Mathfest 2019
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Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform
World Academy of Science, Engineering and Technology International Journal of Computer and Information Engineering Vol:13, No:8, 2019 Fast and Efficient Algorithms for Evaluating Uniform and Nonuniform Lagrange and Newton Curves Taweechai Nuntawisuttiwong, Natasha Dejdumrong Abstract—Newton-Lagrange Interpolations are widely used in quadratic computation time, which is slower than the others. numerical analysis. However, it requires a quadratic computational However, there are some curves in CAGD, Wang-Ball, DP time for their constructions. In computer aided geometric design and Dejdumrong curves with linear complexity. In order to (CAGD), there are some polynomial curves: Wang-Ball, DP and Dejdumrong curves, which have linear time complexity algorithms. reduce their computational time, Bezier´ curve is converted Thus, the computational time for Newton-Lagrange Interpolations into any of Wang-Ball, DP and Dejdumrong curves. Hence, can be reduced by applying the algorithms of Wang-Ball, DP and the computational time for Newton-Lagrange interpolations Dejdumrong curves. In order to use Wang-Ball, DP and Dejdumrong can be reduced by converting them into Wang-Ball, DP and algorithms, first, it is necessary to convert Newton-Lagrange Dejdumrong algorithms. Thus, it is necessary to investigate polynomials into Wang-Ball, DP or Dejdumrong polynomials. In this work, the algorithms for converting from both uniform and the conversion from Newton-Lagrange interpolation into non-uniform Newton-Lagrange polynomials into Wang-Ball, DP and the curves with linear complexity, Wang-Ball, DP and Dejdumrong polynomials are investigated. Thus, the computational Dejdumrong curves. An application of this work is to modify time for representing Newton-Lagrange polynomials can be reduced sketched image in CAD application with the computational into linear complexity. -
Polygon Review and Puzzlers in the Above, Those Are Names to the Polygons: Fill in the Blank Parts. Names: Number of Sides
Polygon review and puzzlers ÆReview to the classification of polygons: Is it a Polygon? Polygons are 2-dimensional shapes. They are made of straight lines, and the shape is "closed" (all the lines connect up). Polygon Not a Polygon Not a Polygon (straight sides) (has a curve) (open, not closed) Regular polygons have equal length sides and equal interior angles. Polygons are named according to their number of sides. Name of Degree of Degree of triangle total angles regular angles Triangle 180 60 In the above, those are names to the polygons: Quadrilateral 360 90 fill in the blank parts. Pentagon Hexagon Heptagon 900 129 Names: number of sides: Octagon Nonagon hendecagon, 11 dodecagon, _____________ Decagon 1440 144 tetradecagon, 13 hexadecagon, 15 Do you see a pattern in the calculation of the heptadecagon, _____________ total degree of angles of the polygon? octadecagon, _____________ --- (n -2) x 180° enneadecagon, _____________ icosagon 20 pentadecagon, _____________ These summation of angles rules, also apply to the irregular polygons, try it out yourself !!! A point where two or more straight lines meet. Corner. Example: a corner of a polygon (2D) or of a polyhedron (3D) as shown. The plural of vertex is "vertices” Test them out yourself, by drawing diagonals on the polygons. Here are some fun polygon riddles; could you come up with the answer? Geometry polygon riddles I: My first is in shape and also in space; My second is in line and also in place; My third is in point and also in line; My fourth in operation but not in sign; My fifth is in angle but not in degree; My sixth is in glide but not symmetry; Geometry polygon riddles II: I am a polygon all my angles have the same measure all my five sides have the same measure, what general shape am I? Geometry polygon riddles III: I am a polygon. -
Mials P
