst n a structure. Prary objectve of the force ethod s to deterne the chosen set

### Non-degenerate Hilbert Cubes in Random Sets

Journal de Théore des Nombres de Bordeaux 00 (XXXX), 000 000 Non-degenerate Hlbert Cubes n Random Sets par Csaba Sándor Résumé. Une légère modfcaton de la démonstraton du lemme des cubes de Szemeréd donne

### Least Squares Fitting of Data

Least Squares Fttng of Data Davd Eberly Geoetrc Tools, LLC http://www.geoetrctools.co/ Copyrght c 1998-2016. All Rghts Reserved. Created: July 15, 1999 Last Modfed: January 5, 2015 Contents 1 Lnear Fttng

### General Auction Mechanism for Search Advertising

General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

### Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

### Introduction to Differential Algebraic Equations

Dr. Abebe Geletu Ilmenau Unversty of Technology Department of Smulaton and Optmal Processes (SOP) Wnter Semester 2011/12 4.1 Defnton and Propertes of DAEs A system of equatons that s of the form F (t,

### Rotation and Conservation of Angular Momentum

Chapter 4. Rotaton and Conservaton of Angular Momentum Notes: Most of the materal n ths chapter s taken from Young and Freedman, Chaps. 9 and 0. 4. Angular Velocty and Acceleraton We have already brefly

### LOOP ANALYSIS. The second systematic technique to determine all currents and voltages in a circuit

LOOP ANALYSS The second systematic technique to determine all currents and voltages in a circuit T S DUAL TO NODE ANALYSS - T FRST DETERMNES ALL CURRENTS N A CRCUT AND THEN T USES OHM S LAW TO COMPUTE

### Some geometric probability problems involving the Eulerian numbers

Some geometrc probablty problems nvolvng the Euleran numbers Frank Schmdt Rodca Smon Department of Mathematcs The George Washngton Unversty Washngton, DC 20052 smon@math.gwu.edu Dedcated to Herb Wlf on

### Enabling P2P One-view Multi-party Video Conferencing

Enablng P2P One-vew Mult-party Vdeo Conferencng Yongxang Zhao, Yong Lu, Changja Chen, and JanYn Zhang Abstract Mult-Party Vdeo Conferencng (MPVC) facltates realtme group nteracton between users. Whle P2P

### POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and

POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n

### Numerical Methods 數 值 方 法 概 說. Daniel Lee. Nov. 1, 2006

Numercal Methods 數 值 方 法 概 說 Danel Lee Nov. 1, 2006 Outlnes Lnear system : drect, teratve Nonlnear system : Newton-lke Interpolatons : polys, splnes, trg polys Approxmatons (I) : orthogonal polys Approxmatons

### Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

### Multicomponent Distillation

Multcomponent Dstllaton need more than one dstllaton tower, for n components, n-1 fractonators are requred Specfcaton Lmtatons The followng are establshed at the begnnng 1. Temperature, pressure, composton,

### APPLICATIONS OF VARIATIONAL PRINCIPLES TO DYNAMICS AND CONSERVATION LAWS IN PHYSICS

APPLICATIONS OF VAIATIONAL PINCIPLES TO DYNAMICS AND CONSEVATION LAWS IN PHYSICS DANIEL J OLDE Abstract. Much of physcs can be condensed and smplfed usng the prncple of least acton from the calculus of

### Extending Probabilistic Dynamic Epistemic Logic

Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

### A random variable is a variable whose value depends on the outcome of a random event/experiment.

Random varables and Probablty dstrbutons A random varable s a varable whose value depends on the outcome of a random event/experment. For example, the score on the roll of a de, the heght of a randomly

### Part 1: quick summary 5. Part 2: understanding the basics of ANOVA 8

Statstcs Rudolf N. Cardnal Graduate-level statstcs for psychology and neuroscence NOV n practce, and complex NOV desgns Verson of May 4 Part : quck summary 5. Overvew of ths document 5. Background knowledge