##### NP-Complete Problems (Fun part)

the first NP-complete problem. S. A. Cook in 1971 Won Turing prize for this work. Significance: If Satisfiability problem can be solved in polynomial time, then ALL problems in class NP can be solved in polynomial time. If you want to solve PNP, then you should work on NPC/as follows: (Directly connect two nodes in R based on the visited order of the traversal.) 7 1 2 3 4 5 6 8 9 10 From triangle inequality, w(ST)≤w(traversal) ≤2OPT(I). ..........(2) Inequality(2) says that the cost of the spanning tree ST/

##### 1 The Theory of NP-Completeness 2 NP P NPC NP: Non-deterministic Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete P=NP? X = P.

Polynomial P: Polynomial NPC: Non-deterministic Polynomial Complete P=NP? X = P 3 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class of decision problem which can be solved by a non-deterministic polynomial algorithm. NP-hard: the class/as not in P No known deterministic algorithms that are polynomially bounded for many problems in NP So, “ does P = /) -FAILURE v -SUCCESS (7) SUCCESS (8) x(1)=7 (9) x(2)≠7 (10) 30 Satisfiable with the following assignment: i=1/

##### FA to Classroom Practice

), or product (answer when multiplying) of two polynomials will always be. 18 18 I can define “closure” as it applies to polynomials. I can tell what it means for polynomials to be closed under the operations of addition, /Class attendance Punctuality of assignments Class behavior or attitude Progress made 78 Case Study 9: Grading Practices that Inhibit Learning Does Figure Intro.14 reflect any practices you used in the past? Page 35 What grading issues arise from this case study? 79 Grading Issues Basis for/

##### College Algebra Prerequisite Topics Review Quick review of basic algebra skills that you should have developed before taking this class 18 problems that.

for squaring a binomial: Problem 3 Perform the indicated operation: Answer: Dividing a Polynomial by a Polynomial First write each polynomial/polynomials as being prime We will also completely factor polynomials by writing them as a product of prime polynomials Importance of Factoring If you don’t learn to factor polynomials you can’t pass college algebra or more advanced math classes/completely: Answer: Problem 9 Factor completely: Answer: Rational Expression A ratio of two polynomials where the denominator /

##### The Remainder and Factor Theorems Check for Understanding 2.3 – Factor polynomials using a variety of methods including the factor theorem, synthetic division,

4 2 –4 –7 –13 –10 8 16 36 92 2 4 9 23 82 Time for Class work Time for Class work Evaluate each function at the given value. The Factor Theorem The binomial (x – a) is a factor of the polynomial f(x) if and only if f(a) = 0. The Factor Theorem/the Factor Theorem, determine if f(x) is a factor of p(x) The Factor Theorem Time for Class work Time for Class work Using the Factor Theorem, factor fully each of the following polynomials: Using the Factor Theorem, determine if f(x) is a factor of p(x) The Rational /

##### Week 4. Due for this week…  Homework 4 (on MyMathLab – via the Materials Link)  Monday night at 6pm.  Prepare for the final (available tonight 10pm.

in the same session to return to a question.  There is no time limit to the exam (except for 11:59pm Saturday night after the last class).  You will not have the following help that exists in homework:  Online sections of the textbook  /checking Check: Divide the expression and check the result. Try some of Q: 9-14 Slide 66 Copyright © 2009 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE Solution Dividing polynomials The quotient is 2x + 4 with remainder −4, which also can be written/

##### Computational Methods for Management and Economics Carla P. Gomes Introduction to Complexity Theory Module 11.

3 4 2 3 2 Size of instance  Number of cities and roads A particular instance: Size of instance: 6 nodes 9 edges The size of an instance is given by the size of its input. An instance of problem 5: max 3 / there is a polynomial time algorithm A for solving X. We denote by P the class of problems solvable in polynomial time. Hard problems --- everything else. Any problem for which there is no polynomial time algorithm is an intractable problem.. Class P Class of Problems solvable in Polynomial Time Finding a /