Euclid's Algorithm 171 Euclid's Algorithm Horner’s method is a special case of Euclid's Algorithm which constructs, for given polyno- mials p and h =6 0, (unique) polynomials q and r with deg r<deg h so that p = hq + r: For variety, here is a nonstandard discussion of this algorithm, in terms of elimination. Assume that d h(t)=a0 + a1t + ···+ adt ;ad =06 ; and n p(t)=b0 + b1t + ···+ bnt : Then we seek a polynomial n−d q(t)=c0 + c1t + ···+ cn−dt for which r := p − hq has degree <d. This amounts to the square upper triangular linear system adc0 + ad−1c1 + ···+ a0cd = bd adc1 + ad−1c2 + ···+ a0cd+1 = bd+1 . adcn−d−1 + ad−1cn−d = bn−1 adcn−d = bn for the unknown coefficients c0;:::;cn−d which can be uniquely solved by back substitution since its diagonal entries all equal ad =0.6 19aug02 c 2002 Carl de Boor 172 18. Index Rough index for these notes 1-1:-5,2,8,40 cartesian product: 2 1-norm: 79 Cauchy(-Bunyakovski-Schwarz) 2-norm: 79 Inequality: 69 A-invariance: 125 Cauchy-Binet formula: -9, 166 A-invariant: 113 Cayley-Hamilton Theorem: 133 absolute value: 167 CBS Inequality: 69 absolutely homogeneous: 70, 79 Chaikin algorithm: 139 additive: 20 chain rule: 153 adjugate: 164 change of basis: -6 affine: 151 characteristic function: 7 affine combination: 148, 150 characteristic polynomial: -8, 130, 132, 134 affine hull: 150 circulant: 140 affine map: 149 codimension: 50, 53 affine polynomial: 152 coefficient vector: 21 affine space: 149 cofactor: 163 affinely independent: 151 column map: -6, 23 agrees with y at Λt:59 column space: 29 algebraic dual: 95 column -
Formulas Involving Polygons - Lesson 7-3
you are here > Class Notes – Chapter 7 – Lesson 7-3 Formulas Involving Polygons - Lesson 7-3 Here’s today’s warmup…don’t forget to “phone home!” B Given: BD bisects ∠PBQ PD ⊥ PB QD ⊥ QB M Prove: BD is ⊥ bis. of PQ P Q D Statements Reasons Honors Geometry Notes Today, we started by learning how polygons are classified by their number of sides...you should already know a lot of these - just make sure to memorize the ones you don't know!! Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon 11 Undecagon 12 Dodecagon 13 Tridecagon 14 Tetradecagon 15 Pentadecagon 16 Hexadecagon 17 Heptadecagon 18 Octadecagon 19 Enneadecagon 20 Icosagon n n-gon Baroody Page 2 of 6 Honors Geometry Notes Next, let’s look at the diagonals of polygons with different numbers of sides. By drawing as many diagonals as we could from one diagonal, you should be able to see a pattern...we can make n-2 triangles in a n-sided polygon. Given this information and the fact that the sum of the interior angles of a polygon is 180°, we can come up with a theorem that helps us to figure out the sum of the measures of the interior angles of any n-sided polygon! Baroody Page 3 of 6 Honors Geometry Notes Next, let’s look at exterior angles in a polygon. First, consider the exterior angles of a pentagon as shown below: Note that the sum of the exterior angles is 360°. -
CS321-001 Introduction to Numerical Methods
CS321-001 Introduction to Numerical Methods Lecture 3 Interpolation and Numerical Differentiation Professor Jun Zhang Department of Computer Science University of Kentucky Lexington, KY 40506-0633 Polynomial Interpolation Given a set of discrete values, how can we estimate other values between these data The method that we will use is called polynomial interpolation. We assume the data we had are from the evaluation of a smooth function. We may be able to use a polynomial p(x) to approximate this function, at least locally. A condition: the polynomial p(x) takes the given values at the given points (nodes), i.e., p(xi) = yi with 0 ≤ i ≤ n. The polynomial is said to interpolate the table, since we do not know the function. 2 Polynomial Interpolation Note that all the points are passed through by the curve 3 Polynomial Interpolation We do not know the original function, the interpolation may not be accurate 4 Order of Interpolating Polynomial A polynomial of degree 0, a constant function, interpolates one set of data If we have two sets of data, we can have an interpolating polynomial of degree 1, a linear function x x1 x x0 p(x) y0 y1 x0 x1 x1 x0 y1 y0 y0 (x x0 ) x1 x0 Review carefully if the interpolation condition is satisfied Interpolating polynomials can be written in several forms, the most well known ones are the Lagrange form and Newton form. Each has some advantages 5 Lagrange Form For a set of fixed nodes x0, x1, …, xn, the cardinal functions, l0, l1,…, ln, are defined as 0 if i j li (x j ) ij -
