E(c 1 )E(c 0 ) …using an “additive” encryption scheme E(x) * E(y) = E(x+y) [Paillier’**99**] Step 2: Alice encrypts her **polynomial**… Alice’s encrypted **polynomial** p(x) E(c 4 )E(c 3 )E(c 2 )E(c 1 )E(c 0 ) …using an “additive” encryption/ Solution for Two Parties: Yao’s Encrypted Circuit Alice’s logical circuit C Bob’s input x 0 1 1 Encrypted circuit Oblivious transfer **of** keys A More General Solution for N Parties: Secure Multi-Party Computation Split the computation into logical steps (ANDs, ORs, NOTs) or algebraic/

Szegedy] 1992 MAX-3SAT Desired Way to disguise instances **of** any NP problem as instances **of** MAX-3SAT s.t. “Yes” instances turn into satisfiable formulae “No” instances turn into formulae in which < 0.**99** fraction **of** clauses can be simultaneously satisfied “Gap” Cook-Levin/ MIP = NEXPTIME [Babai, Fortnow, Lund ’90] 1 st “PCP Theorem” Dinur [05]’s proof uses random walks on expander graphs instead **of** **polynomials**. Håstad’s 3-bit Theorem (and “fourier method”) NP = PCP(lg n, 1) T1T1 T2T2 c bits 1 bit YES instances/

**of** tour found Could derive a lower bound by rounding values down and keeping these values when computing cost **of** tour found IJCAI-05 TutorialBounding Techniques76 Modifying objective function coefficients Frieze’s **polynomial**-time STSP algorithm [SIAM Computing 1987] Frieze’s **polynomial**/range equal to number **of** cities Random ATSP graphs with cost range equal to number **of** cities Found close to half **of** the backbone arcs and **99**% **of** the fat arcs Found close to half **of** the backbone arcs and **99**% **of** the fat arcs /

˜ ( n R A S A ) or O ˜ ( S A 1+ ) if char K = 0 Texas A&M University ACA2004 14 RUR Complexity Analysis n [Rojas **99**] l Evaluating Pert ( u ) requires O ˜ ( n R A 2 S A ) or O ˜ ( S A 1+ ) if char K = 0 Texas A&M/ Texas A&M University ACA2004 29 RUR Future Work n RUR l Faster sparse resultant algorithms l Take advantages **of** sparseness **of** matrices [Emiris and Pan 97] l Faster univariate **polynomial** operations Texas A&M University ACA2004 30 RUR Thank you for listening! n Contact l Koji Ouchi, kouchi@/

, Jerrum ’02). Glauber dynamics does not mix in **polynomial** time for 6-regular bipartite graphs (example: union **of** 6 random matchings) (Dyer, Frieze, Jerrum ’02). = maximum degree **of** G counting/sampling independent sets in bipartite graphs: A different view **of** independent sets in bipartite graphs **polynomial** time sampler for 5 (Dyer,Greenhill ’00, Luby,Vigoda’**99**, Weitz’06). no **polynomial** time sampler (unless NP=RP) for 25/

double errors –all errors **of** an odd length –all bursts **of** 16-bits or less, **99**.997% **of** 17-bits, and **99**.998% **of** 18-bits and longer. 7/2/2015Errors27 The Algorithm To compute the checksum –Append n 0s to the end **of** the message, where n is the number **of** bits in the checksum –The resulting value is divided by the generator **polynomial** –Each division step is/

**polynomials** Multiply each term **of** the first **polynomial** by each term **of** the second **polynomial**, and then combine like terms. Multiplying **Polynomials** Martin-Gay, Prealgebra & Introductory Algebra, 3ed 44 Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply each **of**/25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 Multiplying **Polynomials** Example Martin-Gay, Prealgebra & Introductory Algebra, 3ed **99** Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x 2 + x –/

