**Polynomials** Operations with Rational Expressions Chapter 5 Section 1 The Basics of Factors Just the Fact-or Ma’am. or The O’Righty Factor (I’m getting the puns out of my system). Primes are your friend… A prime number is a rock bottom, hard end, stop playing with the **math**/ Example 2 page 330 Identifying special **products**… a) x 2 -14x+49/**in** your head… and bango… (3x-2y)(2x-y) ** Ex 69-72** Example 5 page 348 Factoring Completely (Doing **what** it takes) Factor each **polynomial**/sum of two squares **does** not factor. 3./

x n / 2 coefficients n values of x **Does** not meet precondition! FFT nlogn Time Fast Fourier Transformation/ of unity (**in** a field) if n = 1 (There should be n solutions of this **polynomial**) Fermat’s / 6 9 … 3n-3... But that’s still O(N 2 ) !! The Ugly **Math** for the FFT Just watch! For example, if N=8 and I use the N roots of unity/x 1 ± x 5, x 3 ± x 7 Oh my! Half the columns are gone. **What**’s next? (x 0 + x 4 ) + (x 2 + x 6 ) + / bit blocks Pad with zero 2N bits to hold **product** n blocks where n is a power of 2/

**Polynomials** and **Polynomials** Operations Grade 9 **Math** Bedford Junior High **Polynomials**: Monomials, Binomials, and Trinomials A monomial is a constant or numerical coefficient, a variable or literal coefficient, or the **product** of a constant and one or more variables. A **polynomial**/answer is 5x 17 Distributive Property Means that **what** is inside the bracket is affected by / “3” affects each item inside the bracket. It **does** not matter how many terms are inside the bracket. / Construct a rectangle **in** the bottom right/

D) k(x) Functions GHSGT **Math** Review For **what** x value **does** f(x) = 3? A) -2 B) 0 C) 2 D) 6 Functions GHSGT **Math** Review Transformations of Functions ReflectionStretch or Shrink /**in** this diagram. The area of this particular Georgia flag is 60 square feet. **What** is the length of x? A) 2 ft. B) 4 ft. C) 2 √5 ft. D) 2 √15 ft **Polynomial** Applications GHSGT **Math**/.” Truth Value GHSGT **Math** Review Logic Statements Which of these true statements also has a true inverse? A.If the **product** of integers a and/

slope standard coordinate plane transversal trapezoid vertex x-intercept y-intercept **Math** Vocabulary 7 area of a circle—A = π r /**does** Joelle earn for a week **in** which she works 42 hours? A. $126.00 B. $315.00 C. $322.50 D. $378.00 E. $472.50 6. Which of the following mathematical expressions is equivalent to the verbal expression “A number, x, squared is 39 more than the **product**/– When you can’t factor a **polynomial** cleanly. You can always use the quadratic formula – **In** x 2 +7x+15=0, **what** is a, b, and c? /

function as a group. **What** qualities **does** your identity have that will help you be an effective **MATH** group? An effective college/**in** the CW Section of your binder: Students will REVIEW and CRITIQUE work and solutions from other students. **What** **does** /CW: Pg. 279/ #81-88 Test #1 WEDNESDAY MAT-055 Multiplying **Polynomials** Multiplying BI-nomials >> TRI-nomials – F-O-I-L: FIRST, /100 Test WEDNESDAY MAT-055 CW: Factoring WSheet ‘Books…’ Factoring Special **Products** HW: Pg. 303/#1-42 Due FRIDAY QUIZ WEDNESDAY – Simplifying/

Test 2 – **Math** 96 Flash Cards Test 2 Functions: Domain & Range Systems of Equations Word Problems – Mixture, investment & DRT **Polynomials**: add, subtract, multiply, & divide Factor **Does** the graph represents a function. State the domain and the range. Vertical line / common factors, show **what**’s left inside ONE set of parenthesis Identify the number of terms. If there was a common factor, then look at the number of terms left inside the parenthesis. Do NOT remove parentheses already given **in** the expression. If /

failure…. From LEGO Bricks to **Math** 560 pieces for $120, or 21 cents per piece (cpp). From LEGO Bricks to **Math** 292 pieces for $65, /**what** the `right sum of a **product** of three binomial coefficients is? One difficulty is you have to figure out **whats** the right triple! – To solve the roulette recurrence from the video involves finding the roots of a **polynomial** of degree 5; sadly **in**/ xn.We saw Horners algorithm **does** significantly better than brute force, standard **polynomial** evaluation. **What** if instead we used fast /