##### Computational Complexity Theory Lecture 1: Intro; Turing machines; Class P and NP Indian Institute of Science.

hard functions for secure encryptions. Basic Course Info Course title: Computational Complexity Theory Credits: 3:1 Instructor: Chandan Saha TA: Vineet Nair ( vineet.nair@csa.iisc.ernet.in )vineet.nair@csa.iisc.ernet.in Class timings : Tuesday, Thursday: 9:30-11am./. Class NP captures the set of decision problems whose solutions are efficiently verifiable. Nondeterministic polynomial-time Complexity Class NP u is called a certificate or witness for x (w.r.t L and M) if x ∈ L Complexity Class NP Class NP/

##### Summary  NP-hard and NP-complete  NP-completeness proof  Polynomial time reduction  List of NP-complete problems  Knapsack problem  Isomprphisim.

literal to be true.  3-SAT although seems straight forward but presently we don’t know an algorithm to solve it in polytime algorithm.. 5 Class of NP problems  Def: A Decision Prbl X is associated with a set of strings or input, to a problem X {language}, such that /P.  Equivalently if P  NP, then no any NPC is in P. Therefore if X is NPC, there can be no polynomial time solution for problem X. 9 NP-hard and NP-complete Then the roof of NP is collapsing into P.  Equivalently if P  NP, then no any NPC /

##### BCH_Code 2004/5/5Yuh-Ming Huang, CSIE, NCNU1 BCH Code a larger class of powerful random error-correcting cyclic codes a remarkable generalization of the.

BCH Code a larger class of powerful random error-correcting cyclic codes a remarkable generalization of the Hamming codes for multiple-error correction Def : Let q and m be given and let α be any element of GF(q m ) of order n. Then for any positive integer /(z) = z 2 +z+2 α α 2 α 4 α 8 α 16  α α 3 α 6 α 12 α 24 α 48  α 3  α 9 Exponential Notation Polynomial Notation Binary Notation Decimal Notation Minimal Polynomial 00000 α0α0 1011x + 1 α1α1 z104x 2 + x + 2 α2α2 z + 2126x 2 + x + 3 α3α3 3z + 23214x 2 + /

##### Cs3102: Theory of Computation Class 24: NP-Completeness Spring 2010 University of Virginia David Evans.

Short Dilbert on Decidability Complexity Classes Recap Polynomial time: languages that can be decided by a deterministic TM in  (N k ) steps. Nondeterministic Polynomial time: languages that can be decided by a nondeterministic TM in  (N k ) steps. NP does not stand for “Non-Polynomial” time. If you say so/is in L(NPDA) but not L(DPDA) According to a 2002 poll of CSists, 61 think Option 1 is right vs. 9 for Option 2 Problems in NP All problems we know are in P Some problems we don’t know if they are in P /

##### MTH 209 Week 1 Third. Due for this week…  Homework 1 (on MyMathLab – via the Materials Link)  The fifth night after class at 11:59pm.  Read Chapter.

Polynomials Multiplying Polynomials Multiplying Monomials A monomial is a number, a variable, or a product of numbers and variables raised to natural number powers. To multiply monomials, we often use the product rule for exponents. Example Multiply. a.b. Solution a.b. Try Q: 9/ Try Q: 41 pg 349 Due for this week…  Homework 1 (on MyMathLab – via the Materials Link)  The fifth night after class at 11:59pm.  Read Chapter 6.1-6.4  Do the MyMathLab Self-Check for week 1.  Learning team planning /

##### Do Now: Solve the inequality. Academy Algebra II/Trig 5.1: Polynomial Functions and Models HW: p.340 (12, 13, 17-20, 40, 41, 43, 45-47 – parts a,d,e only)

order of exponents. Degree = the highest exponent. Common Polynomial Functions DegreeTypeExample 0constantf(x)=14 1linearf(x)=5x – 7 2quadraticf(x)=x 2 +x-9 3cubicf(x)=x 3 -x 2 4quarticf(x)=x 4/ the zero-product property. Example: Find the real zeros of the function: Real Zeros For the polynomial: 7 is a zero of multiplicity 1 because the exponent on the factor of x /– part f: like class work) Graph the polynomial. Label intercepts, determine turning points, and end behavior. (May use graphing calculator/