**of** tour found Could derive a lower bound by rounding values down and keeping these values when computing cost **of** tour found AAAI-05 TutorialBounding Techniques75 Modifying objective function coefficients Frieze’s **polynomial**-time STSP algorithm [SIAM Computing 1987] Frieze’s **polynomial**/range equal to number **of** cities Random ATSP graphs with cost range equal to number **of** cities Found close to half **of** the backbone arcs and **99**% **of** the fat arcs Found close to half **of** the backbone arcs and **99**% **of** the fat arcs Time/

(quadratic, cubic) are implicated by the distribution **of** residuals. How to Determine Order 12 **Polynomial** vs. Power Transformation **of** X Power Transformations **of** X **Polynomial** Regression 13 How does number **of** electives taken in an area predict interest in /paragraph optional in some cases] The overall effect **of** Number **of** Electives was significant, F(2,97) = **99**.62, p<.0001, with Number **of** Electives accounting for 67.3% **of** the total variance in Interest. The linear effect **of** Electives was significant, B= 1.2, /

Algorithm Idea: Find algebraic explanation Find algebraic explanation **of** all points. **of** all points. Stare at it! Stare at it! Factor the **polynomial**! **of** 31 09/19/2011UIUC: Communication & Computation15 Decoding Algorithm Fact: There is always a degree 2n **polynomial** thru n points Fact: There is always a degree 2n **polynomial** thru n points Can be found in **polynomial** time (solving linear system). Can be found in/

The Complexity **of** the Network Design Problem Networks, 1978 Classic Paper Reading **99**.12 Outline Introduction NDP is NP-complete SNDP is NP-complete Conclusion 2 Introduction B96902094 傅莉雯 Combinatorial optimization is/3,4,5},{1,2,6}} No! Yes! Reduction Given any instance **of** EXACT 3-COVER, we define an instance **of** SNDP as follows: 31 = r Illustration **of** reduction 32 S T Illustration **of** reduction 33 R Illustration **of** reduction 34 S T R **polynomial**-time reduction Claim has a feasible solution if and only if/

RCHESTRA, one kind **of** annotation (provenance **polynomials**) can support many kinds **of** trust models, ranking,... – Compute propagation **of** annotations just once Extends to recursive mappings Analysis **of** previous provenance models: – All special cases **of** framework – None/equivalence with where-provenance [Tan 03] Answering queries using views [Levy+ 95], [Chaudhuri+ 95], [Cohen+ **99**], [Afrati+ **99**],... View adaptation [Gupta+ 95], mapping adaptation [Velegrakis+ 03] 48 We studied an important practical problem – /

and GLUT contain **polynomial** approximations **of** quadrics 4 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 One-Dimensional Evaluators Evaluate a Bernstein **polynomial** **of** any degree at a set **of** specified values Can evaluate a variety **of** variables Points along/ } must draw in both directions 13 Angel: Interactive Computer Graphics 4E © Addison-Wesley 2005 Rendering with Quadrilaterals for(j=0; j<**99**; j++) { glBegin(GL_QUAD_STRIP); for(i=0; i<100; i++) { glEvalCoord2f ((float) i/100.0, (float) j/100./

, a function F:N -> N is said to be one-way function if it is easily computable - in **polynomial** time - but any computation **of** its inverse is infeasible. A one-way permutation is a 1-1 one-way function. easy x f(x) /is that decoding seems to be infeasible. Example If A = (74, 82,94, 83, 39, **99**, 56, 49, 73, **99**) and B = (1100110101) then ABT = Public-key cryptography Design **of** knapsack cryptosystems IV054 Design **of** knapsack cryptosystems 1. Choose a superincreasing vector X = (x1,…,xn). 2. Choose m, u such/