**Math** for CS Further Examples: The key is to know **what** formula to use. **In** a National Lottery a twice-weekly drawing takes place **in** which 6 numbers are randomly drawn from the numbers 1..49. Problem: Find the probability of winning. Analysis: Without replacement and order **does**/ the running times for n = 1, 10, 100 and 1000. **polynomial** timeexponential timelinear time **Polynomial** time algorithms with the same degree are **in** the same ballpark. Discrete **Math** for CS Time Complexity Functions: Suppose f(n) and g(n) /

GRADE **MATH** Unit 8 Quadratic Expressions and Equations Template by Bill Arcuri, WCSD JEOPARDY! 100 200 300 400 500 Add and subtract **polynomials** Multiply **polynomials** Factoring /, WCSD Topic 1 300 Points Write the expression **in** standard form. Identify the Leading Coefficient and Degree of the **polynomial**. -3 + 3x - x + 5x 33 2 Template by/integer. **What** is the **product** of the next Two consecutive integers? Template by Bill Arcuri, WCSD Topic 6 200 Points Answer Let x be an integer. **What** is the **product** of /

are retained INRIA Sophia-Antipolis Colloquium Schwarz **polynomials** l **Polynomials** of Schwarz projections that are combinations of/ Falgout (LLNL) to support research, prototyping, and **production** parallel solutions of operator equations **in** message-passing environments; now joined by seven additional staff/ do you know **what** your solver is up to? Has your solver not been updated **in** the past five /://www.**math**.odu.edu/~keyes l SciDAC initiative http://www.science.**doe**.gov/scidac l TOPS project http://www.**math**.odu.edu/

**polynomials** Why **does** anyone care about Zernike **polynomials**? A little history about their development. Definitions and **math** - **what** are they? How do they make certain questions easy to answer? A couple of practical applications **What** will Zernikes do for me? Widely used **in** industry/ –Similar to modes of a circular drum head Real surface is constructed of linear combination of basis shapes or modes **Polynomials** are a **product** of a radial and azimuthal part –Radial orders are positive, integers (n), 0, 1,2, 3, 4,/

**Math**/ends **in** 0. 370 is divisible by 10 because it ends **in** 0. 7003 is not divisible by 10 because it **does** not end **in** 0/For example: **What** is 30 percent of 250? X = 30 / 100 x 250 To find a percentage increase or decrease, divide the change **in** value by/are roots. ab = a x b The square root of a **product** is the **product** of the roots. 3 www.topmath.info/m5 Best links on / – 2 = 9x 4 + 5x 2 – 5y 2 – 2x To multiply a **polynomial** by a number, multiply each term **in** the **polynomial** by the number: 5 x (x 2 - 4xy + 3y 2 –2) = /

Outreach with Elementary School My Qualifications: – Know **math**, know Legos. – More forgiving than Darth Vader, less than the /**what** the `right sum of a **product** of three binomial coefficients is? One difficulty is you have to figure out **whats** the right triple! – To solve the roulette recurrence from the video involves finding the roots of a **polynomial** of degree 5; sadly **in**/.We saw Horners algorithm **does** significantly better than brute force, standard **polynomial** evaluation. **What** if instead we used/

is full. After messing around with lots of cases it occurred to me to ask for the minimal **polynomial** of M Example 3. Related Matrices (1993-2006) > linalg[minpoly](M(12),t); > linalg[/**production** is £800 and his profit is £200. Underline the number 200. 5. Teaching **Maths** **In** 2008 A logger cuts down a beautiful forest because he is a totally selfish and inconsiderate bastard and cares nothing for the habitat of animals or the preservation of our woodlands. He **does** this so he can make a profit of £200. **What**/

**products** of for-profit corporations Goal is to teach kids that there are some people **in** society who are good at **math** /**polynomial** divided by a trinomial, they won’t be able to do it. Secondly, synthetic division is based on long division, seeing **what**/**what** the Common Core responded by saying. They said "First of all you cant field test standards, you can field test implementation of standards but you cant field test the standards themselves. So when people say that they have not been field tested, it **does**/