##### NP-Completness Turing Machine. Hard problems There are many many important problems for which no polynomial algorithms is known. We show that a polynomial-time.

the function f : X → {0,1}, defined by f(x) = 1 for x ∈ Y and f(x) = 0 for x ∈ XY. Class P The class of all decision problems for which there is a polynomial-time algorithm is denoted by P. In other words, a member of P is a/. A nondeterministic algorithm for a decision problem always answers “no” for a no- instance, and for each yes-instance there is a chance that it answers “yes”. Class NP Proposition 9.3 A decision problem belongs to NP if and only if it has a polynomial-time nondeterministic algorithm. /

##### 1 Ch 10 - NP-completeness Tractable and intractable problems Decision/Optimization problems Deterministic/NonDeterministic algorithms Classes P and NP.

of integers S –Output: An element in S of highest frequency What is the algorithm to solve this problem? How much does it cost? 9 Coloring a Graph Decision Problem: Coloring –Input: G=(V,E) undirected graph and k , k > 0. –Question: Is G k-/to an answer of no for a certain guess, that it does not accept it? 18 The Class NP Definition: The class of decision problems NP consists of those decision problems for which there exists a nondeterministic algorithm that runs in polynomial time. –The running time /

##### May 9, 2001Applied Symbolic Computation1 Applied Symbolic Computation (CS 680/480) Lecture 6: Multiplication, Interpolation, and the Chinese Remainder.

the CRT. May 9, 2001Applied Symbolic Computation18 Polynomial Algebra mod a Polynomial A(x)  B(x) (mod f(x))  f(x)|(A(x) - B(x)) This equivalence relation partitions polynomials in F[x] into equivalence classes where the class [A(x)] consists of the set {A(x) + k(x)f(x), where k(x) and f(x) are in F[x]}. Choose a representative for [A(x/

##### CS 154 Formal Languages and Computability May 10 Class Meeting Department of Computer Science San Jose State University Spring 2016 Instructor: Ron Mak.

9 Computer Science Dept. Spring 2016: May 10 CS 154: Formal Languages and Computability © R. Mak 10 Efficient Algorithms  Do we always have efficient (non-exponential growth) solutions to problems?  For many problems, the answer is No.  Worse: For a large class / Mak 26 The Traveling Salesman Problem, cont’d  With a nondeterministic computer, we can solve the yes-no problem in polynomial time.  Check all possible subsets of graph edges in parallel. Verify the edges form a cycle, Verify that each vertex/

##### Warm-Up Exercises Warm-up: Homework: Worksheet given in class. #1-20 all.

Warm-Up Exercises Warm-up: Homework: Worksheet given in class. #1-20 all. Warm-Up Exercises Monomial- a number, a variable or the product of a number and one or more variables with whole number exponents/ as well as the leading coefficient of the polynomial. 1. You Do for Examples 1,2, and 3 3yx 3 – 2y 2 +5y + 9 Classification: 4 th degree polynomial Degree: 3 Number of terms: 4 Leading Coefficient: 3 ANSWER Tell whether y 3 – 4y is a polynomial. If it is a polynomial, find its degree and classify it by the/

##### CS 583: Algorithms All pairs shortest path Ch. 25 NP Completeness Ch. 34 Review for final.

are an interesting class of problems whose status is unknown No polynomial-time algorithm has been discovered for an NP-Complete problem No one has been able to prove that no-polynomial algorithm exists for any of them No super-polynomial (e.g. exponential) low bound has been shown for these problems / have seen, however, it can be solved with dynamic programming 0-1 Knapsack problem: a picture W = 20 wiwi bibi 10 9 8 5 5 4 4 3 3 2 WeightBenefit value This is a knapsack Max weight: W = 20 Items 0-1 Knapsack /

