creative capacities (writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in expressing oneself orally/structure and properties of the real number system. Topics covered include linear equation, graphs, functions, **polynomials**, **polynomial** functions, rational expressions, rational equations, and radical expressions. 11 Faculty Weekend Course Overview Course /

Multiplication Cache Aware algorithms/ Cache oblivious Algorithms Static Van Emde Boas Layout 5.3 Counting Inversions **Music** site tries to match your song preferences with others. –You rank n songs. –**Music** site consults database to find people with similar tastes. Similarity metric: number of inversions between /n-1 … a 1 a 0 and b = b n-1 … b 1 b 0, compute their product c = a b. Convolution algorithm. –Form two **polynomials**. –Note: a = A(2), b = B(2). –Compute C(x) = A(x) B(x). –Evaluate C(2) = a b. /

algorithms/ Cache oblivious Algorithms Static Van Emde Boas Layout FFTs Streaming Algorithms 5.3 Counting Inversions **Music** site tries to match your song preferences with others. –You rank n songs. –**Music** site consults database to find people with similar tastes. Similarity metric: number of inversions between two/n-1 … a 1 a 0 and b = b n-1 … b 1 b 0, compute their product c = a b. Convolution algorithm. –Form two **polynomials**. –Note: a = A(2), b = B(2). –Compute C(x) = A(x) B(x). –Evaluate C(2) = a b. /

Multiplication Cache Aware algorithms/ Cache oblivious Algorithms Static Van Emde Boas Layout 5.3 Counting Inversions **Music** site tries to match your song preferences with others. –You rank n songs. –**Music** site consults database to find people with similar tastes. Similarity metric: number of inversions between /n-1 … a 1 a 0 and b = b n-1 … b 1 b 0, compute their product c = a b. Convolution algorithm. –Form two **polynomials**. –Note: a = A(2), b = B(2). –Compute C(x) = A(x) B(x). –Evaluate C(2) = a b. /

of the Shortest Rick Trebino School of Physics Georgia Institute of Technology Atlanta, GA 30332 The **Musical** Score, the Fundamental Theorem of Algebra, Events Ever Created The **Musical** Score In order to measure an event in time, you need a shorter one. To study/ but changes the phase, yielding an ambiguity! So 1D phase retrieval is impossible! Phase Retrieval & the Fund Thm of Algebra 2 **polynomial**! 2D Phase Retrieval and the Fundamental Theorem of Algebra The Fourier transform {F 0,0, …, F N-1,N-1 } /

one or two variables. Graphing calculators will be used as a tool for factoring and for confirming algebraic factorizations. Performing Operations on **Polynomials** 2 Students need additional practice applying the laws of exponents when multiplying expressions. Suggested Practice for SOL A.2a Simplify: a. b/this situation? a. b. c. d. Extension: How many days would it take 60 workers to set up for the Summer **Music** Festival? Answer: It would take days or 1 day and 16 hours. SOL A.9 The student, given a set of /

Course Information World Languages: Mrs. Stern, Department Chairperson Family and Consumer Sciences Technology Education Art Education **Music** Education Humanities and Other Electives Are You Ready For 9 th Grade? Things to Consider: –High School/ Covered: Review of solving equations, Solving Systems of Linear Equations and Inequalities, Functions, Working with **Polynomials**, Factoring **Polynomials**, Exponents, Rational Expressions, Radical Expressions. Students will take the Algebra 1 Keystone Exam in May/

capacities (writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in expressing /, graphs, and linear equations as applied to problems in economics and business. Topics include mathematics of finance, linear, **polynomial**, exponential, and logarithmic functions. 11 Faculty Evening Course Overview Course Materials Required Textbook and Materials Lial, M. L.,/

writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in/Hospitality Community 15 Faculty Weekend Course Schedule SessionLoc.Class TitleAssignments 03 AF2FApplications of Quadratic Functions **Polynomial** Functions Rational Functions Exponential Functions Applications of Exponential Functions Logarithmic Functions Logarithmic and Exponential Equations Sections/

capacities (writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in expressing / Hospitality Community 14 Student Evening Course Schedule SessionLoc.Class TitleAssignments 05 A F2FIntroduction to **polynomials** **Polynomial** functions Multiplication of **polynomials** Factoring trinomials Homework review In-class quiz on sections 2.4 and 2.5 Practice problems/

creative capacities (writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in expressing oneself orally/structure and properties of the real number system. Topics covered include linear equation, graphs, functions, **polynomials**, **polynomial** functions, rational expressions, rational equations, and radical expressions. 8 Student Weekend Course Overview Course /