Fall 2003 I. INTRODUCTION: **Math** Clinic Fall 2003 A. Why fuzzy sets? - Modeling with uncertainty requires more than probability theory - There are problems where boundaries are gradual EXAMPLES: **What** is the boundary of the USA? Is the boundary a mathematical curve? **What** is the area of USA? Is the area a real number? Where **does** a tumor begin **in** the transition? **What** is the habitat of/

**Math** Vocabulary (Rubenstein, 2007) Some words are… found only **in** **math** (e.g., denominator, hypotenuse, **polynomial**, histogram) found only **in** **math** (e.g., denominator, hypotenuse, **polynomial**,/ **productive** and enjoyable? 10. How can I help students use the word/concept **in** meaningful ways **in** multiple context? 20 **What** you can do (Allen, 1999) Repeat words **in**/) Creating a glossary and memorization **does** not help students learn. (Allen, 1999) Creating a glossary and memorization **does** not help students learn. (Allen/

a long division problem **in** basic **math** Example Dividing **Polynomial** Function by “x – k” Using Synthetic Division First write each **polynomial** **in** descending powers If /**In** a previous example: we found f(2) by synthetic division: Notice bottom row. **What** **does** this tell us about zeros? Example of Boundedness Theorem Using the same function: find f(-3) by synthetic division: Notice bottom row. **What** **does**/and inversely with others **In** such cases, we write y as being equal to a constant, k, times the **product** of the other /

**what** we now call “**polynomial**-time Turing reductions”: A → B if problem A can be solved **in** **polynomial** time when given a subroutine for solving B that provides answers **in**/tree is graceful (verified for all trees with |V| ≤ 35). Biggest Remaining Open Problem **Does** P = NP? Solution is worth $1,000,000 Opinion Polls Bill Gasarch, “Guest /**in** reasonable running times. Executables and source code can be downloaded from http://www.**math**.uwaterloo.ca/tsp/concorde/index.html Concorde & the TSP N = 1000 Running times (**in**/

particular state of the catalyst) that **does** not change significantly during this non-steady/residues **in** multi-dimensional complex analysis”, Nauka, Novosibirsk, 1979 Tsikh, A.K., Multidimensional residues and their applications, Trans. **Math**./ is a linear **polynomial** regarding its parameters **in** difference from the non/**products**) 3. Coincidences Surprising properties of the simple kinetic models; **in** particular, A->B->C. Coincidences (cont’d) Solutions (known before) Coincidences (cont’d) New problem is posed: **what**/

**Math** 112 Prepared by Richard Cangelosi and Mariamma Varghese 2 If the sum of two numbers is 14, and their difference is 2, **what** is the **product** of the two numbers? A.24 B.28 C.40 D.45 E.48 F.None of these. 3 Compare the quantity **in** Column A with the quantity **in**/same route home, but average speed is 60 miles per hour, how long **does** it take her to get home? (A)2 hours (B)2 hours/below? (A). (B). (C). (D). (E). 33 If P(x) is a **polynomial** of degree, then the degree of the quotient of is (A) Less than 2 (B) Exactly/

University of California Berkeley cs h196 -- Polynomials2 **Polynomials** force basic decisions **in** the design of a CAS Computationally, nearly every “applied **math**” object is either a **polynomial** or a ratio of **polynomials** **Polynomial** representation(s) can make or break a system/, by interpolation, a **polynomial** of degree 12 by 13 values, H(0), …, H(12). Thus we compute the **product** H=F*G cs h196 -- Polynomials19 **What** **does** evaluation cost? Horner’s rule evaluates a degree d **polynomial** **in** d adds and multiplies./

variables ‘x’ is the variable **What** number **does** ‘x’ represent? **What** numbers do ‘x’ and ‘y’ represent? “Coefficient” 3x + 4y - 2 The number **in** front of a variable **in** an expression or an equation. / the sum. “Factors” “**Product**” 2 x 3 = 6 The numbers that are multiplied together to get the answer. 2 x 3 = 6 “**Product**” The answer when factors are/ **in** words before you write it **in** mathematical symbols. **Polynomials** POLY nomials many terms each with the same variable raised to an positive INTEGER power. **Polynomials** /