The Horned Frog in Play Descriptions of the Important at Election—Hickey Number 39
■-Jf <<-"■' - -+#" CAMPUS NEWS COMPLETELY COVERED BY DEPARTMENT OF ifOURNALISM STUDENTS Attend the Dance Tomorrow Night Of filial Student Body Publication of Texas Christian University VOLUME 34 FORT WORTH, TEXAS, SEPTEMBER 27, 1935. NUMIiER 2 162 Honor Roll Saam To Announce Announcer. Conference Games Boooe to Present Feature Story Band Chooses Students Listed Student Will Give Description Football Team Of The Cowans Helen Moody of Rice—L. S. U. Tilt Traces History For Spring Term Saturday Night, At Stag Dinner For Third Time The story of a former T. C. U. Byrum JSaam, juhior in the Uni- journalism instructor and his wife, a Seniors Lead Classes versity and announcer for radio sta- Mayor Jarvis to Intro- graduate of the University who won Bacus Made President With 57—Freshmen tion KTAT, will broadcast play-by- duce Guests—Invoca- beauty pages in the Horned Frog in play descriptions of the important at Election—Hickey Number 39. her junior and senior years in the Southwest Conference football games tion by Gresham. University, is one of the features of Vice-president. over the Southwest Broadcasting' last week's issue of. Editor and Pub- System network, according, to an an- 19 Make "A" Grades lisher, the outstanding trade journal nouncement made last week. " 2 Prices for Dinner Uniforms Ordered in the field of journalism. The first of these broadcasts, the 11 Fort Worth Students The couple are Mr. and Mrs. Jos- Board of Trustees Announces Rice-L. S. U. game, will be broadcast School Spirit Demonstration to Included in Perfect eph B. Cowan. Cowan was an in- New Suits —. -
Delphian Anddentralqlewsdypists
T H E 1 9 4 4 DELPHIAN Published by THE SENIOR CLASS NEW PHILADELPHIA HIGH SCHOOL NEW PHILADELPHIA, OHIO D ale D u n n * * * Business Manager Jane Patrick * * * E ditor In this period of conflict and dis order, the Delphian Staff feels that it is fitting and proper to pay tribute to the men and women who are so gal lantly defending our rights against our enemies. ☆☆ It is, therefore, w ith great pride that we dedicate this book to the students and faculty of our school who have made sacrifices to uphold the cause of liberty wherever it is being threatened. "Maj estic monarck or "I pledge allegiance to tke Flag tk e sky of tke United States of America Tke sign of kope and and to tke Repuklic for wkick it triumpk kigk. stands, one Nation indivisikle, pause to salute you! witk likerty and justice for all. ☆ ☆ ☆ ☆ This year in publishing the Delphian, we have tried to in clude the things which you would want to remember. If in a few years you look back through your Delphian and are able to recall happy mem ories of your days in Senior High , we will, in a small way, have attained our purpose. ☆☆☆ NTS ☆ ☆ ☆ ☆ o DSDICA1I0N O ADMINISTRATION DIVISION o CLASSOS DIVISION O ACTIVITIES DIVISION oATOLTTICS DIVISION J ( M ? Ol Q s t j J oum a& ajcflwt/ a t f o j l v q DELPHIAN C~ 8 -3 DELPHIAN J. B. RUDY, Principal B. S. in Ed., Wooster College M. A., Ohio State University E il e e n G r e e n Secretary to the Principal g- 9 ~3 Left to right—-Mr. -
Chess As a Key to Solving Nabokov's Korol', Dama, Valet