Theory Classical Off-the-Grid Method: Prony (1795) Uniform time samples Off-the-grid frequencies Robust variants: Pisarenko (1973), **MUSIC** (1986), ESPRIT (1989), Matrix pencil (1990)... Atomic norm (2011) Main inspiration: Finite-Rate-of-Innovation (FRI) [/multiplication annihilating filter convolution Can recover edge set when it is the zero-set of a 2-D trigonometric **polynomial** [Pan et al., 2014] “FRI Curve” 25x25 coefficients13x13 coefficients Multiple curves & intersections Non-smooth points Approximate/

capacities (writing, inventing, designing, performing in art, **music**, drama, etc.) Gaining a broader understanding and appreciation of intellectual/ cultural activity (**music**, science, literature, etc.) Developing skill in expressing /, graphs, and linear equations as applied to problems in economics and business. Topics include mathematics of finance, linear, **polynomial**, exponential, and logarithmic functions. 11 Faculty Evening Course Overview Course Materials Required Textbook and Materials Lial, M. L.,/

age recognition GMM/SVM supervector approach for acoustic speech features Detection task and pseudo-NIST evaluation procedure Rank and **polynomial** rank normalization Knowledge-driven feature selection Classification methods for independent “bag of observations” features Valid application-independent evaluation /yes 1.32432 no -0.3212 no 1.8463 no -2.5773 yes 0.00132 no 2.20122 emergency vehicle ? **music** ? talking ? laughing ? phone ? no event ? The system performance is evaluated by presenting it with a set of/

Lewandowski, N., Bengio, Y., and Vincent, P. “Modeling Temporal Dependencies in High-Dimensional Sequences: Application to Polyphonic **Music** Generation and Transcription,” Proc. ICML, 2012. Bourlard, H. and Morgan, N., Connectionist Speech Recognition: A Hybrid /recognition,” Neurocomputing, Vol 106, 148-157, 2013. Siniscalchi, M., Li, J., and Lee, C. “Hermitian **Polynomial** for Speaker Adaptation of Connectionist Speech Recognition Systems,” IEEE Transactions on Audio, Speech, and Language Processing, Vol. 21,/

show that a certain class of functions can be learned in **polynomial** time. Negative results show that certain classes cannot be learned in **polynomial** time. https://store.theartofservice.com/the-machine-learning-toolkit.html Machine/search) https://store.theartofservice.com/the-machine-learning-toolkit.html **Music** Information Retrieval - Statistics and Machine Learning *Software for **music** information retrieval — Semantic Web and **musical** digital objects, intelligent agents, collaborative software, web-based /

people believe that P NP? NP-completeness I can appreciate good **music** (being able to verify solutions) doesn’t mean that I can compose good **music** (to find solutions). (from wikipedia) "Proof by contradiction. Assume P = NP. Let y be a proof that P = NP. The proof y can be verified in **polynomial** time by a competent computer scientist, the existence of which/

feet The markings that are found on dice are called "pips." A newly hatched fish is called a "fry." The **music** band UB40 got its name from an unemployment form in England Armadillos can be housebroken The Eisenhower interstate system requires that one/Mahal was brought in from various parts of India by a fleet of 1000 elephants Chapter 5 Project Unit map Topic: **Polynomials** and **Polynomial** Functions Key Learning(s): Every function has an inverse, but not every inverse has a function Unit Essential Question (UEQ/