##### Next-Generation Bioinformatics Systems Jelena Kovačević Center for Bioimage Informatics Department of Biomedical Engineering Carnegie Mellon University.

for a specific protein 50-58 sets per class 50-58 sets per class 14-24 2D slices per set 14-24 2D slices per set Resolution 0.049 x 0.049 x 0.2 μm Resolution 0.049 x 0.049 x 0.2 μm Huang & Murphy, Journal of Biomedical Optics 9/ frames TFtight frames use orthogonal polynomials use orthogonal polynomials construct a polynomial transform construct a polynomial transform construct the closest orthogonal polynomial transform construct the closest orthogonal polynomial transform ENequal norm ENequal norm use/

##### 1 Bart Jansen Polynomial Kernels for Hard Problems on Disk Graphs Accepted for presentation at SWAT 2010.

which is not planar So there are disk graphs with Class of (unit)disk graphs Closed under vertex deletion Not closed under edge deletion Not closed under edge contraction 9 TRIANGLE PACKING AND H-MATCHING Structure theory and kernels 10 /subset of what another red vertex sees Same for the blue vertices Structural theorems show that in such colored graphs the sizes of the color classes are polynomially related So size of the largest class is polynomial in the size of smallest class Hence |V| = |R| + |/

##### David Evans CS200: Computer Science University of Virginia Computer Science Class 38: Intractable Problems (Smiley Puzzles.

(8.9/10), 7 (8.6/10) and 8 (8.1/10) We’ve learned how to show some problems are undecidable –Harder (average on question 4 = 7.7) Today: reasoning about the complexity of some problems 21 April 2004CS 200 Spring 20043 Complexity Classes Class P: problems that can be solved in polynomial time by a deterministic TM. O (n k ) for some constant/

##### STAT 424/524 Statistical Design for Process Improvement

Codes for the Tensile Example: Fitting An Orthogonal Polynomial with Orthogonal Contrasts DATA tensile; do percent = 15 to 35 by 5; do i = 1 to 5; input strength@@; drop I; output; end; end; cards; 7 7 15 11 9 12/ is impossible. Without replication, there is little choice but to adopt an additive model. Producing Interaction Diagrams Using Cell Means proc glm; class machinist machine; model time = machinist|machine; means machinist|machine; ods output means = outmeans; run; proc print data = outmeans; /

##### Redefining Developmental Math for Non-Algebra Core Math Courses Dr. Daryl Stephens Murray Butler East Tennessee State.

an algebra-based course (such as college algebra, precalculus, calculus) for graduation. These students take MATH 1530, Probability and Statistics, as their core math class. About 90% of our students do NOT take an algebra-based course/and Polynomials 5. Exponents and Polynomials 5.1 Exponents 5.1 Exponents 9. Inequalities and Absolute Value 9. Inequalities and Absolute Value 9.1 Compound Inequalities 9.1 Compound Inequalities 9.4 Linear Inequalities in Two Variables and Systems of Linear Inequalities 9./

##### NP Completeness Tractability Polynomial time Computation vs. verification Power of non-determinism Encodings Transformation & reducibilities P vs. NP “Completeness”

parallelism (via non-determinism); 8.Enabled vast, deep, and general studies of other “completeness” theories; 9.Helps explain why verifying proofs seems to be easier than constructing them; 10.Illuminates the fundamental nature of /.  “Karp” reduction (transformation) Polynomial-time Turing reduction: solves A by polynomially-many calls to “oracle” for B.  “Cook” reduction B A Open: do polynomial-time-bounded many-one and Turing reductions yield the same complexity classes? (NP, co-NP, NP-complete/

##### Computability and Complexity 13-1 Computability and Complexity Andrei Bulatov The Class NP.

polynomial p(x) such that, for any w  L, there is a certificate c with |c|  p(|w|) All problems from the previous slide have a polynomial time verifier Computability and Complexity 13-5 The Class NP Definition The class of languages that have polynomial time verifiers is called NP Definition The class of languages that have polynomial/= Not Composite) Computability and Complexity 13-9 Problems not in NP Instance: A positive integer n. Question: Is there a winning strategy for whites in a checker game on n /