CHESS AS A KEY TO SOLVING NABOKOV’S KOROL’, DAMA, VALET James Murray Slater A thesis submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the requirements for the degree of Master of Arts in the Department of Slavic Languages and Literatures. Chapel Hill 2009 Approved by: Radislav Lapushin Madeline Levine Christopher Putney ABSTRACT James Murray Slater: Chess as a Key to Solving Nabokov’s Korol’, dama, valet (Under the direction of Professor Radislav Lapushin) This thesis examines the role of chess in Nabokov’s novel, Korol’, dama, valet , and how the struggle between the narrator and the protagonist, Martha-Marta , can be justified by using chess as a key. The first portion of my analysis is based on the role of the diegetic narrator in an attempt to subvert Martha’s movements toward autonomy. In explaining the narrator’s subversion of Martha, I explore the nature of the protagonists, the movement of the narrative, and the structure and parameters of the novel in relation to it being a figurative game of chess. Furthermore, Martha’s attempts at dominance are examined in connection to the concept of past-reconstruction. I conclude my analysis by ushering in unifying intersections of plot and characterization in Nabokov’s later work. ii TABLE OF CONTENTS Chapter I. INTRODUCTION……………….……………………………….………1 II. NARRATIVE DIEGESIS, PROTAGONISTS AND CHESS…………..11 III. BLADVAK VINOMORI AS THE AUTHOR INCARNATE….………28 IV. MARTHA’S STRUGGLE FOR AUTHORIAL CONTROL…...………32 V. NABOKOV & REINVENTION……………………………….………..40 APPENDICES…………………………………………………………………………...46 WORKS CITED OR CONSULTED ………………………………………………....…52 iii CHAPTER 1 INTRODUCTION Бархатный стук в голове : это ходят фигуры резные . -
Lagrange & Newton Interpolation
Lagrange & Newton interpolation In this section, we shall study the polynomial interpolation in the form of Lagrange and Newton. Given a se- quence of (n +1) data points and a function f, the aim is to determine an n-th degree polynomial which interpol- ates f at these points. We shall resort to the notion of divided differences. Interpolation Given (n+1) points {(x0, y0), (x1, y1), …, (xn, yn)}, the points defined by (xi)0≤i≤n are called points of interpolation. The points defined by (yi)0≤i≤n are the values of interpolation. To interpolate a function f, the values of interpolation are defined as follows: yi = f(xi), i = 0, …, n. Lagrange interpolation polynomial The purpose here is to determine the unique polynomial of degree n, Pn which verifies Pn(xi) = f(xi), i = 0, …, n. The polynomial which meets this equality is Lagrange interpolation polynomial n Pnx=∑ l j x f x j j=0 where the lj ’s are polynomials of degree n forming a basis of Pn n x−xi x−x0 x−x j−1 x−x j 1 x−xn l j x= ∏ = ⋯ ⋯ i=0,i≠ j x j −xi x j−x0 x j−x j−1 x j−x j1 x j−xn Properties of Lagrange interpolation polynomial and Lagrange basis They are the lj polynomials which verify the following property: l x = = 1 i= j , ∀ i=0,...,n. j i ji {0 i≠ j They form a basis of the vector space Pn of polynomials of degree at most equal to n n ∑ j l j x=0 j=0 By setting: x = xi, we obtain: n n ∑ j l j xi =∑ j ji=0 ⇒ i =0 j=0 j=0 The set (lj)0≤j≤n is linearly independent and consists of n + 1 vectors. -
( ) Methods for Construction of Odd Number Pointed
Daniel DOBRE METHODS FOR CONSTRUCTION OF ODD NUMBER POINTED POLYGONS Abstract: The purpose of this paper is to present methods for constructing of polygons with an odd number of sides, although some of them may not be built only with compass and straightedge. Odd pointed polygons are difficult to construct accurately, though there are relatively simple ways of making a good approximation which are accurate within the normal working tolerances of design practitioners. The paper illustrates rather complicated constructions to produce unconstructible polygons with an odd number of sides, constructions that are particularly helpful for engineers, architects and designers. All methods presented in this paper provide practice in geometric representations. Key words: regular polygon, pentagon, heptagon, nonagon, hendecagon, pentadecagon, heptadecagon. 