(1) = 2(1) 3 – (1) 2 + 3(1) – 4= 0. Since P(1) = 0, x – 1 is a factor. Table **of** Contents **Polynomials**: The Remainder and Factor Theorems Slide 3 Try:For the **polynomial**, P(x) = x 3 – x 2 + x – 6, (a) find P(5) using the remainder theorem, (b)use the factor theorem to show /that x – 2 is a factor. (a) 1- 1 1- 6 5 20 105 5 | 1 4 21 **99** = P(5) (b)P(2) = (2) 3 – (2) 2 + (2) – 6 = 0 Since P(2) = 0, x – 2 is a factor. Table **of** Contents **Polynomials**: The Remainder and Factor Theorems

the Regression Errors Normally Distributed? Principles **of** Econometrics, 4t h EditionPage 75 Chapter 4: Prediction, Goodness-**of**-fit, and Modeling Issues 4.4 **Polynomial** Models Principles **of** Econometrics, 4t h EditionPage 76 Chapter 4: Prediction, Goodness-**of**-fit, and Modeling Issues In addition / is: 4.5.4 A Generalized R 2 Measure 4.5 Log-linear Models Principles **of** Econometrics, 4t h EditionPage **99** Chapter 4: Prediction, Goodness-**of**-fit, and Modeling Issues For the wage equation, the general R 2 is: – /

**99** Erice 2005, the Analysis **of** Patterns. Grammatical Inference **99** Definitions Let be the length-lex ordering over * Let Pref(L) be the set **of** all prefixes **of** strings in some language L. 10 0 Erice 2005, the Analysis **of** / Erice 2005, the Analysis **of** Patterns. Grammatical Inference 126 Properties (4) **Polynomial** aspects **Polynomial** characteristic sets **Polynomial** update time But not necessarily a **polynomial** number **of** mind changes 12 7 Erice 2005, the Analysis **of** Patterns. Grammatical Inference 127 Extensions/

Pearson Education, Inc. Publishing as Prentice Hall. To Multiply Two **Polynomials** Multiply each term **of** the first **polynomial** by each term **of** the second **polynomial**, and then combine like terms. Multiplying **Polynomials** Martin-Gay, Developmental Mathematics, 2e 55 Copyright © 2011 Pearson / – 2z) = 25x 2 – 10xz – 10xz + 4z 2 = 25x 2 – 20xz + 4z 2 Example Martin-Gay, Developmental Mathematics, 2e **99** Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall. Multiply (2x 2 + x – 1)(x 2 + 3x + 4). (2x 2/

2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rational expressions can be written as quotients (fractions) **of** two **polynomials**. Examples include: Basic Concepts Slide 45 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE/radical expressions Q 77-88 Rationalizing the denominator Q 89-98 Using a conjugate to rationalize the denominator Q **99**-102 Rationalizing the denominator Q 103-108 Rationalizing the denominator having a cube root Q 113-116 Slide /

array containing zero The latter is used in the implementation For convenience, an instance variable is used to store the degree **of** the Poly /** OVERVIEW: Polys are immutable **polynomials** with integer coefficients. A typical Poly is c0 + c1x + c2x^2 +... */ public class Poly { private int[] trms/above rep, if an integer i between 0 and **99** is in the set, we just set els[i] to be true All integers > **99** are stored in otherEls For efficiency purpose, we store the size **of** the set in sz This will be an /

**of** data with the same seeds: 1, New Forward + Old Backward Simulate the REAL DeltaP reconstruction on HRS-R (δp wrong ); 2, New Forward + New Backward Simulate the CORRECTED DeltaP reconstruction (δp right ). but wrong! E08-014 Data Analysis -DeltaP HRS Optics **Polynomials**/ Cross Sections Cuts & Efficiencies VDC One-Track-Only Cuts (> **99**%); Trigger Cuts (>**99**% since GC included); PID Cuts (very loose cuts, **99**% for GC, **99**% for Calo); Focal Plane Variables Cuts (removing big-angle events); /

by Factoring Step 1: Write the equation in standard form, ax 2 + bx + c = 0. Step 2: Factor the **polynomial** on the left side **of** the equation. Step 3: Set each factor found in Step 2 equal to zero using the Zero-Product Property Step 4: Solve /Write a formula s(t) that models the height **of** the baseball after t seconds. b) How high is the baseball after 2 seconds? c) Find the maximum height **of** the baseball. Support your answer graphically. Solution a) b) Baseball is **99** feet high after 2 seconds. c) Because a is/