/images: –8 X 10 inch photo, 400 X400 pixels/inch. 38.4MB. Voice: PCM/ADPCM, 4kHz, 64kbps (this as well the followings called stream) **Music**/Audio: –MPEG/MP3, 16-24 kHz, 512-748 kbps. Video: a sequence of pictures (moving pictures) –H.261, 176 X 144 pixels/frame, 10/ as follows: ) 11010101011 1111 1011 1100 1011 1111 1011 1000 1011 11 The remainder is not zero, so error occurs. CIS, IUPUI36 Typical standard CRC **polynomials** CRC-8: x 8 + x 2 + x + 1 ATM header error check CRC-16: x 16 +x 12 +x 5 +1 HDLC, /

priori – not clear why he should have been able to decide right/wrong. **Polynomial** time learning not possible in our model of “helpful Alice”. **Polynomial** time learning not possible in our model of “helpful Alice”. Better definitions can be/Amazon Buy something on Amazon Intellectual Curiosity: Intellectual Curiosity: Learning/Teaching Learning/Teaching Listening to **music**, watching movies Listening to **music**, watching movies Coming to this talk Coming to this talk Searching for alien intelligence Searching for/

priori – not clear why he should have been able to decide right/wrong. **Polynomial** time learning not possible in our model of “helpful Alice”. **Polynomial** time learning not possible in our model of “helpful Alice”. Better definitions can be/Amazon Buy something on Amazon Intellectual Curiosity: Intellectual Curiosity: Learning/Teaching Learning/Teaching Listening to **music**, watching movies Listening to **music**, watching movies Coming to this talk Coming to this talk Searching for alien intelligence Searching for/

heard one. n Ace an exam. Verify TAs solutions. Imagine the wealth of a society that produces optimal planes, bridges, rockets, theorems, art, **music**, wine, jokes. 16 The Main Question Does P = NP? n Is the original DECISION problem as easy as CERTIFICATION? If yes, then: n/ X if and only if y is a YES instance of Y. n We require |y| to be of size **polynomial** in |x|. **Polynomial** transformation is **polynomial** reduction with just one call to oracle for Y, exactly at the end of the algorithm for X. Note: all /

every string in {0, 1}*. Class P P = {L ⊆ {0, 1}* : there exists an algorithm A that decides L in **polynomial** time} **Polynomial**-time verification For example, suppose that for a given instance 〈G, u, v, k〉 of the decision problem PATH, We are also given a/-oh-oh, find the longest path! Woh-oh-oh-oh, find the longest path. Lyrics. Copyright © 1988 by Daniel J. Barrett. **Music**. Sung to the tune of The Longest time by Billy Joel. Recorded by Dan Barrett while a grad student at Johns Hopkins during a difficult/

to be well suited for describing the physical universe. Defining characteristic is precision. It shares some similarities with **music**. The Uncertainty Principle of Language Breadth vs. precision Mathematics is not suitable for expressing certain ideas (for example/ to CDs in 1982. They have been generalized in many ways, including algebraic geometry codes in which the **polynomials** are replaced by functions on a curve over F. Roomba and computational geometry Robotic vacuum cleaner. Navigation/

Vector Spaces Colour Error Correcting Codes Linear Transformations Integrating Changing Basis Fourier Transformation (sine) Fourier Transformation (JPEG) Fourier Transformation (**Polynomials**) Other Algebra Taylor Expansions Generating Functions Primes Numbers Fields A Field has: A universe U of values Two operations: + and/ time y(t) Find the contribution of each frequency Sine &Cosine Basis If this is the dominate **musical** note of frequency = 2 /T, then all the other basis functions are its harmonics /

in our algorithm needs commutativity 87 Examples of Noncommutative Additive Semigroups Positive even integers Negative integers Real numbers **Polynomials** Planar vectors Boolean functions Line segments 88 Mathematical Naming Conventions If a set has one binary operation that is/ innovation in the 18 th and 19 th centuries… 155 Golden Age of German Culture (18 th and 19 th centuries) **Music** – Bach, Händel, Haydn, Mozart, Beethoven, Schubert, Schumann, Brahms, Wagner, Mahler, Richard Strauss Literature – Goethe,/

original creator has put original idea in concrete form creator has put original idea in concrete form e.g. literary works, **musical** works, dramatic works, pantomimes and choreographic works, pictorial, graphic, and sculptural works, motion pictures and other audiovisual works, sound recordings/No fewer than k of them can derive any knowledge of S All k of them can reconstruct S Shamir’s **Polynomial** SS **Polynomial** of degree k can be specified By k+1 coefficients By k+1 coefficients By k+1 distinct points By k+/