1. INTRODUCTION n – number of sides; Ln – length of a side; For a specialist inured to engineering graphics, plane an – apothem (radius of inscribed circle); and solid geometries exert a special fascination. Relying R – radius of circumscribed circle. on theorems and relations between linear and angular sizes, geometry develops students' spatial imagination so 2. THREE - POINT GEOMETRY necessary for understanding other graphic discipline, descriptive geometry, underlying representations in The construction for three point geometry is shown engineering graphics. in fig. 1. Given a circle with center O, draw the diameter Construction of regular polygons with odd number of AD. From the point D as center and, with a radius in sides, just by straightedge and compass, has preoccupied compass equal to the radius of circle, describe an arc that many mathematicians who have found ingenious intersects the circle at points B and C. -
Mathematical Topics
A Mathematical Topics This chapter discusses most of the mathematical background needed for a thor ough understanding of the material presented in the book. It has been mentioned in the Preface, however, that math concepts which are only used once (such as the mediation operator and points vs. vectors) are discussed right where they are introduced. Do not worry too much about your difficulties in mathematics, I can assure you that mine a.re still greater. - Albert Einstein. A.I Fourier Transforms Our curves are functions of an arbitrary parameter t. For functions used in science and engineering, time is often the parameter (or the independent variable). We, therefore, say that a function g( t) is represented in the time domain. Since a typical function oscillates, we can think of it as being similar to a wave and we may try to represent it as a wave (or as a combination of waves). When this is done, we have the function G(f), where f stands for the frequency of the wave, and we say that the function is represented in the frequency domain. This turns out to be a useful concept, since many operations on functions are easy to carry out in the frequency domain. Transforming a function between the time and frequency domains is easy when the function is periodic, but it can also be done for certain non periodic functions. The present discussion is restricted to periodic functions. 1IIIt Definition: A function g( t) is periodic if (and only if) there exists a constant P such that g(t+P) = g(t) for all values of t (Figure A.la). -
Volume 2 Shape and Space
Volume 2 Shape and Space Colin Foster Introduction Teachers are busy people, so I’ll be brief. Let me tell you what this book isn’t. • It isn’t a book you have to make time to read; it’s a book that will save you time. Take it into the classroom and use ideas from it straight away. Anything requiring preparation or equipment (e.g., photocopies, scissors, an overhead projector, etc.) begins with the word “NEED” in bold followed by the details. • It isn’t a scheme of work, and it isn’t even arranged by age or pupil “level”. Many of the ideas can be used equally well with pupils at different ages and stages. Instead the items are simply arranged by topic. (There is, however, an index at the back linking the “key objectives” from the Key Stage 3 Framework to the sections in these three volumes.) The three volumes cover Number and Algebra (1), Shape and Space (2) and Probability, Statistics, Numeracy and ICT (3). • It isn’t a book of exercises or worksheets. Although you’re welcome to photocopy anything you wish, photocopying is expensive and very little here needs to be photocopied for pupils. Most of the material is intended to be presented by the teacher orally or on the board. Answers and comments are given on the right side of most of the pages or sometimes on separate pages as explained. This is a book to make notes in. Cross out anything you don’t like or would never use. Add in your own ideas or references to other resources.