models, use R 2 (adj) to compare their effectiveness. 5.Use diagnostic plots to check for violation **of** the regression assumptions. Copyright ©2006 Brooks/Cole A division **of** Thomson Learning, Inc. A **Polynomial** Model quadraticWhen k = 2, the model is quadratic: A response y is related to a single independent /0.000 x2 866.7 305.3 2.84 0.022 x1x2 260.13 87.06 2.**99** 0.017 S = 201.344 R-Sq = **99**.2% R-Sq(adj) = 98.9% Analysis **of** Variance Source DF SS MS F P Regression 3 42108777 14036259 346.24 0.000 Residual /

one that yields the largest max eigenvalue. Amounts to solving (n − k) generalized eigenvalue problems **of** size k + 1. **Polynomial** runtime: 17 **Polynomial** Time Heuristic Approaches Truncation Method (Fogarasi et al 2012) Compute unconstrained solution then use k heaviest dimensions/, EDD – Earliest Due Date) Job size 510152030405075100 % outperf **99**.9100 **99**.599.299.699.398.698.8 38 Thesis II.3 Further improving HNN Smart HNN (SHNN) Use the result **of** Largest Weighted Path First (LWPF) as starting point for HNN /

MIT32 Root Bound A root bound for an expression E is a value such that n E.g., the Cauchy’s bound says that because is a root **of** the **polynomial** x 4 0 x 2 1. n Root bit-bound is defined as log(b) b > 0 b = Oct 18/ routinely construct robust programs * n Current Libraries: Real/Expr [Yap-Dube’94] LEDA real [Burnikel et al’**99**] Core Library [Karamcheti et al’**99**] Oct 18, 2001Talk @ MIT37 Core Library n An EGC Library y C++, compact (200 KB) y Focus on “Numerical Core/

Let (C,t) be a k(.)-round intractability assumption where k is a **polynomial**. If there exists a PPT reduction R for basing security **of** any **of** previously mentioned schemes, on the hardness **of** (C,t), then there exists a PPT attacker B that breaks (C,t/can find (recursively) find one proof where nesting depth is “small” Use Techniques reminiscent **of** Concurrent ZK a la [RK’**99**], [CPS’10] x2x2 x3x3 rewinding here: redo work **of** nested sessions w2w2 w3w3 w1w1 General Reductions: Problem II Problem: R might not only /

- special case **of** “PCP Spot Checkers” PCPPs - special case **of** “PCP Spot Checkers” [EKR ’**99**] [EKR ’**99**] PCPP – extension **of** Property Testing PCPP – extension **of** Property Testing [RS ’92, GGR ’96] [RS ’92, GGR ’96] Assignment Testers **of** [DR ’03] similar to PCPPs. Assignment Testers **of** [DR ’03]/f ) < d and l – line, then f restricted to line l is a univariate **polynomial** **of** degree < d. If deg( f ) < d and l – line, then f restricted to line l is a univariate **polynomial** **of** degree < d. P r [ f ( x ) = g ( x )] · /

2/16) log 2 (2/8) - (6/16) log 2 (6/8) = 0.88 bit Mutual Information I(f;x 1 )= 0.**99** - 0.88 = 0.11 bit Information theoretical measures x 1 x 2 x 3 x 4 0 0 1 1 0 1 0 0 1 1 /will be applied to Galois Logic Shannon entropy Information theoretic criterion in minimization **of** **polynomial** expressions in GF(4) Linearly Independent Expansions in any Logic New Idea Shannon entropy Information theoretic criterion in minimization **of** trees, lattices and flattened forms in this logic Shannon decomposition Shannon entropy /

Nearly Tight Smoothed Bounds for 2D: Many Moments Three or More Objectives Not So Tight Yet: But **Polynomial** Smoothed Bound for Fixed Dimensions This Talk Part I: Overview **of** Smoothed Analysis Part II : Multi-objective Optimization Part III: Machine Learning Part VI: Games, Markets / LEGIT x4x4 YES NO YESSPAM x5x5 YES NOYES SPAM [Valiant84] P.A.C Learning Poly-time learning algorithm –Succeed with prob. ≥ 1- (e.g. 0.**99**) –m = # examples = poly(n/ε) Output: h: X → {–1,+1} with err(h) = Pr x←D [ h(x)≠f(x) ] /