**polynomial** time (contains factoring). NP – Problems where solutions can be verified in **polynomial** time (contains factoring). PSPACE – Problems solvable in **polynomial** space (quite infeasible for Bob to solve on his own). PSPACE – Problems solvable in **polynomial**/Buy something on Amazon Intellectual Curiosity: Intellectual Curiosity: Learning/Teaching Learning/Teaching Listening to **music**, watching movies Listening to **music**, watching movies Coming to this talk Coming to this talk Searching for alien intelligence/

and Biological, Symbolic, Sub-symbolic and Agent- based - importance of the diversity of intelligences ( logical, linguistic, spatial, **musical**, kinesthetic, interpersonal, intrapersonal, …) for both natural and artifactual intelligence. Lessons Learned: Thinking The Biggest Picture and the Smallest./ 29 30 Is there any decision problem that can be solved by a non-deterministic Turing machine in **polynomial** time (belongs to complexity class NP), but not by a deterministic Turing machine, (meaning it does /

works by oversampling a **polynomial** constructed from the data. The **polynomial** is evaluated at several points, and these values are sent or recorded. Sampling the **polynomial** more often than is necessary makes the **polynomial** over- determined. As long/ noise by canceling it with digitally ________(generate) antinoise. Another interesting DSP application is artificial reverberation in **music** preparation. If the individual channels are simple added together, the resulting piece ______(sound) frail and diluted/

)(1+x 2 +x 3 ) (?) (Hint: 1 = -1, (1+x) 2 = 1+x 2.) The set of all **polynomials** in R 7 which are multiples of 1+x+x 3 forms a linear code with 16 codewords. This is “ essentially the same ” code as / technique, some “ average ” is constructed using the valid data around an error. This average is then substituted in for the erroneous data. Since most **music** (with the possible exception of heavy metal!) is continuous -- this method works well for concealing relatively short errors. Muting( 消音 ): Muting is a/

2.72 exp(10) = 2.20 104 (daily salesman trip) exp(100) = 2.69 1043 (monthly salesman planning) exp(500) = 1.40 10217 (**music** band worldwide tour) exp(250,000) = 10108,573 (fedex, postal services) Fastest computer = 1012 operations/second So… In general, exponential-complexity problems cannot be solved/g is o(h), then f is o(h) if f is o(g) and g is O(h), then f is o(h) **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). NP P AC0 NC1 NC P complete NP complete/

2.72 exp(10) = 2.20 10 4 (daily salesman trip) exp(100) = 2.69 10 43 (monthly salesman planning) exp(500) = 1.40 10 217 (**music** band worldwide tour) exp(250,000) = 10 108,573 (fedex, postal services) Fastest computer = 10 12 operations/second CS 561, Lectures 3-5 76 So… In general/then f is o(h) if f is o(g) and g is O(h), then f is o(h) CS 561, Lectures 3-5 83 **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). AC 0 NC 1 NCP completeNP complete P NP PH AC 0/

2.72 exp(10) = 2.20 10 4 (daily salesman trip) exp(100) = 2.69 10 43 (monthly salesman planning) exp(500)= 1.40 10 217 (**music** band worldwide tour) exp(250,000)= 10 108,573 (fedex, postal services) Fastest computer = 10 12 operations/second CS 561, Lectures 3-5 60 So… In general,/ then f is o(h) if f is o(g) and g is O(h), then f is o(h) CS 561, Lectures 3-5 66 **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). AC 0 NC 1 NCP completeNP complete P NP PH AC 0/