**polynomials** Statistics and Eyeglasses There are lots of glasses and contact lenses **in** your classroom. Great source of data for introductory statistics. **In** fact, more than 150 million Americans use corrective eyewear, spending $15 billion annually on eyewear. **What**/ only zeros and ones: 00, 01, 10, 11. Why **does** each fail? __ __ 0s 1s Ten Digit Challenge Possible three-digit/ ten digits. More **Math** Maps (p. 14) Book Signatures (**math** maps, p.15) Pages for a book are traditionally printed **in** multiples of four (/

**polynomials** by factoring Modeling view – solving real world problems modeled by **polynomial** functions which almost never factor Solving **Polynomial** Equations- Algebraic Method Set **Polynomial** Equation equal to zero. Factor the resulting **polynomial** expression into a **product**/**math**.wvu.edu/~mays/ AVdemo/Labs/Lab01/Lab01- 04.htm Question: **What** happens to the y-values as the table approaches a solution to the **polynomial** equation? Question: For a general **polynomial** equation, **what** are some difficulties **in**/ **does** it/

July 2010 **Math** III Standard MM3A3 Students will solve a variety of equations and inequalities. a. Find real and complex roots of higher degree **polynomial** equations using /**Polynomial** Equations Solve Steps **in** the Analytic Solution Process ▫ Put the equation **in** standard form ▫ Factor the **polynomial** into linear and quadratic terms ▫ Use the zero-**product** property **Polynomial** Equations Solve analytically (symbolically). Standard form Factor by grouping Factor difference of squares zero **product** property **Polynomial**/

**Math** II **Polynomial** Review Quick Practice “Quiz” 1. A rectangular sheet of metal 36 inches wide is to be made into a trough by turning up sides of equal length (x) perpendicular to the sheet. Find the value of x that will produce the maximum cross sectional area of the trough. **What**/**polynomial** with integral coefficients if two of the roots are at 2 - 3i and 1 + 2 A) **in** factored form. B) **in** expanded form. 7. How many x intercepts **does**/- 11x + 30 is 2. Find the **product** of the other two roots. Quick Practice “/

+ 6t – 8)(t – 6) Find the **product**. Boss/Secretary Practice Students will work **in** pairs. per pair Materials: One worksheet, one calculator, one pencil per pair Students will be given one of two roles (you will alternate these roles): – Boss – Boss: Tells the secretary how to solve the problem, is the only one talking **in** the pair, **does** not write anything – Secretary – Secretary: Listens/

November 2010 Vincenzo Innocente MOTIVATIONS 1825 November 2010 Vincenzo Innocente Formula Translation: **what** was the author’s intention? 19 // Energy loss and variance / Vincenzo Innocente 32 APPROXIMATE (FAST) **MATH** 25 November 2010 Vincenzo Innocente 33 /Log, the **polynomial** expansion works **in** a very limited range (and adding more terms **does** not help/2, 3, 0, 1)); return _mm_add_ps(mul, swp); #endif } Vector **product** 25 November 2010 Vincenzo Innocente 52 // cross (just 3x3) inline __m128 _mm_cross_ps(__m128/

**Polynomial** dynamical systems Note: {0, 1} = k has a field structure (1+1=0). Fact: Any Boolean function **in** n variables can be expressed uniquely as a **polynomial** function **in**/**Math**. Biol. 55 (1993) 973-991 Key model features: Time delays of different lengths for different variables are important Positive and negative feedback loops are important Model description Basic structure of logical models: 1.Sets of variables x 1, …, x n ; X 1, …, X n (X i = genes and x i = gene **products**/ and at **what** levels they /

is a complex quantity, so **what** can be its significance for /**in** a solid (hence phonons) Electromagnetic field modes (hence photons), even though this field **does** not obey exactly the same Schrődinger equation You will do another, more elegant, solution method (no series or Hermite **polynomials**/-8 (Images from http://odin.**math**.nau.edu/~jws/dpgraph/Yellm.html/**in** postion Mean momentum: Mean-squared uncertainty **in** momentum: **In** fact, can show that this form of wavepacket (“Gaussian wavepacket”) minimizes the **product**/

**polynomial**, Discrete Mathematics and Theoretical Computer Science 6 (2003) 69-90. P(G; x, y) is a **polynomial** **in** variables x, y, which simultaneously generalizes the chromatic **polynomial**, the matching **polynomial**, and the independence **polynomial**/ independence **polynomials** whose **product** is /**polynomials** of trees **does** not directly implies the unimodality of independence **polynomials** of forests ! Consequence Hence, independence **polynomials**/” Conjecture is valid for (G) > 11 **What** about (G) > 11 ? Still open …/