**of** a joint) and continuous second derivatives (the rate **of** change in the slope **of** the line will not change across a joint). A spline **of** degree n will have continuous derivatives across the points up to order n – 1. R Mathematical Explanation A smoothly joined piecewise **polynomial** **of**/trees (b, e, h), each **of** which accounted for > **99**% **of** the total ss. COMPARISON **OF** CART AND GLMs ANOVA is powerful technique but as number **of** predictor variables and complexity **of** data increase (interactions, unbalanced designs, /

hours (samples were weighed at 15 minute intervals), the total available 154 values **of** specific humidity **of** grain for various temperatures and times. From these values only the **99** data were used for training the network, which had two neurons in the input/points is approximated by regression **polynomial** **of** degree k, where k<2N w. Value **of** this regression **polynomial** in the point x i substitutes original value y i. Number **of** data N=1024, width **of** window N w =50, quadratic **polynomial**. The SG filtration is /

) Z t {Z t } WN(0, 2 ), where (B) and (B) are different AR **polynomials** **of** orders p and P, respectively; and (B) and (B) are different MA **polynomials** **of** orders q and Q, respectively. The idea here is to try to model the seasonal behavior via the 12 ARMA, (B s )Y/ 2 f t W t, t=1,…,120. where f t =0, 1 ≤ t ≤ 98, and f t =1, t ≥ **99** (file SBLIN.TSM). Steps in ITSM2000: n Regression > Specify > Poly Regression, order 0 > Include Auxiliary Variables Imported from File > SBLIN.TSM. 22 /

2,..., a n are constant coefficients.evaluatingargument Degrees **of** **Polynomial** Functions Constant function such as f(x)=a has a degree **of** zero. Linear functions, f(x) = mx+b, have a degree **of** one. Quadratic functions have a degree **of** two. Quadratic Functions: Quadratic Functions All quadratic/ one **of** the following percent is possible as the percent **of** 25 words a student defined correctly. Which one is it. A)**99**% B)80% C)69% D)45% E)26% Writing in Standard Form Write the standard form **of** the equation **of** the /

70% accuracy Simple quadratic 90% accuracy Simple linear 90% accuracy Simple quadratic 95% accuracy Simple linear **99**% accuracy Simple quadratic **99**% accuracy One Solution: Charge Penalty for complex models For example, for the simple {**polynomial**} classifier, we could charge 1% for every increase in the degree **of** the **polynomial** 10 123456789 1 2 3 4 5 6 7 8 9 123456789 1 2 3 4/

sa n-2 = 0, Which can be associated with x 2 – rx - s This **polynomial** is called the characteristic **polynomial** **of** the recurrence relation The Characteristic **Polynomial** Its roots are called the characteristic **polynomial** roots **of** the recurrence relation. x 2 – rx - s x 2 – 5x + 6 and characteristic /S = n/2 [2a + (n – 1)d] S = 100/2 [2(-17) + (100 – 1)5] S = 50 [-34 + (**99**)5] S = 23,050 Arithmetic Sequences The 100th term **of** this sequence is a n = a + (n – 1)d a 100 = a + (n – 1)d a 100 = -17 + (100 – 1/

holds, but breaks around n = 10 C = 1000? C = 1000? 10n 2 +4n+2 < 1000n 10n 2 n n < **99**.5 If n = 1 through **99**, inequality holds, but breaks around n = 100 Can’t find a n 0 for which it holds for all n >= n 0 Big/many comparisons the minIndex function performs, so the same number **of** spots in the array are always examined for a given input size. However, reverse sorted data does force more assignments (everything is a new min). Worst Case Selection Adding **Polynomials**: Adding **Polynomials**: 3x 3 + 2x + 1 + 2x 3 + /