2.72 exp(10) = 2.20 10 4 (daily salesman trip) exp(100) = 2.69 10 43 (monthly salesman planning) exp(500) = 1.40 10 217 (**music** band worldwide tour) exp(250,000) = 10 108,573 (fedex, postal services) Fastest computer = 10 12 operations/second CS 561, Lectures 3-5 73 So… In general/then f is o(h) if f is o(g) and g is O(h), then f is o(h) CS 561, Lectures 3-5 80 **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). AC 0 NC 1 NCP completeNP complete P NP PH AC 0/

2.72 exp(10) = 2.20 10 4 (daily salesman trip) exp(100) = 2.69 10 43 (monthly salesman planning) exp(500) = 1.40 10 217 (**music** band worldwide tour) exp(250,000) = 10 108,573 (fedex, postal services) Fastest computer = 10 12 operations/second DCP 1172, Ch. 3 72 So… In general, /h), then f is o(h) if f is o(g) and g is O(h), then f is o(h) DCP 1172, Ch. 3 79 **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). AC 0 NC 1 NCP completeNP complete P NP PH AC 0 /

to assign grades in Pinnacle. Mathematics –Adding **Polynomials** MM1A2—Students will simplify and operate with radical expressions, **polynomials**, and rational expressions. c.Add, subtract, multiply, and divide **polynomials** Pre-requisite skill—substituting values into expressions But what/SSEPF6ELA11LSV1 Unit review – Vocabulary Cards Unique – October monthly check point cards JB-Sitting in chair 9:00- **Music** This allows you to hit the students IEP goals and objectives, GPS standards, and the new Evaluation system/

72 exp(10) = 2.20 10 4 (daily salesman trip) exp(100) = 2.69 10 43 (monthly salesman planning) exp(500) = 1.40 10 217 (**music** band worldwide tour) exp(250,000) = 10 108,573 (fedex, postal services) Fastest computer = 10 12 operations/second SE 420 66 So… In general, exponential-complexity/ o(h), then f is o(h) if f is o(g) and g is O(h), then f is o(h) SE 420 73 **Polynomial**-time hierarchy From Handbook of Brain Theory & Neural Networks (Arbib, ed.; MIT Press 1995). AC 0 NC 1 NCP completeNP complete P NP PH AC/

Interactive challenges Lesson Plans CCSSM Content Standards: Math in **Music** Arithmetic with **Polynomials** and Rational Functions A-APR Perform arithmetic operations on **polynomials** Use **polynomial** identities to solve problems Rewrite rational functions Creating Equations A/ one variable CCSSM Content Standards: Math in Videogames Functions A-APR Perform arithmetic operations on **polynomials** Rewrite rational functions Creating Equations A-CE Create equations that describe numbers or relationships Reasoning with/

D Point Set Matching: T: (t 1,t 2,…,t n ) P: (p 1,p 2,…,p m ) 2-D Point Set Matching – Searching in **Music**: Notations: Length of text: N Length of Pattern: M Number of “1”s in text: n Number of “1”s in pattern: m. Idea: Map /Analysis: We could continue this process recursively and analyze probability of clash of masked element with an element that is already there but… More elegant solution: **Polynomials** over a finite field Consider indices as elements in F 2 L. In F 2 L : x+y = x y. Length Reduction: Every /

x) – g(x) = x 3 + x + 1 f(x) x g(x) = x 5 + 3x 2 – 2x + 2 **Polynomial** Arithmetic with Modulo Coefficients when computing value of each coefficient do calculation modulo some value could be modulo any prime but we are most interested in mod 2/ all that apply) 68 1. Send spam and phishing emails. 2. Harvest and sell email addresses and passwords. 3. Illegally distribute **music**, movies and software. 4. Distribute child pornography. 5. Infect other systems. 6. Hide programs that launch attacks on other computers. /

A plane in this quadratic space is equivalent to a quadratic surface in our original space. Problem & Solution If dimensionality of space is high, lots of calculations For a high **polynomial** space, combinations of coordinates explodes Need to do all these calculations for all training points, and for each testing point Infinite dimensional spaces impossible Nonlinear surfaces can be used without/