Science and Applied **Math** Dept The /2D grids with local communication between neighbors 15 Practical limits on **what** we can do: We can use linear space to store /find a vector u **in** the kernel of a singular A: Theorem: Let p(x) be the characteristic **polynomial** of the matrix A./Schimmler mesh sorting algorithm three times to complete each matrix- vector **product**. For a mxm matrix, this requires about 24m steps. We/ 1/7 1/15 1/6 of 45,000. 29 **Does** it always work? We simulated the algorithm thousands of time with/

Cochran’s Q test An APA Results Section Proportions of student scoring high **in** **math**, reading and writing were .22, .24, and .25, respectively/ Syntax:- GLM y1 y2 y3 y4 /WSFACTOR=a 4 **Polynomial** /METHOD=SSTYPE(3) Only the additional outputs are presented/select two variables then >a*b> for the **product** term. 178 Factorial logistic regression Define categorical variables 179/the documentation to know find out just **what** the algorithm **does** **in** the software you are using - **in** particular, whether it has a stopping rule/

**What** is Computational Chemistry? Use of computer to help solving chemical problems Chemical Problems Computer Programs Physical Models **Math**/**in** the valleys on a PES Energetics of reactions can be calculated from the energies or altitudes of the minima for reactants and **products** A reaction path connects reactants and **products**/ to equilibrium geometries **Does** not take electronic interactions / r 0 )] 2 a cubic **polynomial** has wrong asymptotic form, but a quartic **polynomial** is a good fit for bond length of/

**product** of A x B. This is a relation The domain is the set of all x values **in** the relation The range is the set of all y values **in**/assigned to anything Comparing to our example, a student **in** **maths** must receive a grade Set A is the domain 1/ then is: All real numbers x such that x ≠ 2. means **does** not equal illegal if this is zero Note: There is nothing wrong with/are **polynomial** functions. The domain of f is the set of all real numbers except the values of x that make the denominator h (x) zero. **In** **what** follows/

**polynomials** Top down, youll know, all these confusing systems Cause when you come up **in** the class, you blankin’ out Got **math** **in**/**in** front of the parenthesis and making sure of the exponents. ~Dividing **Polynomials**: Long Division~ Example: Set up the division where you would normally put the divisor and dividend To know **what** to put for the quotient, divide the first terms of the dividend and the divisor and use the **product**/1 ± 4/1 ± 8/1. However, that **does** not mean these will all work, the ones that work are the/

**in** **polynomial** identities and equations Vector and Matrix /**What** is the most efficient way to solve the problem?”, “**Does** this make sense?”, and “Can I solve the problem **in**/**in** using procedures flexibly as they solve contextual and mathematical problems. 7.Support **productive** struggle **in** learning mathematics. Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage **in** **productive**/ provided. **What** is the **math** involved **in** this problem/

**Polynomials** for 800 When (c + 8) and (c - 5) are multiplied, **what** type of **polynomial** do we get as the **product**? **What** is its degree? Answer: The **product** is a trinomial (c² + 3c – 40) of degree two. **Polynomials** for 900 **What**/ rate. **What** type of slope **does** this line have? Answer: The slope of this line is negative. Functions for 800 Organize the following function **in** descending order/ Definitions for 800 Algebraic Equation Answer: An equation is a **math** sentence that says that 2 things are equal. An equation always/

To “Decimal” Positional Weighting Addition of the **products** of each digit by its respective “column weight” /# Biased Notation for Exponents Two’s complement **does** not work for exponent Most negative exponent: 00000001/ 0 Value: Not a Number **What** is the result of: sqrt/num: b = 2 -149 b a0 + - Gap! 0 + - Rounding **Math** on real numbers => rounding Rounding also occurs when converting types – Double single /**in** the positive and negative domains we use the Two’s Complement Number System Here is the **polynomial**/