sa n-2 = 0, Which can be associated with x 2 – rx - s This **polynomial** is called the characteristic **polynomial** **of** the recurrence relation The Characteristic **Polynomial** Its roots are called the characteristic **polynomial** roots **of** the recurrence relation. x 2 – rx - s x 2 – 5x + 6 and characteristic /S = n/2 [2a + (n – 1)d] S = 100/2 [2(-17) + (100 – 1)5] S = 50 [-34 + (**99**)5] S = 23,050 Arithmetic Sequences The 100th term **of** this sequence is a n = a + (n – 1)d a 100 = a + (n – 1)d a 100 = -17 + (100 – 1/

reliability Cons –Complicated tool development and support ($$) –Tool testing, integration, and training C. Chen, ICCAD ‘**99** Embedded Tutorial, Session 12A High-Speed Circuit-Sizing Techniques based on Lagrangian Relaxation Good Tuning Algorithm Fast Optimality /) –No need for sensitization vectors –Solves for all critical paths in a **polynomial** formulation –False paths; Potentially inaccurate modeling **of** slopes **of** input excitation Dynamic Sizing –Simulation based –More accurate –No false path problems /

ε >0 Exact version: find a solution **of** weight exactly K Pseudo-**polynomial** time: **polynomial** in K Problems satisfy condition (1): shortest path, minimum spanning tree, matching, knapsack. Our Results If μ is a threshold function, maximizing E[ μ (w(S))] is equivalent to maximizing Pr[w(S)<1] minimizing overflow prob. [Kleinberg, Rabani, Tardos. STOC’97] [Goel, Indyk. FOCS’**99**] chance-constrained stochastic optimization problem [Swamy/

s 0,2 s 1,2 s 1,3 s 2,3 s 2,0 s 3,0 s 3,1 Secret Key Systems - AES Addition: modulo 2 addition (xor) **of** **polynomials** **of** maximum degree 7 Examples: (x 6 +x 4 +x 2 +x 1 +1) + (x 7 +x 1 +1) = x 7 +x 6 +x 4 +x 2/ c1 1d 9e e e1 f8 98 11 69 d9 8e 94 9b 1e 87 e9 ce 55 28 df f 8c a1 89 0d bf e6 42 68 41 **99** 2d 0f b0 54 bb 16 Secret Key Systems - AES The Inverse S-Box: y 0 1 2 3 4 5 6 7 8 9 a b c d e/

Technology Evaluation FpVTE 2003 48,105 sets, 25,309 individuals, 393,370 distinct fingerprints Verification results **99**.4% true accept rate @ 0.01% false accept rate **99**.9% true accept rate @ 1.0% false accept rate Challenges Poor-quality images Database size /, add equal norm tight ENMR Construction by seeding Construction by seeding 01n-1 Tools: **Polynomial** algebras and transforms Tools: **Polynomial** algebras and transforms m S 79 Invariance **of** Frame Properties FA B is FA, B invertible 0 0 MR F A is MRFA/

N>10) Wouter Verkerke, UCSB Properties **of** the Gaussian distribution Mean and Variance Integrals **of** Gaussian 68.27% within 1s 90% 1.645s 95.43% within 2s 95% 1.96s **99**.73% within 3s **99**% 2.58s **99**.9% 3.29s Wouter Verkerke, / 0.96156 -0.681 0.127 -0.895 1.000 Wouter Verkerke, UCSB Mitigating fit stability problems -- **Polynomials** Warning: Regular parameterization **of** **polynomials** a0+a1x+a2x2+a3x3 nearly always results in strong correlations between the coefficients ai. Fit stability problems, inability to/

8 Third differences: 0.6 1 0.6 0.8 0.8Close Check It Out! Example 2 Continued Step 2 Determine the degree **of** the **polynomial**. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on/)2 – 858.**99**(6) + 693.88 Based on the model, the opening value was about $2563.18 in 2000. Check It Out! Example 3 The table below shows the opening value **of** a stock index on the first day **of** trading in various years. Use a **polynomial** model to estimate /