##### 1 Querying Big Data: Theory and Practice Theory –Tractability revisited for querying big data –Parallel scalability –Bounded evaluability Techniques –Parallel.

polynomial-time queries are BD-tractable 15 16 Polynomial hierarchy revised Tractability revised for querying big data NP and beyond P P BD-tractable not BD-tractable Parallel polylog time 16 17 What can we get from BD-tractability? Guidelines for the following. Why we need to study theory for querying big data What query classes/more processors, the “better”? 19 D Using 10000 SSD of 6G/s, a linear scan of D might take: D 1.9 days/10000 = 16 seconds when D is of 1PB (10 15 B) D 5.28 years/10000 = 4.63 days /

##### Preference Elicitation in Combinatorial Auctions: An Overview Tuomas Sandholm [For an overview, see review article by Sandholm & Boutilier in the textbook.

e.g. side constraints –New query policies New polynomially elicitable valuation classes Using models of how costly it is to answer different queries [Hudson & S. AMEC-02] Decision-theoretic elicitation using priors Elicitors for markets beyond combinatorial auctions –(Combinatorial) reverse auctions / bound-approximation queries with order queries Integrated as before –Computationally more expensive agents items 345678 9 2 10 23456 queries 1 10 100 1000 40 80 120 160 queries Full revelation Total/

##### 1 Intractable Problems Time-Bounded Turing Machines Classes P and NP Polynomial-Time Reductions.

to “an O(T(n)) running-time algorithm.” 3 The class P uIf a DTM M is T(n)-time bounded for some polynomial T(n), then we say M is polynomial-time (“polytime ”) bounded. uAnd L(M) is said to be in the class P. uImportant point: when we talk of P, it doesn’t/ and “false” for all other differences. uInduction: To consider i j, start with a new table, initially all false. uThen, set k to true if, in the old table, there is a value m that was true, and k is either m+i j or m-i j. 9 Knapsack – (3) uSuppose /

##### 1 Polynomial Church-Turing thesis A decision problem can be solved in polynomial time in a reasonable sequential model of computation if and only if it.

computation if and only if it can be solved in polynomial time by a Turing machine. P is the class of these decision problems 2 Search Problems: NP L is in NP iff there is a language L’ in P and a polynomial p so that: 3 Intuition The y-strings are the/9 NPC A language L 2 NP that is NP-hard is called NP-complete. NPC := the class of NP-complete problems. Proposition: L 2 NPC ) [L 2 P iff P=NP]. 10 L is in NP means: There is a language L’ in P and a polynomial p so that L 1 · L 2 means: For some polynomial/

##### 88- 1 Chapter 8 The Theory of NP-Completeness. 88- 2 P: the class of problems which can be solved by a deterministic polynomial algorithm. NP : the class.

x 2 & -x 1 v x 2 & -x 1 v -x 2 88- 9 Definition of the satisfiability problem: Given a Boolean formula, determine whether this formula is satisfiable or/13 Semantic tree In a semantic tree, each path from the root to a leaf node represents a class of assignments. If each leaf node is attached with a clause, then it is unsatisfiable. 88-/NP-hard: Every NP algorithm for problem A can be transformed in polynomial time to SAT [Horowitz 1998] such that SAT is satisfiable if and only if the answer for A is “ YES ”./

##### Circuits CSE 373 Data Structures Lecture 22. 3/12/03Circuits - Lecture 222 Readings Reading ›Sections 9.6.3 and 9.7.

starts and ends at the same vertex (Problem 2) 3/12/03Circuits - Lecture 229 An Euler Circuit for Puzzle A 1 2 3 4 5 6 7 8 9 10 3/12/03Circuits - Lecture 2210 Euler Circuit Property A graph has an Euler circuit if and only /finite (but unbounded) time this fact. 3/12/03Circuits - Lecture 2227 The Class NP NP : Non-deterministic Polynomial Time The class NP is the set of all problems for which a given solution can be checked in polynomial time Check via a non-deterministic algorithm i.e., one that is free to/