1970; J. Thatcher 1970),, ga S NPVP PRO he VBZ enjoys NP VBG listening VP P to NP SBAR **music** karekikuongakuowadaisukidesugano Original input:Final output:,,,,,,,, Top-Down Tree Transducer (W. Rounds 1970; J. Thatcher 1970) To get total/ composition Inclusivenesscapture any transformation that a string-based FST can Teachabilitygiven input/output pairs, find locally optimal rule probabilities in low- **polynomial** time Top-down Tree Transducers S x0x2x1 S VP x2 x0 LNT T – top-down L – linear (non-copying) N/

sets the standard of objective truth for all intellectual endeavors, science and technology bear witness to its practical usefulness. Besides language and **music**, it is one of the primary manifestations of the free creative power of the human mind, and it is the universal / the constant A is the following function continuous for all real x? Solution: Solution: Since Ax+5 and are both **polynomials**, it follows that f(x) will be continuous everywhere except possibly at x=1. According to the three criteria for /

upon pattern will give a remainder of zero. We see this is so in the calculation at the right. Real applications use longer **polynomials** to cover larger information strings. –Some of the standard poly- nomials are listed in the text. 2.8 Error Detection and Correction /hundred years. Other media are good for only a decade-- at best. 7.5 Optical Disks 430 CD-ROMs were designed by the **music** industry in the 1980s, and later adapted to data. This history is reflected by the fact that data is recorded in a single/

in figure 5. The evaluator chose number 12 as her favorite creation. Although judging art was not as sanity challenging as the **music** example. Figure 4. The initial 16 fractal art pieces initialized by the GA. Figure 5. The final 16 fractal art / has weights that assume the coefficients of a binomial **polynomial** with five coefficients. Binomial coefficients of an (N-1)th-order **polynomial**, or binomial weights of an N element array, are the coefficients of the **polynomial** given by the Nth row of Pascal’s triangle/

? The best place to start is with a good title. Then build a song around it. (Wisdom of country **music**) System 16 bit counter & 1 Sequence of 2 16 bits Sea of gates Technical University Tallinn, ESTONIA Copyright 2000-/in BIST Pseudorandom Test generation by LFSR: Using special LFSR registers Several proposals: –BILBO –CSTP Main characteristics of LFSR: –**polynomial** –initial state –test length Technical University Tallinn, ESTONIA Copyright 2000-2003 by Raimund Ubar 270 Linear Feedback Shift Register (LFSR/

closely the course textbook “ Pattern Recognition and Machine Learning ” by Christopher Bishop and the slides “ Machine Learning and **Music** ” by Prof. Douglas Eck 免责声明 CV PR Feature Extraction Pattern Classification ML Statistical Symbol Contents IPSP Probability Theory Information Theory/ some serious challenges and is an important factor influencing the design of pattern recognition techniques. Curse of Dimensionality **Polynomial** curve fitting, M = 3 1.5 Decision Theory Determination of p(x, t) from a set /

data ● Programs and data (documents) ● Network cards (Ethernet card) – Used to connect several computers ● CD/DVD drives – Read-only storage devices (**music**, video, data) ● Video Card (3D specialized hardware) – Driven by the CPU to transmit data on screen(s) – Contains a Graphic processor (GPU)/is NP-C if: – it is in NP and it is NP-hard (every other problem in NP is reducible in **polynomial** time to it) – A reduction is a transformation of one problem into another problem. ● Intuitively, if problem A is /

data ● Programs and data (documents) ● Network cards (Ethernet card) – Used to connect several computers ● CD/DVD drives – Read-only storage devices (**music**, video, data) ● Video Card (3D specialized hardware) – Driven by the CPU to transmit data on screen(s) – Contains a Graphic processor (GPU)/is NP-C if: – it is in NP and it is NP-hard (every other problem in NP is reducible in **polynomial** time to it) – A reduction is a transformation of one problem into another problem. ● Intuitively, if problem A is /

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