**in** everything we do and **in** every **product** we design” 1 1 Jack Welch - General Electric website at ge.com **What** are Statistically Designed Experiments? Purposeful, systematic changes **in** the inputs **in** order to observe corresponding changes **in** the outputs Results **in**/ order **polynomials** with ability to detect nonlinearities Designs also included additional confirmation points to confirm the empirical **math** model **in** the test envelope Results: Original CFD study envisioned 1556 runs **DOE** optimized parameters **in** 84 /

pieces, 2x + 12x This now gives 3x2 + 2x + 12x + 8 Now we have a FOUR-TERM **polynomial**. **What** **does** this suggest?? You got it: Factoring by grouping! 3x2 + 2x + 12x + 8 Factoring the first pair /last coefficients of the trinomial together: 3*8 = 24 Now look for two factors of 24 that add up to 15. **product**: sum: 24 = 1*24 1+ 24 = 25 24 = 2*12 2 + 12 = 14 24 = 3/ class tomorrow. Lab hours **in** 203: Mondays through Thursdays 8:00 a.m. to 7:30 p.m. Please remember to sign **in** on the **Math** 110 clipboard by the front/

Signals? Option 1: Taylor series represents any function using **polynomials**. **Polynomials** are not the best - unstable and not very physically meaningful. (they still have some applications) Easier to talk about “signals” **in** terms of its “frequencies” (how fast/often signals change/ **in** Engineering A Sum of Sinusoids Our building block: Add enough of them to get any signal f(x) you want! How many degrees of freedom? **What** **does** each control? Which one encodes the coarse vs. fine structure of the signal? **Math** /

(or its normal vector) can be found **in** **polynomial** time. (The derandomization approach is beyond the /produce better approximations for max cut? **What** other problems have good approximations based / inner **products** x i ¢ x j. Triangle constraints At most two edges are cut **in** a triangle/can be generalized to show that even best hyperplane **does** not give approximation ratio better than . Summary Max/ Sudan [JACM 98]. Independent set: Alon and Kahale [**Math**. Programming 98]. Vertex cover: Charikar [SODA 02]. Max/

It [1989 NCTM Standards ] **does** not hold the properties of /**polynomials** or quotients of **polynomials** no discussion of things as basic as the standard exponent rules.” An Evaluation of CMP, R James Milgram Teacher Concerns With NCTM **Math**/**in** **math**, and before motherhood worked under contract to the US Navy as an analyst (mathematician, really), designing complex **math** models for dynamic simulation. If the two of us cannot solve a third-grade **math** problem, just **what** the heck is going on with this "wonderful" **math**/

Unit 5 – **Polynomials** (Sections 5.1 – 5.6) Unit 6 – Linear Equations & Inequalities (Sections 6.1-6.5) Unit 4 – Linear Relations (Sections 4.1 – 4.5) Unit 8 – Circles Geometry (Sections 8.1 – 8.4) **Math** 90 Course Outline **Math** 90 Plus Unit 9/**in** which we multiply **does** not matter: (2 x 2) x 2 = 2 x (2 x 2) = 2 x 2 x 2 How would you write this **product** as a power? **What** **does** the word **product** mean? **What** **does** the word quotient mean? 2.4 Exponent Laws I **Product** of PowersProduct as Repeated Multiplication **Product**/

of Computer Images and Graphics, **Math** Dept. ZJU October 26 Outline 1/**in** the appropriate continuity conditions, similar to **what** is done with the B-Spline.” Introduction Bézier Curves B-Spline Curves Triangular Bézier patches **What**? Desirable Control Scheme Attributes Piecewise **polynomial** of a fixed degree Individual piecewise **polynomials**/ Be able to model discontinuities like sharp edges or corners as well( tensor **product** B-spline will produce a discontinuity curve across the whole patch) Application(3)/

**Does** the US need Top-Tier **Math** Circles? erosion of capacity for new scientific and technological breakthroughs deteriorating **math** and science education **in**/**polynomials**. K-1 ∞ The paper invokes some advanced concepts: Recursive relations Generating functions Functional analysis Partial derivatives Algebraic numbers Ramanujan-type continued franctions For the Die-Hards:Conway Checkers John Conway Question: **What**/functions Dirichlet **product** (convolution of functions) Mobius inversion formula Build a **math** circle /

Trace formats, coordinate transformations Annotation text and XML **What** the Computer Sees Usual Character Reco. Methods Smooth and/ dictionary Difficulties Having many similar characters (e.g. for **math**) means comparison against all possible symbol models is slow./RMS **does** not mean similar shape. Problem 1. On-Line Ink The main problem: **In** handwriting/ 2. Shape vs Variation Legendre-Sobolev Basis Life **in** an Inner **Product** Space With the Legendre-Sobolev inner **product** we have – Low dimensional rep for curves /

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