##### Chapter 10 P and NP.

logarithmic (n log n) (n2 log n) ... Polynomial (exponent is constant) (1) – Sub-linear (n0.001) – Sub-linear (n0.5) – Sub-linear (n) - Linear (n2) (n3) ... (n100) Classes of Algorithms Exponential (bases make a big difference) / Example) Find the shortest path from A back to A that visits every vertex. 9 B 2 1 C 5 4 A 6 D 3 2 ... 4 7 1 / evidence that P != NP More problems = more opportunity to find a polynomial algorithm for an NP-Complete problem All you have to do is prove two reductions One/

##### 1 Polynomial Space The classes PS and NPS Relationship to Other Classes Equivalence PS = NPS A PS-Complete Problem.

it makes at most c p(n) steps on input of length n, for some constant c and polynomial p. uSay L(M) is in the class EP. uIf M is an NTM instead, say L(M) is in the class NEP (nondeterministic exponential polytime ). 6 More Class Relationships uP  NP  PS  EP, and at least one of /n)+1+c. uWe can count to the maximum number of ID’s on a separate tape using base t+1 and p(n)+1+c cells – a polynomial. 9 Proof PS  EP – (2) uRedesign M to have a second tape and to count on that tape to sp(n)t p(n). uThe new TM/

##### Elementary Algebra Exam 4 Material Exponential Expressions & Polynomials.

Problems Section:4.3 Page:278 Problems:Odd: 1 – 9, 13 – 49, 63 – 75 MyMathLab Section 4.3 for practice MyMathLab Homework Quiz 4.3 is due for a grade on the date of our next class meeting Review of Terminology of Algebra Constant – A specific number/1 – 55, 59 – 69, 73 – 77 MyMathLab Section 4.4 for practice MyMathLab Homework Quiz 4.4 is due for a grade on the date of our next class meeting Multiplying Polynomials To multiply polynomials: –Get rid of parentheses by multiplying every term of the first by /

##### Quadratic Equation- Session1. Session Objective 1.Definition of important terms (equation,expression,polynomial, identity,quadratic etc.) 2. Finding roots.

of the equation (x+a) 2 =0 _H001 Identity Identity : Equation true for all values of the variable (x+1) 2 = x 2 +2x+1 Equation holds true for all real x _H001 Polynomial identity If a polynomial equation of degree n satisfies for the values more than n it is an identity Example: (x-1) 2/+ b + c) 1 2 + 2 (a + b).1 + (a + b – c) = 0  1 is a root, which is rational  so other root will be rational. Class Exercise 9 Solution: D = 0 4 (ac+bd) 2 - 4 (a 2 +b 2 )(c 2 +d 2 )= 0 a 2 c 2 + b 2 d 2 + 2abcd = (a 2/

##### Cryptosystems for Social Organizations based on TSK( Tsujii-Shamir-Kasahara ) ー MPKC Shigeo Tsujii Kohtaro Tadaki Masahito Gotaishi Ryo Fujita Hiroshi.

and Function of PQ type TSK-MPKC 7.Simulation Result 8.Considerations for Security 9.Conclusion 1 Introduction In secret communication, such as between a local government/, 2,....m; random quadratic polynomial (only u i is linear for all i) g i (u m+1, u m+2,..., u 2m ); random quadratic n-variate polynomial for all i. Perturbed TSK-MPKC/ Proposed System Combining two TSK together (p and q term) Residue Class Ring is used Above polynomial system is the central map and public key is generated by applying /

##### Rings,Fields TS. Nguyễn Viết Đông 1. 1. Rings, Integral Domains and Fields, 2. Polynomial and Euclidean Rings 3. Quotient Rings 2.