8 Third differences: 0.6 1 0.6 0.8 0.8Close Check It Out! Example 2 Continued Step 2 Determine the degree **of** the **polynomial**. Because the third differences are relatively close, a cubic function should be a good model. Step 3 Use the cubic regression feature on/)2 – 858.**99**(6) + 693.88 Based on the model, the opening value was about $2563.18 in 2000. Check It Out! Example 3 The table below shows the opening value **of** a stock index on the first day **of** trading in various years. Use a **polynomial** model to estimate /

Circular encryption.... Attacks outside standard model: Timing attacks [Kocher 96] Fault detection [BDL 97, BS 97] Power analysis [KJJ **99**] Cache attacks [OST 05] Memory attacks [HSHCPCFAF 08]... Adversarial Models Attacks - standard model: Chosen-plaintext attacks Chosen-ciphertext attacks Composition/A function f:{0,1} n → {0,1} n is called a (t,ε) one-way function, if f is a **polynomial**-time computable function (independent **of** t) for every t-time algorithm A, Prob[A(f(x)) f -1 (f(x)) ] ≤ ε Where /

Informally, a function F:N -> N is said to be one-way function if it is easily computable - in **polynomial** time - but any computation **of** its inverse is infeasible. A one-way permutation is a 1-1 one-way function. easy x f(x) computation/seems to be infeasible. Example If A = (74, 82,94, 83, 39, **99**, 56, 49, 73, **99**) and B = (1100110101) then ABT = Public-key cryptography Another view **of** the knapsack problem IV054 Another view **of** the knapsack problem Each knapsack vector A = (a1,…,an) defines an integer valued /

Pearson Education, Inc. Publishing as Pearson Addison-Wesley The product **of** two binomials can be shown in terms **of** geometry. 35 5x5x 7x7xx2x2 Length width = Sum **of** the areas **of** the four internal rectangles Combine like terms. Slide 5- 98 Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Objective 3 Multiply **polynomials**. Slide 5- **99** Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison/

We introduce infinite dimensional algebra (purely abstract device) (fg)h=f(gh) Algebra **of** formal power series deg f: Scalar **polynomial** cx 0 Monic **polynomial** f n = 1. Theorem 1: f, g nonzero **polynomials** over F. Then 1.fg is nonzero. 2.deg(fg)=deg f + deg/. This is the Lagrange interpolation formula. –This follows from Theorem 15. P.**99**. (a->f, f i ->L i,a i ->P i ) Example: Let f = x j. Then Bases The change **of** basis matrix is invertible (The points are distinct.) Vandermonde matrix Linear algebra isomorphism/

16 0.01 0.68 0.31 0.34 0.66 0.01 0.**99** 53% 0.85 0.15 0.01 0.68 0.31 0.34 0.66 0.01 0.**99** 67% 0.85 0.15 0.01 0.68 0.31 0.34 /0.66 0.01 0.**99** 80% 0.84 0.16 0.01 0.67 0.32 0.33 0.66 0.01 0.**99** Statistic Numbers ( 0 K) CNRFC 1200-1800 1800-0000 0000-0600 0600-1200/ 1.97 Larger bias and error at 18-00Z and 06-12Z 3-day (April 14-16, 2011) daily cycle **of** basin averaged hourly temperature (generated by GES-DISC Giovanni) Try **polynomial** fit 00-06, 06-12, and 12-18Z: T ave = a*T des + b*T asc + c /

**of** the collocation points: Interpolation **of** Runge‘s function R(x) using 6 th -order **polynomials** and Gauss-Lobatto-Legendre collocation points [ roots **of** (1-x 2 )Lo N-1 = completed Lobatto **polynomial** ] interpolant We should use the GLL points as collocation points for the Lagrange **polynomials**. SPECTRAL-ELEMENT METHOD: General Concept Example: GLL Lagrange **polynomials** **of**/ with the upper layer **of** elements … … and PREM below boundary between the upper 2 layers **of** elements lon=142.74° lat=-5.**99**° d=80 km SA08/

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