1.1.1. Show that ( Z n,+, ・ ) is a commutative ring, where addition and multiplication on congruence classes, modulo n, are defined by the equations [x] + [y] = [x + y] and [x] ・/ as [12] 26  [7  12 + 3] 26 = [87] 26 = [9] 26 which corresponds to I. Conversely I is decrypted as [9] 26  [15  (9 – 3) ] 26 = [90] 26 = [12] 26 which corresponds to M. Now/i. Two polynomials f (x) and g(x) are called equal when they are identical, that is, when the coefficient of x n is the same in each polynomial for every n. /

##### Algebra tiles How can we use it? PEC 2015 What topics can we use it for? by Chizuko Matsumoto & Sweeny Term.

2 + 6x + ____ Activity × Part of the square is missing. How many one-tiles do you need to complete it?. 9 = (x + 3) 2 X＋3X＋3 X+3 Try This Topic 7. Use algebra tiles to model the expression. Add / form. x 2 + 8x Marshall Island High School Open Class Grade 10 Thank you for joining. See you again. For printing Equation(Topic 1&2) MODEL ALGEBRA MODEL Polynomial addition/subtraction (Topic 3) Answer _____________________________ Polynomial Multiplication/Division or factorization (Topic 4~8) MODEL ALGEBRA ×/

##### Warm up 1 Determine algebraically whether each of the following functions is even, odd or neither. Write the equation for transformation of.

x + 9 quartic Y = 3x 4 – 2x 3 + 8x 2 – 6x + 5 quintic Graphs of Polynomial Functions The polynomial functions that / polynomial. 12 Zeros of Polynomial Functions It can be shown that for a polynomial /polynomial functions. Find and use zeros of polynomial functions as sketching aids. Find a polynomial equation given the zeros of the function. Homework 8 2.2 Page 130 – 1-8 all (matching) – 13-18(left and right behavior), all – 27-41 odds (finding zeros-verify with a calculator) – 47-55 odds Quiz next class/

##### 5.1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Chapter 5 Introduction to Modern Symmetric-key Ciphers.

3 in decimal. The result of squaring is 9, which is 1001 in binary. Solution 5.1.5 Two Classes of Product Ciphers 5.58 Figure 5.16 Improvement of the previous Feistel design 5.1.5 Two Classes of Product Ciphers Improvement : We want the/5.2.1 Synchronous Stream Ciphers 5.98 Example 5.20 The characteristic polynomial for the LFSR in Example 5.19 is (x 4 + x + 1), which is a primitive polynomial. Table 4.4 (Chapter 4) shows that it is an irreducible polynomial. This polynomial also divides (x 7 + 1) = (x 4 + x/

##### Today’s class Spline Interpolation Quadratic Spline Cubic Spline Fourier Approximation Numerical Methods Lecture 21 Prof. Jinbo Bi CSE, UConn 1.

CSE, UConn 9 Quadratic Splines Numerical Methods Lecture 21 Prof. Jinbo Bi CSE, UConn 10 Cubic Spline Function Each two neighboring points are connected or interpolated by a 3 rd -order (Cubic) polynomial. If # /. Jinbo Bi CSE, UConn 13 Second Cubic Spline Functions Second derivative is a line Lagrange interpolating polynomial for second derivative Integrate twice to get f i (x) Numerical Methods Lecture 21 Prof. Jinbo Bi/, UConn 32 Next class Review Numerical Methods Lecture 21 Prof. Jinbo Bi CSE, UConn 33

##### MAT 125 – Applied Calculus 1.1 Review I. Today’s Class  We will be reviewing the following concepts:  The Real Number Line  Intervals  Exponents and.

Erickson 1.1 Review I 24 Example 8  Find the correct factorization for the problems below. Dr. Erickson 1.1 Review I 25 Roots of Polynomial Expressions  A polynomial equation of degree n in the variable x is an equation of the/c = 0 (a ≠ 0) are given by Dr. Erickson 1.1 Review I 29 Example 9  Solve the equation using the quadratic formula. Dr. Erickson 1.1 Review I 30 Next Class  We will continue reviewing the following concepts:  Rational Expressions  Other Algebraic Fractions  Rationalizing /

##### Polynomial Operations Hanna Jusufi Julia Ly Karl Bryan Ly Huynh Girl : I keep losing stuff when I try to multiply three binomials. Boy : I do, too, but.

all have different ways of doing each polynomial so watch out for your signs! When you see something like (x+4)+(x-9) you add just like that. However, that is completely different from something like (x+4)(x-9). The first one is adding straight / Makes You Beautiful- One Direction Youre insecure, dont know what for, Polynomial Operations leave you wanting mo-o-re, Whether to subtract, or maybe add, Multiply, and or divi-i-de, Everyone else in the class can do it, Everyone including you! Baby it aint hard /

##### CSE 024: Design & Analysis of Algorithms Chapter 9: NP Completeness Sedgewick Chp:40 David Luebke’s Course Notes / University of Virginia, Computer Science.

. 7) 5.Searching (Brassard & Bratley, Chap. 9) 6.Graph Algorithms (Weiss, Chap. 9) 7.Randomized Algorithms (Weiss, Chap. 10) 8.String Searching (Sedgewick, Chap. 19) 9.NP Completeness (Sedgewick, Chap. 40) 25 October 2015CS/polynomial time O(n c ) for some constant c non-polynomial time Polynomial Time Algorithms Are there problems solvable in polynomial time? Answer is yes: every algorithm we’ve studied provides polynomial-time solution to some problems Define P: the class of problems solvable in polynomial/

##### Scientific Computing General Least Squares. Polynomial Least Squares Polynomial Least Squares: We assume that the class of functions is the class of all.

polynomial 0.0857 + 0.4 x + 1.4286 x 2. Class Project Write a Matlab function that will take a vector of x values and a vector of y-values and will return the vector of coefficients for the best quadratic fit to the data. Class Project 2 Exercise 9.15/ in Pav: Write a Matlab function that will find the coefficients (a,b) for the function a e x + b e -x that best approximates/

##### Technical University Tallinn, ESTONIA Overview 1.Introduction 2.Testability measuring 3.Design for testability 4.Built in Self-Test.

7101510 86510165320 Technical University Tallinn, ESTONIA Deterministic Synthesis of LFSR Generation of the polynomial and seed for the given test sequence (1) 100x0 (2) x1010 (3) 10101 (/ Extended Fault Models Defect Extensions of the parallel critical path tracing for two large general fault classes for modeling physical defects: 01010101 Conditional fault Pattern fault Constrained SAF Single/2 0 1 0 0 0 1 4.1 1 0 0 1 1 0 9.2 (9)mod3 = 0 (2)mod3 = 2 Error! Technical University Tallinn, ESTONIA Error/

##### Lecture 8 Overview. Analysis of Algorithms Algorithms – Time Complexity – Space Complexity An algorithm whose time complexity is bounded by a polynomial.

NP-hard problem implies the existence of polynomial solutions for every problem in NP NP-complete problems are the NP-hard problems that are also in NP 6 CS 450/650 Lecture 8: Algorithm Background Relationships between different classes NP P NP-complete NP-hard 7 CS 450/650 Lecture 8: Algorithm Background Lecture 9 Rivest-Shamir-Adelman (RSA) CS 450/650 Fundamentals of/

##### 1. 2 Lecture outline Basic definitions: Basic definitions: P, NP complexity classes P, NP complexity classes the notion of a certificate. the notion of.

complexity classes Def: A Decision problem for a language L  {0,1} * is to decide whether a given string x belongs to the language L. Def: P is the class of languages (decision problems) that can be recognized by a deterministic polynomial time Turing machine. Def: NP is the class of/L : – Guess y of proper polynomial size – Call M * L to check if (x,y)  R L – Accept x if M * L accepts (x,y) M L is a non-deterministic TM for L M L is a non-deterministic TM for L 9 Search Problems Def: A search problem /